Biseparating maps between operator algebras

Abstract

We prove that a biseparating map between spaces B (E), and some other Banach algebras, is automatically continuous and an algebra isomorphism.1991 Mathematics Subject Classification. Primary 47L10; Secondary 46H40, 46B28.

Mappings preserving zero products

“…Here, it is not necessary to assume that S contains the identity or is closed in any topology. Recently, Araujo and Jarosz [2] showed that every bijective linear operator between two unital standard operator algebras which preserves zero products in both directions is a scalar multiple of an algebra isomorphism. However, in the nonbijective case it becomes a very difficult task without assuming continuity.…”

“…In fact, θ preserves zero products if and only if θ(f ) = hf •σ, where h can be zero somewhere and σ is a general continuous map (see, e.g., [21]). In [1,2,12,16,18,20,21], to name a few, the above relations are extended to the case where θ is not bijective, or is not continuous, or maps between vector-valued function spaces. We are interested in the question if the zero-product preserving property still plays an important role in a general algebraic setting.…”

On Maps Preserving Zero Jordan Products

“…Let R and B be rings with 1 2 , A ¼ M n ðRÞ where n 5 3, and : A ! B an additive map which preserves 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.…”

Maps on idempotent operators

“…This may result in a stronger conclusion than one expected at the beginning. For example, Araujo and Jarosz [1] obtained a characterization of linear biseparating maps (maps preserving zero products) on B(X). Using the above suggested approach a non-linear extension of their result was given in [25].…”