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On the continuity of derivations over locally regular Banach algebras

Felipe I. Flores

Abstract

We study the problem of continuity of derivations over Banach algebras. More specifically, we consider a class of Banach algebras that contain a dense '$C^*$-like' subalgebra. We discuss applications to $L^p$-crossed products and symmetrized $L^p$-crossed products. As an example, our results imply that every derivation over the $L^p$-crossed product $F^p(G,X,α)$ is continuous, provided that $G$ is infinite, finitely generated, has polynomial growth, and acts freely on the compact Hausdorff space $X$.

On the continuity of derivations over locally regular Banach algebras

Abstract

We study the problem of continuity of derivations over Banach algebras. More specifically, we consider a class of Banach algebras that contain a dense '-like' subalgebra. We discuss applications to -crossed products and symmetrized -crossed products. As an example, our results imply that every derivation over the -crossed product is continuous, provided that is infinite, finitely generated, has polynomial growth, and acts freely on the compact Hausdorff space .

Paper Structure

This paper contains 3 sections, 12 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications to --crossed products

Key Result

Theorem 1.1

Let $\mathfrak{A}\subset\mathfrak{B}$ be a locally regular inclusion and let $D:\mathfrak{B}\to\mathcal{X}$ be a derivation into a Banach $\mathfrak{B}$-bimodule. Further suppose that $\mathfrak{A}$ is unital and that ${\rm C^*}(\mathfrak{A})$ has no proper closed two-sided ideals of finite codimens

Theorems & Definitions (24)

  • Theorem 1.1: see Theorem \ref{['main-simple']}
  • Theorem 1.2: see Theorem \ref{['main']}
  • Corollary 1.3: see Corollary \ref{['cor2']}
  • Corollary 1.4: see Corollary \ref{['cor1']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 14 more