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

Caspers, Martijn

doi:10.2140/apde.2020.13.1

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

Martijn Caspers

Vol. 13 (2020), No. 1, 1–28

DOI: 10.2140/apde.2020.13.1

Abstract

For a right-angled Coxeter system $(W, S)$ and $q > 0$ , let $ℳ_{q}$ 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 $| S | \geq 3$ it was proved by Garncarek (J. Funct. Anal. 270:3 (2016), 1202–1219) that $ℳ_{q}$ is a factor (of type II $_{1}$ ) for a range $q \in [ρ, ρ^{- 1}]$ and otherwise $ℳ_{q}$ is the direct sum of a II $_{1}$ -factor and $ℂ$ .

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

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Keywords

Hecke von Neumann algebras, approximation properties, Cartan subalgebras

Mathematical Subject Classification 2010

Primary: 47L10

Milestones

Received: 10 November 2016

Revised: 2 May 2018

Accepted: 12 February 2019

Published: 6 January 2020

Authors

Martijn Caspers
Utrecht University Utrecht Netherlands

Vol. 13, No. 1, 2020