March 3, 2026Open Access

Nonlinear maps preserving bi-skew Lie triple product on factor von Neumann algebras

Key Points

Bijective maps preserving the bi-skew lie triple product indicate profound algebraic structures between two factor von Neumann algebras.
If a map satisfies the required algebraic relationship, it can be a linear or conjugate linear isomorphism under certain conditions.
The proofs use algebraic identities related to the bidirectional relationship between factors in von Neumann algebras.
Implications extend to understanding the nature and transformation of algebraic structures in quantum mechanics.

Abstract

Abstract Let A A and B B be two factor von Neumann algebras with dimensions greater than 1. It is proved that if a bijective map Φ: A → B: A B satisfies Φ ([ A, B ⋄, C ] ⋄) = [ Φ (A), Φ (B) ⋄, Φ (C) ] ⋄ ([A, B, C]) =[{ (A), { (B) }, (C) ]} for all A, B, C ∈ A A, B, C A, then Φ is a linear ∗-isomorphism, or a conjugate linear ∗-isomorphism, or the negative of a linear ∗-isomorphism, or the negative of a conjugate linear ∗-isomorphism, where A, B ⋄ = AB * − BA *.

Nonlinear maps preserving bi-skew Lie triple product on factor von Neumann algebras

Key Points

Abstract

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