Twisted convolution algebras with coefficients in a differential subalgebra

Abstract

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

Theorems & Definitions (27)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Example 2.1
• Example 2.2
• Remark 2.3
• Definition 2.4
• Lemma 2.5
• proof
• Definition 2.6