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

Felipe I. Flores

TL;DR

The paper develops a systematic theory for $(k,p,q)$-differential subalgebras, Banach $^*$-subalgebras of a $C^*$-algebra governed by the inequality $ orm{a^{k}}_{ rak A} le C orm{a}_{ rak A}^{p} orm{a}_{ rak B}^{q}$, and shows they inherit C$^*$-like harmonic-analysis features. It builds a Dixmier– Baillet-type smooth functional calculus via a regular commutative algebra $ rak D_ au$, proving spectral stability and representation-compatible functional calculus, along with symmetry, $^*$-regularity, and the Wiener property. The framework encompasses a wide array of examples, including normable ideals, full Hilbert algebras, domains of closed derivations, and weighted Beurling-type algebras (Fell bundles, groups with subexponential growth, and rapid-decay groups), as well as matrix and integral-operator algebras with decay. It further derives automatic-continuity results for intertwining operators and derivations, yielding general criteria and recovering classical theorems such as Kissin–Shul’man and Longo in this broader context. Overall, the results unify and extend known harmonic-analytic properties for diverse algebras, providing new automatic-continuity tools and spectral-analytic insights for generalized differential 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.

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

TL;DR

The paper develops a systematic theory for -differential subalgebras, Banach -subalgebras of a -algebra governed by the inequality , and shows they inherit C-like harmonic-analysis features. It builds a Dixmier– Baillet-type smooth functional calculus via a regular commutative algebra , proving spectral stability and representation-compatible functional calculus, along with symmetry, -regularity, and the Wiener property. The framework encompasses a wide array of examples, including normable ideals, full Hilbert algebras, domains of closed derivations, and weighted Beurling-type algebras (Fell bundles, groups with subexponential growth, and rapid-decay groups), as well as matrix and integral-operator algebras with decay. It further derives automatic-continuity results for intertwining operators and derivations, yielding general criteria and recovering classical theorems such as Kissin–Shul’man and Longo in this broader context. Overall, the results unify and extend known harmonic-analytic properties for diverse algebras, providing new automatic-continuity tools and spectral-analytic insights for generalized differential subalgebras.

Abstract

For appropriate parameters , we introduce and systematically study the class of -differential subalgebras. This is a vast class of Banach -algebras defined by their relation with their -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 , and properties of automatic continuity.

Paper Structure

This paper contains 17 sections, 32 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. The class of --differential subalgebras
  3. Normable ideals
  4. Full Hilbert algebras
  5. Domains of closed --derivations
  6. Weighted --algebras of Fell bundles over discrete groups with polynomial growth
  7. Weighted subalgebras of groups with strong subexponential growth
  8. Sobolev algebras of groups with the rapid decay property
  9. Matrix algebras with polynomial off-diagonal decay
  10. Algebras of localized integral operators
  11. Aspects of harmonic analysis
  12. Symmetry
  13. Stability under the functional calculus of -
  14. Preservation of spectra
  15. Minimal ideals of a given hull
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\mathfrak{A}$ be a $(k,p,q)$-differential subalgebra of the $C^*$-algebra $\mathfrak{B}$. Then there exists a $\tau\in(0,1)$ such that, for every $a=a^*\in \mathfrak{A}$, one has Furthermore, the following are true for all $f\in\mathfrak D_\tau$.

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • ...and 60 more