Matthew J. Heath
Compactness of derivations from commutative Banach algebras.
Mon, 11/02/2009 - 16:01 — SomNambulAuthors: Matthew J. Heath
Subjects: Functional Analysis
AbstractWe 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.Characterising derivations from the disc algebra to its dual.
Fri, 09/11/2009 - 19:12 — SomNambulAuthors: Yemon Choi, Matthew J. Heath
Subjects: Functional Analysis
AbstractWe characterize the bounded derivations from the disc algebra to its dual in
terms of a natural `symbol' function. This is the first non-trivial uniform
algebra for which such a characterisation has been obtained.As an immediate corollary we show that all such derivations are automatically
compact, resolving a question raised by S. E. Morris. We also give the first
construction of explicit "Pietsch control measures" for such derivations, thus
obtaining an independent proof that they are 2-summing.
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