Compactness of derivations from commutative Banach algebras

Details Compactness of derivations from commutative Banach algebras Compactness of derivations from commutative Banach algebras

2009-10-30

arxiv.org/abs/0910.5902v1

Mathematics

Functional Analysis

Scientific paper

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.

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