Simplicity of the reduced C-*-algebras of certain Coxeter groups

TL;DR

This paper proves the simplicity and uniqueness of the trace for the reduced C*-algebras of certain Coxeter groups, and establishes a Haagerup inequality for all finitely generated Coxeter groups, advancing understanding of their operator algebras.

Contribution

It demonstrates the simplicity and unique trace property for reduced C*-algebras of specific Coxeter groups and proves a Haagerup inequality for all finitely generated Coxeter groups.

Findings

Reduced C*-algebra of certain Coxeter groups is simple with a unique trace.

Established a Haagerup inequality for all finitely generated Coxeter groups.

Provided bounds on convolution norms related to group element length.

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-*-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>0 and a natural number L such that for a function f in l^2(G) supported on elements of length n with respect to the generating set S: || f * h || <= C(n+1)^{3/2 L} || f || || h ||, forall h in l^2(G).