Twisted convolution algebras with coefficients in a differential subalgebra

Abstract

Let $(\mathsf{G},\alpha ,\omega ,\mathfrak{B})$ be a measurable twisted action of the locally compact group $\mathsf{G}$ on a Banach ${}^{\ast }$-algebra $\mathfrak{B}$ and $\mathfrak{A}$ a differential Banach ${}^{\ast }$-subalgebra of $\mathfrak{B}$, which is stable under said action. We observe that $L_{\alpha ,\omega }^1(\mathsf{G},\mathfrak{A})$ is a differential subalgebra of $L_{\alpha ,\omega }^1(\mathsf{G},\mathfrak{B})$. We use this fact to provide new examples of groups with symmetric Banach ${}^{\ast }$-algebras. In particular, we prove that discrete rigidly symmetric extensions of compact groups are symmetric or that semidirect products $\mathsf{K}⋊\mathsf{H}$, with $\mathsf{H}$ symmetric and $\mathsf{K}$ compact, are symmetric.

arXiv comment9 pages. Some typos are fixed. The title is changed