Original scientific paper

https://doi.org/10.3336/gm.57.1.05

Approximately orthogonality preserving mappings on Hilbert C_{0}(Z)-modules

Mohammad B Asadi ; School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran
Zahra Hassanpour Yakhdani ; School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran
Fatemeh Olyaninezhad ; Department of Mathematics, University of Guilan, Rasht, Guilan, Iran
Abbas Sahleh ; Department of Mathematics, University of Guilan, Rasht, Guilan, Iran

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Abstract

In this paper, we will use the categorical approach to
Hilbert $C^{\ast }$-modules over a commutative $C^{\ast }$-algebra
to investigate the approximately orthogonality preserving mappings
on Hilbert $C^{\ast }$-modules over a commutative
$C^{\ast }$-algebra.

Indeed, we show that if $\mathrm{\Psi }:\mathrm{\Gamma }\to {\mathrm{\Gamma }}^{\prime }$ is a nonzero $C_0(Z)$-linear
$(\delta ,\epsilon )$-orthogonality preserving mapping
between the continuous fields of Hilbert spaces on a locally
compact Hausdorff space $Z$, then $\mathrm{\Psi }$ is injective, continuous
and also for every $x,y\in \mathrm{\Gamma }$ and $z\in Z$,
$$|⟨\mathrm{\Psi }(x),\mathrm{\Psi }(y)⟩(z)-{\phi }^2(z)⟨x,y⟩(z)|\le \frac{4(\epsilon -\delta )}{(1-\delta )(1+\epsilon )}\Vert \mathrm{\Psi }(x)\Vert \Vert \mathrm{\Psi }(y)\Vert ,$$
where $\phi (z)=sup\{\Vert \mathrm{\Psi }(u)(z)\Vert :u\text{is a unit vector in}\mathrm{\Gamma }\}$.

Keywords

Approximately orthogonality preserving, Hilbert$C^{\ast }$-module, Continuous field of Hilbert spaces

Hrčak ID:

279800

URI

https://hrcak.srce.hr/279800

Publication date:

28.6.2022.

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