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.
Categories: math.OA