Asymptotic representations of the reduced C*-algebra of a free group: an example

TL;DR

This paper presents a non-trivial asymptotic representation of the reduced C*-algebra of a free group, enabling analysis of tensor norms and extension semi-invertibility, contributing to operator algebra theory.

Contribution

It provides a concrete example of asymptotic representation for the reduced C*-algebra of a free group, illustrating its applications in tensor norm evaluation and extension properties.

Findings

Evaluates asymptotic tensor C*-norms of certain elements

Shows semi-invertibility of a specific non-invertible extension

Provides a new example of asymptotic representation in operator algebras

Abstract

We give an example of a non-trivial asymptotic representation of the reduced C*-algebra of a free group. This example allows to evaluate the asymptotic tensor C*-norm of some elements in tensor product C*-algebras and to show semi-invertibility of the non-invertible extension of $C^*_r(\mathbb F_2)$ considered by Haagerup and Thorbjornsen.