Strong commutativity preserving maps on triangular rings

Abstract

Let U = Tri(A ,M ,B) be a triangular ring. It is shown, under some mild assumption, that every surjective strong commutativity preserving map Φ : U →U (i.e. [Φ(T ),Φ(S)]= [T,S] for all T,S ∈ U ) is of the form Φ(T ) = ZT + f (T ) , where Z is in Z (U ) , the center of U , Z2 = I and f is a map from U into Z (U ) . As an application, a characterization of general surjective maps that preserve the strong commutativity on the nest algebras of Banach space operators is given. Mathematics subject classification (2010): 47L35, 16U80.