2-Local $${*}$$ ∗ -Lie isomorphisms of operator algebras

Li, Changjing; Lu, Fangyan

doi:10.1007/s00010-016-0411-5

Cited by 17 publications

(1 citation statement)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let A be a * -algebra over the complex field C. For A, B ∈ A, define the skew Lie product of A and B by [A, B] * = AB − BA * and the Jordan * -product of A and B by A • B = AB + BA * . The skew Lie product and the Jordan * -product are fairly meaningful and important in some research topics (see [10][11][12][13][14]25]). They were extensively studied because they naturally arise in the problem of representing quadratic functionals with sesquilinear functionals (see [17][18][19]) and in the problem of characterizing ideals (see [2,16]).…”

Section: Introductionmentioning

confidence: 99%

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

Zhao

2021

Filomat

View full text Add to dashboard Cite

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.

show abstract