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].

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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.

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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.

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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.

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This dissertation is concerned with the study of the structure of certain deformations of operator algebras associated with Coxeter groups. These operator algebras, called Hecke C*-algebras and Hecke-von Neumann algebras, are operator algebraic completions of Iwahori-Hecke algebras. They occur as natural abstractions of certain endomorphism rings occurring in the representation theory of Lie groups and play a role in knot theory, combinatorics, the theory of buildings, quantum group theory, non-commutative geometry, and the local Langlands program. In this thesis we mainly focus on the ideal structure of Hecke C*-algebras, on approximation properties, and the rigidity of Hecke-von Neumann algebras. On our way we encounter and study several other concepts such as (Khintchine inequalities of) graph products of operator algebras, topological dynamics associated with boundaries and compactifications of graphs and (Coxeter) groups, C*-simplicity methods, the relative Haagerup property of sigma-finite unital inclusions of von Neumann algebras, approximation properties of operator algebras, and the rigidity theory of von Neumann algebras.@en

Graph products of groups were introduced by Green in her thesis [32]. They have an operator algebraic counterpart introduced and explored in [14]. In this paper we prove Khintchine type inequalities for general C⁎-algebraic graph products which generalize results by Ricard and Xu [50] on free products of C⁎-algebras. We apply these inequalities in the context of (right-angled) Hecke C⁎-algebras, which are deformations of the group algebra of Coxeter groups (see [22]). For these we deduce a Haagerup inequality which generalizes results from [33]. We further use this to study the simplicity and trace uniqueness of (right-angled) Hecke C⁎-algebras. Lastly we characterize exactness and nuclearity of general Hecke C⁎-algebras.@en