Harmonic analysis and automatic continuity in the context of generalized differential subalgebras

For appropriate parameters $k,p,q$, we introduce and systematically study the class of $(k,p,q)$-differential subalgebras. This is a vast class of Banach ${}^{\ast }$-algebras defined by their relation with their $C^{\ast }$-envelopes. Some examples are given by normable two-sided ${}^{\ast }$-ideals, domains of closed ${}^{\ast }$-derivations, full Hilbert algebras, and some weighted convolution algebras of various kinds. We prove that this class of algebras possesses various interesting properties, such as closedness under a functional calculus based on smooth functions, ${}^{\ast }$-regularity, Wiener’s property $(W)$, and properties of automatic continuity.