## Absence of Cartan subalgebras for right-angled Hecke von Neumann algebras.(English)Zbl 1447.46047

Summary: For a right-angled Coxeter system $$(W,S)$$ and $$q>0$$, let $$\mathcal{M}_q$$ 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 $$\vert S \vert \geq 3$$, it was proved by Ł. Garncarek [J. Funct. Anal. 270, No. 3, 1202–1219 (2016; Zbl 1397.20012)] that $$\mathcal{M}_q$$ is a factor (of type $$\mathrm{II}_1)$$ for a range $$q \in [\rho, \rho^{-1}]$$ and otherwise $$\mathcal{M}_q$$ is the direct sum of a $$\mathrm{II}_1$$-factor and $$\mathbb{C}$$.
In this paper we prove (under the same natural conditions as Garncarek) that $$\mathcal{M}_q$$ is noninjective, that it has the weak-$$\ast$$ completely contractive approximation property and that it has the Haagerup property. In the hyperbolic factorial case, $$\mathcal{M}_q$$ is a strongly solid algebra and consequently $$\mathcal{M}_q$$ cannot have a Cartan subalgebra. In the general case, $$\mathcal{M}_q$$ need not be strongly solid. However, we give examples of nonhyperbolic right-angled Coxeter groups such that $$\mathcal{M}_q$$ does not possess a Cartan subalgebra.

### MSC:

 46L10 General theory of von Neumann algebras 20F55 Reflection and Coxeter groups (group-theoretic aspects) 20C08 Hecke algebras and their representations

Zbl 1397.20012
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