Additive maps preserving products equal to fixed elements on Cayley-Dickson algebras

Abstract

– In this paper, we study additive maps f : A → A on an alternative Cayley-Dickson algebra A satisfying the product preserving property: f ( x ) f ( y ) = m whenever xy = k, where k and m are constant elements of A . We prove that f may be written as a multiple of a Jordan homomorphism. Unlike in the associative case, Jordan homomorphisms of strictly alternative division rings need not be homomorphisms or anti-homomorphisms, but we provide necessary and sufficient conditions for this to occur. We then extend these ideas to split alternative Cayley-Dickson algebras of characteristic not 2 and present some open questions when k or m is noninvertible. The final section handles the problem k = m = 0 and generalizes to biadditive maps that preserve the zero product.