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arXiv · 0910.5902

Compactness of derivations from commutative Banach algebras

Matthew J. Heath
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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.

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Matthew J. Heath. 2009-10-30. Compactness of derivations from commutative Banach algebras. https://arxiv.org/abs/0910.5902

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