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

Let $\mathcal{H}$ be a Hilbert space and $E$ be a Hilbert $\mathcal{C}^*$-module over a $C^*$-algebra $\mathcal{A}$ of compact operators on $\mathcal{H}$ and $\mathcal{L}(E)$ the set of all adjointable maps on $E$. In this talk, we show that if $\varphi:\mathcal{L}(E)\to \mathcal{L}(E)$ preserves partial-isometry in both directions, then there are unitary operators $U, V\in \mathcal{L}(E)$ such that \begin{eqnarray*} \varphi(T)=UTV ~~\textsl{or} ~~\varphi(T)=UT^{tr}V \end{eqnarray*} or \begin{eqnarray*} \varphi(T)=V^*TU^* ~~\textsl{or}~ ~\varphi(T)=V^*T^{tr}U^*, \end{eqnarray*} where $T^{tr}$ is the transpose of $T$ with respect to an arbitrary but fixed orthonormal basis of $E$. \end{abstract}}