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.
DOI: http://dx.doi.org/10.7900/jot.2012jul12.1966
Keywords: orthogonality preservers, Hilbert $C^*$-modules, Uhlhorn
theorem, auto continuity
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