Jordan product commuting maps with $\lambda$-Aluthge transform

TL;DR

This paper characterizes bijective maps on operator algebras that preserve a specific relation involving the $ ext{lambda}$-Aluthge transform and Jordan product, showing they are implemented by unitary conjugation.

Contribution

It provides a complete characterization of maps preserving a $ ext{lambda}$-Aluthge transform-related relation, revealing they are essentially unitary conjugations.

Findings

Such maps are exactly unitary conjugations.

The preservation condition characterizes the structure of these maps.

The result links $ ext{lambda}$-Aluthge transform properties with algebraic structure.

Abstract

Let H and K be two complex Hilbert spaces and B(H) be the algebra of bounded linear operators from H into itself. The main purpose in this paper is to obtain a characterization of bijective maps $\Phi$ : B(H) $\rightarrow$ B(K) satisfying the following condition $\Delta$ $\lambda$ ($\Phi$(A) $\bullet$ $\Phi$(B)) = $\Phi$($\Delta$ $\lambda$ (A $\bullet$ B)) for all A, B $\in$ B(H), where $\Delta$ $\lambda$ (T) stands the $\lambda$-Aluthge transform of the operator T $\in$ B(H) and A $\bullet$ B = 1 2 (AB + BA) is the Jordan product of A and B. We prove that a bijective map $\Phi$ satisfies the above condition, if and only if there exists an unitary operator U : H $\rightarrow$ K, such that $\Phi$ has the form $\Phi$(A) = UAU * for all A $\in$ B(H).