Simplicity of the reduced $C^{\ast}$-algebras of certain Coxeter groups

Abstract

Let $(G,S)$ be a finitely generated Coxeter group such that the Coxeter system is indecomposable and the canonical bilinear form is indefinite but non-degenerate. We show that the reduced $C^{\ast}$-algebra of $G$ is simple with unique normalised trace.

For an arbitrary finitely generated Coxeter group we prove the validity of a Haagerup inequality: There exist constants $C \gt 0$ and $\Lambda\in \mathbb{N}$ such that, for any function $f\in l^2(G)$ supported on elements of length $n$ with respect to the generating set $S$ and for all $h\in l^2(G)$, $\|\,f\ast h\,\| \leq C(n+1)^{\frac{3}{2}{\Lambda}}\|\,f\,\|$.