Maps preserving quasi-isometries on Hilbert $C^*$-modules

Abstract

Let $\mathcal {K}(\mathcal {H})$ be the $C^*$-algebra of compact op\-erators on a Hilbert space $\mathcal {H}$. Let $E$ be a Hilbert $\mathcal {K}(\mathcal {H})$-mod\-ule and $\mathcal {L}(E)$ the $C^*$-algebra of all adjointable maps on $E$. In this paper, we prove that, if $\varphi :\mathcal {L}(E)\to \mathcal {L}(E)$ is a unital surjective bounded linear map, which preserves quasi-isometries in both directions, then there are unitary oper\-ators $U, V\in \mathcal {L}(E)$ such that \[ \varphi (T)=UTV\quad \mbox {or}\quad \varphi (T)=UT^{tr }V \] for all $T\in \mathcal {L}(E)$, where $T^{tr }$ is the transpose of $T$ with re\-spect to an arbitrary but fixed orthonormal basis of $E$.