#### Vol. 13, No. 1, 2020

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

### Martijn Caspers

Vol. 13 (2020), No. 1, 1–28
##### Abstract

For a right-angled Coxeter system $\left(W,S\right)$ and $q>0$, let ${\mathsc{ℳ}}_{q}$ be the associated Hecke von Neumann algebra, which is generated by self-adjoint operators ${T}_{s}$, $s\in S$, satisfying the Hecke relation $\left(\sqrt{q}\phantom{\rule{2.22144pt}{0ex}}{T}_{s}-q\right)\left(\sqrt{q}\phantom{\rule{2.22144pt}{0ex}}{T}_{s}+1\right)=0$, as well as suitable commutation relations. Under the assumption that $\left(W,S\right)$ is irreducible and $|S|\ge 3$ it was proved by Garncarek (J. Funct. Anal. 270:3 (2016), 1202–1219) that ${\mathsc{ℳ}}_{q}$ is a factor (of type II${}_{1}$) for a range $q\in \left[\rho ,{\rho }^{-1}\right]$ and otherwise ${\mathsc{ℳ}}_{q}$ is the direct sum of a II${}_{1}$-factor and $ℂ$.

In this paper we prove (under the same natural conditions as Garncarek) that ${\mathsc{ℳ}}_{q}$ is noninjective, that it has the weak-$\ast$ completely contractive approximation property and that it has the Haagerup property. In the hyperbolic factorial case ${\mathsc{ℳ}}_{q}$ is a strongly solid algebra and consequently ${\mathsc{ℳ}}_{q}$ cannot have a Cartan subalgebra. In the general case ${\mathsc{ℳ}}_{q}$ need not be strongly solid. However, we give examples of nonhyperbolic right-angled Coxeter groups such that ${\mathsc{ℳ}}_{q}$ does not possess a Cartan subalgebra.

##### Keywords
Hecke von Neumann algebras, approximation properties, Cartan subalgebras
Primary: 47L10