Spectral invariance of Besov–Bessel subalgebras

Using principles of the theory of smoothness spaces, we give systematic constructions of scales of inverse-closed subalgebras of a given Banach algebra with the action of a dd-parameter automorphism group. In particular, we obtain the inverse-closedness of Besov algebras, Bessel potential algebras and approximation algebras of polynomial order in their defining algebra. By a proper choice of the group action, these general results can be applied to algebras of infinite matrices and yield inverse-closed subalgebras of matrices with off-diagonal decay of polynomial order. Besides alternative proofs of known results we obtain new classes of inverse-closed subalgebras of matrices with off-diagonal decay.This work is a continuation and extension of results presented by Gröchenig and Klotz (2010) [23].