Maps completely preserving commutativity and maps completely preserving Jordan zero-product

Abstract

Let X, Y be real or complex Banach spaces with infinite dimension, and let A, B be standard operator algebras on X and Y , respectively. In this paper, we show that every map completely preserving commutativity from A onto B is a scalar multiple of either an isomorphism or (in the complex case) a conjugate isomorphism. Every map completely preserving Jordan zero-product from A onto B is a scalar multiple of either an isomorphism or (in the complex case) a conjugate isomorphism.

Maps completely preserving the quadratic operators

“…Completely invertibility preserving maps were characterized in [5]. Subsequently, in [6] completely idempotent and completely square-zero preserving maps and in [7] completely commutativity and completely Jordan zero product preserving maps were discussed.…”