Mappings preserving sum of products a∘b+ba∗ on factor von Neumann algebras

Abstract

Let A and B be two factor von Neumann algebras. In this paper, we proved that a bijective mapping Φ:A→B satisfies Φ(a∘b+ba∗)=Φ(a)∘Φ(b)+Φ(b)Φ(a)∗ (where ∘ is the special Jordan product on A and B, respectively), for all elements a,b∈A, if and only if Φ is a ∗-ring isomorphism. In particular, if the von Neumann algebras A and B are type I factors, then Φ is a unitary isomorphism or a conjugate unitary isomorphism.

Mappings preserving sum of products a◊b + b*a (resp., a*◊b + ab*) on ∗-algebras

“…We will establish the proof of Lemma 2.3 in a series of Properties, based on the techniques used in [1], [2], [3], [4] and [6]. We begin, though, with a well-known result that will be used throughout this paper.…”

“…These products play an important role in some research topics and their studies have recently attracted the attention of some authors (for example, see [1], [4], [5] and [6] and for other products see [2], [3], [7] and [8]). In particular, the authors in [1], [4] and [6] studied bijective mappings preserving the new products mentioned above.…”