Kazhdan's Property T and C*-algebras

TL;DR

This paper explores the extension of Kazhdan's property T to C*-algebras, demonstrating that nuclear C*-algebras with this property are essentially finite-dimensional, with a complex proof compared to the group case.

Contribution

It extends Kazhdan's property T to the C*-algebra setting and proves that nuclear C*-algebras with property T are finite-dimensional.

Findings

Nuclear C*-algebras with property T are finite-dimensional.

The proof is more complex than in the discrete group case.

Abstract

Kazhdan's notion of property T has recently been imported to the C$^*$-world by Bekka. Our objective is to extend a well known fact to this realm; we show that a nuclear C$^*$-algebra with property T is finite dimensional (for all intents and purposes). Though the result is not surprising, the proof is significantly more complicated than the discrete group case.