## The isomorphism problem for semigroup $$C^*$$-algebras of right-angled Artin monoids.(English)Zbl 1351.46051

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This paper contains a complete classification of $$C^*$$-algebras associated to right-angled Artin monoids, both in terms of graph-theoretic data associated directly to the graph defining the Artin monoid, and in terms of $$K$$-theoretic data associated to the $$C^*$$-algebra.
Let $$\Gamma=(V,E)$$ be a countable, symmetric, anti-reflexive graph, that is, $$V$$ is a countable set of vertices and $$V\subseteq E\times E$$ is a symmetric set of edges not containing any loops $$(v,v)$$ for $$v\in V$$. The associated semigroup $$C^*$$-algebra $$C^*(A^+_\Gamma)$$ was introduced in [J. Crisp and M. Laca, J. Aust. Math. Soc. 72, 223–245 (2002; Zbl 1055.20028)]. It is defined as the universal $$C^*$$-algebra generated by isometries $$s_v$$, for $$v\in V$$, subject to the following relations:
$$\bullet$$
$$s_vs_w=s_ws_v$$ and $$s_v^*s_w=s_ws_v^*$$ whenever $$(v,w)\in E$$; and
$$\bullet$$
$$s_v^*s_w=0$$ if $$v\neq w$$ and $$(v,w)\notin E$$.
The graph-theoretic data used for the classification is based on the notion of co-irreducible components. A graph $$\Gamma$$ is called co-irreducible if there is no non-trivial partition $$V=V_1\cup V_2$$ with $$V_1\times V_2\subseteq E$$. Every graph $$\Gamma$$ decomposes into its co-irreducible components $$\Gamma_i$$, and one has $$C^*(A_\Gamma^+) \cong \bigotimes_i C^*(A_{\Gamma_i}^+)$$.
To each graph $$\Gamma$$, the following invariants are associated: The number $$t(\Gamma)$$ of co-irreducible components of $$\Gamma$$ that have exactly one vertex; and for each integer $$n$$, the number $$N_n(\Gamma)$$ of co-irreducible components of $$\Gamma$$ with Euler characteristic $$n$$ that have at least one, but finitely many, vertices.
Topological $$K$$-theory is a $$2$$-periodic cohomology theory. Therefore, whenever $$I$$ and $$J$$ are (closed, two-sided) ideals in a $$C^*$$-algebra satisfying $$I\subseteq J$$, one obtains a six-term exact sequence $\begin{tikzcd} K_0(I) \ar[r] & K_0(J) \ar[r] & K_0(J/I) \ar[d] \\ K_1(J/I) \ar[u] & K_1(J) \ar[l] & K_1(I)\ar[l] \end{tikzcd}$ The (ordered) filtered $$K$$-theory of a $$C^*$$-algebra $$A$$, denoted by $$\mathrm{FK}_+(A)$$, is defined as the data consisting of the lattice of ideals in $$A$$, and for each inclusion of ideals $$I\subseteq J$$ in $$A$$ the six-term exact sequence in $$K$$-theory associated to the inclusion $$J/I\subseteq A/I$$, together with the order in the $$K_0$$-groups.
The main result of the paper (Theorem 1.1) states that for finite, countable, symmetric, anti-reflexive graphs $$\Gamma$$ and $$\Lambda$$, the following statements are equivalent:
$$\bullet$$
The $$C^*$$-algebras $$C^*(A^+_\Gamma)$$ and $$C^*(A^+_\Lambda)$$ are isomorphic.
$$\bullet$$
We have $$t(\Gamma)=t(\Lambda)$$; and $$N_{-n}(\Gamma)+N_n(\Gamma)=N_{-n}(\Lambda)+N_n(\Lambda)$$ for all integers $$n$$; and $$N_0(\Gamma)>0$$ or $$\sum_{n<0}N_n(\Gamma)=\sum_{n<0}N_n(\Lambda)$$ modulo $$2$$.
$$\bullet$$
The filtered $$K$$-theories $$\mathrm{FK}_+(C^*(A^+_\Gamma))$$ and $$\mathrm{FK}_+(C^*(A^+_\Lambda))$$ are isomorphic.
There is also a version for infinite graphs. It should be noted that the graph-theoretic invariants used for the classification are very easy and straightforward to compute.
Given a directed graph $$\Gamma$$, there is a different way to associate a $$C^*$$-algebra to $$\Gamma$$, called the graph $$C^*$$-algebra $$C^*(\Gamma)$$. Very recently, the first and last author, together with G. Restorff and A. Sørensen, obtained a classification of graph $$C^*$$-algebras of finite graphs by filtered $$K$$-theory [“Geometric classification of graph $$C^*$$-algebras over finite graphs”, Preprint (2016), arXiv:1604.05439]. However, these remarkable recent results do not supersede the results of the paper under review.
Indeed, $$C^*(A^+_\Gamma)$$ is in general not isomorphic to a graph $$C^*$$-algebra (in particular $$C^*(A^+_\Gamma)$$ need not be isomorphic to $$C^*(\Gamma)$$). In Theorem 6.7, the authors characterize when $$C^*(A^+_\Gamma)$$ is isomorphic to a graph $$C^*$$-algebra (of some possibly different graph). This is then used to address the problem of when $$C^*(A^+_\Gamma)$$ is semiprojective.

### MSC:

 46L05 General theory of $$C^*$$-algebras 46L35 Classifications of $$C^*$$-algebras 46L80 $$K$$-theory and operator algebras (including cyclic theory) 20F36 Braid groups; Artin groups 20M05 Free semigroups, generators and relations, word problems

Zbl 1055.20028
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