On surjective linear maps preserving commutativity

Abstract

We describe surjective linear maps preserving commutativity from (symmetric elements of) any algebra (with involution) onto (symmetric elements of) a prime algebra (with involution) not satisfying polynomial identities of low degree. Bijective commutativity preservers on skew elements of centrally closed prime algebras with involution of the first kind are also investigated.

Commutativity preserving maps revisited

“…In the 80's these results have been extended to various operator algebras, and finally in the 90's the treatment has moved to ring theory. We refer the reader to two recent papers [8,20] for more references and historic details.…”

On Maps Preserving Zero Jordan Products

“…Later it was sequentially studied for matrix algebras in [2,25,26] and for operator algebras in [19,23,24,27,32]. Applying the powerful techniques on functional identities (see surveys [10,12]) the problem of character-izing maps preserving zero Lie products was solved for many classes of rings [4,5,8,11,14,21], in particular prime rings [8,11] and matrix rings over unital rings [4].…”

Strong commutativity preserving generalized derivations on right ideals

“…research area in matrix theory, operator theory and ring theory (see for instance [2,6,9,[11][12][13]15,16,20,32,[41][42][43][44][45]47]). In [5] Bell and Daif investigated a certain kind of commutativity preserving maps as follows: Let S be a subset of R. A map f : S → R is called strong commutativity preserving (SCP) on S if [ f (x), f (y)] = [x, y] for all x, y ∈ S. Precisely, they proved that if a semiprime ring R admits a derivation which is SCP on a right ideal ρ, then ρ ⊆ Z (R).…”