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

Published 19 Dec 2019 in math.OA | (1912.09061v2)

Abstract: Graph products of groups were introduced by Green in her thesis. They have an operator algebraic counterpart introduced and explored by Fima and the first-named author. In this paper we prove Khintchine type inequalities for general C$^{{\ast}$-algebraic} graph products which generalize results by Ricard and Xu on free products of C$^{{\ast}$-algebras.} We apply these inequalities in the context of (right-angled) Hecke C$^{{\ast}$-algebras,} which are deformations of the group algebra of Coxeter groups. For these we deduce a Haagerup inequality. We further use this to study the simplicity and trace uniqueness of (right-angled) Hecke C$^{{\ast}$-algebras.} Lastly we characterize exactness and nuclearity of general Hecke C$^{{\ast}$-algebras.}