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 (W,S) and q > 0, let q be the associated Hecke von Neumann algebra, which is generated by self-adjoint operators Ts, s S, satisfying the Hecke relation (qTsq)(qTs+1) = 0, as well as suitable commutation relations. Under the assumption that (W,S) is irreducible and |S| 3 it was proved by Garncarek (J. Funct. Anal. 270:3 (2016), 1202–1219) that q is a factor (of type II1) for a range q [ρ,ρ1] and otherwise q is the direct sum of a II1-factor and .

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

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