Product commuting maps with the $\lambda$-Aluthge transform

TL;DR

This paper characterizes bijective maps on operator algebras that preserve the $ ext{lambda}$-Aluthge transform structure, showing they are implemented by unitary conjugation.

Contribution

It provides a complete characterization of bijective maps preserving the $ ext{lambda}$-Aluthge transform, revealing they are precisely unitary conjugations.

Findings

Bijective maps preserving the $ ext{lambda}$-Aluthge transform are unitary conjugations.

The characterization holds for operators on Hilbert spaces with the $ ext{lambda}$-Aluthge transform.

The result generalizes known structure-preserving maps in operator theory.

Abstract

Let H and K be two Hilbert spaces and B(H) be the algebra of all bounded linear operators from H into itself. The main purpose of this paper is to obtain a characterization of bijective maps $\Phi$ : B(H) $\rightarrow$ B(K) satisfying the following condition $\Delta$ $\lambda$ ($\Phi$(A)$\Phi$(B)) = $\Phi$($\Delta$ $\lambda$ (AB)) f orall A, B $\in$ B(H), where $\Delta$ $\lambda$ (T) stands the $\lambda$-Aluthge transform of the operator T $\in$ B(H). More precisely, we prove that a bijective map $\Phi$ satisfies the above condition, if and only , if $\Phi$(A) = U AU * for all A $\in$ B(H), for some unitary operator U : H $\rightarrow$ K.