Characterizations of isomorphisms and derivations of some algebras

Let ϕ be a zero-product preserving bijective bounded linear map from a unital algebra A onto a unital algebra B such that ϕ(1)=k. We show that if A is a CSL algebra on a Hilbert space or a J-lattice algebra on a Banach space then there exists an isomorphism ψ from A onto B such that ϕ=kψ. For a nest algebra A in a factor von Neumann algebra, we characterize the linear maps on A such that δ(x)y+xδ(y)=0 for all x,y∈A with xy=0.