A note on finite-dimensional quotients and the problem of automatic continuity for twisted convolution algebras
Felipe I. Flores
TL;DR
The paper studies automatic continuity for twisted convolution Banach algebras $L^1_{\boldsymbol{\alpha},\omega}({\rm G},\mathfrak{A})$ arising from a twisted action $( {\rm G}, \boldsymbol{\alpha}, \omega, \mathfrak{A})$. It proves that all closed cofinite ideals yield semisimple quotients by constructing a finite-dimensional $^*$-representation of the quotient via an extension to the multiplier algebra and a similarity transform using the multiplier group $\Gamma_{\rm G,\mathfrak{A}}$, showing the quotient is $^*$-isomorphic to a $C^*$-algebra. This semisimplicity, together with Willis and Dales–Willis results, yields broad automatic-continuity phenomena: for certain $\mathfrak{B}$, including twisted group algebras $L^1_\omega({\rm G})$ and convolution algebras $\ell^1_{\boldsymbol{\alpha}}({\rm G},\mathfrak{A})$, automatic continuity holds for intertwiners with finite-dimensional range when $\rm G$ is nilpotent or amenable with nuclear $\mathfrak{A}$. The work also shows that cofinite ideals possess bounded approximate identities, enabling equivalence results for finite-dimensional range homomorphisms and derivations (Dales–Willis). Overall, the paper broadens automatic continuity results beyond compact-generation assumptions and provides new examples in twisted harmonic analysis.
Abstract
In this note, we will show that the twisted convolution algebra $L^1_{α,ω}({\sf G},\mathfrak A)$ 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.