On isomorphisms of standard operator algebras

Abstract

Abstract. The aim of this paper is to show that between standard operator algebras every bijective map with a certain multiplicativity property related to the Jordan triple isomorphisms of associative rings is automatically additive.

MAPPING PRESERVING NUMERICAL RANGE OF OPERATOR PRODUCTS ON C*-ALGEBRAS

“…Also according to Theorem 3.5, the map ψ = u 2 φ, is multiplicative and ψ(1) = 1. Consequently, by [14] it is additive. Finally, we have shown that ψ is an algebra isomorphism which preserves self-adjoint elements.…”

“…The question of when a multiplicative map is additive was attacked by several authors. For instance, if ψ is a bijective map on a standard operator algebra, Molnàr showed in [14] that if φ satisfies ψ(ABA) = ψ(A)ψ(B)ψ(A), then ψ is additive. Hence, based on the aforesaid, when the algebras A and B are the algebras of all bounded linear operators acting on some Hilbert spaces, Theorem 3.5 can be refined as follows.…”

Additivity of Jordan Triple Product Homomorphisms on Generalized Matrix Algebras

“…For operator algebras, the same problem was treated in [1,14,21]. In the papers [2,3,7,12,13,15,19], the additivity of maps on operator algebras which are multiplicative with respect to other products, such as the Jordan product, the Jordan triple product or Jordan triple product homomorphisms were investigated. Also, the papers [6,9,20] studied the similar questions for elementary maps and Jordan elementary maps on rings or operator algebras.…”

“…Kuzma [7] described the forms of Jordan triple product homomorphisms on matrix algebras and Molnár [19] obtained the exact forms of Jordan triple product homomorphisms between standard operator algebras.…”

Jordan elementary maps on rings

“…Recently, we extended Martindale's results to elementary maps of rings [4]. The paper [7] described the form of a bijective Jordan triple map of standard operator algebras on Banach spaces of dimensions at least 3; it turns out that such a map is additive. As remarked in the end of this reference, it is a challenging problem to generalize the "additive result" to general rings.…”