Mappings preserving zero products

Abstract

Abstract. Let θ : M → N be a zero-product preserving linear map between algebras. We show that under some mild conditions θ is a product of a central element and an algebra homomorphism. Our result applies to matrix algebras, standard operator algebras, C * -algebras and W * -algebras.

Zero product preserving maps of operator-valued functions

“…By results in [10] (see also [7]), for each y in Y we have that θ π (1)(y) = H y (1) is in the center of N and…”

“…For example, two abelian C * -algebras are *-isomorphic if there exists a bijective linear map between them preserving zero products ( [13], [11], [14]). On the other hand, bounded bijective linear zero product preservers of nonabelian C * -algebras also provide algebraic *-isomorphisms [7]. Recently, Araujo and Jarosz [2] showed that the existence of a bijective linear map of standard operator algebras preserving zero products in both ways also implies that they are isomorphic as Banach algebras.…”

On Maps Preserving Zero Jordan Products

“…In case à is the ordinary multiplication, a similar problem, namely that of describing maps preserving zero products, was studied in [1,15,18,30,31]. It turns out that the question of preservers on zero products is more difficult than that on zero Lie products.…”

“…So a reasonable approach to the question of preservers on zero products is to impose on the rings under consideration the existence of some nontrivial zero-divisors. In our recent papers maps preserving zero products were described for rings generated by idempotents [18] and for prime rings containing nontrivial idempotents [15].…”

Partially defined σ-derivations on semisimple Banach algebras

“…Recently, there have been much work concerning maps preserving zero products in the literature (see [8,10,11,14,18,28]). Applying Theorem 1.1 we obtain the continuity of linear maps which satisfy the σ-derivation expansion formula on zero products.…”