On the lack of inverses to C*-extensions related to property T groups

TL;DR

This paper investigates the invertibility properties of certain C*-extensions associated with property T groups, demonstrating non-invertibility and non-semi-invertibility through advanced algebraic techniques.

Contribution

It provides new examples and proofs showing that some C*-extensions related to property T groups are not invertible or semi-invertible, extending understanding of their algebraic structure.

Findings

A non-invertible C*-extension example is identified.

A modified example is shown to be not invertible up to homotopy.

The vanishing of a specific element in the asymptotic tensor product is demonstrated.

Abstract

Using ideas of S. Wassermann on non-exact $C^*$-algebras and property T groups, we show that one of his examples of non-invertible C*-extensions is not semi-invertible. To prove this, we show that a certain element vanishes in the asymptotic tensor product. We also show that a modification of the example gives a C*-extension which is not even invertible up to homotopy.