Nonlinear Maps Preserving Mixed Product on Factors
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Nonlinear Maps Preserving Mixed Product on Factors Yuanyuan Zhao1 · Changjing Li1
· Quanyuan Chen2
Received: 2 May 2020 / Revised: 11 July 2020 / Accepted: 20 July 2020 © Iranian Mathematical Society 2020
Abstract Let A and B be two factors with dimA > 4. In this article, it is proved that a bijective map : A → B satisfies ([A • B, C]) = [(A)•(B), (C)] for all A, B, C ∈ A if and only if is a linear ∗-isomorphism, or a conjugate linear ∗-isomorphism, or the negative of a linear ∗-isomorphism, or the negative of a conjugate linear ∗isomorphism. Keywords Jordan ∗-product · Isomorphism · Factors Mathematics Subject Classification 47B48 · 46L10
1 Introduction Let A and B be two algebras. Recall that a map : A → B preserves product or is multiplicative if (AB) = (A)(B) for all A, B ∈ A. The question of when a multiplicative map is additive was discussed in [16]. Motivated by this, many authors pay more attention to the maps on algebras preserving Lie product [A, B] = AB − B A (for example, see [13–15,17,18]), or the skew Lie product [A, B]∗ = AB − B A∗ (for example, see [1,3,7,11]), or the Jordan ∗-product A • B = AB + B A∗ (for example,
Communicated by Shirin Hejazian.
B
Changjing Li [email protected] Yuanyuan Zhao [email protected] Quanyuan Chen [email protected]
1
School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, People’s Republic of China
2
College of Information, Jingdezhen Ceramic Institute, Jingdezhen 333403, People’s Republic of China
123
Bulletin of the Iranian Mathematical Society
see [4,8,12,22]). These results show that the (skew) Lie product or Jordan ∗-product structure is enough to determine the algebraic structure. Recently, maps preserving the products of the mixture of (skew) Lie product and Jordan ∗-product have received a fair amount of attention. For example, Yang and Zhang [19] studied the nonlinear maps preserving the mixed skew Lie triple product [[A, B]∗ , C] on factors. Li et al. studied the nonlinear maps preserving the skew Lie triple product [[A, B]∗ , C]∗ (for example, see [6,10]) and the Jordan triple ∗-product A • B • C (for example, see [9,21]) on von Neumann algebras. In the present paper, we will establish the structure of the nonlinear maps preserving the mixed product [A • B, C] on factors. Let R and C denote, respectively the real field and complex field. A von Neumann algebra A is a weakly closed, self-adjoint algebra of operators on a Hilbert space H containing the identity operator I . A is a factor means that its center is CI . It is well known that the factor A is prime, in the sense that AAB = 0 for A, B ∈ A implies either A = 0 or B = 0.
2 Additivity The main result in this section is the following. Theorem 2.1 Let A and B be two factors. Suppose that a bijective map : A → B satisfies ([A • B, C]) = [(A) • (B), (C)] for all A, B, C ∈ A. Then is additive. Proof We will complete the proof by proving several claims.
Claim 1 (0) = 0. For every A ∈ A, we have (0) = ([0 • 0, A]) = [(0) • (0), (A)]. Since is
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