Compactness of derivations from commutative Banach algebras

TL;DR

This paper investigates the conditions under which derivations from commutative Banach algebras into their dual modules are compact, providing characterizations and examples, especially focusing on the convolution algebra (\u2115_+).

Contribution

It establishes that the absence of compact derivations in a commutative Banach algebra implies the absence in all symmetric bimodules and characterizes compact derivations for (\u2115_+).

Findings

No compact derivations from A imply none into any symmetric bimodule.

Characterization of compact derivations from (_+) to its dual.

Existence of a non-compact bounded derivation in a uniform algebra.

Abstract

We consider the compactness of derivations from commutative Banach algebras into their dual modules. We show that if there are no compact derivations from a commutative Banach algebra, $A$, into its dual module, then there are no compact derivations from $A$ into any symmetric $A$-bimodule; we also prove analogous results for weakly compact derivations and for bounded derivations of finite rank. We then characterise the compact derivations from the convolution algebra $\ell^1(\Z_+)$ to its dual. Finally, we give an example (due to J. F. Feinstein) of a non-compact, bounded derivation from a uniform algebra $A$ into a symmetric $A$-bimodule.