Factoriality of Hecke-von Neumann algebras of right-angled Coxeter groups

TL;DR

This paper investigates when Hecke-von Neumann algebras associated with right-angled Coxeter groups are factors, identifying parameter ranges for trivial centers and decompositions for nontrivial centers.

Contribution

It determines the parameter ranges for which these algebras are factors and provides a decomposition method when the center is nontrivial.

Findings

Identifies parameter ranges where $ abla_q(W)$ is a factor.

Provides a decomposition of $ abla_q(W)$ into finite direct sums when not a factor.

Establishes criteria for trivial and nontrivial centers of these algebras.

Abstract

The Hecke algebra $\mathbb{C}_q[W]$ of a Coxter group $W$, associated to parameter $q$, can be completed to a von Neumann algebra $\mathcal{N}_q(W)$. We study such algebras in case where $W$ is right-angled. We determine the range of $q$ for which $\mathcal{N}_q(W)$ is a factor, i.e. has trivial center. Moreover, in case of nontrivial center, we prove a result allowing to decompose $\mathcal{N}_q(W)$ into a finite direct sum of factors.