Abstract
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.
Highlights
A graph product of groups, first introduced in [32], is a group theoretic construction that generalizes both free products and Cartesian products of groups. It associates to a simplicial graph with discrete groups as vertices a new group by taking the free product of the vertex groups, with added relations depending on the graph
In the final part of this paper, we prove that Hecke C∗-algebras are exact and characterize their nuclearity in terms of the properties of the underlying group
A right-angled Coxeter group can be seen as a graph product where all groups Gv, v ∈ V Γ are equal to Z2
Summary
A graph product of groups, first introduced in [32], is a group theoretic construction that generalizes both free products and Cartesian products of groups. Applying Theorem 0.1 we deduce a Haagerup inequality for right-angled Hecke algebras that generalizes Haagerup’s inequality for free groups, see [33] and [43, Section 9.6] These are inequalities that estimate the operator norm of an operator of length d with the L2-norm up to some polynomial bound.
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