Classifying right-angled Hecke \(\mathrm{C}^\ast\)-algebras via \(K\)-theoretic invariants (Q2166336)

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 \textit{P. Fima} and \textit{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.