On Maps that Preserve the Lie Products Equal to Fixed Elements

TL;DR

This paper characterizes bijective linear maps on matrix and operator algebras that preserve specific Lie product relations, revealing their general forms and structural properties.

Contribution

It provides a comprehensive description of linear maps preserving fixed Lie product relations in finite and infinite-dimensional settings.

Findings

Characterized maps on matrix algebras preserving fixed Lie products.

Extended the characterization to bounded operators on Hilbert spaces.

Identified the structural form of such maps in both finite and infinite dimensions.

Abstract

This work characterizes the general form of a bijective linear map $\Psi:\mathscr{M}_n(\mathbb{C}) \to \mathscr{M}_n(\mathbb{C})$ such that $[\Psi(A_1),~\Psi(A_2)]=D_2$ whenever $[A_1,~A_2]=D_1$ where $D_1~\text{and}~D_2$ are fixed matrices. Additionally, let $\mathscr{H}_1$ and $\mathscr{H}_2$ be the infinite-dimensional complex Hilbert spaces. We characterize the bijective linear map $\Psi: \mathscr{B}(\mathscr{H}_1) \to \mathscr{B}(\mathscr{H}_2)$ where $\Psi(A_1) \circ ~\Psi(A_2)=D_2$ whenever $A_1\circ ~A_2=D_1$ and $D_1~\text{and}~D_2$ are fixed operators.