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

Abstract

For a right-angled Coxeter system $\left(W,S\right)$ 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 $\left(\sqrt{q}{T}_s-q\right)\left(\sqrt{q}{T}_s+1\right)=0$, as well as suitable commutation relations. Under the assumption that $\left(W,S\right)$ is irreducible and $\left|S\right|\ge 3$ it was proved by Garncarek (J. Funct. Anal. 270:3 (2016), 1202–1219) that ${\mathcal{ℳ}}_q$ is a factor (of type II${}_1$) for a range $q\in \left[\rho ,{\rho }^{-1}\right]$ and otherwise ${\mathcal{ℳ}}_q$ is the direct sum of a II${}_1$-factor and $ℂ$.

In this paper we prove (under the same natural conditions as Garncarek) that ${\mathcal{ℳ}}_q$ is noninjective, 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 nonhyperbolic right-angled Coxeter groups such that ${\mathcal{ℳ}}_q$ does not possess a Cartan subalgebra.