Maps Preserving k-Jordan Products on Operator Algebras

Abstract

For any positive integer k, the k-Jordan product of a , b in a ring R is defined by { a , b } k = { { a , b } k − 1 , b } 1 , where { a , b } 0 = a and { a , b } 1 = a b + b a . A map f on R is k-Jordan zero-product preserving if { f ( a ) , f ( b ) } k = 0 whenever { a , b } k = 0 for a , b ∈ R ; it is strong k-Jordan product preserving if { f ( a ) , f ( b ) } k = { a , b } k for all a , b ∈ R . In this paper, strong k-Jordan product preserving nonlinear maps on general rings and k-Jordan zero-product preserving additive maps on standard operator algebras are characterized, generalizing some known results.