We introduce the relative Haagerup approximation property for a unital, expected inclusion of arbitrary von Neumann algebras and show that if the smaller algebra is finite then the notion only depends on the inclusion itself, and not on the choice of the conditional expectation. Several variations of the definition are shown to be equivalent in this case, and in particular the approximating maps can be chosen to be unital and preserving the reference state. The concept is then applied to amalgamated free products of von Neumann algebras and used to deduce that the standard Haagerup property for a von Neumann algebra is stable under taking free products with amalgamation over finite-dimensional subalgebras. The general results are illustrated by examples coming from q-deformed Hecke-von Neumann algebras and von Neumann algebras of quantum orthogonal groups.

For a real Hilbert space H_R and −1 < q < 1 Bozejko and Speicher introduced the C^∗-algebra A_q(H_R) and von Neumann algebra M_q(H_R) of qGaussian variables. We prove that if dim(H_R) = ∞ and −1 < q < 1, q ∕= 0 then M_q(H_R) does not have the Akemann-Ostrand property with respect to A_q(H_R). It follows that A_q(H_R) is not isomorphic to A₀(H_R). This gives an answer to the C^∗-algebraic part of Question 1.1 and Question 1.2 in raised by Nelson and Zeng [Int. Math. Res. Not. IMRN 17 (2018), pp. 5486–5535].

We introduce and study certain topological spaces associated with connected rooted graphs. These spaces reflect combinatorial and order theoretic properties of the underlying graph and relate in the case of hyperbolic graphs to Gromov’s hyperbolic compactification. They are particularly tractable in the case of Cayley graphs of finite rank Coxeter systems and are intimately related to the corresponding Iwahori–Hecke algebras. We study this connection by considering dynamical properties of the induced action of the Coxeter group.

By exploiting properties of boundaries associated with Coxeter groups we obtain a complete characterization of simple right-Angled multi-parameter Hecke C$^{\ast }$-Algebras. This extends previous results by Caspers, Larsen, and the author. Based on a work by Raum and Skalski, we further describe the center and the character space of right-Angled Hecke C$^{\ast }$-Algebras.