Twisted convolution algebras with coefficients in a differential subalgebra

TL;DR

This paper studies twisted convolution algebras with coefficients in a differential subalgebra, showing how certain group extensions and semidirect products lead to symmetric Banach *-algebras, expanding the class of known symmetric structures.

Contribution

It demonstrates that twisted convolution algebras with differential subalgebras are themselves differential, and applies this to identify new classes of symmetric Banach *-algebras from group extensions.

Findings

L^1_{α,ω}(G, A) is a differential subalgebra of L^1_{α,ω}(G, B).

Discrete rigidly symmetric extensions of compact groups are symmetric.

Semidirect products with a symmetric group and a compact group are symmetric.

Abstract

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