Maps preserving zero products
Volume 193 / 2009
Abstract
A linear map $T$ from a Banach algebra $A$ into another $B$ preserves zero products if $T(a)T(b)=0$ whenever $a,b\in A$ are such that $ab=0$. This paper is mainly concerned with the question of whether every continuous linear surjective map $T\colon A\rightarrow B$ that preserves zero products is a weighted homomorphism. We show that this is indeed the case for a large class of Banach algebras which includes group algebras. Our method involves continuous bilinear maps $\phi\colon A\times A\rightarrow X$ (for some Banach space $X$) with the property that $\phi(a,b)=0$ whenever $a,b\in A$ are such that $ab=0$. We prove that such a map necessarily satisfies $\phi(a\mu,b)=\phi(a,\mu b)$ for all $a,b\in A$ and for all $\mu$ from the closure with respect to the strong operator topology of the subalgebra of $\mathcal{M}(A)$ (the multiplier algebra of $A$) generated by doubly power-bounded elements of $\mathcal{M}(A)$. This method is also shown to be useful for characterizing derivations through the zero products.
