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2020 Absence of Cartan subalgebras for right-angled Hecke von Neumann algebras
Martijn Caspers
Anal. PDE 13(1): 1-28 (2020). DOI: 10.2140/apde.2020.13.1

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, sS, 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.

Citation

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Martijn Caspers. "Absence of Cartan subalgebras for right-angled Hecke von Neumann algebras." Anal. PDE 13 (1) 1 - 28, 2020. https://doi.org/10.2140/apde.2020.13.1

Information

Received: 10 November 2016; Revised: 2 May 2018; Accepted: 12 February 2019; Published: 2020
First available in Project Euclid: 16 January 2020

zbMATH: 07171987
MathSciNet: MR4047640
Digital Object Identifier: 10.2140/apde.2020.13.1

Subjects:
Primary: 47L10

Keywords: approximation properties , Cartan subalgebras , Hecke von Neumann algebras

Rights: Copyright © 2020 Mathematical Sciences Publishers

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