The Isomorphism Problem for Semigroup C^{∗}-Algebras

of Right-Angled Artin Monoids

Søren Eilers, Xin Li, and Efren Ruiz

Received: February 2, 2016

Communicated by Joachim Cuntz

Abstract. Semigroup C*-algebras for right-angled Artin monoids were introduced and studied by Crisp and Laca. In the paper at hand, we are able to present the complete answer to their question of when such C*-algebras are isomorphic. The answer to this question is presented both in terms of properties of the graph defining the Artin monoids as well as in terms of classification by K-theory, and is obtained using recent results from classification of non-simple C*- algebras. Moreover, we are able to answer another natural question: Which of these semigroup C*-algebras for right-angled Artin monoids are isomorphic to graph algebras? We give a complete answer, and note the consequence that many of the C*-algebras under study are semiprojective.

2010 Mathematics Subject Classification: Primary: 46L05. Secondary: 20F36, 20M05, 46L35.

1 Introduction

Semigroup C*-algebras for right-angled Artin monoids were introduced and studied by Crisp and Laca in [CL02] and [CL07]. In [CL07], the authors ask how to classify these semigroup C*-algebras up to *-isomorphism. We now present the complete answer to their question. The Artin monoids studied here are given by countable, symmetric and antireflexive graphs Γ = (V, E) as

A_{Γ}^{+} := ⟨{σ_{v}: v ∈ V } |σ_{v}σ_{w} = σ_{w}σ_{v} if (v, w) ∈ E⟩^{+} .

The corresponding right-angled Artin groups, defined by the same generators and relations, are special cases of Artin groups, which form an important class of examples of groups. We refer the reader to [CL02, CL07] and the references therein for more details. Semigroup C*-algebras of left cancellative semigroups, generated by the left regular representation of the semigroup, have been studied for a long time. Recently, there has been a renewed interest in this topic (see [Li12, Li13] and the references therein). By [CL02], the semigroup C*-algebras C^{∗}(A_{Γ}^{+}) attached to right-angled Artin monoids are given as the universal C*-algebras for

⟨\{s_{v}: v ∈ V \} | \begin{matrix}[s_{v}, s_{w}] = [s_{v}, s_{w}^{∗}] = 0 if (v, w) ∈ E \\ s_{v}^{∗}s_{w} = δ_{v,w} if (v, w) ∉ E \end{matrix} ⟩

We answer the question of when two graphs Γ, Λ produce C*-algebras that are isomorphic. Although we emphasize that our results cover the full range of such graphs, it is instructive to state our main results in the case of finite graphs. This is a specialization of the combination of Theorems 4.2 and 5.2.

Theorem 1.1 Let Γ and Λ be finite undirected graphs with no loops. The following are equivalent

1. C^{∗}(A_{Γ}^{+}) ≅ C^{∗}(A_{Λ}^{+})

2. (a) t(Γ) = t(Λ)

(b) N_{k}(Γ) + N_{−k}(Γ) = N_{k}(Λ) + N_{−k}(Λ) for all k ∈ ℤ

(c) N_{0}(Γ) > 0 or

∑_{k>0} N_{−k}(Γ) ≡ ∑_{k>0} N_{−k}(Λ) mod 2

3. [F K_{+}(C^{∗}(A_{Γ}^{+})), [1_{C^{∗}(A_{Γ}^{+})}]] ≅ [F K_{+}(C^{∗}(A_{Λ}^{+})), [1_{C^{∗}(A_{Λ}^{+})}]]

In this result, the invariant mentioned in (3) is the standard ordered filtered K-theory – implicitly containing the primitive ideal space – which has been conjectured in [ERR10] to be a complete invariant for a large and important class of C*-algebras. This conjecture is still open, but has been confirmed in a multitude of situations partially overlapping with the case at hand. But the main strength of our result is in fact that the ad hoc invariant of (2) is extremely easy to compute for Γ and Λ. Indeed, as we shall detail below, the numbers t(Γ) and N_{k}(Γ) are obtained by dividing Γ into co-irreducible components and then counting how many of these are singletons, yielding t(Γ), and counting how many of the remaining co-irreducible components have Euler characteristic k, yielding N_{k}(Γ). In Figure 1 this process has been completed for all 34 graphs with five vertices, and we conclude that they define 18 different C*-algebras. When the number of vertices increase, it is possible for two graphs to have different sets of invariants, yet define the same C*-algebra. For instance, defining

The Isomorphism Problem for Semigroup C^{∗}-Algebras 311

N_{−4}=1  N_{−3}=1  N_{−2}=1  N_{−2}=1  N_{−2}=1  N_{−1}=1  N_{−1}=1

N_{−1}=1  N_{−1}=1  N_{−1}=1  N_{−1}=1  N_{−1}=1  N_{0}=1  N_{0}=1

N_{0}=1  N_{0}=1  N_{0}=1  N_{0}=1  N_{0}=1  N_{1}=1  N_{1}=1

N_{−3}=1  N_{−2}=1  N_{−2}=1  N_{−1}=2  N_{−1}=1  N_{−1}=1  N_{−1}=1
t=1  N_{−1}=1  t=1  t=1  t=1  t=1

N_{0}=1  N_{−2}=1  N_{−1}=2  N_{−1}=1  N_{−1}=1  t=5
t=1  t=2  t=1  t=2  t=3

Figure 1: Invariants for all graphs with 5 vertices. Any quantity not mentioned equals zero. a graph Γ′ with 10 vertices having its co-irreducible components chosen among those given in Figure 1 so that

N_{−1}(Γ′) = 2

and Λ′ similarly defined so that

N_{1}(Λ′) = 2,

then C^{∗}(A_{Γ′}^{+}) and C^{∗}(A_{Λ′}^{+}) will be isomorphic. Similarly, we may define Γ′′ and Λ′′ with 15 vertices each so that

N_{−1}(Γ′′) = 1, N_{0}(Γ′′) = 1, N_{1}(Γ′′) = 1
N_{−1}(Λ′′) = 2, N_{0}(Λ′′) = 1

obtaining that C^{∗}(A_{Γ′′}^{+}) ≅ C^{∗}(A_{Λ′′}^{+}). In the general case of possibly infinite graphs, an additional quantity o(Γ) must be introduced to count those co-irreducible components which have an infinite number of vertices, and to address the possibility of having an infinite number of co-irreducible components, but the necessary condition replacing (2) in this general case is not much more complicated than the one given above. We note that the C*-algebras associated via semigroups to undirected and loop-free graphs are not always graph C*-algebras in the usual sense, not only because graph C*-algebras are defined using directed graphs. We provide a complete description of when C^{∗}(A_{Γ}^{+}) is in fact a graph C*-algebra, and note that there is a rather complicated relation between Γ and the G_{Γ} when in fact C^{∗}(A_{Γ}^{+}) ≅ C^{∗}(G_{Γ}). In this case, G_{Γ} is not unique. Our results have surprising consequences for the issue of stable relations (cf. [Lor97]) among sets of isometries of separable Hilbert spaces, subject to commutativity or orthogonality relations as given by the graph Γ, or, which is nearly the same, for the issue of semiprojectivity (cf. [Bla85]) of the C*-algebras C^{∗}(A_{Γ}^{+}). Indeed, it is easy to see by spectral theory that C^{∗}(A_{Γ}^{+}) is semiprojective when Γ is a finite graph with no edges, corresponding to a family of isometries having orthogonal range projections. Similarly, it follows e.g. from considering the celebrated Voiculescu matrices ([Voi83],[EL91]) that when Γ is a complete graph with more than one vertex, C^{∗}(A_{Γ}^{+}) cannot have this property. It is a question attracting a lot of attention (see e.g. [Bla04]) to what extent it is possible to obtain stable relations for commuting sets of stable relations, or to what extent tensor products of semiprojective C*-algebras can themselves be semiprojective. In fact, it was only established recently ([End15]) that O_{3} ⊗ O_{3} is semiprojective, as a consequence of Enders’ sweeping solution to the semiprojectivity problem for Kirchberg algebras. In our setting, because we have found that many settings in which some isometries are required to be orthogonal, and others to commute, give the same C*-algebras as the ones where all are required to be orthogonal, we immediately see that many such settings — for instance the first 12 listed in Figure 1 — provide for stable relations. Involving the notion of graph algebras as outlined above, we will show in Theorem 6.9 semiprojectivity and nonsemiprojectivity for many of the C*-algebras under study, and it follows from our results that exactly those C*-algebras arising from the graphs in Figure 1 in the non-shaded entries are semiprojective. We have not been able to resolve the issue completely as Enders’ methods do not apply directly, the first open case having six vertices and two co-irreducible components each with Euler characteristic −2. All authors acknowledge support by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92), and thank the Department of Mathematical Sciences at the University of Copenhagen, where the initial phases of this work were carried out, for providing excellent facilities. The first named author gratefully acknowledges support from the VILLUM Foundation, and the third named author gratefully acknowledges support from the Simons Foundation (#279369 to Efren Ruiz).

2 Preliminaries

2.1 Semigroup C*-algebras for right-angled Artin monoids

Let Γ be a countable graph. Γ = (V, E) is given by a countable set of vertices V and a set of edges E. We only consider unoriented edges, and given two vertices, there is at most one edge joining these two vertices. In other words, we can think of E as a symmetric subset of V × V not containing elements of the diagonal. Given such a graph Γ = (V, E), the right-angled Artin group A_{Γ} is defined as follows:

A_{Γ} := ⟨{σ_{v}: v ∈ V } |σ_{v}σ_{w} = σ_{w}σ_{v} if (v, w) ∈ E⟩ .

Similarly, the right-angled Artin monoid A_{Γ}^{+} is defined as follows:

A_{Γ}^{+} := ⟨{σ_{v}: v ∈ V } |σ_{v}σ_{w} = σ_{w}σ_{v} if (v, w) ∈ E⟩^{+} .

It turns out that the canonical semigroup homomorphism A_{Γ}^{+} → A_{Γ} is injective, see [Par02]. Moreover, it is shown in [CL02] that A_{Γ}^{+} ⊆ A_{Γ} is quasi-lattice ordered. This means that for every g in A_{Γ}, either (gA_{Γ}^{+}) ∩ A_{Γ}^{+} = ∅ or there exists p ∈ A_{Γ}^{+} with (gA_{Γ}^{+}) ∩ A_{Γ}^{+} = pA_{Γ}^{+}. The (left) reduced semigroup C*-algebra of A_{Γ}^{+} is given by

C_{λ}^{∗}(A_{Γ}^{+}) = C^{∗} ⟨{S_{v}: v ∈ V }⟩ ⊆ L(ℓ^{2}(A_{Γ}^{+})),

where S_{v}e_{x} = e_{σ_{v}x} with {e_{x}} the canonical orthonormal basis. The full semigroup C*-algebra of A_{Γ}^{+} is defined as

C^{∗}(A_{Γ}^{+}) = C_{univ}^{∗} ⟨\{s_{v}: v ∈ V \} | \begin{matrix}[s_{v}, s_{w}] = [s_{v}, s_{w}^{∗}] = 0 if (v, w) ∈ E \\ s_{v}^{∗}s_{w} = δ_{v,w} if (v, w) ∉ E \end{matrix} ⟩

The canonical homomorphism C^{∗}(A_{Γ}^{+}) → C_{λ}^{∗}(A_{Γ}^{+}) is an isomorphism by [CL02]. Hence we do not distinguish between reduced and full versions and simply write C^{∗}(A_{Γ}^{+}) for the semigroup C*-algebra of A_{Γ}^{+}.

2.2 Co-irreducible components

The graph Γ is called co-reducible if there exist non-empty subsets V_{1} and V_{2} of V with V = V_{1} ⊔ V_{2} such that V_{1} × V_{2} ⊆ E. Γ is called co-irreducible if Γ is not co-reducible. In general, we can always decompose Γ into co-irreducible components. This means that there exist co-irreducible graphs Γ_{i} = (V_{i}, E_{i}) such that A_{Γ}^{+} ≅ ⨁_{i} A_{Γ_{i}}^{+} (and also A_{Γ} ≅ ⨁_{i} A_{Γ_{i}}). As explained in [CL07], these co-irreducible components are found by looking at the opposite graph of Γ. For the semigroup C*-algebra, we get C^{∗}(A_{Γ}^{+}) ≅ ⨂_{i} C^{∗}(A_{Γ_{i}}^{+}). Note that if there are (necessarily countably) infinitely many co-irreducible components, the tensor product is defined as an inductive limit of finite tensor products with respect to the canonical unital embeddings as tensor factors. It is shown in [CL07] that for a co-irreducible graph Γ = (V, E) with 1 < |V | < ∞, C^{∗}(A_{Γ}^{+}) has a unique non-trivial ideal isomorphic to the compact operators. It is easy to see the compact operators in the description of C^{∗}(A_{Γ}^{+}) as a concrete C*-algebra on ℓ^{2}(A_{Γ}^{+}): We just have to observe that 1 − ⋁_{v∈V} S_{v}S_{v}^{∗} is the orthogonal projection onto the one-dimensional subspace of ℓ^{2}(A_{Γ}^{+}) corresponding to the identity element of A_{Γ}^{+}. This projection then generates the ideal of compact operators. The corresponding quotient C_{Q}^{∗}(A_{Γ}^{+}) is a (unital) Kirchberg algebra satisfying the UCT. However, if our co-irreducible graph has infinitely many vertices, then C^{∗}(A_{Γ}^{+}) itself is a (unital) Kirchberg algebra satisfying the UCT. That we obtain UCT Kirchberg algebras follows also from [Li13, Corollary 7.23]. The case where Γ consists of only one vertex is easy to understand; in that case, C^{∗}(A_{Γ}^{+}) is canonically isomorphic to the Toeplitz algebra T .

2.3 Primitive ideal space

We can now describe the primitive ideal space of C^{∗}(A_{Γ}^{+}) for arbitrary Γ. Let Γ_{i} = (V_{i}, E_{i}) be the co-irreducible components of Γ. Then by [Bla77, Theorem 4.9], we have an identification

Prim (C^{∗}(A_{Γ}^{+})) ≅ ∏_{i} Prim (C^{∗}(A_{Γ_{i}}^{+})).

Under this identification, an element (I_{i}) of the space on the right hand side corresponds to the primitive ideal I of C^{∗}(A_{Γ}^{+}) which is generated by {⨂_{j} J_{ij}}_{i}, where J_{ij} = C^{∗}(A_{Γ_{j}}^{+}) if j ≠ i and J_{ii} = I_{i}. Since each of the Γ_{i} is co-irreducible, the primitive ideal space Prim (C^{∗}(A_{Γ_{i}}^{+})) is easy to describe because of the results summarized above:

• If Γ_{i} just consists of one point, then Prim (C^{∗}(A_{Γ_{i}}^{+})) is homeomorphic to the primitive ideal space of the Toeplitz algebra. This means that as a set, Prim (C^{∗}(A_{Γ_{i}}^{+})) is the disjoint union of a point and a circle, and the non-empty open sets are given by unions of the point and open subsets in the usual topology of the circle.

• If Γ_{i} has more than one, but finitely many vertices, Prim (C^{∗}(A_{Γ_{i}}^{+})) consists of two points, one of which is closed (the corresponding primitive ideal is the ideal of compact operators) and the other one is dense.

• If Γ_{i} has infinitely many vertices, then Prim (C^{∗}(A_{Γ_{i}}^{+})) consists of only one point.

2.4 K-theory

K-theory for C^{∗}(A_{Γ}^{+}) and the quotients C_{Q}^{∗}(A_{Γ}^{+}) has been computed in [Iva10] in an ad hoc way, and can also be computed using [CEL13]. Let us explain the computation via the latter route. First of all, we need the Euler characteristic of a graph Γ. We view Γ as a simplicial complex by defining for every n = 0, 1, 2, … the set of n-simplices by

K_{n} := {{v_{0}, …, v_{n}} ⊆ V: (v_{i}, v_{j}) ∈ E for all i, j ∈ {0, …, n} , i ≠ j} .

Then we set for a graph Γ with finitely many vertices χ(Γ) := 1 − ∑_{n=0}^{∞}(−1)^{n}|K_{n}|.

Remark 2.1 It is easy to see that there are co-irreducible graphs attaining any integer as its Euler characteristic. Indeed, letting Γ_{−m} denote the graph with m + 1 vertices and no edges, we clearly have

χ(Γ_{−m}) = −m.

Systematically generating positive characteristics is harder; one option is to let Γ_{n^{2}−1} denote the graph with 2n + 2 vertices obtained by deleting one edge from the complete bipartite graph K_{n+1,n+1} and note that

χ(Γ_{n^{2}−1}) = n^{2} − 1

To obtain positive characteristics in {(n − 1)^{2}, …, n^{2} − 1} one may simply add a suitable number of isolated vertices to Γ_{n^{2}−1}.

Now, by [CEL13, Theorem 5.2], we know that we always have K_{∗}(C^{∗}(A_{Γ}^{+})) ≅ K_{∗}(ℂ), and K_{0}(C^{∗}(A_{Γ}^{+})) ≅ ℤ is generated by the class of the unit [1]. Here we use that right-angled Artin groups satisfy the Baum-Connes conjecture with coefficients because these groups have the Haagerup property (see [NR97], and also [AD]). To compute K-theory for the quotient C_{Q}^{∗}(A_{Γ}^{+}) in the case that Γ has (more than one and) finitely many vertices, we consider the short exact sequence 0 → K → C^{∗}(A_{Γ}^{+}) → C_{Q}^{∗}(A_{Γ}^{+}) → 0 and its six term exact sequence in K-theory:

\xymatrix{K_{0}(K) \ar[r] & K_{0}(C^{∗}(A_{Γ}^{+})) \ar[r] & K_{0}(C_{Q}^{∗}(A_{Γ}^{+})) \ar[d] \\ K_{1}(C_{Q}^{∗}(A_{Γ}^{+})) \ar[u] & K_{1}(C^{∗}(A_{Γ}^{+})) \ar[l] & K_{1}(K) \ar[l]}  (1)

Since both K_{1}(K) and K_{1}(C^{∗}(A_{Γ}^{+})) vanish, all we have to do is to compute the homomorphism K_{0}(K) ≅ ℤ → ℤ ≅ K_{0}(C^{∗}(A_{Γ}^{+})). K_{0}(K) ≅ ℤ is generated by the class of any minimal projection. So we can take e = 1 − ⋁_{v∈V} S_{v}S_{v}^{∗}. It is easy to see that in K_{0}, [e] ∈ K_{0}(K) is sent to χ(Γ)[1] ∈ K_{0}(C^{∗}(A_{Γ}^{+})). Therefore, by exactness of (1), we conclude that K_{0}(C_{Q}^{∗}(A_{Γ}^{+})) ≅ ℤ/|χ(Γ)|ℤ and

K_{1}(C_{Q}^{∗}(A_{Γ}^{+})) ≅ \begin{cases}\{0\} if χ(Γ) ≠ 0 \\ ℤ if χ(Γ) = 0 \end{cases} .

3 Extension algebras

We now discuss the C*-algebras associated to co-irreducible graphs and see how they are all isomorphic to either the Toeplitz algebra, the Cuntz algebra O_{∞}, or an extension algebra as specified below.

Theorem 3.1 Consider the following properties for a unital C*-algebra A:

(1) A contains K as an ideal, and A/K is a Kirchberg algebra satisfying the UCT,

(2) K_{0}(A) = ℤ with [1_{A}] = 1.

For each k ∈ ℤ\{0} there is a unique C*-algebra satisfying (1), (2) and

(3) The six-term exact sequence for K and A is given by

\xymatrix{ℤ \ar[r]^{k} & ℤ \ar[r] & ℤ/kℤ \ar[d] \\ 0 \ar[u] & 0 \ar[l] & 0 \ar[l]}

There is also a unique C*-algebra satisfying (1), (2) and

(3’) The six-term exact sequence for K and A is given by

[figure: ℤ; 0; ℤ; ℤ; ℤ; 0; 0]

Proof: Note that K is an essential ideal of A (i.e., every nonzero ideal of A has a nontrivial intersection with K) since A is unital and A/K is simple. Uniqueness follows from [ERR, Corollary 4.22]. For existence, we note that when Γ is a finite and co-irreducible graph with |Γ| > 1 and χ(Γ) = k, all properties are met as noted in Section 2. □ When specifying the map K_{0}(K) → K_{0}(A) above we let the unit of the leftmost copy of ℤ denote the class of a minimal projection of ℤ.

Definition 3.2 The unique C*-algebras satisfying (1),(2) and (3) are denoted E_{|k|+1}^{sgn(k)}. The unique C*-algebra satisfying (1),(2) and (3’) is denoted E_{1}^{0}. The quotient E_{1}^{0}/K is denoted O_{1}.

Our notation has been chosen to fit the notation E_{n}^{k} for the extension algebras of O_{n} studied in [FL07]. With our name O_{1} for the appropriately chosen Kirchberg algebra, we have

\xymatrix{0 \ar[r] & K \ar[r]^{ι} & E_{n}^{k} \ar[r]^{π} & O_{n} \ar[r] & 0}

for any k ∈ {−1, 0, 1} and n ∈ ℕ, provided k = 0 precisely when n = 1.

Lemma 3.3 E_{n}^{k} ≅ E_{n′}^{k′} only when n = n′ and k = k′. E_{n}^{k} ⊗ K ≅ E_{n′}^{k′} ⊗ K precisely when n = n′.

Proof: Since the six-term exact sequences are as specified in (3) or (3’) of Theorem 3.1, stable isomorphism can only occur if n = n′, and hence we only need to check that for n > 0, we have E_{n}^{1} ≅ E̸_{n}^{−1}, yet E_{n}^{1} ⊗ K ≅ E_{n}^{−1} ⊗ K. We note that the only two options for an isomorphism among the six-term exact sequences in this case are given as

\xymatrix{ℤ \ar[r]^{n} \ar[d]^{±1} & ℤ \ar[r] \ar[d]^{∓1} & ℤ/nℤ \ar[d] \\ ℤ \ar[r]_{−n} & ℤ \ar[r] & ℤ/nℤ}

and that we must choose +1 as the left most isomorphism to preserve the positive cone of K_{0}(K). Thus, an isomorphism is ruled out as it would fail to send the class of the unit of E_{n}^{1} to the unit of E_{n}^{−1}, but an isomorphism after stabilization is guaranteed by, e.g., [ERR09]. □ The following result follows directly from § 2.2, § 2.4 and Theorem 3.1.

Theorem 3.4 When Γ is a co-irreducible graph, C^{∗}(A_{Γ}^{+}) is one of the C*- algebras T , E_{n}^{k}, O_{∞} according to

1. If |Γ| = 1, C^{∗}(A_{Γ}^{+}) ≅ T

2. If 1 < |Γ| < ∞, C^{∗}(A_{Γ}^{+}) ≅ E_{1+|χ(Γ)|}^{sgn χ(Γ)}

3. If |Γ| = ∞, C^{∗}(A_{Γ}^{+}) ≅ O_{∞}

We note that by the information already noted on the ideal structures in combination with Lemma 3.3, the C*-algebras appearing are not mutually isomorphic, and hence we have a complete classification by the cardinality of Γ and the Euler characteristic in the co-irreducible case. In preparation for the general case we now study isomorphisms between various tensor products amongst the relevant extension algebras and some of their quotients. For this, we will need:

Theorem 3.5 Let A_{i}, i = 1, 2, be unital C*-algebras whose proper ideals are precisely given by (0), I_{i}, J_{i} and I_{i} ⊕ J_{i}. We assume that I_{i} and J_{i} are UCT Kirchberg algebras, and the quotients A_{i}/(I_{i} ⊕ J_{i}) are also UCT Kirchberg algebras. Let α_{I} : K_{∗}(I_{1}) ≅ K_{∗}(I_{2}), α_{J} : K_{∗}(J_{1}) ≅ K_{∗}(J_{2}), α_{I⊕J} : K_{∗}(I_{1} ⊕ J_{1}) ≅ K_{∗}(I_{2} ⊕ J_{2}), β : K_{∗}(A_{1}) ≅ K_{∗}(A_{2}), γ_{I} : K_{∗}(A/I_{1}) ≅ K_{∗}(A/I_{2}), γ_{J} : K_{∗}(A/J_{1}) ≅ K_{∗}(A/J_{2}), and γ_{I⊕J} : K_{∗}(A_{1}/(I_{1} ⊕ J_{1})) ≅ K_{∗}(A_{2}/(I_{2} ⊕ J_{2})) be isomorphisms, with β preserving the K_{0}-classes of the units and α_{I⊕J} = α_{I}⊕α_{J} (under the canonical isomorphism K_{∗}(I_{i} ⊕ J_{i}) ≅ K_{∗}(I_{i}) ⊕ K_{∗}(J_{i})). Furthermore, we assume that these isomorphism are compatible with the Ktheoretic six term exact sequences attached to

0 → I_{i} → A_{i} → A_{i}/I_{i} → 0, 0 → J_{i} → A_{i} → A_{i}/J_{i} → 0,
0 → I_{i} ⊕ J_{i} → A_{i} → A_{i}/(I_{i} ⊕ J_{i}) → 0, 0 → J_{i} → A_{i}/I_{i} → A_{i}/(I_{i} ⊕ J_{i}) → 0

and

0 → I_{i} → A_{i}/J_{i} → A_{i}/(I_{i} ⊕ J_{i}) → 0.

Then there exists an isomorphism φ : A_{1} ≅ A_{2} which induces α_{I}, α_{J}, α_{I⊕J}, β, γ_{I}, γ_{J} and γ_{I⊕J} in K-theory.

Proof: Combine [Kir00, Folgerung 4.3] and [BK, Theorem 1.3] with [RR07, Theorem 2.1] or [ERR, Theorem 3.3]. □

Lemma 3.6 For every n ≥ 2, we have O_{∞} ⊗ E_{n}^{+1} ≅ O_{∞} ⊗ E_{n}^{−1}.

Proof: Both O_{∞} ⊗E_{n}^{+1} and O_{∞} ⊗E_{n}^{−1} are unital C*-algebras with unique ideal isomorphic to O_{∞} ⊗ K and corresponding quotient isomorphic to O_{∞} ⊗ O_{n} ≅ O_{n}. The K-theoretic six term exact sequences for 0 → O_{∞}⊗K → O_{∞}⊗E_{n}^{+1} → O_{n} → 0 and 0 → O_{∞} ⊗ K → O_{∞} ⊗ E_{n}^{−1} → O_{n} → 0 look as follows:

\xymatrix{ℤ \ar[r] & ℤ \ar[r] & ℤ/(n − 1)ℤ \ar[d] \\ 0 \ar[u] & 0 \ar[l] & 0 \ar[l]}

where ℤ ≅ K_{0}(O_{∞} ⊗ K) is generated by [1 ⊗ e] for a minimal projection e ∈ K and the unit 1 of O_{∞}, and ℤ ≅ K_{0}(O_{∞} ⊗ E_{n}^{±1}) is generated by the class of the unit. The only difference is that for E_{n}^{+1}, the homomorphism ℤ → ℤ is given by z[1 ⊗ e] ↦ (n − 1)z[1], whereas for E_{n}^{−1}, the homomorphism ℤ → ℤ is given by z[1⊗e] ↦ −(n−1)z[1] (for z ∈ ℤ). We now apply [RR07, Theorem 2.2] to I_{i} = O_{∞} ⊗ K, A_{1} = O_{∞} ⊗ E_{n}^{+1}, A_{2} = O_{∞} ⊗ E_{n}^{−1}, Q_{i} = O_{n} and the homomorphisms α = −id_{K_{0}(O_{∞}⊗K)}, β : K_{0}(O_{∞} ⊗ E_{n}^{+1}) → K_{0}(O_{∞} ⊗ E_{n}^{−1}), z[1] ↦ z[1] (for z ∈ ℤ), γ = id_{K_{0}(O_{n})}. It is then obvious that all the assumptions in [RR07] are satisfied, and we conclude that O_{∞} ⊗ E_{n}^{+1} ≅ O_{∞} ⊗ E_{n}^{−1}. □ Now recall that we have introduced the extension algebra E_{1}^{0} in Theorem 3.1. The C*-algebra E_{1}^{0} ⊗ E_{n}^{+1} (n ≥ 2) contains the ideal K ⊗ K ≅ K, and we denote the corresponding quotient by Q^{+}. Obviously, the primitive ideals of Q^{+} are given by K ⊗ O_{n}, O_{1} ⊗ K and K ⊗ O_{n} ⊕ O_{1} ⊗ K. From the six term exact sequence in K-theory for 0 → K → E_{1}^{0} ⊗ E_{n}^{+1} → Q^{+} → 0, we obtain K_{0}(Q^{+}) ≅ ℤ ≅ K_{1}(Q^{+}), where K_{0}(Q^{+}) is generated by [1_{Q^{+}}]. All this also holds for the quotient Q^{−} of E_{1}^{0} ⊗ E_{n}^{−1} by the ideal K ⊗ K ≅ K.

Lemma 3.7 Q^{+} and Q^{−} are isomorphic. Moreover, there exists an automorphism of Q^{+} which induces id_{ℤ} on K_{0} and −id_{ℤ} on K_{1}.

Proof: The first statement is an application of Theorem 3.5 to A_{1} = Q^{+}, A_{2} = Q^{−}, I_{1} = K ⊗ O_{n} ◁ Q^{+}, J_{1} = O_{1} ⊗ K ◁ Q^{+}, I_{2} = K ⊗ O_{n} ◁ Q^{−}, J_{2} = O_{1} ⊗ K ◁ Q^{−}. Namely, it is straightforward to check that it is possible to choose α_{I}, α_{J}, α_{I⊕J}, β, γ_{I}, γ_{J}, and γ_{I⊕J} with all the desired properties in Theorem 3.5. The second statement follows in a similar way by applying Theorem 3.5 to A_{1} = A_{2} = Q^{+}, I_{1} = I_{2} = K ⊗ O_{n} ◁ Q^{+}, J_{1} = J_{2} = O_{1} ⊗ K ◁ Q^{+}. □

Lemma 3.8 For every n ≥ 2, we have E_{1}^{0} ⊗ E_{n}^{+1} ≅ E_{1}^{0} ⊗ E_{n}^{−1}.

Proof: By the previous lemma, we can identify Q^{+} and Q^{−} (we use the same notation as in the previous lemma) so that we can view E_{1}^{0}⊗E_{n}^{+1} and E_{1}^{0}⊗E_{n}^{−1} as extensions of Q^{+}:

0 → K → E_{1}^{0} ⊗ E_{n}^{+1} → Q^{+} → 0  (2)
0 → K → E_{1}^{0} ⊗ E_{n}^{−1} → Q^{+} → 0.  (3)

Again by the previous lemma, we can choose the identification Q^{+} ≅ Q^{−} in such a way that for a fixed choice of isomorphisms K_{1}(Q^{+}) ≅ ℤ, K_{0}(K) ≅ ℤ, the index maps for both extensions (2) and (3) coincide. Now [BD96, Theorem 2] implies that (2) and (3) give the same class in Ext_{s}(Q^{+}). The reason is that Ext(K_{0}(Q^{+}), [1_{Q^{+}}]) is the trivial group as K_{0}(Q^{+}) ≅ ℤ and [1_{Q^{+}}] is a generator of K_{0}(Q^{+}) ≅ ℤ. So the short exact sequence in [BD96, Theorem 2] tells us that two extensions of Q^{+} by K give the same class in Ext_{s}(Q^{+}) if their index maps coincide. But this is the case for (2) and (3) by construction. Hence E_{1}^{0} ⊗ E_{n}^{+1} ≅ E_{1}^{0} ⊗ E_{n}^{−1} by [JT91, § 3.2]. □ For m, n ≥ 2, the C*-algebra E_{m}^{+1} ⊗ E_{n}^{−1} contains the ideal K ⊗ K ≅ K, and we denote the corresponding quotient by Q^{+−}. Obviously, the primitive ideals of Q^{+−} are given by K ⊗ O_{n}, O_{m} ⊗ K and K ⊗ O_{n} ⊕ O_{m} ⊗ K. From the six term exact sequence in K-theory for 0 → K → E_{m}^{+1} ⊗ E_{n}^{−1} → Q^{+−} → 0, we obtain K_{0}(Q^{+−}) ≅ ℤ/(m − 1)(n − 1)ℤ, with the class of the unit being a generator, and K_{1}(Q^{+−}) ≅ {0}. All this also holds for the quotient Q^{−+} of E_{m}^{−1} ⊗ E_{n}^{+1} by the ideal K ⊗ K ≅ K.

Lemma 3.9 Q^{+−} and Q^{−+} are isomorphic.

Proof: As Lemma 3.7, this is an application of Theorem 3.5 to A_{1} = Q^{+−}, A_{2} = Q^{−+}, I_{1} = K ⊗ O_{n} ◁ Q^{+−}, J_{1} = O_{m} ⊗ K ◁ Q^{+−}, I_{2} = K ⊗ O_{n} ◁ Q^{−+}, J_{2} = O_{m} ⊗ K ◁ Q^{−+}. Namely, it is straightforward to check that it is possible to choose α_{I}, α_{J}, α_{I⊕J}, β, γ_{I}, γ_{J}, and γ_{I⊕J} with all the desired properties in Theorem 3.5. □

Lemma 3.10 We have E_{m}^{+1} ⊗ E_{n}^{−1} ≅ E_{m}^{−1} ⊗ E_{n}^{+1}.

Proof: By the previous lemma, we can identify Q^{+−} and Q^{−+} (using the same notation as in the previous lemma) so that we can view E_{m}^{+1} ⊗ E_{n}^{−1} and E_{m}^{−1} ⊗ E_{n}^{+1} as extensions of Q^{+−}:

0 → K → E_{m}^{+1} ⊗ E_{n}^{−1} → Q^{+−} → 0  (4)
0 → K → E_{m}^{−1} ⊗ E_{n}^{+1} → Q^{+−} → 0.  (5)

Since Hom(K_{1}(Q^{+−}), ℤ) = {0}, [BD96, Theorem 2] yields Ext((K_{0}(Q^{+−}), [1]), ℤ) ≅ Ext_{s}(Q^{+−}). Hence (2) and (3) give the same class in Ext_{s}(Q^{+−}). The reason is that the exact sequences in K_{0} for (4) and (5) clearly give rise to the same class in Ext((K_{0}(Q^{+−}), [1]), ℤ). Hence E_{m}^{+1} ⊗ E_{n}^{−1} ≅ E_{m}^{−1} ⊗ E_{n}^{+1} by [JT91, § 3.2]. □ In an entirely analogous way, we get

Lemma 3.11 For all m, n ≥ 2, we have E_{m}^{+1} ⊗ E_{n}^{+1} ≅ E_{m}^{−1} ⊗ E_{n}^{−1}.

4 Classification of semigroup C*-algebras

We are now ready to address the general classification problem for C*-algebras of the form C^{∗}(A_{Γ}^{+}). We begin with notation:

Definition 4.1 Let Γ be a graph with co-irreducible components Γ_{i} = (V_{i}, E_{i}). We set

t(Γ) = # {Γ_{i}: |V_{i}| = 1}
o(Γ) = # {Γ_{i}: |V_{i}| = ∞} ,

and for every n ∈ ℤ

N_{n}(Γ) = # {Γ_{i}: 1 < |V_{i}| < ∞, χ(Γ_{i}) = n} .

Theorem 4.2 Let Γ and Λ be two graphs. The semigroup C*-algebras C^{∗}(A_{Γ}^{+}) and C^{∗}(A_{Λ}^{+}) of the Artin monoids for Γ and Λ are stably isomorphic if and only if the following conditions hold:

(i) t(Γ) = t(Λ);

(ii) N_{−n}(Γ) + N_{n}(Γ) = N_{−n}(Λ) + N_{n}(Λ) for any n ∈ ℤ;

(iii) ∑_{n∈ℤ} N_{n}(Γ) = ∞ or min(o(Γ), 1) = min(o(Λ), 1).

They are isomorphic if and only if further (iv) If ∑_{n∈ℤ} N_{n}(Γ) < ∞, o(Γ) = 0 and N_{0}(Γ) = 0, then

∑_{n=1}^{∞} N_{−n}(Γ) ≡ ∑_{n=1}^{∞} N_{−n}(Λ) mod 2

holds.

Remark 4.3 Note that when (ii) holds, all the conditions in (iii) are symmetric in Γ and Λ. Similarly, when (ii) and (iii) hold, so are the conditions in (iv).

For the proof of Theorem 4.2, we need some preparation. Given a graph Γ with co-irreducible components Γ_{i} = (V_{i}, E_{i}), let Γ′ be the graph we get from Γ by removing all the co-irreducible components Γ_{i} with |V_{i}| = 1 and the corresponding edges. We then have a canonical isomorphism C^{∗}(A_{Γ′}^{+}) ≅ ⨂_{{Γ_{i}: |V_{i}|>1}} C^{∗}(A_{Γ_{i}}^{+}).

Lemma 4.4 There is a primitive ideal I′ of C^{∗}(A_{Γ}^{+}) such that Prim (C^{∗}(A_{Γ}^{+})/I′) does not continuously surject onto Prim (T ) and which is minimal among all the primitive ideals having this property, and we have

C^{∗}(A_{Γ}^{+})/I′ ≅ C^{∗}(A_{Γ′}^{+}).

Proof: Let I be a primitive ideal of C^{∗}(A_{Γ}^{+}). As seen in Section 2, we know that I is generated by {⨂_{j} J_{ij}}_{i}, where J_{ij} = C^{∗}(A_{Γ_{j}}^{+}) for i ≠ j and J_{ii} = I_{i} for

primitive ideals I_{i} of C^{∗}(A_{Γ_{i}}^{+}). It follows that C^{∗}(A_{Γ}^{+})/I ≅ ⨂_{i} C^{∗}(A_{Γ_{i}}^{+})/I_{i}, and hence Prim (C^{∗}(A_{Γ}^{+})/I) ≅ ∏_{i} Prim (C^{∗}(A_{Γ_{i}}^{+})/I_{i}). We now claim that there exists a continuous surjection Prim (C^{∗}(A_{Γ}^{+})/I) ↠ Prim T if and only if there exists a co-irreducible component Γ_{i} of Γ with |V_{i}| = 1 and I_{i} = (0). The direction “⇐” is obvious. For “⇒”, assume that for every co-irreducible component Γ_{i} of Γ with |V_{i}| = 1, I_{i} is a maximal ideal of C^{∗}(A_{Γ_{i}}^{+}) such that C^{∗}(A_{Γ_{i}}^{+})/I_{i} ≅ ℂ. Then Prim (C^{∗}(A_{Γ}^{+})/I) ≅ ∏_{k} X_{k} where X_{k} = {x_{k}, y_{k}} and the open subsets of X_{k} are given by ∅, \{x_{k}\} and X_{k}. This means that \overline{\{x_{k}\}} = X_{k} and \overline{\{y_{k}\}} = \{y_{k}\}. Furthermore, we know that Prim (T ) = \{•\} ⊔ 𝕋, where \overline{\{•\}} = Prim (T ). Let f : ∏_{k} X_{k} → Prim (T ) be a continuous map. We want to show that f cannot be surjective. Let y = (y_{k})_{k} and f(y) = z. For arbitrary x ∈ ∏_{k} X_{k}, we always have y ∈ \overline{\{x\}}. As f^{−1}(\overline{\{f(x)\}}) is closed and contains x, it must also contain y. Hence z = f(y) lies in \overline{\{f(x)\}}. This implies that f(x) = z or f(x) = •. But this holds for every x in ∏_{k} X_{k}. Hence the image of f contains at most 2 points, and thus f cannot be surjective. This shows our claim. Therefore, a primitive ideal I′ of C^{∗}(A_{Γ}^{+}) such that Prim (C^{∗}(A_{Γ}^{+})/I′) does not continuously surject onto Prim (T ) and which is minimal among all the primitive ideals with this property is generated by {⨂_{j} J_{ij}}_{i}, where for a co-

irreducible component Γ_{i} with |V_{i}| = 1, J_{ii} = I_{i} is a maximal ideal of C^{∗}(A_{Γ_{i}}^{+}) with C^{∗}(A_{Γ_{i}}^{+})/I_{i} ≅ ℂ, and for a co-irreducible component Γ_{i} with |V_{i}| > 1, J_{ii} = (0). We conclude that C^{∗}(A_{Γ}^{+})/I′ ≅ C^{∗}(A_{Γ′}^{+}). □

Lemma 4.5 Let Γ and Λ be two graphs.

(1) If C^{∗}(A_{Γ}^{+}) and C^{∗}(A_{Λ}^{+}) are isomorphic, then t(Γ) = t(Λ) and C^{∗}(A_{Γ′}^{+}) ≅ C^{∗}(A_{Λ′}^{+}).

(2) If C^{∗}(A_{Γ}^{+}) ⊗ K and C^{∗}(A_{Λ}^{+}) ⊗ K are isomorphic, then t(Γ) = t(Λ) and C^{∗}(A_{Γ′}^{+}) ⊗ K ≅ C^{∗}(A_{Λ′}^{+}) ⊗ K.

Proof: We first prove (1). Since an isomorphism C^{∗}(A_{Γ}^{+}) ≅ C^{∗}(A_{Λ}^{+}) sends the primitive ideal I to a primitive ideal of C^{∗}(A_{Λ}^{+}) with the analogous property, we conclude that every isomorphism C^{∗}(A_{Γ}^{+}) ≅ C^{∗}(A_{Λ}^{+}) induces an isomorphism C^{∗}(A_{Γ′}^{+}) ≅ C^{∗}(A_{Λ′}^{+}). To prove that t(Γ) = t(Λ), we observe that the primitive ideals of C^{∗}(A_{Γ}^{+}) which are contained in I are in one-to-one correspondence with the subsets of {Γ_{i}: |V_{i}| = 1}. Again, as an isomorphism C^{∗}(A_{Γ}^{+}) ≅ C^{∗}(A_{Λ}^{+}) sends the primitive ideal I to a primitive ideal of C^{∗}(A_{Λ}^{+}) with the analogous property, we conclude that the power sets of {Γ_{i}: |V_{i}| = 1} and {Λ_{j}: |W_{j}| = 1} have the same cardinality. Hence also {Γ_{i}: |V_{i}| = 1} and {Λ_{j}: |W_{j}| = 1} must have the same cardinality (which is either finite or countably infinite). This proves (1).

(2) is proved in a similar way as (1) using the observation that every primitive ideal of B ⊗ K is of the form I ⊗ K, where I is a primitive ideal of B. □

Lemma 4.6 Let A_{i}, i = 1, 2, …, be a countably infinite family of properly infinite unital C*-algebras. Then A = ⨂_{i=1}^{∞} A_{i} is purely infinite.

Proof: We have to show that every non-zero positive element a of A is properly infinite. By [KR02, Lemma 3.3], it suffices to find for every ε > 0 a properly infinite, positive element b ∈ A with ∥a − b∥ < ε and b ≾ a. Since A = ⨂_{i=1}^{∞} A_{i}, there exists a (sufficiently large) natural number n and a positive element x ∈ ⨂_{i=1}^{n} A_{i} with ∥a − x ⊗ 1∥ < \frac{ε}{2}. By [KR02, Lemma 2.2], we have that b := (x − \frac{ε}{2})_{+} ⊗ 1 = (x ⊗ 1 − \frac{ε}{2})_{+} satisfies b ≾ a. Also, we have ∥b − a∥ ≤ ∥b − x ⊗ 1∥ + ∥x ⊗ 1 − a∥ < ε. So it suffices to show that b is properly infinite. By construction, b is of the form c ⊗ 1 for some positive element c ∈ ⨂_{i=1}^{n} A_{i}. Since the unit 1 ∈ A_{n+1} is properly infinite, we can find isometries s and t in A_{n+1} with ss^{∗} ⊥ tt^{∗}. So b = c⊗1 = (c^{1/2}⊗s)^{∗}(c^{1/2}⊗s) ≈ (c^{1/2}⊗s)(c^{1/2}⊗s)^{∗} = c ⊗ ss^{∗}. Similarly, b ≈ c ⊗ tt^{∗}. But since (c ⊗ ss^{∗})(c ⊗ tt^{∗}) = 0, we conclude that b ⊕ b ≈ (c ⊗ ss^{∗}) ⊕ (c ⊗ tt^{∗}) ≈ c ⊗ (ss^{∗} + tt^{∗}) ≤ c ⊗ 1 = b. □

Lemma 4.7 Let Γ be a graph with (countably) infinitely many co-irreducible components Γ_{i} = (V_{i}, E_{i}), i = 1, 2, . . .. Assume that 1 < |V_{i}| < ∞ for all i. Then C^{∗}(A_{Γ}^{+}) is strongly purely infinite, i.e., C^{∗}(A_{Γ}^{+}) ≅ C^{∗}(A_{Γ}^{+}) ⊗ O_{∞}.

Proof: By [CL07, Theorem 8.3], we know that C^{∗}(A_{Γ}^{+}) has the ideal property (the definition can be found in [PR07, Remark 2.1]). Moreover, we know that C^{∗}(A_{Γ}^{+}) ≅ ⨂_{i=1}^{∞} C^{∗}(A_{Γ_{i}}^{+}), and each of the C^{∗}(A_{Γ_{i}}^{+}) is a properly infinite unital C*-algebra. Hence by the previous lemma, we know that C^{∗}(A_{Γ}^{+}) is purely infinite. Therefore, [PR07, Proposition 2.14] tells us that C^{∗}(A_{Γ}^{+}) is strongly purely infinite. And finally, if C^{∗}(A_{Γ}^{+}) is strongly purely infinite, then [KR02, Theorem 9.1] implies that C^{∗}(A_{Γ}^{+}) ≅ C^{∗}(A_{Γ}^{+})⊗O_{∞} because C^{∗}(A_{Γ}^{+}) is nuclear and unital. □ Finally, we are ready for the proof of Theorem 4.2. Proof:[Proof of Theorem 4.2] Let us first of all show that if C^{∗}(A_{Γ}^{+})⊗K ≅ C^{∗}(A_{Λ}^{+})⊗K holds, then conditions (i), (ii) and (iii) must be satisfied. By Lemma 4.5 condition (i) holds and that C^{∗}(A_{Γ′}^{+}) ⊗ K ≅ C^{∗}(A_{Λ′}^{+}) ⊗ K. Hence we may assume that all the co-irreducible components of Γ and Λ have more than one vertex. To prove (ii), we observe that the minimal non-zero primitive ideals of C^{∗}(A_{Γ}^{+}) are of the form I_{i} = ⊗_{j}J_{ij}, where J_{ij} = C^{∗}(A_{Γ_{j}}^{+}) if j ≠ i, and J_{ii} = K◁C^{∗}(A_{Γ_{i}}^{+}) (Γ_{i} consists of only finitely many vertices). For the corresponding quotient, we get C^{∗}(A_{Γ}^{+})/I_{i} ≅ ⨂_{j} Q_{ij}, where Q_{ij} = C^{∗}(A_{Γ_{j}}^{+}) if j ≠ i, and Q_{ii} = C^{∗}(A_{Γ_{i}}^{+})/K. Since K_{0}(C^{∗}(A_{Γ_{i}}^{+})/K) ≅ ℤ/|χ(Γ_{i})|ℤ, it follows that K_{0}(C^{∗}(A_{Γ}^{+})/I_{i}) ≅ ℤ/|χ(Γ_{i})|ℤ. Hence, we have shown that N_{0}(Γ) is the number of minimal non-zero primitive ideals I ⊗ K of C^{∗}(A_{Γ}^{+}) ⊗ K with the property that K_{0}(C^{∗}(A_{Γ}^{+})/I) ≅ ℤ, and that for every n = 1, 2, …, N_{−n}(Γ) + N_{n}(Γ) is the number of minimal non-zero primitive ideals I ⊗ K of C^{∗}(A_{Γ}^{+}) ⊗ K with the property that K_{0}(C^{∗}(A_{Γ}^{+})/I) ≅ ℤ/nℤ. Since these descriptions are invariant under stable isomorphisms of C*-algebras, we conclude that (ii) must hold. Let us now prove (iii) under the assumption of stable isomorphism. If ∑ N_{n}(Γ) = ∞ we are done, so suppose the contrary and note that in this case, C^{∗}(A_{Γ}^{+}) is strongly purely infinite if and only if o(Γ) > 0. The direction “⇒” is clear, since o(Γ) > 0 implies that C^{∗}(A_{Γ}^{+}) has O_{∞} as a tensor factor. To prove “⇐”, we observe that if o(Γ) = 0, then C^{∗}(A_{Γ}^{+}) contains the algebra of compact operators as an ideal, hence cannot be strongly purely infinite. As a consequence, C^{∗}(A_{Γ}^{+}) ⊗ K ≅ C^{∗}(A_{Λ}^{+}) ⊗ K implies that either both o(Γ) > 0 and o(Λ) > 0, or o(Γ) = o(Λ) = 0, as desired. Finally, we assume that C^{∗}(A_{Γ}^{+}) ≅ C^{∗}(A_{Λ}^{+}) and that ∑ N_{n}(Γ) < ∞, that N_{0}(Γ) = 0 and that o(Γ) = 0. The algebra K of compact operators sits inside C^{∗}(A_{Γ}^{+}) as the (unique) minimal non-zero ideal. The inclusion K ↪ C^{∗}(A_{Γ}^{+}) sends in K-theory the K_{0}-class of a minimal projection to (∏_{i} χ(Γ_{i})) · [1], where (∏_{i} χ(Γ_{i})) is the product over all co-irreducible components of Γ (there are only finitely many by assumption) of the Euler characteristics. As N_{0}(Γ) = 0, (∏_{i} χ(Γ_{i})) is a non-zero number, and it is positive if and only if ∑_{n=1}^{∞} N_{−n}(Γ) ≡ 0 mod 2. Since C^{∗}(A_{Γ}^{+}) ≅ C^{∗}(A_{Λ}^{+}), we must have ∑_{n=1}^{∞} N_{−n}(Γ) ≡ ∑_{n=1}^{∞} N_{−n}(Λ) mod 2. Therefore, all in all, condition (iv) follows when the C*-algebras are isomorphic. In the opposite direction, we know from Sections 2 and 3 that

C^{∗}(A_{Γ}^{+}) ≅ T^{⊗t(Γ)} ⊗ O_{∞}^{⊗o(Γ)} ⊗ ⨂_{n=0}^{∞} ⨂_{{i: |χ(Γ_{i})|=n}} E_{1+n}^{sgn(χ(Γ_{i}))}

and

C^{∗}(A_{Λ}^{+}) ≅ T^{⊗t(Λ)} ⊗ O_{∞}^{⊗o(Λ)} ⊗ ⨂_{n=0}^{∞} ⨂_{{i: |χ(Λ_{i})|=n}} E_{1+n}^{sgn(χ(Λ_{i}))}

We note from the outset that whenever o(Γ) > 0 or N_{0}(Γ) > 0 then by repeated application of either Lemma 3.6 or Lemma 3.8 we may simplify these expressions to

C^{∗}(A_{Γ}^{+}) ≅ T^{⊗t(Γ)} ⊗ O_{∞}^{⊗o(Γ)} ⊗ (E_{1}^{0})^{⊗N_{0}(Γ)} ⊗ ⨂_{n=1}^{∞}(E_{1+n}^{+1})^{⊗(N_{−n}(Γ)+N_{n}(Γ))}. (6)

Assume that (i), (ii) and (iii) hold. We begin by noting that in the case ∑ N_{n}(Γ) < ∞ if either o(Γ) > 0 or N_{0}(Γ) > 0, we also have either o(Λ) > 0 or N_{0}(Λ) > 0, and we get C^{∗}(A_{Γ}^{+}) ≅ C^{∗}(A_{Λ}^{+}) by reducing to the form given in (6) and applying (i) and (ii).

When ∑ N_{n}(Γ) = ∞ then we have by (ii) and Lemma 4.7 that both C^{∗}(A_{Γ}^{+}) and C^{∗}(A_{Λ}^{+}) are strongly purely infinite, and hence we have

C^{∗}(A_{Γ}^{+}) ≅ T^{⊗t(Γ)} ⊗ O_{∞} ⊗ (E_{1}^{0})^{⊗N_{0}(Γ)} ⊗ ⨂_{n=1}^{∞}(E_{1+n}^{+1})^{⊗(N_{−n}(Γ)+N_{n}(Γ))}

= T^{⊗t(Λ)} ⊗ O_{∞} ⊗ (E_{1}^{0})^{⊗N_{0}(Λ)} ⊗ ⨂_{n=1}^{∞}(E_{1+n}^{+1})^{⊗(N_{−n}(Λ)+N_{n}(Λ))}
≅ C^{∗}(A_{Λ}^{+}),

since Lemma 3.6 may be applied as above.

It remains to treat the case that o(Γ) = N_{0}(Γ) = 0 and ∑ N_{n}(Γ) < ∞. Again, by (ii) and (iii), we must have o(Λ) = N_{0}(Λ) = 0 and ∑ N_{n}(Λ) < ∞ as well, and we get

C^{∗}(A_{Γ}^{+}) ⊗ K ≅ T^{⊗t(Γ)} ⊗ ⨂_{n=1}^{∞}(E_{1+n}^{+1})^{⊗(N_{−n}(Γ)+N_{n}(Γ))} ⊗ K

= T^{⊗t(Λ)} ⊗ ⨂_{n=1}^{∞}(E_{1+n}^{+1})^{⊗(N_{−n}(Λ)+N_{n}(Λ))} ⊗ K
≅ C^{∗}(A_{Λ}^{+}) ⊗ K

this time appealing to the second half of Lemma 3.3.

Assuming further (iv), we now aim for exact isomorphism, noting that we have already established it when o(Γ) > 0, N_{0}(Γ) > 0 or ∑ N_{n}(Γ) = ∞. We hence assume that o(Γ) = N_{0}(Γ) = 0 and note that also o(Λ) = N_{0}(Λ) = 0 Consider first the case where both ∑_{n=1}^{∞} N_{−n}(Γ) and ∑_{n=1}^{∞} N_{−n}(Λ) are even. We have

C^{∗}(A_{Γ}^{+}) ≅ T^{⊗t(Γ)} ⊗ ⨂_{n=1}^{∞}(E_{1+n}^{+1})^{⊗N_{n}(Γ)} ⊗ ⨂_{n=1}^{∞}(E_{1+n}^{−1})^{⊗N_{−n}(Γ)}

≅ T^{⊗t(Γ)} ⊗ ⨂_{n=1}^{∞}(E_{1+n}^{+1})^{⊗N_{n}(Γ)} ⊗ ⨂_{n=1}^{∞}(E_{1+n}^{+1})^{⊗N_{−n}(Γ)}

≅ T^{⊗t(Λ)} ⊗ ⨂_{n=1}^{∞}(E_{1+n}^{+1})^{⊗(N_{−n}(Γ)+N_{n}(Γ))}

= T^{⊗t(Λ)} ⊗ ⨂_{n=1}^{∞}(E_{1+n}^{+1})^{⊗(N_{−n}(Λ)+N_{n}(Λ))} ≅ C^{∗}(A_{Λ}^{+})

by Lemma 3.11. Now assume both ∑_{n=1}^{∞} N_{−n}(Γ) and ∑_{n=1}^{∞} N_{−n}(Λ) are odd. If there exists χ < 0 such that there are co-irreducible components Γ_{k} and Λ_{l} with χ(Γ_{k}) = χ = χ(Λ_{l}), then we deduce from the previous case that

C^{∗}(A_{Γ}^{+}) ≅ ( ⨂_{Γ_{i}≠Γ_{k}} C^{∗}(A_{Γ_{i}}^{+})) ⊗ C^{∗}(A_{Γ_{k}}^{+})

≅ ( ⨂_{Γ_{i}≠Γ_{k}} C^{∗}(A_{Γ_{i}}^{+})) ⊗ E_{1+|χ|}^{−1}

≅ ( ⨂_{Λ_{j}≠Λ_{l}} C^{∗}(A_{Λ_{j}}^{+})) ⊗ C^{∗}(A_{Λ_{l}}^{+}) ≅ C^{∗}(A_{Λ}^{+}).

If there exists no such χ, then by (ii) there must be χ < 0, ψ < 0 and coirreducible components Γ_{k_{−}}, Γ_{k_{+}}, Λ_{l_{−}}, Λ_{l_{+}} with χ(Γ_{k_{−}}) = χ, χ(Λ_{l_{+}}) = −χ, χ(Γ_{k_{+}}) = −ψ and χ(Λ_{l_{−}}) = ψ. Hence

C^{∗}(A_{Γ}^{+}) ≅ ( ⨂_{Γ_{i}≠Γ_{k_{+}} , Γ_{k_{−}}} C^{∗}(A_{Γ_{i}}^{+})) ⊗ C^{∗}(A_{Γ_{k_{+}}}^{+} ) ⊗ C^{∗}(A_{Γ_{k_{−}}}^{+} )

≅ ( ⨂_{Γ_{i}≠Γ_{k_{+}} , Γ_{k_{−}}} E_{1+|χ(Γ_{i})|}^{sgn(χ(Γ_{i}))}) ⊗ E_{1+|ψ|}^{+1} ⊗ E_{1+|χ|}^{−1}

≅ ( ⨂_{Γ_{i}≠Γ_{k_{+}} , Γ_{k_{−}}} E_{1+|χ(Γ_{i})|}^{sgn(χ(Γ_{i}))}) ⊗ E_{1+|ψ|}^{−1} ⊗ E_{1+|χ|}^{+1}

≅ ( ⨂_{Γ_{i}≠Γ_{k_{+}} , Γ_{k_{−}}} C^{∗}(A_{Γ_{i}}^{+})) ⊗ C^{∗}(A_{Λ_{l_{−}}}^{+} ) ⊗ C^{∗}(A_{Λ_{l_{+}}}^{+} ) ≅ C^{∗}(A_{Λ}^{+}).

In the third step, we used Lemma 3.10, and in the fourth step, we used our argument in the previous case. □

5 The isomorphism problem from the perspective of classification of non-simple C*-algebras

We give an interpretation of Theorem 4.2 from the point of view of classifying non-simple C*-algebras. We let O(Prim(A)) denote the set of open subsets in Prim(A), and 𝕀(A) the lattice of ideals. A lattice map ψ_{A}: O(Prim(A)) → 𝕀(A) given by ψ_{A}(U) = ⋂_{ρ∉U} ρ is then a lattice isomorphism which preserves arbitrary suprema and finite infima. We denote ψ_{A}(U) by A[U]. For every C*-algebra A, we denote the pair

(Prim(A), \{K_{six}^{+}(A/A[U]; A[V]/A[U])\}\begin{matrix}V,U∈𝕆(Prim(A)) \\ U⊆V \end{matrix})

by 𝔉(A), where K_{six}^{+}(B, J) denotes the standard six-term exact sequence associated to an ideal J of a C*-algebra B, considering each K_{0}-group as an ordered group. An isomorphism from 𝔉(A) to 𝔉(B) thus consists of a homeomorphism

ϕ: Prim(A) → Prim(B)

and isomorphisms

α_{U,V} : K_{∗}(A[V]/A[U]) → K_{∗}(B[ϕ(V)]/B[ϕ(U)])

for each U, V ∈ 𝕆(Prim(A)) with U ⊆ V, such that (α_{U,V} , α_{X,U}, α_{X,V}) is an isomorphism from K_{six}^{+}(A/A[U]; A[V]/A[U]) to

K_{six}^{+}(B/B[ϕ(U)]; B[ϕ(V)]/B[ϕ(U)])

in the sense that it makes all squares commute and is an order isomorphism on all even parts of the K-theory. If A and B are unital, we write (𝔉(A), [1_{A}]) ≅ (𝔉(B), [1_{B}]) if 𝔉(A) ≅ 𝔉(B) in such a way that the isomorphism α_{X,∅} sends [1_{A}] in K_{0}(A) to [1_{B}] in K_{0}(B). Note that if ϕ: Prim(A) → Prim(B) is a homeomorphism, there exists a lattice isomorphism from 𝕀(A) to 𝕀(B) given by I ↦ ψ_{B}(ϕ(ψ_{A}^{−1}(I))). Hence, if A and B are separable and ϕ: Prim(A) → Prim(B) is a homeomorphism, then for all U ∈ 𝕆(Prim(A)), we have that A[U] is a primitive ideal of A if and only if B[ϕ(U)] is a primitive ideal of B (because primitive ideals are precisely given by prime ideals for separable C*-algebras). The following easy observation is left to the reader.

Lemma 5.1 Let A and B be separable C*-algebras. Let U ∈ 𝕆(Prim(A)).

(1) If 𝔉(A) ≅ 𝔉(B) via a homeomorphism ϕ: Prim(A) → Prim(B), then

𝔉(A/A[U]) ≅ 𝔉(B/B[ϕ(U)]).

(2) If A and B are unital C*-algebras and (𝔉(A), [1_{A}]) ≅ (𝔉(B), [1_{B}]) via a homeomorphism ϕ: Prim(A) → Prim(B), then

(𝔉(A/A[U]), [1_{A/A[U]}]) ≅ (𝔉(B/B[ϕ(U)]), [1_{B/B[ϕ(U)]}]) .

Theorem 5.2 Let Γ and Λ be two (countable) graphs, and let Γ_{i} = (V_{i}, E_{i}) be the co-irreducible components of Γ, Λ_{j} = (W_{j}, F_{j}) the co-irreducible components of Λ. Then C^{∗}(A_{Γ}^{+}) ≅ C^{∗}(A_{Λ}^{+}) if and only if (𝔉(C^{∗}(A_{Γ}^{+})), [1_{C^{∗}(A_{Γ}^{+})}]) ≅ (𝔉(C^{∗}(A_{Λ}^{+})), [1_{C^{∗}(A_{Λ}^{+})}]).

Proof: The direction “⇒” is obvious. To prove “⇐”, we show that

(𝔉(C^{∗}(A_{Γ}^{+})), [1_{C^{∗}(A_{Γ}^{+})}]) ≅ (𝔉(C^{∗}(A_{Λ}^{+})), [1_{C^{∗}(A_{Λ}^{+})}])

implies (i), (ii), (iii) and (iv) from Theorem 4.2, using the notations from Lemma 4.5. The first step is to prove that (𝔉(C^{∗}(A_{Γ}^{+})), [1_{C^{∗}(A_{Γ}^{+})}]) ≅ (𝔉(C^{∗}(A_{Λ}^{+})), [1_{C^{∗}(A_{Λ}^{+})}]) implies t(Γ) = t(Λ) and (𝔉(C^{∗}(A_{Γ′}^{+})), [1_{C^{∗}(A_{Γ′}^{+} )}]) ≅

(𝔉(C^{∗}(A_{Λ′}^{+})), [1_{C^{∗}(A_{Λ′}^{+} )}]). t(Γ) = t(Λ) follows by Lemma 4.4, because we only use the primitive ideal space and the lattice structure of the set of ideals in this proof. To see that (𝔉(C^{∗}(A_{Γ′}^{+})), [1_{C^{∗}(A_{Γ′}^{+} )}]) ≅ (𝔉(C^{∗}(A_{Λ′}^{+})), [1_{C^{∗}(A_{Λ′}^{+} )}]),

let I′ be a primitive ideal of C^{∗}(A_{Γ}^{+}) stipulated in Lemma 4.4, and let U be an open set of Prim (C^{∗}(A_{Γ}^{+})) such that C^{∗}(A_{Γ}^{+})[U] = I′. Then C^{∗}(A_{Λ}^{+})[ϕ(U)] is an ideal with the analogous property. In the proof of Lemma 4.5, we have seen that C^{∗}(A_{Γ}^{+})/C^{∗}(A_{Γ}^{+})[U] ≅ C^{∗}(A_{Γ′}^{+}). Similarly, we have C^{∗}(A_{Λ}^{+})/C^{∗}(A_{Λ}^{+})[U] ≅ C^{∗}(A_{Λ′}^{+}). Therefore, (2) from the previous lemma tells us that (𝔉(C^{∗}(A_{Γ′}^{+})), [1_{C^{∗}(A_{Γ′}^{+} )}]) ≅ (𝔉(C^{∗}(A_{Λ′}^{+})), [1_{C^{∗}(A_{Λ′}^{+} )}]), as desired. In particular, this implies (i), and we may assume as in the proof of Theorem 4.2 that all the co-irreducible components of Γ and Λ have more than one vertex. Then (ii) follows in exactly the same way as in the proof of Theorem 4.2 because we only use primitive ideal spaces, lattice structures of sets of ideals and K_{0} in this proof. All this can be extracted from the invariant 𝔉. Let us prove (iii). As we have seen in the proof of Theorem 4.2, o(Γ) = 0 implies that K is an ideal of C^{∗}(A_{Γ}^{+}), whereas o(Γ) > 0 implies that C^{∗}(A_{Γ}^{+}) (and hence also every non-zero ideal) is strongly purely infinite. These two cases can be distinguished by the order on K_{0}. Therefore, we see as in the proof of Theorem 4.2 that if (𝔉(C^{∗}(A_{Γ}^{+})), [1_{C^{∗}(A_{Γ}^{+})}]) ≅ (𝔉(C^{∗}(A_{Λ}^{+})), [1_{C^{∗}(A_{Λ}^{+})}]), then either o(Γ) > 0 and o(Λ) > 0 or o(Γ) = 0 and o(Λ) = 0. The proof of (iv) then follows the proof of Theorem 4.2, where we only use lattice structures of sets of ideals, K_{0} and the K_{0}-classes of the units. All this can be extracted from the invariant 𝔉 together with the position of the K_{0}-class of the unit. □

6 Graph algebras and the semiprojectivity question

Apart from semigroup C*-algebras we discussed above, there is another - more traditional - way of constructing a C*-algebra out of a directed graph, possibly allowing for loops. Now we would like to discuss the overlap of these two constructions. In other words, we are interested in the question: Which semigroup C*-algebras for right-angled Artin monoids are isomorphic to graph algebras? We can provide a complete answer to this question.

6.1 Extensions of C*-algebras

We first establish some facts about absorbing extensions and the C*-algebras associated to these extensions. To each injective Busby map τ: A → Q(B), where Q(B) = M(B)/B with M(B) the multiplier algebra of B, associate as usual the extension

[figure: e :; 0; B; E; ψ; A; τ; 0; 0; B; M(B); π; Q(B); 0]

with E = π^{−1}(τ(A)) and ψ(x) = τ^{−1}(π(x)). Note that ψ is a homomorphism since τ is injective. We call τ (and e) unital if A is unital and τ is a unital homomorphism, or, equivalently, if E is a unital C*-algebra. If τ = π ∘ α for some homomorphism α: A → M(B), then τ is called a trivial extension. If A is unital and τ = π ∘ α for some unital homomorphism α: A → M(B), then τ is called strongly unital. Not all unital trivial extensions are strongly unital. Assume that B is stable. The sum τ ⊕ τ′ of two extensions τ, τ′: A → Q(B) is defined as follows. Since B is stable, there exist isometries s_{1}, s_{2} ∈ M(B) with 1_{M(B)} = s_{1}s_{1}^{∗} + s_{2}s_{2}^{∗}. Set

(τ ⊕ τ′)(a) = π(s_{1})τ(a)π(s_{1}^{∗}) + π(s_{2})τ′(a)π(s_{2}^{∗})

for all a ∈ A. Two extensions τ, τ′: A → Q(B) are said to be unitarily equivalent, denoted by τ ∼_{u} τ′, if there exists a unitary u ∈ M(B) such that π(u)τ(a)π(u)^{∗} = τ′(a) for all a ∈ A. Then two extensions τ_{1}, τ_{2} : A → Q(B) define the same element in Ext(A, B) if there exists a unitary u ∈ M(B) and there exist trivial extensions τ′_{1}, τ′_{2} : A → Q(B) such that τ_{1}⊕τ′_{1} ∼_{u} τ_{2}⊕τ′_{2}. If τ_{1} and τ_{2} are unital extensions, then τ′_{1} and τ′_{2} can be chosen to be unital extensions (see [Ror97, Section 5]). For a C*-algebra C, we let C̃ be the unitization of C (adding a new unit if C is a unital C*-algebra) and let ι_{C}: C → C̃ be the embedding of C into C̃ as an ideal. Recall that an ideal I of a C*-algebra A is an essential ideal if every nonzero ideal of A has a nontrivial intersection with I. An extension 0 → I →^{ι} A → B → 0 is essential if ι(I) is an essential ideal of A. It is a well-known fact that an extension 0 → I → A → B → 0 is an essential extension if and only if the Busby invariant of the extension is injective. We now prove in the following proposition that every essential extension 0 → B → E → A → 0 with A a nonunital, separable, nuclear C*-algebra and B a C*-algebra that is isomorphic to either K or a nuclear, purely infinite simple C*-algebra is absorbing.

Before proving the proposition, we show that any absorbing extension must be an essential extension. Hence, the assumption that the extension is essential is necessary. Note that if τ or τ′ is injective, then the sum τ ⊕τ′ is injective. Since B is stable, there exists a unital embedding from O_{2} to M(B) which induces a unital embedding from O_{2} to Q(B). Nuclearity of A gives us an embedding of A into O_{2}, thus the composition gives a trivial essential extension τ_{0} : A → Q(B). Therefore, an absorbing extension τ is an essential extension since τ is unitarily equivalent to τ ⊕ τ_{0}.

Proposition 6.1 Let A be a non-unital, separable, nuclear C*-algebra and let B be a separable C*-algebra that is isomorphic to either K or a nuclear, purely infinite simple C*-algebra. If τ: A → Q(B) is an essential extension, then for every trivial extension τ_{0} : A → Q(B) we have that τ ∼_{u} τ ⊕ τ_{0}. Consequently, if e_{i} : 0 → B → E_{i} → A → 0 is an essential extension for i = 1, 2 and [τ_{e_{1}}] = [τ_{e_{2}}] in Ext(A, B), then E_{1} ≅ E_{2}.

Proof: Let α_{0} : A → M(B) be a homomorphism with τ_{0} = π ∘ α_{0}. Extend τ and α_{0} to the unitization of Ã, and denote these extensions by τ̃: Ã → Q(B) and α̃_{0} : Ã → M(B) respectively.

We claim that τ̃ is injective. Let y ∈ ker (τ̃). Then τ(yx) = τ̃(y)τ̃(ι_{A}(x)) = 0 and τ(xy) = τ̃(ι_{A}(x))τ̃(y) = 0 for all x ∈ A. Since τ is injective, we have that yx = xy = 0 for all x ∈ A. Since A is non-unital, A is an essential ideal of Ã. Hence, y = 0. Thus, proving our claim.

Set E = π^{−1}(τ̃(Ã)) ⊆ M(B). Since τ̃ is injective, we may define a surjective homomorphism ψ: E → Ã by ψ(x) = τ^{−1}(π(x)). Define η: E → M(B) by η(x) = α̃_{0} ∘ ψ(x). Then η is a unital homomorphism such that η(E ∩ B) = {0}. Let s_{1} and s_{2} be isometries such that 1_{M(B)} = s_{1}s_{1}^{∗} + s_{2}s_{2}^{∗}. By [Arv77, Corollary 2] and [Kir, Proposition 7], there exists a unitary u ∈ M(B) such that u(s_{1}xs_{1}^{∗} + s_{2}η(x)s_{2}^{∗})u^{∗} − x ∈ B for all x ∈ E.

We claim u implements a unitary equivalence between τ and τ ⊕τ_{0}. Let a ∈ A. Choose x ∈ E such that π(x) = (τ̃ ∘ ι_{A})(a). Note that

π ∘ η(x) = π ∘ α̃_{0} ∘ ψ(x) = (π ∘ α̃_{0})(τ^{−1}(π(x))) = (π ∘ α̃_{0} ∘ ι_{A})(a).

Then

π(u) (τ(a) ⊕ τ_{0}(a)) π(u)^{∗}
= π(u) (π(s_{1})τ̃(ι_{A}(a))π(s_{1}^{∗}) + π(s_{2})(π ∘ α̃_{0} ∘ ι_{A})(a)π(s_{2})^{∗}) π(u)^{∗}
= π(u(s_{1}xs_{1}^{∗} + s_{2}η(x)s_{2}^{∗})u^{∗})
= π(x)
= (τ̃ ∘ ι_{A})(a)
= τ(a).

Hence, τ ⊕ τ_{0} ∼_{u} τ, proving the first part of the proposition. Suppose e_{i} : 0 → B → E_{i} → A → 0 is an essential extension for i = 1, 2 and [τ_{e_{1}}] = [τ_{e_{2}}] in Ext(A, B). By the discussion before the proposition, there exist trivial extensions τ′_{1}, τ′_{2} : A → Q(B) such that τ_{e_{1}} ⊕ τ′_{1} ∼_{u} τ_{e_{2}} ⊕ τ′_{2}. By the first part of the proposition, we have that τ_{e_{1}} ∼_{u} τ_{e_{1}} ⊕ τ′_{1} and τ_{e_{2}} ⊕ τ′_{2} ∼_{u} τ′_{e_{2}}. Therefore, τ_{e_{1}} ∼_{u} τ_{e_{2}}. By [JT91, § 3.2], E_{1} ≅ E_{2}. □

6.2 Corners of graph algebras

We also need some results involving corners of graph algebras. The general case will be worked out in [AGR]. For the convenience of the reader, we will prove the case that will suit our purposes (see Proposition 6.2). Recall that if E = (E^{0}, E^{1}, r, s) is a graph, the C*-algebra C^{∗}(E) associated to E is the universal C*-algebra generated by {p_{v} : v ∈ E^{0}} ⊔ {s_{e} : e ∈ E^{1}} subject to the relations

(i) p_{v}p_{w} = δ_{v,w}p_{v} for all v, w ∈ E^{0};

(ii) s_{e}^{∗}s_{f} = δ_{e,f}p_{r(e)} for all e, f ∈ E^{1};

(iii) s_{e}s_{e}^{∗} ≤ p_{s(e)} for all e ∈ E^{1}; and

(iv) p_{v} = ∑_{e∈s^{−1}(v)} s_{e}s_{e}^{∗} for all v ∈ E^{0} with 0 < |s^{−1}(v)| < ∞.

A loop in E is a path α = e_{1} ⋯ e_{n} with s(e_{1}) = s(e_{n}) and we say that s(e_{1}) is the base point of α. A simple loop in E is a loop α = e_{1} ⋯ e_{n} such that s(e_{i}) ≠ s(e_{j}) for i ≠ j. We say that E satisfies Condition (K) if every vertex is either the base point of at least two simple loops or is not the base point of a loop. It is well-known that if A is a Cuntz-Krieger algebra, then A is isomorphic to C^{∗}(E), where E is a finite graph with no sinks. If, in addition, A is purely infinite, then E will also satisfy Condition (K).

Proposition 6.2 Let E be a graph with finitely many vertices. Suppose there exists a vertex w in E such that

(i) {w} is a hereditary and saturated subset of E^{0};

(ii) |{e ∈ E^{1} : s(e) = w}| is either equal to 0 or ∞; (iii) for every v ∈ E^{0} \ {w}, there are finitely many edges from v to w and

there exists at least one v ∈ E^{0} \ {w} such that there exists an edge from v to w; and

(iv) every vertex v ∈ E^{0} \ {w} emits finitely many edges and is the base point of at least two loops of length one.

Then for every full projection p ∈ C^{∗}(E) ⊗ K, we have that p(C^{∗}(E) ⊗ K)p is isomorphic to a graph algebra. Consequently, if A is a unital C*-algebra such that A ⊗ K ≅ C^{∗}(E) ⊗ K, then A is isomorphic to a graph algebra.

Proof: Let {e_{ij}} be a system of matrix units for K. Throughout the proof, if

p is a projection in C^{∗}(E) and n ∈ ℕ, then set np = \overbrace{p ⊕ ⋯ ⊕ p}^{n} in C^{∗}(E) ⊗ K. Let {p_{v}, s_{e} : v ∈ E^{0}, e ∈ E^{1}} be a Cuntz-Krieger E-family generating C^{∗}(E). Since the only vertex in E that is a singular vertex, i.e., emits no edges or infinitely many edges, is w, by [HL+14, Theorem 3.4 and Corollary 3.5],

p ∼ (⨁_{v∈S} n_{v}p_{v}) ⊕ n_{1} (p_{w} − ∑_{e∈T_{1}} s_{e}s_{e}^{∗}) ⊕ ⋯ ⊕ n_{k} (p_{w} − ∑_{e∈T_{k}} s_{e}s_{e}^{∗}) , (7)

where n_{v} > 0 for all v ∈ S, n_{i} ≥ 0 for all i, S ⊆ E^{0} \{w}, and T_{i} is a finite (possibly empty) subset of s^{−1}(w) for all i. Arguing as in [AR15, Lemma 4.6], we have that the projection on the right hand side of (7) is Murray-von Neumann equivalent to

q = ⨁_{v∈E^{0}} m_{v}p_{v}

where m_{v} > 0 for all v ∈ E^{0}. We use the fact that if S is a finite subset of s^{−1}(w), then

|S|p_{w} ⊕ (p_{w} − ∑_{e∈S} s_{e}s_{e}^{∗}) ∼ (∑_{e∈S} s_{e}s_{e}^{∗}) ⊕ (p_{w} − ∑_{e∈S} s_{e}s_{e}^{∗}) ∼ p_{w}

and the fact that if v_{0} ∈ E^{0} \ {w} with s^{−1}(v_{0}) ∩ r^{−1}(w) ≠ ∅, then for any n, we have that p_{v_{0}} ∼ np_{w} ⊕ p_{v_{0}} ⊕ (⨁_{v∈E^{0}} m′_{v}p_{v}) for m′_{v} ≥ 0. Now, arguing as in [AR15, Proposition 4.7], we have that q(C^{∗}(E) ⊗ K)q is isomorphic to a graph algebra. Since p ∼ q, we have that p(C^{∗}(E) ⊗ K)p ≅ q(C^{∗}(E) ⊗ K)q. Therefore, p(C^{∗}(E) ⊗ K)p is isomorphic to a graph algebra. For the last part of the proposition note that A ≅ p(C^{∗}(E) ⊗ K)p, where p is the projection given by the image of 1_{A} ⊗ e_{11} under some isomorphism from A ⊗ K to C^{∗}(E) ⊗ K. Since 1_{A} ⊗ e_{11} is full in A ⊗ K, we have that p is full in C^{∗}(E) ⊗ K. □

6.3 Semigroup C*-algebras and graph algebras

We now determine when a C*-algebra associated to an Artin monoid is isomorphic to a graph algebra. To do this, we need to determine when an extension of two graph algebras is isomorphic to a graph algebra. In spite of substantial effort the extension problem for graph algebras has not be completely resolved even for the single non-trivial ideal case. Moreover, the results in the literature are not sufficient for our purposes. The following ad hoc result will give us what we need.

Lemma 6.3 For each i, let A_{i} be a separable, nuclear C*-algebra with an essential ideal I_{i} such that I_{i} is isomorphic to either K or a purely infinite simple C*-algebra with trivial K_{1} group, A_{i}/I_{i} satisfies the Universal Coefficient Theorem, and K_{1}(A_{i}/I_{i}) = {0} or K_{0}(A_{i}/I_{i}) is a free group (possibly K_{0}(A_{i}/I_{i}) = {0}). Suppose there exist isomorphisms β: I_{1} ⊗ K → I_{2} ⊗ K, α: (A_{1}/I_{1}) ⊗ K → (A_{2}/I_{2}) ⊗ K, and η_{∗}: K_{∗}(A_{1} ⊗ K) → K_{∗}(A_{2} ⊗ K) such that (K_{∗}(β), η_{∗}, K_{∗}(α)): K_{six}(A_{1} ⊗ K; I ⊗ K) → K_{six}(A_{2} ⊗ K; I_{2} ⊗ K) is an isomorphism. Then A_{1} ⊗ K ≅ A_{2} ⊗ K.

Proof: Let e_{1} : 0 → I_{2} ⊗K → B_{1} → (A_{1}/I_{1})⊗K → 0 be the extension obtained by pushing forward the extension 0 → I_{1} ⊗ K → A_{1} ⊗ K → (A_{1}/I_{1}) ⊗ K → 0 via the isomorphism β and let e_{2} : 0 → I_{2} ⊗ K → B_{2} → (A_{1}/I_{1}) ⊗ K → 0 be the extension obtained by pulling back the extension 0 → I_{2} ⊗ K → A_{2} ⊗ K → (A_{2}/I_{2}) ⊗ K → 0 via the isomorphism α. Note that there exist isomorphisms ϕ_{1} : A_{1} ⊗ K → B_{1} and ϕ_{2} : B_{2} → A_{2} ⊗ K such that (K_{∗}(β), K_{∗}(ϕ_{1}), K_{∗}(id_{(A_{1}/I_{1})⊗K})): K_{six}(A_{1} ⊗ K; I_{1} ⊗ K) → K_{six}(B_{1}; I_{2} ⊗ K) and (K_{∗}(id_{I_{2}⊗K}), K_{∗}(ϕ_{2}), K_{∗}(α)): K_{six}(B_{2}; I_{2} ⊗ K) → K_{six}(A_{2} ⊗ K; I_{2} ⊗ K) are isomorphisms. Then (K_{∗}(id_{I_{2}⊗K}), K_{∗}(ϕ_{2}^{−1}) ∘ η_{∗} ∘ K_{∗}(ϕ_{1}^{−1}), K_{∗}(id_{(A_{1}/I_{1})⊗K})) is an isomorphism from K_{six}(B_{1}; I_{2} ⊗ K) to K_{six}(B_{2}; I_{2} ⊗ K). We claim that [τ_{e_{1}}] = [τ_{e_{2}}] in Ext((A_{1}/I_{1})⊗K, I_{2}⊗K). Since A_{1}/I_{1} satisfies the Universal Coefficient Theorem, we may identify Ext((A_{1}/I_{1}) ⊗ K, I_{2} ⊗ K) with KK^{1}((A_{1}/I_{1})⊗K, I_{2} ⊗K). Note that Ext_{ℤ}^{1}(K_{1}((A_{1}/I_{1})⊗K), K_{1}(I_{2} ⊗K)) = {0} since K_{1}(I_{2}) = {0}. Suppose K_{1}(A_{1}/I_{1}) = {0}. Then K_{1}(τ_{e_{i}}) = {0}. Since K_{1}(I_{2}) = {0}, we have that K_{0}(τ_{e_{i}}) = {0}. Hence, K_{∗}(τ_{e_{i}}) = {0}. By the Universal Coefficient Theorem, [τ_{e_{i}}] can be identified with the element in Ext_{ℤ}^{1}(K_{0}((A_{1}/I_{1}) ⊗ K), K_{0}(I_{2} ⊗ K)) given by K_{six}(B_{i}; I_{2} ⊗ K). Since (K_{∗}(id_{I_{2}⊗K}), K_{∗}(ϕ_{2}^{−1}) ∘ η_{∗} ∘ K_{∗}(ϕ_{1}^{−1}), K_{∗}(id_{(A_{1}/I_{1})⊗K})) is an isomorphism from K_{six}(B_{1}; I_{2} ⊗ K) to K_{six}(B_{2}; I_{2} ⊗ K) we have that K_{six}(B_{1}; I_{2} ⊗ K) and K_{six}(B_{2}; I_{2}⊗K) induce the same element in Ext_{ℤ}^{1}(K_{0}((A_{1}/I_{1})⊗K), K_{0}(I_{2}⊗K)). Hence, [τ_{e_{1}}] = [τ_{e_{2}}] in Ext((A_{1}/I_{1}) ⊗ K, I_{2} ⊗ K). Suppose K_{0}(A_{1}/I_{1}) is a free group (possibly the zero group). By the Universal Coefficient, [τ_{e_{i}}] is completely determined by K_{∗}(τ_{e_{i}}). Since (K_{∗}(id_{I_{2}⊗K}), K_{∗}(ϕ_{2}^{−1}) ∘ η_{∗} ∘ K_{∗}(ϕ_{1}^{−1}), K_{∗}(id_{(A_{1}/I_{1})⊗K})) is an isomorphism from K_{six}(B_{1}; I_{2} ⊗ K) to K_{six}(B_{2}; I_{2} ⊗ K), we have that K_{∗}(τ_{e_{1}}) = K_{∗}(τ_{e_{2}}). Hence, [τ_{e_{1}}] = [τ_{e_{2}}] in Ext((A_{1}/I_{1}) ⊗ K, I_{2} ⊗ K).

In both cases, we have shown that [τ_{e_{1}}] = [τ_{e_{2}}] in Ext((A_{1}/I_{1}) ⊗ K, I_{2} ⊗ K), proving our claim. By Proposition 6.1, we have that B_{1} ≅ B_{2}. Therefore, A_{1} ⊗ K ≅ A_{2} ⊗ K. □

Lemma 6.4 Let A be a unital, separable, nuclear C*-algebra with an essential ideal I such that I ≅ K or I ≅ O_{∞} ⊗ K and A/I is isomorphic to a purely infinite Cuntz-Krieger algebra. If K_{1}(A/I) = {0} or K_{0}(A/I) is a free group (possibly K_{0}(A/I) = {0}), then A is isomorphic to a graph algebra.

Proof: By [ABK14, Theorem 4.4] and [Res06, Proposition 8.3], there exists a finite graph F such that each vertex of F is the base point of at least two loop of length one and there exists an isomorphism ϕ: C^{∗}(F) ⊗ K → A/I ⊗ K. Let ψ: C^{∗}(G) ⊗ K → I be an isomorphism such that K_{∗}(ψ) = id, where G is the graph {v} with one vertex and no edges if I ≅ K and G is the graph with one vertex {v} with infinitely many edges when I ≅ O_{∞} ⊗K. By [EKTW, Lemma 5.2(r1) and Proposition 5.5], there exists a graph E with the properties that

(1) E^{0} = G^{0} ⊔ F^{0},

(2) E^{1} is the union of G^{1} and F^{1} together with a finite nonzero number of edges from each w ∈ F^{0} to v, and

(3) there exist an isomorphism α_{∗}: K_{∗}(C^{∗}(E)) → K_{∗}(A) with the property that (K_{∗}(ψ), α_{∗}, K_{∗}(ϕ)) is an isomorphism from K_{six}(C^{∗}(E); I_{{v}}) to K_{six}(A; I).

Note that I_{{v}} ⊗K is an essential ideal of C^{∗}(E)⊗K and there exist an isomorphism \overline{α}_{∗}: K_{∗}(C^{∗}(E) ⊗ K) → K_{∗}(A ⊗ K) such that (K_{∗}(ψ ⊗ id_{K}), \overline{α}_{∗}, K_{∗}(ϕ ⊗ id_{K})) is an isomorphism from K_{six}(C^{∗}(E) ⊗ K; I_{{v}⊗K}) to K_{six}(A ⊗ K; I ⊗ K). Also, note that I ≅ I_{{v}} = K or I ≅ I_{{v}} ≅ O_{∞} ⊗ K . By Lemma 6.3, A ⊗ K ≅ C^{∗}(E) ⊗ K. Therefore, A is isomorphic to a graph algebra by Proposition 6.2. □

Lemma 6.5 For each m ∈ ℕ, for each n ≥ 0, the smallest nonzero ideal I of E_{m}^{±1} ⊗ ⨂_{k=1}^{n} E_{2}^{r_{k}} is isomorphic to K and (E_{m}^{±1} ⊗ ⨂_{k=1}^{n} E_{2}^{r_{k}})/I is isomorphic to a Cuntz-Krieger algebra with vanishing K_{1}-group. Consequently, E_{m}^{±1} ⊗ ⨂_{k=1}^{n} E_{2}^{r_{k}} and E_{m}^{±1} ⊗ ⨂_{k=1}^{n} E_{2}^{r_{k}} ⊗ O_{∞} are isomorphic to graph algebras.

Proof: Note that for each m ∈ ℕ, by [EKTW, Theorem 7.2], E_{m}^{+1}, E_{m}^{−1}, E_{m}^{+1}⊗O_{∞}, and E_{m}^{−1}⊗O_{∞} are graph algebras with E_{m}^{±1}/K and (E_{m}^{±1}⊗O_{∞})/(K⊗ O_{∞}) ≅ (E_{m}^{±1}/K)⊗O_{∞} are isomorphic to purely infinite Cuntz-Krieger algebras. Therefore, we may assume that n ≥ 1. For notational convenience, set A = E_{m}^{±1} ⊗ ⨂_{k=1}^{n} E_{2}^{r_{k}}. Note that I = ⨂_{k=1}^{n+1} K. Let J = K⊗⨂_{k=1}^{n} E_{2}^{r_{k}}. Then J is a primitive ideal and A/J ≅ O_{m}⊗⨂_{k=1}^{n} E_{2}^{r_{k}}. We will show that J/I is stably isomorphic to an O_{2}-absorbing Cuntz-Krieger algebra, A/J is isomorphic to a Cuntz-Krieger algebra with vanishing boundary maps, and the boundary maps in K-theory induced by the extension 0 → J/I → A/I → A/J → 0 are zero. We will first prove that J/I is O_{2}-absorbing. Note that it is enough to show that (⨂_{k=1}^{n} E_{2}^{r_{k}}) / (⨂_{k=1}^{n} K) is O_{2}-absorbing since J/I ≅ K ⊗ (⨂_{k=1}^{n} E_{2}^{r_{k}}) / (⨂_{k=1}^{n} K). Since E_{2}^{±}/K ≅ O_{2} which is O_{2}-absorbing by [KP00, Theorem 3.8], we have that (⨂_{k=1}^{n} E_{2}^{r_{k}}) / (⨂_{k=1}^{n} K) is O_{2}-absorbing for n = 1. Suppose (⨂_{k=1}^{m} E_{2}^{r_{k}}) / (⨂_{k=1}^{m} K) is O_{2}-absorbing for 1 ≤ m < n. Consider the extension

0 → (E_{2}^{r_{1}} ⊗ ⨂_{k=2}^{n} K) / (⨂_{k=1}^{n} K) → → (⨂_{k=1}^{n} E_{2}^{r_{k}}) / (⨂_{k=1}^{n} K) → (⨂_{k=1}^{n} E_{2}^{r_{k}}) / (E_{2}^{r_{1}} ⊗ ⨂_{k=2}^{n} K) → 0

Now, (E_{2}^{r_{1}} ⊗⨂_{k=2}^{n} K)/(⊗_{k=1}^{n}K) ≅ (E_{2}^{r_{1}}/K)⊗⨂_{k=2}^{n} K ≅ O_{2} ⊗⨂_{k=2}^{n} K which is O_{2}-absorbing by [KP00, Theorem 3.8]. Since (⨂_{k=1}^{n} E_{2}^{r_{k}})/(E_{2}^{r_{1}} ⊗ ⨂_{k=2}^{n} K) ≅ E_{2}^{r_{1}} ⊗ ((⨂_{k=2}^{n} E_{2}^{r_{k}}) / (⨂_{k=2}^{n} K)) and because of the inductive hypothesis, we have that (⊗_{k=1}^{n}E_{2}^{r_{k}})/(E_{2}^{r_{1}} ⊗⨂_{k=2}^{n} K) is O_{2}-absorbing. Hence, by [KP00, Theorem 3.8] and [TW07, Corollary 4.3], (⨂_{k=1}^{n} E_{2}^{r_{k}}) / (⨂_{k=1}^{n} K) is O_{2}-absorbing. This proves our claim. Since J/I is O_{2}-absorbing and J/I has finitely many ideals, by [Kir00], J/I is stably isomorphic to a Cuntz-Krieger algebra with vanishing boundary maps. This is because for any finite T_{0}-space X, there exists an O_{2}-absorbing Cuntz-Krieger algebra with primitive ideal space X. We also note that the boundary maps in K-theory induced by the extension 0 → J/I → A/I → A/J → 0 are zero since K_{∗}(J/I) = {0}. We now show that A/J is isomorphic to a Cuntz-Krieger algebra with vanishing boundary maps. Recall that A/J ≅ O_{m} ⊗ ⨂_{k=1}^{n} E_{2}^{r_{k}}. Hence, every simple sub-quotient of A/J is isomorphic to O_{m} ⊗ (I_{2}/I_{1}) where I_{1}, I_{2} are ideals of ⨂_{k=1}^{n} E_{2}^{r_{k}} with I_{1} ⊆ I_{2} and I_{2}/I_{1} simple. Note that if I_{1}, I_{2} are ideals of ⨂_{k=1}^{n} E_{2}^{r_{k}} with I_{1} ⊆ I_{2} and I_{2}/I_{1} simple, then I_{2}/I_{1} ≅ ⨂_{k=1}^{n} B_{k} where B_{k} is a simple sub-quotient of E_{2}^{r_{k}}. Hence, every simple sub-quotient of ⨂_{k=1}^{n} E_{2}^{r_{k}} is either isomorphic to ⨂_{k=1}^{n} K or is O_{2}-absorbing. Hence, every simple subquotient of O_{m} ⊗ ⨂_{k=1}^{n} E_{2}^{r_{k}} is either stably isomorphic to O_{m} or O_{2}. So every simple sub-quotient of A/J is stably isomorphic to a Cuntz-Krieger algebra. Consider the extension e : 0 → O_{m} ⊗ I_{1} → O_{m} ⊗ I_{2} → O_{m} ⊗ (I_{2}/I_{1}) → 0 with I_{2}/I_{1} simple. If I_{1} = {0} and I_{2} = ⨂_{k=1}^{n} K, then O_{m}⊗I_{1} = {0} which implies that e has vanishing boundary maps. If I_{2}/I_{1} is O_{2}-absorbing, then K_{∗}(O_{m} ⊗ (I_{2}/I_{1})) = {0} which also implies that e has vanishing boundary maps. By [Ben14, Corollary 3.6], we have that A/J ≅ O_{m} ⊗ ⨂_{k=1}^{n} E_{2}^{r_{k}} has vanishing boundary maps. Therefore, by [Ben14, Corollary 8.2], A/J is isomorphic to a Cuntz-Krieger algebra with vanishing boundary maps. This finishes the proof of the above claim. The above claim shows that all the assumptions in [Ben14, Proposition 3.7, Proposition 3.10, and Corollary 8.4] are satisfied. Thus, A/I is isomorphic to a purely infinite Cuntz-Krieger algebra.

We now show that K_{1}((E_{m}^{±1} ⊗ ⨂_{k=1}^{n} E_{2}^{r_{k}})/I) ≅ {0}. Since 0 → J/I → A/I → A/J → 0 has vanishing boundary maps and the fact that J/I is O_{2}- absorbing, we have that the surjective map A/I → A/J induces an injective map K_{1}(A/I) → K_{1}(A/J). Since every simple sub-quotient of A/J is stably isomorphic to O_{m} or O_{2} and since A/J has finitely many ideals, one can show that K_{1}(A/J) = {0}. Therefore, K_{1}(A/I) = {0}. Lemma 6.4 implies that E_{m}^{±1} ⊗ ⨂_{k=1}^{n} E_{m_{k}}^{r_{k}} and E_{m}^{±1} ⊗ ⨂_{k=1}^{n} E_{m_{k}}^{r_{k}} ⊗ O_{∞} are isomorphic to graph algebras. □

Lemma 6.6 Let m_{1}, m_{2}, … m_{n} ∈ ℕ. Then

(1) ⨂_{k=1}^{n} E_{m_{k}}^{±1} is stably isomorphic to unital graph algebra if and only if whenever there exists an i such that m_{i} ∈ {1}⊔ℤ_{≥3}, we have that m_{j} = 2 for all j ≠ i.

(2) ⨂_{k=1}^{n} E_{m_{k}}^{±1} ⊗ O_{∞} is stably isomorphic to unital graph algebra if and only if whenever there exists an i such that m_{i} ∈ {1} ⊔ ℤ_{≥3}, we have that m_{j} = 2 for all j ≠ i.

Proof: We prove (1). (2) is proved in a similar way. Suppose whenever there exists an i such that m_{i} ∈ {1} ⊔ ℤ_{≥3}, we have that m_{j} = 2 for all j ≠ i. By Lemma 6.5, ⨂_{k=1}^{n} E_{m_{k}}^{±1} is isomorphic to a graph algebra. So, also stably isomorphic to a unital graph algebra. Suppose ⨂_{k=1}^{n} E_{m_{k}}^{±1} is stably isomorphic to graph algebra. Note that E_{m}^{±1}⊗K ≅ E_{m}^{+1} ⊗ K for any m. Therefore, it is enough to prove the case ⨂_{k=1}^{n} E_{m_{k}}^{+1}. Note that ⨂_{k=1}^{n} E_{m_{k}}^{+1} has finitely many ideals. Since ⨂_{k=1}^{n} E_{m_{k}}^{+1} is stably isomorphic to a unital graph algebra C^{∗}(E), we have that C^{∗}(E) has finitely many ideals. Therefore, every sub-quotient of C^{∗}(E) is stably isomorphic to a unital graph algebra with finitely many ideals. Consequently, every sub-quotient of ⨂_{k=1}^{n} E_{m_{k}}^{+1} is stably isomorphic to a unital graph algebra with finitely many ideals. Suppose there exists i and j such that m_{i}, m_{j} ∈ {1} ⊔ ℤ_{≥3}. Let I = ⨂_{k=1}^{n} I_{k} be the ideal of ⨂_{k=1}^{n} E_{m_{k}}^{+1} where I_{k} = K if k ∉ {i, j}, I_{i} = E_{m_{i}}^{+1}, and I_{j} = E_{m_{j}}^{+1}. From the above observation we must have that every sub-quotient of I is stably isomorphic to a unital graph algebra with finitely many ideals. Note that I is stably isomorphic to E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1} and E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1} has a quotient isomorphic to O_{m_{i}} ⊗ O_{m_{j}}. Therefore, O_{m_{i}} ⊗ O_{m_{j}} is stably isomorphic to a graph algebra. Let K ⊗ K be the smallest non-zero ideal of E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1}. By the Künneth formula, K_{0}(E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1}) ≅ ℤ and K_{1}(E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1}) = {0}, and hence the extension 0 → K ⊗ K → E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1} → (E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1})/(K ⊗ K) → 0 induces a six-term exact sequence in K-theory of the form

\xymatrix{ℤ \ar[r] & ℤ \ar[r] & K_{0}((E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1})/(K ⊗ K)) \ar[d] \\ K_{1}((E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1})/(K ⊗ K)) \ar[u] & 0 \ar[l] & 0. \ar[l]}

Documenta Mathematica 21 (2016) 309–343

In particular, K_{0}((E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1})/(K ⊗ K)) and K_{1}((E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1})/(K ⊗ K)) are cyclic groups. Since (E_{m_{i}}^{+1}⊗E_{m_{j}}^{+1})/(K⊗K) is stably isomorphic to a graph algebra with finitely many ideals, (E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1})/(K ⊗ K) has real rank zero. Therefore, the quotient of (E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1})/(K ⊗ K) by the ideal (K ⊗ E_{m_{j}}^{+1} + E_{m_{i}}^{+1} ⊗ K)/(K ⊗ K) induces the following six-term exact sequence

\xymatrix{K_{0}(O_{m_{i}}) ⊕ K_{0}(O_{m_{j}}) \ar[r] & K_{0}((E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1})/(K ⊗ K)) \ar[r] & K_{0}(O_{m_{i}} ⊗ O_{m_{j}}) \ar[d]^{0} \\ K_{1}(O_{m_{i}} ⊗ O_{m_{j}}) \ar[u] & K_{1}((E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1})/(K ⊗ K)) \ar[l] & K_{1}(O_{m_{i}}) ⊕ K_{1}(O_{m_{j}}). \ar[l]} (8)

Using the Künneth formula, we get

K_{0}(O_{m_{i}} ⊗ O_{m_{j}}) = K_{1}(O_{m_{i}} ⊗ O_{m_{j}}) = \begin{cases}ℤ_{gcd(m_{i}−1,m_{j}−1)} & if m_{i}, m_{j} ≥ 3 \\ K_{1}(O_{m_{i}}) ⊕ K_{0}(O_{m_{i}}) & if m_{j} = 1 \\ K_{1}(O_{m_{j}}) ⊕ K_{0}(O_{m_{j}}) & if m_{i} = 1. \end{cases}

Since O_{m_{i}} ⊗ O_{m_{j}} is stably isomorphic to a unital graph algebra, gcd(m_{i} − 1, m_{j} − 1) = 1 if m_{i}, m_{j} ≥ 3 and m_{i} = 1 if and only if m_{j} = 1. Suppose m_{i}, m_{j} ≥ 3. The exactness of Diagram (8) implies that K_{0}((E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1})/(K ⊗ K)) ≅ K_{0}(O_{m_{i}}) ⊕ K_{0}(O_{m_{j}}) ≅ ℤ_{m_{i}−1} ⊕ ℤ_{m_{j}−1} which contradicts the fact that K_{0}(E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1})/(K ⊗ K)) is a cyclic group. Suppose m_{i} = 1. Then m_{j} = 1. Then by the exactness of Diagram (8), K_{1}((E_{m_{i}}^{+1}⊗E_{m_{j}}^{+1})/(K⊗K)) has a sub-group isomorphic to K_{1}(O_{m_{i}})⊕K_{1}(O_{m_{j}}) ≅ ℤ ⊕ ℤ. This can not happen since K_{1}((E_{m_{i}}^{+1} ⊗ E_{m_{j}}^{+1})/(K ⊗ K)) is a cyclic group. □ Let the notation be as in Definition 4.1.

Theorem 6.7 Let Γ be a countable graph. Then C^{∗}(A_{Γ}^{+}) is isomorphic to a graph algebra if and only if one of the following holds

1. t(Γ) = 1, o(Γ) = 0 and N_{k}(Γ) = 0 for all k

2. t(Γ) = 0, N_{−1}(Γ) + N_{1}(Γ) < ∞ and

∑_{|k|≠1} N_{k}(Γ) ≤ 1

Proof: Suppose there exists an isomorphism ψ: C^{∗}(A_{Γ}^{+}) → C^{∗}(E) for some countable directed graph E. Since C^{∗}(A_{Γ}^{+}) is unital, C^{∗}(E) is unital. Let Γ_{i} = (V_{i}, E_{i}) be the co-irreducible components of Γ. To prove (1), let I be the ideal of C^{∗}(A_{Γ}^{+}) generated by {⨂_{j} J_{ij}}_{i} where J_{ij} = C^{∗}(A_{Γ_{j}}^{+}) if j ≠ i and

J_{ii} = \begin{cases}K & if 1 < |V_{i}| < ∞ \\ 0 & otherwise. \end{cases}

Then C^{∗}(A_{Γ}^{+})/I ≅ ⨂_{i} C^{∗}(A_{Γ_{i}}^{+})/J_{ii} where C^{∗}(A_{Γ_{i}}^{+})/J_{ii} is a Kirchberg algebra if |V_{i}| ≥ 2 and C^{∗}(A_{Γ_{i}}^{+})/J_{ii} ≅ T otherwise. In particular,

Prim(C^{∗}(A_{Γ}^{+})/I) ≅ \begin{cases}∏_{k=1}^{t(Γ)} Prim(T ) & if there exists i with |V_{i}| = 1 \\ \{•\} & otherwise. \end{cases}

Note that I is generated by projections. Therefore, ψ(I) is generated by projections and hence is a gauge-invariant ideal of C^{∗}(E). Hence, by [BPRS00, Corollary 3.5 and Theorem 3.6], C^{∗}(E)/ψ(I) is isomorphic to a graph algebra. Since C^{∗}(A_{Γ}^{+})/I ≅ C^{∗}(E)/ψ(I), we have that C^{∗}(A_{Γ}^{+})/I is isomorphic to a unital graph algebra. Note that C^{∗}(A_{Γ}^{+})/I is O_{∞}-absorbing (if there exists i such that |V_{i}| ≥ 2) or C^{∗}(A_{Γ}^{+})/I ≅ ⨂_{k=1}^{t(Γ)} T . Suppose C^{∗}(A_{Γ}^{+})/I is O_{∞}-absorbing. Since any unital O_{∞}-absorbing graph algebra has a finite primitive ideal space, we must have that t(Γ) = 0. Suppose C^{∗}(A_{Γ}^{+})/I is not O_{∞}-absorbing. Then C^{∗}(A_{Γ}^{+})/I ≅ ⨂_{k=1}^{t(Γ)} T . Let J be the ideal generated by {⨂_{j} J_{ij}}_{i} where J_{ij} = T if j ≠ i and J_{ii} = K, then

J is an ideal generated by projections such that (⨂_{k=1}^{t(Γ)} T ) /J ≅ C(𝕋^{t(Γ)}).

Since C^{∗}(A_{Γ}^{+})/I is isomorphic to a graph algebra and every ideal generated by projections in a graph algebra is gauge invariant, by [BPRS00, Corollary 3.5 and Theorem 3.6] every quotient of C^{∗}(A_{Γ}^{+})/I by an ideal generated by projections is isomorphic to a graph algebra. Hence, C(𝕋^{t(Γ)}) ≅ (⨂_{k=1}^{t(Γ)} T ) /J is isomorphic to a unital graph algebra. Since the only unital commutative graph algebra is isomorphic to finite direct sums of ℂ and 𝕋, we must have that t(Γ) = 1. In both cases, we have shown that t(Γ) ≤ 1. Suppose o(Γ) ≠ 0 or N_{k}(Γ) ≠ 0 for some k, then there exists an i such that C^{∗}(A_{Γ_{i}}^{+})/J_{ii} is a Kirchberg algebra. Hence, by [KP00, Theorem 3.15] and [TW07, Corollary 3.4] C^{∗}(A_{Γ}^{+})/I ≅ ⨂_{i} C^{∗}(A_{Γ_{i}}^{+})/J_{ii} is an O_{∞}-absorbing C*-algebra. Since every unital graph algebra that is O_{∞}-absorbing must have finitely many ideals and since

Prim(C^{∗}(A_{Γ}^{+})/I) ≅ \begin{cases}∏_{k=1}^{t(Γ)} Prim(T ) & if there exists i with |V_{i}| = 1 \\ \{•\} & otherwise, \end{cases}

we have that t(Γ) = 0. Hence, we only get a graph algebra in the case t(Γ) = 1 when all other data vanish. Suppose t(Γ) = 0. Note that 1 < |V_{i}| for all i. Thus, C^{∗}(A_{Γ_{i}}^{+}) is a unital properly infinite C*-algebra, and Prim(C^{∗}(A_{Γ_{i}}^{+})) = {x_{i}, y_{i}} with open sets {∅, {x_{i}}, {x_{i}, y_{i}}} when |V_{i}| < ∞ and Prim(C^{∗}(A_{Γ_{i}}^{+})) ≅ {•} when |V_{i}| = ∞. We claim that | {k: N_{k}(Γ) ≠ 0} | < ∞ and N_{k}(Γ) < ∞ for all k. Suppose first | {k: N_{k}(Γ) ≠ 0} | = ∞ or N_{k}(Γ) = ∞ for some k. Then C^{∗}(A_{Γ}^{+}) ≅ ⨂_{i=1}^{∞} C^{∗}(A_{Γ_{i}}^{+}) and C^{∗}(A_{Γ}^{+}) has infinitely many ideals. By Lemma 4.7 , C^{∗}(A_{Γ}^{+}) is O_{∞}-absorbing. Again, using the fact that a unital graph algebra that is O_{∞}-absorbing has finitely many ideals, we have a contradiction. Therefore, | {k: N_{k}(Γ) ≠ 0} | < ∞ and N_{k}(Γ) < ∞ for all k, proving the claims in (2). Note that

C^{∗}(A_{Γ}^{+}) ≅ (E_{1}^{0})^{⊗N_{0}(Γ)} ⊗ ⨂_{n=1}^{∞}(E_{1+n}^{−1})^{⊗N_{−n}(Γ)} ⊗ ⨂_{n=1}^{∞}(E_{1+n}^{+1})^{⊗N_{n}(Γ)} ⊗ (O_{∞})^{⊗o(Γ)}

By Lemma 6.6, (1) and (2) hold. In the other direction, we have in the case (1) that C^{∗}(A_{Γ}^{+}) ≅ T which is isomorphic to a graph algebra. And in case (2) we have that either

C^{∗}(A_{Γ}^{+}) ≅ (E_{2}^{−1})^{⊗N_{−1}(Γ)} ⊗ (E_{2}^{+1})^{⊗N_{1}(Γ)} ⊗ (O_{∞})^{⊗o(Γ)},

C^{∗}(A_{Γ}^{+}) ≅ E_{m}^{+1} ⊗ (E_{2}^{−1})^{⊗N_{−1}(Γ)} ⊗ (E_{2}^{+1})^{⊗N_{1}(Γ)} ⊗ (O_{∞})^{⊗o(Γ)}

for some m ≠ 2, or

C^{∗}(A_{Γ}^{+}) ≅ E_{m}^{−1} ⊗ (E_{2}^{−1})^{⊗N_{−1}(Γ)} ⊗ (E_{2}^{+1})^{⊗N_{1}(Γ)} ⊗ (O_{∞})^{⊗o(Γ)}.

for some m ≠ 2. If o(Γ) ≥ 1, then by [KP00, Theorem 3.15], (O_{∞})^{⊗o(Γ)} ≅ O_{∞}. Hence, by Lemma 6.5, C^{∗}(A_{Γ}^{+}) is isomorphic to a graph algebra. □

Remark 6.8 The relation between a (undirected, loop-free) graph Γ and a directed graph G_{Γ} with C^{∗}(A_{Γ}^{+}) ≅ C^{∗}(G_{Γ}) is somewhat opaque, although the proof given above in principle is constructive. In Figure 2 we present eight graphs presenting the C*-algebras given by five-vertex graphs of Figure 1 in the unshaded regions.

We conclude by establishing semiprojectivity and non-semiprojectivity of C^{∗}(A_{Γ}^{+}) in a number of cases, covering for instance all graphs with 5 or fewer vertices. We note, however, that this theorem does not contain a full answer to the question of which of the C*-algebras under study are semiprojective. The most basic open case has N_{−2} = 2 and may be represented by a graph with 6 vertices.

Theorem 6.9

1. When t(Γ) > 1, C^{∗}(A_{Γ}^{+}) is not semiprojective.

2. When t(Γ) = 1, C^{∗}(A_{Γ}^{+}) is semiprojective if and only if

o(Γ) = ∑_{k} N_{k}(Γ) = 0.

3. When t(Γ) = 0, C^{∗}(A_{Γ}^{+}) is semiprojective when N_{−1}(Γ) + N_{1}(Γ) < ∞

and

∑_{|k|≠1} N_{k}(Γ) ≤ 1

[figure: •; ∘; •; ∘; •; ∘]
N_{−4} = 1  N_{−3} = 1  N_{−2} = 1

[figure: •; ∘; •; •; •; ∘; •; •; ∘]

N_{−1} = 1  N_{0} = 1  N_{1} = 1

[figure: •; •; •; ∘; •; •; •; •; ∘]

N_{−1} = 2  N_{−1} = N_{−2} = 1

Figure 2: Graphs representing cases from Figure 1

.

Proof: We first note that by [End13, Corollary 4.4.16], a C*-algebra of the form A ⊗ T with A unital, nuclear, infinite-dimensional and in the UCT-class can never be semiprojective. This proves (1) and (2) since T itself is trivially semiprojective. For (3), we first apply Theorem 6.7 to see that C^{∗}(A_{Γ}^{+}) in this case is a unital graph algebra. We have seen that when o(Γ) > 0, C^{∗}(A_{Γ}^{+}) is strongly purely infinite, and when o(Γ) = 0, there is a minimal ideal K in C^{∗}(A_{Γ}^{+}) so that C^{∗}(A_{Γ}^{+})/K is strongly purely infinite. In either case, [EK] applies to guarantee that the C*-algebra is semiprojective. □

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Søren Eilers Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen Denmark Xin Li School of Mathematical Sciences Queen Mary University of London Mile End Road London E1 4NS United Kingdom

Efren Ruiz

Department of Mathematics University of Hawaii, Hilo 200 W. Kawili St., Hilo Hawaii, 96720-4091 USA 344

Documenta Mathematica 21 (2016)