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

Eilers, Søren; Li, Xin; Ruiz, Efren

Doc. Math. 21, 309-343 (2016).

[Note: Some diagrams below cannot be displayed correctly in the web version. Please use the PDF version for correct display.]
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.

Reviewer: Hannes Thiel (Münster)

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

Keywords:

\(C^*\)-algebras; Artin monoids; semigroup \(C^*\)-algebras; filtered \(K\)-theory; semiprojectivity; graph \(C^*\)-algebras

Citations:

Zbl 1055.20028