Symmetry and Inverse Closedness for Some Banach $^ *$-Algebras Associated to Discrete Groups

TL;DR

This paper investigates symmetry and inverse-closedness properties of Banach *-algebras associated with discrete groups, extending known results to twisted crossed products and kernels, with implications for operator theory.

Contribution

It demonstrates that for rigidly symmetric groups, twisted crossed products and certain kernel algebras are also symmetric Banach *-algebras, extending previous symmetry results.

Findings

Twisted crossed products preserve symmetry for rigidly symmetric groups.

Extension of symmetry properties to decay conditions beyond -conditions.

Analysis of algebra of twisted kernels in both intrinsic and represented forms.

Abstract

A discrete group $\G$ is called rigidly symmetric if for every $C^*$-algebra $\A$ the projective tensor product $\ell^1(\G)\widehat\otimes\A$ is a symmetric Banach $^*$-algebra. For such a group we show that the twisted crossed product $\ell^1_{\alpha,\o}(\G;\A)$ is also a symmetric Banach $^*$-algebra, for every twisted action $(\alpha,\o)$ of $\G$ in a $C^*$-algebra $\A$\,. We extend this property to other types of decay, replacing the $\ell^1$-condition. We also make the connection with certain classes of twisted kernels, used in a theory of integral operators involving group $2$-cocycles. The algebra of these kernels is studied, both in intrinsic and in represented version.