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

Caspers, Martijn

Anal. PDE 13, No. 1, 1-28 (2020).

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

Keywords:

Hecke von Neumann algebras; approximation properties; Cartan subalgebras

Citations:

Zbl 1397.20012

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