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Graph products of groups were introduced by Green in her thesis. They have anoperator algebraic counterpart introduced and explored by Fima and thefirst-named author. In this paper we prove Khintchine type inequalities forgeneral C$^{\ast}$-algebraic graph products which generalize results by Ricardand Xu on free products of C$^{\ast}$-algebras. We apply these inequalities inthe context of (right-angled) Hecke C$^{\ast}$-algebras, which are deformationsof the group algebra of Coxeter groups. For these we deduce a Haagerupinequality. We further use this to study the simplicity and trace uniqueness of(right-angled) Hecke C$^{\ast}$-algebras. Lastly we characterize exactness andnuclearity of general Hecke C$^{\ast}$-algebras.

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