Nonlinear maps preserving the mixed product [A ● B,C]* on von Neumann algebras | Zendy

Changjing Li | Zendy; Yuanyuan Zhao | Zendy; Fangfang Zhao | Zendy

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Nonlinear maps preserving the mixed product [A ● B,C]* on von Neumann algebras

Author(s) -

Changjing Li,

Yuanyuan Zhao,

Fangfang Zhao

Publication year - 2021

Publication title -

filomat

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.449

H-Index - 34

eISSN - 2406-0933

pISSN - 0354-5180

DOI - 10.2298/fil2108775l

Subject(s) - bijection , mathematics , isomorphism (crystallography) , conjugate , von neumann algebra , abelian group , abelian von neumann algebra , von neumann architecture , combinatorics , pure mathematics , linear map , algebra over a field , jordan algebra , mathematical analysis , algebra representation , chemistry , crystal structure , crystallography

Let A and B be two von Neumann algebras. For A,B ? A, define by [A,B]* = AB-BA* and A ? B = AB + BA* the new products of A and B. Suppose that a bijective map ? : A ? B satisfies ?([A ? B,C]*) = [?(A)? ?(B),?(C)]* for all A,B,C ? A. In this paper, it is proved that if A and B be two von Neumann algebras with no central abelian projections, then the map ?(I)? is a sum of a linear *-isomorphism and a conjugate linear +-isomorphism, where ?(I) is a self-adjoint central element in B with ?(I)2 = I. If A and B are two factor von Neumann algebras, then ? is a linear *-isomorphism, or a conjugate linear *-isomorphism, or the negative of a linear *-isomorphism, or the negative of a conjugate linear *-isomorphism.

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