Linear orthogonality preservers of Hilbert $C^*$-modules over general $C^*$-algebras

TL;DR

This paper generalizes the Uhlhorn theorem to Hilbert $C^*$-modules, showing that orthogonality-preserving module maps are scalar multiples by central positive multipliers, leading to automatic boundedness and module isomorphisms.

Contribution

It establishes that orthogonality-preserving module maps between Hilbert $C^*$-modules are scalar multiples by central positive multipliers, extending the structure theory.

Findings

Orthogonality-preserving maps are scalar multiples by central positive multipliers.

Such maps are automatically bounded and adjointable.

Bijective maps imply module isomorphisms.

Abstract

As a partial generalisation of the Uhlhorn theorem to Hilbert $C^*$-modules, we show in this article that the module structure and the orthogonality structure of a Hilbert $C^*$-module determine its Hilbert $C^*$-module structure. In fact, we have a more general result as follows. Let $A$ be a $C^*$-algebra, $E$ and $F$ be Hilbert $A$-modules, and $I_E$ be the ideal of $A$ generated by $\{\langle x,y\rangle_A: x,y\in E\}$. If $\Phi : E\to F$ is an $A$-module map, not assumed to be bounded but satisfying $$ \langle \Phi(x),\Phi(y)\rangle_A\ =\ 0\quad\text{whenever}\quad\langle x,y\rangle_A\ =\ 0, $$ then there exists a unique central positive multiplier $u\in M(I_E)$ such that $$ \langle \Phi(x), \Phi(y)\rangle_A\ =\ u \langle x, y\rangle_A\qquad (x,y\in E). $$ As a consequence, $\Phi$ is automatically bounded, the induced map $\Phi_0: E\to \overline{\Phi(E)}$ is adjointable, and $\overline{Eu^{1/2}}$ is isomorphic to $\overline{\Phi(E)}$ as Hilbert $A$-modules. If, in addition, $\Phi$ is bijective, then $E$ is isomorphic to $F$.