Symmetry and Spectral Invariance for Topologically Graded C*-Algebras and Partial Action Systems

TL;DR

This paper demonstrates that topologically graded C*-algebras over rigidly symmetric groups contain inverse-closed symmetric Banach *-algebras, extending to dual actions, partial crossed products, and convolution kernels, with various concrete examples.

Contribution

It establishes the existence of inverse-closed symmetric Banach *-algebras within topologically graded C*-algebras over rigidly symmetric groups, including new classes like dual actions and partial crossed products.

Findings

Existence of inverse-closed symmetric Banach *-algebras in graded C*-algebras

Extension to dual actions and partial crossed products

Concrete examples with convolution kernels and decay properties

Abstract

A discrete group $\G$ is called rigidly symmetric if the projective tensor product between the convolution algebra $\ell^1(\G)$ and any $C^*$-algebra $\A$ is symmetric. We show that in each topologically graded $C^*$-algebra over a rigidly symmetric group there is a $\ell^1$-type symmetric Banach $^*$-algebra, which is inverse closed in the $C^*$-algebra. This includes new general classes, as algebras admitting dual actions and partial crossed products. Results including convolution dominated kernels, inverse closedness with respect with ideals or weighted versions of the $\ell^1$-decay are included. Various concrete examples are presented.