Some Properties of the Symmetric

Enveloping Algebra of a Subfactor,

with Applications to Amenability and Property T

Sorin Popa

Received: September 27, 1999

Communicated by Joachim Cuntz

Abstract. We undertake here a more detailed study of the structure and basic properties of the symmetric enveloping algebra M ⊠_{e_{N}} M^{op} as-

sociated to a subfactor N ⊂ M, as introduced in [Po5]. We prove a number of results relating the amenability properties of the standard invariant of N ⊂ M,G_{N,M}, its graph Γ_{N,M} and the inclusion M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op}, notably showing that M ⊠_{e_{N}} M^{op} is amenable

relative to its subalgebra M ∨ M^{op} iff Γ_{N,M} (or equivalently G_{N,M}) is amenable, i.e., ∥Γ_{N,M}∥^{2} = [M : N]. We then prove that the hyperfiniteness of M ⊠_{e_{N}} M^{op} is equivalent to M being hyperfinite and Γ_{N,M} being

amenable. We derive from this a hereditarity property for the amenability of graphs of subfactors showing that if an inclusion of factors Q ⊂ P is embedded into an inclusion of hyperfinite factors N ⊂ M with amenable graph, then its graph Γ_{Q,P} follows amenable as well. Finally, we use the symmetric enveloping algebra to introduce a notion of property T for inclusions N ⊂ M, by requiring M ⊠_{e_{N}} M^{op} to have the property T relative

to M ∨ M^{op}. We prove that this property doesn’t in fact depend on the inclusion N ⊂ M but only on its standard invariant G_{N,M}, thus defining a notion of property T for abstract standard lattices G.

1991 Mathematics Subject Classification: Primary: 46L37, secondary: 46L40

0. Introduction

Let N ⊂ M be an inclusion of type II_{1} factors with finite Jones index, [M : N] < ∞, and extremal. In short, its symmetric enveloping von Neumann algebra M ⊠_{e_{N}} M^{op} is the unique (up to isomorphism) type II_{1} factor S, generated by

mutually commuting copies of M, M^{op} that satisfy M′ ∩S = M^{op}, M^{op}′ ∩S = M and by a projection e_{N} which implements, at the same time, both the trace preserving expectation E_{N} of M onto N and the trace preserving expectation E_{N^{op}} of M^{op} onto N^{op}. One can construct this factor by first taking the C^{∗}-algebra S_{0} generated on the Hilbert space L^{2}(M) by the operators of left and right multiplication by elements in M and by the orthogonal projection of L^{2}(M) onto L^{2}(N), then proving that there exists a unique trace τ on this C^{∗}-algebra and then defining M ⊠_{e_{N}} M^{op} to be the type II_{1} von Neumann factor obtained via the Gelfand-

Naimark-Segal representation for (S_{0}, τ), i.e., M ⊠_{e_{N}} M^{op} =^{def} \overline{π_{τ} (S_{0})}. This con-

struction doesn’t in fact depend on the (binormal) representation of the triple (N ⊂ M, e_{N}, M^{op} ⊃ N^{op}) that one starts with: any M − M bimodule with an e_{N}-type projection on it, instead of L^{2}(M), will do, provided certain obvious compatibility conditions for the commutants are satisfied. The following exemple of symmetric enveloping algebras is quite relevant: if N ⊂ M is an inclusion associated to a finitely generated discrete group G and an outer action σ of G on a type II_{1} factor P (see e.g., 5.1.5 in [Po2]) then (M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op}) is isomorphic to (P ¯⊗P^{op} ⊂ P ¯⊗P^{op} ⋊_{σ⊗σ^{op}} G).

In general, one has an interpretation of the symmetric enveloping inclusion M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op} that is very much the same as this crossed product

situation. The symmetric enveloping algebra M ⊠_{e_{N}} M^{op} and the inclusions M ∨ M^{op} ⊂

M ⊠_{e_{N}} M^{op} were introduced in ([Po5]) in order to provide an additional tool

for studying subfactors of finite index. It proved to be particularily useful for relating the analysis aspects of the theory of subfactors to its combinatorial features. We undertake here a more detailed study of these objects and use them to get more insight into the structure of subfactors, notably proving a number of results on the amenability and the property T for subfactors N ⊂ M and for their associated combinatorial invariants: the standard graph Γ_{N,M} and the standard invariant G_{N,M}. Thus, we prove that G_{N,M} is amenable (by definition this means that its graph Γ_{N,M} is amenable, i.e., it satisfies the Kesten-type condition ∥Γ_{N,M}∥^{2} = [M : N]) if and only if M ⊠_{e_{N}} M^{op} is amenable relative to M ∨ M^{op} in the sense of

[Po8]. In fact, we establish a few more additional equivalent characterizations of the amenability for Γ_{N,M}: a Følner type condition; a local Shannon-McMillan-Breiman type condition; a local bicommutation condition; a characterization in terms of the representations of N ⊂ M. We then study the amenability in the special case of subfactors N ⊂ M for which the algebras N, M involved are assumed amenable (or, equivalently, by Connes theorem [C1], hyperfinite) type II_{1} factors. The key result along this line shows that the algebra M ⊠_{e_{N}} M^{op} is itself amenable if and only if both

G_{N,M} and the single algebras N, M are amenable.

Again, some other characterizations of this situation are proved, notably an ”injectivity”-type condition requiring N ⊂ M to be the range of a norm one projection from its standard representation, or, equivalently, to be the range of a norm one projection from any of its (smooth) representations. Also, it is proved equivalent to an Effros-Lance type condition, requiring S_{0} = C^{∗}(M, e_{N}, M′) ⊂ B(L^{2}(M)) to be a simple C^{∗}-algebra . We call an inclusion of factors N ⊂ M satisfying any of these conditions an amenable inclusion. While proving all these results we also show that if N ⊂ M is amenable (i.e., N, M are hyperfinite and Γ_{N,M} is amenable) then there exists a choice of a tunnel of subfactors M ⊃ N ⊃ P_{1} ⊃ P_{2}..., obtained by taking downward basic constructions for certain induced-reduced algebras in the Jones tower (the choice being dictated by information contained in the standard invariant G_{N,M}) such that the relative commutants P′_{k} ∩ N ⊂ P′_{k} ∩ M exhaust N ⊂ M. This shows in particular that amenable subfactors are completely classified by their standard invariants G_{N,M} (see also [Po16]). Next we derive the result that we regard as the most significant application of the methods developed in this paper, showing that the amenability in the category of inclusions of factors with finite Jones index N ⊂ M, with ”morphisms” given by commuting square embeddings between such inclusions, is a hereditarity property. In the case one takes degenerate inclusions N = M we recover Connes’ hereditarity result for single hyperfinite type II_{1} factors ([C1]). In terms of graphs, the result states that if an extremal inclusion of hyperfinite factors N ⊂ M has amenable graph then any of its sub-inclusions Q ⊂ P (i.e., Q ⊂ P is embedded in N ⊂ M as a commuting square) has amenable graph. It should be noted that the embedding of Q ⊂ P into N ⊂ M does not require [P : Q] = [M : N], nor that [M : P] < ∞ ! This hereditarity property for the amenability of graphs is somewhat surprising and there is little that could appriorically predict it. It only holds true within the class of hyperfinite subfactors, as if we drop the amenability assumption on the ambient single algebras M involved it is no longer valid, in general. Indeed, it is proved in ([Po7]) that given any abstract standard λ- lattice G and any of its sublattices G_{0} ⊂ G, there exist subfactors N ⊂ M and N_{0} ⊂ M_{0} and a commuting square inclusion of N_{0} ⊂ M_{0} into N ⊂ M, such that G_{N,M} = G and G_{N_{0},M_{0}} = G_{0}. But any standard λ-lattice G contains the Temperley-Lieb-Jones standard λ-lattice with graph A_{∞}, which is never amenable if λ^{−1} > 4. Thus if G is taken to be amenable, for instance to have finite depth, then N ⊂ M has amenable graph while N_{0} ⊂ M_{0}, which is embedded into it, doesn’t. The reason is, of course, that in the examples of subfactors N ⊂ M constructed in ([Po7]) the algebras N, M involved are not hyperfinite. One consequence of the hereditarity result is that, for instance, one cannot embed subfactors Q ⊂ P of index α > 4 that are contructed by commuting squares of finite dimensional algebras like in ([Sc], [We]) and having graph Γ_{Q,P} equal to A_{∞} (note that by [H1] Γ_{Q,P} = A_{∞} if α < (5 + \sqrt{13})/2) into hyperfinite subfactors of finite depth and index > α. Also, by ([H2]) there exists a subfactor of index α = (5 + \sqrt{13})/2, constructed from commuting squares of finite dimensional algebras and having graph A_{∞}, which thus, by our result, cannot be embedded into Haagerup’s finite depth subfactor of same index (5 + \sqrt{13})/2 ([H1]). Our last application to the symmetric enveloping algebra approach is the consideration of a notion of property T for standard lattices. Thus, we prove that if a standard lattice G is given then M ⊠_{e_{N}} M^{op} has the property T relative to

M ∨M^{op}, in the sense of ([A-D],[Po8]), for some N ⊂ M for which G_{N,M} = G, if and only if it has this property for any subfactor N ⊂ M for which G_{N,M} = G. If G satisfies these conditions then we say that G has the property T. Note that this definition does not require the ambient factors involved to have the property T in the sense of Connes ([C4,5]). On the other hand, if G is a standard lattice coming from a discrete group G as described above, then G has the property T if and only if the group G has the property T in the classical sense of Kazhdan. Thus, our notion generalizes this notion, from discrete groups to the larger class of group-like objects G. Our main result in this direction shows that if a sublattice G_{0} of a standard lattice G has the property T then G has the property T. As a consequence it follows that, generically, the Temperley-Lieb-Jones standard lattices with graph A_{∞} do not have the property T. Although we only work with type II_{1} factors, many of the considerations in this paper can be suitably carried over to subfactors of type III (see the remarks 1.10.3^{∘}, 2.2.2^{∘}, 2.5.2^{∘}). The corresponding symmetric enveloping type III factors may prove to be a useful tool in the analysis of the Jones-Wassermann subfactors coming from representations of loop groups ([Wa], [Xu]). In a different direction, it would be interesting to relate the symmetric enveloping algebra associated to an extremal II_{1} subfactor to Jones’ affine Hecke algebra associated with that subfactor ([J3,4]). An explicit description of the symmetric enveloping algebras coming from certain special classes of subfactors ([BiH], [BiJ]) would certainly be most illuminating for getting some insight on this and other related problems. The paper is organized in 9 Sections. In the first section we introduce the C^{∗}-analogues of the symmetric enveloping algebras, needed in order to obtain the necessary universality properties and the functoriality of the von Neumann construction. A key ingredient for these considerations is the relative Dixmier property for subfactors of finite index, that we prove in the Appendix A.1. In Sec. 2 we define the actual symmetric enveloping type II_{1} factors (2.1, 2.2) and symmetric enveloping inclusions and prove their basic properties (2.6, 2.7, 2.9, 2.10). Also, we define a more general class of enveloping inclusions, in which to two given subfactors N ⊂ M and Q ⊂ P having the same higher relative commutant picture one associates their concatenation inclusion M ∨ P^{op} ⊂ M ⊠ P^{op} (2.5.1^{∘}). We end that section by introducing a notion of index [G : G_{0}] for sublattices G_{0} of standard lattices G (2.11, 2.12). In Sec. 3 we discuss the example of symmetric enveloping algebras associated to subfactors coming from discrete groups acting outerly on factors, case in which it becomes an actual crossed-product algebra. In Sec. 4 we prove that even for general inclusions N ⊂ M the corresponding symmetric enveloping algebras look very much like crossed products (4.5). Also, we prove some decomposition properties for such algebras, showing for instance that when N, M are hyperfinite then, regardless of whether M ⊠_{e_{N}} M^{op} is hyperfinite or

not, it is a thin factor, i.e., M ⊠_{e_{N}} M^{op} = \overline{sp}R_{1}R_{2}, for some suitable hyperfinite

subfactors R_{1}, R_{2} (4.3). Also, we prove a general ergodicity property for the higher relative commutants of a subfactor which is quite useful in applications (4.8, 4.9). In Sec. 5 and 6 we relate the amenability properties of G_{N,M}, Γ_{N,M} and (M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op}), obtaining a number of equivalent characterizations of the

amenability for standard lattices and graphs (5.3, 6.1, 6.3, 6.4). In Sec. 7 we discuss the case when M ⊠_{e_{N}} M^{op} is hyperfinite, proving this equiv-

alent to the amenability of N ⊂ M and to various other properties of the representation theory of N ⊂ M (7.1). For instance, we show that for hyperfinite subfactors it is enough that the universal graph Γ_{N,M}^{u,rf} be amenable for Γ_{N,M} to follow amenable (7.6). We also prove here the hereditarity property for amenable inclusions (7.5). The proof uses the characterization of the amenability for N ⊂ M by the hyperfiniteness condition on M ⊠_{e_{N}} M^{op}, a fact

that roughly reduces the argument to Connes’ hereditarity of hyperfiniteness for single type II_{1} factors. Sec.8 contains the proof of the Effros-Lance type characterization of amenability (8.1). Finally, in Sec. 9 we introduce the property T for standard lattices and prove some results about this notion. For most notations and general technical background we refer the reader to ([Po2,4,7]). More specific references are made in the text. For the reader’s convenience we included an Appendix which, besides the already mentioned relative Dixmier property for subfactors of finite index, contains a generalized version of Connes’ joint distribution trick needed in the proof of the Følner condition for graphs. The results on amenability in this paper were presented by the author in lectures and seminars, during 1991-1997. A more formal announcement of these results, with sketches of proofs, appeared in [Po5], while a couple of statements on the equivalence of the definition of amenability with representations and the Kesten condition, respectively Følner condition, were already announced in [Po2], resp. [Po4]. A rather complete discussion of the role of amenability within the overall classification of subfactors, with a presentation of most of the results in this paper (including the ones on the property T) appeared in [Po11].

1. Symmetric Enveloping C^{∗}-Algebras

In this and the next section we discuss the definition and basic properties of the symmetric enveloping algebras (C^{∗} in this section and von Neumann in the next) associated to an extremal inclusion of factors with finite index, as introduced in [Po5]. The statements below are similar to the ones in §1 of [Po5], but the proofs, that are only briefly sketched there, are given here in details. So let N ⊂ M be an inclusion of type II_{1} factors with finite Jones index, [M : N] < ∞, which we assume to be extremal, i.e., [pMp : Np] = [M : N]τ(p)^{2}, ∀ p ∈ P(N′ ∩ M). We denote by M ⊂ M_{1} = ⟨M, e_{N}⟩ the (abstract) basic construction for N ⊂ M, e_{N} being the projection implementing the trace preserving conditional expectation E_{N} of M onto N. We first construct the universal C^{∗}-algebra generated by mutually commuting copies of M, M^{op} and an e_{N}-like projection implementing the expectations E_{N}, E_{N^{op}} on them. A representation (π, π′) of (N ⊂ M, e_{N}, M^{op} ⊃ N^{op}) is a pair of unital ∗-representations π : M_{1} → B(H), π′ : M_{1}^{op} → B(H) such that [π(M), π′(M^{op})] = 0, π(e_{N}) = π′(e_{N}^{op}). Two such representations, (π_{1}, π′_{1}) on H_{1} and (π_{2}, π′_{2}) on H_{2}, are equivalent if there exists a unitary U : H_{1} → H_{2} such that Uπ_{1}(x)U^{∗} = π_{2}(x), Uπ′_{1}(x)U^{∗} = π′_{2}(x), ∀ x ∈ M_{1}. A representation (π, π′) on H is cyclic if ∃ξ ∈ H such that \overline{Alg(π(M_{1}), π′(M_{1}^{op}))ξ} = H. Note that if (π, π′) is a representation on H then there exists a representation (π̄, π̄′) on the conjugate Hilbert space H̄ defined by π̄(x) = \overline{π′(x^{∗op})}, π̄′(x^{op}) = \overline{π(x^{∗})}, x ∈ M_{1}, where T ↦ T̄ denotes the antiisomorphism from B(H) to B(H̄) implemented by the conjugation H ∋ ξ ↦ ξ̄ ∈ H̄. We denote by Ĉ the set of all equivalence classes of cyclic representants of (N ⊂ M, e_{N}, M^{op} ⊃ N^{op}) and by C a set of chosen representations for Ĉ such that if (π, π′) ∈ C then (π̄, π̄′) ∈ C.

1.1. Proposition. There exists a unital C^{∗}-algebra U with unital embeddings j : M_{1} ↪ U, j′ : M_{1}^{op} ↪ U such that

a) [j(M), j′(M^{op})] = 0,

b) j(e_{N}) = j′(e_{N}^{op}). and such that given any other unital C^{∗}-algebra U_{0} with unital embeddings j_{0} : M_{1} ↪ U_{0}, j′_{0} : M_{1}^{op} ↪ U_{0} satisfying a), b) (with (j′, j′_{0}) instead of (j, j′)), there exists a unital ∗-algebra morphism π : U → U_{0} such that

(∗)  π(j(x)) = j_{0}(x), π(j′(x^{op})) = j′_{0}(x^{op}),  ∀ x ∈ M_{1}.

Moreover, U is unique (up to an isomorphism (∗)) with these properties. Also, U is generated as a C^{∗}-algebra by j(M), j′(M^{op}), j(e_{N}) (= j′(e_{N}^{op})) and it has a unique antiautomorphism ^{op} such that j(x)^{op} = j′(x^{op}), (j′(x^{op}))^{op} = j(x), ∀ x ∈ M_{1} (so in particular j(e_{N})^{op} = j′(e_{N}^{op})^{op} = j(e_{N})).

Proof. Put

U =^{def} C^{∗} ({ ⨁_{(π,π′)∈C} π(x), ⨁_{(π,π′)∈C} π′(y^{op}) | x, y ∈ M_{1}}) ,
j(x) =^{def} ⨁_{(π,π′)∈C} π(x), j′(x^{op}) =^{def} ⨁_{(π,π′)∈C} π′(x^{op}),  x ∈ M_{1}.

U, j, j′ this way defined clearly satisfy a), b), and (∗) and the uniqueness is then trivial. Then we can define ^{op} on U by:

U ∋ ⨁_{(π,π′)∈C} π(x) ↦ ⨁_{(π,π′)∈C} π̄′(x^{∗op}) ∈ U
U ∋ ⨁_{(π,π′)∈C} π′(x^{op}) ↦ ⨁_{(π,π′)∈C} π̄(x^{∗}) ∈ U

Q.E.D.

1.2. Definition. We denote by C_{u,max}^{∗}(M, e_{N}, M^{op}) the C^{∗}-algebra U constructed in 1.1 and call it the universal symmetric enveloping C^{∗}-algebra. Also we denote by C_{u,bin}^{∗}(M, e_{N}, M^{op}) =^{def} C_{max}^{∗}(M, e_{N}, M^{op})/ ∩ ker π, where the intersection is over all representations π of C_{max}^{∗}(M, e_{N}, M^{op}) for which π(M), π(M^{op}) are von Neumann algebras and call it the universal binormal symmetric enveloping C^{∗}-algebra associated with N ⊂ M (and the trace preserving expectation). We still denote by j, j′ the embeddings of M_{1}, M_{1}^{op} into C_{u,bin}^{∗}(M, e_{N}, M^{op}) resulting from the composition of the embeddings into C_{u,max}^{∗}(M, e_{N}, M^{op}) with the quotient map. Note that, with the notations in the proof of 1.1, if we let C_{bin}^{u} = {(π, π′) ∈ C | π(M), π′(M^{op}) are von Neumann algebras}, then C_{u,bin}^{∗}(M, e_{N}, M^{op}) can alternatively be defined as

C^{∗} ({ ⨁_{C_{bin}^{u}} π(x), ⨁_{C_{bin}^{u}} π′(y^{op}) | x, y ∈ M_{1}})

with

j(x) = ⨁_{C_{bin}^{u}} π(x), j′(x^{op}) = ⨁_{C_{bin}^{u}} π′(x^{op}).

Since (π, π′) ∈ C_{bin}^{u} implies (π̄, π̄′) ∈ C_{bin}^{u}, it follows that ^{op} implements an antiautomorphism on C_{u,bin}^{∗}(M, e_{N}, M^{op}), still denoted ^{op}, satisfying j′(x^{op}) = j(x)^{op}, j′(x^{op})^{op} = j(x).

In addition, C_{u,bin}^{∗}(M, e_{N}, M^{op}) satisfies the following universality property:

1.3. Proposition. Given any binormal representation (π_{o}, π′_{0}) of (N ⊂ M, e_{N}, M^{op} ⊃ N^{op}) on a Hilbert space H_{0} there exists a unital ∗-representation π : C_{u,bin}^{∗}(M, e_{N}, M^{op}) → B(H_{0}) such that π(j(x)) = π_{0}(x), π′(j′(x^{op})) = π′_{0}(x^{op}), ∀ x ∈ M_{1}. Moreover, C_{u,bin}^{∗}(M, e_{n}, M^{op}) has a faithful representation π̃ such that π̃(M), π̃(M^{op}) are von Neumann algebras. Also, C_{u,bin}^{∗}(M, e_{N}, M^{op}) with the embeddings j, j′ is unique (up to isomorphism) satisfying these properties.

Proof. Trivial. Q.E.D.

1.4. Lemma. Let N ⊂ M ⊂^{e_{1}} M_{1} ⊂^{e_{2}} M_{2} ⊂ ⋯ be the Jones tower for N ⊂ M, with e_{1} = e_{N}, and M ⊃^{e_{0}} N ⊃^{e_{−1}} N_{1} ⊃ ⋯ be a choice of a tunnel. Let S_{0} be a unital C^{∗}-algebra with unital ∗-embeddings j_{0} : M_{1} → S_{0}, j′_{0} : M_{1}^{op} → S_{0}, such that [j_{0}(M), j′_{0}(M^{op})] = 0, j_{0}(e_{N}) = j′_{0}(e_{N}). Then j_{0}, j′_{0} extend uniquely to ∗-embeddings of ⋃_{n≥1} M_{n}, ⋃_{n≥1} M_{n}^{op} into S_{0}, still denoted by j_{0}, j′_{0}, such that

j_{0}(e_{n+2}) = j′_{0}(e_{−n}^{op}), j′_{0}(e_{n+2}^{op}) = j_{0}(e_{−n}), n ≥ 0.

Proof. Trivial by the abstract characterization of the basic contruction in ([PiPo2]. [Po2]). Q.E.D.

1.5. Lemma. Let ⋯ ⊂^{e_{−1}} N ⊂^{e_{0}} M ⊂^{e_{1}} M_{1} ⊂ ⋯, S_{0}, j_{0}, j′_{0} be like in 1.4. Then we have

Alg(j_{0}(M_{1}), j′_{0}(M_{1}^{op})) = Alg(j_{0}(M), j_{0}(e_{N}) = j′_{0}(e_{N}), j′_{0}(M^{op}))

= ⋃_{n} sp j′_{0}(M^{op})j_{0}(M_{n})j′_{0}(M^{op})

= ⋃_{n} sp j_{0}(M)j′_{0}(M_{n}^{op})j_{0}(M)

= ⋃_{n} sp j′_{0}(M^{op})j_{0}(M)j_{0}(f_{−n}^{n})j_{0}(M)j′_{0}(M^{op}),

where f_{−n}^{n} is the Jones projection for N_{n−1} ⊂ M ⊂ M_{n} obtained as a scalar multiple of the word of maximal length in e_{−n+2}, …, e_{0}, e_{1}, …, e_{n} (cf. [PiPo2]) and it satisfies j_{0}(f_{−n}^{n}) = j′_{0}((f_{−n}^{n})^{op}). Similarly, for any i ∈ ℤ we have

Alg(j_{0}(M), j_{0}(e_{N}), j′_{0}(M^{op})) = ⋃_{n} sp j′_{0}(M_{i}^{op})j_{0}(M_{n})j′_{0}(M_{i}^{op}),

where M_{i} = N_{−i−1} for i ≤ −1, M_{0} = M, M_{−1} = N.

Proof. It is sufficient to show that

⋃_{n} sp j′(M^{op})j(M)j(f_{−n}^{n})j(M)j(M^{op})

is an algebra. If we denote by f_{−2n}^{0} the Jones projection for N_{2n−1} ⊂ N_{n−1} ⊂ M and by f_{0}^{2n} the one for M ⊂ M_{n} ⊂ M_{2n}, as in [PiPo2], then we have M_{n} = sp Mf_{−n}^{n}M, M = sp N_{n−1}f_{−2n}^{0}N_{n−1} so that we get:

j′(M^{op})j(M)j(f_{−n}^{n})j(M)j′(M^{op})j(f_{−n}^{n})j(M)j′(M^{op})

⊂ sp j′(M^{op})j(M)j(f_{−n}^{n})(j(f_{−2n}^{0})j(N_{n−1}))(j′(N_{n−1}^{op})j′((f_{−2n}^{0})^{op})j′(N_{n−1}^{op}))

· j(f_{−n}^{n})j(M)j′(M^{op})

= sp(j′(M^{op})j′(N_{n−1}^{op}))(j(M)j(N_{n−1}))(j(f_{−n}^{n})j(f_{−2n}^{0})j′((f_{−2n}^{0})^{op})j(f_{−n}^{n}))

· (j(N_{n−1})j(M))(j′(N_{n−1}^{op})j′(M^{op}))

= sp j′(M^{op})J(M)j(f_{−2n}^{2n})j(M)j′(M^{op})

in which we used that [j(N_{n−1}), j(f_{−n}^{n})] = 0, [j′(N_{n−1}^{op}), j(f_{−n}^{n})] = 0 and f_{−2n}^{2n} = λ^{−n}f_{−n}^{n}f_{0}^{2n}f_{−2n}^{0}f_{−n}^{n}, j(f_{0}^{2n}) = j′((f_{−2n}^{0})^{op}), λ = [M : N]^{−1}. Q.E.D.

1.6. Corollary. Let ⋯ ⊂ N_{1} ⊂^{e_{−1}} N ⊂^{e_{0}} M ⊂^{e_{1}} M_{1} ⊂ ⋯, f_{−n}^{n} be as

in 1.5. Then C_{u,max}^{∗}(M, e_{N}, M^{op}) (respectively C_{u,bin}^{∗}(M, e_{n}, M^{op})) is generated, as a C^{∗}-algebra, by j(M), j(f_{−n}^{n}) = j′(f_{−n}^{n}), j′(M^{op}) and there exists a natural isomorphism of C_{u,max}^{∗}(M, e_{N}, M^{op}) (respectively C_{u,bin}^{∗}(M, e_{N}, M^{op})) onto C^{∗}(M, e_{N_{n−1}}, M^{op}) (respectively C_{bin}^{∗}(M, e_{N_{n−1}}, M^{op})), taking the canonical images of the elements in M, M^{op} in one algebra into the corresponding canonical images in the other algebra and j(f_{−n}^{n}) onto j(e_{N_{n−1}}).

Proof. Trivial by definitions and 1.5. Q.E.D.

1.7. Lemma. With the notations of 1.4 and 1.5, assume in addition that j_{0}(M), j_{0}(e_{N}) = j′_{0}(e_{N}), j′_{0}(M^{op}) generate S_{0} as a C^{∗}-algebra, and that the following condition is satisfied:

(∗)  j_{0}(M′ ∩ M_{k}) ⊂ j′_{0}(M^{op}), ∀k ≥ 1

Then j_{0}(M_{i})′ ∩ S_{0} = j′_{0}(M_{−i}^{op}), (j′_{0}(M op_{i}))′ ∩ S_{0} = j_{0}(M_{−i}), ∀ i ∈ ℤ, and for all

k, i in ℤ one has j_{0}(M′_{i} ∩ M_{k}) = j_{0}(M_{i})′ ∩ j_{0}(M_{k}) = j′_{0}(M_{−i}^{op}) ∩ (j′_{0}(M_{−k}^{op}))′ = j′_{0}((M_{−k}^{op})′ ∩ M_{−i}^{op}). Also, if x ∈ M′_{−n} ∩ M_{n} and x′ denotes the canonical conjugate of x (= Jx^{∗}J) ([Po2]), then j_{0}(x′) = j′(x^{op}). Moreover, for each i ∈ ℤ there exist unique conditional expectations E_{i}^{+} : S_{0} → j_{0}(M_{i})′ ∩ S_{0} = j′_{0}(M_{−i}^{op}), E_{i}^{−} : S_{0} → j′_{0}(M_{i}^{op})′ ∩ S_{0} = j_{0}(M_{−i}) such that E_{i}^{+}(j_{0}(x)) = j_{0}(E_{M′_{i}∩M_{n}}(x)), E_{i}^{−}(j′_{0}(x^{op})) = j′_{0}(E_{M′_{i}∩M_{n}}(x)^{op}), ∀ x ∈ M_{n}, n ≥ i, which satisfy E_{i}^{+} = E_{i}^{+}(j_{0}(u) · j_{0}(u^{∗})), E_{i}^{−} = E_{i}^{−}(j′_{0}(u^{op}) · j′_{0}(u^{op})^{∗}), ∀ u ∈ U(M_{i}).

Proof. Since j_{0}(M′ ∩ M_{k}) ⊂ M^{op} and [j_{0}(M_{k}), j′_{0}(M_{−k}^{op}] = 0, it follows that j_{0}(M′ ∩ M_{k}) ⊂ j′_{0}(M_{−k}^{op})′ ∩ j′_{0}(M^{op}) = j′_{0}(M_{−k}^{op}′ ∩ M^{op}). But the two finite dimensional algebras involved in this inclusion have the same dimension, so they actually follow equal. By averaging over unitaries in M_{i} it then follows that j_{0}(M′_{i} ∩ M_{k}) ⊂ j′_{0}(M^{op}), ∀i ≥ 1, giving in a similar way j_{0}(M′_{i} ∩ M_{k}) = j′_{0}(M_{−k}^{op}′ ∩ M_{−i}^{op}). Then by duality isomorphisms these equalities follow for arbitrary i, k ∈ ℤ. By the relative Dixmier property for subfactors of finite index (see the Appendix, A.1), if for x ∈ M_{n} we denote C_{M_{i}}(x) = \overline{co}^{n}\{uxu^{∗} | u ∈ U(M_{i})\}∩M′_{i} ∩M_{n} then C_{M_{i}}(x) = {E_{M′_{i}∩M_{n}}(x)} and ∀ x_{1}, …, x_{k} ∈ M_{n}, ∀ ε > 0, ∃ u_{1}, …, u_{m} ∈ U(M_{i}) such that

∥\frac{1}{m} ∑_{l=1}^{m} u_{l}x_{j}u_{l}^{∗} − E_{M′_{i}∩M_{n}}(x_{j})∥ < ε, j = 1, 2, …, k.

Since, by 1.5 we have

Alg(j_{0}(M_{1}), j′_{0}(M_{1}^{op})) = ⋃_{n} sp j′_{0}(M_{−i}^{op})j_{0}(M_{n})j′_{0}(M_{−i}^{op}),

which is dense in S_{0}, it follows that

j_{0}(M_{i})′ ∩ S_{0} = ⋃_{n} j′_{0}(M_{−i}^{op})j_{0}(M′_{i} ∩ M_{n})j′_{0}(M_{−i}^{op})

= ⋃_{n} j′_{0}(M_{−i}^{op})j′_{0}((M_{−n}^{op})′ ∩ (M_{−i}^{op}))j′_{0}(M_{−i}^{op})

= j′_{0}(M_{−i}^{op}).

Also, it follows that if T = ∑_{l} j′_{0}(y_{1,l}^{op})j_{0}(x_{l})j′_{0}(_{2,l}^{op}), for some y_{1,l}, y_{2,l} ∈ M_{−i}, x_{l} ∈ M_{n}, and we denote by C_{i,S_{0}}(T) = \overline{co}^{n}\{j_{0}(u)T j_{0}(u^{∗}) | u ∈ U(M_{i})\} ∩ (j_{0}(M_{i}))′ ∩ S_{0}, then

C_{i,S_{0}}(T) = {∑_{l} j′_{0}(y_{1,l}^{op})j_{0}(E_{M′_{i}∩M_{n}}(x_{l}))j′_{0}(y_{2,l}^{op})}

is a single point set. Also, C_{i,S_{0}}(αT_{1} + βT_{2}) ⊂ αC_{i,S_{0}}(T_{1}) + βC_{i,S_{0}}(T_{2}) and 1 ≥ T ≥ 0 implies 1 ≥ T′ ≥ 0, ∀ T′ ∈ C_{i,S_{0}}(T). It follows that

Alg{j_{0}(M_{1}), j′_{0}(M_{1}^{op})} ∋ T = ∑_{l} j′_{0}(y_{1,l}^{op})j_{0}(x_{l})j′_{o}(y_{2,l}^{op})

↦ ∑_{l} j′_{0}(y_{1,l}^{op})j_{0}(E_{M′_{i}∩M_{n}}(x_{l}))j′_{0}(y_{2,l}^{op}) ∈ (j_{0}(M_{i}))′ ∩ S_{0} = j′_{0}(M_{−i}^{op})

is a well defined positive linear norm one projection onto j′_{0}(M_{−i}^{op}) and the rest of the statement is then clear by continuity. Q.E.D.

1.8. Definition. We denote C_{max}^{∗}(M, e_{N}, M^{op}) =^{def} C_{u,max}^{∗}(M, e_{N}, M^{op})/ ∩ ker π, where the intersection is over all smooth representations π of C_{u,max}^{∗}(M, e_{N}, M^{op}), i.e., representations satisfying the following smoothness condition (or axiom):

(∗)  π(j(M′ ∩ M_{i})) ⊂ π(j′(M^{op})), i ∈ ℕ.

Note that by 1.7 this condition actually implies π(j(M′_{k} ∩ M_{i})) = π(j′(M_{−i}^{op}′ ∩ M_{−k}^{op})), ∀i, k ∈ ℤ. We call C_{max}^{∗}(M, e_{N}, M^{op}) the symmetric enveloping C^{∗}-algebra associated with N ⊂ M. Similarly, we put C_{bin}^{∗}(M, e_{N}, M^{op}) =^{def} C_{u,bin}^{∗}(M, e_{N}, M^{op})/ ∩ ker π, where the intersection is over all representations π of C_{u,bin}^{∗}(M, e_{N}, M^{op}) such that π(j(M)), π(j′(M^{op})) are von Neumann algebras and such that axiom (∗) is satisfied. We call it the binormal symmetric enveloping C^{∗}-algebra associated with N ⊂ M. Note that, since B(L^{2}(M)), with the representation of M and M^{op} as operators of left and right multiplication by elements in M and e_{N} = proj_{L^{2}(N)} ∈ B(L^{2}(M)), does satisfy the condition (∗), both these symmetric enveloping C^{∗}-algebras are non-degenerate.

We still denote by j, j′ the canonical embeddings of M_{1}, M_{1}^{op} in C_{max}^{∗}(M,e_{N}, M^{op}) and C_{bin}^{∗}(M, e_{N}, M^{op}). Note that the same argument as in 1.2 shows that the antiautomorphism ^{op} on C_{u,max}^{∗}(M, e_{N}, M^{op}) (respectively ^{op} on C_{u,bin}^{∗}(M, e_{N}, M^{op})) implements an antiautomorphism, still denoted by ^{op}, on C_{max}^{∗}(M, e_{N}, M^{op}) (resp. ^{op} on C_{bin}^{∗}(M, e_{N}, M^{op})). Also, by universality properties of C_{u,max}^{∗}(M, e_{N}, M^{op}) and C_{u,bin}^{∗}(M, e_{N}, M^{op}) and the definitions, it follows that given any C^{∗}-algebra S_{0} generated by copies of M, M^{op}, e_{N} satisfying 1.1 a), b), such that the corresponding tunnel-towers {M_{i}}_{i}, {M_{j}^{op}}_{j} (cf. 1.4) satisfy the smoothness axiom 1.8 (∗), there exists a natural ∗-morphism of C_{max}^{∗}(M, e_{N}, M^{op}) onto S_{0} carrying j(M_{i}), j′(M_{j}^{op}) onto the corresponding images of M_{i}, M_{j}^{op} (⊂ S_{0}). If in addition S_{0} ⊂ B(H_{0}) is so that the images of M, M^{op} are weakly closed, then this morphism factors to a ∗-morphism of C_{bin}^{∗}(M, e_{N}, M^{op}). The above can be regarded as the universality property satisfied by C_{max}^{∗}(M, e_{N}, M^{op}) and C_{bin}^{∗}(M, e_{N}, M^{op}). Moreover, as a consequence of the prior results and definitions, if we denote by S either of these two algebras, then the following properties hold true: 1.9.1. j(M_{i})′ ∩ S = j′(M_{−i}^{op}), j′(M_{−i}^{op})′ ∩ S = j(M_{i}), ∀ i ∈ ℤ. 1.9.2. If x ∈ M′_{−n} ∩ M_{n} and x′ denotes the canonical conjugate of x (= Jx^{∗}J) then j(x′) = j′(x^{op}). 1.9.3. There exist unique conditional expectations E_{i}^{−} : S → j(M_{−i}), E_{i}^{+} : S → j′(M_{−i}^{op}) such that E_{i}^{−}(j′(x^{op})) = j′(E_{M′_{i}∩M_{n}}(x)^{op}), E_{i}^{+}(j′(x)) = j(E_{M′_{i}∩M_{n}}(x)), ∀ x ∈ M_{n}, n ≥ i. Also, these expectations satisfy E_{i}^{−} = E_{i}^{−}(j′(u^{op}) · j′(u^{op∗})), E_{i}^{+} = E_{i}^{+}(j(u) · j(u^{∗})), ∀ u ∈ U(M_{i}). 1.9.4. C_{max}^{∗}(M, e_{N_{n−1}}, M^{op}) (resp. C_{bin}^{∗}(M, e_{N_{n−1}}, M^{op})) naturally identifies with C_{max}^{∗}(M, e_{N}, M^{op}) (resp. C_{bin}^{∗}(M, e_{N}, M^{op})), as in 1.6.

1.10. Remarks. 1^{∘}. Note that the smoothness condition 1.8 (∗) is redundant if M′ ∩ M_{n} = Alg{1, e_{1}, e_{2}, ..., e_{n}}, ∀n, i.e., in the case the graph of N ⊂ M is of the form Γ_{N,M} = A_{n} for some n ≤ ∞. 2^{∘}. In the case S_{0} ⊂ B(H) is so that j_{0}(M), j′_{0}(M^{op}) are von Neumann algebras (e.g., if S_{0} = C_{bin}^{∗}(M, e_{N}, M^{op})) then one can give another proof to Lemma 1.7, which doesn’t use the relative Dixmier property, as follows: if M is weakly separable (i.e., M has separable predual) then take R ⊂ M to be a hyperfinite subfactor such that R′ ∩ M_{∞} = M′ ∩ M_{∞} (cf. [Po2,9]), so in particular R′ ∩M_{n} = M′ ∩M_{n}, ∀ n (here M_{∞} = \overline{∪M_{n}}^{w} as usual). Then denote by Φ the conditional expectation of B(H) onto j_{0}(R)′ ∩ B(H), obtained by averaging over a suitable amenable subgroup of U(R). Then clearly Φ|_{S_{0}} = E_{0}^{+} and the other expectations are obtained similarly. If M is not separable one can still apply [Po2,9] to get that ∀ B ⊂ ∪j(M_{n}) countably generated, ∃ R ⊂ M such that E_{R′∩M_{n}}(B) = E_{M′∩M_{n}}(B), ∀ n, and the rest of the proof is then similar. 3^{∘}. The considerations in this section are easily seen to cary over to the case when instead of an extremal inclusion of type II_{1} factors N ⊂ M (with trace preserving expectation) we take an extremal inclusion of factors of type III, N ⊂ M ([Po3]). However, in this more general case, some adjustements of the argument in 1.7 are needed, depending on the nature of the inclusion. Then, if E denotes the expectation of minimal index of M onto N, an argument similar to 2^{∘} above can be used to prove the existence of a unique conditional expectation E^{0} from C_{bin}^{∗}(M, e_{N} , M^{op}) onto its C^{∗}-subalgebra generated by M and M^{op}.

2. Symmetric Enveloping type II_{1} Factors

2.1. Theorem. There exists a unique trace state tr on C_{max}^{∗}(M, e_{N}, M^{op}) and the corresponding ideal trace I_{tr} = {x ∈ C_{max}^{∗}(M, e_{N}, M^{op}) | tr(x^{∗}x) = 0} is the unique maximal ideal in C_{max}^{∗}(M, e_{N}, M^{op}). In particular, there exists a unique state τ_{0} on each quotient C^{∗}-algebra S_{0} of C_{max}^{∗}(M, e_{N}, M^{op}) (in particular on C_{bin}^{∗}(M, e_{N}, M^{op})) and its ideal is the unique maximal ideal of S_{0}.

Proof. By the uniqueness properties of the expectations E_{i}^{+}, i ∈ ℤ, of a C^{∗}- algebra S_{0} generated by j_{0}(M_{1}), j′_{0}(M_{1}^{op}) onto j′_{0}(M_{−i}^{op}) like in 1.6, it follows

that E_{i}^{+} = E_{j′_{0}(N_{i−1}^{op})}^{j′_{0}(M^{op})} ∘ E_{0}^{+}. Let τ be the trace on j_{0}(M) and τ′ the trace on

j′_{0}(M^{op}) and define τ_{0} = τ′ ∘ E_{0}^{+} on S_{0}. Since E_{j′_{0}(N_{i−1}^{op})}^{j′_{0}(M^{op})} is τ′ preserving, we have for i ≥ 0, x ∈ S_{0}:

τ_{0}(x) = τ′(E_{0}^{+}(x)) = τ′ (E_{j′_{0}(N_{i−1}^{op})}^{j′_{0}(M^{op})}(E_{0}^{+}(x))) = τ′ ∘ E_{i}^{+}(x).

If k ≥ i, u ∈ U(j_{0}(M_{i})), x ∈ j_{0}(M_{k}), y′, y′′ ∈ j′_{0}(N_{i−1}^{op}) then we have:

τ_{0}(uy′xy′′u^{∗}) = τ_{0}(y′uxu^{∗}y′′) = τ′(E_{i}^{+}(y′uxu^{∗}y′′)) = τ′(y′E_{i}^{+}(uxu^{∗})y′′)

= τ′(y′E_{j_{0}(M′_{i}∩M_{k})}(uxu^{∗})y′′) = τ′(y′E_{i}^{+}(x)y′′) = τ′(E_{i}^{+}(y′xy′′))

= τ_{0}(y′xy′′).

Thus, by 1.6 it follows that τ_{0}(uT u^{∗}) = τ_{0}(T), ∀ T ∈ S_{0}, ∀ u ∈ U(M_{i}). Also, if u′ ∈ j′_{0}(M^{op}) is a unitary element and x ∈ j_{0}(M_{k}), y′, y′′ ∈ j′_{0}(M^{op}) then we get:

τ_{0}(u′y′xy′′u′^{∗}) = τ′(E_{0}^{+}(u′^{∗}y′xy′′u′)) = τ′(u′y′E_{0}^{+}(x)y′′u′^{∗})
= τ′(y′E_{0}^{+}(x)y′′) = τ′(E′_{0}(y′xy′′)) = τ_{0}(y′xy′′).

This shows that τ_{0}(u′T u′^{∗}) = τ_{0}(T), ∀ T ∈ S_{0}, ∀ u′ ∈ U(j′_{0}(M^{op})), by virtue of 1.6. Since the centralizer of τ_{0} is an algebra and it contains both U(j_{0}(M_{i})), U(j′_{0}(M^{op})), with i ≥ 1, τ_{0} has all S_{0} = C^{∗}(j_{0}(M_{i}), j′_{0}(M^{op})) in its centralizer, thus, it is a trace. If τ_{1} is another trace on S_{0} and (π_{τ_{1}}, H_{τ_{1}}, E_{τ_{1}}) is the corresponding GNS construction, then let S_{0} = \overline{π_{τ_{1}}(S_{0})}^{w}. Since the unit ball of π_{τ_{1}}(M_{k}) is complete in the norm given by ∥π_{τ_{1}}(x)ξ_{τ_{1}}∥ (because the unit ball of j_{0}(M_{k}) is complete in the norm τ_{1}(x^{∗}x)^{1/2}, by the uniqueness of the trace on the factor M_{k}) it follows that \overline{π_{τ_{1}}(M_{k})} = π_{τ_{1}}(M_{k}). Thus, for x ∈ j_{0}(M_{k}), y′, y′′ ∈ j′_{0}(M^{op}) we get:

τ_{1}(y′xy′′) = ⟨E_{π_{τ_{1}} (j_{0}(M))′∩S_{0}}(π_{τ_{1}}(y′xy′′))ξ_{τ_{1}}, ξ_{τ_{1}}⟩
= ⟨(π_{τ_{1}}(y′)E_{π_{τ_{1}} (j_{0}(M))′∩S_{0}}(π_{τ_{1}}(j_{0}(x))π_{τ_{1}}(y′′)ξ_{τ_{1}}, ξ_{τ_{1}}⟩
= ⟨π_{τ_{1}}(y′E_{j_{0}(M′∩M_{k})}(x)y′′)ξ_{τ_{1}}, ξ_{τ_{1}}⟩
= τ′(y′E_{j_{0}(M′∩M_{k})}(x)y′′)
= τ_{0}(y′xy′′).

with the last part following from the uniqueness of the trace on j′_{0}(M^{op}). This shows that S_{0} has a unique trace τ_{0} and also that if I_{0} ⊂ S_{0} is a two sided closed proper ideal then S_{1} = S_{0}/I_{0} has a trace, which thus composed with the quotient map gives the trace on S_{0}. Thus, I_{0} ⊂ I_{τ_{0}}, so I_{τ_{0}} is the unique maximal ideal of S_{0}. Q.E.D.

2.2. Remarks. 1^{∘}. Let

C_{min}^{∗}(M, e_{N}, M^{op}) =^{def} C_{max}^{∗}(M, e_{N}, M^{op})/I_{tr} (≃ π_{tr}(C_{max}^{∗}(M, e_{N}, M^{op}))),

where I_{tr} is the trace ideal (= maximal ideal) of C_{max}^{∗}(M, e_{N}, M^{op}) corresponding to the unique trace tr, as given by 2.1. From the previous theorem and its proof if follows that C_{min}^{∗}(M, e_{N}, M^{op}) is simple, has a unique trace, still denoted tr, and has the Dixmier property, i.e., \overline{co}^{n}\{uxu^{∗} | u ∈ U(C_{min}^{∗}(M, e_{N}, M^{op}))} ∩ ℂ1 = {tr(x)1}, ∀ x ∈ C_{min}^{∗}(M, e_{N}, M^{op}). In fact, by 2.1 any C^{∗}-algebra S_{0} generated by mutually commuting copies of M, M^{op} and a projection e_{N} such that N ⊂ M ⊂^{e_{N}} Alg(M, e_{N}) and N^{op} ⊂ M^{op} ⊂^{e_{N}} Alg(M^{op}, e_{N}) are basic constructions and such that the smoothness condition 1.8 (∗) is satisfied, has a unique trace tr, I_{tr} is its unique maximal ideal and S_{0}/I_{tr} = C_{min}^{∗}(M, e_{N}, M^{op}). Also it should be noted that if N = M then C_{max}^{∗}(M, e_{N}, M^{op}) coincides with M ⊗_{max} M^{op}, C_{bin}^{∗}(M, e_{N}, M^{op}) with M ⊗_{bin} M^{op} (as considered in [EL]) and

C_{min}^{∗}(M, e_{N}, M^{op}) with M ⊗_{min} M^{op}.

2^{∘}. Let N ⊂ M be an extremal inclusion of von Neumann factors of type III, with the conditional expectation of minimal index E, as in 1.10.3^{∘}. The construction analoguous to 2.1 is then as follows: one first considers the expectation E^{0} given by 1.10.3^{∘}; one takes a normal faithful state φ on M such that φ ∘ E = φ; instead of the trace tr, one defines a state ψ on C_{bin}^{∗}(M, e_{N} , M^{op}) by ψ = (φ ⊗ φ^{op}) ∘ E^{0}.

2.3. Corollary.

S = π_{tr}(C_{max}^{∗}(M, e_{N}, \overline{M^{op}))}^{w} ≃ π_{tr}(C_{bin}^{∗}(M, e_{N}, \overline{M^{op}))}^{w}

is a type II_{1} factor with embeddings π_{tr} ∘ j : M_{1} → S, π_{tr} ∘ j′ : M_{1}^{op} → S and an antisymmetry ^{op} such that

a) [π_{tr}(j(M)), π_{tr}(j′(M^{op}))] = 0

b) π_{tr}(j(e_{N})) = π_{tr}(j′(e_{N}))

c) S = vN(π_{tr}(j(M)), π_{tr}(j(e_{N})), π_{tr}(j′(M^{op})))

d) π_{tr}(j(x))^{op} = π_{tr}(j′(x^{op})), ∀ x ∈ M, π_{tr}(j(e_{N}))^{op} = π_{tr}(j(e_{N})). Moreover, if S_{0} is another type II_{1} factor with embeddings j_{0} : M_{1} → S_{0}, j′_{0} : M_{1}^{op} → S_{0} satisfying conditions a), b) (with j_{0} instead of π_{tr} ∘ j and j′_{0} instead of π_{tr} ∘ j′) and such that j_{0}(M′ ∩ M_{n}) ⊂ j′_{0}(M^{op}), ∀n ≥ 1, then there exists a unique isomorphism σ of S into S_{0} such that j_{0} = σ ∘ π_{tr} ∘ j, j′_{0} = σ ∘ π_{tr} ∘ j′. And if in addition S_{0} = vN(j_{0}(M), j_{0}(e_{N}), j′_{0}(M^{op})), then σ is onto.

Proof. Trivial by 2.1. Q.E.D.

2.4. Definition. We denote by M ⊠_{e_{N}} M^{op} the type II_{1} factor S =

π_{tr}(C_{max}^{∗}(M, e_{N}, \overline{M^{op}))} in the previous corollary and call it the symmetric enveloping type II_{1} factor associated with N ⊂ M. Also, we call M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op} the symmetric enveloping inclusion associated with N ⊂ M. We will

identify M, e_{N}, M^{op} with their corresponding canonical images in M ⊠_{e_{N}} M^{op},

more generally we will identify M_{n}, M_{n}^{op}, e_{n} with their canonical images via π_{tr} ∘ j, π_{tr} ∘ j′ (cf 1.4), whenever some tunnel for N ⊂ M is chosen. We’ve seen in 2.3 that M ⊠_{e_{N}} M^{op} has an antisymmetry^{op} and that it satisfies a universality

and uniqueness property. Also, from now on we will use the notation τ for the unique trace on the factor M ⊠_{e_{N}} M^{op} (as in fact for any generic factor).

2.5. Remarks. 1^{∘}. As one can see from 2.1-2.3, the symmetric enveloping type II_{1} factor M ⊠_{e_{N}} M^{op} associated to an inclusion N ⊂ M can be constructed out

of any C^{∗}-algebra S_{0} generated by copies of M and M^{op}, satisfying M′ ∩ S_{0} = M^{op}, and by a projection e_{N}, implementing the expectations E_{N} on M and E_{N^{op}} on M^{op}: just put M ⊠_{e_{N}} M^{op} to be the completion of the algebra S_{0}/I_{0}

in the strong topology given by its unique trace, I_{0} being the maximal ideal of S_{0} or, alternatively, the ideal corresponding to the unique trace on S_{0}. In particular, M ⊠_{e_{N}} M^{op} = \overline{C^{∗}(M, e_{N}, J_{M}MJ_{M})/I_{0}}. But one can also construct

M ⊠_{e_{N}} M^{op} by defining directly the Hilbert space of its standard representation.

In order to show this, we will in fact consider a more general construction. Thus, let N ⊂ M and Q ⊂ P be extremal inclusions with the same extended higher relative commutant picture (or extended standard invariant), i.e., ˜→G_{N,M} = tilde → G_{Q,P} = {A_{ij}}_{i,j∈ℤ}. The concatenation algebra associated to these two inclusions is then the unique (up to isomorphism) type II_{1} factor S generated by commuting copies of M, P^{op} and by a projection e, implementing both E_{N} and E_{Q^{op}}, such that M′ ∩ S = P^{op}. This algebra is denoted by M ⊠_{e_{N} =e_{Q}} P^{op} (or simply M ⊠ P^{op}, when no confusion is possible). Its uniqueness follows the same way as the uniqueness of M ⊠_{e_{N}} M^{op} above. To prove its existence, we

consider the following construction: Take {m_{j}}_{j∈J} to be be an orthonormal basis of A_{−∞,∞} over A_{−∞,0} ∨ A_{0,∞} and identify A_{−∞,0} ∨ A_{0,∞} with its image in M ¯⊗P^{op} (through the choice of tunnels in M and P); note that the m_{j}’s can be chosen of bounded norm and such that the set of indices J can be written as ∪_{n}J_{n}, where each J_{n} is finite and such that ∑_{j∈J_{n}} m_{j}M ∨ M^{op} is a M ∨ M^{op}- bimodule of finite dimension (equivalently, ∑_{j∈J_{n}} m_{j}P ∨ P^{op} is a P ∨ P^{op}- bimodule of finite dimension) See 4.5 below for how to get these m_{j}’s. Then let H_{n} =^{def} ∑_{j∈J_{n}} m_{j}L^{2}(M ∨ P^{op}) and H =^{def} ∨_{n}H_{n}, the scalar product on H being defined by ⟨m_{j}ξ, m_{i}η⟩ = ⟨E_{A_{−∞,0}∨A_{0,∞}}(m_{i}^{∗}m_{j})ξ, η⟩, ∀ξ, η ∈ L^{2}(M ∨P^{op}). Finally, we let M, P^{op} and e = e_{N} = e_{P^{op}} act on H as follows: M (and similarily P^{op}) acts on each H_{n} by multiplication to the left, according to the relations Mm_{i} ⊂ ∑_{j∈J_{n}} m_{j}(M ∨ A_{0,∞}), ∀i ∈ J_{n}, with the latter vector space being identified with a subset of H_{n}; e acts also by multiplication to the left, by regarding H as a left A_{−∞,∞} module in the obvious way and letting e = e_{1}. Then M ⊠_{e_{N} =e_{Q}} P^{op} is simply the von Neumann algebra generated by M, P^{op}, e_{1}

on H. It is easy to check that these actions of M, P^{op}, e on H are well defined, that they satisfy M′ ∩ C^{∗}(M, e, P^{op}) = P^{op}, exe = E_{N}(x)e, eye = E_{P^{op}}(y)e, for x ∈ M, y ∈ P^{op}, and that ⟨ · ˆ1, ˆ1⟩ defines a trace on C^{∗}(M, e, P^{op}). This shows the existence of the concatenation algebra. Note that, by using the same proofs as for M ⊠_{e_{N}} M^{op}, it follows that the con-

catenation algebra has similar properties as the ones the symmetric enveloping algebras are shown to have in this section and in Sec. 4. Obviously, when (Q ⊂ P) ≃ (N ⊂ M) this algebra coincides with the symmetric enveloping type II_{1} factor associated with N ⊂ M. Note that any extremal hyperfinite subfactor N ⊂ R gives rise to a canonical non-separable concatenation algebra as follows: Let ω be a free ultrafilter on ℕ and denote by R^{ω} the corresponding ultrapower algebras associated to the hyperfinite factor R. Then (R′ ∩ R^{ω})′ ∩ R^{ω} = R and more generally (N′_{k} ∩ R^{ω})′ ∩ R^{ω} = N_{k}, ∀k, where R ⊃ N ⊃ N_{1} ⊃ ... is a tunnel for R ⊃ N (cf. [C1]). Thus, if we denote P^{0} = R′∩R^{ω} and Q^{0} to be the downward basic construction for P_{1}^{0} = N′ ∩R^{ω} ⊃ R′ ∩R^{ω} = P^{0} and put (Q ⊂ P ⊂ P_{1}) ≃ (Q^{0} ⊂ P^{0} ⊂ P_{1}^{0})^{op} then N ⊂ M and Q ⊂ P have the same higher relative commutant pictute (extended standard invariant) and the von Neumann algebra S generated by R and P_{1}^{op} = N′ ∩ R^{ω} is isomorphic to the concatenation of (N ⊂ R) and (Q ⊂ P) (see also Remark 2.11, 1^{∘} in [Po3], with caution to the obvious typos there...). 2^{∘}. For an extremal inclusion of type III factors N ⊂ M like in 1.10.3^{∘}, 2.2.2^{∘}, one defines its symmetric enveloping von Neumann algebra as π_{ψ}(C_{bin}^{∗}(M, e_{N} , \overline{M^{op}))}. It is easy to see that this algebra does not in fact depend on the normal faithful state φ, with φ = φ ∘ E, taken on M.

The next proposition summarizes the main properties of the factor M ⊠_{e_{N}} M^{op}

and its canonical subalgebras:

2.6. Proposition. M ⊠_{e_{N}} M^{op} with its subalgebras M_{i}, M_{i}^{op} projections e_{k},

i, j, k ∈ ℤ, and antisymmetry ^{op} satisfy the conditions:

a) [M, M^{op}] = 0;

b) e_{1}^{op} = e_{1} = e_{N}, e_{n}^{op} = e_{−n+2}, n ∈ ℤ, and ⋯ ⊂ N_{1} ⊂^{e_{−1}} N ⊂^{e_{0}} M ⊂^{e_{1}} M_{1} ⊂^{e_{2}}

M_{2} ⊂ ⋯ is a Jones tower-tunnel for N ⊂ M, where M_{0} = M, M_{−1} = N, M_{−n} = N_{n−1}, n ≥ 2.

c) ⋃_{n≥1} M_{i}M_{n}^{op}M_{i} = ⋃_{n≥1} M_{j}^{op}M_{n}M_{j}^{op} = Alg(M, e_{N} = e_{1}, M^{op}), ∀ i, j ∈ ℤ and

it is a dense ∗-subalgebra in M ⊠_{e_{N}} M^{op}.

d) M′_{j} ∩ M ⊠_{e_{N}} M^{op} = M_{−j}^{op} and (M_{j}^{op})′ ∩ M ⊠_{e_{N}} M^{op} = M_{−j}, ∀ j ∈ ℤ.

Proof. Clear by 1.9 and the definition of M ⊠_{e_{N}} M^{op}. Q.E.D.

The bicommutant relations in d) above can actually be taken as an abstract characterization of the symmetric enveloping algebra:

2.7. Proposition. Let N ⊂ M be an extremal inclusion and S be a type II_{1} von Neumann algebra containing M. If (M′∩S ⊂ N′∩S) ≃ (M^{op} ⊂ M_{1}^{op}), and S is generated by M and N′ ∩ S then M ∨ M′ ∩ S ⊂ S is naturally isomorphic

to M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op}

Proof. Let e_{0} ∈ M be a Jones projection for N ⊂ M and {m_{j}}_{j} an orthonormal basis of N over N_{1} = {e_{0}}′ ∩ N such that one of the m_{j}’s equals 1. For x ∈ S define E(x) = Σ_{j}m_{j}e_{0}xe_{0}m_{j}^{∗} ∈ S. Note that if x ∈ M′ ∩ S then m_{j} and e_{0} commute with x so E(x) = x. Also, if x ∈ N′ ∩ S then for each y ∈ M we have

yE(x) = y ∑_{j} m_{j}e_{0}xe_{0}m_{j}^{∗} = λ^{−1} ∑_{i,j} m_{i}e_{0}E_{N}^{M}(e_{0}m_{i}^{∗}ym_{j}e_{0})xe_{0}m_{j}^{∗}

= λ^{−1} ∑_{i,j} m_{i}e_{0}xE_{N}^{M}(e_{0}m_{i}^{∗}ym_{j}e_{0})e_{0}m_{j}^{∗} = ∑_{i} m_{i}e_{0}xe_{0}m_{i}^{∗}y = E(x)y

showing that [E(x), y] = 0. Thus E(x) ∈ M′ ∩ S. This shows that E is a norm one projection of N′ ∩ S onto M′ ∩ S so by Tomiyama’s theorem it is a conditional expectation. Also, if x ∈ N′ ∩ S then we have

τ(E(x)) = τ(Σ_{j}m_{j}e_{0}xe_{0}m_{j}^{∗}) = τ(xe_{0}Σ_{j}m_{j}^{∗}m_{j}e_{0})

= Σ_{j}τ(xe_{0}E_{N_{1}}^{N}(m_{j}^{∗}m_{j})) = Σ_{j}τ(E_{N′_{1}∩S}(xe_{0}E_{N_{1}}^{N}(m_{j}^{∗}m_{j}))) = λ^{−1}τ(xe_{0}) = τ(x)

Thus E is trace preserving as well, so it must coincide with the unique trace preserving expectation of N′ ∩S onto M′ ∩S. Also, from the definition of E(x), if x ∈ N′ ∩ S then e_{0}xe_{0} = E(x)e_{0}. Thus, if e_{1} ∈ N′ ∩ S is a Jones projection for (M′ ∩ S ⊂ N′ ∩ S) = (M^{op} ⊂ M_{1}^{op}) then e_{0}e_{1}e_{0} = λe_{0}. But

τ(e_{0})^{−1} = [M : N] = [M^{op} : N^{op}] = [M_{1}^{op} : M^{op}] = τ(e_{1})^{−1},

so that τ(e_{0}) = τ(e_{1}). Together with e_{0}e_{1}e_{0} = λe_{0}, this implies that e_{1}e_{0}e_{1} = λe_{1}. Thus, if x, y ∈ N then e_{1}(xe_{0}y)e_{1} = λxye_{1} = E_{N}^{M}(xe_{0}y)e_{1}, showing that e_{1} implements the conditional expectation E_{N}^{M}. But, by its definition, e_{1} also implements the conditional expectation of M′ ∩ S onto {e_{1}}′ ∩ (M′ ∩ S). Since we also have the isomorphism (M′∩S ⊂ N′∩S) ≃ (M^{op} ⊂ M_{1}^{op}), which in turn implements an isomorphism ({e_{1}}′ ∩M′ ∩S ⊂ M′ ∩S ⊂ N′ ∩S) ≃ (N^{op} ⊂ M^{op} ⊂ M_{1}^{op}), 2.3 applies to yield (M ∨M′ ∩S ⊂ S) ≃ (M ∨M^{op} ⊂ M ⊠_{e_{N}} M^{op})

. Q.E.D.

Note that from the above proposition and [Po2] it follows that if M is hyperfnite and the graph Γ_{N,M} of N ⊂ M is strongly amenable (see [Po2] for the definitions) then the inclusion M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op} is isomorphic to the in-

clusion M ∨M′ ∩M_{∞} ⊂ M_{∞}. The inclusions M ∨M′ ∩M_{∞} ⊂ M_{∞} for N ⊂ M hyperfinite with finite depth, i.e., with finite (thus strongly amenable) graph, were considered and extensively studied by Ocneanu ([Oc], see also [EvKa]). Note that if M is an arbitrary type II_{1} factor and N ⊂ M is a subfactor of finite depth and we denote by Q ⊂ P the standard model N^{st} ⊂ M^{st} then G_{Q,P} = G_{N,M} and M ∨ M′ ∩ M_{∞} ⊂ M_{∞} naturally identifies with the ”concatenation” inclusion considered in 2.5.1^{∘}, i.e., with M ∨ P^{op} ⊂ M ⊠ P^{op}. The next lemma provides some useful localization properties relating the Jones projections, the relative commutants and the antiisomorphism ^{op}. They are reminiscent of some well known facts (see e.g., [PiPo1] page 83, [Bi1] page 205).

2.8. Lemma. Let N ⊂ M be an extremal inclusion, N ⊂ M ⊂^{e_{N}} M_{1} its basic construction and ^{op} the canonical antiautomorphism of N′ ∩ M onto M′ ∩ M_{1} (so x^{op} = J_{M}x^{∗}J_{M}, x ∈ N′ ∩ M).

a) If x ∈ N′∩M then xe_{N} = x^{op}e_{N} and x^{op} is the unique element y′ ∈ M′∩M_{1} such that y′e_{N} = xe_{N}.

b) e_{N}xy^{op}e_{N} = τ(xy)e_{N} and τ(xy^{op}e_{N}) = λτ(xy), ∀ x, y ∈ N′ ∩ M, where λ = [M : N]^{−1}.

c) If q ∈ P(N′ ∩ M), q ≠ 0, then Nqq^{op} ⊂ qMqq^{op} ⊂ qq^{op}M_{1}qq^{op} is a basic construction with Jones projections equal to

τ(q)^{−1}qq^{op}e_{N}qq^{op} = τ(q)^{−1}qe_{N}q = τ(q)^{−1}q^{op}e_{N}q^{op}.

d) E_{M∨M′∩M_{1}}(e_{N}) = λ ∑_{i,j,k} τ(f_{jj}^{k})^{−1}f_{ij}^{k}f_{ji}^{k}^{op}, where {f_{ij}^{k}} is a matrix unit for N′ ∩ M.

Proof. a) If y ∈ M̂ then e_{N}(ŷ) = { \atop E_{N}(y)} so that

xe_{N}(ŷ) = { \atop xE_{N}(y)} = { \atop E_{N}(y)x} = x^{op}e_{N}(ŷ).

The uniqueness is clear because y′e_{N} = 0 implies e_{N}y′^{∗}y′e_{N} = 0 so that E_{N^{op}}(y′^{∗}y′) = 0, thus y′ = 0.

b) By a) we have

e_{N}xy^{op}e_{N} = e_{N}xye_{N} = E_{N}(xy)e_{N} = E_{N′∩N}(xy) = τ(xy)e_{N},

whenever x, y ∈ N′ ∩ M. The second part is then trivial.

c) Since

qq^{op}B(L^{2}(M))qq^{op} = B(L^{2}(qMq))

and

(Nqq^{op}) ∩ qq^{op}B(L^{2}(M))qq^{op} = qq^{op}M_{1}qq^{op},

it follows that

Nqq^{op} ⊂ qMqq^{op} ⊂ qq^{op}M_{1}qq^{op}

is a basic construction. Also, if e = τ(q)^{−1}qq^{op}e_{N}qq^{op} then by a) we have

e = τ(q)^{−1}qe_{N}q = τ(q)^{−1}q^{op}e_{N}q^{op}.

Also, the range of e = τ(q)^{−1}qq^{op}e_{N}qq^{op} is clearly

L^{2}(Nq) = L^{2}(qNq) = qL^{2}(N)q = L^{2}(N)q

and so, since e is a projection we get e = proj_{L^{2}(N)q} = proj_{L^{2}(Nqq^{op})} as an element in B(L^{2}(qMq)).

d) To prove this it is sufficient to show that τ(xy^{op}e_{N}) = τ(xy^{op}a), ∀ x, y ∈ N′ ∩ M, where a = λ ∑_{i,j,k} τ(f_{jj}^{k})^{−1}f_{ij}^{k}f_{ji}^{k}^{op}. It is then enough to check this for x = f_{rs}^{k′}, y = f_{s′r′}^{k′′}. For the left hand side, by b) we have:

τ(f_{rs}^{k′}f_{s′r′}^{k′′}^{op}e_{N}) = λδ_{k′k′′}δ_{ss′}δ_{rr′}τ(f_{rr}^{k′}).

For the right hand side we have:

τ(f_{rs}^{k′}f_{s′r′}^{k′′}^{op}a) = λ ∑_{i,j,k} τ(f_{jj}^{k})^{−1}τ(f_{ij}^{k}f_{ji}^{k}^{op}f_{rs}^{k′}f_{s′r′}^{k′′}^{op})

= λ ∑_{i,j,k} τ(f_{rr}^{k′})^{−1}δ_{kk′}δ_{kk′′}δ_{jr}δ_{jr′}τ(f_{is}^{k})τ(f_{is′}^{k})

= λτ(f_{rr}^{k′})^{−1}δ_{k′k′′}δ_{rr′}δ_{ss′}τ(f_{ss}^{k′})^{2}

= λδ_{k′k′′}δ_{ss′}δ_{rr′}τ(f_{rr}^{k′}).

Q.E.D.

2.9. Proposition. Let N ⊂ M be an extremal inclusion of type II_{1} factors. Then we have

a) M ⊠_{e_{N_{n−1}}} M^{op} naturally identifies with M ⊠_{e_{N}} M^{op}, by letting e_{N_{n−1}} ↦ f_{−n}^{n}.

b) The inclusion M_{1} ∨N^{op} ⊂ M ⊠_{e_{N}} M^{op} naturally identifies with the reduced by

e_{1}^{op} of the symmetric enveloping inclusion of M ⊂ M_{1}, M_{1}∨M_{1}^{op} ⊂ M_{1} ⊠_{e_{M}} M_{1}^{op}.

More generally, M_{n}∨N_{n−1}^{op} ⊂ M ⊠_{e_{N}} M^{op} is isomorphic to the reduced by (f_{−n}^{n})^{op}

of M_{n} ∨ M_{n}^{op} ⊂ M_{n} ⊠_{e_{M}} M_{n}.

c) If p ∈ P(N′ ∩ M) and we denote by L ⊂ K the inclusion Np ⊂ pMp then K ⊠_{e_{L}} K^{op} is naturally embedded in M ⊠_{e_{N}} M^{op} as the weakly closed ∗-subalgebra

generated by pp^{op}(M ∨ M^{op})pp^{op} and by

e′_{L} =^{def} σ(p)^{−1}pp^{op}e_{N}pp^{op}.

Also, the ientity of this algebra is pp^{op}.

d) If T ⊂ S denotes the symmetric enveloping inclusion associated with N ⊂ M and T_{0} ⊂ S_{0} the symmetric enveloping inclusion associated with some other extremal inclusion of type II_{1} factors N_{0} ⊂ M_{0}, then the symmetric enveloping inclusion associated with N ¯⊗N_{0} ⊂ M ¯⊗M_{0} is naturally isomorphic to T ¯⊗T_{0} ⊂ S ¯⊗S_{0}.

Proof. a) Is clear by 1.6, 1.9.4 and 2.4.

b) follows then immediately, from 2.1, 2.3 and the fact that (f_{−n}^{n})^{op}M_{n}^{op}(f_{−n}^{n})^{op} = N_{n−1}^{op}(f_{−n}^{n})^{op} ≃ N_{n−1}^{op}. To prove c) note that if π is the canonical representation of C^{∗}(M, e_{N}, JMJ) into M ⊠_{e_{N}} M^{op} then the C^{∗}-algebra generated by pp^{op}(M ∪ M^{op})pp^{op} and

e′_{L} is the image under π of C^{∗}(pJpJ(M ∪ JMJ)pJpJ, pJpJe_{N}pJpJ) which naturally identifies with the C^{∗}-algebra generated by K, J_{K}KJ_{K} and e_{L} in B(L^{2}(K)), where L = Np ⊂ pMp = K. Since this representation of C_{u,max}^{∗}(K, e_{L}, K^{op}) is smooth, it follows that π implements a smooth representation of C_{u,max}^{∗}(K, e_{L}, K^{op}) into pp^{op}(M ⊠_{e_{N}} M^{op})pp^{op}.

Since the latter has a trace, it follows by 2.1, 2.2 that K ⊠_{e_{L}} K^{op} =

(C^{∗}(pp^{op}Mpp^{op}, e′_{L}, pp^{op}M^{op}p\overline{p^{op}))}^{w} ⊂ M ⊠_{e_{N}} M^{op}.

d) follows trivially from any of the characterizing universality properties of the symmetric enveloping algebras (e.g., from 2.7). Q.E.D.

2.10. Proposition. Let N ⊂ M be an extremal inclusion of type II_{1} factors.

a) If Q ⊂ N is an extremal subfactor of N then M ⊠_{e_{N}} M^{op} is unitally embedded

as a subfactor of M ⊠_{e_{Q}} M^{op}, by taking e_{N} ↦ ∑_{j} m_{j}e_{Q}m_{j}^{∗} (= ∑_{j} m_{j}^{∗}^{op}e_{Q}m_{j}^{op}),

where {m_{j}}_{j} is a orthonormal basis of N over Q. Moreover, if there exists a tunnel M ⊃ N ⊃ N_{1} ⊃ ⋯ for N ⊂ M such that N_{k} ⊂ Q for some k, then this unital embedding is in fact an equality. b) If Q ⊂ P is an extremal inclusion of factors embedded in N ⊂ M as a commuting square, such that [P : Q] = [M : N] and P′∩P_{n} ⊂ M′∩M_{n}, ∀ n then P ⊠_{e_{Q}} P^{op} is unitally embedded in M ⊠_{e_{N}} M^{op}, by taking P ↪ M, P^{op} ↪ M^{op}

and e_{Q} ↦ e_{N}. Also, P ⊠_{e_{Q}} P^{op} ⊂ M ⊠_{e_{N}} M^{op} has finite index iff P ⊂ M has

finite index, with the estimate [M ⊠_{e_{N}} M^{op} : P ⊠_{e_{Q}} P^{op}] ≤ [M : P]^{2}. Moreover,

if P′ ∩ P_{n} = M′ ∩ M_{n}, then this embedding implements the nondegenerate commuting square:

M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op}
∪  ∪
P ∨ P^{op} ⊂ P ⊠_{e_{Q}} P^{op}.

Proof. a) It is easy to check by direct computation that

e_{N} = ∑_{j} m_{j}e_{Q}m_{j}^{∗} = ∑_{j}(Jm_{j}J)e_{Q}(Jm_{j}^{∗}J)

in B(L^{2}(M)), so a) follows from 2.2-2.6. The last part of a) then follows from 2.7 a) and the first part.

b) To prove the first part we only need to show that the representation of C_{u,bin}^{∗}(P, e_{Q}, P^{op}) in M ⊠_{e_{N}} M^{op} satisfies the faithfulness condition 1.8 (∗), i.e.,

we need to show that P′ ∩ P_{n} = Q_{n−1}^{op}′ ∩ P^{op}, ∀ n. By 2.1–2.4, it is sufficient to check this equality in a representation of C_{bin}^{∗}(M, e_{N}, M^{op}), and we’ll choose C^{∗}(M, e_{N}, JMJ) ⊂ B(L^{2}(M)) to do this. Let e_{P}^{M} ∈ B(L^{2}(M)) be the orthogonal projection of L^{2}(M) onto L^{2}(P). Note that all the elements in C^{∗}(P, e_{N}, J_{M}P J_{M}) ⊂ B(L^{2}(M)) commute with e_{P}^{M} and that if x ∈ JMJ then x ∈ JP J iff [x, e_{P}^{M}] = 0. Now, if x ∈ P′ ∩ P_{n} then x ∈ M′ ∩ P_{n} by hypothesis, so x ∈ JMJ ∩ P_{n}. Also, [x, e_{P}^{M}] = 0, because P_{n} ⊂ C^{∗}(P, e_{N}, JP J). Thus, x ∈ JP J ∩ P_{n}. But P_{n} ⊂ M_{n} = JN_{n−1}J′ ∩ B(L^{2}(M)) ⊂ JQ_{n−1}J′. Thus, x ∈ JP J ∩ (JQ_{n−1}J)′. This proves the first part of b). Further on, assume [M : P] < ∞ and take {m_{j}}_{j} to be a finite orthonormal basis of M over P. Note that M ⊠_{e_{N}} M^{op} = \overline{sp}(M ∨ M^{op})vN\{e_{j}\}_{j∈ℤ} and

P ⊠_{e_{Q}} P^{op} = \overline{sp}(P ∨ P^{op})vN\{e_{j}\}_{j∈ℤ}, with \{e_{j}\}_{j} ⊂ P_{∞} being the Jones projec-

tions for a tower-tunnel for Q ⊂ P (see 4.1, 4.2), and thus for N ⊂ M as well. Since M ∨ M^{op} = Σ_{i,j}m_{i}m_{j}^{∗op}P ∨ P^{op} it thus follows that M ⊠_{e_{N}} M^{op} =

Σ_{i,j}m_{i}m_{j}^{∗op}P ⊠_{e_{Q}} P^{op}, showing that M ⊠_{e_{N}} M^{op} is a finitely generated left module

over P ⊠_{e_{Q}} P^{op}, with the estimate [M ⊠_{e_{N}} M^{op} : P ⊠_{e_{Q}} P^{op}] ≤ [M : P]^{2} as a bonus.

For the last part, we have that ⋃_{n} spP^{op}P_{n}P^{op} is so-dense in P ⊠_{e_{Q}} P^{op}

and writing P_{n} as spP f_{−n}^{n}P we get E_{M∨M^{op}}(P ⊠_{e_{Q}} P^{op}) = \overline{sp} ⋃_{n}((P ∨

P^{op})E_{M∨M^{op}}(f_{−n}^{n})(P ∨ P^{op})). But since P′ ∩ P_{n} = M′ ∩ M_{n} and Q′_{n−1} ∩ P = N′_{n−1} ∩ M, ∀ n, it follows that E_{M∨M^{op}}(f_{−n}^{n}) = E_{P ∨P^{op}}(f_{−n}^{n}), proving the desired commuting square condition. Q.E.D.

Let us end this section by considering a notion of index for sublattices of standard λ-lattices (see [Po7] for the definition of abstract standard lattices and for the notations and results used hereafter). We relate this notion with the content of this section by showing that the index of a sublattice coincides with the index of a certain canonically associated inclusion of symmetric enveloping type II_{1} factors. This latter result will be used in Sections 5 and 8.

2.11. Definition. Let G = (A_{ij})_{0≤i≤j} be a standard λ-lattice and G_{0} =

(A_{ij}^{0})_{0≤i≤j} a sublattice. We define the index of G_{0} in G by [G : G_{0}] =^{def} lim_{n→∞} IndE_{A_{0n}^{0}}^{A_{0n}} = IndE_{A_{0,∞}^{0}}^{A_{0,∞}}, where Ind(E) denotes as usual the index

([PiPo1]) of the conditional expectation E and A_{i,∞} = \overline{∪_{n}A_{in}}, A_{i,∞}^{0} = \overline{∪_{n}A_{in}}.

Let us make right away some comments on this definition. By (1.1.6 in [Po3]), if {m_{j}}_{j} is an orthonormal basis of A_{0,∞} over A_{0,∞}^{0} (apriorically made up of square summable operators) then ∥Σ_{j}m_{j}m_{j}^{∗}∥ = IndE_{A_{0,∞}^{0}}^{A_{0,∞}}. But both A_{1,∞} ⊂

A_{0,∞} and A_{1,∞}^{0} ⊂ A_{0,∞}^{0} are λ-Markov inclusions (see 1.1.5 in [Po2] for the definition), so the commuting square embedding of the latter into the former is nondegenerate (1.5,1.6 in [Po2]). Thus, by (1.6 in [Po2]) any orthonormal basis of A_{1,∞} over A_{1,∞}^{0} is an orthonormal basis of A_{0,∞} over A_{0,∞}^{0}. Thus, in the above we may assume that {m_{j}}_{j} lies in A_{1,∞}. On the other hand, if bounded, Σ_{j}m_{j}m_{j}^{∗} belongs to the center of A_{0,∞} (see e.g. 1.1.5 in [Po3]), thus Σ_{j}m_{j}m_{j}^{∗} ∈ Z(A_{0,∞}) ∩ A_{1,∞} = Z(A_{0,∞}) ∩ Z(A_{1,∞}). But by (Corollary 1.4.2 in [Po2]) this latter intersection is in fact equal to the scalar multiples of the identity. Thus, Σ_{j}m_{j}m_{j}^{∗} ∈ ℂ1. Altogether, this shows that we may as well

take [G : G_{0}] =^{def} ∥Σ_{j}m_{j}m_{j}^{∗}∥ = Σ_{j}m_{j}m_{j}^{∗}, {m_{j}}_{j} being an arbitrary orthonormal basis of A_{i,∞} over A_{i,∞}^{0}, for some i ≥ 0. The next proposition gives more ways to look at this index.

2.12. Proposition. Let G be a standard λ-lattice with a sublattice G_{0}. Let Q_{0} be a non-atomic finite von Neumann algebra with a faithful trace and N^{G}(Q_{0}) ⊂ M^{G}(Q_{0}), respectively N^{G_{0}}(Q_{0}) ⊂ M^{G_{0}}(Q_{0}) be the associated extremal inclusions of type II_{1} factors having G, respectively G_{0} as standard invariants, given by the universal construction in ([Po7]). Let

N^{G}(Q_{0}) ⊂ M^{G}(Q_{0})
∪  ∪
N^{G_{0}}(Q_{0}) ⊂ M^{G_{0}}(Q_{0}).

be the corresponding commuting square like in ([Po7]). Then we have

[G : G_{0}] = [M^{G}(Q_{0}) : M^{G_{0}}(Q_{0})] = [S : S_{0}], where S and respectively S_{0} denote the symmetric enveloping algebras of N^{G}(Q_{0}) ⊂ M^{G}(Q_{0}) and respectively N^{G_{0}}(Q_{0}) ⊂ M^{G_{0}}(Q_{0}).

Proof. Recall from [Po7] that M_{∞}^{G}(Q_{0}) identifies with the free product with amalgamation Q_{0} ¯⊗A_{1,∞} ∗_{A_{1,∞}} A_{0,∞}, with M_{∞}^{G_{0}}(Q_{0}) identifying with the subalgebra generated by Q_{0} and A_{0,∞}^{0}. By the resulting commuting square relations for these inclusions (see [Po7], pages 435 and 438), it follows that any orthonormal basis of A_{1,∞} over A_{1,∞}^{0} is an orthonormal basis of Q_{0} ¯⊗A_{1,∞} ∗_{A_{1,∞}} A_{0,∞} over Q_{0} ¯⊗A_{1,∞}^{0}∗_{A_{1,∞}^{0}} A_{0,∞}^{0}, thus of M_{∞}^{G}(Q_{0}) over M_{∞}^{G_{0}}(Q_{0}). But the commuting square embedding of M^{G_{0}}(Q_{0}) ⊂ M^{G}(Q_{0}) into M_{∞}^{G_{0}}(Q_{0}) ⊂ M_{∞}^{G}(Q_{0}) is nondegenarate (cf. [Po7]), so that in the end, if {m_{j}} denotes an orthonormal basis of A_{1,∞} over A_{1,∞}^{0}, we get [M^{G}(Q_{0}) : M^{G_{0}}(Q_{0})] = [M_{∞}^{G}(Q_{0}) : M_{∞}^{G_{0}}(Q_{0})] = Σ_{j}m_{j}m_{j}^{∗} = [G : G_{0}]. Finally, from the universality properties of the symmteric enveloping algebras and the definition of N^{G}(Q_{0}) ⊂ M^{G}(Q_{0}) and N^{G_{0}}(Q_{0}) ⊂ M^{G_{0}}(Q_{0}), we see that, if we denote by N ⊂ M and N_{0} ⊂ M_{0} these two inclusions then S_{0} ⊂ S identifies with the inclusion Q_{0} ¯⊗N_{0}^{op} ∗_{N_{0}^{op}} M_{0}^{op} ⊂ Q_{0} ¯⊗N^{op} ∗_{N^{op}} M^{op}. But from the above we have that any orthonormal basis of N^{op} over N_{0}^{op} will be an orthonormal basis of S over S_{0}. Q.E.D.

3. A Class of Examples

Let Q be a type II_{1} factor and σ_{1}, …, σ_{n} a n-tuple of automorphisms of Q. Let N ⊂ M be the locally trivial inclusion of factors associated with σ_{1}, …, σ_{n} (see e.g. [Po2] ), i.e., M = Q⊗M_{n+1}(ℂ), N = {∑_{i=0}^{n} σ_{i}(x)⊗e_{ii} | x ∈ Q ≃ Q⊗ℂ1}, where σ_{0} = id_{Q} and {e_{ij}}_{0≤i,j≤n} is a matrix unit for M_{n+1}(ℂ). We still denote by σ_{i} the automorphism of M = Q ⊗ M_{n+1}(ℂ) defined by σ_{i}(x ⊗ e_{kl}) = σ_{i}(x) ⊗ e_{kl}, ∀ x ∈ Q, 0 ≤ k, l ≤ n. Denote by G the discrete group generated by σ_{1}, …, σ_{n} in Aut(M)/Int(M). Also, we let σ : G → Aut(M)/Int(M) be the corresponding faithful G-kernel. Then note that the faithful G-kernel σ⊗σ^{op} on M ¯⊗M^{op} has vanishing H^{3}(G, 𝕋) cohomology obstruction ([J5]), so that it can be viewed as a (properly outer) cocycle action of G on M ¯⊗M^{op}. In this section we show that, with the above notations, we have

(M ∨ M^{op} ⊂ M ⊠_{e_{Q}} M^{op}) ≃ (M ¯⊗M^{op} ⊂ (M ¯⊗M^{op}) ⋊_{σ⊗σ^{op}} G),

in which the cross product is associated with the cocycle action σ ⊗ σ^{op} as in (4.1 of [J5]). Since by the previous sections M ∨ M^{op} ⊂ M ⊠_{e_{Q}} M^{op} is the

(weak closure of the) quotient of C^{∗}(M, JMJ) ⊂ C^{∗}(M, e_{N}, JMJ), it will be sufficient to study this latter inclusion of algebras. So let U_{i} be the unitary element acting on L^{2}(M, τ), defined on the dense subset

M̂ ⊂ L^{2}(M, τ) by U_{i}(x̂) = { \atop σ_{i}(x)}, x ∈ M, 0 ≤ i ≤ n. Note that U_{i}xU_{i}^{∗} = σ_{i}(x),

∀ x ∈ M, 0 ≤ i ≤ n, and [J, U_{i}] = 0. In particular, since σ_{i}(e_{kl}) = e_{kl}, 0 ≤ k, l ≤ n, we also have [U_{i}, e_{kl}] = 0, [U_{i}, Je_{kl}J] = 0, ∀ i, k, l.

3.1. Lemma.

a) e_{N} = \frac{1}{n + 1} ∑_{i,j=0}^{n} U_{j}U_{i}^{∗}e_{ji}Je_{ji}J.

b) U_{j} = (n + 1) ∑_{k,l=0}^{n} Je_{lj}Je_{kj}e_{N}e_{0k}Je_{0l}J.

Proof. a) If x = ∑ x_{ij} ⊗ e_{ij} ∈ Q ⊗ M_{n+1}(ℂ) = M, then

{ \atop E_{N}(x)} = \frac{1}{n + 1} ∑_{i,j}(σ_{j}σ_{i}^{−1}(x_{ii}) ⊗ e_{jj}) = \frac{1}{n + 1} ∑_{i,j=0}^{n} U_{j}U_{i}^{∗}e_{ji}Je_{ji}J(x̂)

proving the first formula.

b) By a) we have e_{jj}e_{N}Je_{00}J = \frac{1}{n + 1}U_{j}e_{j0}Je_{j0}J, so that U_{j}e_{00}Je_{00}J = (n +

1)e_{0j}Je_{0j}Je_{jj}e_{N}Je_{00}J. Thus we get

U_{j} = ∑_{k,l=0}^{n} e_{k0}Je_{l0}J(U_{j}e_{00}Je_{00}J)Je_{0l}Je_{0k} = (n + 1) ∑_{k,l=0}^{n} e_{kj}Je_{lj}Je_{N}Je_{0l}Je_{0k}.

Q.E.D.

3.2. Corollary. C^{∗}(M, e_{N}, JMJ) = C^{∗}(M, {U_{i}}_{i≤n}, JMJ). In fact, C^{∗}(M_{n+1}(ℂ), e_{N}, JM_{n+1}(ℂ)J) = C^{∗}(M_{n+1}(ℂ), {U_{i}}_{0≤i≤n}, JM_{n+1}(ℂ)J).

Proof. Trivial by the previous lemma. Q.E.D. Describing M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op} as a cross product is now an immediate consequence of the previous lemma and of 2.1, once we notice that U_{j}(xJy^{∗}J)U_{j}^{∗} = σ_{i}(x)Jσ_{i}(y^{∗})J. To write the corresponding isomorphism in more specific terms, denote by u_{i} the image of U_{i} in M ⊠_{e_{N}} M^{op} (cf. 2.1) and by g_{i} the image of σ_{i} as an element of the group G. 3.3 Theorem. There exists a unique isomorphism γ, of (M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op}) onto (M ¯⊗M^{op} ⊂ M ¯⊗M^{op} ⋊_{σ⊗σ^{op}} G), satisfying:

a) γ(xy^{op}) = x ⊗ y^{op}, x, y ∈ M.

b) u_{g_{i}} =^{def} γ(u_{i}) are unitary elements in the cross product M ¯⊗M^{op} ⋊_{σ⊗σ^{op}} G which implement the automorphism σ ⊗ σ^{op}(g_{i}), 0 ≤ i ≤ n.

c) γ(e_{N}) = \frac{1}{n + 1} ∑_{i,j=0}^{n} u_{g_{j}}u_{g_{i}}^{∗}e_{ji} ⊗ e_{ij}^{op}.

d) γ^{−1}(u_{g_{j}}) = u_{j} = (n + 1) ∑_{k,l=0}^{n} e_{kj}e_{jl}^{op}e_{N}e_{0k}e_{l0}^{op}, 0 ≤ j ≤ n.

Proof. Trivial by 2.1 and 3.1. Q.E.D.

Documenta Mathematica 4 (1999) 665–744

In the next section we will see that even for arbitrary extremal subfactors N ⊂ M the resulting inclusion M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op} can be interpreted as a

’cross-product’-type structure.

3.4 Remarks. 1^{∘}. As one knows (see e.g. 5.1.5 in [Po2]), the standard invariant G_{N,M} of the above locally trivial subfactor N ⊂ M only depends on the cohomology obstruction in H^{3}(G, 𝕋) ([J5]) of the corresponding G-kernel σ on Q. Thus, if we take another G-kernel σ′, on another type II_{1} factor Q′ but with the same H^{3}(G, 𝕋)-obstruction as σ, and denote the similar locally trivial inclusion (corresponding to the same generators of G) by N′ ⊂ M′, then G_{N′,M′} = G_{N,M} and we can thus consider the concatenation algebra 2.5.1^{∘} associated with these two inclusions. Then M ∨ M′^{op} ⊂ M ⊠ M′^{op} is isomorphic to a cocycle cross product M ¯⊗M′^{op} ⊂ (M ¯⊗M′^{op}) ⋊_{σ⊗σ′^{op}} G. 2^{∘}. Let G = (A_{ij})_{0≤i≤j} be the standard λ-lattice associated to the locally trivial subfactor N ⊂ M, constructed from the automorphisms σ_{1}, ..., σ_{n} acting on the factor Q as above, with G denoting the group generated by the σ_{i}’s in Aut(Q)/Int(Q) (and with the corresponding generators denoted hereafter by g_{1}, ..., g_{n}). Let G_{0} = (A_{ij}^{0})_{i,j} be a sublattice of G with the property that A_{01}^{0} is a maximal abelian subalgebra of A_{01}. Note that this amounts to saying that G_{0} has same ”generators” but possibly lesser ”relations” than G. Now take Q_{0} to be an arbitrary finite von Neumann algebra without atoms. With Q_{0} as ”initial data”, do the universal construction [Po7] of subfactors N^{G}(Q_{0}) ⊂ M^{G}(Q_{0}) and N^{G_{0}}(Q_{0}) ⊂ M^{G_{0}}(Q_{0}) with higher relative commutants picture given by G respectively G_{0}, like at the end of Sec. 2, thus obtaining the non-degenerate commuting square of inclusions:

N^{G}(Q_{0}) ⊂ M^{G}(Q_{0})
∪  ∪
N^{G_{0}}(Q_{0}) ⊂ M^{G_{0}}(Q_{0}).

One can then show that the above algebras and the inclusions involved can be alternatively described in terms of the following objects: a). A type II_{1} factor Q′ with a faithful G kernel σ′ on it such that if N ⊂ M denotes the locally trivial subfactor constructed out of this G-kernel and the generators g_{1}, ..., g_{n}, like at the beginning of this section, then (N ⊂ M) ≃ (N^{G}(Q_{0}) ⊂ M^{G}(Q_{0})); b). An irreducible regular (in the sense of [D1]) subfactor Q′_{0} ⊂ Q′, a group G_{0} with generators g′_{1}, ..., g′_{n} and a G_{0}-kernel σ′_{0} on Q′_{0} such that if N_{0} ⊂ M_{0} denotes the associated locally trivial subfactor, constructed from this G_{0}-kernel and the generators g′_{1}, ..., g′_{n}, like at the beginning of this section, then (N_{0} ⊂ M_{0}) ≃ (N^{G_{0}}(Q_{0}) ⊂ M^{G_{0}}(Q_{0})); c). A group morphism ρ of G_{0} onto G such that ρ(g′_{i}) = g_{i} and such that if H = ker(ρ) denotes the corresponding kernel group then H is isomorphic to N_{Q′}(Q′_{0})/U(Q′_{0}) (so that Q′ is a cocycle cross-product of Q′_{0} by H), in such a way that if we denote by {u_{h}}_{h∈H} a set of unitaries in N_{Q′}(Q′_{0}) that give a cross-section for H then, modulo perturbations by inner automorphisms, σ′ and σ′_{0} are related as follows: σ′(ρ(g′))(u_{h}a′_{0}) = u_{hgh^{−1}}σ′_{0}(g′)(a′_{0}), ∀a′_{0} ∈ Q′_{0}, h ∈ H, g′ ∈ G_{0}. Moreover, through these identifications, N_{0} ⊂ M_{0} is embedded in N ⊂ M by the inclusion M_{0} = Q′_{0} ⊗ M_{n+1}(ℂ) ⊂ Q′ ⊗ M_{n+1}(ℂ) = M, and the corresponding commuting square is isomorphic to the above commuting square. Thus, in this exemple the sublattice G_{0} of the lattice G (which was associated to the group G) corresponds to a ”covering” group G_{0} of the group G. Note that, with these identifications, we have that the index of G_{0} in G equals the order of the group H, [G : G_{0}] = |H|. Finally, let us see what the symmetric enveloping algebras become in this case: if we extend the atomorphisms σ′(g), σ′_{0}(g′) to M, M_{0} by putting them to act as the identity on M_{n+1}(ℂ), then the symmetric enveloping algebras S, S_{0} of N ⊂ M respectively N_{0} ⊂ M_{0}, and the corresponding inclusion S_{0} ⊂ S (cf. 2.10, b)), are given by

S_{0} = M_{0} ¯⊗M_{0}^{op} ⋊_{σ′_{0}⊗σ′_{0}^{op}} G_{0} ⊂ M ¯⊗M^{op} ⋊_{σ′⊗σ′^{op}} G = S

with the inclusion being described similarily to c).

4. Thinness and Quasi-Regularity Properties

We’ve already seen that sp ⋃_{n} MM_{n}^{op}M = sp ⋃_{n} M^{op}M_{n}M^{op} is a ∗-subalgebra

which is dense in M ⊠_{e_{N}} M^{op} in the weak (or strong) operator topology. Let

{e_{n}}_{n∈ℤ} be the Jones projections for the Jones tunnel-tower ⋯ N_{1} ⊂ N ⊂ M ⊂ M_{1} ⊂ ⋯, with e_{N} = e_{1}, as in Sections 1–2, and denote by P the von Neumann algebra they generate in M ⊠_{e_{N}} M^{op}. Fix n ≥ 0 and choose an

orthonormal basis {m_{j}}_{j} of M over N_{n−1} that belongs to vN{e_{k}}_{k≤0} ⊂ P and an orthonormal basis {m_{k}^{n}}_{k} of M_{n} over M that belongs to vN{e_{k}}_{k≤n} ⊂ P. Thus we have

MM_{n}^{op}M ⊂ M (∑_{k} M^{op}m_{k}^{n}^{op}) (∑_{j} N_{n−1}m_{j}^{∗}) = ∑_{j,k} MM^{op}N_{n−1}m_{k}^{n}^{op}m_{j}^{∗}

= ∑_{j,k} MM^{op}m_{k}^{n}^{op}m_{j}^{∗} ⊂ sp MM^{op}P.

Thus we obtain sp ⋃_{n} MM_{n}^{op}M ⊂ sp(M ∨ M^{op})P. Similarly, since

∑_{j,k} Mm_{k}^{n}^{op}m_{j}^{∗} ⊂ M_{∞}, we get sp ⋃_{n} MM_{n}^{op}M ⊂ sp M^{op}M_{∞}, giving us

the following:

4.1. Proposition. With the above notations we have:

S =^{def} M ⊠_{e_{N}} M^{op} = \overline{sp}(M ∨ M^{op})P = \overline{sp} M_{∞}M^{op}
= \overline{sp}(M ∨ M^{op})(Alg\{f_{−n}^{n}\}_{n})(M ∨ M^{op})

the closure being taken in either of the wo, so or ∥ ∥_{2} topologies in S.

Proof. Since sp(M ∨M^{op})P, M_{∞}M^{op} and sp(M ∨M^{op})(Alg{f_{−n}^{n}}_{n})(M ∨M^{op}) contain sp ⋃_{n} MM_{n}^{op}M, which is a dense ∗-subalgebra in S = M ⊠_{e_{N}} M^{op}, we are

done. Q.E.D.

Note that if M is hyperfinite then M ∨ M^{op}, M^{op}, M_{∞} are all hyperfinite. Thus, in this case M ⊠_{e_{N}} M^{op} can be written as a “product” of two hyperfi-

nite subfactors. Recall from ([Po5]) that such situation is singled out by the following:

4.2. Definition. A type II_{1} factor S for which there exist two hyperfinite type II_{1} subfactors R_{1}, R_{2} ⊂ S such that S = \overline{sp} R_{1}R_{2}, the closure being taken in ∥ ∥_{2}, is called a thin type II_{1} factor.

With this terminology the above observation takes the form:

4.3. Corollary. If N ⊂ M is a extremal inclusion of hyperfinite type II_{1} factors then S = M ⊠_{e_{N}} M^{op} is a thin type II_{1} factor.

From the above, the previous section and Connes’ fundamental theorem ([C1]) we can already conclude:

4.4. Corollary. If N ⊂ M is an inclusion of factors associated to a faithful G-kernel σ on a hyperfinite type II_{1} factor R like on Section 3, where G is a finitely generated discrete group, then M ⊠_{e_{Q}} M^{op} ≃ R ⊗ R^{op} ⋊_{σ⊗σ^{op}} G is thin

but it is hyperfinte iff G is amenable.

More precise statements along these lines will be obtained in Sec.5 and 7. Let us note now that the Hilbert space K_{n} obtained as the closure of

(spM^{op}M_{n}M^{op}) = (spMM_{n}^{op}M)

in L^{2}(M ⊠_{e_{N}} M^{op}, τ) is invariant to multiplication from left and right by both

M and M^{op}, thus by T = M ∨ M^{op}. Thus K_{n} is a T -T bimodule. Since sp MM_{n}^{op}M = sp ∑_{j,k} m_{k}m_{j}^{∗}^{op}f_{−n}^{n} MM^{op} = sp ∑_{j,k} MM^{op}f_{−n}^{n}m_{j}^{op}m_{k}^{∗}, it follows that K_{n} has finite dimension both as a left and as a right T module. Thus, if p_{n} is the orthogonal projection of L^{2}(S, τ) onto K_{n} the p_{n} commutes with the operators of left and right multiplication by elements in T, i.e., p_{n} ∈ T′ ∩ ⟨S, T ⟩. Also, since \overline{⋃_{n} K_{n}} = L^{2}(S, τ), we have p_{n} ↗ 1 and the above shows

that Tr p_{n} < ∞, ∀ n, where Tr = Tr_{⟨S,T ⟩} denotes the unique trace on ⟨S, T ⟩ satisfying Tr(e_{T}) = 1. Thus, T′ ∩⟨S, T ⟩ is generated by finite projections of ⟨S, T ⟩ and the inclusion of factors T p_{n} ⊂ p_{n}⟨S, T ⟩p_{n} has finite index for all n. Since T′ ∩ S = ℂ1 (cf. 2.3), by (1.8 in [PiPo1]) we can already conclude that Tr p ≥ 1, ∀ p ∈ T′ ∩ ⟨S, T ⟩ (so in particular T′ ∩ ⟨S, T ⟩ is atomic) and that the multiplicity of any minimal projection p in T′ ∩ ⟨S, T ⟩ is ≤ Tr p. In fact we have the following more precise statement:

4.5. Theorem. Let N ⊂ M be an extremal inclusion of type II_{1} factors and denote S = M ⊠_{e_{N}} M^{op}, T = M ∨ M^{op} ⊂ S.

a) If {H_{k}}_{k∈K} denotes the set of irreducible M-M bimodules corresponding to the set of even vertices of the standard graph Γ_{N,M} of N ⊂ M then L^{2}(S, τ) is isomorphic as a T -T bimodule with ⨁_{k∈K} H_{k} ¯⊗H̄_{k}^{op} and K_{n} with ⨁_{k∈K_{n}} H_{k} ¯⊗H̄_{k}^{op}

(in which T ≃ M ¯⊗M^{op}).

b) If L^{2}(S, τ) is identified with ⨁_{k∈K} H_{k} ¯⊗H̄_{k}^{op} as in a) and s_{k} denotes the

orthogonal projection of L^{2}(S, τ) onto its direct summand H_{k} ¯⊗H̄_{k}^{op} then s_{k} is a minimal projection in T′ ∩ ⟨S, T ⟩, p_{n} = ∑_{k∈K_{n}} s_{k} and T′ ∩ ⟨S, T ⟩ = vN{s_{k}}_{k∈K} ≃ ℓ^{∞}(K). Moreover, (Tr s_{k})^{2} = [s_{k}⟨S, T ⟩s_{k} : T s_{k}] = v_{k}^{4}, where v⃗ = (v_{k})_{k∈K} is the standard vector giving the weights at the even vertices of Γ_{N,M}.

c) The antiautomorphism ^{op} on S leaves T invariant and thus implements an antiautomorphism on ⟨S, T ⟩, still denoted by ^{op}. We have (T′ ∩ ⟨S, T ⟩)^{op} = T′ ∩ ⟨S, T ⟩, the projection s_{k}^{op} coincides with J_{S}s_{k}J_{S} and the corresponding bimodule is (H_{k} ¯⊗H̄_{k}^{op})^{−} = H̄_{k} ¯⊗H_{k}^{op}.

Proof. Let k ∈ K_{1} and choose q = q_{k} ∈ N′_{1} ∩ M to be a minimal projection in the direct summand labeled by k. Denote v′_{q} = (λτ(q))^{1/2}qq^{op}e_{1}e_{0}e_{0}^{op} and v_{q} = λ^{−2}E_{N′∩M_{1}}(v′_{q}). Note that f = v′_{q}v′_{q}^{∗} is the Jones projection for the irreducible inclusion q^{op}qN_{1} ⊂ q^{op}qMq ⊂ q^{op}qM_{2}qq^{op} (cf. 2.8.b) and 2.8.c)). Note also that by applying twice the ”push down lemma” (1.2 in [PiPo1]) and using the above definitions we get:

v_{q}e_{0}^{op}e_{0} = λ^{−2}E_{N′∩M_{1}}(v′_{q})e_{0}^{op}e_{0}
= λ^{−1}E_{N′∩M_{2}}(λ^{−1}E_{N′_{1}∩M_{1}}(v′_{q})e_{0}^{op})e_{0} = λ^{−1}E_{N′∩M_{2}}(v′_{q})e_{0} = v′_{q},

implying that:

v_{q}e_{0}e_{0}^{op}v_{q}^{∗} = v′_{q}e_{0}e_{0}^{op}v′_{q}^{∗} = v′_{q}v′_{q}^{∗} = f ≤ q^{op}q.

Step I. We first prove that L^{2}(spMv_{q}M) ≃ H_{k} and that L^{2}(spM^{op}v_{q}M^{op}) ≃ H̄_{k}^{op}. Indeed, since e_{0}^{op} = e_{2}, by the definition of H_{k} we have H_{k} =

L^{2}(∑_{j} m_{j}M), where {m_{j}}_{j} ⊂ M_{1} = ⟨M, e_{1}⟩ (⊂ M ⊠_{e_{N}} M^{op}) are so that

{m_{j}e_{0}^{op}m_{j}^{∗}}_{j} are mutually orthogonal projections with ∑_{j} m_{j}e^{op}m_{j}^{∗} = q^{op} ∈ M′ ∩ M_{2}. Since v_{q} ∈ M_{1} and v_{q}e_{0}^{op} = q^{op}v_{q}e_{0}^{op}, it follows that v_{q} ∈ ∑ m_{j}M. Thus Mv_{q}M ⊂ ∑_{j} m_{j}M, so that L^{2}(spMv_{q}M) ⊂ H_{k}. Since H_{k} is irreducible and L^{2}(spMv_{q}M) is a M-M bimodule, we actually have the equality L^{2}(spMv_{q}M) = H_{k}. To prove the second isomorphism, note that given any T − T (resp. M − M) bimodule H ⊂ L^{2}(S, τ), its conjugate T − T (resp. M − M) bimodule H̄ can be identified with (H)^{∗} = {ξ^{∗} | ξ ∈ H} and its opposite T^{op} − T^{op} (resp.

M^{op}−M^{op}) bimodule H^{op} can be identified with (H)^{op} = {ξ^{op} | ξ ∈ H} (all this is trivial by the definitions). As a consequence, we also have H̄^{op} ≃ ((H)^{∗})^{op}. By taking into account that (v_{q}^{∗})^{op} = v_{q} and that M^{∗} = M, from the isomorphism L^{2}(spMv_{q}M) ≃ H_{k} and the above remark it thus follows that L^{2}(spM^{op}v_{q}M^{op}) ≃ H̄_{k}^{op} as well. Step II. We now prove that H_{k} ¯⊗H̄_{k}^{op} ≃ L^{2}(spMM^{op}v_{q}MM^{op}). To see this, by Step I it is sufficient to prove that there exists α ∈ ℂ such that:

⟨x_{1}x_{2}^{op}v_{q}y_{1}y_{2}^{op}, x_{3}x_{4}^{op}v_{q}y_{3}y_{4}^{op}⟩ = α⟨x_{1}v_{q}y_{1}, x_{3}v_{q}y_{3}⟩⟨x_{2}^{op}v_{q}y_{2}^{op}, x_{4}^{op}v_{q}y_{4}^{op}⟩,

∀ x_{i}, y_{j} ∈ M, 1 ≤ i, j ≤ 4. By denoting a = x_{3}^{∗}x_{1}, b = y_{1}y_{3}^{∗}, c = x_{2}x_{4}^{∗}, d = y_{4}^{∗}y_{2}, it follows that it is sufficient to prove that

⟨av_{q}b, c^{∗op}v_{q}d^{∗op}⟩ = α⟨av_{q}b, v_{q}⟩⟨c^{op}v_{q}d^{op}, v_{q}⟩,

∀ a, b, c, d ∈ M. Writing b = b_{1}e_{0}b_{2}, d = d_{1}e_{0}d_{2} for b_{1,2}, d_{1,2} ∈ N and using that

⟨av_{q}b, c^{∗op}v_{q}d^{∗op}⟩ = ⟨b_{2}ab_{1}v_{q}e_{0}, d_{2}^{∗op}c^{∗op}d_{1}^{∗op}v_{q}e_{0}^{op}⟩,
⟨av_{q}b, v_{q}⟩ = ⟨b_{2}ab_{1}v_{q}e_{0}, v_{q}⟩,
⟨c^{op}v_{q}d^{op}, v_{q}⟩ = ⟨d_{1}^{op}c^{op}d_{2}^{op}v_{q}e_{0}^{op}, v_{q}⟩,

by putting a for b_{2}ab_{1} and c for d_{2}cd_{1}, it follows that we only need to check that:

⟨av_{q}e_{0}, c^{∗op}v_{q}e_{0}^{op}⟩ = α⟨av_{q}e_{0}, v_{q}⟩⟨c^{op}v_{q}e_{0}^{op}, v_{q}⟩,

∀ a, c ∈ M. But

⟨av_{q}e_{0}, c^{∗op}v_{q}e_{0}^{op}⟩ = τ(ac^{op}v_{q}e_{0}v_{q}^{∗}) = τ(ac^{op}f)

and also

⟨av_{q}e_{0}, v_{q}⟩ = ⟨av_{q}e_{0}, v_{q}e_{0}⟩ = τ(av_{q}e_{0}v_{q}^{∗}) = λ^{−1}τ(av_{q}e_{0}e_{0}^{op}v_{q}^{∗})
= λ^{−1}τ(af),

and similarily ⟨c^{op}v_{q}e_{0}^{op}, v_{q}⟩ = λ^{−1}τ(c^{op}f), where f = v_{q}e_{0}e_{0}^{op}v_{q}^{∗} is the Jones projection for the irreducible inclusion N_{1}qq^{op} ⊂ qMqq^{op} ⊂ q^{op}qM_{2}qq^{op}. Since E_{M∨M^{op}}(f) = v_{k}^{−2}qq^{op} (where v⃗ = (v_{k})_{k∈K} is the standard vector as usual), we have τ(ac^{op}f) = α_{0}τ(aq)τ(c^{op}q^{op}), ∀ a, c ∈ M, for some constant α_{0} ∈ ℝ_{+}. Also, we have τ(af) = α_{1}τ(aq), τ(c^{op}f) = α_{1}τ(c^{op}q^{op}), ∀ a, c ∈ M for some constant α_{1} ∈ ℝ_{+}. This ends the proof. Step III. We next show that

L^{2}(spMM^{op}e_{1}MM^{op}) = ∑_{k∈K_{1}} L^{2}(spMM^{op}v_{q_{k}}MM^{op}).

To see this let q′ ∈ N′_{1} ∩ M be a minimal projection in the same simple direct summand as q and u ∈ U(N′_{1} ∩ M) such that u^{∗}qu = q′. Let v′′ = λ^{−1}E_{M_{1}}(u^{op}v_{q}u^{∗op}) ∈ N′ ∩ M_{1} and note that v′′e_{0}^{op}v^{′′∗} = u^{op}(v_{q}e_{0}^{op}v_{q}^{∗})u^{∗op} ≤ q^{′op}. By the same reasoning as in Step I, it follows that L^{2}(spMv′′M) = L^{2}(∑_{j} m′_{j}M), where {m′_{j}}_{j} ⊂ M_{1} is an orthonormal system such that ∑ m′_{j}e_{0}^{op}m_{j}^{′∗} = q^{′op}. But v′′ ∈ spMM^{op}v_{q}M^{op}M, because u^{op}v_{q}u^{∗op} ∈ spM^{op}v_{q}M^{op} and E_{M_{1}}(u^{op}v_{q}u^{∗op}) = λ ∑_{j} b_{j}^{op}(u^{op}v_{q}u^{∗op})b_{j}^{∗op} ∈ spM^{op}v_{q}M^{op} as well, where {b_{j}^{∗}}_{j} is an orthonormal basis of N over N_{1}. Thus we have spMM^{op}v_{q}M^{op}M ⊃ spMM^{op}v_{q′}MM^{op}, ∀ q′ chosen this way. Thus, if {m_{j}^{k}}_{j} ⊂ M_{1} is a orthonormal system such that ∑_{j} m_{j}^{k}e_{0}^{op}m_{j}^{k} is the central support of q^{op} in M′ ∩ M_{2} then spM^{op}(∑_{j} m_{j}^{k}M)M^{op} = spMM^{op}v_{q}M^{op}M. Summing up over k and using that ∑_{k} ∑_{j} m_{j}^{k}M = M_{1} = spMe_{1}M, the statement follows.

Step IV. We now derive that

L^{2}(spMM^{op}f_{−n}^{n}MM^{op}) ≃ ⨁_{k∈K_{n}} H_{k} ¯⊗H̄_{k}^{op}

and then

L^{2} (M ⊠_{e_{N}} M^{op}) = ⨁_{k∈K} H_{k} ¯⊗H̄_{k}^{op}.

To see this, note first that H_{k} ¯⊗H̄_{k}^{op} ≃ H_{k′} ¯⊗H̄_{k′}^{op} if and only if H_{k} ≃ H′_{k}. This fact follows immediately by interpreting H_{k} as irreducible representation of M ⊗ M^{op}, according to Connes’ alternative view on correspondences (see [C4], [Po8]). Since by Steps II and III we have ∨_{k∈K_{1}}H_{k} ¯⊗H̄_{k}^{op} = L^{2}(spMM^{op}e_{1}MM^{op}), with H_{k} ¯⊗H̄_{k}^{op} mutually nonisomorphic, the first part of the statement follows

for n = 1. By using this fact for N_{n−1} ⊂ M ⊂^{f_{−n}^{n}} M_{n}, n ≥ 1, we get it for any n ≥ 1. The last part is now clear, since ∪_{n}spMM^{op}f_{−n}^{n}MM^{op} is dense in

M ⊠_{e_{N}} M^{op}.

Step V. We finally show that if s_{k} denotes the minimal projection in T′∩⟨S, T ⟩ labeled by k ∈ K then Tr s_{k} = v_{k}^{2}. This fact can be checked directly by using a similar strategy as in Step III. Instead, we will use the following more elegant argument: Since

v_{k}^{4} = (Tr_{⟨S,T ⟩}s_{k})(Tr_{T′}s_{k}) = (Tr_{⟨S,T ⟩}s_{k})(Tr_{⟨S,T ⟩}J_{S}s_{k}J_{S})

(cf. [J1]), we only need to show that T′ ∩ ⟨S, T ⟩ ∋ s_{k} ↦ J_{S}s_{k}J_{S} ∈ T′ ∩ ⟨S, T ⟩ is Tr_{⟨S,T ⟩}-preserving. To see this note that since ^{op} acts on S leaving T invariant, it implements a Tr_{⟨S,T ⟩}-preserving anti-automorphism on ⟨S, T ⟩, thus a Tr_{⟨S,T ⟩}-preserving automorphism on the commutative algebra T′ ∩ ⟨S, T ⟩. Moreover, if we put s_{k}L^{2}(S, τ) = L^{2}(spT v_{q}T) as in Steps I and II and use that v_{q}^{op} = v_{q}^{∗}, then we have

s_{k}^{op}L^{2}(S, τ) = (L^{2}(spT v_{q}T))^{op}
= L^{2}(spT^{op}v_{q}^{op}T^{op}) = L^{2}(spT v_{q}^{op}T) = L^{2}(spT v_{q}^{∗}T)
= L^{2}(spT^{∗}v_{q}^{∗}T^{∗}) = L^{2}(spT v_{q}T)^{∗} = J_{S}s_{k}J_{S}L^{2}(S, τ)

Thus, s_{k}^{op} = J_{S}s_{k}J_{S} so that Tr(s_{k}) = Tr(s_{k}^{op}) = Tr(J_{S}s_{k}J_{S}). Q.E.D.

Note that the above theorem agrees with the exemples in Section 3. Indeed, if σ ∈ Aut(P) is an automorphism of a type II_{1} factor P and H_{σ} = L^{2}(σ) denotes the P −P bimodule associated with σ as in [Po8] then an easy calculation shows that H̄_{σ}^{op} = H_{σ^{op}}.

4.6. Corollary. Let N ⊂ M be an extremal inclusion. Then N ⊂ M has finite depth if and only if [M ⊠_{e_{N}} M^{op} : M ∨ M^{op}] < ∞. Moreover if these

conditions are satisfied then M ∨ M^{op} has finite depth in M ⊠_{e_{N}} M^{op}.

Proof. With the notations used in 4.5 and its proof, if we assume that N ⊂ M has finite depth then K is finite so that by 4.5 we have dim(S′ ∩ ⟨S, T ⟩) < ∞ and each of the local indices is finite. But then, by Jones’ formula ([J1]), it follows that [S : T] < ∞. Conversely, if [S : T] < ∞ then dim(S′ ∩ ⟨S, T ⟩) < ∞, so that K follows finite, i.e., N ⊂ M has finite depth. Moreover, we see from 4.5 that if [S : T] < ∞ then the set of all T −T irreducible bimodules generated by L^{2}(S, τ) under Connes’ tensor product (fusion) are contained in the set of bimodules {H_{k} ¯⊗H̄_{k′}^{op}}_{k,k′∈K} and is thus finite, i.e., T ⊂ S has finite depth. Q.E.D.

4.7. Remark. As mentioned before, if M is hyperfinite and N ⊂ M is a subfactor of finite depth then by ([Po15]) we have M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op}

is isomorphic to the inclusion M ∨ M′ ∩ M_{∞} ⊂ M_{∞} of [Oc]. This latter inclusion was already shown to have finite depth in [Oc] and in fact all its standard invariant (paragroup) has been calculated ([Oc], see also [EvKa]). In particular, for this class of symmetric enveloping inclusions, part b) of 4.5 can be recovered from ([Oc]). If N ⊂ M is a finite depth subfactor with M not necessarily hyperfinite, then it is imediate to see that M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op}

has the same standard invariant (paragroup) as P ∨ P^{op} ⊂ P ⊠_{e_{Q}} P^{op} where

Q ⊂ P denotes the standard model for N ⊂ M, which is thus an inclusion of hyperfinite factors. Thus, for any N ⊂ M with finite depth the standard invariant (paragroup) of M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op} can be recovered from these

results. Theorem 4.5 shows that in the case (T ⊂ S) = (M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op}) then

L^{2}(S, τ) is spanned by T −T bimodules which are finitely generated both as left and right T -modules. Equivalently, T ⊂ S is such that T′ ∩ ⟨S, T ⟩ is generated by finite projections of ⟨S, T ⟩. Inclusions T ⊂ S verifying this latter condition are called discrete in [ILP]. We’ll introduce here a new terminology for such subfactors based on the former, more intrinsic characterization.

4.8. Definition. Let T ⊂ S be an irreducible inclusion of type II_{1} factors. We

denote by qN_{S}(T) =^{def} {x ∈ S | ∃ x_{1}, x_{2}, …, x_{n} ∈ S such that xT ⊂ ∑_{i=1}^{n} T x_{i}

and T x ⊂ ∑_{i=1}^{n} x_{i}T }. We call qN_{S}(T) the quasi-normalizer of T in S.

Note that the condition “xT ⊂ ∑ T x_{i}, T x ⊂ ∑ x_{i}T ” is equivalent to “T xT ⊂ (∑_{i=1}^{n} T x_{i}) ∩ (∑_{i=1}^{n} x_{i}T)” and also to “spT xT is finitely generated both as left and as a right T -module.” It then follows readily that sp(qN_{S}(T)) is a ∗-algebra. Thus P =^{def} \overline{sp}(qN_{S}(T)) = qN_{S}(T)′′ is a subfactor of S containing T. Note also that L^{2}(P) = ∨{H | H ⊂ L^{2}(P), H is a T -T bimodule, dim(_{T} H) < ∞, dim(H_{T}) < ∞} and that the orthogonal projection e_{P}, of L^{2}(S) onto L^{2}(P), satisfies e_{P} = ∨{f ∈ T′ ∩ ⟨S, T ⟩ | Trf < ∞, TrJ_{S}fJ_{S} < ∞}. All these facts are just reformulations of some results in [PiPo1] and [ILP], but can also be proved as exercises. The terminology we wanted to introduce is then as follows:

4.9. Definition. Let T ⊂ S be an irreducible inclusion. If qN_{S}(T)′′ = S, we say that T is quasi-regular in S. From the above remarks we see that an irreducible inclusion T ⊂ S is discrete (as defined in [ILP]) iff T is quasi-regular in S.

Thus, from 4.5 it follows that if N ⊂ M is an extremal inclusion of type II_{1} factors then M ∨ M^{op} is quasi-regular in M ⊠_{e_{N}} M^{op}. Note that, even more,

we showed that each irreducible T -T bimodule in L^{2}(S) (where T = M ∨ M^{op}, S = M ⊠_{e_{N}} M^{op}) has multiplicity 1 and its (finite) dimension as a left T

module coincides with its dimension as a right T -module. Thus, our symmetric enveloping inclusions have very similar properties to the inclusions given by cross-products of factors by outer actions of discrete groups. We wanted to emphasize even more this aspect by choosing the terminology “quasi-normalizer”, “quasi-regular” in analogy with Dixmier’s notions of “normalizer” and “regularity” for an irreducible subfactor ([D1]). This is particularily justified by noticing that exemples of quasi-regular subfactors T ⊂ S can be obtained by requiring S to be generated by unitary elements u such that uT u^{∗} is included in T and has finite index in it (see the Appendix in [ILP] for a concrete exemple of such a situation). Let us end this section with a result showing that the extended sequence of Jones projections in a tunnel-tower associated to a subfactor N ⊂ M has a certain general ergodicity property with respect to the higher relative commutants that is very useful in applications (see e.g. 2.2 and 2.3 in [GePo]). We’ll refer to this result as the Ergodicity Theorem for Higher Relative Commutants.

4.10. Theorem. Let N ⊂ M be a subfactor with finite index (but not necessarily extremal). Let {M_{j}}_{j∈ℤ} be a tunnel-tower for N ⊂ M, where M_{0} = M, M_{−1} = N, and {e_{j}}_{j∈ℤ} be its corresponding Jones projections. Denote A_{ij} = M′_{i} ∩ M_{j} and A_{−∞,i} = \overline{⋃_{n≤i} A_{ni}}, A_{−∞,∞} = \overline{⋃_{i} A_{−∞,i}}. Then we have:

a) {e_{j}}′_{j∈ℤ} ∩ A_{−∞,∞} = ℂ. In particular, A_{−∞,∞} is a factor.

b) If M has separable predual then the tunnel {M_{j}}_{j≤0} can be chosen such that {e_{j}}′_{j≤k} ∩ M_{n} ⊂ A_{−∞,n}, ∀k ≤ n in ℤ.

c) If N ⊂ M is extremal and its tunnel is chosen to satisfy condition b) then

{e_{j}}′_{j∈ℤ} ∩ M ⊠_{e_{N}} M^{op} = ℂ.

Proof. a). Let θ be the trace preserving automorphism on A_{−∞,∞} implemented by the duality isomorphism (1.5 of [PiPo1] or 1.3.3 of [Po2]), i.e., θ satisfies θ(A_{ij}) = A_{i+2,j+2}, θ(e_{k}) = e_{k+2}, ∀i, j, k ∈ ℤ, with θ_{|A_{ij}} being defined as the restriction to M′_{i} ∩ M_{j} = M_{i}^{α}′ ∩ M_{j}^{α} of σ′_{ij} : (M_{i} ⊂ M_{i+1} ⊂ ... ⊂ M_{j})^{α} → (M_{i+2} ⊂ ... ⊂ M_{j+2}), where σ′_{ij}((x_{rs})_{r,s}) = λ^{i−j+1} ∑_{r,s} m_{r}e_{i+2}e_{i+3}...e_{j+2}x_{rs}e_{j+2}...e_{i+2}m_{s}^{∗}, in which {m_{r}}_{r} is an orthonormal basis of vN{e_{n}}_{n≤i+1} over vN{e_{n}}_{n≤i} and λ^{−1} = [M : N]. We first show that this automorphism satisfies the identity θ(z)e_{j+2} = ze_{j+2} for all z ∈ {e_{k}}′_{k≤j} ∩ A_{−∞,j}. To this end let ε > 0 and i ≤ j be so that ∥E_{A_{ij}}(z) − z∥_{2} < ε. Put z_{0} = E_{A_{ij}}(z) ∈ {e_{i+2}, ..., e_{j}}′ ∩ A_{ij}. From the above local formula for θ we have

e_{j+2}θ(z_{0})e_{j+2} = λ^{i−j+1}e_{j+2}(∑_{r} m_{r}e_{i+2}...e_{j+2}z_{0}e_{j+2}...e_{i+2}m_{r}^{∗})e_{j+2}

= λ^{i−j+3} ∑_{r} m_{r}e_{i+2}...e_{j}(z_{0}e_{j+2})e_{j}...e_{i+2}m_{r}^{∗}

= ∑_{r} m_{r}(z_{0}e_{i+2}e_{j+2})m_{r}^{∗} = (∑_{r} m_{r}(z_{0}e_{i+2})m_{r}^{∗})e_{j+2}.

By taking into account that the orthonormal basis {m_{r}}_{r} can be taken to be made up of no more than [M : N] + 1 elements, we thus get the estimates:

∥θ(z)e_{j+2} − ze_{j+2}∥_{2}

≤ ∥θ(z) − θ(z_{0})∥_{2} + ∥z − z_{0}∥_{2} + ∥e_{J+2}θ(z_{0})e_{j+2} − z_{0}e_{j+2}∥_{2}
≤ 2ε + ∥Σ_{r}m_{r}(z_{0}e_{i+2})m_{r}^{∗}e_{j+2} − z_{0}e_{j+2}∥_{2}

≤ 2ε + ∑_{r} ∥[m_{r}e_{i+2}, z_{0}]∥_{2}

≤ 2ε + ([M : N] + 1)^{2}ε.

Letting ε tend to 0, we get the desired identity. Now to prove part a) of the statement let z ∈ vN{e_{n}}′_{n∈ℤ}∩A_{−∞,∞} with τ(z) = 0 and take z_{0} = E_{A_{−∞,j}}(z) for some j. Note that τ(z_{0}) = 0 as well. For such a z_{0}, and in fact for any z_{0} in vN{e_{n}}′_{n≤j} ∩ A_{−∞,j}, we then have the estimates:

∥(z − z_{0})e_{j+2}∥_{2}^{2} = τ((z − z_{0})^{∗}(z − z_{0})e_{j+2})

= τ(E_{{e_{l}}′_{l≥j+2}∩A_{−∞,∞}}((z − z_{0})^{∗}(z − z_{0})e_{j+2}))
= τ((z − z_{0})^{∗}(z − z_{0})E_{{e_{l}}′_{l≥j+2}∩A_{−∞,∞}}(e_{j+2})) = λ∥z − z_{0}∥_{2}^{2},

in which we used that by Jones ergodicity theorem we have E_{{e_{l}}′_{l≥j+2}∩A_{−∞,∞}}(e_{j+2}) = λ1. Since z_{0}e_{j+2} = θ(z_{0})e_{j+2}, we get similarily:

∥(z − z_{0})e_{j+2}∥_{2}^{2} = ∥(z − θ(z_{0}))e_{j+2}∥_{2}^{2}
= τ((z − θ(z_{0})^{∗}(z − θ(z_{0}))e_{j+2})
= τ(E_{{e_{l}}′_{l≤j+2}∩A_{−∞,∞}}((z − θ(z_{0}))^{∗}(z − θ(z_{0}))e_{j+2}))
= τ((z − θ(z_{0}))^{∗}(z − θ(z_{0}))E_{{e_{l}}′_{l≤j+2}∩A_{−∞,∞}}(e_{j+2}))
= λ∥z − θ(z_{0})∥_{2}^{2},

in which we used the fact that θ(z_{0}) commutes with vN{e_{l}}_{l≤j+2} and that by Jones ergodicity theorem we have E_{{e_{l}}′_{l≤j+2}∩A_{−∞,∞}}(e_{j+2}) = λ1. Altogether, the above shows that ∥z − z_{0}∥_{2} = ∥z − θ(z_{0})∥_{2} and by applying this recursively n times we get ∥z − z_{0}∥_{2} = ∥z − θ^{n}(z_{0})∥_{2}, ∀n ≥ 1. On the other hand θ^{n}(A_{ij}) = A_{i+2n,j+2n} and so, if n is so that 2n > j − i then τ(z_{1}θ^{n}(z_{1})) = τ(z_{1})^{2}, ∀z_{1} ∈ A_{ij}, showing that θ is mixing on A_{−∞,∞} = \overline{⋃_{i,j} A_{ij}}. Thus, for z_{0} ∈ A_{−∞,j} with τ(z_{0}) = 0 we have

lim_{n→∞} ∥z_{0} − θ^{n}(z_{0})∥_{2}^{2} = 2∥z_{0}∥_{2}^{2}.

Since ∥z_{0} −θ^{n}(z_{0})∥_{2} ≤ ∥z_{0} −z∥_{2} +∥z −θ^{n}(z_{0})∥_{2} and since for z ∈ vN{e_{n}}′_{n∈ℤ} ∩ A_{−∞,∞} we proved that ∥z − z_{0}∥_{2} = ∥z − θ^{n}(z_{0})∥_{2}, ∀n ≥ 1, in which z_{0} = E_{−∞,j}(z), it follows that for each j we have the estimate:

2∥z_{0}∥_{2}^{2} = lim_{n→∞} ∥z_{0} − θ^{n}(z_{0})∥_{2}^{2} ≤ 4∥z − z_{0}∥_{2}^{2}.

Now, letting j tend to infinity we get ∥z − z_{0}∥_{2} tend to 0 and ∥z_{0}∥_{2} tend to ∥z∥_{2}, which from the above estimate forces z = 0. This ends the proof of a). b). Let {x_{n}}_{n≥1} ⊂ M be a sequence of elements dense in the unit ball of M in the so-topology. We construct recursively a sequence of integers 0 < k_{1} < k_{2} < ... and a tunnel M ⊃ N ⊃ N_{1}... ⊃ N_{k_{1}} ⊃ ... ⊃ N_{k_{n}} ⊃ ... for N ⊂ M such that if {e_{n}}_{n≤0} are the corresponding Jones projections and we denote by B_{n} = Alg{e_{j}}_{−k_{n}+1≤j≤−k_{n−1}−1} then we have:

∥E_{B′_{n}∩M}(x_{j}) − E_{N′_{k_{n−1}} ∩M}(x_{j})∥_{2} < 2^{−n}, ∀j ≤ n

Assume we have this up to some n. By ([Po1]) there exists a hyperfinite subfactor R ⊂ N_{k_{n}} such that E_{R′∩M}(x) = E_{N′_{kn} ∩M}(x), ∀x ∈ M. On the other hand, by Jones ergodicity theorem, we can regard R as being generated by a sequence of Jones λ-projections e_{j} indexed over the integers ≤ −k_{n} − 1. Thus, there will exist a sufficiently large k_{n+1} such that if we denote B_{n+1} = Alg{e_{j}}_{−k_{n+1}+1≤j≤−k_{n}−1} then

∥E_{B′_{n+1}∩M}(x_{j}) − E_{N′_{kn} ∩M}(x_{j})∥_{2} < 2^{−n−1}, ∀j ≤ n + 1.

Now choose a Jones projection e_{−k_{n}} for N_{k_{n}} ⊂ N_{k_{n}−1} such that it commutes with e_{j} ∈ B_{n+1} for j ≤ −k_{n} − 2 and such that it satisfies the Jones-Temperley-Lieb relation for j = −k_{n} − 1 (see the proof of 4.4 on page 33 of [Po15]), i.e., e_{−k_{n}}e_{−k_{n}−1}e_{−k_{n}} = λe_{−k_{n}}, and then simply define the corresponding tunnel N_{k_{n}} ⊃ N_{k_{n}+1} ⊃ ... ⊃ N_{k_{n+1}} as given by these newly chosen Jones projections e_{j} with −k_{n+1} + 1 ≤ j ≤ −k_{n}. Thus, if we take A_{n} = \overline{∪_{m≥n}B_{m}} ⊂ vN\{e_{j}\}_{j≤−k_{n}−1} then it follows from the above that E_{A′_{n}∩M}(x) ∈ \overline{∪_{k}N′_{k} ∩ M} for all x ∈ \{x_{j}\}_{j} and thus by density for all x ∈ M. Thus even more so vN\{e_{j}\}′_{j≤−m} ∩ M ⊂ \overline{∪_{k}N′_{k} ∩ M} for m = k_{n} and thus in fact for all m ≥ 0. Finally, if x ∈ M_{n} for some n ≥ 0 then for any ε > 0 there exists k ≤ 0 and x′ ∈ sp((Alg{e_{j}}_{k≤j≤n})M) such that ∥x − x′∥_{2} < ε. But then x′′ = E_{\{e_{l}\}′_{l≤k−2}∩M_{n}}(x′) belongs to sp((Alg(\{e_{j}\}_{k≤j≤n})\overline{∪_{i}N′_{i} ∩ M}) which in turn is

icluded into \overline{∪_{i}N′_{i} ∩ M_{n}} and we have:

∥E_{{e_{l}}′_{l≤k−2}∩M_{n}}(x) − x′′∥_{2} = ∥E_{{e_{l}}′_{l≤k−2}∩M_{n}}(x − x′)∥_{2} ≤ ∥x − x′∥_{2} ≤ ε.

Letting ε go to 0 and j to −∞ we get

lim_{j→−∞} E_{{e_{l}}′_{l≤j−2}∩M_{n}}(x) = E_{A_{−∞,n}}(x), ∀x ∈ M_{n}

This ends the proof of b) and c) follows then imediately, by taking into account that ⋃_{n} spMM_{n}^{op}M is dense in M ⊠_{e_{N}} M^{op} and applying a) and b). Q.E.D.

4.11. Corollary. Let N ⊂ M be an extremal inclusion of type II_{1} factors with separable preduals. There exists a choice of a tunnel {M_{j}}_{j≤0} for N ⊂ M such that if we denote M_{n} = (M_{−n})^{op}′ ∩ M ⊠_{e_{N}} M^{op}, n ≥ 1, M_{∞} = \overline{∪_{n}M_{n}},

M_{∞}^{op} = \overline{∪_{n}M_{n}^{op}} and A_{−∞,∞} = \overline{∪_{n}M′_{−n} ∩ M_{n}} then M_{∞}, M_{∞}^{op} ⊂ M ⊠_{e_{N}} M^{op}

satisfy the conditions:

a) \overline{sp}M_{∞}M_{∞}^{op} = M ⊠_{e_{N}} M^{op}.

b) M_{∞} ∩ M_{∞}^{op} = A_{−∞,∞} and E_{M_{∞}}E_{M_{∞}^{op}} = E_{A_{−∞,∞}}.

c) A′_{−∞,∞} ∩ M ⊠_{e_{N}} M^{op} = ℂ1.

Proof.. Conditions a) and b) are actually valid for any choice of the tunnel while 4.9 clearly implies c). Q.E.D.

5. Relating the Amenability Properties

of Γ_{N,M}, G_{N,M} and (M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op})

In [Po8] one considers a notion of relative amenability for inclusions of finite von Neumann algebras T ⊂ S by requiring the existence of norm one projections from ⟨S, T ⟩ onto S, equivalently of Connes-type S-hypertraces on ⟨S, T ⟩. In the case S = T ⋊ G for some discrete group G this condition on the inclusion T ⊂ S is equivalent to the amenability of the group G. As we have seen in the previous section, when T = M ∨M^{op} ⊂ M ⊠_{e_{N}} M^{op} = S,

for N ⊂ M a locally trivial subfactor associated to some faithful G-kernel σ, with G a finitely generated discrete group, then (T ⊂ S) ≃ (T ⊂ T ⋊_{σ⊗σ^{op}} G). Thus, the relative amenability of the inclusion T ⊂ S is equivalent in this case to the amenability of G. On the other hand, one of the equivalent characterizations of the amenability of G is Kesten’s condition requiring that the Cayley graph of G, Γ, corresponding to some finite, self-adjoint set of generators g_{0} = 1, g_{1}, …, g_{n}, satisfies ∥Γ∥ = n + 1. Recalling from [Po2,5] that the standard graph of a subfactor Γ_{N,M} is called amenable if it satisfies the Kesten-type condition ∥Γ_{N,M}∥^{2} = [M : N] and that its standard invariant G_{N,M} is called amenable if Γ_{N,M} is amenable, and noticing that for the locally trivial subfactor N ⊂ M corresponding to the above (G; g_{0}, …, g_{n}; σ) the Cayley graph Γ coincides with the standard graph Γ_{N,M}, while [M : N] = (n + 1)^{2}, it follows that in this case the amenability of G (thus, the relative amenability of T ⊂ S) is equivalent to the amenability of G_{N,M}. We prove in this section that in fact even for arbitrary extremal subfactors of finite index N ⊂ M the relative amenability condition on T = M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op} = S is equivalent to the amenability of the standard lattice G_{N,M}.

Along the lines, we will obtain some other related characterizations of the amenability of G_{N,M}, thus of Γ_{N,M}. Before stating the result, recall some terminology and notations from [Po2].

So let (N ⊂^{E} M) = ⨁((N ⊗P^{op})^{∗∗} ⊂^{(E⊗id)^{∗∗}} (M ⊗P^{op})^{∗∗}), the sum being taken over all isomorphism classes of type II_{1} factors P that can be embedded with finite index in some amplification of M, i.e., factors P that are weakly stably equivalent to M in the sense of 1.4.3 in [Po8] (like for instance P = M). Then

take first the atomic part of this inclusion, (N ⊂^{E} M)_{at}, and next the binormal

part of the latter inclusion, ((N ⊂^{E} M)_{at})_{bin} (i.e., the largest direct summand in which both M and P^{op} sit as von Neumann algebras), which we denote by

N^{u} ⊂^{E^{u}} M^{u}, and call the universal atomic (binormal) representation of N ⊂ M. Also, the inclusion graph (or matrix) of N^{u} ⊂ M^{u} is denoted by Γ_{N,M}^{u} and called the universal graph (or matrix) of N ⊂ M.

Finally one defines (N^{st} ⊂^{E^{st}} M^{st}) to be the minimal direct summand of N^{u} ⊂ M^{u} (or, equivalently, of (N ⊂^{E} M)_{at}) containing the standard representation of M ⊗ M^{op}, B(L^{2}(M)) and call it the standard representation of N ⊂ M. It is easy to see (cf. e.g., [Po2]) that the commuting square embedding

N^{st}  ⊂^{E^{st}} M^{st}
∪  ∪
N ⊗ M^{op} ⊂^{E⊗id} M ⊗ M^{op}

can be identified with the embedding

⨁_{ℓ∈L} B(K_{ℓ}) ⊂^{E^{st}} ⨁_{k∈K} B(H_{k})
∪  ∪
N ⊗ M^{op} ⊂^{E⊗id} M ⊗ M^{op}

in which {H_{k}}_{k∈K} (respectively, {K_{ℓ}}_{ℓ∈L}) is the list of all irreducible M-M (resp. N-M) bimodules appearing as direct summands in L^{2}(M_{j}), j = 0, 1, 2, …, and M ⊗ M^{op} (resp, N ⊗ M^{op}) is represented on each H_{k} (resp. K_{ℓ}) by operators of left and right multiplication by elements in M (respectively, right multiplication by elements in N and right multiplication by elements in M). Moreover, the inclusion matrix (or graph) for

N^{st} ⊂ ⨁_{ℓ∈L} B(K_{ℓ}) ⊂ ⨁_{k∈K} B(H_{k}) = M^{st}

(which is thus a direct summand of the universal graph Γ_{N,M}^{u}) is given by (Γ_{N,M})^{t}, while E^{st} is the unique expectation that preserves the trace Tr on M^{st} = ⨁_{k∈K} B(H_{k}) given by the weight vector v⃗ = (v_{k})_{k∈K}, with

v_{k} = dim(_{M}H_{k}_{M})^{1/2}. Finally, note that N ⊂ M is in fact embedded in the smaller inclusion

N^{st,f} :=^{def} (1 ⊗ M^{op})′ ∩ N^{st} ⊂^{E^{st,f}} (1 ⊗ M^{op})′ ∩ M^{st} =:^{def} M^{st,f}

where E^{st,f} is the restriction of E^{st} to M^{st,f}.

5.1. Definition. The commuting square embedding:

N^{st,f} ⊂^{E^{st,f}} M^{st,f}
∪  ∪
N ⊂^{E_{N}} M

is called the finite (or reduced) standard representation of N ⊂ M.

5.2. Lemma. N^{st,f}, M^{st,f} are finite type II_{1} von Neumann algebras with atomic centers Z(N^{st,f}) = Z(N^{st}) ≃ ℓ^{∞}(L), Z(M^{st,f}) = Z(M^{st}) ≃ ℓ^{∞}(K). Moreover, the inclusion N^{st,f} ⊂ M^{st,f} is a matricial inclusion having inclusion matrix (or graph) (Γ_{N,M})^{t}. Also, E^{st,f} is the unique conditional expectation of M^{st,f} onto N^{st,f} preserving the trace Tr on M^{st,f} given by the weights {v_{k}^{2}}_{k∈K} on the center Z(M^{st,f}) ≃ ℓ^{∞}(K).

Proof. The first part is trivial, by the definition of N^{st,f} ⊂ M^{st,f} and the properties of N^{st} ⊂ M^{st}. Then the last part is an immediate consequence of the first part and of 2.7 in [PiPo2]. Q.E.D.

5.3. Theorem. Let N ⊂ M be an extremal inclusion of type II_{1} factors. The following conditions are equivalent:

1) G_{N,M} is amenable. 1′) Γ_{N,M} is amenable, i.e., Γ_{N,M} satisfies the Kesten type condition ∥Γ_{N,M}∥^{2} = [M : N].

2) (Γ_{N,M}, v⃗) satisfies the Følner-type condition: ∀ ε > 0, ∃F ⊂ K finite such that

∑_{k∈∂F} v_{k}^{2} < ε ∑_{k∈F} v_{k}^{2},

where

∂F = {k ∈ K \ F | ∃ k_{0} ∈ F such that (Γ_{N,M}Γ_{N,M}^{t})_{kk_{0}} ≠ 0}.

3) There exists a state ψ_{0} on ℓ^{∞}(K) ≃ T′ ∩ ⟨S, T ⟩ such that ψ_{0} ∘ E has S = M ⊠_{e_{N}} M^{op} in its centralizer, where E is the unique Tr-preserving conditional

expectation of ⟨S, T ⟩ onto T′ ∩ ⟨S, T ⟩.

4) M ⊠_{e_{N}} M^{op} is amenable relative to M ∨ M^{op}.

5) There exists a norm one projection from (N^{st,f} ⊂^{E^{st,f}} M^{st,f}) onto (N ⊂^{E_{N}} M).

5′) There exists a (N ⊂ M)-hypertrace on (N^{st,f} ⊂^{E^{st,f}} M^{st,f}).

Proof. 1) ⟺ 1′) is clear by the definitions. To prove 1′) ⟹ 2) let Φ = λV^{−1}ΓΓ^{t}V, where V is the diagonal matrix over K with entries (v_{k})_{k∈K}. Note that Φ defines a bounded positive linear operator from P =^{def} T′ ∩ ⟨S, T ⟩ ≃ ℓ^{∞}(K) into itself such that Φ(1) = 1. Note also that the trace Tr on P inherited from ⟨S, T ⟩ has weights (v_{k}^{2})_{k∈K} as a measure on K, i.e., if b ∈ P ≃ ℓ^{∞}(K) then

∥b∥_{1,Tr} = ∑_{k∈K} |b_{k}|v_{k}^{2}.

For a, b : K → ℂ, at least one of which has finite support, we denote ⟨a, b⟩ = ∑_{k∈K} a_{k}b̄_{k}. For each b ∈ P ≃ ℓ^{∞}(K) with finite support we then have:

Tr(Φ(b)) = ⟨Φ(b), V^{2}(1)⟩ = ⟨b, λV ΓΓ^{t}V^{−1}V^{2}(1)⟩
= ⟨b, λV ΓΓ^{t}V (1)⟩ = ⟨b, V^{2}(1)⟩ = Tr(b).

Thus Tr ∘ Φ = Tr. In particular, by Kadison’s inequality, this implies ∥Φ(a)∥_{2,Tr} ≤ ∥a∥_{2,Tr}, ∀ a ∈ L^{2}(T, Tr). Since ∥λΓΓ^{t}∥ = 1, it follows that ∀ δ > 0 ∃ F_{0} ⊂ K finite such that T_{0} = F_{0}(λΓΓ^{t})_{F_{0}} satisfies 1 ≥ ∥T_{0}∥ ≥ 1 − δ^{2}/2. By the classical Perron-Frobenius theorem applied to T_{0} (which is a finite symmetric matrix with nonnegative entries) it follows that there exists b_{0} ∈ ℓ^{∞}(K) ≃ P, supported in the set F_{0}, with b_{0}(k) ≥ 0, ∀ k, and ⟨b_{0}, b_{0}⟩ = 1, such that T_{0}b_{0} ≥ (1 − δ^{2}/2)b_{0}. Thus, λΓΓ^{t}b_{0} ≥ (1 − δ^{2}/2)b_{0}. Let then b =^{def} V^{−1}(b_{0}) ∈ ℓ^{∞}(K) ≃ P and note that

∥b∥_{2,tr}^{2} = ⟨V^{−1}(b_{0}), V^{2}V^{−1}(b_{0})⟩ = ⟨b_{0}, b_{0}⟩ = 1.

Moreover, we have:

∥Φ(b) − b∥_{2,Tr}^{2} ≤ 2 − 2Tr(Φ(b)b)
= 2 − 2⟨λV^{−1}ΓΓ^{t}(b_{0}), V (b_{0})⟩
= 2 − 2⟨λΓΓ^{t}(b_{0}), b_{0}⟩
≤ 2 − 2(1 − δ^{2}/2) = 2δ^{2}/2 = δ^{2}.

Thus ∥b − Φ(b)∥_{2,Tr} < δ and ∥Φ(b)∥_{2,Tr} ≥ 1 − δ, while ∥b∥_{2,Tr} = 1. By Theorem A.2 it follows that if δ < 10^{−4} then there exists a finite spectral projection e of b such that ∥Φ(e) − e∥_{2,Tr} < δ^{1/4}∥e∥_{2,Tr}. In particular we have:

∥(1 − e)Φ(e)∥_{2,Tr}^{2} ≤ ∥(1 − e)Φ(e)∥_{2,Tr}^{2} + ∥e − eΦ(e)∥_{2,Tr}^{2}
= ∥e − Φ(e)∥_{2,Tr}^{2} < δ^{1/4}∥e∥_{2,Tr}^{2}.

Let F ⊂ K be the support set of e ∈ ℓ^{∞}(K) ≃ P. By the first 3 lines of the proof of Lemma 3.2 on page 281 of [Po3], we have v_{k}^{−1}v_{k_{0}} ≥ λ for all k_{0}, k ∈ K for which (ΓΓ^{t})_{kk_{0}} ≠ 0. Thus we have

(Φ)_{kk_{0}} = λv_{k}^{−1}v_{k_{0}} ∑_{l∈L} a_{kl}a_{k_{0}l} ≥ λ^{2},

forall k, k_{0} ∈ K for which the entry (k, k_{0}) of Φ is nonzero. In particular, this shows that Φ(e)(1 − e) ≥ λ^{2}χ_{∂F}, where χ_{∂F} ∈ ℓ^{∞}(K) is the characteristic function of ∂F ⊂ K. Thus we have

λ^{4} ∑_{k∈∂F} v_{k}^{2} = ∥λ^{2}χ_{F} ∥_{2,Tr}^{2}

≤ ∥(1 − e)Φ(e)∥_{2,Tr}^{2} < δ^{1/4}∥e∥_{2,Tr}^{2}

= δ^{1/4} ∑_{k∈F} v_{k}^{2}.

Thus, if ε > 0 was given and we take δ = (λ^{4}ε)^{4} then

∑_{k∈∂F} v_{k}^{2} < ε ∑_{k∈F} v_{k}^{2}

thus proving 1′) ⟹ 2). Proof of 2) ⟹ 3). By 2), for each ε = 2^{−n} there exisits a finite subset F_{n} ⊂ K such that

∑_{k∈∂F_{n}} v_{k}^{2} < 2^{−n} ∑_{k∈F_{n}} v_{k}^{2}

or, equivalently,

( ∑_{k∈ΓΓ^{t}F_{n}} v_{k}^{2} − ∑_{k∈F_{n}} v_{k}^{2}) < 2^{−n} ∑_{k∈F_{n}} v_{k}^{2}.

Let f_{n} ∈ T′ ∩ ⟨S, T ⟩ ≃ ℓ^{∞}(K) be the support projection of F_{n}. Let ω be a free ultrafilter on ℕ ≃ K and define ψ_{0} on ℓ^{∞}(K) ≃ T′ ∩ ⟨S, T ⟩ by

ψ_{0} = lim_{n→ω} Tr(·f_{n})/Trf_{n}.

Let ψ =^{def} ψ_{0} ∘ E and note that ψ = lim_{n→ω} Tr(·f_{n})/Trf_{n} on ⟨S, T ⟩ as well. Note that for each n we have that Tr(·f_{n})/Trf_{n} has T in its centralizer and it is a normal state on ⟨S, T ⟩. Since T′ ∩ S = ℂ this implies that Tr(·f_{n})/Trf_{n} coincides with the trace τ when restricted to S. Thus, ψ|_{S} = τ and ψ has T in its centralizer. Let us show that ψ also has e_{N} in its centralizer. To do this, it is sufficient to prove that

lim_{n→∞}(∥f_{n}e_{N} − e_{N}f_{n}∥_{1,Tr}/Trf_{n}) = 0.

Let f′_{n} ∈ T′ ∩ ⟨S, T ⟩ ≃ ℓ^{∞}(K) be the support projection of F_{n} ∪ ∂F_{n} and note that we have

lim_{n→∞}(∥f′_{n} − f_{n}∥_{1,Tr}/Trf_{n}) = lim_{n→∞} ( ∑_{k∈∂F_{n}} v_{k}^{2}/ ∑_{k∈F_{n}} v_{k}^{2}) = 0.

Also, we have

∥f_{n}e_{N} − e_{N}f_{n}∥_{1,Tr} ≤ ∥e_{N}f_{n} − f′_{n}e_{N}f_{n}∥ + 2∥f′_{n} − f_{n}∥_{1,Tr}.

So, to prove that [e_{N}, ψ] = 0, it is in fact sufficient to prove that f′_{n}e_{N}f_{n} = e_{N}f_{n}, ∀ n. We will show that, more generally, we have s_{F′}e_{N}s_{F} = e_{N}s_{F}, ∀ F ⊂ K, where F′ = F ∪ ∂F and s_{F} = ∑_{k∈F} s_{k}, s_{F′} = ∑_{k∈F′} s_{k}. To this end, it is clearly sufficient to do it for single element sets F = {k_{1}}. It then amounts to show that if k_{2} ∈ K \ F′, then s_{k_{2}}e_{N}s_{k_{1}} = 0. By the proof of 4.5 we thus need to show that if k_{1}, k_{2} ∈ K_{n−1} for some n, with k_{2} ∉ {k_{1}} ∪ ∂{k_{1}} and we take a minimal projection q_{i} in the direct summand labeled by k_{i} in N′_{2n−1} ∩ M, for each i = 1, 2, then we have MM^{op}v_{q_{2}} ⊥ e_{1}MM^{op}v_{q_{1}}MM^{op}, where e_{1} = e_{N} and v_{q_{i}} = E_{N′_{n−1}∩M_{n}}(q_{i}q_{i}^{op}f_{−n}^{n}f_{−2n}^{0}f_{0}^{2n}), i = 1, 2. Before proving this, note that for such q_{1}, q_{2} we have q_{2}e_{N}q_{1} = 0 and in fact q_{2}(N′_{2n−1} ∩ M_{1})q_{1} = 0. Now, if we take x_{1,2} ∈ M, y_{1,2} ∈ M^{op}, x, x_{0} ∈ N_{n−1}, y, y_{0} ∈ N_{n−1}^{op}, then we get

τ(v_{q_{2}}^{∗}y_{0}^{op}y_{2}^{op}xx_{2}e_{1}x_{1}y_{1}^{op}v_{q_{1}}x_{0}f_{−2n}^{0}xy^{op}f_{0}^{2n}y_{0}^{op})
= τ(E_{N′_{n−1}∩M_{n}}(f_{0}^{2n}f_{−2n}^{0}f_{−n}^{n}q_{2}q_{2}^{op})y_{0}^{op}y_{2}^{op}xx_{2}e_{1}x_{1}x_{0}y_{1}^{op}y^{op}
· E_{N′_{n−1}∩M_{n}}(q_{1}q_{1}^{op}f_{−n}^{n}f_{0}^{2n}f_{−2n}^{0})f_{−2n}^{0}f_{0}^{2n}).

Taking the conditional expectation onto N′_{2n−1} ∩ (M ⊠_{e_{N}} M^{op}) and denoting

Y_{1}^{op} = y_{0}^{op}y_{2}^{op} ∈ M^{op}, Y_{2}^{op} = y_{1}^{op}y^{op} ∈ M^{op}, X′ = E_{N′_{2n−1}}(xx_{2}e_{1}x_{1}x_{0}) ∈ N′_{2n−1} ∩ M_{1}, we thus obtain that the above is equal to:

τ(f_{0}^{2n}f_{−2n}^{0}f_{−n}^{n}q_{2}^{op}q_{2}Y^{op}X′Y_{2}^{op}q_{1}q_{1}^{op}f_{−n}^{n}f_{0}^{2n}f_{−2n}^{0})

= τ(f_{0}^{2n}f_{−2n}^{0}f_{−n}^{n}q_{2}^{op}Y_{1}^{op}(q_{2}X′q_{1})Y_{2}^{op}q_{1}^{op}f_{−n}^{n})

= 0

in which we first used that v_{q_{i}}f_{−2n}^{0}f_{0}^{2n} = q_{i}q_{i}^{op}f_{−n}^{n}f_{0}^{2n}f_{−2n}^{0} and then we used that q_{2}X′q_{1} = 0. Since the elements of the form x_{0}f_{−2n}^{0}x with x, x_{0} ∈ N_{n−1} span all M, this finishes the proof of the fact that e_{N} is in the centralizer of ψ. Since ψ is equal to the trace on S = M ⊠_{e_{N}} M^{op} and has in its centralizer the weakly dense

∗-subalgebra generated by T = M ∨ M^{op} and e_{N} in S, by [C3] it follows that ψ has all S in its centralizer. This ends the proof of 2) ⟹ 3). The proof of 3) ⟹ 4) is then trivial, since the relative amenability of T = M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op} = S merely requires the existence of a state on ⟨S, T ⟩

which has S in its centralizer, while condition 3) provides very special such states. To prove 4) ⟹ 5) we need the following lemma.

5.4. Lemma. Let M_{0} = vN(M ∪ J_{S}MJ_{S}), N_{0} = vN(N ∪ J_{S}MJ_{S}) and Φ_{0} : B(L^{2}(S)) → B(L^{2}(S)) be defined by Φ_{0}(T) = λ ∑_{j} m_{j}^{op}T m_{j}^{op}^{∗}, where {m_{j}^{op}}_{j} is an orthonormal basis of M_{1}^{op} over M^{op} and λ = [M : N]^{−1} as usual. Then Φ_{0}(M_{0}) = N_{0}, E_{0} = Φ_{0}|_{M_{0}} is a conditional expectation and in fact

\begin{matrix}N_{0} & ⊂^{E_{0}} & M_{0} \\ ∪ & & ∪ \\ N & ⊂^{E_{N}} & M \end{matrix}

is a commuting square embedding of N ⊂ M, which is isomorphic to the standard representation of N ⊂ M. Moreover, if N_{0}^{f} = J_{S}MJ′_{S} ∩ N_{0} ⊂ J_{S}MJ′_{S} ∩ M_{0} = M_{0}^{f}, then

\begin{matrix}N_{0}^{f} & ⊂^{E_{0}^{f}} & M_{0}^{f} \\ ∪ & & ∪ \\ N & ⊂ & M \end{matrix}

is a commuting square embedding isomorphic to the finite standard representation of N ⊂ M.

Proof. By construction, we see that N_{0} ⊂ M_{0} is a direct summand of (N ⊗

M^{op})^{∗∗} ⊂^{(E⊗id)^{∗∗}} (M ⊗M^{op})^{∗∗}. Also, since N_{0} ⊂ (M_{1}^{op} ∪J_{S}M^{op}J_{S})′ ∩B(L^{2}(S)),

we have

M′_{0} ∩ N_{0} ⊂ (M ∪ J_{S}MJ_{S})′ ∩ (M_{1}^{op} ∪ J_{S}M^{op}J_{S})′

= vN(M ∪ M_{1}^{op} ∪ J_{S}MJ_{S} ∪ J_{S}M^{op}J_{S})′ ∩ B(L^{2}(S))

= (M ⊠_{e_{N}} M^{op} ∪ J_{S}(M ∨ M^{op})J_{S})′

= J_{S}((M ∨ M^{op})′ ∩ M ⊠_{e_{N}} M^{op})J_{S} = ℂ1.

Thus Z(M_{0}) ∩ Z(N_{0}) = ℂ. But if p_{0} denotes the projection of L^{2}(S) onto L^{2}(M) then clearly p_{0}M_{0}p_{0} = M_{0}p_{0} is isomorphic to B(L^{2}(M)) as a M ⊗M^{op} representation. Thus, N_{0} ⊂ M_{0} must in fact coincide with N^{st} ⊂ M^{st}. The last part is now clear, since this isomorphism sends 1 ⊗ M^{op} onto J_{S}MJ_{S}. Q.E.D.

Proof of 4) ⟹ 5) ⟺ 5′). The equivalence of 5) and 5′) was proved in [Po2], the argument being identical to Connes’ single algebra analogue statement. Let us then prove 4) ⟹ 5′). So let ψ be a S-hypertrace on ⟨S, T ⟩ = J_{S}T J′_{S} ∩ B(L^{2}(S)). Since T = M ∨M^{op} and M_{0} ⊂ J_{S}M^{op}J′_{S} ∩B(L^{2}(S)) (we’ve already noticed this in the above lemma) it follows that

M_{0}^{f} = (J_{S}MJ_{S})′ ∩ M_{0} ⊂ (J_{S}(vN(M ∪ M^{op}))J_{S})′ ∩ B(L^{2}(S))
= (J_{S}T J_{S})′ ∩ B(L^{2}(S)) = ⟨S, T ⟩.

Thus ψ restricts to a state ϕ on M_{0}^{f} which has M in its centralizer (since ψ has S in its centralizer and S contains M). Note now that if T ∈ M_{0}^{f} then

ψ(e_{N}T) = ψ(u^{op}(e_{N}T)u^{op∗}) = ψ((u^{op}e_{N}u^{op}^{∗})T), ∀ u^{op} ∈ U(M^{op}).

Averaging by unitaries in U(M^{op}) and using that \overline{co}^{n}\{u^{op}e_{N}u^{op}^{∗} | u^{op} ∈ U(M^{op})} ∩ ℂ1 = {λ1} (see the Appendix A.1), it follows that

ψ(e_{N}T) = λψ(T) = λϕ(T).

But e_{N} ∈ S is in the centralizer of ψ so

ψ(e_{N}T) = ψ(e_{N}T e_{N}) = ψ(E_{0}(T)e_{N}).

By the same argument as above, the latter equals

λψ(E_{0}(T)) = λϕ(E_{0}(T)).

Thus ϕ = ϕ ∘ E_{0}^{f} showing that ϕ is a (N ⊂ M)-hypertrace on N_{0}^{f} ⊂^{E_{0}^{f}} M_{0}^{f} thus

on N_{0}^{st,f} ⊂^{E_{0}^{st,f}} M_{0}^{st,f}, proving 5′).

Proof of 5′) ⟹ 1). Since N_{0}^{stf} ⊂^{E_{0}^{stf}} M_{0}^{st,f} has inclusion matrix (Γ_{N,M})^{t} and the trace Tr on M^{st,f} defined in 5.2 is preserved by E^{st,f}, it follows by the general result in [Po13] that ∥Γ_{N,M}∥^{2} = ∥Γ_{N,M}^{t}∥^{2} = [M : N]. This ends the proof of the theorem. Q.E.D.

5.5 Remarks. 1^{∘}. Of all the equivalent characterisations of amenability for standard graphs, the Kesten-type amenability condition ∥Γ_{N,M}∥^{2} = [M : N] seems to remain the easiest to check in practice. For instance, it immediately implies that if [M : N] ≤ 4 then Γ_{N,M} is amenable, and it is the condition that was used by Bisch and Haagerup to construct many examples of infinite depth subfactors with amenable graphs, by taking compositions between a fixed point algebra inclusion and a cross product inclusion, corresponding to actions of finite groups ([BiH]). Nevertheless, each of the other equivalent characterizations of amenability provided in [Po2-5] and in this paper has its own role in understanding various combinatorial and functional analytical aspects of this concept. The main interest in this notion of amenability comes from the fact that the hyperfinite subfactors having amenable graphs are precisely those that can be recovered from their standard invariants and are thus, in particular, completely classified by this invariant (see 7.1, 7.2 later in this paper, and also [Po16]). 2^{∘}. Note that in the proof of the Følner condition 5.3.2 for Γ_{N,M}, from the Kesten-type condition ∥Γ_{N,M}∥^{2} = [M : N] (taken as the definition of the amenability for a graph) we do not actually use the fact that Γ_{N,M} is standard, i.e., the fact that it comes from a subfactor. Indeed, the proof goes the same for any weighted bipartite graph (see [Po14] for more comments on this). However, by using the ergodicity property 4.8 of the standard invariant and of its subalgebra generated by the Jones projections, one can prove an interesting sharper Følner type condition for standard graphs. This will be discussed in a forthcoming paper.

5.6 Corollary. (a). Let G be a standard λ-lattice and G_{0} a sublattice. If G_{0} is amenable then G is amenable. Conversely, if [G : G_{0}] < ∞ and G is amenable then G_{0} is amenable. (b). Let G_{k} = {A_{ij}^{k}}_{i,j≥0} be standard λ_{k}-lattices with corresponding graphs Γ_{G_{k}} = Γ_{k}, k = 1, 2. Let G denote the system of finite dimensional algebras A_{ij} =^{def} A_{ij}^{1} ⊗ A_{ij}^{2}, i, j ≥ 0, with the tensor product trace. Then G is a standard λ_{1}λ_{2}- lattice, its graph Γ is naturally identified with the tensor product of the graphs Γ_{k} (regarded as matrices) and we have that G is amenable if and only if both G_{1} and G_{2} are amenable.

Proof. (a). The first part follows trivially from the (Kesten-type) definition of amenability, since G_{0} ⊂ G implies ∥Γ_{G}∥ ≥ ∥Γ_{G_{0}}∥. The second part follows from 2.11, 5.3.4 and [Po8]. (b). The first part follows imediately from the axiomatization of standard lattices in [Po7]. The second part follows from the definition of the amenability, because we have ∥Γ∥ = ∥Γ_{1}∥∥Γ_{2}∥, so that (λ_{1}λ_{2})^{−1} = ∥Γ∥^{2} iff λ_{1}^{−1} = ∥Γ_{1}∥^{2} and λ_{2}^{−1} = ∥Γ_{2}∥^{2}. Note that, by using 2.9.d) and [Po8], this part is an imediate consequence of 5.3.1 as well. Q.E.D.

6. Some More Characterizations of the Amenability for Γ_{N,M} and G_{N,M}

In this section we prove several more equivalent characterizations of the amenability for standard graphs and lattices, which clarify some of the results and ideas of the approach to amenability in [Po2,12]. We mention that, while related in spirit with the rest of the paper, the present section will not make explicit use of the symmetric enveloping algebras. So, in this respect, it can be regarded as a digression. To state the first result, recall that if B ⊂ A is an inclusion of von Neumann subalgebras of an ambient type II_{1} factor then H(A | B) denotes its Connes-Størmer relative entropy. By [PiPo1], if N ⊂ M is an extremal inclusion of type II_{1} factors then H(M | N) = ln([M : N]). Also, if N ⊂ M ⊂ M_{1} ⊂ ⋯ is the Jones tower associated to N ⊂ M then

H(M′ ∩ M_{k+1} | M′ ∩ M_{k}) ≤ H(M_{k+1} | M_{k}) = ln([M_{k+1} : M_{k}]) = ln([M : N]),

for all k ≥ 0. More generally, if p is a projection in M′ ∩ M_{k} then by [PiPo1] we have

H(p(M′ ∩ M_{k+1})p |p(M′ ∩ M_{k})p)
≤ H(pM_{k+1}p | pM_{k}p) = ln([pM_{k+1}p : pM_{k}p])
= ln([M_{k+1} : M_{k}]) = ln([M : N]) = H(M | N).

Similarly, if N^{st} ⊂ M^{st} denotes as usual the “model” inclusion generated by the higher relative commutants, as in [Po2], then the same remark as above shows that H(pM^{st}p | pN^{st}p) ≤ H(M | N), ∀ p ∈ P(N^{st}). The result that follows states that this “upper bound” for the “local relative entropies” is attained precisely when G_{N,M} (equivalently Γ_{N,M}) is amenable. Since H(pM′ ∩ M_{k+1}p | pM′ ∩ M_{k}p) also represents the conditional entropy from step k to step k + 1 of the restriction to the support set of p (in K or L) of the random walk on the graph Γ ∘ Γ^{t} ∘ Γ ∘ Γ^{t} ⋯, for Γ = Γ_{N,M}, with transition probabilities determined by v = (v_{k})_{k∈K}, this maximality condition on the entropy can be interpreted as a local Shanon-McMillan-Breimann type condition, in the same spirit as 5.3.5 in [Po2].

6.1. Theorem. Let N ⊂ M be an extremal inclusion of type II_{1} factors. The following conditions are equivalent.

1) G_{N,M} is amenable.

2) ∀ ε > 0, ∃ n ≥ 1 and p ∈ P(M′ ∩ M_{n}) such that

∥E_{(pM′∩M_{n+1}p)′∩(pM′∩M_{n+2}p)}(e_{n+2}p) − λp∥_{2} < ε∥p∥_{2}.

3) ∀ ε > 0, ∃ p ∈ P(N_{1}^{st}) such that

∥E_{pN^{st}p′∩pM^{st}p}(e_{0}p) − λp∥_{2} < ε∥p∥_{2}.

4)

lim_{k} sup_{p∈P(M′∩M_{k})} H(pM′ ∩ M_{k+1}p | pM′ ∩ M_{k}p)

= H(M | N) = ln([M : N]).

5)

sup_{p∈P(N^{st})} H(pM^{st}p | pN^{st}p) = H(M | N).

Proof. First of all, note that since by [Po2] we have that Γ_{N,M} is amenable if and only if Γ_{M,M_{1}} is amenable, it is suficient to prove the above equivalences in the case n is even in condition 2) and the k’s are taken odd in condition 4).

1) ⟹ 2). If G_{N,M} is amenable then by 5.3 its graph Γ_{N,M} verifies the Følner condition 5.3.2). Thus, ∀ ε > 0, ∃ F ⊂ K finite non-empty such that

∑_{k∈∂F} v_{k}^{2} < (ε/2) ∑ v_{k}^{2}.

Let n_{0} ≥ 1 be such that F′ =^{def} F ∪ ∂F is included in K_{n}, ∀ n ≥ n_{0}. For each n ≥ n_{0} let {p_{k}^{n}}_{k∈K_{n}} be the list of minimal central projections of M′ ∩ M_{2n}. Note that ∀ k ∈ K we have

lim_{n→∞} dim(M′ ∩ M_{2n}p_{k}^{n}) = ∞.

Let δ > 0. Let {m_{k}}_{k∈F′} be positive integers such that

(∗)  |\frac{m_{k}}{m_{k′}} − \frac{v_{k}}{v_{k′}} | < δmin\{v_{r}/v′_{r} | r, r′ ∈ F′\},  ∀ k, k′ ∈ F′.

Fix n ≥ n_{0} large enough such that

dim(M′ ∩ M_{2n}p_{k}^{n}) ≥ m_{k}^{2}, ∀ k ∈ F′.

Then for each k ∈ F′ choose q_{k} ∈ P(M′ ∩ M_{2n}p_{k}^{n}) such that dim(q_{k}M′ ∩ M_{2n}q_{k}) = m_{k}^{2}. Let p =^{def} ∑_{k∈F′} q_{k}. We will show that, for δ > 0 small enough, p satisfies condition 2). To this end denote by G = Γ^{t}F′ the set of simple summands of pM′ ∩ M_{2n+1}p and by {q̄_{l}}_{l∈G} the corresponding minimal central projections. Let also {s_{k}}_{k∈F′}, {t_{l}}_{l∈G} denote the traces of the minimal projections in pM′ ∩ M_{2n}p and respectively pM′ ∩ M_{2n+1}p. Thus, if Γ = (a_{kl})_{k∈K,l∈L} then for each k ∈ F, l ∈ G with a_{kl} ≠ 0 we have t_{l} = λ ∑_{k′∈K} a_{k′l}s_{k′} = λ ∑_{k′∈F′} a_{k′l}s_{k′}. Also, if we denote by n_{l}^{2} = dim(q̄_{l}M′∩ M_{2n+1}q̄_{l}) and m′_{k}^{2} = dim(q′_{k}M′ ∩ M_{2n+2}q′_{k}), where {q′_{k}}_{k∈F′′}, F′′ = F′ ∪ ∂F′, are the minimal central projections of pM′∩M_{2n+2}p, then n_{l} = ∑_{k′∈F′} a_{k′l}m_{k′} and m′_{k} = ∑_{k′′∈F′} b_{kk′′}m_{k′′}, where (b_{kk′})_{k,k′∈K} = ΓΓ^{t}. From (∗) it follows that for k ∈ F and l ∈ G with a_{kl} ≠ 0 we have the estimates:

|\frac{t_{l}}{s_{k}} − \frac{n_{l}}{m′_{k}} | = |λ ∑_{k′∈F′} a_{k′l}s_{k′}/s_{k} − ∑_{k′∈F′} a_{k′l}m_{k′}/ ∑_{k′′∈F′} b_{kk′′}m_{k′′} |

≤ |λ ∑_{k′∈F′} a_{k′l}s_{k′}/s_{k} − ∑_{k′∈F′} a_{k′l}s_{k′}/ ∑_{k′′∈F′} b_{kk′′}s_{k′′} | + f(δ)

= |λ ∑_{k′∈F′} a_{k′l}s_{k′}/s_{k} − ∑_{k′∈F′} a_{k′l}s_{k′}/λ^{−1}s_{k}| + f(δ)

= f(δ)

where f(δ) → 0 as δ → 0 and in which we used that for k ∈ F we have

(ΓΓ^{t}(s_{k′′})_{k′′∈F′})_{k} = λ^{−1}s_{k}.

With these estimates in mind recall that, with the above notations, we have (see e.g. Sec.6 in [PiPo1]):

E_{(pM′∩M_{2n+1}p)′∩(pM′∩M_{2n+2}p)}(e_{2n+2}p) = ∑_{k∈F′′, l∈G}(τ(q′_{k}q̄_{l})^{2}/a_{kl}^{2}τ(q′_{k})τ(q̄_{l}))q̄_{l}q′_{k}

= ∑_{k∈F′′, l∈G}(λs_{k}n_{l}/m′_{k}t_{l})q̄_{l}q′_{k}.

But from the above estimates we see that for all k ∈ F and l ∈ G with a_{kl} ≠ 0 we have:

|λs_{k}n_{l}/m′_{k}t_{l} − λ| < f′(δ)

where f′(δ) → 0 as δ → 0. This would finish the proof if we could show that the trace of the sum of the projections q̄_{l}q′_{k} for l ∈ G and k ∈ F′′ \ F is small with respect to the trace of p. To show this, it is sufficient to show that ∑_{k∈F′′−F} τ(q′_{k}) is small with respect to τ(p). To this end, note first that we have

∑_{k∈F} τ(q′_{k}) = ∑_{k∈F} λm′_{k}t_{k} = λ ∑_{k∈F} ( ∑_{k′′∈F′} b_{kk′′}m_{k′′}) t_{k}

≥ λ ∑_{k∈F} ∑_{k′′∈F′} b_{kk′′}m_{k}t_{k′′} − λδ ∑_{k∈F, k′′∈F′} b_{kk′′}m_{k}t_{k}

≥ ∑_{k∈F} τ(q_{k}) − λ^{−2}δ ∑_{k∈F} τ(q_{k})

in which we first used (∗) and then the fact that ∑_{k′′∈K} b_{kk′′} ≤ λ^{−3}, ∀ k ∈ K (see e.g., [Po3], page 281). Thus we get:

∑_{k∈F′′−F} τ(q′_{k}) = ∑_{k∈F′′} τ(q′_{k}) − ∑_{k∈F} τ(q′_{k})

= τ(p) − ∑_{k∈F} τ(q′_{k}) ≤ τ(p) − ∑_{k∈F} τ(q_{k}) + λ^{−2}δ ∑_{k∈F} τ(q_{k})

≤ ∑_{k′∈∂F} τ(q_{k′}) + λ^{−2}δτ(p).

But by applying (∗) again we also have

∑_{k′∈∂F} τ(q_{k′})/τ(p) = ∑_{k′∈∂F} τ(q_{k′})/ ∑_{k∈F′} τ(q_{k}) ≤ ∑_{k′∈∂F} τ(q_{k′})/ ∑_{k∈F} τ(q_{k})

= ∑_{k′∈∂F} m_{k′}s_{k′}/ ∑_{k∈F} m_{k}s_{k} = ∑_{k′∈∂F} (s_{k′}/ ∑_{k∈F} \frac{m_{k}}{m_{k′}} s_{k})

≤ (1 − δ)^{−1} ∑_{k′∈∂F} (s_{k′}/ ∑_{k∈F} \frac{v_{k}}{v_{k′}} · s_{k})

= (1 − δ)^{−1} ∑_{k′∈∂F} v_{k′}^{2}/ ∑_{k∈F} v_{k}^{2} < (1 − δ)^{−1}ε/2.

Altogether we get:

∥E_{(pM′∩M_{2n+1}p)′∩(pM′∩M_{2n+2}p)}(e_{2n+2}p) − λp∥_{2}^{2}
≤ f′(δ)^{2}τ(p) + λ^{−2}δτ(p) + ((1 − δ)^{−1}ε^{2}/2)τ(p)
= (f′′(δ) + ε^{2}/2)τ(p).

Thus, if δ is chosen sufficiently small to make f′′(δ) < ε^{2}/2 then the above is majorized by ε^{2}τ(p), thus finishing the proof of 1) ⟹ 2).

Now, 2) ⟺ 3) is trivial by the definition of N^{st} ⊂ M^{st}. Also 4 ⟺ 5) is clear from the continuity properties of the relative entropy under commuting square conditions ([PiPo1]). Then 2) ⟹ 4) follows from (4.2 in [PiPo1]). Finally, to prove 4) ⟹ 1), recall from [PiPo2] that if B ⊂ A is an inclusion of finite dimensional algebras with inclusion matrix T then Ind(E_{A}^{B}) ≥ ∥T ∥^{2} ≥ exp(H(B | A)). Since the inclusion matrix T of pM′∩M_{2n+1}p ⊂ pM′∩M_{2n+2}p is a restriction of Γ_{N,M}, we have

∥Γ_{N,M}∥^{2} ≥ ∥T ∥^{2} ≥ exp(H(pM′ ∩ M_{2n+2}p | pM′ ∩ M_{2n+1}p)).

Thus, if the right hand side term can be made arbitrarily close to exp(H(M | N)) = [M : N] then we obtain ∥Γ_{N,M}∥^{2} = [M : N], i.e., G_{N,M} follows amenable. Q.E.D.

6.2. Notation. We denote by M̃ the bicommutant of M in its enveloping algebra M_{∞}, i.e., M̃ = (M′ ∩M_{∞})′ ∩M_{∞}. Similarily we put Ñ = (N′ ∩M_{∞})′ ∩ M_{∞} and more generally M̃_{i} = (M′_{i}∩M_{∞})′∩M_{∞}, i ∈ ℤ, {M_{i}}_{i∈ℤ} being as usual a Jones tunnel-tower for N ⊂ M and M_{0} = M, M_{−1} = N, M_{−n} = N_{n−1}, n ≥ 2. Note that there exists a unique conditional expectation Ẽ from M̃ onto Ñ defined by Ẽ(X) = λΣ_{j}m_{j}Xm_{j}^{∗}, for X ∈ M̃, {m_{j}}_{j} being any orthonormal basis of N′ ∩ M_{∞} over M′ ∩ M_{∞} (e.g., an orthonormal basis of vN{e_{k}}_{k≥1} over vN{e_{k}}_{k≥2} will do, as the definition of Ẽ is anyway easily seen to be independent of the choice of {m_{j}}_{j}) and that Ẽ is implemented by e_{1}, i.e.,

e_{1}Xe_{1} = Ẽ(X)e_{1} (see Sec. 2.2 in [Po2] or 6.9 in [Po6]). The inclusion Ñ ⊂^{Ẽ} M̃ is in fact homogeneous λ-Markov in the sense of (1.2.3 and 1.2.11 of [Po3]) and we have a non-degenerate commuting square

\begin{matrix}Ñ & ⊂^{Ẽ} & M̃ \\ ∪ & & ∪ \\ N & ⊂^{E_{N}} & M \end{matrix}

It should also be noted that, while Ẽ(Y_{1}e_{0}Y_{2}) = λY_{1}Y_{2}, ∀Y_{1,2} ∈ Ñ (this relation can in fact be taken as the definition of Ẽ), in general Ẽ is not trace preserving. In fact, one can easily show (see the proof of 6.4 hereafter) that it is trace preserving if and only if M̃ = M, i.e., when the bicommutant relation holds true, (M′ ∩ M_{∞})′ ∩ M_{∞} = M, equivalently when Γ_{N,M} is strongly amenable (cf. 5.3.1 in [Po2]). The Jones tower-tunnel of the above commuting square is obtained by defining the conditional expectations Ẽ_{i} from M̃_{i−1} onto M̃_{i−2} in a similar manner with Ẽ. Recalling from ([Po2]) that a representation N ⊂^{E} M of N ⊂ M is smooth if N′ ∩ M_{n} ⊂ N′ ∩ M_{n}, ∀n, note that by its construction, Ñ ⊂ M̃ is obviously a smooth representation of N ⊂ M.

6.3. Theorem. Let N ⊂ M be an extremal inclusion of type II_{1} factors. The following conditions are equivalent:

1) Γ_{N,M} is amenable.

2) There exists a (possibly singular) trace ψ on M̃ such that ψ ∘ Ẽ = ψ.

3) There exists a norm one projection of Ñ ⊂^{Ẽ} M̃ onto N ⊂ M.

4) If

\begin{aligned}N & ⊂ & M \\ ∪ & & ∪ \\ N & ⊂ & M \end{aligned}

is a smooth representation of N ⊂ M such that there exists a norm one projection of M onto M (equivalently, a M-hypertrace on M), then there exists

a norm one projection of N ⊂^{E} M onto N ⊂ M (equivalently, a N ⊂ M-

hypertrace on N ⊂^{E} M).

5) For any smooth representation of N ⊂ M into an inclusion of type II_{1} von

Neumann algebras N ⊂^{E} M, there exists a norm one projection of N ⊂^{E} M onto

N ⊂ M (equivalently, a N ⊂ M-hypertrace on N ⊂^{E} M).

Proof. 1) ⟹ 2) By Theorem 6.1 (see condition 6.1.3) applied to M ⊂ M_{1} and the anti-isomorphism between N_{1}^{st} ⊂ N^{st} ⊂^{e_{0}} M^{st} and M′ ∩ M_{∞} ⊂ N′ ∩ M_{∞} ⊂^{e_{0}} N′_{1} ∩ M_{∞}, it follows that there exist projections p_{n} ∈ M′ ∩ M_{∞} such that

∥E_{(p_{n}N′∩M_{∞}p_{n})′∩(p_{n}N′_{1}∩M_{∞}p_{n})}(e_{0}p_{n}) − λp_{n}∥_{2}/∥p_{n}∥_{2} ≤ 2^{−n}, ∀ n.

We then define on M_{∞} the state φ =^{def} lim_{n→ω} τ(p_{n})^{−1}τ( · p_{n}). Note that, since p_{n} ∈ M′∩M_{∞}, we have [p_{n}, (M′∩M_{∞})′∩M_{∞}] = 0, in other words [p_{n}, M̃] = 0. Thus, [φ, M̃] = 0, in particular φ|M̃ is a trace. Moreover, by noting that τ( · p_{n}) = τ(E_{p_{n}Bp_{n}}( ·)p_{n}) for any von Neumann subalgebra B ⊂ M_{∞} with p_{n} ∈ B, taking B = (p_{n}N′ ∩ M_{∞}p_{n})′ ∩ p_{n}M_{∞}p_{n} and using the above and the Cauchy-Schwartz inequality it follows that for all x, y ∈ Ñ we have:

|τ(xe_{0}yp_{n})/τ(p_{n}) − λτ(xyp_{n})/τ(p_{n})|
= |τ(p_{n}yxe_{0}p_{n})/τ(p_{n}) − τ(p_{n}xyλp_{n})/τ(p_{n})|
= |τ(E_{p_{n}Bp_{n}}(p_{n}yxe_{0}p_{n})/τ(p_{n}) − τ(p_{n}xyλp_{n})/τ(p_{n})|
= τ(p_{n}yx(E_{p_{n}Bp_{n}}(e_{0}p_{n}) − λp_{n}))/τ(p_{n}))
≤ ∥p∥_{2}∥yx∥ ∥E_{p_{n}Bp_{n}}(e_{0}) − λp_{n}∥_{2}/τ(p_{n})
≤ 2^{−n}∥yx∥.

Since Ẽ(xe_{0}y) = λxy, ∀ x, y ∈ Ñ, and spÑe_{0}Ñ = M̃, it follows that

lim_{n→∞} ∥τ(Xp_{n})/τ(p_{n}) − τ(Ẽ(X)p_{n})/τ(p_{n})| = 0 ∀ X ∈ M̃.

Thus, φ(X) = φ(Ẽ(X)), ∀ X ∈ M̃. All this shows that ψ =^{def} φ|M̃ is both a trace and satisfies ψ = ψ ∘ Ẽ.

2) ⟹ 3). Since ψ is a trace on M̃, it is in particular a M-hypertrace and ψ =

ψ ∘ Ẽ implies it is actually a (N ⊂ M)-hypertrace on (Ñ ⊂^{Ẽ} M̃); equivalently, there exists a conditional expectation of Ñ ⊂ M̃ onto N ⊂ M.

3) ⟹ 4). If there exists a conditional expectation Φ of M onto M then by amplification it follows that there exist conditional expectations Φ_{2n} of M_{2n} onto M_{2n}, ∀ n ≥ 0. Let F_{2n} : ∪_{k}M_{k} → M_{2n} be the conditional expectation implemented by ⋯ ∘ E_{2n+2} ∘ E_{2n+1} and denote Ψ_{2n} : ∪_{k}M_{k} → M_{∞} the aplications defined by Ψ_{2n}(X) = Φ_{2n} ∘ F_{2n}(X) ∈ M_{2n} ⊂ M_{∞}. Note that Ψ_{2n} is M_{2n}-M_{2n} linear. Finally, we put Ψ(X) =^{def} lim_{n→ω} ψ_{2n}(X), for X ∈ ∪_{k}M_{k}, where ω is a free ultrafilter on ℕ. Thus, Ψ(1) = 1 and Ψ is M_{2n}-M_{2n} linear ∀ n. Since the representation of N ⊂ M into N ⊂ M is smooth, M′ ∩ M_{j} ⊂ M′ ∩ M_{j}, ∀ j. Thus, if X ∈ M then [X, M′ ∩ M_{j}] = 0 and by applying ψ we get [Ψ(X), M′ ∩ M_{j}] = 0. Thus, ψ(X) ⊂ (∪_{j}M′ ∩ M_{j})′ ∩ M_{∞} = M̃. Similarly, we obtain that if X ∈ N then Ψ(X) ∈ Ñ. But by 3) we have a conditional expectation of M̃ onto M, say Ψ_{0}, such that Ψ_{0}(Ñ) = N. We then define Ψ_{1} : M → M by Ψ_{1}(X) = Ψ_{0}(Ψ(X)), which is a conditional expectation and satisfies Ψ_{1}(N) = Ψ_{0}(ψ(N)) ⊂ Ψ_{0}(Ñ) ⊂ N.

4) ⟹ 5). Since M has projections p ∈ Z(M) such that Mp is finite, it follows that there is a conditional expectation of Mp onto Mp ≃ M, thus of M onto M and so 4) applies.

5) ⟹ 1) If 5) holds true then in particular there exists a norm one projection from the finite standard representation onto N ⊂ M, so by Theorem 5.3 we have 1). Q.E.D.

Recall from [Po2] that a standard λ-graph (Γ, s⃗) is called ergodic if s⃗ is the unique s⃗-bounded eigenvector for ΓΓ^{t} corresponding to the eigenvalue λ^{−1}, equivalently, if Z(A_{0,∞}) = ℂ, where A_{0,∞} is the finite von Neumann algebra obtained as an inductive limit with the Bratteli diagram given by Γ, Γ^{t}, Γ, ..., starting from the even vertex ∗ of Γ, and having trace given by s⃗ = (s_{k})_{k∈K}. Note that if N ⊂ M is a subfactor having standard graph (Γ, s⃗) then the algebra A_{0,∞} equals M′ ∩ M_{∞}, where N ⊂ M = M_{0} ⊂ M_{1} ⊂ ... is the Jones tower for N ⊂ M and M_{∞} = (∪_{n}M_{n})^{−} as usual. In what follows we’ll call the standard λ-graph almost ergodic if dimZ(A_{0,∞}) < ∞. This is equivalent to the fact that, up to scalar multiples, there are only finitely many s⃗-bounded eigenvectors for ΓΓ^{t} corresponding to the eigenvalue λ^{−1} (see the proof of 1.4.2 in [Po2]). Note that Haagerup constructed extremal hyperfinite subfactors of index λ^{−1} = 2 · 4cos^{2}π/5 = 3 + \sqrt{5} which have almost ergodic, but not ergodic, standard graph. The following consequence of 6.3 shows that this cannot happen if Γ is amenable.

6.4. Corollary. If an amenable, extremal standard graph (Γ, s⃗) is almost ergodic then it is ergodic, and thus it is strongly amenable.

Proof. Let N ⊂ M be a subfactor having (weighted) standard graph equal to (Γ, s⃗). Denote like in 6.2 by Ñ_{1} = (N′_{1} ∩ M_{∞})′ ∩ M_{∞}, Ñ = (N′ ∩ M_{∞})′ ∩ M_{∞}, M̃ = (M′ ∩ M_{∞})′ ∩ M_{∞} and by F̃ the expectation from M̃ onto Ñ_{1} defined as in 6.2 (so that in fact, with the notations there, we also have F̃ = Ẽ_{−1} ∘ Ẽ_{0}). Note that Z(M̃) = Z(M′ ∩ M_{∞}). Since N_{1} ⊂ M has amenable graph (= ΓΓ^{t}), by 6.3 there exists a trace τ′ on M̃ such that τ′ ∘ F̃ = τ′. Since dimZ(M̃) = dimZ(M′ ∩ M_{∞}) < ∞, it follows that there exists a ∈ Z(M̃)_{+} such that τ′(X) = τ(Xa), ∀X ∈ M̃. Since Ẽ is τ′-preserving, this implies that a = F̃(a) ∈ F̃(Z(M̃)) = Z(Ñ_{1}). Thus a ∈ Z(M̃) ∩ Z(Ñ_{1}) = ℂ1, so a = 1 and τ′ = τ. Thus F̃ coincides with the trace preserving expectation F of M̃ onto Ñ. In particular, this implies that E_{(N′_{1}∩M_{∞})′∩M_{∞}}(f) = F (f) = λ^{2}1, where f ∈ M is the Jones projection for N_{1} ⊂ M. By duality it follows that E_{(M′_{2j}∩M_{∞})′∩M_{∞}}(f_{j}) = λ^{2}1 for any j ∈ ℤ, where f_{j} is the Jones projection for the inclusion M_{2j} ⊂ M_{2j+2}. By (5.3 in [Po2]) it follows that M = (M′ ∩ M_{∞})′ ∩ M_{∞}, so in particular M′ ∩ M_{∞} is a factor, i.e., (Γ, s⃗) is ergodic. Q.E.D.

We now examine the effect of amenability on the universal graph Γ_{N,M}^{u}. To

this end, let us denote, like in [Po2], by N^{u,f} ⊂^{E^{u,f}} M^{u,f} the direct summand of N^{u} ⊂ M^{u} given by all the irreducible representations B(H) of M ⊗ P^{op} which, when regarded as M −P bimodules, have finite dimension, dim(_{M}H_{P}) < ∞, P denoting here a generic ”dummy” type II_{1} factor weakly stably equivalent to M (in the sense of 1.4.3 in [Po8], i.e., P can be embedded with finite index in the amplification by some α > 0 of M). Let Γ_{N,M}^{u,f} denote its inclusion graph (or matrix). Recall from [Po2] that Γ_{N,M}^{u,f} is in a natural way a weighted bipartite graph, the weights being given by the vector ((dim_{M,P} H)^{1/2}), which in fact also gives the weights of an E^{u,f}-invariant trace on M^{u,f}.

6.5. Theorem. Let N ⊂ M be an extremal inclusion of type II_{1} factors. The following conditions are equivalent:

1) The standard graph Γ_{N,M} is amenable, i.e., ∥Γ_{N,M}∥^{2} = [M : N].

2) The graph Γ_{N,M}^{u,f} is amenable, i.e., ∥Γ_{N,M}^{u,f}∥^{2} = [M : N].

3) Each irreducible component Γ of Γ_{N,M}^{u} satisfies ∥Γ∥^{2} = [M : N].

4) For any ε > 0 there exists a subfactor Q ⊂ N, with [N : Q] < ∞, such that the inclusion matrix T_{0} = T_{Q′∩N⊂Q′∩M} satisfies ∥T_{0}∥^{2} ≥ [M : N] − ε. 4′) For any ε > 0 there exists a factor P containing M with [P : M] < ∞, such that ∥T_{M′∩P ⊂N′∩P} ∥^{2} ≥ [M : N] − ε.

Proof. 3) ⟹ 2) is trivial.

2) ⟹ 1). For simplicity of notations, we let (N ⊂^{E} M) = (N^{u,f} ⊂^{E^{u,f}} M^{u,f}). Let K′ be the set of simple summands of M and T T^{t} be the inclusion matrix of M ⊂ M_{2}. It follows that ∀ ε > 0, ∃ k_{0} ∈ K′ such that

lim_{n→∞} ∥(T T^{t})^{n}δ_{k_{0}}∥^{1/n} ≥ ∥T T^{t}∥ − ε = [M : N] − ε.

But if for each n ≥ 0 we denote by p_{2n} the minimal central projection in M_{2n} corresponding to k_{0} ∈ K′, then

∥(T T^{t})^{n}δ_{k_{0}}∥^{2} = dim(Mp′_{0} ∩ p_{0}M_{2n}p_{0}) = dim(Mp′_{2n} ∩ M_{2n}p_{2n}).

Moreover, by using that if R ⊂ Q ⊂ P are inclusions of type II_{1} factors with finite index then dim R′ ∩ P ≤ ([P : Q]) dim R′ ∩ Q (because the inclusion matrix of R′ ∩ Q ⊂ R′ ∩ P has square norm ≤ [P : Q]), it follows that, with the notations M^{0} = P′ ∩ M, M_{k}^{0} = P′ ∩ M_{k}, k ≥ 0, we have:

dim(Mp′_{2n} ∩ M_{2n}p_{2n}) ≤ dim(Mp′_{2n} ∩ M_{2n}^{0}p_{2n})
≤ ([M_{2n}^{0}p_{2n} : M_{2n}p_{2n}]) dim(Mp′_{2n} ∩ M_{2n}p_{2n})
= ([M^{0}p_{0} : Mp_{0}]) dim(M′ ∩ M_{2n})
= ([M^{0}p_{0} : Mp_{0}])∥(Γ_{N,M}Γ_{N,M}^{t})^{n}δ_{∗}∥^{2}.

Thus,

lim_{n→∞} ∥(T T^{t})^{n}δ_{k_{0}}∥^{1/n} ≤ lim_{n→∞}([Mp_{0} : Mp_{0}])^{1/2n}∥(ΓΓ^{t})^{n}δ_{∗}∥^{1/n} = ∥ΓΓ^{t}∥

showing that ∥Γ∥^{2} ≥ [M : N] − ε. Since ε was arbitrary, ∥Γ_{N,M}∥^{2} = [M : N].

1) ⟹ 3). Let k′ be any of the labels corresponding to an even vertex of Γ and let q ∈ M_{2n}^{u} be a minimal central projection corresponding to that same label. Then dim((M^{u}q)′ ∩ M_{2n}^{u}q) = ∥(ΓΓ^{t})^{n}δ_{k′}∥^{2}. But by smoothness, (M′ ∩ M_{2n})q ⊂ (M^{u}q)′ ∩ M_{2n}^{u}q, thus ∥ΓΓ^{t}∥ ≥ ∥Γ_{N,M}Γ_{N,M}^{t}∥.

1) ⟹ 4). This is clear, by simply taking Q = N_{k}, a subfactor in a Jones tunnel, with k large enough.

4) ⟺ 4′). This follows immediately by taking into account that if Q ⊂ N(⊂ M) is a subfactor of finite index in N and we denote Q ⊂ N ⊂ M ⊂ M_{1} ⊂ Q_{1} its basic construction, then T_{Q′∩N⊂Q′∩M} = T_{M′_{1}∩Q_{1}⊂M′∩Q_{1}}. 4′) ⟹ 2). If P ⊃ M is as in condition 4′) for some ε then let p ∈ Z(M^{u,f}) be the central projection supporting all the N − M bimodules appearing as direct summands in _{N}L^{2}(P)_{M}. Then clearly ∥Γ∥ ≥ ∥Γ_{p}∥ ≥ ∥T_{M′∩P ⊂N′∩P} ∥. Q.E.D.

6.6. Corollary. Let Q ⊂ N ⊂ M be inclusions of II_{1} factors with finite index (not necessarily extremal). (i). If Q ⊂ M has amenable graph then Q ⊂ N and N ⊂ M have amenable graphs. (ii). If N ⊂ M has amenable graph and p ∈ N′ ∩ M is a projection, then Np ⊂ pMp has amenable graph.

Proof. By [L], there exist extremal inclusions Q_{0} ⊂ N_{0} ⊂ M_{0} such that: a). The higher relative commutants of Q_{0} ⊂ N_{0}, N_{0} ⊂ M_{0} and respectively Q_{0} ⊂ M_{0} are algebraically isomorphic to those of Q ⊂ N, N ⊂ M and respectively Q ⊂ M; so, in particular, the graphs of the induced-reduced algebras in the Jones towers of the corresponding subfactors are equa. b). [N_{0} : Q_{0}] = IndE_{min}^{Q,N}, [M_{0} : N_{0}] = IndE_{min}^{N,M} and [M_{0} : Q_{0}] = IndE_{min}^{Q,M} and the local indices in the Jones tower for the inclusions Q_{0} ⊂ N_{0}, N_{0} ⊂ M_{0}, respectively Q_{0} ⊂ M_{0} are the same as for the initial incusions Q ⊂ N, N ⊂ M, respectively Q ⊂ M. With these in mind, let us prove (i) and (ii). (i). Let Γ_{Q,N}^{u,f}, Γ_{N,M}^{u,f} be as in [Po2] the inclusion matrices describing the inclusions (Q ⊗ M^{op})_{at,f}^{∗∗} ⊂ (N ⊗ M^{op})_{at,f}^{∗∗} ⊂ (M ⊗ M^{op})_{at,f}^{∗∗}. Recall from ([Po2]) that Γ_{Q,M}^{u,f} = Γ_{Q,N}^{u,f} ∘ Γ_{N,M}^{u,f}. Thus, if ∥Γ_{Q,M}^{u,f}∥^{2} = IndE_{min}^{Q,M} then we get

IndE_{min}^{Q,N} · IndE_{min}^{N,M} = IndE_{min}^{Q,M} = ∥Γ_{Q,M}^{u,f}∥^{2} ≤ ∥Γ_{Q,N}^{u,f}∥^{2}∥Γ_{N,M}^{u,f}∥^{2}
≤ IndE_{min}^{Q,N} · IndE_{min}^{N,M},

forcing the equalities ∥Γ_{Q,N}^{u,f}∥^{2} = IndE_{min}^{Q,N}, ∥Γ_{N,M}^{u,f}∥^{2} = IndE_{min}^{N,M}. But by the above considerations and 6.5 this implies Γ_{Q,N} and Γ_{N,M} are amenable. (ii). This can be easily deduced from 6.5, by using the universal graphs as in the proof of (i) above. Instead, we’ll use the following simpler argument: By the first part of the proof, we may assume N ⊂ M is extremal. Then by 2.9.c) it follows that the finite standard representation of Np ⊂ pMp is given

by N^{st,f}p ⊂^{E} pM^{st,f}p, where E is defined by E(pXp) = τ(p)^{−1}E^{st,f}(pXp), for X ∈ M^{st,f}. But then, if Φ is a conditional expectation from M^{st,f} onto N^{st,f} sending M onto N then clearly Φ also sends pM^{st,f}p onto pMp and N^{st,f}p onto Np. By 5.3, this implies that Np ⊂ pMp has amenable graph. Q.E.D.

We mention one last hereditarity property for the amenability of the graphs of subfactors, which has a self-contained and rather elementary proof.

6.7. Proposition. Let

\begin{aligned}N & ⊂ & M \\ ∪ & & ∪ \\ Q & ⊂ & P \end{aligned}

be a nondegenerate commuting square of inclusions of type II_{1} factors with finite index (thus, [M : N] = [P : Q] < ∞, [M : P] = [N : Q] < ∞). Then we have:

a) ∥Γ_{N,M}∥ = ∥Γ_{Q,P} ∥, H(M | N) = H(P | Q), IndE_{min}^{N,M} = IndE_{min}^{Q,P} and E_{Q′∩P} (e_{0}) = E_{N′∩M}(e_{0}), where e_{0} ∈ P is a Jones projection for Q ⊂ P (and thus for N ⊂ M as well).

b) N^{st} ⊂ M^{st} has atomic centers iff Q^{st} ⊂ P^{st} has atomic centers.

c) G_{N,M} is amenable (resp. strongly amenable, resp. has finite depth) iff G_{Q,P} is amenable (resp. strongly amenable, resp. has finite depth).

Proof. Let

… N_{1} ⊂ N ⊂ M
∪ ∪ ∪
… Q_{1} ⊂ Q ⊂ P

be a tunel for the given commuting square. Then dim N′_{k} ∩M ≤ dim Q′_{k} ∩M ≤ [M : P] dim Q′_{k} ∩ P, so that

∥Γ_{N,M}∥^{2} = lim_{k→∞}(dim N′_{k} ∩ M)^{1/k} ≤ lim_{k→∞}(dim Q′_{k} ∩ P)^{1/k} = ∥Γ_{Q,P} ∥^{2}.

Taking

⟨N, Q⟩ ⊂ ⟨M, P ⟩
∪  ∪
N ⊂ M

and using that ⟨N, Q⟩ ⊂ ⟨M, P ⟩ is an amplified of Q ⊂ P (so that Γ_{⟨N,Q⟩,⟨M,P ⟩} = Γ_{Q,P}), by the first part we also get ∥Γ_{Q,P} ∥ ≥ ∥Γ_{N,M}∥, thus ∥Γ_{N,M}∥ = ∥Γ_{Q,P} ∥. Now remark that E_{P} (Q′_{k} ∩ M) = Q′_{k} ∩ P and E_{P} (Z(Q′_{k} ∩ M)) ⊂ Z(Q′_{k} ∩ P). Also, we have

Ind(E_{Q′_{k}∩P}^{Q′_{k}∩M}) ≤ [M : P],
Ind(E_{N′_{k}∩M}^{Q′_{k}∩M}) ≤ [Q′_{k} : N′_{k}] = [N_{k} : Q_{k}] = [M : P].

It follows that if we denote R = \overline{∪_{k}Q′_{k} ∩ M} then Ind(E_{P^{st}}^{R}) ≤ [M : P], Ind(E_{M^{st}}^{R}) ≤ [M : P]. Thus, P^{st} has atomic center iff R has atomic center iff M^{st} has atomic center. Also, the above shows that

sup_{n} dim Z(N′_{k} ∩ M) < ∞ ⟺ sup_{n} dim Z(Q′_{k} ∩ M) < ∞

⟺ sup_{k} dim Z(Q′_{k} ∩ P) < ∞.

Thus, N ⊂ M has finite depth iff Q ⊂ P has finite depth. Since Q ⊂ P is embedded as a commuting square in N ⊂ M, by the definition of relative entropy we have H(P | Q) ≤ H(M | N) ≤ H(⟨M, P ⟩ | ⟨N, Q⟩) = H(P | Q), thus, H(M | N) = H(P | Q). Next, if e_{0} ∈ P is a Jones projection then

E_{N′∩M}(e_{0}) = E_{N′∩M}(E_{Q′∩M}(e_{0})) = E_{N′∩M}(E_{Q′∩P} (e_{0}))

so that ∥E_{N′∩M}(e_{0})∥_{2} ≤ ∥E_{Q′∩P} (e_{0})∥_{2} with equality iff E_{N′∩M}(e_{0}) = E_{Q′∩P} (e_{0}). But N ⊂ M is embedded as a commuting square in ⟨N, Q⟩ ⊂ ⟨M, P ⟩ which is an amplified of Q ⊂ P, so we get similarly ∥E_{Q′∩P} (e_{0})∥_{2} ≤ ∥E_{N′∩M}(e_{0})∥_{2} giving E_{N′∩M}(e_{0}) = E_{Q′∩P} (e_{0}). To prove the statemnt about the minimal index, note from the formula of the Jones projection in ([PiPo1], page 83-84) that E_{min}^{N,M} = E_{N}^{M}(b^{1/2} ·b^{1/2}) with b ∈ Alg{E_{N′∩M}(e_{0})} = Alg{E_{Q′∩P} (e_{0})}. Thus, b ∈ P and E_{min}^{N,M}(P) = Q, implying that Ind(E_{min}^{N,M}) ≥ Ind(E_{min}^{Q,P}). Similarily, Ind(E_{min}^{N,M}) ≤ Ind(E_{min}^{⟨N,Q⟩,⟨M,P ⟩}) Thus, IndE_{min}^{M,N} =IndE_{min}^{P,Q}. From the above, it follows in particular that IndE_{min}^{N,M} = ∥Γ_{N,M}∥^{2} iff IndE_{min}^{Q,P} = ∥Γ_{Q,P} ∥^{2} so Γ_{N,M} is amenable iff Γ_{Q,P} is amenable (without the extremality assumtion required).

If Γ_{N,M} is amenable and M^{st} is a factor (i.e., G_{N,M} is strongly amenable) then Γ_{Q,P} is amenable and P^{st} has finite dimensional center. Thus 6.4 applies to get that Γ_{Q,P} follows strongly amenable. Alternatively, and in order to keep the proof of this Proposition elementary and self-contained, note that the same proof as on (pages 235 and 183 of [Po2]) can be used to get the same conclusion, i.e., that P^{st} follows a factor and thus Γ_{Q,P} strongly amenable. Q.E.D.

7. Hyperfiniteness of M ⊠_{e_{N}} M^{op} and Hereditarity

of the Amenability for Subfactors

Recall from [Po2] that an inclusion of factors N ⊂ M is called amenable if it is the range of a norm one projection from any of its smooth representations, equivalently, if the algebras N, M are themselves amenable (i.e., hyperfinite by [C1]) and the graph Γ_{N,M} is amenable ([Po2,3,4]), i.e., ∥Γ_{N,M}∥^{2} = IndE_{min}^{N,M}. In this section we will show that, in the case the inclusion N ⊂ M is extremal, the amenability of N ⊂ M is in fact equivalent to the hyperfiniteness of its symmetric enveloping algebra. We will then derive that the amenability of an inclusion is inherited by its ”sub-inclusions”

7.1. Theorem. Let N ⊂ M be an extremal inclusion of type II_{1} factors. The following conditions are equivalent:

1) N ⊂ M is amenable.

2) Γ_{N,M} is amenable and M is hyperfinite.

3) ∀ x_{1}, …, x_{m} ∈ M, ∀ ε > 0, ∃ n, a projection f in N′ ∩ M_{n}, a subfactor P ⊂ N such that P f ⊂ Nf ⊂ fM_{n}f is a basic construction and a finite dimensional subfactor Q_{0} ⊂ P such that

x_{i} ∈_{ε} Q_{0} ∨ (P′ ∩ M), i = 1, 2, …, m.

4) ∀ x_{1}, …, x_{m} ∈ M, ∀ ε > 0, ∃ Q ⊂ N with [N : Q] < ∞ such that

x_{i} ∈_{ε} Q′ ∩ M, i = 1, 2, …, m.

5) M ⊠_{e_{N}} M^{op} is isomorphic to the hyperfinite type II_{1} factor.

6) There exists a M ⊠_{e_{N}} M^{op}-hypertrace on B(L^{2}(M ⊠_{e_{N}} M^{op}).

7) There exists a (N ⊂ M)-hypertrace on N^{st} ⊂ M^{st} (equivalently, a norm one projection of N^{st} ⊂ M^{st} onto N ⊂ M).

Proof. We will prove 1) ⟹ 7) ⟹ 2) ⟹ 3) ⟹ 4) ⟹ 5) ⟹ 6) ⟹ 7) and 2) ⟹ 1). The implication 1) ⟹ 7) is trivial, as N^{st} ⊂ M^{st} is just a particular case of a smooth representation. If 7) is satisfied then by [Po13] we have ∥Γ_{N,M}∥^{2} = [M : N] and N, M follow amenable (as ranges of norm one projections from the amenable von Neumann algebras N^{st}, M^{st}). Thus we have 7) ⟹ 2).

2) ⟹ 3). This is essentially (4.4.1 in [Po2]), or the proof of (4.1 in [Po4], up to Step VI on page 291), with some changes and additional considerations that we explain below. Like in the proof of 1) ⟹ 2) in Theorem 6.1, we let F be an ε′-Følner set for Γ_{N_{1},N} (by 2) we have that Γ_{N,M} is amenable, equivalently Γ_{N_{1},N} is amenable), then we choose a large n and some integers m_{k} ≤ (dim N′ ∩ M_{2n+1}p_{k}^{n+1})^{1/2} such that

(1)  |\frac{m_{k}}{m_{k′}} − \frac{v_{k}}{v_{k′}} | < δ  ∀ k, k′ ∈ F,

where {p_{k}^{n+1}}_{k} is now the list of minimal central projections in N′ ∩M_{2n+1} and v⃗ = (v_{k})_{k∈K} is the standard vector of local indices (at even levels) for Γ_{N_{1},N}. We then take q_{k} ∈ P(N′ ∩ M_{2n+1}p_{k}^{n+1}) such that dim(q_{k}N′ ∩ M_{2n+1}q_{k}) = m_{k}^{2} and define p = ∑_{k∈F} q_{k}. Let then P_{0} ⊂ N be a downward basic construction for Np ⊂ pM_{2n+1}p. By the choice of F (i.e., satisfying ∑_{k∈F ∪∂F} v_{k}^{2} ≤ (1 + ε′) ∑_{k∈F} v_{k}^{2}) it follows that if {x̄_{j}}_{j} is an orthonormal basis of N over P_{0} ∨ P′_{0} ∩ N then {x̄_{j}}_{j} is almost an orthonormal basis of M over P_{0} ∨ P′_{0} ∩ M as well. Moreover, by the choice of integers {m_{k}}_{k∈F} it follows that

(2)  ∑_{j} τ(E_{P ∨P′∩N}(x̄_{j}^{∗}x̄_{j})p̄_{k_{0}})/τ(p̄_{k_{0}}) ≈ ∑_{k∈F} v_{k}^{2},  ∀ k_{0} ∈ F,

{p̄_{k}}_{k∈F} being the minimal central projections in P′_{0} ∩ N. Also, since P_{0} is a type II_{1} factor, we may assume E_{P_{0}∨P′_{0}∩N}(x̄_{j}^{∗}x̄_{j})p̄_{k} ∈ P_{0}p̄_{k}, ∀ k ∈ F. But then, by using first the approximate innerness of Np ⊂ pM_{2n+1}p then the central freeness of P_{0} ⊂ M, like in (Steps I, II, III in the proof of 4.1 in [Po4]), we obtain a conjugate of P_{0} by a unitary element in N, say P_{1}, such that we have the type of estimates (a)–(f) on page 285 of [Po4] with P_{1} instead of N_{m_{0}}. Then we go through Step IV on pages 286–288 of [Po4], noting that due to the condition (2) above, we don’t need to take a further tunnel and that taking P_{1} for N_{m} will do. Then Step V on page 289 can be taken unchanged. Altogether, after doing all this we end up obtaining the following: ∀ x_{1}, …, x_{l} ∈ M, ∀ ε > 0, if F ⊂ Even(Γ_{N_{1},N}) is a ε-Følner set, n is sufficiently large and {m_{k}}_{k∈F} satisfy (1) with δ suficiently small, then there exists a choice of a downward basic construction P_{1} ⊂ N for Np ⊂ pM_{2n+1}p, where p = ∑_{k∈F} q_{k} as above, and a projection s_{0} ∈ P_{1}, such that for all 1 ≤ i ≤ l we have

(3)  ∥[s_{0}, x_{i}]∥_{2} < f(ε′)∥s_{0}∥_{2},
∥s_{0}x_{i}s_{0} − E_{s_{0}(P_{1}∨P′_{1}∩M)s_{0}}(s_{0}x_{i}s_{0})∥_{2} < f(ε′)∥s_{0}∥_{2}

where f(ε′) → 0 as ε′ → 0. Arguing like in Step VI on page 290 of [Po4] we obtain a family of such choices of downward basic constructions (P_{i})_{i∈I} with projections (s_{i})_{i∈I}, s_{i} ∈ P_{i}, such that (P_{i}, s_{i}) satisfy (3) and ∑_{i} s_{i} = 1. But then there exists a downward basic construction P ⊂ N for Np ⊂ pM_{2n+1}p such that s_{i} ∈ P, ∀ i, and s_{i}P s_{i} = s_{i}P_{i}s_{i}, ∀ i. Thus P will satisfy

∥E_{P ∨P′∩M}(x_{i}) − x_{i}∥_{2} < f(ε′), 1 ≤ i ≤ n.

Since P is hyperfinite, by taking ε′ so that f(ε′) < ε we get 7.1.3).

3) ⟹ 4) is trivial, by simply taking Q = Q′_{0} ∩ P in 3).

4) ⟹ 5). since Alg(M, e_{N}, M^{op}) is so-dense in M ⊠_{e_{N}} M^{op} it is sufficient to

prove that ∀ x_{1}, x_{2}, …, x_{n} ∈ M, ∀ ε > 0, ∃ B ⊂ M ⊠_{e_{N}} M^{op} finite dimensional

such that

∥E_{B}(x_{i}) − x_{i}∥_{2} < ε
∥E_{B}(e_{N}) − e_{N}∥_{2} < ε
∥E_{B}(x_{i}^{op}) − x_{i}^{op}∥_{2} < ε.

By 4) there exists Q ⊂ N with [N : Q] < ∞ such that x_{i} ∈_{ε} Q′ ∩ M. But

[Q^{op}′ ∩ M ⊠_{e_{N}} M^{op} : M^{op}′ ∩ M ⊠_{e_{N}} M^{op}] ≤ [M^{op} : Q^{op}] < ∞

and thus

[Q^{op}′ ∩ M ⊠_{e_{N}} M^{op} : Q] ≤ [M : Q]^{2} < ∞,

implying that B =^{def} Q′ ∩(Q^{op}′ ∩M ⊠_{e_{N}} M^{op}) has finite dimension. Since e_{N} ∈ B

and Q′ ∩ M, Q^{op}′ ∩ M^{op} ⊂ B, we are done.

5) ⟹ 6) is trivial, because hyperfinite algebras are amenable, so they have hypertraces.

6) ⟹ 7) By 5.2 we have M^{st} = vN(M, J_{S}MJ_{S}) ⊂ B(L^{2}(S)), N^{st} = vN(N, J_{S}MJ_{S}), where S = M ⊠_{e_{N}} M^{op}. Let then Φ : B(L^{2}(S)) → S be a

conditional expectation. Since Φ is S-S linear and [M^{st}, M^{op}] = 0 it follows that Φ(M^{st}) ⊂ M^{op}′ ∩ S = M. Similarly, since [M^{st}, M_{1}^{op}] = 0, we get Φ(N^{st}) ⊂ M_{1}^{op}′ ∩ S = N. 2 ⟹ 1). If

\begin{aligned}N & ⊂ & M \\ ∪ & & ∪ \\ N & ⊂ & M \end{aligned}

is an arbitrary smooth representation of N ⊂ M then, M being hyperfinite, it follows that there exists a conditional expectation of M onto M. By Theorem

5.7 it then follows that there exists a conditional expectation of N ⊂^{E} M onto N ⊂ M. Q.E.D.

7.2. Remarks. 1^{∘}. Note that by using condition 7.1.3 one can easily proceed to construct recursively a sequence of appropriate downward basic constructions for suitable local inclusions in the Jones tower-tunnel, say M ⊃ N ⊃ P ⊃ P_{1} ⊂ ..., such that if we let Q = ∩_{n}P_{n} and (N^{0,st} ⊂ M^{0,st}) = (\overline{∪_{n}P′_{n} ∩ N} ⊂ \overline{∪_{n}P′_{n} ∩ M}) then (N ⊂ M) = (Q ¯⊗N^{0,st} ⊂ Q ¯⊗M^{0,st}), with the isomorphism class of N^{0,st} ⊂ M^{0,st} only depending on G_{N,M}. Indeed, from the proof of 7.1.2) ⇒ 7.1.3) we see that, up to conjugacy by a unitary in N, the choice of the subfactor P = P_{1} (equivalently, the choice of the projection p ∈ N′ ∩ M_{2n+1}) is determined by a choice of the ε-Følner set F and by a choice of the integers {m_{k}}_{k∈F}, which in turn both depend on ε. Similarily, each time one goes from step n to step n + 1, one uses the Følner-type amenability condition for P_{n} ⊂ M and some ε = ε_{n+1} to get the next subfactor P_{n+1} (up to conjugation by a unitary element in P_{n}), from a downward basic construction that only depends on some choice of an ε_{n+1}- Følner set F_{n+1} and of some integers {m_{j}^{n+1}}_{j∈F_{n+1}}. Thus, if we let for instance ε = 2^{−n}, ∀n, and make the choice of the F_{n}’s and m_{j}^{n}’s this way, once for all, then the isomorphism class of {P′_{n} ∩ N ⊂ P′_{n} ∩ M}_{n} will only depend on G_{N,M}, as all the above choices can be ”read” from this object through its amenability properties. In particular, the isomorphism class of N^{0,st} ⊂ M^{0,st} will only depend on G_{N,M}. Thus, when complemented with this remark, we see that condition 7.1.3 in the above theorem shows that hyperfinite subfactors with amenable graphs are completely classified by their standard invariants (for more on this, see [Po16]). 2^{∘}. Recently, F. Hiai and M. Izumi have further investigated our notion of amenability for standard lattices and weighted graphs coming from subfactors and obtained two more equivalent characterizations ([HiIz]): the first one requires the existence of invariant means on the (weighted) fusion algebra of all M −M bimodules in the Jones tower of N ⊂ M; the second one is a ratio limit condition on the weight vector v⃗, stating that the (weighted) graph (Γ_{N,M}, v⃗) is amenable if and only if for every vertex k ∈ K one has

lim_{n→∞} \frac{⟨(ΓΓ^{t})^{n}δ_{∗}, δ_{k}⟩}{⟨(ΓΓ^{t})^{n}δ_{∗}, δ_{∗}⟩} = v_{k},

where Γ = Γ_{N,M}. This ”ratio limit” result for group-like objects coming from subfactors, which generalizes in a non-trivial way a prior result of Avez for discrete groups ([Av]), shows that in fact the projections q_{k} ∈ (N′∩M_{2n+1})p_{k}^{n+1} in the proof of 2) ⟹ 3) of Theorem 7.1 can be taken equal to p_{k}^{n+1}. It also shows that the standard weight vector v⃗ of an amenable standard λ-lattice G can be completely recovered from its graph Γ. It should be noted however that there exist no known examples of standard graphs Γ which admit two distinct standard weights, say v⃗_{1}, v⃗_{2}, for the same value of the index, i.e., such that (Γ, v⃗_{1}) ≄ (Γ, v⃗_{2}). Whether such examples exist or not seems to be an interesting problem. In order to prove the hereditarity result in its largest generality, namely without assuming that the inclusions involved are extremal, we’ll need the following:

7.3. Lemma. Let N ⊂ M be an inclusion of type II_{1} factors, of finite index (but not necessarily extremal). Let B be a C^{∗}-algebra containing M, such that B = C^{∗}(M, ⋃_{k} N′_{k} ∩ B) and such that B has a state ϕ with [φ, M] = 0. Let

(π_{φ}, H_{φ}, ξ_{φ}) be the GNS representation for (B, φ). Then, as a M-M Hilbert bimodule, H_{φ} is a direct sum of irreducible bimodules H_{φ} = ⨁_{j} H′_{j} with each

H′_{j} isomorphic to a bimodule in the list {H_{k}}_{k∈K}.

Proof. Note first that if {K_{l}^{op}}_{l∈L} denotes the list of all irreducible M-N bimodules contained in ⨁_{k∈K} _{M}H_{k}_{N} (see the beginning of Section 5) and

H′_{0} ≃ ⨁_{j} K_{l_{j}}^{op} is a M-N Hilbert bimodule contained in some M − M bimodule

H, (i.e., H′_{0} ⊂ _{M}H_{N}) then \overline{sp}MH′_{0}M ≃ ⨁_{i} H_{k_{i}}.

Then note that ∪_{k}spM(N′_{k} ∩ B)ξ_{φ} is dense in H_{φ}. Indeed, we have

spM(N′_{k} ∩ B)M(N′_{k} ∩ B)M = spM(N′_{k} ∩ B)N_{k}f_{−k−1}^{0}N_{k}
(N′_{k} ∩ B)M = spM(N′_{k} ∩ B)f_{−k−1}^{0}(N′_{k} ∩ B)M
⊂ spM(N′_{2k+1} ∩ B)M.

showing that

Alg(M, ∪_{k}N′_{k} ∩ B) = ∪_{k}spM(N′_{k} ∩ B)M = ∪_{k}spM(N′_{k} ∩ B)R,

where R = \overline{∪_{k}N′_{k} ∩ M}, the closure being taken in the norm ∥ · ∥_{2} in M. But ∪_{k}(N′_{k} ∩M)ξ_{φ} is dense in Rξ_{φ} (because φ implements τ on M), so ∪_{k}spM(N′_{k} ∩ B)ξ_{φ} is dense in ∪_{k}spM(N′_{k} ∩ B)Rξ_{φ} = ∪_{k}spM(N′_{k} ∩ B)Mξ_{φ} which is dense in H_{φ}. Let then H′_{φ} =^{def} ∨{H′ ⊂ H_{φ} | ∃ k ∈ K such that H′ ≃ H_{k} as M-M bimodules}. Assume H′_{φ} ≠ H_{φ}. Thus, there exists ξ ∈ M(N′_{k} ∩ B)ξ_{φ} such that ξ ∉ H′_{φ}. Let ξ = X_{0}Y′_{0}ξ_{φ}, for some X_{0} ∈ M, Y′_{0} ∈ N′_{k} ∩ B. It follows that if x ∈ M, y ∈ N_{k} then

⟨xξy, ξ⟩ = ⟨xX_{0}Y′_{0}ξ_{φ}y, X_{0}Y′_{0}ξ_{φ}⟩ = ⟨X_{0}^{∗}xX_{0}yY′_{0}ξ_{φ}, Y′_{0}ξ_{φ}⟩.

But the state on M defined by ψ(X) = ⟨XY′_{0}ξ_{φ}, Y′_{0}ξ_{φ}⟩, X ∈ M has N_{k} in its centralizer so by A.1 it is automatically normal and of the form ψ(X) = τ(Xa) for some a ∈ N′_{k} ∩ M. Thus we get

⟨xξy, ξ⟩ = τ(X_{0}^{∗}xX_{0}ya) = ⟨x(X_{0}a^{1/2}ξ_{τ})y, (X_{0}a^{1/2}ξ_{τ})⟩

so if we define ξ′ = X_{0}a^{1/2}ξ_{τ} ∈ Mξ_{τ} ⊂ L^{2}(M) then the above shows that H′_{0} = \overline{sp}MξN_{k} is a M-N_{k} bimodule isomorphic to a sub-bimodule of _{M}L^{2}(M)_{N_{k}}. By the first part applied to N = N_{k} it follows that \overline{sp}MH′_{0}M is a sub-bimodule of (⨁_{k∈K} H_{k})^{n} for some multiplicity n ≤ ∞, giving a contradiction. Q.E.D.

7.4. Corollary. Let N ⊂ M be an inclusion of type II_{1} factors with finite index. Assume that for any ε > 0 there exists an amenable type II_{1} von Neumann algebra B containing M such that

∥E_{(N′∩B)′∩B}(e_{0}) − λ1∥_{2} < ε.

Then there exists a norm one projection from N^{st} ⊂ M^{st} onto N ⊂ M.

Proof. If B ⊃ M satisfies the condition in the hypothesis for some ε, then there exist some finite many unitary elements u_{1}, …, u_{n} ∈ N′ ∩ B such that

∥\frac{1}{n} ∑_{i=1}^{n} u_{i}e_{0}u_{i}^{∗} − λ1∥_{2} < ε.

Thus, by taking instead of B the von Neumann algebra generated by M and {u_{1}, …, u_{n}}, it follows that we may assume B is separable in the norm ∥ ∥_{2}. Let then H be the M-M Hilbert bimodule obtained by summing up countably many copies of each H_{k}, k ∈ K. By 7.3 we have L^{2}(B) ⊂ H, for each B as in the hypothesis, where L^{2}(B) has the M-M bimodule structure given by left-right multiplication by elements of M. For each ε = 1/n we choose an algebra B_{n} satisfying the hypothesis. We let Φ_{n} : B(L^{2}(B_{n})) → B_{n} be norm one projections and define the state φ on B(H) by

φ(T) = lim_{n→ω} τ_{n} ∘ Φ_{n}(p_{n}T |_{L^{2}(B_{n})})

where p_{n} = proj_{L^{2}(B_{n})}, τ_{n} is the trace on B_{n} and ω is a free ultrafilter on ℕ. Since each τ_{n}∘Φ_{n} is a M-hypertrace, φ follows a M-hypertrace. Moreover, if we identify M^{st} with the von Neumann algebra generated in B(H) by the operators of left and right multiplication by M and N^{st} with its von Neumann subalgebra generated by the operators of left multiplication by N and right multiplication by M, then M^{st} = spN^{st}e_{0}N. Let Y ∈ N^{st}, y ∈ N. We want to show that φ = φ ∘ E^{st} on M^{st}, thus we need to show that φ(Y e_{0}y) = λφ(Y y). But [p_{n}, M^{st}] = 0 and [N^{st}p_{n}, N′ ∩ B_{n}] = 0, so that Φ_{n}(N^{st}p_{n}) ⊂ (N′ ∩ B_{n})′ ∩ B_{n}. Thus Φ_{n}((Y e_{0}y)p_{n}) = Φ_{n}(Y p_{n})e_{0}y = y′e_{0}y, with y′ ∈ (N′ ∩ B_{n})′ ∩ B_{n}. Thus τ_{n}(y′e_{0}y) = τ_{n}(E_{(N′∩B_{n})′∩B_{n}}(y′e_{0}y)) = τ_{n}(y′E_{(N′∩B_{n})′∩B_{n}}(e_{0})y). It follows that

|τ_{n} ∘ Φ_{n}((Y e_{0}y)p_{n})−λτ_{n} ∘ Φ_{n}((Y y)p_{n})|

= |τ_{n}(y′E_{(N′∩B_{n})′∩B_{n}}(e_{0})y) − λτ_{n}(y′y)|

≤ ∥y′∥ ∥y∥ ∥E_{(N′∩B_{n})∩B_{n}}(e_{0}) − λ1∥_{2}

≤ \frac{1}{n}∥Y ∥ ∥y∥.

This proves that indeed φ(Y e_{0}y) = λφ(Y y), so φ = φ ∘ E^{st} on M^{st}. Q.E.D.

We can now prove the announced hereditarity property for amenable inclusions.

7.5. Theorem. Let N ⊂ M be an extremal inclusion of hyperfinite type II_{1} factors with amenable graph (equivalently, with amenable standard invariant G_{N,M}), i.e., ∥Γ_{N,M}∥^{2} = [M : N]. Assume Q ⊂ P is an inclusion of factors embedded in N ⊂ M as commuting squares (i.e., such that E_{N}(P) = Q), but without necessarily being extremal and not necessarily having the same index as N ⊂ M. Then Γ_{Q,P} is amenable (equivalently, G_{Q,P} is amenable), i.e., ∥Γ_{Q,P} ∥^{2} equals the minimal index of Q ⊂ P.

Proof. By Theorem 7.1, S = M ⊠_{e_{N}} M^{op} follows amenable so in particular the

von Neumann algebra B generated in S by P and Q′ ∩ S is also amenable. Let e_{0} ∈ P be a Jones projection for Q ⊂ P. Thus, E_{Q}(e_{0}) = E_{N}(e_{0}) = λ_{0}1 = [P : Q]^{−1}1. Since (N′ ∩ S)′ ∩ S = N it follows that E_{(N′∩S)′∩S}(e_{0}) = λ_{0}1. Since N′ ∩ S ⊂ Q′ ∩ S = Q′ ∩ B, this implies that E_{(Q′∩B)′∩B}(e_{0}) = λ_{0}1 as well. Thus Q ⊂ P satisfies the conditions in the hypothesis of 7.4, so there exists a norm one projection from the standard representation Q^{st} ⊂ P^{st} onto Q ⊂ P. By [Po13] this implies ∥Γ_{Q,P} ∥^{2} = ∥T_{Q^{st}⊂P^{st}}∥^{2} = IndE_{min}^{Q,P}. Q.E.D.

In Sec. 6 we’ve seen that for extremal inclusions of arbitrary type II_{1} factors N ⊂ M the condition ∥Γ_{N,M}^{u,f}∥^{2} = [M : N] is sufficient to insure the amenability of the standard graph Γ_{N,M}. We now show that for inclusions of hyperfinite factors the weaker condition ∥Γ_{N,M}^{u,rf}∥^{2} = [M : N] is enough, where Γ_{N,M}^{u,rf} denotes the inclusion graph of the direct summand N^{u,rf} ⊂ M^{u,rf} of N^{u} ⊂ M^{u}, in which M^{u,rf} consists of all irreducible representations B(H) of M ⊗ M^{op}, with H having finite right dimension over M, i.e., dim(H_{M}) < ∞, but leaving the left dimensions dim(_{M}H) arbitrary.

7.6. Theorem. Let N ⊂ M be an extremal inclusion of hyperfinite type II_{1} factors. The following conditions are equivalent:

1) N ⊂ M has amenable graph, i.e., ∥Γ_{N,M}∥^{2} = [M : N].

2) ∀ ε > 0, ∃ P a hyperfinite factor containing M, such that dim M′ ∩ P < ∞ and ∥T_{M′∩P ⊂N′∩P} ∥^{2} ≥ [M : N] − ε.

3) ∥Γ_{N,M}^{u,rf}∥^{2} = [M : N].

Proof. 1) ⟹ 3) is trivial because Γ_{N,M}^{u,rf} ⊃ Γ_{N,M}.

3) ⟹ 2) By 3) there exists a direct summand N ⊂ M = ⊕_{l∈L′}B(K′_{l}) ⊂ ⊕_{k∈K′}B(H′_{k}) of N^{u,rf} ⊂ M^{u,rf} such that its inclusion graph Γ is connected and ∥Γ∥^{2} > [M : N] − ε. Take K′_{0} ⊂ K′ finite and sufficiently large so that we still have ∥Γ_{K′_{0}}^{t}∥^{2} > [M : N] − ε.

By the definition of the universal representation N^{u,rf} ⊂ M^{u,rf}, if Q = M′∩N then Q is a factor of type II_{1}, N = N ∨ Q ⊂ M ∨ Q = M and Q has finite coupling constant in each direct summand B(H′_{k}) of M. But then, if one takes P = Q′ ∩ B(⊕_{k∈K′_{0}}H′_{k}) then ∥T_{M′∩P ⊂N′∩P} ∥^{2} ≥ ∥Γ_{K′_{0}}^{t}∥^{2} ≥ [M : N] − ε.

2) ⟹ 3) follows by noticing that M^{u,rf} contains the von Neumann algebra generated by the operators of left multiplication by M and right multiplication by P on L^{2}(P) as a direct summand (by taking M^{op} ≃ P^{op}, both being hyperfinite factors). Thus T_{M′∩P ⊂N′∩P} will be a restriction of the graph Γ_{N,M}^{u,rf} 2) ⟹ 1). Let ε > 0 and choose P a hyperfinite II_{1} factor satisfying 2) for ε^{32}. Denote by T the bipartite graph describing the inclusions M′ ∩ P ⊂ N′ ∩ P ⊂ N′_{1} ∩ P ⊂ N′_{2} ∩ P ⊂ · · ·. Thus ∥T ∥^{2} ≥ [M : N] − ε^{32} and there is a positive vector w = (w_{j})_{j∈J} such that T T^{t}w = λ^{−1}w, giving the traces on {N′_{k} ∩P }_{k≥1}. But then A.2 applies the same way as in the proof of 1) ⟹ 2) in Theorem 5.3 to get a finite set F ⊂ J such that ∑_{j∈∂F} w_{j}^{2} < ε ∑_{j∈F} w_{j}^{2} (see 5.5.2^{∘} and [Po14]). Arguing like in the proof of 1) ⟹ 2) in Theorem 6.1 it then follows that there exist k ≥ 1 and a projection p ∈ N′_{2k} ∩ P such that

∥E_{(pN′_{2k+1}∩P p)′∩pP p}(e_{−2k−1}p) − λp∥_{2} < ε∥p∥_{2}.

By taking an α = [M : N]^{k+1}-amplification of the inclusion N_{2k+2} ⊂ N_{2k+1} ↪ pP p and using that (N_{2k+2} ⊂ N_{2k+1})^{α} = (N ⊂ M) it follows that there exists a hyperfinite type II_{1} factor P_{0} ≃ (pP p)^{α} such that N ⊂ M ⊂ P_{0} and

∥E_{(N′∩P_{0})′∩P_{0}}(e_{0}) − λ1∥_{2} < ε.

But then 7.3 applies to get that Γ_{N,M} is amenable. Q.E.D

8. An Effros-Lance Type Characterization of Amenability

We will prove in this section yet another equivalent characterization for the amenability of a subfactor N ⊂ M, in terms of simplicity properties of the C^{∗}- algebra C_{bin}^{∗}(M, e_{N}, M^{op}). In the case N = M our result reduces to the implication “C_{bin}^{∗}(M, M^{op}) simple ⟹ ∃ conditional expectations of B(L^{2}(M)) onto M”, which is one of the well known results of Effros and Lance in [EL], relating various amenability conditions for single von Neumann algebras (semidiscreteness, injectivity, etc).

8.1. Theorem. Let N ⊂ M be an extremal inclusion of type II_{1} factors. The following conditions are equivalent: 1^{∘}. N ⊂ M is amenable. 2^{∘}. C_{bin}^{∗}(M, e_{N}, M^{op}) is simple. 3^{∘}. C^{∗}(M, e_{N}, JMJ) is simple. 4^{∘}. C_{bin}^{∗}(M, e_{N}, M^{op}) ≃ C^{∗}(M, e_{N}, JMJ) ≃ C_{min}^{∗}(M, e_{N}, M^{op}), with the isomorphisms being given by the natural quotient maps.

Proof. 1^{∘} ⟹ 2^{∘}. Let C_{bin}^{∗}(M, e_{N}, M^{op}) ↪ B(H) be a faithful representation of C_{bin}^{∗}(M, e_{N}, M^{op}) such that M and M^{op} are von Neumann algebras in B(H). It is sufficient to prove that if

x ∈ Alg(M, e_{N}, M^{op}) = ⋃_{k} spM^{op}M_{k}M^{op} ⊂ C_{bin}^{∗}(M, e_{N}, M^{op}) ⊂ B(H),

then ∥x∥_{B(H)} ≤ ∥x∥_{min}, where ∥x∥_{min} is the norm of (the image of) x in C_{min}^{∗}(M, e_{N}, M^{op}). For such x ∈ Alg(M, e_{N}, M^{op}) let k be large enough such that

x = ∑_{i=n}^{n} y_{i}^{op}z_{i}x_{i}^{op} ∈ spM^{op}M_{k}M^{op},

for some x_{i}, y_{i} ∈ M, z_{i} ∈ M^{op}, 1 ≤ i ≤ n. We will prove that x can be approximated in the so topology on B(H) by elements x′ ∈ Alg(M, e_{N}, M^{op}) such that ∥x′∥_{B(H)} ≤ ∥x∥_{min}. By the inferior semicontinuity of the norm ∥ ∥_{B(H)} with respect to the so-topology on B(H), this will show that ∥x∥_{B(H)} ≤ ∥x∥_{min} and will thus end the proof of 1^{∘} ⟹ 2^{∘}. To prove this approximation, let us first note that ∀ ξ_{1}, …, ξ_{p} ∈ H, ∀ ε > 0, ∃ δ > 0 such that if z′_{i} ∈ M_{k}, with ∥z′_{i} − z_{i}∥_{2} < δ, ∥z′_{i}∥ ≤ ∥z_{i}∥ then

x′ =^{def} ∑_{i=1}^{n} y_{i}^{op}z′_{i}x_{i}^{op}

satisfies ∥(x − x′)ξ_{j}∥ < ε, ∀ 1 ≤ j ≤ p. Indeed we have:

∥(x − x′)ξ_{j}∥ ≤ ∑_{i=1}^{n} ∥y_{i}^{op}∥ ∥(z_{i} − z′_{i})x_{i}^{op}ξ_{j}∥

and since the so-topology on the ball of radius ∥z_{i}∥ (in the uniform norm) in M_{k} coincides with the topology given by the norm ∥ ∥_{2} on this ball, it follows that there exists δ > 0 such that if ∥z_{i} − z′_{i}∥_{2} < δ then ∥(z_{i} − z′_{i})x_{i}^{op}ξ_{j}∥ < ε/n∥y_{i}∥, ∀ i. But then we have

∥(x − x′)ξ_{j}∥ < ∑_{i=1}^{n} ∥y_{i}^{op}∥ε/n∥y_{i}^{op}∥ = ε.

Now, if we assume N ⊂ M is amenable then M ⊂ M_{k} follows amenable and by [Po2] we get that ∀ δ > 0, ∃ finitely many tunnels {N_{k}^{r}}_{1≤k≤n_{r}}, r = 1, …, m, and projections p_{r} ∈ P(N_{n_{r}}^{r}′ ∩ M), r = 1, …, m, such that {p_{r}}_{r} are mutually orthogonal, Σ_{r}p_{r} = 1 and

z′_{i} =^{def} ∑_{r=1}^{m} p_{r}E_{N^{r} ′_{nr} ∩M_{k}}^{M_{k}}(z_{i})p_{r}

satisfies ∥z′_{i} − z_{i}∥_{2} < δ. Also, by its definition, z′_{i} checks ∥z′_{i}∥ ≤ ∥z_{i}∥. Furthermore, since p_{r} ∈ M commute with x_{i}^{op}, y_{i}^{op} ∈ M^{op}, ∀ r, i, it follows that if we let x′ = Σ_{i}y_{i}^{op}z′_{i}x_{i}^{op} as above, then ∥x′∥_{B(H)} = sup_{r} ∥x′p_{r}∥_{B(H)}. But since

x′p_{r} = ∑_{i} y_{i}^{op}p_{r}E_{N_{nr}^{r}′∩M_{k}}(z_{i})p_{r}x_{i}^{op} = p_{r} (∑_{i=1}^{n} y_{i}^{op}E_{N_{nr}^{r}′∩M_{k}}(z_{i})x_{i}^{op}) p_{r},

each ∥x′p_{r}∥_{B(H)} is majorized by

∥∑_{i=1}^{n} y_{i}^{op}E_{N_{nr}^{r}′∩M_{k}}(z_{i})x_{i}^{op}∥_{B(H)} = ∥∑_{i=1}^{n} y_{i}^{op}E_{N_{nr}^{r}′∩M_{k}}(z_{i})x_{i}^{op}∥_{M_{nr}^{op}}

= ∥∑_{i=1}^{n} y_{i}^{op}E_{N_{nr}^{r}′∩M_{k}}(z_{i})x_{i}^{op}∥_{C_{min}^{∗}(M,e_{N} ,M^{op})}

≤ ∥x∥_{min},

with the last inequality following from the fact that, in the algebra C_{min}^{∗}(M, e_{N}, M^{op}), the element Σ_{i}y_{i}^{op}E_{N_{nr}^{r}′∩M_{k}}(z_{i})x_{i}^{op} is the image of x under a conditional expectation. Thus, from the the above remarks, if we take δ sufficiently small, we are done. 2^{∘} ⟹ 3^{∘} is trivial, since by the definition of C_{bin}^{∗}(M, e_{N}, M^{op}), C^{∗}(M, e_{N}, JMJ) is its quotient. 3^{∘} ⟹ 1^{∘}. If C^{∗}(M, e_{N}, JMJ) is simple then there exists an isomorphism

φ : S^{0} =^{def} C_{min}^{∗}(M, e_{N}, M^{op}) ≃ C^{∗}(M, e_{N}, JMJ) ⊂ B(L^{2}(M)).

Since S^{0} ⊂ S ⊂ B(L^{2}(S)), where S = M ⊠_{e_{N}} M^{op} as usual, by Arveson’s theo-

rem φ can be extended to a completely positive map Φ from all B(L^{2}(S)) to B(L^{2}(M)). (Note that in fact we only use here a particular case of Arveson’s theorem stating that if B ⊂ A are unital C^{∗}-algebras and π_{0} : B → B(H_{0}) is a representation of B then there exists a Hilbert space H ⊃ H_{0} and a representation π : A → B(H) such that π_{0}(b) = proj_{H_{0}}π(b)|_{H_{0}}, ∀ b ∈ B. See 2.10.2 in [D2]). Since Φ is a unital ∗-homomorphism when restricted to S^{0}, it follows that it is a S^{0}-S^{0} bimodule map. In particular, if x_{1,2}^{op} ∈ M^{op} ⊂ S^{0} (⊂ B(L^{2}(S))) then Φ(x_{1}^{op}T x_{2}^{op}) = φ(x_{1}^{op})Φ(T)φ(x_{2}^{op}), ∀ T ∈ B(L^{2}(S)). Thus, if T satisfies T x^{op} − x^{op}T = 0, ∀ x^{op} ∈ M^{op}, then Φ(T)φ(x^{op}) = φ(x^{op})Φ(T), ∀ x^{op} ∈ M^{op}. Thus we have Φ((M^{op})′ ∩ B(L^{2}(S)) = φ(M^{op})′ ∩ B(L^{2}(M)). But φ(M^{op}) = JMJ and JMJ′ ∩ B(L^{2}(M)) = M, so that Φ((M^{op})′ ∩ B(L^{2}(S))) = M. Similarily we get

Φ((M_{1}^{op})′ ∩ B(L^{2}(S))) = φ(M_{1}^{op})′ ∩ B(L^{2}(M))
= JM_{1}J′ ∩ B(L^{2}(M))
= N.

But from 5.3 we have that M^{st} ⊂ (M^{op})′ ∩ B(L^{2}(S)) and N^{st} ⊂ (M_{1}^{op})′ ∩ B(L^{2}(S)), so Φ implements a positive unital M-M bimodule map from M^{st} onto M carrying N^{st} onto N. This shows that there exists a conditional expectation of (N^{st} ⊂ M^{st}) onto (N ⊂ M), so N ⊂ M follows amenable.

All this shows that the conditions 1^{∘}–3^{∘} are equivalent. Since clearly 4^{∘} ⟺ 2^{∘}, all the conditions 1^{∘}–4^{∘} follow equivalent. Q.E.D.

8.2. Remarks. 1^{∘}. Note that when applied to the case N = M the above proof of the implication 3^{∘} ⇒ 1^{∘} in Theorem 8.1 reduces to a very short and elementary proof to one of the results in ([EL]). 2^{∘}. Recall from ([Bi3]) that C^{∗}(M, e_{N}, JMJ) contains the ideal K of compact operators over the Hilbert space L^{2}(M, τ) if and only if N contains no nontrivial central sequences of M. Thus, 8.1 implies that amenable inclusions always have non-trivial cental sequences contained in the subfactor (because if C^{∗}(M, e_{N}, JMJ) is simple then it cannot contain the ideal K). In fact, 7.1.4 shows that there even exist non-commuting such central sequences, so that amenable inclusions split off the hyperfinite type II_{1} factor (this is, of course, a consequence of the classification result 7.2.1^{∘} as well).

9. Property T for Subfactors and Standard Lattices

In this section we introduce a notion of property T for standard λ-lattices G (or, equivalently, for paragroups). When restricted to the class of standard lattices associated with subfactors coming from finitely generated discrete groups, our notion coincides with the classical property T of Kazhdan, which it thus generalizes, from discrete groups to the larger class of group-like objects G. In order to define this notion, we will use a strategy similar to the approach to amenability in Section 5. Thus, the property T for a standard λ-lattice G will be defined by requiring M ⊠_{e_{N}} M^{op} to have the property T relative to M ∨ M^{op}

in the sense of ([A-D], [Po8]), where N ⊂ M is an extremal subfactor with G_{N,M} = G. This definition however depends on proving that such a property does not in fact depend on the subfactor N ⊂ M one takes. We do prove this in the next few lemmas. First of all, let us recall the definition of the relative property T, as introduced in ([A-D], [Po8]):

(∗). Let U be a type II_{1} factor and B ⊂ U a von Neumann subalgebra of U. Then we say that U has the property T relative to B if there exists ε > 0 and x_{1}, x_{2}, ..., x_{n} ∈ U such that whenever H is a given U − U bimodule with a vector ξ ∈ H satisfying ∥ξ∥ = 1, [ξ, B] = 0, ∥[ξ, x_{i}]∥ < ε, it follows that H must contain a non-zero vector ξ_{0} satisfying [ξ_{0}, U] = 0.

Note that in the case B = ℂ the above definition reduces to Connes’ definition of property T for single type II_{1} factors U. In general though, the definition (∗) does not require the ambient algebra U to have the property T. Instead, note that by ([A-D], [Po8]), if V is a type II_{1} factor and G is a discrete group acting outerly on V, then U = V ⋊ G has the property T relative to V if and only if the group G has Kazhdan’s property T. With this in mind, let us proceed with some technical results.

9.1. Lemma. Let V ⊂ U be an inclusion of type II_{1} factors with V′ ∩ U = ℂ1. Then U has the property T relative to V if and only if ∀ ε > 0 ∃ δ > 0 and x_{1}, …, x_{n} ∈ U such that if φ : U → U is completely positive, unital, trace preserving, with φ(v_{1}xv_{2}) = v_{1}φ(x)v_{2}, ∀ v_{1}, v_{2} ∈ V, x ∈ U, and ∥φ(x_{i})−x_{i}∥_{2} < δ then ∥φ(x) − x∥_{2} < ε, ∀ x ∈ U, ∥x∥ ≤ 1.

Proof. If U has the property T relative to V then the condition on completely positive maps holds true by 4.1.4 in [Po8]. Conversely, if this latter condition holds, then let H be a U-U bimodule with

ξ ∈ H, ∥ξ∥ = 1, vξ = ξv, ∀ v ∈ V, ∥x_{i}ξ − ξx_{i}∥ < δ′ =^{def} δ/2 ∑ ∥x_{i}∥_{2}. Let

φ : U → U be defined by τ(yφ(x)) = ⟨xξy, ξ⟩, x, y ∈ V as in [C4] (see [Po8]). Then φ is a well defined completely positive map and τ(φ(x)) = ⟨xξ, ξ⟩. Since ξ = vξv^{∗}, ∀ v ∈ U(V), one gets

⟨vxv^{∗}ξ, ξ⟩ = ⟨vxξv^{∗}, ξ⟩ = ⟨vxξ, ξv⟩ = ⟨vxξ, vξ⟩
= ⟨v^{∗}vxξ, ξ⟩ = ⟨xξ, ξ⟩

for all x ∈ U. Averaging by unitaries v ∈ U(V) like in [Po1] and using that V′ ∩ U = ℂ1 and E_{V′∩U}(x) = τ(x)1, it follows that

⟨xξ, ξ⟩ = ⟨τ(x)ξ, ξ⟩ = τ(x), ∀ x ∈ U.

Similarly, we obtain that

τ(yφ(1)) = ⟨ξy, ξ⟩ = τ(y), ∀ y ∈ U.

Thus φ(1) = 1. Also, if x, y ∈ U, v_{1}, v_{2} ∈ V then

τ(yφ(v_{1}xv_{2})) = ⟨v_{1}xv_{2}ξy, ξ⟩ = ⟨xξv_{2}y, v_{1}^{∗}ξ⟩⟨xξv_{2}y, ξv_{1}^{∗}⟩
= ⟨xξv_{2}yv_{1}, ξ⟩ = τ(v_{2}yv_{1}φ(x)) = τ(y(v_{1}φ(x)v_{2})).

This shows that φ(v_{1}xv_{2}) = v_{1}φ(x)v_{2}. Finally, since ∥x_{i}ξ − ξx_{i}∥_{2} < δ′ we have

∥φ(x_{i}) − x_{i}∥_{2}^{2} = τ(φ(x_{i}^{∗})φ(x_{i})) = τ(x_{i}^{∗}x_{i}) − 2Re τ(x_{i}^{∗}φ(x_{i}))
≤ τ(φ(x_{i}^{∗}x_{i})) + τ(x_{i}^{∗}x_{i}) − 2Re τ(x_{i}^{∗}φ(x_{i}))
= 2τ(x_{i}^{∗}x_{i}) − 2Re τ(x_{i}^{∗}φ(x_{i}))
= 2⟨ξx_{i}, ξx_{i}⟩ − 2Re⟨x_{i}ξ, ξx_{i}⟩
≤ 2∥x_{i}ξ − ξx_{i}∥ ∥ξx_{i}∥ ≤ 2δ′∥x_{i}∥_{2} < δ^{2}.

Thus, φ this way defined satisfies the required condition, so ∥φ(x) − x∥_{2} < ε, ∀ x ∈ U, ∥x∥ ≤ 1. In particular, we have

∥φ(u) − u∥_{2} < ε, ∀ u ∈ U(U).

Thus,

∥ξu − uξ∥^{2} = 2 − 2Re⟨ξu, ξu⟩ = 2 − 2Re⟨u^{∗}ξu, ξ⟩
= 2 − 2Re τ(φ(u)u^{∗}) = 2Re(τ((u − φ(u))u^{∗}))
≤ 2∥φ(u) − u∥_{2} ≤ 2ε.

Thus, if ε < 1/2 then ∥u^{∗}ξu − ξ∥ < 1, ∀ u ∈ U(U). But then ∃ ξ_{0} ∈ H, ∥ξ_{0} − ξ∥ < 1, such that uξ_{0} = ξ_{0}u, ∀ u (see e.g., [Po1]). Thus H has a nonzero vector commuting with U, showing that U has the property T relative to V Q.E.D.

9.2. Lemma. Let

\begin{aligned}V & ⊂ & U \\ ∪ & & ∪ \\ Q & ⊂ & P \end{aligned}

be a nondegenerate commuting square of type II_{1} von Neumann algebras with a countable set X = {f_{n}}_{n} ⊂ P such that sp QX Q is ∥ ∥_{2}-dense in P and sp V X V is ∥ ∥_{2}-dense in U. Let φ : P → P be a unital, τ-preserving, completely positive map such that

φ(q_{1}xq_{2}) = q_{1}φ(x)q_{2}, ∀ q_{1}, q_{2} ∈ Q, x ∈ P

and assume that ∀ n ≥ 1, ∃ {m_{j}}_{j} ⊂ L^{2}(V, τ) orthonormal basis of V over Q such that [m_{j}, f_{n}] = 0, [m_{j}, φ(f_{n})] = 0, ∀ j. Then there exists a unique unital, τ-preserving, completely positive map φ̃ : U → U such that φ̃|_{P} = φ and φ̃(v_{1}xv_{2}) = v_{1}φ(x)v_{2}, ∀ v_{1}, v_{2} ∈ V, x ∈ P.

Proof. Let e = e_{P}^{U}. Let {m_{j}}_{j} ⊂ L^{2}(V) be a fixed orthonormal basis of V over Q and note that any element in ⟨U, P ⟩ can be written in the form Σ_{i,j}m_{i}p_{ij}em_{j}^{∗}, with p_{ij} ∈ P (see Ch.1 in [Po2]). We first define an application φ̃ : ⟨U, e⟩ → ⟨U, e⟩ by

φ̃ (∑_{i,j} m_{i}p_{ij}em_{j}^{∗}) = ∑_{i,j} m_{i}φ(p_{ij})em_{j}^{∗}, p_{ij} ∈ P.

It is easy to see that φ̃ this way defined is completely positive and Tr-preserving and satisfies φ̃(1) = 1, φ̃(Y_{1}XY_{2}) = Y_{1}φ̃(X)Y_{2}, ∀ Y_{1}, Y_{2} ∈ ⟨V, e⟩, X ∈ ⟨U, e⟩. Let us next show that φ̃ does not depend on the choice of the orthonormal basis {m_{j}} of V over Q. So let {m′_{j}}_{j} ⊂ L^{2}(V, τ) be another such orthonormal basis. Then m_{i} = ∑_{k} m′_{k}E_{P}^{U}(m_{k}^{′∗}m_{i}) so that if p ∈ P then m_{i}pem_{j}^{∗} = ∑_{k,l} m′_{k}E_{P}^{U}(m_{k}^{′∗}m_{i})pE(m_{j}^{∗}m′_{l})em_{l}^{′∗} (note that the sums do make sense in L^{2}(U, τ), with convergence in ∥ ∥_{2}, respectively so-topologies). By definition we thus have φ̃(m_{i}pem_{j}^{∗}) = m_{i}φ(p)em_{j}^{∗} and since E_{P}^{U}(m_{k}^{′∗}m_{i}) ∈ Q and

φ(E_{P}^{U}(m_{k}^{′∗}m_{i})pE_{P}^{U}(m_{j}^{∗}m′_{l})) = E_{P}^{U}(m_{k}^{′∗}m_{i})φ(p)E_{P}^{U}(m_{j}^{∗}m′_{l}),

we further get

m_{i}φ(p)em_{j}^{∗} = ∑_{k,l} m′_{k}(E_{P}^{U}(m_{k}^{′∗}m_{i})φ(p)E_{P}^{U}(m_{j}^{∗}m′_{l}))em_{l}^{′∗}

= ∑_{k,l} m′_{k}φ(E_{P}^{U}(m_{k}^{′∗}m_{i})pE_{P}^{U}(m_{j}^{∗}m′_{l}))em_{l}^{′∗}.

Taking linear combinations and limits, this shows that if

∑_{i,j} m_{i}p_{ij}em_{j}^{∗} = ∑_{i,j} m′_{i}p′_{ij}em_{j}^{′∗}

then

∑_{i,j} m_{i}φ(p_{ij})em_{j}^{∗} = ∑_{i,j} m′_{i}φ(p′_{ij})em_{j}^{′∗},

showing that φ̃ does not depend on {m_{j}}_{j}. We will now show that φ̃(U) = U and that φ̃|_{P} = φ. To this end, let us first note that φ̃(f_{n}) = φ(f_{n}), ∀ n. Indeed, we have f_{n} = f_{n}1 = f_{n} ∑_{i} m_{i}em_{i}^{∗} in which we may assume [m_{j}, f_{n}] = 0, ∀ j ( by the hypothesis and the above). Thus we get f_{n} = ∑_{i} m_{i}f_{n}em_{i}^{∗}. According to the definition of φ̃ we get φ̃(f_{n}) = ∑_{i} m_{i}φ(f_{n})em_{i}^{∗}. But by the hypothesis we may also assume [m_{i}, φ(f_{n})] = 0 so that we get ∑_{j} m_{j}φ(f_{n})em_{j}^{∗} = φ(f_{n}) ∑_{j} m_{j}em_{j}^{∗} = φ(f_{n}). Since φ̃ is V -bilinear (being ⟨Q, e⟩-bilinear) it follows that φ̃(V X V) = V φ̃(X)V = V φ(X)V ⊂ U. In particular φ̃_{|QX Q} = φ. The rest of the statement thus follows by continuity. Q.E.D.

9.3. Corollary. Let

\begin{aligned}N & ⊂ & M \\ ∪ & & ∪ \\ N_{0} & ⊂ & M_{0} \end{aligned}

be a nondegenerate commuting square of type II_{1} factors with N ⊂ M, N_{0} ⊂ M_{0}

extremal and N′ ∩ M_{j} = N′_{0} ∩ M_{0j}, ∀ j. Let T = M ∨ M^{op} ⊂ M ⊠_{e_{N}} M^{op} = S

and T_{0} = M_{0} ∨ M_{0}^{op} ⊂ M_{0} ⊠_{e_{N_{0}}} M_{0}^{op} = S_{0}. If S has the property T relative to T

then S_{0} has the property T relative to T_{0}.

Proof. By 2.5 we have T′_{0} ∩ S_{0} = ℂ, T′ ∩ S = ℂ. Also, by 2.8 S_{0} is naturally included in S and we have a nondegenerate commuting square

\begin{aligned}T & ⊂ & S \\ ∪ & & ∪ \\ T_{0} & ⊂ & S_{0}. \end{aligned}

Let {N_{0,m}}_{m} be some tunnel for N_{0} ⊂ M_{0} and N_{m} be the corresponding tunnel for N ⊂ M and denote by f_{n} = f_{−n}^{n} ∈ M_{0,n} the Jones projection for N_{0,n−1} ⊂ M_{0} ⊂ M_{0,n}. By 4.1.4 in [Po8], since S has the property T relative to T and sp∪_{n}T f_{n}T contains the dense ∗-subalgebra ∪_{n}MM_{n}^{op}M in S (cf. 4.1), it follows that ∀ ε > 0 there exists n and δ such that if φ : S → S is unital, trace preserving, completely positive, T -T bimodule map with ∥φ(f_{n}) − f_{n}∥_{2} < δ then ∥φ(x) − x∥_{2} < ε, ∀ x ∈ S, ∥x∥ ≤ 1. Since

f_{m} ∈ (N_{0,m} ∨ N_{0,m}^{op})′ ∩ M_{0} ⊠_{e_{N_{0}}} M_{0}^{op} = (N_{m} ∨ N_{m}^{op})′ ∩ M ⊠_{e_{N}} M^{op}, ∀ m,

it follows that ∀ k, ∃ {m_{j}^{k}}_{j} ⊂ N_{k} ∨ N_{k}^{op} an orthonormal basis of N_{k} ∨ N_{k}^{op} over N_{0,k} ∨ N_{0,k}^{op} (which will therefore be an orthonormal basis of T over T_{0} as well). Thus [m_{j}^{k}, f_{k}] = 0, ∀j.

Let φ_{0} : S_{0} → S_{0} be a unital, trace preserving, completely positive, T_{0}-T_{0} bimodule map satisfying ∥φ_{0}(f_{n}) − f_{n}∥_{2} < δ. Since φ_{0} is T_{0} − T_{0} bilinear and since [f_{k}, N_{0,k} ∨ N_{0,k}^{op}] = 0, we get [φ_{0}(f_{k}), N_{0,k} ∨ N_{0,k}^{op}] = 0. Thus we also have:

φ_{0}(f_{m}) ∈ (N_{0,m} ∨ N_{0,m}^{op})′ ∩ M_{0} ⊠_{e_{N_{0}}} M_{0}^{op} = (N_{m} ∨ N_{m}^{op})′ ∩ M ⊠_{e_{N}} M^{op}, ∀ m.

So we may apply Lemma 9.2 to get φ : S → S unital, τ-preserving, completely positive T -T bimodule map with φ|_{S_{0}} = φ_{0}. Thus ∥φ(f_{n}) − f_{n}∥_{2} = ∥φ_{0}(f_{n}) − f_{n}∥_{2} < δ, implying that ∥φ(x) − x∥_{2} < ε, ∀ x ∈ S, ∥x∥ ≤ 1. In particular, ∥φ_{0}(x) − x∥_{2} < ε ∀ x ∈ S_{0}. By Lemma 9.1, this is suficient to ensure that S_{0} has the property T relative to T_{0}. Q.E.D.

9.4. Proposition. Let

\begin{aligned}N & ⊂ & M \\ ∪ & & ∪ \\ N_{0} & ⊂ & M_{0} \end{aligned}

be a nondegenerate commuting square of type II_{1} factors with N_{0} ⊂ M_{0}, N ⊂ M extremal and N′_{0} ∩ M_{0,j} ⊂ N′ ∩ M_{j}, ∀ j (i.e., N_{0} ⊂ M_{0} is smoothly embedded in N ⊂ M, in the sense of [Po2]). Let T_{0} ⊂ S_{0}, T ⊂ S be the corresponding symmetric enveloping inclusions. If S_{0} has the property T relative to T_{0}, then S has the property T relative to T.

Proof. By [Po8], ∀ ε > 0, ∃ δ > 0 and x_{1}, …, x_{n} ∈ S_{0} such that if H_{0} is a S_{0}-S_{0} bimodule with a unit vector ξ_{0} ∈ H_{0} satisfying [y, ξ_{0}] = 0, ∀ y ∈ T_{0}, and ∥[x_{i}, ξ_{0}]∥ < δ, ∀ i, then there exists ξ_{1} ∈ H_{0} satisfying [x, ξ_{1}] = 0, ∀ x ∈ S_{0}, and ∥ξ_{1} − ξ_{0}∥ < ε. Let then H be a S-S bimodule with a unit vector ξ_{0} ∈ H such that [y, ξ_{0}] = 0, ∀ y ∈ T, and ∥[x_{i}, ξ_{0}]∥ < δ, ∀ i. Regarding H as a S_{0}-S_{0} bimodule it follows that there exists ξ′_{0} ∈ H such that [x, ξ′_{0}] = 0, ∀ x ∈ S_{0}, and ∥ξ′_{0} − ξ_{0}∥ < ε. Denote

K = {ξ ∈ H | xξ = ξx, ∀ x ∈ S_{0}},
K_{0} = {η_{0} ∈ H | [y, η_{0}] = 0, ∀ y ∈ T = M ∨ M^{op}},
K_{1} = {η_{1} ∈ H | [y, η_{1}] = 0, ∀ y ∈ M_{1} ∨ N^{op}}.

With these notations, it follows that ξ_{0} ∈ K_{0} and ξ′_{0} ∈ K. We then need to construct some positive contractions A, B ∈ B(H) such that 0 ≤ A, B ≤ 1, Aξ = ξ = Bξ, ∀ ξ ∈ K, AK_{0} ⊂ K_{1}, BK_{1} ⊂ K_{0}. For if we have such A and B, then

∥(BA)^{n}ξ_{0} − ξ′_{0}∥ = ∥(BA)^{n}ξ_{0} − (BA)^{n}ξ′_{0}∥ ≤ ∥ξ_{0} − ξ′_{0}∥ < ε

so that if ξ′′_{0} is a weak limit point of {(1/n) ∑_{k=1}^{n}(BA)^{k}ξ_{0}}_{n} then BAξ′′_{0} = ξ′′_{0}, ξ′′_{0} ∈ K_{0} (because all (BA)^{k}ξ_{0} belong to K_{0}) and ∥ξ′′_{0} −ξ′_{0}∥ < ε. But 0 ≤ A ≤ 1, 0 ≤ B ≤ 1, BAξ′′_{0} = ξ′′_{0} implies that Aξ′′_{0} = ξ′′_{0}, Bξ′′_{0} = ξ′′_{0}, so that ξ′′_{0} ∈ K_{0} ∩K_{1}. Thus e_{1}ξ′′_{0} = ξ′′_{0}e_{1}, yξ′′_{0} = ξ′′_{0}y, ∀ y ∈ T, and since T and e_{1} generate S we get xξ′′_{0} = ξ′′_{0}x, ∀ x ∈ S. This shows that H has a nonzero vector commuting with S. Finally, in order to construct A, B with the required properties, let {p_{j}^{n}}_{1≤j≤k_{n}} ⊂ M_{0,1} = ⟨M_{0}, e_{1}⟩, {q_{i}^{k}}_{1≤i≤m} ⊂ M_{0}^{op} be partitions of the identity such that if p^{n}, respectively q^{k} denote the spectral projection of | ∑_{j} p_{j}^{n}e_{2}p_{j}^{n} − λ1|, respectively | ∑_{i} q_{i}^{k}e_{1}q_{i}^{k} − λ1|, corresponding to the interval [ε, ∞], where λ = [M_{0} : N_{0}]^{−1} = [M : N]^{−1}, then τ(p^{n}) < (1/n) min_{j} τ(p_{j}^{n}) and τ(q^{k}) < (1/k) min_{i} τ(q_{i}^{k}) (cf. the Appendix in [Po2], or [Po9]). We claim that if A is a weak limit of the sequence of operators {∑_{j} p_{j}^{n} ·p_{j}^{n}}_{n} ⊂ B(H) and B is a weak limit of {∑_{i} q_{i}^{k} · q_{i}^{k}}_{k} ⊂ B(H) then A, B do satisfy the required conditions. Indeed, since p_{j}^{n}, q_{i}^{k} ∈ S_{0} we have

∑_{j} p_{j}^{n}ξp_{j}^{n} = ξ, ∑_{i} q_{i}^{k}ξq_{i}^{k} = ξ, ∀ ξ ∈ K, ∀ k, n.

Thus, Aξ = ξ = Bξ, ∀ ξ ∈ K. Since p_{j}^{n} ∈ ⟨M_{0}, e_{N}⟩ ⊂ N^{op′} ∩ S it follows that if [y, η_{0}] = 0, ∀ y ∈ T = M ∨ M^{op}, then

[x^{op}, ∑_{j} p_{j}^{n}η_{0}p_{j}^{n}] = 0, ∀ x^{op} ∈ N^{op}.

Thus,

[x^{op}, Aη_{0}] = 0, ∀ x^{op} ∈ N^{op}, ∀ η_{0} ∈ K_{0}.

Let η_{0} ∈ K_{0} with ∥η_{0}∥ = 1 and note that, since η_{0} commutes with T and T′ ∩ S = ℂ, η_{0} follows a trace vector for S. Let also ξ ∈ H, and x ∈ M_{1} = ⟨M, e_{1}⟩ and note that

∥λ^{−1} ∑_{j} p_{j}^{n}E_{M}^{M_{1}}(p_{j}^{n}xp_{i}^{n}) − xp_{i}^{n}∥_{2}

= λ^{−1/2} ∥λ^{−1} ∑_{j} p_{j}^{n}e_{2}p_{j}^{n}xp_{i}^{n}e_{2} − xp_{i}^{n}e_{2}∥_{2}

≤ λ^{−1/2} ∥(1 − p^{n}) (λ^{−1} ∑_{j} p_{j}^{n}e_{2}p_{j}^{n} − 1)∥ ∥xp_{i}^{n}e_{2}∥_{2} + λ^{−1/2}∥p^{n}∥_{2}

≤ 2λ^{−1/2}(∥x∥/n)∥p_{i}^{n}∥_{2}.

Thus we get

∥x ∑_{i} p_{i}^{n}η_{0}p_{i}^{n} − λ^{−1} ∑_{i,j} p_{j}^{n}E_{M}^{M_{1}}(p_{j}^{n}xp_{i}^{n})η_{0}p_{i}^{n}∥^{2}

= ∑_{i} ∥xp_{i}^{n}η_{0}p_{i}^{n} − λ^{−1} ∑_{j} p_{j}^{n}E_{M}^{M_{1}}(p_{j}^{n}xp_{i}^{n})η_{0}p_{i}^{n}∥^{2}

≤ ∑_{i} ∥xp_{i}^{n} − λ^{−1} ∑_{j} p_{j}^{n}E_{M}^{M_{1}}(p_{j}^{n}xp_{i}^{n})∥_{2}^{2}

≤ 4λ^{−1}(∥x∥^{2}/n^{2}) ∑_{j} ∥p_{i}^{n}∥_{2}^{2} = 4λ^{−1}∥x∥^{2}/n^{2}.

Similarly we get

∥∑_{i} p_{i}^{n}η_{0}p_{i}^{n}x − λ^{−1} ∑_{i,j} p_{j}^{n}η_{0}E_{M}^{M_{1}}(p_{j}^{n}xp_{i}^{n})p_{i}^{n}∥^{2} ≤ 4λ^{−1}∥x∥^{2}/n^{2}

as well. But since yη_{0} = η_{0}y, ∀ y ∈ M ⊂ M ∨ M^{op}, we have

∑_{i} (∑_{j} p_{j}^{n}E_{M}^{M_{1}}(p_{j}^{n}xp_{i}^{n})) η_{0}p_{i}^{n} = ∑_{j} p_{j}^{n}η_{0} (∑_{i} E_{M}^{M_{1}}(p_{j}^{n}xp_{i}^{n})p_{i}^{n})

so by the above estimates we get

∥x ∑_{i} p_{i}^{n}η_{0}p_{i}^{n} − ∑_{j} p_{j}^{n}η_{0}p_{j}^{n}x∥ < 8λ^{−1/2}∥x∥/n → 0.

Since A is a weak limit of {∑_{i} p_{i}^{n} · p_{i}^{n}}_{n} it follows that [x, Aη_{0}] = 0, thus AK_{0} ⊂ K_{1}. Similar calculations show that BK_{1} ⊂ K_{0} and A, B are thus constructed. As we have previously shown, this was sufficient to ensure that H has a nonzero vector commuting with S. Thus S has the property T relative to T. Q.E.D.

We can now conclude with the following:

9.5. Theorem. Let N_{0} ⊂ M_{0} be an extremal inclusion of type II_{1} factors such that M_{0} ⊠_{e_{N_{0}}} M_{0}^{op} has the property T relative to M_{0} ∨ M_{0}^{op}. Then M ⊠_{e_{N}} M^{op} has the property T relative to M ∨ M^{op} for any extremal inclusion N ⊂ M with G_{N,M} = G_{N_{0},M_{0}}

Proof. Since N_{0} ⊂ M_{0} is embedded smoothly in N_{0}^{ω} ⊂ M_{0}^{ω} and the two subfactors have the same higher relative commutants, by 9.4 it follows that M_{0}^{ω} ⊠_{e_{N^{ω}_{0}}} M_{0}^{ωop} has the property T relative to M_{0}^{ω} ∨M_{0}^{ωop}. But by [Po9], the in-

clusion of factors N^{G}(R) ⊂ M^{G}(R), where G = G_{N_{0},M_{0}} and R is the hyperfinite II_{1} factors, is also embedded as a commuting square with same higher relative commutants in N_{0}^{ω} ⊂ M_{0}^{ω}. Thus, by 9.2 it follows that M^{G}(R) ⊠(M^{G}(R))^{op} has the property T relative to M^{G}(R) ∨ (M^{G}(R))^{op}. But N^{G}(R) ⊂ M^{G}(R) is included in N^{ω} ⊂ M^{ω} as well ([Po9]), so M^{ω} ⊠_{e_{N^{ω}}} M^{ωop} has the property T

relative to M^{ω} ∨ M^{ωop}, by 9.4. Then, again by 9.2 it follows that M ⊠_{e_{N}} M^{op}

has the property T relative to M ∨ M^{op}. Q.E.D.

9.6. Definition. We say that a standard λ-lattice G has the property T if M ⊠_{e_{N}} M^{op} has the property T relative to M ∨ M^{op} for some (and thus all!)

subfactor N ⊂ M with G_{N,M} = G.

The following class of examples shows that our notion of property T agrees with Kazhdan’s classical notion for groups.

9.7. Proposition. Let G be the standard λ-lattice of a locally trivial subfactor associated to some faithful G-kernel on some type II_{1} factor. Then G has the property T if and only if the group G has the property T.

Proof. Let P be the factor on which G acts and σ be the G-kernel on P. By Section 3 and 9.6, G has the property T iff P ¯⊗P^{op} ⋊_{σ⊗σ^{op}} G has the property T relative to P ¯⊗P^{op}. By ([A-D], [Po8]) this is equivalent to G having the property T. Q.E.D.

Let us next note some simple properties of this notion.

9.8. Proposition. (i) G is both amenable and has the property T if and only if it has finite depth. (ii) If G = G_{1}×G_{1} (see part (b) in 5.6 for the definition) then G has the property T if and only if both G_{1} and G_{2} have the property T. (iii) G has the property T if and only if G^{op} has it. (iv) If N ⊂ M is an extremal inclusion {M_{i}}_{i} is its tower, then G_{N,M} has the property T iff G_{M_{i},M_{j}} has the property T for some i < j iff G_{M_{i},M_{j}} has the property T for all i < j.

Proof. To prove (i), let N ⊂ M be an extremal inclusion such that G_{N,M} = G. Then G is both amenable and has the property T iff M ⊠_{e_{N}} M^{op} is both amenable

and has the property T relative to M ∨ M^{op}. And by (4.1.4 in [Po8]) this is further equivalent to [M ⊠_{e_{N}} M^{op} : M ∨ M^{op}] < ∞. But by 4.6, [M ⊠_{e_{N}} M^{op} :

M ∨ M^{op}] < ∞ is equivalent to N ⊂ M having finite depth.

To prove (ii) let N_{j} ⊂ M_{j}, j = 1, 2, be such that G_{N_{j},M_{j}} = G_{j} and note that G_{N,M} = G where G = G_{1} × G_{2}, N = N_{1} ¯⊗N_{2} ⊂ M_{1} ¯⊗M_{2} = M. Then (T ⊂ S) = (T_{1} ¯⊗T_{2} ⊂ S_{1} ¯⊗S_{2}), where T ⊂ S, T_{1} ⊂ S_{1}, T_{2} ⊂ S_{2} are the symmetric enveloping inclusions associated with N ⊂ M, N_{1} ⊂ M_{1}, respectively N_{2} ⊂ M_{2}. If T ⊂ S has the relative property T and {x_{i}}_{1≤i≤n} ⊂ S is its critical set for some ε > 0, then by [Po8] we may assume x_{i} are in the algebraic tensor product S_{1} ⊗ S_{2}, i.e., x_{i} = ∑_{j} x_{j}^{i} ⊗ y_{j}^{i}, x_{j}^{i} ∈ S_{1}, y_{j}^{i} ∈ S_{2}. Let H_{1} be a S_{1}-S_{1} bimodule with a unit vector ξ_{1} ∈ H_{1} commuting with T_{1} and δ_{1}-commuting with {x_{j}^{i}}_{i,j}. Denote by H = H_{1} ¯⊗L^{2}(S_{2}), ξ = ξ_{1} ⊗ ˆ1 and note that if δ_{1} is sufficiently small then ξ ε-commutes with {x_{i}}_{i}. It follows that there exists ξ′ ∈ H, commuting with S at distance Kε from ξ (see [Po8]), where K is a universal constant. But then, if Kε < 1, the projection ξ′′ of ξ′ onto H_{1} ⊗ ℂ1 ≃ H_{1} is a nonzero vector commuting with S_{1}. This shows that S_{1} has the property T relative to T_{1}. Similarly, S_{2} has the property T relative to T_{2}. Conversely, if S_{i} has property T relative to T_{i} for i = 1, 2 and H is a S-S bimodule with ξ ∈ H a unit vector commuting with {x_{i} ⊗ y_{j}}_{i,j}, where {x_{i}}_{i} ⊂ S_{1}, {y_{j}}_{j} ⊂ S_{2} are the critical sets for T_{1} ⊂ S_{1}, respectively T_{2} ⊂ S_{2} then it follows that

∥uξ − ξu∥ < ε, ∀ u ∈ U(S_{1} ⊗ 1) ∪ U(1 ⊗ S_{2}).

Thus,

∥(u ⊗ v)ξ − ξ(u ⊗ v)∥ < 2ε, ∀ u ∈ U(S_{1}), v ∈ U(S_{2}).

A simple convexity argument in Hilbert space, or Ryll-Nardjewski’s fixed point theorem then shows that there exists ξ′ ∈ H, ∥ξ′ −ξ∥ < 2ε, commuting with all elements in the group F = {u⊗v | u ∈ U(S_{1}), v ∈ U(S_{2})}. Since sp F ⊃ S_{1}⊗S_{2} it follows that ξ′ commutes with S = S_{1} ¯⊗S_{2}. Taking ε < 1/2, this shows that H has a nonzero vector commuting with S, so S has the property T relative to T. To prove (iii) we only need to remark that the symmetric enveloping inclusions associated to N ⊂ M and N^{op} ⊂ M^{op} are identical, so that 9.5 applies to get that G_{N,M} has T iff G_{N^{op},M^{op}} (= (G_{N,M})^{op}) has this property. Finally, to prove (iv) recall from [Po8] that if V_{0} ⊂ V ⊂ U are inclusions of factors and [V : V_{0}] < ∞ then U has the property T relative to V iff U has the property T relative to V_{0}. Thus, if N ⊂ M is an extremal inclusion and we put U = M ⊠_{e_{N}} M^{op}, V = M ∨ M^{op}, V_{0} = M ∨ N^{op}, V_{1} = M_{1} ∨ N^{op}, it follows that

U has the property T relative to V iff U has the property T relative V_{1}. But V_{1} ⊂ U is a reduced of the symmetric enveloping inclusion for M ⊂ M_{1} (cf. 2.6) so, by [Po8] again, it has the relative property T iff M_{1}∨M_{1}^{op} ⊂ M_{1} ⊠_{e_{N}} M_{1}^{op}

has relative propert T. Thus, G_{N,M} has T iff G_{M,M_{1}} has T. The rest follows from 2.6 a). Q.E.D.

We do not have more examples of property T standard λ-lattices other than the ones coming from groups (in 9.7) or the obvious ones that can be constructed by using jointly 9.7 and 9.8. For example, we do not know whether there exist standard lattices with the property T that come from irreducible subfactors. As for the minimal standard lattices generated by the Jones projections only, i.e., the so-called Temperley-Lieb-Jones standard lattices, we will prove below that generically they do not have the property T. This will in fact be an imediate corollary of the following more important consequence of 9.4:

9.9. Theorem. Let G be a standard λ-lattice and G_{0} be a sublattice of G. If G_{0} has the property T then G has the property T. Conversely, if [G : G_{0}] < ∞ and G has T then G_{0} has T.

Proof. By [Po7] there exists a commuting square

\begin{aligned}N & ⊂ & M \\ ∪ & & ∪ \\ N_{0} & ⊂ & M_{0} \end{aligned}

such that G_{0} = G_{N_{0},M_{0}} and G = G_{N,M}. By 9.4 and the definition of property T for standard latices 9.6, it follows that if G_{N_{0},M_{0}} has T then G_{N,M} has this property as well. The last part is trivial, by [Po8], 2.7, 2.9 and 2.10. Q.E.D.

9.10. Corollary. If a standard λ-lattice G_{0} is a sublattice of an amenable standard λ-lattice with infinite graph then G_{0} doesn’t have the property T. In particular, if there exists an amenable subfactor of index λ^{−1} and infinite depth then the Temperley-Lieb-Jones standard lattice of graph A_{∞} and index λ^{−1} does not have the property T.

Proof. Trivial by 9.9. Q.E.D.

Let us end by mentioning a problem which at this point seems of interest:

9.11. Problem. Is it true that the property T for a standard lattice G only depends on its graph, i.e., if G, G_{0} have the same (weighted) graph Γ and G has T, does it follow that G_{0} has T ? Note that in all the examples of property T standard lattices that we have in this paper (obtained by combining 9.7 with 9.8) this is indeed the case. We strongly believe that this question has a positive answer. If this would be indeed the case, then one would have a notion of property T for standard graphs. We mention that in the combinatorial theory of groups there has been a steady interest towards generalizing the property T from groups to more general objects, in particular to (certain classes of) graphs. Since the standard lattices do generalize discrete groups and certain classes of Kac algebras and compact quantum groups ([Ba]), our definition of property T does provide a generalization along these lines.

Appendix

A.1. Relative Dixmier Property for Subfactors of Finite Index

We prove in this section a version for inclusions of type II_{1} factors with finite Jones index of Dixmier’s classical result on the norm closure of ”averaging” elements by unitaries, as follows:

Theorem. Let N ⊂ M be an inclusion of factors of finite index. Then N ⊂ M has the relative Dixmier property, i.e., for any x ∈ M, we have \overline{co}^{n}\{uxu^{∗} | u unitary element in N} ∩ N′ ∩ M = {E_{N′∩M}(x)}.

Proof. For x ∈ M denote C_{N}(x) = \overline{co}^{n}\{uxu^{∗} | u ∈ U(N)\}. Since E_{N′∩M}(uxu^{∗}) = E_{N′∩M}(x), ∀u ∈ U(N), it follows that E_{N′∩M}(y) = E_{N′∩M}(x), ∀y ∈ C_{N}(x). Thus, if for some x ∈ M we have C_{N}(x) ∩ N′ ∩ M ≠ ∅ then C_{N}(x) ∩ N′ ∩ M = {E_{N′∩M}(x)}. By replacing if necessary x by x − E_{N′∩M}(x), it follows that it is sufficient to check that 0 ∈ C_{N}(x) for all x ∈ M with E_{N′∩M}(x) = 0. Moreover, by arguing like in the single algebra case ([D3]), it is sufficient to check this property for selfadjoint such elements x. We will proceed by contradiction, assuming there exists an element x = x^{∗} in M, with E_{N′∩M}(x) = 0, such that 0 ∉ C_{N}(x). By the Hahn-Banach theorem there exists a functional Φ = Φ^{∗} ∈ M^{∗} and ε_{0} > 0 such that Φ(y) ≥ ε_{0}, ∀y ∈ C_{N}(x). It follows that Ψ(x) ≥ ε_{0}, ∀Ψ ∈ co{Φ(u · u^{∗}) | u ∈ U(N)} so that

Ψ(x) ≥ ε_{0}, ∀Ψ ∈ C_{N}(Φ) =^{def} \overline{co}^{σ(M^{∗},M)}\{Φ(u · u^{∗}) | u ∈ U(N)\} and in fact

Ψ(y) ≥ ε_{0}, ∀y ∈ C_{N}(x) as well. To get to a contradiction we first show that there exists Ψ in C_{N}(Φ) which can be written as Ψ = Ψ_{1} − Ψ_{2}, with Ψ_{1,2} positive functionals on M which are scalar multiples of the trace τ when restricted to N. To this end let Φ = Φ_{1}−Φ_{2} be the polar decomposition of Φ, into its positive and negative parts. Let V = {F ⊂ (N)_{1} | F finite}. By Dixmier’s classical Theorem ∀F ∈ V, ∃u_{F} = (u_{1}^{F} , ..., u_{n_{F}}^{F}) ⊂ U(N) such that ∥T_{u_{F}} (y) − τ(y)1∥ < 1/|F |, ∀y ∈ F,

where for X ∈ M we denote T_{u_{F}} (X) =^{def} (n_{F})^{−1}Σ_{j}u_{j}^{F} Xu_{j}^{F ∗}. Then let ω be a free ultrafilter majorizing the filter V and for each i = 1, 2 define Ψ_{i}(X) = lim_{F →ω} Φ_{i}(T_{u_{F}} (X)), the limit being taken in the usual Banach sense. Then we clearly have Ψ_{i|N} = c_{i}τ_{|N}, where c_{i} = Φ_{i}(1), i = 1, 2. Also, if we let Ψ = Ψ_{1} − Ψ_{2} then Ψ(X) = lim_{F →ω} Φ(T_{u_{F}} (X)), ∀X ∈ M and since Φ(T_{u_{F}} ( )) ∈ C_{N}(Φ), ∀F, it follows that Ψ belongs to C_{N}(Φ). Thus Ψ = Ψ_{1} − Ψ_{2} satisfies the desired conditions. But by [PiPo1] we have E_{N}(X) ≥ λX, ∀X ∈ M_{+}, so by applying Ψ_{1,2} to both sides we get c_{i}τ(X) = c_{i}τ(E_{N}(X)) = Ψ_{i}(E_{N}(X)) ≥ λΨ_{i}(X), implying that Ψ_{i} ≤ λ^{−1}c_{i}τ, i = 1, 2. Thus Ψ_{1,2} actually follow normal on all M and so does Ψ. By Sakai’s Radon-Nykodim type theorem there exists a = a^{∗} ∈ M such that Ψ(X) = τ(aX), ∀X ∈ M. Putting this into the relation that Ψ satisfies gives τ(ya) ≥ ε_{0}, ∀y ∈ C_{N}(x).

In particular, from the last relation and the trace property we get τ(xu^{∗}au) = τ(uxu^{∗}a) ≥ ε_{0}. By taking convex combinations of elements of the form u^{∗}au and weak limits and using that \overline{co}^{w}\{u^{∗}au | u unitary element in N\} ∩ N′ ∩ M = {E_{N′∩M}(a)} (cf. [Po1]), we deduce that τ(xE_{N′∩M}(a)) ≥ ε_{0}. But τ = τ ∘ E_{N′∩M} and, since E_{N′∩M}(xE_{N′∩M}(a)) = E_{N′∩M}(x)E_{N′∩M}(a) and x was assumed to satisfy E_{N′∩M}(x) = 0, we obtain 0 ≥ ε_{0} a contradiction which ends the proof of the theorem. Q.E.D.

A.2. A generalized version of Connes’ perturbation theorem

In [C1] A. Connes proved a technical result about Hilbert norm perturbations of square integrable operators in semifinite von Neumann algebras. We will use here a slight modification of his argument (essentially, of his ”joint distribution trick”) to derive the following version of his result, needed in the proof of Theorem 5.4:

A.2.1. Theorem. Let P be a semifinite von Neumann algebra with a normal semifinite faithful trace denoted by Tr. Let Φ be a positive map on P satisfying the conditions:

(1) Φ(1) = 1, Tr ∘ Φ ≤ Tr.

(2) sup{∥Φ(x)∥_{2,Tr} | x ∈ P, ∥x∥_{2,Tr} ≤ 1} ≤ 1. Let δ > 0 be such that δ < (5)^{−4} and b ∈ P_{+} satisfy the conditions:

(3) ∥b∥_{2,Tr} = 1, ∥Φ(b)∥_{2,Tr} ≥ 1 − δ.

(4) ∥b − Φ(b)∥_{2,Tr} < δ. Then there exists s > 0 such that ∥e_{s}(b) − Φ(e_{s}(b))∥_{2,Tr} < δ^{1/4}∥e_{s}(b)∥.

Proof. Like in [C1], let X = ℝ_{+}^{2} \ {0} and H_{0}(x, y) = x, H_{1}(x, y) = y. As on page 77 in [C1] it then follows that

μ(A_{0} × A_{1}) =^{def} Tr(e_{A_{0}}(b)Φ(e_{A_{1}}(b)))

for A_{i} ⊂ ℝ_{+} Borel sets such that either 0 ≠ Ā_{0} or 0 ≠ Ā_{1}, defines a Radon measure μ on X, which satisfies the properties:

a) ∥f(H_{i})∥_{1,μ} = Tr(Φ_{i}(|f|(b))) (respectively ∥f(H_{i})∥_{2,μ}^{2} = Tr(Φ_{i}(|f|^{2}(b))) ≤ ∥f(b)∥_{2,Tr}^{2}), for all f : [0, ∞) → ℂ Borel function with f(0) = 0 and f(b) ∈ L^{1}(P, Tr) (respectively f(b) ∈ L^{2}(P, Tr)), i = 0, 1, where Φ_{0} = id, Φ_{1} = Φ.

b) ∫_{X} f_{0}(H_{0})\overline{f_{1}(H_{1})}dμ = Tr(f_{0}(b))Φ(f̄_{1}(b))), for all f_{i} : [0, ∞) → ℂ Borel with

f_{i}(0) = 0 and f_{i}(b) ∈ L^{2}(P, Tr), i = 0, 1.

c) ∥f_{0}(H_{0}) − f_{1}(H_{1})∥_{2,μ} ≥ ∥f_{0}(b) − Φ(f_{1}(b))∥_{2,Tr}, ∀ f_{i} as in b).

d) ∥H_{0} − H_{1}∥_{2,μ}^{2} = Tr(b^{2}) + Tr(Φ(b^{2})) − 2Tr(bΦ(b)) ≤ 6δ. Indeed, a) and b) are clear by the proof of I.1 in [C1] and the definition of μ.

Further on, by a), b), (1), and Kadison’s inequality we get:

∥f_{0}(H_{0}) − f_{1}(H_{1})∥_{2,μ}^{2} = ∥f_{0}(H_{0})∥_{2,μ}^{2} + ∥f_{1}(H_{1})∥_{2,μ}^{2}
− 2Re ∫_{X} f_{0}(H_{0})\overline{f_{1}(H_{1})}dμ
= Tr(f_{0}(b)^{∗}f_{0}(b) + Tr(Φ(f_{1}(b)^{∗}f_{1}(b)))
− 2Re Tr(f_{0}(b))Φ(\overline{f_{1}}(b)))
≥ Tr(f_{0}(b)^{∗}f_{0}(b_{0}))
+ Tr(Φ(f_{1}(b))^{∗}Φ(f_{1}(b)))
− 2Re Tr(f_{0}(b))Φ(f_{1}(b))^{∗})
= ∥f_{0}(b_{0}) − Φ(f_{1}(b))∥_{2,Tr}^{2}.

This proves c). Then d) is clear by noticing that the hypothesis and the Cauchy-Schwartz inequality imply:

Tr(b^{2}) + Tr(Φ(b^{2})) − 2Tr(bΦ(b))
≤ Tr(b^{2}) + Tr(b^{2}) − 2Tr(bΦ(b))
= 2 − 2Tr(bΦ(b))
≤ 2 − 2Tr(b^{2}) + 2δ
≤ 2(1 − (1 − δ)^{2}) + 2δ ≤ 6δ

Remark now that we have, like in proof of 1.2.6 in [C1], the estimate:

∫_{ℝ_{+}^{∗}} ∥e_{t^{1/2}}(H_{0}) − e_{t^{1/2}}(H_{1})∥_{2,μ}^{2}dt
= ∥H_{0}^{2} − H_{1}^{2}∥_{1,μ} ≤ ∥H_{0} − H_{1}∥_{2,μ}∥H_{0} + H_{1}∥_{2,μ}.

But d) implies ∥H_{0}−H_{1}∥_{2,μ} ≤ (6δ)^{1/2} and a) implies ∥H_{0}+H_{1}∥_{2,μ} ≤ ∥H_{0}∥_{2,μ}+ ∥H_{1}∥_{2,μ} ≤ ∥b∥_{2,Tr} + ∥b∥_{2,Tr} = 2. Thus, by applying c) to the function f = χ_{[t^{1/2},∞)}, for each t > 0, we obtain

∫_{ℝ_{+}^{∗}} ∥e_{t^{1/2}}(b) − Φ(e_{t^{1/2}}(b))∥_{2,Tr}^{2}dt

≤ 2(6δ)^{1/2} = 2(6δ)^{1/2} ∫_{ℝ_{+}^{∗}} ∥e_{t^{1/2}}(b_{0})∥_{2,Tr}^{2}dt. (∗)

This implies that if we denote by D the set of all t > 0 for which

g(t) =^{def} ∥e_{t^{1/2}}(b_{0}) − Φ(e_{t^{1/2}}(b))∥_{2,Tr}^{2}dt < δ^{1/4}∥e_{t^{1/2}}(b)∥_{2,Tr}^{2}

then

∫_{D} ∥e_{t^{1/}}(b)∥_{2,Tr}^{2}dt ≥ 1 − 5δ^{1/4}.

Indeed, for if ∫_{D} ∥e_{t^{1/2}}(b_{0})∥_{2,Tr}^{2}dt < 1 − 5δ^{1/4}, by taking into account that g(t) ≥ δ^{1/4}∥e_{t^{1/2}}(b_{0})∥_{2,Tr}^{2} for t ∈ ℝ_{+}^{∗} \ D, we would get:

∫_{ℝ_{+}^{∗}} g(t)dt ≥ ∫_{ℝ_{+}^{∗}\D} g(t)dt
≥ δ^{1/4} ∫_{ℝ_{+}^{∗}\D} ∥e_{t^{1/2}}(b_{0})∥_{2,Tr}^{2}dt
≥ 5δ^{1/2} > 2(6δ)^{1/2}.

which is in contradiction with (∗). In particular, since δ < 5^{−4}, we have 1 − 5δ^{1/4} > 0 so that D ≠ ∅. Thus, any s > 0 with s^{2} ∈ D will satisfy the condition in the conclusion. Q.E.D.

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[H2] U. Haagerup: Private communication.

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[HiIz] F. Hiai and M. Izumi: Amenability and strong amenability for fusion algebras with applications to subfactor theory, preprint 1996.

[ILP] M. Izumi, R. Longo and S. Popa: A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras, J. Funct. Analysis, 155 (1998), 25-63.

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[PiPo1] M. Pimsner and S. Popa: Entropy and index for subfactors, Ann. Sci. Ecole Norm. Sup., 19 (1986), 57–106.

[PiPo2] M. Pimsner and S. Popa: Finite dimensional approximation for pairs of algebras and obstructions for the index, J. Funct. Anal., 98 (1991), 270–291.

[PiPo3] M. Pimsner and S. Popa: Iterating the basic construction, Trans. Amer. Math. Soc., 310 (1988), 127-133.

[Po1] S. Popa: On a problem of R. V. Kadison on maximal abelian *- subalgebras in factors, Invent. Math., 65 (1981), 269-281.

[Po2] S. Popa: Classification of amenable subfactors of type II, Acta Math., 172 (1994), 163–255.

[Po3] S. Popa: Classification of subfactors and their endomorphisms, CBMS Lecture Notes, 86 (1995).

[Po4] S. Popa: Approximate innerness and central freeness for subfactors: A classification result. In “Subfactors”, World Scientific, Singapore-New Jersey-London-Hong Kong, 1994, pp. 274–293.

[Po5] S. Popa: Symmetric enveloping algebras, amenability and AFD properties for subfactors, Math. Res. Lett., 1 (1994), 409–425.

[Po6] S. Popa: Markov traces on universal Jones algebras and subfactors of finite index, Invent. Math., 111 (1993), 375-405.

[Po7] S. Popa: An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math., 120 (1995), 427–445.

[Po8] S. Popa: Correspondences, INCREST preprint 1986.

[Po9] S. Popa: Free independent sequences in type II_{1} factors and related problems, Astérisque 232 (1995), 187-202.

[Po10] S. Popa: The relative Dixmier property for inclusions of von Neumann algebras, preprint 1997.

[Po11] S. Popa: Amenability in the theory of subfactors, in ”Operator Algebras and Quantum Field Theory”, International Press, Editors S. Doplicher et al, pp 199-211.

[Po12] S. Popa: Classification of hyperfinite subfactors of type II and III_{λ}, 0 < Λ ≤ 1, UCLA preprint 1991, unpublished.

[Po13] S. Popa: Biduals associated to subfactors, hypertraces and restrictions for the Jones index, preprint 1997.

[Po14] S. Popa: On A. Connes’ joint distribution trick, preprint 1997.

[Po15] S. Popa: Classification of subfactors: the reduction to commuting squares, Invent. Math., 101 (1990), 19-43.

[Po16] S. Popa: The classification of hyperfinite subfactors with amenable graph, preprint 1997.

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[Xu] F. Xu: Jones-Wassermann subfactors from disconnected intervals, preprint 1997.

Sorin Popa Math. Dept. University of California Los Angeles, CA 90095-155505 USA popa@math.ucla.edu