Orthogonality preserving pairs of operators on Hilbert C₀(Z)-modules

Abstract

Suppose that Z is a locally compact Hausdorff space and $\mathrm{\Psi },\mathrm{\Phi }:E⟶F$ are $C_0\left(Z\right)$-module maps between Hilbert $C_0\left(Z\right)$-modules such that for every $x,y\in E$, $x\perp y$ implies $\mathrm{\Psi }\left(x\right)\perp \mathrm{\Phi }\left(y\right)$. Then there exists a bounded complex-valued function φ on Z that is continuous on $Z_E=\{z\in Z:⟨x,x⟩(z)\ne 0\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}x\in E\}$ and satisfies $⟨\mathrm{\Psi }\left(x\right),\mathrm{\Phi }\left(y\right)⟩=\phi \cdot ⟨x,y⟩onZ,\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}x,y\in E$.

Keywords:

• Orthogonality preserving map
• Hilbert $C^{\ast }$-module
• continuous field of Hilbert spaces

2010 Mathematics Subject Classifications:

• 46L08
• 46L05
• 47B49

Acknowledgments

The research of the first author was in part supported by a grant from IPM (No. 99460118).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of the first author was in part supported by a grant from IPM [grant number 99460118].