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Journal of Operator Theory

Volume 67, Issue 1, Winter 2012  pp. 3-20.

Automatic continuity and $C_0(\Omega)$-linearity of linear maps between $C_0(\Omega)$-modules

Authors Chi-Wai Leung (1), Chi-Keung Ng (2), 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

Summary:  If $\Omega$ is a locally compact Hausdorff space, we show that any local $\mathbb{C}$-linear map between Banach $C_0(\Omega)$-modules is ``nearly $C_0(\Omega)$-linear'' and ``nearly bounded''. Thus, any local $\mathbb{C}$-linear map $\theta$ between Hilbert $C_0(\Omega)$-modules is $C_0(\Omega)$-linear. If, in addition, $\Omega$ contains no isolated point, any $C_0(\Omega)$-linear map between Hilbert $C_0(\Omega)$-modules is bounded. Moreover, if $\theta$ is a bijective ``biseparating'' map from a full essential Banach $C_0(\Omega)$-module $E$ to a full Hilbert $C_0(\Delta)$-module $F$, then $\theta$ is ``nearly bounded'' and there is a homeomorphism $\sigma: \Delta \rightarrow \Omega$ with $\theta(e\cdot \varphi) = \theta(e)\cdot \varphi\circ \sigma$ ($e\in E, \varphi\in C_0(\Omega)$).

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