A note on finite-dimensional quotients and the problem of automatic continuity for twisted convolution algebras

TL;DR

This paper demonstrates that certain twisted convolution algebras associated with group actions on C*-algebras have the property that their finite-codimension quotients are semisimple, and uses this to identify classes with automatic continuity.

Contribution

It establishes a new property of twisted convolution algebras regarding their finite-codimension quotients and applies this to find classes with automatic continuity.

Findings

Finite-codimension quotients of these algebras are semisimple.

The property helps extend previous results on automatic continuity.

Provides new examples of algebras with automatic continuity.

Abstract

In this note, we will show that the twisted convolution algebra $L^1_{\alpha,\omega}({\sf G},\mathfrak A)$ associated to a twisted action of a locally compact group ${\sf G}$ on a $C^*$-algebra $\mathfrak A$ has the following property: Every quotient by a closed two-sided ideal of finite codimension produces a semisimple algebra. Afterward, we use this property, together with results by H. Dales and G. Willis, to extend previous results by the author and to produce large classes of examples of algebras with automatic continuity properties.