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- W1660147478 abstract "As a partial generalisation of the Uhlhorn theorem to Hilbert $C^*$-modules, we show in this article that the module structure and the orthogonality structure of a Hilbert $C^*$-module determine its Hilbert $C^*$-module structure. In fact, we have a more general result as follows. Let $A$ be a $C^*$-algebra, $E$ and $F$ be Hilbert $A$-modules, and $I_E$ be the ideal of $A$ generated by ${langle x,yrangle_A: x,yin E}$. If $Phi : Eto F$ is an $A$-module map, not assumed to be bounded but satisfying $$ langle Phi(x),Phi(y)rangle_A = 0quadtext{whenever}quadlangle x,yrangle_A = 0, $$ then there exists a unique central positive multiplier $uin M(I_E)$ such that $$ langle Phi(x), Phi(y)rangle_A = u langle x, yrangle_Aqquad (x,yin E). $$ As a consequence, $Phi$ is automatically bounded, the induced map $Phi_0: Eto overline{Phi(E)}$ is adjointable, and $overline{Eu^{1/2}}$ is isomorphic to $overline{Phi(E)}$ as Hilbert $A$-modules. If, in addition, $Phi$ is bijective, then $E$ is isomorphic to $F$." @default.
- W1660147478 created "2016-06-24" @default.
- W1660147478 creator A5058192882 @default.
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- W1660147478 date "2010-07-26" @default.
- W1660147478 modified "2023-09-22" @default.
- W1660147478 title "Linear orthogonality preservers of Hilbert $C^*$-modules over general $C^*$-algebras" @default.
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