Symmetry and spectral invariance for topologically graded C∗$C^*$‐algebras and partial action systems

Abstract

A discrete group G ${\sf G}$ is called rigidly symmetric if the projective tensor product between the convolution algebra ℓ 1 ( G ) $\ell ^1({\sf G})$ and any C ∗ $C^*$ -algebra A $\mathcal {A}$ is symmetric. We show that in each topologically graded C ∗ $C^*$ -algebra over a rigidly symmetric group there is a ℓ 1 $\ell ^1$ -type symmetric Banach ∗ $^*$ -algebra, which is inverse closed in the C ∗ $C^*$ -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 ℓ 1 $\ell ^1$ -decay are included. Various concrete examples are presented.