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Journal of Operator Theory

Volume 71, Issue 2, Spring 2014  pp. 571-584.

Linear orthogonality preservers of Hilbert $C^*$-modules

Authors:  Chi-Wai Leung (1), Chi-Keung Ng (2), and Ngai-Ching Wong (3)
Author institution: (1) Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
(2) Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
(3) Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan, R.O.C.

Summary:  We show in this paper that the module structure and the orthogonality structure of a Hilbert $C^*$-module determine its inner product structure. Let $A$ be a $C^*$-algebra, and $E$ and $F$ be Hilbert $A$-modules. Assume $\Phi : E\to F$ is an $A$-module map satisfying $ \langle \Phi(x),\Phi(y)\rangle_A = 0\mbox{ whenever }\langle x,y\rangle_A = 0. $ Then $\Phi$ is automatically bounded. In case $\Phi$ is bijective, $E$ is isomorphic to $F$. More precisely, let $J_E$ be the closed two-sided ideal of $A$ generated by the set $\{\langle x,y\rangle_A : x,y\in E\}$. We show that there exists a unique central positive multiplier $u\in M(J_E)_+$ such that $ \langle \Phi(x), \Phi(y)\rangle_A = u \langle x, y\rangle_A\ (x,y\in E). $ As a consequence, 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.

Keywords:  orthogonality preservers, Hilbert $C^*$-modules, Uhlhorn theorem, auto continuity

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