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

TL;DR

This paper introduces and analyzes a broad class of Banach $^*$-algebras called $(k,p,q)$-differential subalgebras, exploring their properties and relations to $C^*$-envelopes, with implications for automatic continuity and functional calculus.

Contribution

It systematically studies the properties of $(k,p,q)$-differential subalgebras, including their stability under functional calculus and automatic continuity, expanding understanding of their structure and applications.

Findings

Proves closedness under smooth functional calculus.

Establishes $^*$-regularity and Wiener's property $(W)$ for these algebras.

Demonstrates automatic continuity in the context of these subalgebras.

Abstract

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 $^*$-algebras defined by their relation with their $C^*$-envelopes. Some examples are given by normable two-sided $^*$-ideals, domains of closed $^*$-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, $^*$-regularity, Wiener's property $(W)$, and properties of automatic continuity.