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

TL;DR

This paper establishes Khintchine inequalities for graph product C*-algebras, applies them to Hecke C*-algebras to derive Haagerup inequalities, and investigates their structural properties like simplicity, nuclearity, and exactness.

Contribution

It generalizes Khintchine inequalities to graph product C*-algebras and applies these results to analyze structural properties of Hecke C*-algebras.

Findings

Proved Khintchine inequalities for graph product C*-algebras.

Derived Haagerup inequalities for Hecke C*-algebras.

Characterized conditions for simplicity, nuclearity, and exactness of Hecke C*-algebras.

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.