Skip to main content
SpringerLink
Log in

Orthogonality preserving maps and pro-\(C^{*}\)-modules

  • Original Research Paper
  • Published:
The Journal of Analysis Aims and scope Submit manuscript

Abstract

The aim of this paper is to study orthogonality preserving mappings between inner product pro-\(C^{*}\)-modules V and W over a pro-\(C^{*}\)-algebra A. We prove the analogue of the result of Ilišević and Turnšek in the set up of inner product pro-\(C^{*}\)-module that an orthogonality preserving mappings between inner product pro-\(C^{*}\)-modules turn out to be scalar multiple of isometry if \(K(H)\subset A\subset L(H)\), where H is a locally Hilbert space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chmieliński, J. 2005. Linear mappings approximately preserving orthogonality. Journal of Mathematical Analysis and Applications 304: 158–169.

    Article  MathSciNet  MATH  Google Scholar 

  2. Fragoulopoulou, M. 2005. Topological algebras with involution. Vol. 200, North-Holland mathematics studies. Amsterdam: Elsevier Science B. V. [MR2172581 (2006m:46067)].

  3. Fragoulopoulou, M. 1981. Spaces of representations and enveloping l.m.c \(^*\)-algebras. Pacific Journal of Mathematics 95 (1): 61–73. (MR631659).

    Article  MathSciNet  MATH  Google Scholar 

  4. Giles, J. 1967. Classes of semi-inner-product spaces. Transactions of the American Mathematical Society 129 (3): 436–446. (MR0217574).

    Article  MathSciNet  MATH  Google Scholar 

  5. Inoue, A. 1971. Locally \(C^*\)-algebras. Vol. 25, Memoirs of the Faculty of Science, Kyushu University (series A), 197–235 [MR0305089 (46 #4219)].

  6. Ilišević, D., and A. Turnšek. 2008. Approximately orthogonality preserving mappings on \(C^*\)-modules. Journal of Mathematical Analysis and Applications 341: 298–308.

    Article  MathSciNet  MATH  Google Scholar 

  7. Joita, M. 2000. On Hilbert modules over locally \(C^*\)-algebras. Analele Universităţii Bucureşti. Matematică 49 (1): 41–51 [MR1898359 (2003c:46076)].

    MathSciNet  MATH  Google Scholar 

  8. Joita, M. 2001. Hilbert modules over locally \(C^*\)-algebras: theorem of Stinespring. Mathematical Reports (Bucureşti) 53 (3): 21–27 [MR1887180 (2002m:46090)].

    MathSciNet  MATH  Google Scholar 

  9. Joita, M. 2002. On the bounded part of a Hilbert module over a locally \(C^*\)-algebra. Periodica Mathematica Hungarica 45 (1–2): 81–85.

    Article  MathSciNet  MATH  Google Scholar 

  10. Joita, M. 2004. Tensor products of Hilbert modules over locally \(C^*\)-algebras. Czechoslovak Mathematical Journal 54 (129): 727–737.

    Article  MathSciNet  MATH  Google Scholar 

  11. Joita, M. 2004. The stabilisation theorem for Hilbert modules over locally \(C^*\)-algebras. Acta Universitatis Ouluensis. Series A. Scientiae Rerum Naturalium 408: 118–127 [MR2070461 (2005c:46083)].

    MATH  Google Scholar 

  12. Karia, D.J. 1993. Pro(jective limits of) \(C^*\)-algebras. PhD thesis, Sardar Patel University, Vallabh Vidyanagar

  13. Karia, D.J., and Y.M. Parmar. 2015. Operators on lcally Hilbert space. Journal of Analysis 23: 59–73.

    MathSciNet  MATH  Google Scholar 

  14. Koehler, D., and P. Rosenthal. 1970. On isometries of normed linear spaces. Studia Mathematica 36: 213–216.

    Article  MathSciNet  MATH  Google Scholar 

  15. Lance, E.C. 1995. Hilbert \(C^*\)-modules—a toolkit for operator algberaists. Vol. 210, London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press [MR1325694 (96k:46100)].

  16. Murphy, G.J. 1990. \(C^*\)-algebras and operator theory. San Diego: Academic Press [MR1074574 (91m:46084)].

  17. Manuilov, V.M., E.V. Troitsky. 2005. Hilbert \(C^*\)-modules. Vol. 226, Translation of mathematical monographs. Providence, RI: American Mathematical Society.

  18. Paschke, W.L. 1973. Inner product modules over \(B^*\)-algebras. Transactions of the American Mathematical Society 182: 443–468.

    MathSciNet  MATH  Google Scholar 

  19. Phillips, N.C. 1988. Inverse limits of \(C^*\)-algebras. Journal of Operator Theory 19 (1): 159–195. [MR950831 (90c:46090)].

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the UGC support under UGC-SAP-DRS programme F-510/5/DRS/2004 (SAP-II) as well as F-510/3/DRS/2009 (SAP-III) to the Department of Mathematics, Sardar Patel University.

Author information

Authors and Affiliations

  1. Department of Mathematics, Sardar Patel University, Vallabh Vidhyanagar, Gujarat, 388120, India

    Dinesh J. Karia

  2. Department of Mathematics, Government Science College, Gandhinagar, Gujarat, 382016, India

    Yogita M. Parmar

Authors
  1. Dinesh J. Karia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yogita M. Parmar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yogita M. Parmar.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Karia, D.J., Parmar, Y.M. Orthogonality preserving maps and pro-\(C^{*}\)-modules. J Anal 26, 61–70 (2018). https://doi.org/10.1007/s41478-017-0069-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41478-017-0069-y

Keywords

Mathematics Subject Classification

Search

Navigation