Glasnik Matematicki, Vol. 57, No. 1 (2022), 63-71. \( \)

APPROXIMATELY ORTHOGONALITY PRESERVING MAPPINGS ON HILBERT \(C_{0}(Z)\)-MODULES

Mohammad B. Asadi, Zahra Hassanpour Yakhdani, Fatemeh Olyaninezhad and Abbas Sahleh

School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran
and
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran
e-mail:mb.asadi@ut.ac.ir

School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran
e-mail:z.hasanpour@ut.ac.ir

Department of Mathematics, University of Guilan, Rasht, Guilan, Iran
e-mail:olyaninejad_f@yahoo.com

Department of Mathematics, University of Guilan, Rasht, Guilan, Iran
e-mail:sahlehj@guilan.ac.ir

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 \(\Psi:\Gamma \rightarrow \Gamma^{\prime} \) is a nonzero \( C_{0}(Z) \)-linear \(( \delta , \varepsilon)\)-orthogonality preserving mapping between the continuous fields of Hilbert spaces on a locally compact Hausdorff space \(Z\), then \(\Psi\) is injective, continuous and also for every \( x, y \in \Gamma \) and \(z \in Z\), \[ \vert \langle \Psi(x),\Psi(y) \rangle(z) - \varphi^2(z) \langle x,y \rangle(z) \vert \leq \frac{4(\varepsilon - \delta)}{(1-\delta)(1+\varepsilon)} \Vert \Psi(x) \Vert \Vert \Psi(y) \Vert, \] where \(\varphi(z) = \sup \{ \Vert \Psi(u)(z) \Vert : u ~ \text{is a unit vector in} ~ \Gamma \}\).

2020 Mathematics Subject Classification.   46L08, 47B49

Key words and phrases.   Approximately orthogonality preserving, Hilbert \(C^*\)-module, Continuous field of Hilbert spaces

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

References:

1. M. B. Asadi and F. Olyaninezhad, Orthogonality preserving pairs of operators on Hilbert \( C_{0}(Z) \)-modules, published online in Linear and Multilinear algebra.
   CrossRef

2. J. Chmieliński, Linear mappings approximately preserving orthogonality, J. Math. Anal. Appl. 304 (2005), 158–169.
   MathSciNet    CrossRef

3. J. Chmieliński, Stability of the orthogonality preserving property in finite-dimensional inner product spaces, J. Math. Anal. Appl. 318 (2006), 433–443.
   MathSciNet    CrossRef

4. M. Frank, A. S. Mishchenko and A. A. Pavlov, Orthogonality-preserving, \(C^{\ast}\)-conformal and conformal module mappings on Hilbert \(C^{\ast}\)-modules, J. Funct. Anal. 260 (2011), 327–339.
   MathSciNet    CrossRef

5. C. Heunen and M. L. Reyes, Frobenius structures over Hilbert \( C^{*}\)-modules, Comm. Math. Phys. 361 (2018), 787–824.
   MathSciNet    CrossRef

6. D. Ilišević and A. Turnšek, Approximately orthogonality preserving mappings on \( C^{*} \)-modules, J. Math. Anal. Appl. 341 (2008), 298–308.
   MathSciNet    CrossRef

7. C. W. Leung, C. K. Ng and N. C. Wong, Linear orthogonality preservers of Hilbert bundles, J. Aust. Math. Soc. 89 (2010), 245–254.
   MathSciNet    CrossRef

8. C. W. Leung, C. K. Ng and N. C. Wong, Linear orthogonality preservers of Hilbert \(C^*\)- modules, J. Operator Theory 71 (2014), 571–584.
   MathSciNet    CrossRef

9. V. M. Manuilov and E. V. Troitsky, Hilbert \(C^*\)-modules, Amer. Math. Soc., Providence, 2005.
   MathSciNet    CrossRef

10. M. S. Moslehian and A. Zamani, Mapping preserving approximately orthogonality in Hilbert \(C^*\)-modules, Math. Scand. 122 (2018), 257–276.
    MathSciNet    CrossRef

11. R. G. Swan, Vector bundles and projective modules, Trans. Amer. Math. Soc. 105 (1962), 264–277.
    MathSciNet    CrossRef

12. A. Turnšek, On mappings approximately preserving orthogonality, J. Math. Anal. Appl. 336 (2007), 625–631.
    MathSciNet    CrossRef

13. P. Wójcik, On certain basis connected with operator and its applications, J. Math. Anal. Appl. 423 (2015), 1320–1329.
    MathSciNet