Graph product Khintchine inequalities and Hecke C*-algebras: Haagerup inequalities, (non)simplicity, nuclearity and exactness

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.