Abstract

The main focus of this thesis is to 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 also presented. Finally the symmetry of Fell bundles over groupoids is studied, where in the case of a discrete groupoid the notions of rigid symmetry and hypersymmetry coincide.