Weighted Jordan homomorphisms

Abstract

Let A and B be unital rings. An additive map T : A → B is called a weighted Jordan homomorphism if c = T (1) is an invertible central element and cT (x 2 ) = T (x) 2 for all x ∈ A. We provide assumptions, which are in particular fulfilled when A = B = Mn(R) with n ≥ 2 and R any unital ring with 1 2 , under which every surjective additive map T : A → B with the property that T (x)T (y) + T (y)T (x) = 0 whenever xy = yx = 0 is a weighted Jordan homomorphism. Further, we show that if A is a prime ring with char(A) = 2, 3, 5, then a bijective additive map T : A → A is a weighted Jordan homomorphism provided that there exists an additive map S : A → A such that S(x 2 ) = T (x) 2 for all x ∈ A.

Maps preserving two-sided zero products on Banach algebras

“…In fact, [3,Theorem 3.3] does not require that Φ satisfies (1.1) but only that for all a, b ∈ A, (1.2) ab = ba = 0 =⇒ Φ(a) • Φ(b) = 0. This condition was also considered in the recent algebraic paper [8]. Observe also that it is more general than the condition that Φ preserves zero Jordan products (a [9].…”