Zero product preserving maps of operator-valued functions

Abstract:

Let $X,Y$ be locally compact Hausdorff spaces and ${\mathcal M}$, ${\mathcal N}$ be Banach algebras. Let $\theta : C_0(X,{\mathcal M}) \to C_0(Y, {\mathcal N})$ be a zero product preserving bounded linear map with dense range. We show that $\theta$ is given by a continuous field of algebra homomorphisms from ${\mathcal M}$ into ${\mathcal N}$ if ${\mathcal N}$ is irreducible. As corollaries, such a surjective $\theta$ arises from an algebra homomorphism, provided that ${\mathcal M}$ is a $W^*$-algebra and ${\mathcal N}$ is a semi-simple Banach algebra, or both ${\mathcal M}$ and ${\mathcal N}$ are $C^*$-algebras.