Linear orthogonality preservers of Hilbert $C^{\ast }$-modules over general $C^{\ast }$-algebras

Abstract

As a partial generalisation of the Uhlhorn theorem to Hilbert $C^{\ast }$-modules, we show in this article that the module structure and the orthogonality structure of a Hilbert $C^{\ast }$-module determine its Hilbert $C^{\ast }$-module structure. In fact, we have a more general result as follows. Let $A$ be a $C^{\ast }$-algebra, $E$ and $F$ be Hilbert $A$-modules, and $I_E$ be the ideal of $A$ generated by $\{⟨x,y{⟩}_A:x,y\in E\}$. If $\Phi :E\to F$ is an $A$-module map, not assumed to be bounded but satisfying $$⟨\Phi (x),\Phi (y){⟩}_A=0\text{whenever}⟨x,y{⟩}_A=0,$$ then there exists a unique central positive multiplier $u\in M(I_E)$ such that $$⟨\Phi (x),\Phi (y){⟩}_A=u⟨x,y{⟩}_A(x,y\in E).$$ As a consequence, $\Phi$ is automatically bounded, the induced map ${\Phi }_0:E\to \overline{\Phi (E)}$ is adjointable, and $\overline{E{u}^{1/2}}$ is isomorphic to $\overline{\Phi (E)}$ as Hilbert $A$-modules. If, in addition, $\Phi$ is bijective, then $E$ is isomorphic to $F$.

15 pages

References in corpus (1)

• Linear orthogonality preservers of Hilbert $C^{\ast }$-modules over $C^{\ast }$-algebras with real rank zero