Absence of Cartan subalgebras for right-angled Hecke von Neumann algebras

Abstract

For a right-angled Coxeter system $(W,S)$ and $q>0$, let ${\mathcal{M}}_q$ be the associated Hecke von Neumann algebra, which is generated by self-adjoint operators $T_s,s\in S$ satisfying the Hecke relation $(\sqrt{q}{T}_s-q)(\sqrt{q}{T}_s+1)=0$ as well as suitable commutation relations. Under the assumption that $(W,S)$ is irreducible and $|S|\ge 3$ it was proved by Garncarek that ${\mathcal{M}}_q$ is a factor (of type II${}_1$) for a range $q\in [\rho ,{\rho }^{-1}]$ and otherwise ${\mathcal{M}}_q$ is the direct sum of a II${}_1$-factor and $\mathbb{C}$. In this paper we prove (under the same natural conditions as Garncarek) that ${\mathcal{M}}_q$ is non-injective, that it has the weak-$\ast$ completely contractive approximation property and that it has the Haagerup property. In the hyperbolic factorial case ${\mathcal{M}}_q$ is a strongly solid algebra and consequently ${\mathcal{M}}_q$ cannot have a Cartan subalgebra. In the general case ${\mathcal{M}}_q$ need not be strongly solid. However, we give examples of non-hyperbolic right-angled Coxeter groups such that ${\mathcal{M}}_q$ does not possess a Cartan subalgebra.

To appear in Analysis and PDE

References in corpus (1)

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

Cited by in corpus (2)

• The CCAP for graph products of operator algebras
• Rigidity for von Neumann algebras of graph product groups. I. Structure of automorphisms