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• July 11, 2008, Friday - 15h15 (Room P 3.10):

Compact failure of multiplicativity for linear maps between Banach algebras

  Matthew Heath

(Instituto Superior Técnico, U.T. Lisboa)

Abstract. The definition of compactness (and that of weak compactness) for a linear map between normed spaces may be extended to multilinear maps in a fairly natural way. We treat compactness as a sort of "smallness" condition for multilinear maps. For Banach algebras A and B we call a linear map, T: A -> B , a cf-homomorphism (meaning "compact from a homomorphism") if the bilinear map S :  A x A  ->  B ,  S(a,b) = T(a)T(b) - T(ab)   (i.e. if the "failure to be multiplicative") is a compact bilinear map. We give general theorems showing that such maps are rather well behaved as well as numerous examples. In particular we characterise the pairs of compact, Hausdorff spaces X and Y for which cf-isomorphisms from C(X) to C(Y) are automatically multiplicative.

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