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

A discrete group G${\sf G}$ is called rigidly symmetric if the projective tensor product between the convolution algebra ℓ1(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 ℓ1$\ell ^1$‐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 ℓ1$\ell ^1$‐decay are included. Various concrete examples are presented.