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Linear orthogonality preservers of Hilbert $ C^*$-modules over $ C^*$-algebras with real rank zero


Authors: Chi-Wai Leung, Chi-Keung Ng and Ngai-Ching Wong
Journal: Proc. Amer. Math. Soc. 140 (2012), 3151-3160
MSC (2010): Primary 46L08, 46H40
DOI: https://doi.org/10.1090/S0002-9939-2012-11260-2
Published electronically: January 6, 2012
MathSciNet review: 2917088
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Abstract: Let $ A$ be a $ C^*$-algebra with real rank zero. Let $ E$ and $ F$ be Hilbert $ A$-modules with $ E$ being full. Suppose that $ \theta : E\to F$ is a linear map preserving orthogonality, i.e.,

$\displaystyle \langle \theta (x),\theta (y)\rangle \ =\ 0$$\displaystyle \quad \text {whenever}\quad \langle x,y\rangle \ =\ 0. $

We show in this article that if $ \theta $ is an $ A$-module map (not assumed to be bounded), then there exists a central positive multiplier $ u\in M(A)$ such that

$\displaystyle \langle \theta (x), \theta (y)\rangle \ =\ u \langle x, y\rangle \qquad (x,y\in E). $

In the case when $ A$ is a standard $ C^*$-algebra, when $ A$ is a real rank zero properly infinite unital $ C^*$-algebra, or when $ A$ is a $ W^*$-algebra, we also get the same conclusion with the assumption of $ \theta $ being an $ A$-module map weakened to being a local map.

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Additional Information

Chi-Wai Leung
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong SAR, People’s Republic of China
Email: cwleung@math.cuhk.edu.hk

Chi-Keung Ng
Affiliation: Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
Email: ckng@nankai.edu.cn

Ngai-Ching Wong
Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan, Republic of China
Email: wong@math.nsysu.edu.tw

DOI: https://doi.org/10.1090/S0002-9939-2012-11260-2
Keywords: Hilbert $C^{*}$-modules, orthogonality preservers, module maps, local maps, real rank zero $C^{*}$-algebras
Received by editor(s): May 25, 2010
Received by editor(s) in revised form: March 22, 2011
Published electronically: January 6, 2012
Additional Notes: The authors are supported by The Chinese University of Hong Kong Direct Grant (2060389), National Natural Science Foundation of China (10771106), and Taiwan NSC grant (NSC96-2115-M-110-004-MY3).
Communicated by: Marius Junge
Article copyright: © Copyright 2012 American Mathematical Society

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