Classifying right-angled Hecke \(\mathrm{C}^\ast\)-algebras via \(K\)-theoretic invariants. (English) Zbl 1497.19003
This article computes the \(K\)-theory, traces and the trace pairing for Hecke \(C^*\)-algebras of right-angled Coxeter systems. In particular, it is shown that \(K_0\) is the free Abelian group generated by specific projections coming from cliques in the commutation graph and that \(K_1\) vanishes. It is remarkable that the method used here also gives explicit generators for \(K_0\). A key point is a recent computation of the \(KK\)-theory of amalgamated free products by P. Fima and E. Germain [Adv. Math. 369, Article ID 107174, 34 p. (2020; Zbl 1455.46074)]. The \(C^*\)-algebras in question are written as such amalgamated free products for certain Coxeter subsystems. The general theory of amalgamated free products also implies that the Hecke \(C^*\)-algebras for different deformation parameters are all \(KK\)-equivalent and satisfy the UCT (Corollary 3.3). The pairing of the traces with \(K_0\) is studied to explore to what extent the unordered Elliott invariant is able to distinguish Hecke \(C^*\)-algebras for different deformation parameter. The outcome is that it can distinguish some deformation parameters, but cannot distinguish all nonisomorphic ones.
Reviewer: Ralf Meyer (Göttingen)
MSC:
| 19K99 | \(K\)-theory and operator algebras |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 20C08 | Hecke algebras and their representations |
| 46L09 | Free products of \(C^*\)-algebras |
| 19K35 | Kasparov theory (\(KK\)-theory) |
Keywords:
right-angled Coxeter group; Hecke algebra; \(K\)-theory; KK-equivalence; UCT; amalgamated free product; Elliott invariantCitations:
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