K-POTENT PRESERVING LINEAR MAPS

Let B(X) be the Banach algebra of all bounded linear operators on a complex Banach space X. Let k ≥ 2 be an integer and φ a weakly continuous linear surjective map from B(X) into itself. It is shown that φ is k-potent preserving if and only if it is k-th-power preserving, and in turn, if and only if it is either an automorphism or an antiautomorphism on B(X) multiplied by a complex number λ satisfying λk-1= 1. Let A be a von Neumann algebra and B be a Banach algebra, it is also shown that a bounded surjective linear map from A onto B is k-potent preserving if and only if it is a Jordan homomorphism multiplied by an invertible element with (k - l)-th power I.

K-POTENT PRESERVING LINEAR MAPS

Let B(X) be the Banach algebra of all bounded linear operators on a complcx Banach space X. Let k ≥ 2 be an integer and φ a weakly continuous linear surjective map from B(X) into itself. It is shown that φ is k-potent preserving if and only if it is k-th-power preserving, and in turn, if and only if it is either an automorphism or an antiautomorphism on B(X) multiplied by a complex number λ satisfying λk-1 = 1. Let A be avon Neumann algebra andB be a Banach algebra, it is also shown that a bounded surjective linear map from A onto B is k-potent preserving if and only if it is a Jordan homomorphism multiplied by an invertible element with (k - 1)-th power I.