On the Uniqueness of the Injective III_{1} Factor

Uffe Haagerup

Received: August 2, 2016

Communicated by Joachim Cuntz

Abstract. We give a new proof of a theorem due to Alain Connes, that an injective factor N of type III_{1} with separable predual and with trivial bicentralizer is isomorphic to the Araki–Woods type III_{1} factor R_{∞}. This, combined with the author’s solution to the bicentralizer problem for injective III_{1} factors provides a new proof of the theorem that up to ∗-isomorphism, there exists a unique injective factor of type III_{1} on a separable Hilbert space.

2010 Mathematics Subject Classification: 46L36

Preamble by Alain Connes

Uffe Haagerup solved the hardest problem of the classification of factors, namely the uniqueness problem for injective factors of type III_{1}. The present paper, taken from his unpublished notes, presents a direct proof of this uniqueness by showing that any injective factor of type III_{1} is an infinite tensor product of type I factors so that the uniqueness follows from the Araki–Woods classification. The proof is typical of Uffe’s genius, the attack is direct, and combines his amazing control of completely positive maps and his sheer analytical power, together with his solution to the bicentralizer problem. After his tragic death, Hiroshi Ando volunteered to type the manuscript^{1}. Some pages were missing from the notes, but eventually Cyril Houdayer and Reiji Tomatsu suggested a missing proof of Lemma 3.4 and Theorem 3.1. We heartily thank Hiroshi, Cyril and Reiji for making the manuscript available to the community. We also thank Søren Haagerup for giving permission to publish his father’s paper.

1The manuscript is typed by Hiroshi Ando (Chiba University) in cooperation with Cyril Houdayer (Université Paris-Sud), Toshihiko Masuda (Kyushu University), Reiji Tomatsu (Hokkaido University), Yoshimichi Ueda (Kyushu University) and Wojciech Szymanski (University of Southern Denmark).

Contents

1 Introduction 1194

2 Preliminaries 1196 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196 2.2 Connes–Woods’ characterization of ITPFI factors . . . . . . . . 1197 2.3 Bicentralizers on type III_{1} factors . . . . . . . . . . . . . . . . . 1198 2.4 Almost unitary equivalence in Hilbert N-bimodules . . . . . . . 1199

3 Completely positive maps from m × m-matrices into an injective factor of type III_{1} 1200

4 ℚ-stable states on III_{1} factors 1208

5 Proof of Main Theorem 1213

1 Introduction

The problem, whether all injective factors of type III_{1} on a separable Hilbert space are isomorphic, has been settled affirmatively. The proof of the uniqueness of injective III_{1} factors falls in two parts, namely (see §2.3 for the definition of the bicentralizer):

Theorem 1.1 ([Con85]). Let M be an injective factor of type III_{1} on a separable Hilbert space, such that the bicentralizer B_{φ} is trivial (i.e., B_{φ} = ℂ1) for some normal faithful state φ on M, then M is ∗-isomorphic to the Araki–Woods factor R_{∞}.

Theorem 1.2 ([Haa87]). For any normal faithful state φ on an injective factor M of type III_{1} on a separable Hilbert space, one has B_{φ} = ℂ1.

In this paper we give an alternative proof of Theorem 1.1 above, based on the technique of our simplified proof [Haa85] of Connes’ Theorem [Con76] “injective⇒hyperfinite” in the type II_{1} case^{2}. The key steps in our proof of Theorem 1.1 are listed below:

\underline{Step 1} By use of continuous crossed products, we prove that the identity map on an injective factor N of type III_{1} has an approximate factorization

\xymatrix{ & R \ar[dr]^{T_{λ}} \\ N \ar[rr]_{_{id_{N}}} \ar@{-->}[ur]^{S_{λ}} &  & N}

2Typewriter’s note: Haagerup used this technique to give a new proof of the uniqueness of injective type III_{λ} (0 < λ < 1) factor. This result has been published in [Haa89].

Documenta Mathematica 21 (2016) 1193–1226

Uniqueness of the Injective III_{1} Factor 1195

through the hyperfinite factor R of type II_{1}, such that (S_{λ})_{λ∈Λ} and (T_{λ})_{λ∈Λ} are nets of normal unital completely positive maps, and for a fixed normal faithful state φ on N (chosen prior to S_{λ} and T_{λ}), there exist normal fatihful states (ψ_{λ})_{λ∈Λ} on R, such that for all t ∈ ℝ and λ ∈ Λ,

φ ∘ T_{λ} = ψ_{λ}, ψ_{λ} ∘ S_{λ} = φ,
σ_{t}^{φ} ∘ T_{λ} = T_{λ} ∘ σ_{t}^{ψ_{λ}},
σ_{t}^{ψ_{λ}} ∘ S_{λ} = S_{λ} ∘ σ_{t}^{φ},

and ∥S_{λ} ∘ T_{λ}(x) − x∥_{φ} →^{λ→∞} 0 for all x ∈ N, where ∥y∥_{φ} := φ(y^{∗}y)^{\frac{1}{2}} (y ∈ N).

\underline{Step 2} From Step 1, we deduce that a certain normal faithful state φ (ℚ-stable state defined in §4) on an injective factor N of type III_{1} has the following property: for any finite set of unitaries u_{1}, …, u_{n} in N and for every γ, δ > 0, there exists a finite-dimensional subfactor F of N such that

φ = φ|_{F} ⊗ φ|_{F^{c}},

and such that there exist unitaries v_{1}, …, v_{n} in F and a unital completely positive map T: F → N such that

φ ∘ T = φ,
∥σ_{t}^{φ} ∘ T − T ∘ σ_{t}^{φ|_{F}} ∥ ≤ γ|t|, t ∈ ℝ,

and

∥T (v_{k}) − u_{k}∥_{φ} < δ, k = 1, …, n.

\underline{Step 3} We prove that if N, φ, F, u_{1}, …, u_{n}, v_{1}, …, v_{n} are as in Step 2, then for every σstrong neighborhood V of 0 in N, there exists a finite set of operators a_{1}, …, a_{p} in N such that

(a) ∑_{i=1}^{p} a_{i}^{∗}a_{i} ∈ 1 + V and ∑_{i=1}^{p} a_{i}^{∗}a_{i} ≤ 1,

(b) ε_{F,φ} (∑_{i=1}^{p} a_{i}a_{i}^{∗}) ∈ 1 + V and ε_{F,φ} (∑_{i=1}^{p} a_{i}a_{i}^{∗}) ≤ 1,

(c) ∑_{i=1}^{p} ∥a_{i}ξ_{φ} − ξ_{φ}a_{i}∥^{2} < δ′,

(d) ∑_{i=1}^{p} ∥a_{i}u_{k} − v_{k}a_{i}∥_{φ}^{2} < δ′, k = 1, …, n.

Here ξ_{φ} denotes the unique representing vector of φ in a natural cone. The above δ′ > 0 depends on γ and δ in Step 2, and δ′ is small when γ and δ are small. Here, ε_{F,φ} is the φ-invariant conditional expectation of N onto F. Moreover in (c), the standard Hilbert space H of N is regarded as a Hilbert N-bimodule, by putting ηa := Ja^{∗}Jη (a ∈ N, η ∈ H).

Assume now that the bicentralizer of any normal faithful state on N is trivial. Then by an averaging argument, we can exchange (b) by

(b’) ∑_{i=1}^{p} a_{i}a_{i}^{∗} ∈ 1 + V and ∑_{i=1}^{p} a_{i}a_{i}^{∗} ≤ 1.

\underline{Step 4} From (a), (b’), (c) and (d) above, we derive that there exists a unitary operator w ∈ N such that

∥wξ_{φ} − ξ_{φ}w∥ < ε

and

∥wu_{k} − v_{k}w∥_{φ} < ε, k = 1, …, n,

where ε is small when δ′ is small and V is a small σ-strong neighborhood of 0 in N. The key part of Step 4 is a theorem about general Hilbert N-bimodules, which was proved in [Haa89].

\underline{Step 5} From Step 4, we get that for every finite set of unitaries u_{1}, …, u_{n} ∈ N and every ε > 0, there exists a finite dimensional subfactor F_{1} (namely w^{∗}F w) of N and n unitaries v′_{1}, …, v′_{n} in F_{1} (namely w^{∗}v_{k}w, k = 1, …, n), such that

(i) ∥v′_{k} − v_{k}∥_{φ} < ε

and

(ii) ∥φ − φ|_{F_{1}} ⊗ φ|_{F_{1}^{c}}∥ < ε.

The last inequality follows from the fact, that when w almost commutes with ξ_{φ}, it almost commutes with φ too. The properties (i) and (ii) above show that φ satisfies the product condition of Connes–Woods [CW85] and thus N is an ITPFI factor. But it is well-known that R_{∞} is the only ITPFI factor of type III_{1} (cf. [AW68] and [Con73]).

2 Preliminaries

2.1 Notation

We use M, N, … to denote von Neumann algebras and ξ, η, … to denote vectors in a Hilbert space. Let M be a von Neumann algebra. U(M) denotes the unitary group of M. For a faithful normal state φ on M, we denote by Δ_{φ} (resp. J_{φ}) the modular operator (resp. modular conjugation operator) associated with φ, and the modular automorphism group of φ is denoted by σ^{φ}. The norm ∥x∥_{φ} = φ(x^{∗}x)^{\frac{1}{2}} defines the strong operator topology (SOT) on the unit ball of M. The centralizer of φ is denoted by M_{φ}.

2.2 Connes–Woods’ characterization of ITPFI factors

Recall that a von Neumann algebra M with separable predual is called hyperfinite if there exists an increasing sequence M_{1} ⊂ M_{2} ⊂ ⋯ of finite-dimensional *-subalgebras such that M = (⋃_{n=1}^{∞} M_{n})′′. A factor M is called an Araki– Woods factor or an ITPFI (infinite tensor product of factors of type I) factor, if it is isomorphic to the factor of the form

⨂_{i∈I}(M_{i}, φ_{i}),

where I is a countable infinite set and each M_{i} (resp. φ_{i}) is a σ-finite type I factor (resp. a faithful normal state). Araki and Woods classified most ITPFI factors:

Theorem 2.1 ([AW68]). There exists a unique ITPFI factor with separable predual for each type I_{∞}, II_{1}, II_{∞} and III_{λ}, λ ∈ (0, 1]. In particular, all ITPFI factors of type III_{1} are isomorphic to

R_{∞} := ⨂_{n∈ℕ}(M_{3}(ℂ), Tr(ρ · )),

where ρ := \frac{1}{1+λ+μ}diag(1, λ, μ) and 0 < λ, μ satisfies \frac{log λ}{log μ} ∉ ℚ.

It is clear that an ITPFI factor with separable predual is hyperfinite. The converese is also true for factors not of type III_{0}, but false in general. Namely, Connes–Woods [CW85] characterized hyperfinite factors of type III_{0} with separable predual which are isomorphic to ITPFI factors by the approximate transitivity of their flow of weights, while the existence of hyperfinite factors of type III_{0} with separable predual which are not isomorphic to ITPFI factors had been shown in [Con72]. Let N be a von Neumann algebra, and let F be a finite dimensional subfactor of N with relative commutant F^{c} := F′ ∩ N in N. Then it is elementary to check, that the map

∑_{i=1}^{n} x_{i} ⊗ y_{i} ↦ ∑_{i=1}^{n} x_{i}y_{i}, x_{i} ∈ F, y_{i} ∈ F^{c} (1 ≤ i ≤ n)

is an isomorphism of F ⊗ F^{c} onto N. If ω_{1} is a normal state on F and ω_{2} is a normal state on F^{c}, we let ω_{1} ⊗ ω_{2} denote the corresponding state on N, i.e.,

(ω_{1} ⊗ ω_{2})(xy) = ω_{1}(x)ω_{2}(y), x ∈ F, y ∈ F^{c}.

In our proof of

[N injective III_{1} and B_{φ} = ℂ1] ⇒ N ≅ R_{∞},

we shall need the following criterion for a factor to be ITPFI:

Proposition 2.2 ([CW85, Lemma 7.6]). Let N be a factor on a separable Hilbert space. Then N is ITPFI if and only if N admits a normal faithful state φ with the following property: for every finite set x_{1}, …, x_{n} of operators in N, for every ε > 0, and every strong* neighborhood V of 0 in N, there exists a finite dimensional subfactor F of N, such that

x_{k} ∈ F + V, k = 1, …, n

and

∥φ − φ|_{F} ⊗ φ|_{F^{c}}∥ < ε.

2.3 Bicentralizers on type III_{1} factors

In this subsection, we recall Connes’ bicentralizers. Let M be a σ-finite von Neumann algebra, and let φ be a normal faithful state on M. We denote by AC(φ) the set of all norm-bounded sequences (x_{n})_{n=1}^{∞} in M such that lim_{n→∞} ∥φx_{n} − x_{n}φ∥ = 0 holds.

Definition 2.3 (Connes). The bicentralizer of φ is the set B_{φ} of all x ∈ M such that lim_{n→∞} ∥xa_{n} − a_{n}x∥_{φ} = 0 holds for all (a_{n})_{n=1}^{∞} ∈ AC(φ).

Since B_{φ} is a von Neumann subalgebra of M [Haa87, Proposition 1.3], it holds that lim_{n→∞} ∥xa_{n} − a_{n}x∥_{φ}^{♯} = 0. It was conjectured by Connes that for all factors of type III_{1} with separable predual, the bicentralizer B_{φ} of any normal faithful state φ on M is trivial, i.e., B_{φ} = ℂ1 holds. This is still an open problem. We will need the following result on type III_{1} factors, known as the Connes–Størmer transitivity:

Theorem 2.4 ([CS78]). Let M be a type III_{1} factor with separable predual. Then for every faithful normal states φ, ψ on M and ε > 0, there exists a unitary u ∈ U(M) such that ∥uφu^{∗} − ψ∥ < ε holds.

Connes showed that by the Connes-Størmer transitivity, for a type III_{1} factor M with separable predual, the triviality of B_{φ} for one fixed faithful normal state φ on M implies the triviality of B_{ψ} for every faithful normal state ψ (see [Haa87, Corollary 1.5] for the proof). He also showed that the triviality of the bicentralizer is equivalent to the following property (the proof is given in [Haa87, Proposition 1.3 (2)]):

Proposition 2.5 (Connes). Let M be a von Neumann algebra with a normal faithful state φ. Then B_{φ} = ℂ1 holds, if and only if the following condition is satisfied: for every a ∈ M and δ > 0,

\overline{conv}\{u^{∗}au; u ∈ U(M), ∥uφ − φu∥ ≤ δ\} ∩ ℂ1 ≠ ∅,

where \overline{conv} is the closure of the convex hull in the σ-weak topology.

We will use the following variant of Proposition 2.5.

Proposition 2.6. Let M be a type III_{1} factor with separable predual, and let φ be a normal faithful state on M whose modular automorphism group σ^{φ} leaves a finite-dimensional subfactor F globally invariant. Let ε_{F,φ}: M → F be the normal faithful φ-preserving conditional expectation. Assume that B_{φ} = ℂ1. Then for every δ > 0 and a ∈ M, we have

ε_{F,φ}(a) ∈ \overline{conv}\{u^{∗}au; u ∈ U(F^{c}), ∥uξ_{φ} − ξ_{φ}u∥ ≤ δ\}.  (1)

Here, ξ_{φ} is the representing vector of φ in the natural cone.

Proof. The proof is essentially the same as Proposition 2.5, so we only indicate the outline. Note that by Araki-Powers-Størmer inequality, for every u ∈ U(M) one has:

∥ξ_{φ} − uξ_{φ}u^{∗}∥^{2} ≤ ∥φ − uφu^{∗}∥ ≤ ∥ξ_{φ} − uξ_{φ}u^{∗}∥ · ∥ξ_{φ} + uξ_{φ}u^{∗}∥.

Therefore in the arguments below, we may replace the condition “∥uξ_{φ}−ξ_{φ}u∥ ≤ δ” in Proposition 2.6 with the condition “∥uφ − φu∥ ≤ δ”, as we take δ > 0 to be arbitrarily small. As was pointed out in [Haa87, Remark 1.4], it follows from the proof of Proposition 2.5 that the condition B_{φ} = ℂ1 is equivalent to the next condition that for all a ∈ M and δ > 0,

φ(a)1 ∈ ⋂_{δ>0} \overline{conv}\{u^{∗}au; u ∈ U(M), ∥uξ_{φ} − ξ_{φ}u∥ < δ\}.  (2)

Let a ∈ M. Since M ≅ F ⊗ F^{c} with φ = φ|_{F} ⊗ φ|_{F^{c}}, we may now apply (2) to F^{c}(≅ M) and φ|_{F^{c}} to obtain

ε_{F,φ}(a) = id_{F} ⊗ φ|_{F^{c}}(a) ∈ \overline{conv}\{u^{∗}au; u ∈ U(F^{c}), ∥uξ_{φ} − ξ_{φ}u∥ ≤ δ\}.

Note that we used the fact that ∥φu − uφ∥ = ∥ψu − uψ∥, where ψ := φ|_{F^{c}} and u ∈ U(F^{c}) thanks to the existence of a normal faithful φ-preserving conditional expectation from M onto F^{c}.

2.4 Almost unitary equivalence in Hilbert N-bimodules

We recall a result about almost unitary equivalence in Hilbert bimodules established in [Haa89] which is a generalization of [Haa85, Theorem 4.2]. Let N be a von Neumann algbera, and H be a normal Hilbert N-bimodule, i.e., H is a Hilbert space on which there are defined left and right actions by elements from N:

(x, ξ) ↦ xξ, (x, ξ) ↦ ξx, x ∈ N, ξ ∈ H

such that the above maps N × H → H are bilinear and

(xξ)y = x(ξy), x, y ∈ N, ξ ∈ H.

Moreover, x ↦ L_{x}, where L_{x}ξ := xξ (ξ ∈ H) is a normal unital ∗- homomorphism, and x ↦ R_{x}, where R_{x}ξ := ξx (ξ ∈ H) is a normal unital ∗-antihomomorphism.

Definition 2.7. Let N be a von Neumann algebra, let (N, H) be a normal Hilbert N-bimodule, and let δ ∈ ℝ_{+}. Two n-tuples (ξ_{1}, …, ξ_{n}) and (η_{1}, …, η_{n}) of unit vectors in H are called δ-related, if there exists a family (a_{i})_{i∈I} of operators in N, such that

∑_{i∈I} a_{i}^{∗}a_{i} = ∑_{i∈I} a_{i}a_{i}^{∗} = 1

and

∑_{i∈I} ∥a_{i}ξ_{k} − η_{k}a_{i}∥^{2} < δ, k = 1, …, n.

We will use the following result which relates the δ-relatedness to approximate unitary equivalence in Hilbert N-bimodules:

Theorem 2.8 ([Haa89, Theorem 2.3]). For every n ∈ ℕ and ε > 0, there exists a δ = δ(n, ε) > 0, such that for all von Neumann algebra N and δrelated n-tuples (ξ_{1}, …, ξ_{n}) and (η_{1}, …, η_{n}) of unit vectors in a normal Hilbert N-bimodule H, there exists a unitary u ∈ U(N) such that

∥uξ_{k} − η_{k}u∥ < ε, k = 1, …, n.

Remark 2.9. As can be seen in the proof of [Haa89, Theorem 2.3], in order to show that the conclusion of Theorem 2.8 holds, it suffices to show the following: for every σ-strong neighborhood V of 0 in N, there exist a_{1}, …, a_{p} ∈ N such that

∑_{i=1}^{p} ∥a_{i}ξ_{k} − η_{k}a_{i}∥^{2} < δ,  k = 1, …, n  (3)

∑_{i=1}^{p} a_{i}^{∗}a_{i} ≤ 1, ∑_{i=1}^{p} a_{i}a_{i}^{∗} ≤ 1  (4)

∑_{i=1}^{p} a_{i}^{∗}a_{i} ∈ 1 + V, ∑_{i=1}^{p} a_{i}a_{i}^{∗} ∈ 1 + V.  (5)

This is because we can obtain the conclusions of [Haa89, Lemma 2.5] out of (3),

(4) and (5), which is enough to prove Theorem 2.8. We will use this variant in the proof of Lemma 5.6.

3 Completely positive maps from m×m-matrices into an injective factor of type III_{1}

The main result of this section is:

Theorem 3.1. Let N be an injective factor of type III_{1} with separable predual, and let φ be a faithful normal state on N. Then for every finite set u_{1}, …, u_{n} of unitaries in N, and every ε, δ > 0, there exists m ∈ ℕ, a unital completely positive map T: M_{m}(ℂ) → N, and n unitaries v_{1}, …, v_{n} in M_{m}(ℂ), such that ψ = φ ∘ T is a normal faithful state on M_{m}(ℂ), and

∥σ_{t}^{φ} ∘ T − T ∘ σ_{t}^{ψ}∥ ≤ δ|t|, t ∈ ℝ,
∥T (v_{k}) − u_{k}∥_{φ} < ε, k = 1, …, n.

In the following we let M = N ⋊_{σ^{φ}} ℝ be the crossed product of N by σ^{φ} with generators π_{σ^{φ}}(x) (x ∈ N) and λ(s) (s ∈ ℝ). We identify π_{σ^{φ}}(x) with x ∈ N. Let a be the (unbounded) self-adjoint operator for which λ(s) = exp(isa) (s ∈ ℝ). For f ∈ L^{1}(ℝ), we define the Fourier transform f̂ by

f̂(s) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−ist}f(t) dt, s ∈ ℝ.

In the sequel, von Neumann algebra-valued integrals are understood to be the σ-weak sense. Let (θ_{s}^{φ})_{s∈ℝ} be the dual action of σ^{φ} on M. By [Haa79-2], there exists a normal faithful semifinite operator-valued weight P: M_{+} → N̂_{+} (N̂_{+} is the extended positive part of N) given by

P (x) = ∫_{−∞}^{∞} θ_{s}^{φ}(x) ds, x ∈ M_{+}.  (6)

Following [CT77], if we put

𝔪 := span {x ∈ M_{+}; sup_{c>0} ∥∫_{−c}^{c} θ_{t}^{φ}(x) dt∥ < ∞} ,

then the formula (6) for x ∈ 𝔪 makes sense and P (x) ∈ N. Moreover, 𝔪 ∋ x ↦ P (x) ∈ N defines a positive linear map. For all x ∈ 𝔪, the σ-weak integral ∫_{−c}^{c} θ_{t}^{φ}(x) dt is σ-strongly convergent as c → ∞. The range of P is contained in π_{σ^{φ}}(N), because π_{σ^{φ}}(N) is the fixed point algebra in M under the dual action.

Lemma 3.2. Let t ↦ x(t) be a σ-strongly* continuous function from ℝ to N such that t ↦ ∥x(t)∥ is in L^{1}(ℝ) ∩ L^{∞}(ℝ). Put

x := ∫_{−∞}^{∞} λ(t)x(t) dt ∈ M.

Then x^{∗}x ∈ 𝔪, and

P (x^{∗}x) = 2π ∫_{−∞}^{∞} x(t)^{∗}x(t) dt.

Proof. Note first, that

x^{∗}x = ∫∫_{ℝ^{2}} x(s)^{∗}λ(t − s)x(t) dsdt
= ∫∫_{ℝ^{2}} x(s)^{∗}λ(t)x(s + t) dsdt.

Put f_{n}(s) = e^{−s^{2}/(4n)} (s ∈ ℝ), and

g_{n}(t) = \frac{1}{2π} ∫_{−∞}^{∞} f_{n}(s)e^{−its} ds = (\frac{n}{π})^{\frac{1}{2}} e^{−nt^{2}} (t ∈ ℝ).

Using that θ_{s}^{φ}(y) = y (s ∈ ℝ, y ∈ N), θ_{s}^{φ}(λ(t)) = e^{−ist}λ(t) (s, t ∈ ℝ)^{3} and the Fubini Theorem, we have for every ψ ∈ M_{∗},

⟨ψ, ∫_{−∞}^{∞} θ_{u}^{φ}(x^{∗}x)f_{n}(u) du⟩ =
= ∫_{−∞}^{∞} ∫_{−∞}^{∞} ∫_{−∞}^{∞} e^{−itu}f_{n}(u)ψ(x(s)^{∗}λ(t)x(s + t)) dsdtdu
= ∫_{−∞}^{∞} g_{n}(t) (∫_{−∞}^{∞} 2πψ(x(s)^{∗}λ(t)x(s + t))ds) dt.

Since t ↦ ∫_{−∞}^{∞} ψ(x(s)^{∗}λ(t)x(s + t))ds is in C_{0}(ℝ) and g_{n} →^{n→∞} δ_{0} (weak^{∗} in C_{0}(ℝ)^{∗}), we have

lim_{n→∞}⟨ψ, ∫_{−∞}^{∞} θ_{u}^{φ}(x^{∗}x)f_{n}(u) du⟩ = ⟨ψ, 2π ∫_{−∞}^{∞} x(s)^{∗}x(s) ds⟩.

Since ψ ∈ M_{∗} is arbitrary, θ_{u}^{φ}(x^{∗}x) ≥ 0 and f_{n} ↗ 1 uniformly on compact sets, it follows that

lim_{n→∞} ∫_{−∞}^{∞} θ_{u}^{φ}(x^{∗}x)f_{n}(u)du = 2π ∫_{−∞}^{∞} x(s)^{∗}x(s) ds (σ-strongly).

Therefore x^{∗}x ∈ 𝔪, and P (x^{∗}x) = 2π ∫_{−∞}^{∞} x(t)^{∗}x(t) dt.

Lemma 3.3. Let a be the (unbounded) self-adjoint operator affiliated with M for which exp(ita) = λ(t) (t ∈ ℝ) holds. Let α > 0, and let e_{α} be the spectral projection of the operator a corresponding to the interval [0, α]. Then for each x ∈ N, one has e_{α}xe_{α} ∈ 𝔪 and

P (e_{α}xe_{α}) = ∫_{−∞}^{∞} σ_{t}^{φ}(x)\frac{1 − cos αt}{πt^{2}} dt, x ∈ N.  (7)

Proof. It is sufficient to consider the case x ≥ 0, so we can assume that x = y^{∗}y (y ∈ N). For f ∈ L^{1}(ℝ) ∩ L^{∞}(ℝ) ∩ C(ℝ), we have

yf̂(a) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} yλ(−t)f(t)dt = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} λ(−t)σ_{t}^{φ}(y)f(t) dt

= \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} λ(t)σ_{−t}^{φ}(y)f(−t) dt.

3Typewriter’s note: Haagerup used the convention θ_{s}^{φ}(λ(t)) = e^{ist}λ(t). However, since the negative sign convention is widely accepted, we decided to change the definition.

Documenta Mathematica 21 (2016) 1193–1226

Hence by Lemma 3.2, f̂(a)^{∗}xf̂(a) ∈ 𝔪, and

P (f̂(a)^{∗}xf̂(a)) = ∫_{−∞}^{∞} σ_{t}^{φ}(x)|f(t)|^{2} dt.

For n > \frac{2}{α}, let g_{n} be the continuous function on ℝ for which

g_{n}(t) = 0, t ≤ 0, t ≥ α,
g_{n}(t) = 1, t ∈ [\frac{1}{n}, α − \frac{1}{n}],

and for which the graph is a straight line on [0, \frac{1}{n}] and [α − \frac{1}{n}, \frac{1}{n}]. Since a has no point spectrum, g_{n}(a) ↗ e_{α} (n → ∞). It is elementary to check that each g_{n} is of the form g_{n} = f̂_{n} for a function f_{n} ∈ L^{1}(ℝ) ∩ L^{∞}(ℝ) ∩ C(ℝ) (use, for example, the fact that g_{n} = n1_{[0, \frac{1}{n}]} ∗ 1_{[0,α− \frac{1}{n}]}). Hence g_{n}(a)^{2} ∈ 𝔪, and by the

Plancherel Theorem, we get

P (g_{n}(a)^{2}) = P (f̂_{n}(a)^{∗}f̂_{n}(a)) = ∥f_{n}∥_{2}^{2}1 = ∥f̂_{n}∥_{2}^{2}1.

Since sup_{n} ∥f̂_{n}∥_{2}^{2} = α < ∞, we have e_{α} ∈ 𝔪 and P (e_{α}) = α1. Therefore e_{α}Me_{α} ⊆ 𝔪, and the restriction of P to e_{α}Me_{α} is a positive normal map. Hence for x ∈ N,

P (e_{α}xe_{α}) = lim_{n→∞} P (g_{n}(a)xg_{n}(a)) = lim_{n→∞} ∫_{−∞}^{∞} σ_{t}^{φ}(x)|f_{n}(t)|^{2} dt (σ-strongly).

Since ∥g_{n} −1_{[0,α]}∥_{2} →^{n→∞} 0, it follows that f_{n} converges in L^{2}(ℝ) to the function

f(t) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} 1_{[0,α]}(s)e^{ist} ds

= −\frac{i}{t\sqrt{2π}} (e^{iαt} − 1).

Hence |f_{n}|^{2} →^{n→∞} |f|^{2} in L^{1}(ℝ), with |f|^{2}(t) = \frac{1}{πt^{2}} (1−cos αt) (t ∈ ℝ). Therefore (7) holds.

Lemma 3.4. Let N be an injective factor of type III_{1} with separable predual, φ be a faithful normal state on N and let R be the hyperfinite II_{1} factor with tracial state τ. For every finite set u_{1}, …, u_{n} of unitaries in N and every ε > 0, there exist x_{1}, …, x_{n} in the unit ball of R, a normal unital completely positive map T: R → N, such that ψ = φ ∘ T is a normal faithful state on R, and

σ_{t}^{φ} ∘ T = T ∘ σ_{t}^{ψ},  t ∈ ℝ,  (8)
∥T (x_{k}) − u_{k}∥_{φ} < ε,  k = 1, …, n.  (9)

Moreover, the spectrum of h = dψ/dτ is a closed interval [λ_{1}, λ_{2}], 0 < λ_{1} < λ_{2} < ∞, and h has no eigenvalues.

Proof. ^{4} Let M = N ⋊_{σ^{φ}} ℝ. By [Tak73-2], M has a normal faithful semifinite trace τ, such that

τ ∘ θ_{s}^{φ} = e^{−s}τ (s ∈ ℝ).

The trace τ can be constructed in the following way: Let φ̃ be the dual weight of φ on M (cf. [Haa79]). Let a be the self-adjoint operator for which exp(ita) = λ(t) (t ∈ ℝ). Then a is affiliated with the centralizer Mφ̃ of φ̃ and

τ = φ̃(e^{−a} · )

in the sense of Pedersen-Takesaki [PT73]. By [Haa79-2], φ̃ is on the subspace 𝔪 given by

φ̃(x) = φ ∘ P (x), x ∈ 𝔪.

Let α > 0, and let e_{α} = 1_{[0,α]}(a). Then by Lemma 3.3, e_{α} ∈ 𝔪 and P (e_{α}) = α1. Hence φ̃|_{e_{α}Me_{α}} is a positive normal functional, and φ̃(e_{α}) = α. Finally,

τ(e_{α}) = φ̃(e^{−a}e_{α}) = ∫_{0}^{α} e^{−t} dt = 1 − e^{−α} < ∞,

because

e^{−a}e_{α} = ∫_{0}^{α} e^{−λ} de(λ),

where

a = ∫_{−∞}^{∞} λ de(λ)

is the spectral resolution of a, and dφ̃(e(λ)) = dλ. Since N is of type III_{1}, M is a type II_{∞} factor, and therefore e_{α}Me_{α} is a II_{1} factor. Moreover, the injectivity of N implies that M is also injective, so that e_{α}Me_{α} is isomorphic to the hyperfinite factor R of type II_{1} by [Con76].

Claim. For any x ∈ N, we have

lim_{α→∞} ∥\frac{1}{α}P (e_{α}xe_{α}) − x∥_{φ} = 0.

This follows from a basic property of the Fejér kernel (see e.g., [Kat68, Chapters I and VI]), but we include the proof for completeness. Let ε > 0. Choose t_{0} > 0 small enough so that ∥σ_{t}^{φ}(x) − x∥_{φ} ≤ ε for all t ∈ [−t_{0}, t_{0}]. Moreover,

by |\frac{1 − cos(αt)}{παt^{2}} | ≤ \frac{2}{πα} · \frac{1}{t^{2}} , we have

lim_{α→∞} ∫_{|t|≥t_{0}} \frac{1 − cos(αt)}{παt^{2}} dt = 0.

4Typewriter’s note: Since some pages were missing from the original notes, we could not find all parts of the proofs of Lemma 3.4 and Theorem 3.1. We include the following proof for the reader’s convenience.

Documenta Mathematica 21 (2016) 1193–1226

By Lemma 3.3, we have

lim sup_{α→∞} ∥\frac{1}{α}P (e_{α}xe_{α}) − x∥_{φ} ≤ lim sup_{α→∞} ∫_{−∞}^{∞} ∥σ_{t}^{φ}(x) − x∥_{φ} \frac{1 − cos(αt)}{παt^{2}} dt

≤ ε + 2∥x∥_{φ} lim sup_{α→∞} ∫_{|t|≥t_{0}} \frac{1 − cos(αt)}{παt^{2}} dt

= ε.

Since ε > 0 is arbitrary, we obtain the conclusion.

Let n ≥ 1, u_{1}, …, u_{n} ∈ U(N) and ε, δ > 0 be given. By the above claim, we may choose α > 0 large enough so that

∥\frac{1}{α}P (e_{α}u_{k}e_{α}) − u_{k}∥_{φ} < ε,  1 ≤ k ≤ n.  (10)

Define T := α^{−1}P |_{e_{α}Me_{α}}: R = e_{α}Me_{α} → N and

ψ := φ ∘ T = \frac{1}{α}φ ∘ P (e_{α} · e_{α}) = \frac{1}{α}φ̃(e_{α} · e_{α}).

Then T is a normal unital completely positive map, and ψ is a normal faithful state on R. By (10) we have

∥T (x_{k}) − u_{k}∥_{φ} < ε, 1 ≤ k ≤ n,

where x_{k} := e_{α}u_{k}e_{α} (1 ≤ k ≤ n) are in the unit ball of R. Moreover, since

e_{α} ∈ Mφ̃ and since σ_{t}^{φ} ∘ P = P ∘ σφ̃ ; t (t ∈ ℝ), we have σ_{t}^{φ} ∘ T = T ∘ σ_{t}^{ψ} (t ∈ ℝ).

By construction, we have

h := \frac{dψ}{dτ} = \frac{1 − e^{−α}}{α} exp(a)e_{α},

which has no atoms and the spectrum of h is a closed bounded interval in ℝ_{+}^{∗} = (0, ∞).

Lemma 3.5. Let B ⊂ A be an inclusion of unital C^{∗}-algebras and E: A → B be a unital completely positive map. Let h ∈ A be a self-adjoint element with σ(h) ⊂ [λ_{1}, λ_{2}], where σ( · ) denotes the spectrum and λ_{1} < λ_{2} are reals. Then σ(E(h)) ⊂ [λ_{1}, λ_{2}].

Proof. Let λ < λ_{1}. Then h−λ is positive and invertible. Take a nonzero x ∈ A such that (h − λ)^{\frac{1}{2}} x = 1, so that E((h − λ)^{\frac{1}{2}} xx^{∗}(h − λ)^{\frac{1}{2}} ) = 1. The left hand side is dominated by ∥x∥^{2}E(h − λ), whence E(h − λ) ≥ ∥x∥^{−2}1, showing that E(h) − λ1 is invertible. Thus λ ∉ σ(E(h)). Similarly, σ(E(h)) ∩ (λ_{2}, ∞) = ∅ holds. Therefore σ(E(h)) ⊂ [λ_{1}, λ_{2}].

Proof of Theorem 3.1. We may assume that 0 < ε < 1. By Lemma 3.4, there exist a normal unital completely positive map T: R → N and x_{1}, …, x_{n} in the unit ball of R satisfying σ_{t}^{φ} ∘ T = T ∘ σ_{t}^{ψ}, (t ∈ ℝ), where ψ := φ ∘ T: R → N and

∥T (x_{k}) − u_{k}∥_{φ} < \frac{ε^{2}}{16}, 1 ≤ k ≤ n.  (11)

Let h := dψ/dτ ∈ R_{+}. Then by Lemma 3.4, σ(h) = [λ_{1}, λ_{2}] for some positive reals λ_{1} < λ_{2} and h does not have a point spectrum. Since log( · ) is continuous on [λ_{1}, λ_{2}], continuous functional calculus guarantees that there exists δ′ > 0 such that for all a, b ∈ R_{+}, we have the following implication

σ(a), σ(b) ⊂ [λ_{1}, λ_{2}] and ∥a − b∥ < δ′ ⇒ ∥ log a − log b∥ < \frac{δ}{4}.  (12)

By using the spectral decomposition of h, we can choose a partition of unity {p_{i}}_{i=1}^{ℓ} in R and {μ_{i}}_{i=1}^{ℓ} in ℝ_{+}^{∗} such that

τ(p_{i}) = \frac{1}{ℓ}, hp_{i} = p_{i}h,

∥(log h)p_{i} − (log μ_{i})p_{i}∥ < \frac{1}{4}δ,
∥hp_{i} − μ_{i}p_{i}∥ < δ′,

for all (1 ≤ i ≤ ℓ). Let h_{0} := ∑_{i=1}^{ℓ} μ_{i}p_{i}, and we have

∥h − h_{0}∥ < δ′ and ∥ log h − log h_{0}∥ < \frac{1}{4}δ.  (13)

Moreover, we may arrange {μ_{i}}_{i=1}^{ℓ} so that h_{0} = ∑_{i=1}^{ℓ} μ_{i}p_{i} satisfies

σ(h_{0}) ⊂ [λ_{1}, λ_{2}].  (14)

Since R is hyperfinite, there exists a type I subfactor F of R so that p_{i} ∈ F (1 ≤ i ≤ ℓ) and

∥x_{k} − E_{F} (x_{k})∥_{φ} < \frac{ε^{2}}{16}, 1 ≤ k ≤ n,  (15)

where E_{F} : R → F denotes the τ-preserving conditional expectation. Put T_{F} := T |_{F} : F → N and y_{k} := E_{F} (x_{k}) (1 ≤ k ≤ n). Combining (11) and (15), for all 1 ≤ k ≤ n, we have (use the Schwarz inequality for completely positive maps)

∥T_{F} (y_{k}) − u_{k}∥_{φ} ≤ ∥T (y_{k}) − T (x_{k})∥_{φ} + ∥T (x_{k}) − u_{k}∥_{φ}
≤ ∥y_{k} − x_{k}∥_{ψ} + ∥T (x_{k}) − u_{k}∥_{φ}
< \frac{ε^{2}}{8} .  (16)

Then we follow the argument of [Haa89, Lemma 6.2]. Take v_{1}, …, v_{k} ∈ U(F) such that

y_{k} = v_{k}|y_{k}|, |y_{k}| := (y_{k}^{∗}y_{k})^{\frac{1}{2}} , 1 ≤ k ≤ n.

Then again by the Schwarz inequality for completely positive maps and (16),

∥y_{k}∥_{ψ} ≥ ∥T_{F} (y_{k})∥_{φ} > ∥u_{k}∥_{φ} − \frac{ε^{2}}{8} .

Since ∥(y_{k}^{∗}y_{k})^{\frac{1}{2}} ∥_{ψ} = ∥y_{k}∥_{ψ} and |y_{k}|^{2} + (1 − |y_{k}|)^{2} ≤ 1 (because 0 ≤ |y_{k}| ≤ 1), we have

∥v_{k} − y_{k}∥_{ψ}^{2} = ∥1 − |y_{k}|∥_{ψ}^{2} ≤ 1 − ∥ |y_{k}| ∥_{ψ}^{2}

< 1 − (1 − \frac{ε^{2}}{8} )^{2}

< \frac{ε^{2}}{4} .

Therefore since ε^{2} < ε,

∥T_{F} (v_{k}) − u_{k}∥_{φ} ≤ ∥T_{F} (v_{k} − y_{k})∥_{φ} + ∥T_{F} (y_{k}) − u_{k}∥_{φ}

< ∥v_{k} − y_{k}∥_{ψ} + \frac{ε^{2}}{8}

< ε.

Next, set χ := τ(h_{0} · ) ∈ (R_{∗})_{+}. Note that σ_{t}^{χ|_{F}} = σ_{t}^{χ}|_{F} (t ∈ ℝ), since h_{0} ∈ F. Then by (13), we have

∥h^{it} − h_{0}^{it}∥ = ∥∫_{0}^{1} \frac{d}{ds}e^{ist log h}e^{i(1−s)t log h_{0}} ds∥
≤ ∫_{0}^{1} ∥e^{ist log h}(t log h − t log h_{0})e^{i(1−s)t log h_{0}}∥ ds
≤ ∥ log h − log h_{0}∥ |t|
≤ \frac{δ|t|}{4}.  (17)

On the other hand, h_{F} := dψ|_{F} /dτ|_{F} ∈ F_{+} is equal to E_{F} (h). Therefore by Lemma 3.5, σ(h_{F}) ⊂ [λ_{1}, λ_{2}]. Moreover, since E_{F} (h_{0}) = h_{0}, we have

∥h_{F} − h_{0}∥ = ∥E_{F} (h − h_{0})∥ ≤ ∥h − h_{0}∥ < δ′.

This shows by (12) and (14) that ∥ log h_{F} −log h_{0}∥ < \frac{δ}{4}. Therefore by the same argument, we have

∥h_{F}^{it} − h_{0}^{it}∥ ≤ \frac{δ|t|}{4}, t ∈ ℝ.  (18)

For all t ∈ ℝ and x ∈ F,

∥σ_{t}^{φ} ∘ T_{F} (x) − T_{F} ∘ σ_{t}^{ψ|_{F}} (x)∥ = ∥T (σ_{t}^{ψ}(x) − σ_{t}^{ψ|_{F}} (x))∥
≤ ∥σ_{t}^{ψ}(x) − σ_{t}^{ψ|_{F}} (x)∥
≤ ∥σ_{t}^{ψ}(x) − σ_{t}^{χ}(x)∥ + ∥σ_{t}^{χ|_{F}} (x) − σ_{t}^{ψ|_{F}} (x)∥.

By (17), we have

∥σ_{t}^{ψ}(x) − σ_{t}^{χ}(x)∥ = ∥h^{it}xh^{−it} − h_{0}^{it}xh_{0}^{−it}∥

≤ (∥h^{it} − h_{0}^{it}∥ + ∥h^{−it} − h_{0}^{−it}∥)∥x∥

≤ \frac{1}{2}δ|t|∥x∥.

Similarly, by (18),

∥σ_{t}^{χ|_{F}} (x) − σ_{t}^{ψ|_{F}} (x)∥ ≤ (∥h_{0}^{it} − h_{F}^{it}∥ + ∥h_{0}^{−it} − h_{F}^{−it}∥)∥x∥

≤ \frac{1}{2}δ|t|∥x∥.

These altogether imply that ∥σ_{t}^{φ} ∘ T_{F} (x) − T_{F} ∘ σ_{t}^{ψ|_{F}} (x)∥ ≤ δ|t|∥x∥.

4 ℚ-stable states on III_{1} factors

For technical reasons we shall consider a special class of normal faithful states, which we call ℚ-stable states, because they have nice properties with respect to certain operations involving rationals.

Definition 4.1. A normal faithful state φ on a von Neumann algebra N is called ℚ-stable, if for every m ∈ ℕ, there exist m isometries u_{1}, …, u_{m} ∈ N with orthogonal range projections, such that

∑_{i=1}^{m} u_{i}u_{i}^{∗} = 1,

φu_{i} = \frac{1}{m}u_{i}φ,  i = 1, …, m.

Theorem 4.2. Every factor of type III_{1} with separable predual has a ℚ-stable normal faithful state.

For the proof of Theorem 4.2, we shall need two lemmas:

Lemma 4.3. The Araki–Woods factor R_{∞} has a ℚ-stable normal faithful state.

Proof. Let R_{λ} (0 < λ < 1) be the Powers factor of type III_{λ}, and let φ_{λ} be the product state on R_{λ}. Then φ_{λ} is normal and faithful, and σ^{φ_{λ}} has period −2π/ log λ. Then the centralizer (R_{λ})_{φ_{λ}} is a type II_{1} factor (cf. [Con73, Théorème 4.2.6]), and there exists an isometry u ∈ R_{λ} such that

σ_{t}^{φ_{λ}}(u) = λ^{it}u, t ∈ ℝ.

This implies that σ_{t}^{φ_{λ}}(uu^{∗}) = uu^{∗} (t ∈ ℝ), i.e., uu^{∗} ∈ (R_{λ})_{φ_{λ}}. Moreover, by [Tak73, Lemma 1.6], we have

φ_{λ}u = λuφ_{λ},

and hence φ_{λ}(uu^{∗}) = (φ_{λ}u)(u^{∗}) = λφ_{λ}(u^{∗}u) = λ. Assume now, that λ = \frac{1}{m}, m ∈ ℕ, m ≥ 2. Then we can choose m equivalent orthogonal projections p_{1}, …, p_{m} ∈ (R_{λ})_{φ_{λ}} with sum 1, such that p_{1} = uu^{∗}. Next, choose partial isometries v_{1}, …, v_{m} ∈ (R_{λ})_{φ_{λ}} such that

v_{i}^{∗}v_{i} = p_{1}, v_{i}v_{i}^{∗} = p_{i}, i = 1, …, m.

Put u_{i} = v_{i}u, i = 1, …, m. Then u_{1}, …, u_{m} are m isometries in R_{λ}, such that ∑_{i=1}^{m} u_{i}u_{i}^{∗} = 1, and φ_{λ}u_{i} = λu_{i}φ_{λ}, i = 1, …, m. Put now

(P, φ) = ⨂_{m=2}^{∞}(R_{\frac{1}{m}} , φ_{\frac{1}{m}}).

Then it is clear from the above computations, that φ is a ℚ-stable normal faithful state on P (observe that it is sufficient to consider m ≥ 2 case in Definition 4.1). Moreover, P is an ITPFI factor for which the the asymptotic ratio set r_{∞}(P) contains \{\frac{1}{m}; m ∈ ℕ\}. Since r_{∞}(P) ∩ ℝ_{+} is a closed subgroup of ℝ_{+}, we have r_{∞}(P) ⊇ ℝ_{+}. Therefore by Araki–Woods’ Theorem [AW68, Theorem 7.6], P ≅ R_{∞} holds.

Lemma 4.4. Let N be a factor of type III_{1} with separable predual. Then there exists a normal faithful conditional expectation of N onto a subfactor P isomorphic to R_{∞}.

Proof. ^{5} We can write R_{∞} as an infinite tensor product

R_{∞} = ⨂_{k=1}^{∞}(P_{k}, ω_{k}),

where each P_{k} is a copy of the 2 × 2 matrices M_{2}(ℂ) and (ω_{k})_{k=1}^{∞} is a sequence of normal faithful states on M_{2}(ℂ). Let ψ be a fixed normal faithful state on N. Since N is properly infinite, we have N ⊗ M_{2}(ℂ) ≅ N. Moreover, by Connes-Størmer transitivity theorem [CS78], we can choose a ∗-isomorphism Φ: N ⊗ M_{2}(ℂ) → N such that

∥(ψ ⊗ ω_{1}) ∘ Φ^{−1} − ψ∥ < \frac{1}{2}.

Put F_{1} = Φ(ℂ ⊗ M_{2}(ℂ)), and φ_{1} = (ψ ⊗ ω_{1}) ∘ Φ^{−1}. Then F_{1} is a type I_{2} subfactor of N. Moreover, it holds that N ≅ F_{1} ⊗ F_{1}^{c}, where F_{1}^{c} = F′_{1} ∩ N is the relative commutant, and φ_{1} = φ_{1}|_{F_{1}} ⊗ φ_{1}|_{F_{1}^{c}}. Moreover, we have

(F_{1}, φ_{1}|_{F_{1}}) ≅ (P_{1}, ω_{1}).

5Typewriter’s note: this result has been extended by Haagerup–Musat [HM09, Theorem 3.5], where the authors study more general embeddings of ITPFI type III factors into type III factors as the range of normal faithftul conditional expectations.

Documenta Mathematica 21 (2016) 1193–1226

Using the same arguments to the type III_{1} factor F_{1}^{c}, we can find a type I_{2}- subfactor F_{2} ⊂ F_{1}^{c}, a normal faithful state φ′_{2} on F_{1}^{c}, such that ∥φ′_{2} − φ_{1}|_{F_{1}^{c}}∥ < \frac{1}{4},

φ′_{2} = φ′_{2}|_{F_{2}} ⊗ φ′_{2}|_{(F_{1}⊗F_{2})^{c}},

and (F_{2}, φ′_{2}|_{F_{2}}) ≅ (P_{2}, ω_{2}). Thus, if we put φ_{2} = φ_{1}|_{F_{1}} ⊗ φ′_{2}, we have ∥φ_{1} − φ_{2}∥ < \frac{1}{4},

φ_{2} = φ_{2}|_{F_{1}} ⊗ φ_{2}|_{F_{2}} ⊗ φ_{2}|_{(F_{1}⊗F_{2})^{c}},

and

(F_{1}, φ_{2}|_{F_{1}}) ≅ (P_{1}, ω_{1}),
(F_{2}, φ_{2}|_{F_{2}}) ≅ (P_{2}, ω_{2}).

Proceeding in this way, we obtain a sequence (F_{k})_{k=1}^{∞} of mutually commuting type I_{2}-subfactors of N, and a sequence (φ_{k})_{k=1}^{∞} of normal faithful states on N, such that

∥φ_{k} − φ_{k−1}∥ < 2^{−k}, k ≥ 2,

and such that for fixed m ∈ ℕ:

φ_{m} = φ_{m}|_{F_{1}} ⊗ φ_{m}|_{F_{2}} ⊗ … ⊗ φ_{m}|_{F_{m}} ⊗ φ_{m}|_{(F_{1}⊗...⊗F_{m})^{c}},

and

(F_{i}, φ_{m}|_{F_{i}}) ≅ (P_{i}, ω_{i}) i = 1, …, m.

Let φ be the norm limit in N_{∗} of the sequence (φ_{k})_{k=1}^{∞}. Then φ is a normal state, but it can fail to be faithful. From the properties of φ_{k}, we have for all m ∈ ℕ,

φ = φ|_{F_{1}} ⊗ φ|_{F_{2}} ⊗ … ⊗ φ|_{F_{m}} ⊗ φ_{m}|_{(F_{1}⊗...⊗F_{m})^{c}},

and

(F_{m}, φ|_{F_{m}}) ≅ (P_{m}, ω_{m}).

Let r_{k} be the ratio between the largest and the smallest eigenvalues of dω_{k}/dTr. Let u ∈ U(P_{k}). We may assume that ω_{k} = Tr(h_{k} · ), h_{k} :=

\frac{1}{1+r_{k}} diag(r_{k}, 1) (r_{k} ≥ 1). Then if a = (\begin{matrix}x & y \\ z & w \end{matrix}) ∈ P_{k} is positive, then

uω_{k}u^{∗}(a) = Tr(h_{k}u^{∗}au) = Tr((u^{∗}au)^{\frac{1}{2}} h_{k}(u^{∗}au)^{\frac{1}{2}} )

≥ \frac{1}{1 + r_{k}} Tr(u^{∗}au) = \frac{x + w}{1 + r_{k}}

≥ r_{k}^{−1}(\frac{r_{k}x}{1 + r_{k}} + \frac{w}{1 + r_{k}})

= r_{k}^{−1}ω_{k}(a).

Similarly, uω_{k}u^{∗}(a) ≤ r_{k}ω_{k}(a) holds. This shows that

r_{k}^{−1}ω_{k} ≤ uω_{k}u^{∗} ≤ r_{k}ω_{k}.

Hence for all u ∈ U(F_{k}),

r_{k}^{−1}φ ≤ uφu^{∗} ≤ r_{k}φ.

Thus, φ and uφu^{∗} have the same support projection in N, i.e., with e = supp(φ), we have

ueu^{∗} = e, u ∈ U(F_{k}), k ∈ ℕ.

This shows that e ∈ (⋃_{k=1}^{∞} F_{k})′ ∩ N. Put G_{k} = eF_{k}. Then (G_{k})_{k=1}^{∞} is a sequence of commuting subfactors of eNe. Moreover, the restriction φ_{e} of φ to eNe is a normal faithful state on eNe, and

(G_{1} ⊗ … ⊗ G_{m}, φ_{e}|_{G_{1}⊗...⊗G_{m}}) ≅ ⨂_{k=1}^{m}(P_{k}, ω_{k})

for all m ∈ ℕ. Let P be the von Neumann algebra generated by ⋃_{k=1}^{∞} G_{k}. Then

(P, φ|_{P}) ≅ ⨂_{k=1}^{∞}(P_{k}, ω_{k}).

In particular, P ≅ R_{∞}. Moreover, since

φ_{e} = φ_{e}|_{G_{1}} ⊗ … φ_{e}|_{G_{m}} ⊗ φ_{e}|_{(G_{1}⊗...⊗G_{m})^{c}},

where (G_{1} ⊗ ⋯ ⊗ G_{m})^{c} denotes the relative commutant of ⋃_{k=1}^{m} G_{k} in eNe, we have

σ_{t}^{φ_{e}}(G_{1} ⊗ ⋯ ⊗ G_{m}) = G_{1} ⊗ ⋯ ⊗ G_{m}, t ∈ ℝ

for all m ∈ ℕ, and hence also σ_{t}^{φ_{e}}(P) = P, t ∈ ℝ. Thus by [Tak72], there exists a normal faithful conditional expectation of eNe onto P. This completes the proof, since eNe is isomorphic to N.

Proof of Theorem 4.2. Let N be a type III_{1} factor with separable predual. By Lemmata 4.3 and 4.4, we can choose a normal faithful conditional expectation E of N onto a subfactor P of N isomorphic to R_{∞}. Moreover, we can choose a ℚ-stable normal faithful state ω on P. Put φ = ω ∘ E. Then it follows from the bimodule property of conditional expectations [Tom58, Theorem 1] that φ is a ℚ-stable normal faithful state on N.

Theorem 4.5. Let φ be a ℚ-stable normal faithful state on a von Neumann algebra N, let m ∈ ℕ, and let q_{1}, …, q_{m} be m positive rational numbers with sum 1. Then there exists a type I_{m} subfactor F of N, such that

(a) φ = φ|_{F} ⊗ φ|_{F^{c}}.
(b) φ|_{F^{c}} is ℚ-stable.
(c) dφ|_{F} /dTr_{F} has eigenvalues (q_{1}, …, q_{m}).

Here, Tr_{F} denotes the trace on F for which Tr_{F} (1) = m.

We prove first:

Lemma 4.6. Let φ be a ℚ-stable normal faithful state on a von Neumann algebra N, and let q_{1}, …, q_{m} be positive rational numbers with sum 1. Then there exist isometries u_{1}, …, u_{m} ∈ N with orthogonal ranges, such that

∑_{i=1}^{m}u_{i}u_{i}^{∗} = 1,
φu_{i} = q_{i}u_{i}φ,  i = 1, …, m.

Proof. We can choose integers p, p_{1}, …, p_{m} ∈ ℕ such that

q_{i} = \frac{p_{i}}{p} , i = 1, …, m.

Note that ∑_{i=1}^{m} p_{i} = p. By Definition 4.1, for each i ∈ {1, …, m} we can choose p_{i} isometries v_{i1}, …, v_{ip_{i}} in N with orthogonal ranges, such that

∑_{j=1}^{p_{i}} v_{ij}v_{ij}^{∗} = 1 and φv_{ij} = \frac{1}{p_{i}} v_{ij}φ, j = 1, …, p_{i}.

Moreover, since the set {(i, j); 1 ≤ i ≤ m, 1 ≤ j ≤ p_{i}} contains ∑_{i=1}^{m} p_{i} = p elements, we can also find isometries w_{ij} ∈ N (1 ≤ i ≤ m, 1 ≤ j ≤ p_{i}) with orthogonal ranges, such that

∑_{i=1}^{m} ∑_{j=1}^{p_{i}} w_{ij}w_{ij}^{∗} = 1 and φw_{ij} = \frac{1}{p}w_{ij}φ, 1 ≤ i ≤ m, 1 ≤ j ≤ p_{i}.

Put now

u_{i} := ∑_{j=1}^{p_{i}} w_{ij}v_{ij}^{∗}, i = 1, …, m.

Then

u_{i}^{∗}u_{i} = ∑_{j=1}^{p_{i}} v_{ij}v_{ij}^{∗} = 1,

∑_{i=1}^{m} u_{i}u_{i}^{∗} = ∑_{i=1}^{m} ∑_{j=1}^{p_{i}} w_{ij}w_{ij}^{∗} = 1,

and since φw_{ij} = \frac{1}{p}w_{ij}φ and v_{ij}^{∗}φ = \frac{1}{p_{i}} φv_{ij}^{∗} for all (i, j), we get

φu_{i} = ∑_{j=1}^{p_{i}} φw_{ij}v_{ij}^{∗} = ∑_{j=1}^{p_{i}} \frac{p_{i}}{p} w_{ij}v_{ij}^{∗}φ = q_{i}u_{i}φ.

This proves Lemma 4.6.

Proof of Theorem 4.5. Choose m isometries u_{1}, …, u_{m} ∈ N satisfying the conditions in Lemma 4.6. We can define a ∗-isomorphism Φ of N ⊗ M_{m}(ℂ) onto N by

Φ ( ∑_{i,j=1}^{m} x_{ij} ⊗ e_{ij}) := ∑_{i,j=1}^{m} u_{i}x_{ij}u_{j}^{∗},

where (e_{ij})_{i,j=1}^{m} are the matrix units in M_{m}(ℂ). Then using φu_{i} = λ_{i}u_{i}φ, we get

(φ ∘ Φ) ( ∑_{i,j=1}^{m} x_{ij} ⊗ e_{ij}) = ∑_{i,j=1}^{m} φ(u_{i}x_{ij}u_{j}^{∗})

= ∑_{i,j=1}^{m} q_{i}φ(x_{ij}u_{j}^{∗}u_{i})

= ∑_{i=1}^{m} q_{i}φ(x_{ii}).

Hence

φ ∘ Φ = φ ⊗ ω,

where ω is the state on M_{m}(ℂ) for which

\frac{dω}{dTr} = (\begin{matrix}q_{1} & 0 & ⋯ & 0 \\ 0 & q_{2} & & ⋮ \\ ⋮ & & ... & 0 \\ 0 & ⋯ & 0 & q_{m} \end{matrix}) .

Put now F := Φ(ℂ ⊗ M_{m}(ℂ)). Then the relative commutant of F in N is Φ(N ⊗ ℂ). Since φ ∘ Φ = φ ⊗ ω, φ itself is a tensor product state with respect to the decomposition

N = F · F^{c} ≅ F ⊗ F^{c}.

Moreover, dφ|_{F} /dTr_{F} has eigenvalues (q_{1}, …, q_{m}). Let Φ_{0} be the isomorphism of N onto F^{c} given by

Φ_{0}(x) = Φ(x ⊗ 1), x ∈ N.

Then φ|_{F^{c}} ∘ Φ_{0} = φ. Therefore φ|_{F^{c}} is a ℚ-stable normal faithful state on F^{c}.

5 Proof of Main Theorem

In this section we prove the main result of the paper:

Theorem 5.1. Every injective factor N of type III_{1} on a separable Hilbert space is isomorphic to the Araki–Woods factor R_{∞}.

We need preparations. In this section, for each von Neumann algebra N, we fix a standard form (N, H, J, P^{♮}). For each φ ∈ (N_{∗})_{+}, we denote by ξ_{φ} the unique representing vector in P^{♮} [Haa75].

Lemma 5.2. Let N be a properly infinite factor with separable predual and with a normal faithful state φ, let F be a finite dimensional σ^{φ}-invariant subfactor of N, and let T: F → N be a unital completely positive map, such that φ ∘ T = φ and

∥σ_{t}^{φ} ∘ T − T ∘ σ_{t}^{φ|_{F}} ∥ ≤ δ|t|,  t ∈ ℝ,  (19)

where δ > 0 is a constant. Then there exists a norm-continuous map a: ℝ → N such that

(a) ∫_{−∞}^{∞} a(t)^{∗}a(t) dt = 1 (σ-strongly),

(b) ∫_{−∞}^{∞} e^{−t}ε_{F,φ}(a(t)a(t)^{∗}) dt = 1 (σ-strongly),

(c) ∫_{−∞}^{∞} ∥a(t)ξ_{φ} − e^{−t/2}ξ_{φ}a(t)∥^{2} dt < \frac{δ}{8},

(d) ∥T (x) − ∫_{−∞}^{∞} a(t)^{∗}xa(t) dt∥ ≤ δ^{\frac{1}{2}} ∥x∥, x ∈ F,

where ε_{F,φ} is the normal faithful conditional expectation of N onto F that leaves the state φ invariant.

Proof. Let f be the function

f(t) := (πδ)^{− \frac{1}{4}} exp (−\frac{1}{2δ} t^{2}) , t ∈ ℝ,

and let g be the Fourier-transformed of f:

g(s) := (\frac{δ}{π} )^{\frac{1}{4}} exp (−\frac{δ}{2}s^{2}) , s ∈ ℝ.

Note that

∫_{−∞}^{∞} f(t)^{2}dt = ∫_{−∞}^{∞} g(s)^{2} ds = 1.

By [Haa85, Proposition 2.1], there exists an operator a ∈ N such that

T (x) = a^{∗}xa, x ∈ F.

In particular, a^{∗}a = 1, i.e., a is an isometry. Put

a(t) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−is(t−δ/4)}g(s)σ_{s}^{φ}(a) ds, t ∈ ℝ.

Since t ↦ e^{−is(t−δ/4)}g(s) is a continuous map from ℝ to L^{1}(ℝ), the map t ↦ a(t) is a norm-continuous map from ℝ to N. Using the Plancherel formula in L^{2}(ℝ, H), we get

∫_{−∞}^{∞} ∥a(t)ξ∥^{2} dt = ∫_{−∞}^{∞} g(s)^{2}∥σ_{s}^{φ}(a)ξ∥^{2} ds = ∥ξ∥^{2}

for all ξ ∈ H. Hence

∫_{−∞}^{∞} a(t)^{∗}a(t) dt = 1 (σ-weakly).

Since the convergence of the integral is monotone, we get (a). Using again the Plancherel formula, we get for ξ, η ∈ H and x ∈ F,

∫_{−∞}^{∞}⟨xa(t)ξ, a(t)η⟩ dt = ∫_{−∞}^{∞} g(s)^{2}⟨xσ_{s}^{φ}(a)ξ, σ_{s}^{φ}(a)η⟩ ds
= ∫_{−∞}^{∞} g(s)^{2}⟨σ_{s}^{φ} ∘ T ∘ σ_{−s}^{φ}(x)ξ, η⟩ ds.

Hence for x ∈ F,

∫_{−∞}^{∞} a(t)^{∗}xa(t) dt = ∫_{−∞}^{∞} g(s)^{2}σ_{s}^{φ} ∘ T ∘ σ_{−s}^{φ}(x) ds.  (20)

Note that the left hand side of (20) converges σ-strongly, because F = span(F_{+}) and for x ∈ F_{+}, the integral converges σ-weakly and the convergence is monotone. Therefore by (19), for each x ∈ F we get

∥T (x) − ∫_{−∞}^{∞} a(t)^{∗}xa(t)dt∥ ≤ δ∥x∥ ∫_{−∞}^{∞} |s|g(s)^{2} ds

= (\frac{δ}{π} )^{\frac{1}{2}} ∥x∥

≤ δ^{\frac{1}{2}} ∥x∥.

This proves (d). Since g(s) has the analytic extension to the function g: ℂ → ℂ, and since the integrals

∫_{−∞}^{∞} |g(s + iu)| ds = (\frac{4π}{δ} )^{\frac{1}{4}} e^{\frac{δ}{2} u^{2}}, u ∈ ℝ

are uniformly bounded for u on bounded subsets of ℝ, it follows that a(t) is analytic with respect to σ^{φ} (in the sense of [PT73]) and that

σ_{α}^{φ}(a(t)) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−i(s−α)(t− \frac{δ}{4} )}g(s − α)σ_{s}^{φ}(a) ds,

for all α ∈ ℂ. To prove (c), we use the equality

ξ_{φ}a(t) = J_{φ}a(t)^{∗}ξ_{φ} = Δφ^{\frac{1}{2}}a(t)ξ_{φ} = σ_{−i/2}^{φ}(a(t))ξ_{φ}.

Hence

e^{−t/2}ξ_{φ}a(t) = \frac{e^{− \frac{δ}{8}}}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−is(t− \frac{δ}{4} )}g(s + \frac{i}{2})σ_{s}^{φ}(a)ξ_{φ} ds.

Using the Plancherel formula, we get

∫_{−∞}^{∞} ∥a(t)ξ_{φ} − e^{−t/2}ξ_{φ}a(t)∥^{2} dt = ∫_{−∞}^{∞} |g(s) − e^{− \frac{δ}{8}} g(s + \frac{i}{2})|^{2}∥aξ_{φ}∥^{2} ds.

On the other hand, g(s + \frac{i}{2}) is the Fourier–Plancherel transformed of e^{t/2}f(t). Therefore the above integral is equal to

∫_{−∞}^{∞} f(t)^{2} (1 − e^{− \frac{δ}{8} + \frac{t}{2}} )^{2} dt.

It is easy to compute that for γ ∈ ℝ,

∫_{−∞}^{∞} f(t)^{2}e^{γt} dt = exp(\frac{1}{4}γ^{2}δ).

Therefore

∫_{−∞}^{∞} f(t)^{2} (1 − e^{− \frac{δ}{8} + \frac{t}{2}} )^{2} dt = 2(1 − e^{− \frac{δ}{16}} )

< \frac{δ}{8}.

This proves (c). Put now

A(t) := e^{−t}ε_{F,φ}(a(t)a(t)^{∗}), t ∈ ℝ.

Since ε_{F,φ} is a normal faithful conditional expectation of N onto F that leaves φ invariant, we have for x ∈ F, that

φ(A(t)x) = e^{−t}φ(a(t)a(t)^{∗}x).

By the KMS-condition, it follows that if a, b ∈ N and a is σ^{φ}-analytic, then

φ(ab) = φ(bσ_{−i}^{φ}(a))

(cf. [Haa79, Theorem 3.2]). Hence for x ∈ F,

φ(A(t)x) = e^{−t}φ(a(t)^{∗}xσ_{−i}^{φ}(a(t))).

Using

e^{−t}σ_{−i}^{φ}(a(t)) = \frac{e^{− \frac{δ}{4}}}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−is(t− \frac{δ}{4} )}g(s + i)σ_{s}^{φ}(a) ds,

we get by the Plancherel formula, that

∫_{−∞}^{∞} φ(A(t)x) dt = ∫_{−∞}^{∞}⟨x(e^{−t}σ_{−i}^{φ}(a(t)))ξ_{φ}, a(t)ξ_{φ}⟩ dt
= e^{− \frac{δ}{4}} ∫_{−∞}^{∞} g(s + i)\overline{g(s)}⟨xσ_{s}^{φ}(a)ξ_{φ}, σ_{s}^{φ}(a)ξ_{φ}⟩ ds.

Since φ ∘ T = φ, it holds that

⟨xσ_{s}^{φ}(a)ξ_{φ}, σ_{s}^{φ}(a)ξ_{φ}⟩ = φ ∘ σ_{s}^{φ} ∘ T ∘ σ_{−s}^{φ}(x) = φ(x),

Hence

∫_{−∞}^{∞} φ(A(t)x) dt = e^{− \frac{δ}{4}} φ(x) ∫_{−∞}^{∞} g(s + i)\overline{g(s)} ds.

Since g(s + i) is the Fourier–Plancherel transformed of f(t)e^{t}, we get

∫_{−∞}^{∞} g(s + i)\overline{g(s)} ds = ∫_{−∞}^{∞} |f(t)|^{2}e^{t} dt = e^{\frac{δ}{4}} .

Since F is finite-dimensional and φ is faithful on F, every ψ ∈ F_{∗} is of the form φ( · x), x ∈ F. This shows that

∫_{−∞}^{∞} ψ(A(t)) dt = ψ(1), ψ ∈ F_{∗},

that is, we have

∫_{−∞}^{∞} A(t) dt = 1 (σ-weakly).

This proves (b).

Lemma 5.3. Let N, φ, F and ε_{F,φ} be as in Lemma 5.2. Let λ > 0 and assume that c_{1}, …, c_{s} are operators in F^{c} = F′ ∩ N such that

φc_{i} = λc_{i}φ,  i = 1, …, s,
∑_{i=1}^{s}c_{i}^{∗}c_{i} = 1.

Then for all x ∈ N,

ε_{F,φ} (∑_{i=1}^{s} c_{i}xc_{i}^{∗}) = λε_{F,φ}(x).

Proof. It is sufficient to check the formula for x ∈ N of the form x = ab, a ∈ F, b ∈ F^{c}. For z ∈ F^{c}, ε_{F,φ}(z) commutes with every element in F. Hence ε_{F,φ}(z) is a scalar multiple of the identity. Using that ε_{F,φ} leaves φ invariant, we get

ε_{F,φ}(z) = φ(z)1, z ∈ F^{c}.

Therefore

ε_{F,φ} (∑_{i=1}^{s} c_{i}xc_{i}^{∗}) = ε_{F,φ} (a (∑_{i=1}^{s} c_{i}bc_{i}^{∗}))

= φ (∑_{i=1}^{s} c_{i}bc_{i}^{∗}) a

= λφ (∑_{i=1}^{s} bc_{i}^{∗}c_{i}) a

= λφ(b)a = λε_{F,φ}(x).

Lemma 5.4. Let φ be a ℚ-stable normal faithful state on an injective factor N of type III_{1} with separable predual. Let u_{1}, …, u_{n} ∈ U(N), let δ > 0. Then there exist a finite dimensional σ^{φ}-invariant subfactor F of N and unitaries v_{1}, …, v_{n} ∈ U(F), such that for every σ-strong neighborhood V of 0 in N, there exists a finite set b_{1}, …, b_{r} of operators in N with the following properties:

(a) ∑_{i=1}^{r} b_{i}^{∗}b_{i} ∈ 1 + V and ∑_{i=1}^{r} b_{i}^{∗}b_{i} ≤ 1.

(b) ε_{F,φ} (∑_{i=1}^{r} b_{i}b_{i}^{∗}) ∈ 1 + V and ε_{F,φ} (∑_{i=1}^{r} b_{i}b_{i}^{∗}) ≤ 1.

(c) ∑_{i=1}^{r} ∥b_{i}ξ_{φ} − ξ_{φ}b_{i}∥^{2} < δ.

(d) ∑_{i=1}^{r} ∥b_{i}u_{k} − v_{k}b_{i}∥_{φ}^{2} < δ, k = 1, …, n.

Proof. Put δ_{1} = min(δ^{2}/16, δ). By Theorem 3.1, there exist m ∈ ℕ, a unital completely positive map T_{0} : M_{m}(ℂ) → N and unitaries w_{1}, …, w_{n} ∈ M_{m}(ℂ) such that ψ := φ ∘ T_{0} ∈ M_{m}(ℂ)_{∗} satisfies

∥σ_{t}^{φ} ∘ T_{0} − T_{0} ∘ σ_{t}^{ψ}∥ ≤ \frac{δ_{1}}{2} |t|, t ∈ ℝ,
∥T_{0}(w_{k}) − u_{k}∥_{φ} < ε, k = 1, …, n.

Let {q′_{1}, …, q′_{m}} be the spectrum of dψ/dTr ∈ M_{m}(ℂ)_{+} where the multiplicity is taken into account. Let {q_{1}, …, q_{m}} be positive rationals with sum 1, and let χ on M_{m}(ℂ)_{+} such that dχ/dTr has the same spectral projections as dψ/dTr but the eigenvalues replaced by {q_{1}, …, q_{m}}. Since ∥e^{ia} − e^{ib}∥ ≤ ∥a − b∥ for self-adjoint operators a, b, (cf. (17) in Theorem 3.1), we may arrange q_{i}’s so that the following inequality holds:

∥σ_{t}^{ψ} − σ_{t}^{χ}∥_{M_{m}(ℂ)} ≤ \frac{δ_{1}}{2} |t|, t ∈ ℝ.

Since φ is ℚ-stable and q_{i}’s are rationals, by Theorem 4.5, there exists a finite-dimensional subfactor F ⊂ N and a state-preserving ∗-isomorphism Φ: (M_{m}(ℂ), χ) → (F, φ|_{F}) such that φ|_{F^{c}} is ℚ-stable. Define T := T_{0} ∘ Φ^{−1} : F → N and v_{k} := Φ_{0}(w_{k}) ∈ U(F)(1 ≤ k ≤ n). Then if x = Φ(y) (y ∈ M_{m}(ℂ)) and t ∈ ℝ, we have

∥σ_{t}^{φ} ∘ T (x) − T ∘ σ_{t}^{φ|_{F}} (x)∥ = ∥σ_{t}^{φ} ∘ T_{0}(y) − T_{0} ∘ Φ^{−1} ∘ σ_{t}^{φ|_{F}} ∘ Φ(y)∥
= ∥σ_{t}^{φ} ∘ T_{0}(y) − T_{0} ∘ σ_{t}^{χ}(y)∥
≤ ∥σ_{t}^{φ} ∘ T_{0}(y) − T_{0} ∘ σ_{t}^{ψ}(y)∥ + ∥T (σ_{t}^{ψ}(y) − σ_{t}^{χ}(y))∥
≤ δ_{1}|t|∥y∥.

Therefore we obtain

T (1) = 1, φ ∘ T = φ|_{F} ,
∥σ_{t}^{φ} ∘ T − T ∘ σ_{t}^{φ|_{F}} ∥ ≤ δ_{1}|t|,  t ∈ ℝ,
∥T (v_{k}) − u_{k}∥_{φ} < δ1^{\frac{1}{2}} ,  k = 1, …, n.

Choose now a norm-continuous function t ↦ a(t) of ℝ into N, such that the conditions (a), (b), (c) and (d) in Lemma 5.2 are satisfied with respect to δ_{1} instead of δ. Then using (d), we have

∥u_{k} − ∫_{−∞}^{∞} a(t)^{∗}v_{k}a(t) dt∥_{φ} < 2δ1^{\frac{1}{2}} ≤ \frac{δ}{2}

for k = 1, …, n. Using that ∫_{−∞}^{∞} a(t)^{∗}a(t) dt = 1, it follows that

∫_{−∞}^{∞} ∥a(t)u_{k} − v_{k}a(t)∥_{φ}^{2} dt = 2 − 2Re ∫_{−∞}^{∞}⟨a(t)u_{k}ξ_{φ}, v_{k}a(t)ξ_{φ}⟩ dt
= 2 − 2Re (⟨u_{k}ξ_{φ}, ∫_{−∞}^{∞} a(t)^{∗}v_{k}a(t)ξ_{φ} dt⟩)
≤ 2 ∥u_{k} − ∫_{−∞}^{∞} a(t)^{∗}v_{k}a(t) dt∥_{φ}
< δ.

Let now V be a σ-strong neighborhood of 0 in N. It is no loss of generality to assume that V is open. For sufficiently large γ ∈ ℝ_{+}, we have:

(a’) ∫_{−γ}^{γ} a(t)^{∗}a(t)dt ∈ 1 + V and ∫_{−γ}^{γ} a(t)^{∗}a(t) dt ≤ 1.

(b’) ∫_{−γ}^{γ} e^{−t}ε_{F,φ}(a(t)a(t)^{∗})dt ∈ 1 + V and ∫_{−γ}^{γ} e^{−t}ε_{F,φ}(a(t)a(t)^{∗}) dt ≤ 1.

(c’) ∫_{−γ}^{γ} ∥a(t)ξ_{φ} − e^{−t/2}ξ_{φ}a(t)∥^{2} dt < \frac{δ_{1}}{8} ≤ δ.

(d’) ∫_{−γ}^{γ} ∥a(t)u_{k} − v_{k}a(t)∥_{φ}^{2} dt < δ.

Since t ↦ a(t) is norm-continuous, we can approximate (in norm) the above N-valued Riemann integrals over [−γ, γ] to get the following statements: there exists an h_{0} > 0 such that when 0 < h < h_{0}, the operators

a_{j} = h^{− \frac{1}{2}} a(jh), j ∈ ℤ

satisfy the following relations:

(a”) ∑_{j=−p}^{p} a_{j}^{∗}a_{j} ∈ 1 + V.

(b”) ∑_{j=−p}^{p} e^{−jh}ε_{F,φ}(a_{j}a_{j}^{∗}) ∈ 1 + V.

(c”) ∑_{j=−p}^{p} ∥a_{j}ξ_{φ} − e^{− \frac{1}{2} jh}ξ_{φ}a_{j}∥^{2} < δ.

(d”) ∑_{j=−p}^{p} ∥a_{j}u_{k} − v_{k}a_{j}∥_{φ}^{2}dt < δ,

where p is the largest integer smaller than γ/h_{0}. Moreover, since the Riemann sum is norm-convergent, by multiplying a scalar c > 0 to a_{j}’s which is sufficiently close to 1 if necessary, we may moreover assume that

∑_{j=−p}^{p} a_{j}^{∗}a_{j} ≤ 1  (21)

∑_{j=−p}^{p} e^{−jh}ε_{F,φ}(a_{j}a_{j}^{∗}) ≤ 1.  (22)

Choose now h ∈ (0, h_{0}), such that exp(h) ∈ ℚ. This implies that the numbers q_{j} = e^{−jh}, j ∈ ℤ are rational. Since the restriction of φ to F^{c} is ℚ-stable, there exists for each j ∈ ℤ a finite set of operators c_{j1}, …, c_{js(j)} in F^{c} such that

φc_{ji} = e^{−jh}c_{ji}φ, i = 1, …, s(j)

Uniqueness of the Injective III_{1} Factor 1221

and

∑_{i=1}^{s(j)} c_{ji}^{∗}c_{ji} = 1.

Here we use Lemma 4.6 together with the fact that φ = φ|_{F} ⊗ φ|_{F^{c}}. Put

b_{ji} = c_{ji}a_{j}, |j| ≤ p, 1 ≤ i ≤ s(j).

Then by (21),

(a’”) ∑_{j=−p}^{p} ∑_{i=1}^{s(j)} b_{ji}^{∗}b_{ji} = ∑_{j=−p}^{p} a_{j}^{∗}a_{j} ∈ 1+V and ∑_{j=−p}^{p} ∑_{i=1}^{s(j)} b_{ji}^{∗}b_{ji} ≤

1, and by (22) and Lemma 5.3,

(b’”) ε_{F,φ} ( ∑_{j=−p}^{p} ∑_{i=1}^{s(j)} b_{ji}b_{ji}^{∗}) = ∑_{j=−p}^{p} e^{−jh}ε_{F,φ}(a_{j}a_{j}^{∗}) ∈ 1 + V,

and ε_{F,φ} ( ∑_{j=−p}^{p} ∑_{i=1}^{s(j)} b_{ji}b_{ji}^{∗}) ≤ 1.

The equality φc_{ji} = e^{−jh}c_{ji}φ implies that

ξ_{φ}c_{ji} = e^{− \frac{1}{2} jh}c_{ji}ξ_{φ}.

Therefore

(c’”) ∑_{j=−p}^{p} ∑_{i=1}^{s(j)} ∥b_{ji}ξ_{φ} − ξ_{φ}b_{ji}∥^{2} = ∑_{j=−p}^{p} ∑_{i=1}^{s(j)} ∥c_{ji}(a_{j}ξ_{φ} − e^{− \frac{1}{2} jh}ξ_{φ}a_{j})∥^{2}

= ∑_{j=−p}^{p} ∥a_{j}ξ_{φ} − e^{− \frac{1}{2} jh}ξ_{φ}a_{j}∥^{2}

< ε.

Finally, using that v_{k} ∈ F and c_{ji} ∈ F^{c}, we get

(d’”) ∑_{j=−p}^{p} ∑_{i=1}^{s(j)} ∥b_{ji}u_{k} − v_{k}b_{ji}∥_{φ}^{2} = ∑_{j=−p}^{p} ∑_{i=1}^{s(j)} ∥c_{ji}(a_{j}u_{k} − v_{k}a_{j})∥_{φ}^{2}

= ∑_{j=−p}^{p} ∥a_{j}u_{k} − v_{k}a_{j}∥_{φ}^{2}

< δ.

This completes the proof of Lemma 5.4.

Documenta Mathematica 21 (2016) 1193–1226

In the proof of the following lemma, it is essential that injective type III_{1} factors (on a separable Hilbert space) have trivial bicentralizers.

Lemma 5.5. Let φ be a ℚ-stable normal faithful state on an injective factor N of type III_{1} with separable predual. Let u_{1}, …, u_{n} ∈ U(N), and let δ > 0. Then there exists a finite dimensional σ^{φ}-invariant subfactor F of N and v_{1}, …, v_{n} ∈ U(F), such that for every σ-strong neighborhood V of 0 in N, there exists a finite set a_{1}, …, a_{p} of operators in N with the following properties:

(a) ∑_{i=1}^{p} a_{i}^{∗}a_{i} ∈ 1 + V and ∑_{i=1}^{p} a_{i}^{∗}a_{i} ≤ 1.

(b) ∑_{i=1}^{p} a_{i}a_{i}^{∗} ∈ 1 + V and ∑_{i=1}^{p} a_{i}a_{i}^{∗} ≤ 1.

(c) ∑_{i=1}^{p} ∥a_{i}ξ_{φ} − ξ_{φ}a_{i}∥^{2} < δ.

(d) ∑_{i=1}^{p} ∥a_{i}u_{k} − v_{k}a_{i}∥_{φ}^{2} < δ, k = 1, …, n.

Proof. Choose an F and v_{1}, …, v_{n} ∈ U(F) satisfying the properties of Lemma 5.4 with respect to (u_{1}, …, u_{n}, δ), and let V be a σ-strongly open neighborhood of 0 in N. By Lemma 5.4, there exists b_{1}, …, b_{r} ∈ N such that

(a’) ∑_{i=1}^{r} b_{i}^{∗}b_{i} ∈ 1 + V and ∑_{i=1}^{r} b_{i}^{∗}b_{i} ≤ 1.

(b’) ε_{F,φ} (∑_{i=1}^{r} b_{i}b_{i}^{∗}) ∈ 1 + V and ε_{F,φ} (∑_{i=1}^{r} b_{i}b_{i}^{∗}) ≤ 1.

(c’) ∑_{i=1}^{r} ∥b_{i}ξ_{φ} − ξ_{φ}b_{i}∥^{2} < δ.

(d’) ∑_{i=1}^{r} ∥b_{i}u_{k} − v_{k}b_{i}∥_{φ}^{2} < δ, k = 1, …, n.

Let δ′ > 0 and h denote the operator ∑_{i=1}^{r} b_{i}b_{i}^{∗}. Since B_{φ} = ℂ1, by Proposition 2.6, we have

ε_{F,φ}(h) ∈ \overline{conv}\{whw^{∗}; w ∈ U(F^{c}), ∥wξ_{φ} − ξ_{φ}w∥ < δ′\}.  (23)

Here, \overline{conv} in (23) denotes the σ-strong closure. Hence there exist w_{1}, …, w_{s} ∈ U(F^{c}), and scalars λ_{1}, …, λ_{s} ∈ ℝ_{+}, with sum 1, such that

∥w_{j}ξ_{φ} − ξ_{φ}w_{j}∥ < δ′, j = 1, …, s

Uniqueness of the Injective III_{1} Factor 1223

and

∑_{j=1}^{s} λ_{j}w_{j}hw_{j}^{∗} ∈ 1 + V.

Put

a_{ij} := λj^{\frac{1}{2}} w_{j}b_{i}, i = 1, …, r, j = 1, …, s.

Then

(a”) ∑_{i=1}^{r} ∑_{j=1}^{s} a_{ij}^{∗}a_{ij} = ∑_{i=1}^{r} b_{i}^{∗}b_{i} ∈ 1 + V and ∑_{i=1}^{r} ∑_{j=1}^{s} a_{ij}^{∗}a_{ij} ≤ 1.

(b”) ∑_{i=1}^{r} ∑_{j=1}^{s} a_{ij}a_{ij}^{∗} = ∑_{j=1}^{s} λ_{j}w_{j}hw_{j}^{∗} ∈ 1 + V and ∑_{i=1}^{r} ∑_{j=1}^{s} a_{ij}a_{ij}^{∗} ≤ 1.

Moreover, using

a_{ij}ξ_{φ} − ξ_{φ}a_{ij} = λj^{\frac{1}{2}} w_{j}(b_{i}ξ_{φ} − ξ_{φ}b_{i}) + λj^{\frac{1}{2}} (w_{j}ξ_{φ} − ξ_{φ}w_{j})b_{i},

we obtain

(c”) (∑_{i=1}^{r} ∑_{j=1}^{s} ∥a_{ij}ξ_{φ} − ξ_{φ}a_{ij}∥^{2})^{\frac{1}{2}} ≤

≤ (∑_{i=1}^{r} ∑_{j=1}^{s} λ_{j}∥b_{i}ξ_{φ} − ξ_{φ}b_{i}∥^{2})^{\frac{1}{2}} + δ′ (∑_{i=1}^{r} ∑_{j=1}^{s} λ_{j}∥b_{i}∥^{2})^{\frac{1}{2}}

= (∑_{i=1}^{r} ∥b_{i}ξ_{φ} − ξ_{φ}b_{i}∥^{2})^{\frac{1}{2}} + δ′ (∑_{i=1}^{r} ∥b_{i}∥^{2})^{\frac{1}{2}} .

Finaly, since v_{k} ∈ F and w_{j} ∈ F^{c}, we have

(d”) ∑_{i=1}^{r} ∑_{j=1}^{s} ∥a_{ij}u_{k} − v_{k}a_{ij}∥_{φ}^{2} = ∑_{i=1}^{r} ∑_{j=1}^{s} λ_{j}∥w_{j}(b_{i}u_{k} − v_{k}b_{i})∥_{φ}^{2}

= ∑_{i=1}^{r} ∥b_{i}u_{k} − v_{k}b_{i}∥_{φ}^{2}

< δ.

Since δ′ > 0 was arbitrary (independent of δ, V, and b_{1}, …, b_{r}), we can assume that

(∑_{i=1}^{r} ∥b_{i}ξ_{φ} − ξ_{φ}b_{i}∥^{2})^{\frac{1}{2}} + δ′ (∑_{i=1}^{r} ∥b_{i}∥^{2})^{\frac{1}{2}} < δ^{\frac{1}{2}} .

This proves Lemma 5.5.

Documenta Mathematica 21 (2016) 1193–1226

Lemma 5.6. Let N be an injective factor of type III_{1} with separable predual, and let φ be a ℚ-stable normal faithful state on N. Let u_{1}, …, u_{n} ∈ U(N) and let ε > 0. Then there exist a σ^{φ}-invariant finite dimensional subfactor F of N, v_{1}, …, v_{n} ∈ U(F) and a unitary w ∈ U(N) such that

∥wξ_{φ} − ξ_{φ}w∥ < ε,

and

∥w^{∗}v_{k}w − u_{k}∥_{φ} < ε, k = 1, …, n.

Proof. Let δ(n, ε) > 0 be the function in Theorem 2.8, and put δ_{1} = \frac{1}{16}δ(n + 1, ε/2). Choose F and v_{1}, …, v_{n} ∈ U(F), such that the conditions of Lemma 5.5 are satisfied with respect to (u_{1}, …, u_{n}, δ_{1}). Put

ξ_{k} = u_{k}ξ_{φ}, η_{k} = v_{k}ξ_{φ}, k = 1, …, n.

For every σ-strong neighborhood V of 0 in N, there exist a_{1}, …, a_{p} ∈ N, such that (a), (b), (c) and (d) in Lemma 5.5 are satisfied. Since

a_{i}ξ_{k} − η_{k}a_{i} = (a_{i}u_{k} − v_{k}a_{i})ξ_{φ} + v_{k}(a_{i}ξ_{φ} − ξ_{φ}a_{i}),

we have

(∑_{i=1}^{p} ∥a_{i}ξ_{k} − η_{k}a_{i}∥^{2})^{\frac{1}{2}} ≤ (∑_{i=1}^{p} ∥a_{i}u_{k} − v_{k}a_{i}∥_{φ}^{2})^{\frac{1}{2}} + (∑_{i=1}^{p} ∥a_{i}ξ_{φ} − ξ_{φ}a_{i}∥^{2})^{\frac{1}{2}}

< 2δ1^{\frac{1}{2}} .

Moreover,

(∑_{i=1}^{p} ∥a_{i}ξ_{φ} − ξ_{φ}a_{i}∥^{2})^{\frac{1}{2}} < δ1^{\frac{1}{2}} < 2δ1^{\frac{1}{2}} .

Since ∑_{i=1}^{p} a_{i}^{∗}a_{i} ∈ 1+V, ∑_{i=1}^{p} a_{i}^{∗}a_{i} ≤ 1, ∑_{i=1}^{p} a_{i}a_{i}^{∗} ∈ 1+V and ∑_{i=1}^{p} a_{i}a_{i}^{∗} ≤ 1, the two (n+1)-tuples (ξ_{1}, …, ξ_{n}, ξ_{φ}) and (η_{1}, …, η_{n}, ξ_{φ}) satisfies the conditions

of Remark 2.9 with 4δ1^{\frac{1}{2}} instead of δ, so that the two (n + 1)-tuples are 16δ_{1}- related, or equivalently they are δ(n + 1, \frac{ε}{2})-related in the sense of Remark 2.9. Hence by Theorem 2.8, there exists a unitary operator w ∈ U(N), such that

∥wξ_{k} − η_{k}w∥ < \frac{ε}{2}, k = 1, …, n.

and

∥wξ_{φ} − ξ_{φ}w∥ < \frac{ε}{2}.

Therefore

∥w^{∗}v_{k}w − u_{k}∥_{φ} = ∥w^{∗}(v_{k}w − wu_{k})ξ_{φ}∥
= ∥(wu_{k} − v_{k}w)ξ_{φ}∥
= ∥(wξ_{k} − η_{k}w) + v_{k}(ξ_{φ}w − wξ_{φ})∥
< ε,

which completes the proof of Lemma 5.6.

Now we are ready to prove the main theorem of the paper.

Proof of Theorem 5.1. By [AW68, Theorem 7.6], it is sufficient to show that N is an ITPFI-factor. Let φ be a ℚ-stable normal faithful state on N, let u_{1}, …, u_{n} ∈ U(N), and let ε > 0. Choose now F, v_{1}, …, v_{n} ∈ U(F) and w ∈ U(N) as in Lemma 5.6. Put

w_{k} = w^{∗}v_{k}w, k = 1, …, n.

Then F_{1} := w^{∗}F w is a finite-dimensional subfactor of N, w_{1}, …, w_{n} ∈ U(F_{1}) and

∥w_{k} − u_{k}∥_{φ} < ε, k = 1, …, n.

Hence if we put φ_{1} = w^{∗}φw, then by φ = φ|_{F} ⊗ φ|_{F^{c}}, we have

φ_{1} = φ_{1}|_{F_{1}} ⊗ φ_{1}|_{F_{1}^{c}}.

Since the representing vector of φ_{1} in P_{N}^{♮} is w^{∗}ξ_{φ}w, we have

∥φ − φ_{1}∥ ≤ ∥ξ_{φ} − w^{∗}ξ_{φ}w∥∥ξ_{φ} + w^{∗}ξ_{φ}w∥
≤ 2∥wξ_{φ} − ξ_{φ}w∥
< 2ε.

This shows that φ satisfies the product condition in Proposition 2.2, and thus N is an ITPFI factor.

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Uffe Haagerup

Department of Mathematics and Computer Science The University of Southern Denmark Campusvej 55, DK–5230 Odense M, Denmark