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

Abstract: In this note we 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. We use this property, together with results of H. Dales and G. Willis, to build up on previous results of the author and produce large classes of examples of algebras with properties of automatic continuity.