For a right-angled Coxeter system and , let be the associated Hecke von Neumann algebra, which is generated by self-adjoint operators , , satisfying the Hecke relation , as well as suitable commutation relations. Under the assumption that is irreducible and it was proved by Garncarek (J. Funct. Anal. 270:3 (2016), 1202–1219) that is a factor (of type II) for a range and otherwise is the direct sum of a II-factor and .
In this paper we prove (under the same natural conditions as Garncarek) that is noninjective, that it has the weak- completely contractive approximation property and that it has the Haagerup property. In the hyperbolic factorial case is a strongly solid algebra and consequently cannot have a Cartan subalgebra. In the general case need not be strongly solid. However, we give examples of nonhyperbolic right-angled Coxeter groups such that does not possess a Cartan subalgebra.
"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