Compactness of derivations from commutative Banach algebras

Volume 91 / 2010

Matthew J. Heath Banach Center Publications 91 (2010), 191-198 MSC: Primary 46J10; Secondary 46H25, 46J05. DOI: 10.4064/bc91-0-11

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 $\ell^1({\mathbb Z}_+)$ 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.

Authors

  • Matthew J. HeathDepartamento de Matemática
    Instituto Superior Técnico
    Av. Rovisco Pais
    1049-001 Lisboa, Portugal
    e-mail

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