Functional Analysis and Applications Seminar 

11/07/2008, 15:15 — 16:00 — Room P3.10, Mathematics Building
Matthew Heath, Instituto Superior Técnico, U.T. Lisboa

Compact failure of multiplicativity for linear maps between Banach algebras

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, $T: A \rightarrow B$ , a cf-homomorphism (meaning "compact from a homomorphism") if the bilinear map $S : A \times A \rightarrow B$ , $S(a,b) = T(a)T(b) - T(a b)$ (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.