Preservers of $\lambda$-Aluthge transforms

Abstract

Let $M$ and $N$ be arbitrary von Neumann algebras. For any $a$ in $M$ or in $N$, let ${\Delta }_{\lambda }(a)$ denote the $\lambda$-Aluthge transform of $a$. Suppose that $M$ has no abelian direct summand. We prove that every bijective map $\Phi :M\to N$ satisfying $$\Phi ({\Delta }_{\lambda }(a\circ b^{\ast }))={\Delta }_{\lambda }(\Phi (a)\circ \Phi (b{)}^{\ast }),for\; alla,b\in M,$$ (for a fixed $\lambda \in [0,1]$), maps the hermitian part of $M$ onto the hermitian part of $N$ (i.e. $\Phi (M_{sa})=N_{sa}$) and its restriction $\Phi {|}_{M_{sa}}:M_{sa}\to N_{sa}$ is a Jordan isomorphism. If we also assume that $\Phi (x+iy)=\Phi (x)+\Phi (iy)$ for all $x,y\in M_{sa}$, then there exists a central projection $p_c$ in $M$ such that $\Phi {|}_{p_{c}M}$ is a complex linear Jordan ${}^{\ast }$-isomorphism and $\Phi {|}_{(\text{1}-p_c)M}$ is a conjugate linear Jordan ${}^{\ast }$-isomorphism. Given two complex Hilbert spaces $H$ and $K$ with dim$(H)\ge 2$, we also establish that every bijection $\Phi :\mathcal{B}(H)\to \mathcal{B}(K)$ satisfying $$\Phi ({\Delta }_{\lambda }(a{b}^{\ast }))={\Delta }_{\lambda }(\Phi (a)\Phi (b{)}^{\ast }),for\; alla,b\in \mathcal{B}(H),$$ must be a complex linear or a conjugate linear ${}^{\ast }$-isomorphism.