ε-orthogonality preserving pairs of mappings on Hilbert C*-modules

Document Type : Research Paper

Authors

1 Department of Mathematic, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran.

2 Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

Abstract

Let $ \mathcal{A} $ be a standard $ C^{*} $-algebra. In this paper, we will study the continuity of $ \varepsilon $-orthogonality preserving mappings between Hilbert $ \mathcal{A} $-modules. Moreover, we will show that a local mapping between Hilbert $ \mathcal A $-modules is  $ \mathcal A $-linear. Furthermore, we will prove that for a pair of nonzero $ \mathcal A $-linear mappings $ T,S:E\longrightarrow F $, between Hilbert  $ \mathcal A $-modules, satisfying  $ \varepsilon $-orthogonality preserving property, there exists $ \gamma \in \mathbb{C} $,
\begin{align*}
    \Vert \langle T(x), S(y) \rangle-\gamma \langle x, y \rangle\Vert \leq \varepsilon \Vert T\Vert \Vert S\Vert \Vert x\Vert \Vert y \Vert,  \quad x, y \in E.
\end{align*}
Our results generalize the known ones in the context of Hilbert spaces.

Keywords

Journal of Algebra and Related Topics
Volume 10, Issue 2
December 2022
Pages 51-60

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