The Structure of Hecke Operator Algebras

Abstract

This dissertation is concerned with the study of the structure of certain deformations of operator algebras associated with Coxeter groups. These operator algebras, called Hecke C*-algebras and Hecke-von Neumann algebras, are operator algebraic completions of Iwahori-Hecke algebras. They occur as natural abstractions of certain endomorphism rings occurring in the representation theory of Lie groups and play a role in knot theory, combinatorics, the theory of buildings, quantum group theory, non-commutative geometry, and the local Langlands program. In this thesis we mainly focus on the ideal structure of Hecke C*-algebras, on approximation properties, and the rigidity of Hecke-von Neumann algebras. On our way we encounter and study several other concepts such as (Khintchine inequalities of) graph products of operator algebras, topological dynamics associated with boundaries and compactifications of graphs and (Coxeter) groups, C*-simplicity methods, the relative Haagerup property of sigma-finite unital inclusions of von Neumann algebras, approximation properties of operator algebras, and the rigidity theory of von Neumann algebras.