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LINEAR ORTHOGONALITY PRESERVERS OF HILBERT BUNDLES

Part of: Topological algebras, normed rings and algebras, Banach algebras Linear function spaces and their duals Selfadjoint operator algebras

Published online by Cambridge University Press:  05 November 2010

CHI-WAI LEUNG ,
CHI-KEUNG NG  and
NGAI-CHING WONG
CHI-WAI LEUNG*
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Hong Kong (email: [email protected])
CHI-KEUNG NG
Affiliation:
Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, PR China (email: [email protected])
NGAI-CHING WONG
Affiliation:
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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A ℂ-linear map θ (not necessarily bounded) between two Hilbert C*-modules is said to be ‘orthogonality preserving’ if 〈θ(x),θ(y)〉=0 whenever 〈x,y〉=0. We prove that if θ is an orthogonality preserving map from a full Hilbert C0(Ω)-module E into another Hilbert C0(Ω) -module F that satisfies a weaker notion of C0 (Ω) -linearity (called ‘localness’), then θ is bounded and there exists ϕ∈Cb (Ω)+ such that 〈θ(x),θ(y)〉=ϕ⋅〈x,y〉 for all x,y∈E.

Keywords

automatic continuityorthogonality preserving mapsmodule homomorphismslocal mapsHilbert C*-modulesHilbert bundles

MSC classification

Secondary: 46L08: $C^*$-modules 46H40: Automatic continuity 46E40: Spaces of vector- and operator-valued functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 89 , Issue 2 , October 2010 , pp. 245 - 254
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The authors were supported by a Hong Kong RGC Research Grant (2160255), the National Natural Science Foundation of China (10771106), and a Taiwan NSC grant (NSC96-2115-M-110-004-MY3).

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