Zero product preserving additive maps on triangular algebras

Abstract

Suppose that $Tri(A,M,B)$ is a unital triangular algebra, where $M$ is a faithful $(A,B)$-bimodule, and $U$ is a unital algebra. Let $\theta :Tri(A,M,B)\to U$ be a bijective zero product preserving additive map. We show that under some mild conditions θ is a product of a central invertible element and a ring isomorphism. Our result applies to block upper triangular matrix algebras, nest algebras on Banach spaces and nest subalgebras of von Neumann algebras.