A note on strong skew Jordan product preserving maps on von Neumann algebras
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A note on strong skew Jordan product preserving maps on von Neumann algebras Ali Taghavi1 · Farzaneh Kolivand1
© Akadémiai Kiadó, Budapest, Hungary 2017
Abstract Let A be a von Neumann algebra acting on the complex Hilbert space H and : A −→ A be a surjective map that satisfies the condition (T )(P) + (P)(T )∗ = T P + P T ∗ for all T and all projections P in A. We characterize the concrete form of on selfadjoint elements of A. Also when A is a factor von Neumann algebra, it is shown that is either of the form (T ) = T + iτ (T )I or of the form (T ) = −T + iτ (T )I , where τ : A −→ R is a real map. Keywords Factor von Neumann algebra · Strong commutativity · Skew Jordan product Mathematics Subject Classification 47B48 · 46L10
1 Introduction and statement of the main results Let A be an algebra. For A, B ∈ A, A ◦ B = AB + B A is the Jordan product and [A, B] = AB − B A is the Lie product. The map : A −→ A is Lie product preserving if [A, B] = [(A), (B)] for all A, B ∈ A. This kind of maps are closely related to Lie homomorphisms (linear maps preserving Lie products) which have been studied by many authors. For example see [2,5,7,10,11] and their references. Šemrl [17] characterized bijective maps preserving the Lie product on B(X ). Zhang and Zhang [19] characterized the structure of bijective maps preserving the Lie product on factor von Neumann algebras, with a different method. Also
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Ali Taghavi [email protected] Farzaneh Kolivand [email protected]
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Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, P.O. Box 47416-1468, Babolsar, Iran
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A. Taghavi, F. Kolivand
the problem of characterizing maps preserving the commutativity (maps preserving zero Lie products), studied intensively (see [1,2,5,7,13,17] and references therein). In [1], Bell and Daif gave the conception of strong commutativity preserving maps. We say that a map : A −→ A is strong commutativity preserving if [(A), (B)] = [A, B], for all A, B ∈ A. Note that a strong commutativity preserving map must be commutativity preserving, but the reverse is not true generally. These maps are also called strong Lie product preserving maps. Brešar and Miers [5] proved that every strong commutativity preserving additive map on a semiprime ring R is of the form (A) = λA + μ(A), where λ ∈ C , the extended centroid of R, λ2 = 1, and μ: R −→ C is an additive map. In [9], Lin and Liu obtained a similar result on a noncentral Lie ideal of a prime ring. In [14,15], Qi and Hou gave a complete characterization of strong commutativity preserving surjective maps (without the assumption of additivity) on prime rings and triangular algebras, respectively. Let A be a *-ring. For any A, B ∈ A, [A, B]∗ = AB − B A∗ denotes the skew Lie product of A and B. This kind of product is found playing a more and more important role in some research topics such as representing quadratic functionals with sesquilinear functionals, and its study has attracted many authors attention (see [3,6,18] and the references therein). M
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