Skip to main content
SpringerLink
Log in

Conditionally positive definite kernels in Hilbert \(C^*\)-modules

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

We investigate the notion of conditionally positive definite in the context of Hilbert \(C^*\)-modules and present a characterization of the conditionally positive definiteness in terms of the usual positive definiteness. We give a Kolmogorov type representation of conditionally positive definite kernels in Hilbert \(C^*\)-modules. As a consequence, we show that a \(C^*\)-metric space (S, d) is \(C^*\)-isometric to a subset of a Hilbert \(C^*\)-module if and only if \(K(s,t)=-d(s,t)^2\) is a conditionally positive definite kernel. We also present a characterization of the order \(K'\le K\) between conditionally positive definite kernels.

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. Amyari, M., Chakoshi, M., Moslehian, M.S.: Quasi-representations of Finsler modules over \(C^*\)-algebras. J. Oper. Theory 70(1), 181–190 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arambasić, L., Bakić, D., Moslehian, M.S.: A treatment of the Cauchy–Schwarz inequality in \(C^*\)-modules. J. Math. Anal. Appl. 381, 546–556 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barreto, S.D., Bhat, B.V.R., Liebscher, V., Skeide, M.: Type I product systems of Hilbert modules. J. Funct. Anal. 212(1), 121–181 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions. Graduate Texts in Mathematics, 100. Springer-Verlag, New York (1984)

    Book  MATH  Google Scholar 

  5. Bhatia, R., Sano, T.: Loewner matrices and operator convexity. Math. Ann. 344(3), 703–716 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Donoghue, W.F.: Monotone matrix functions and analytic continuation, Die Grundlehren der mathematischen Wissenschaften, vol. 207. Springer-Verlag, Berlin (1974)

    Book  Google Scholar 

  7. Evans, D.E., Lewis, J.T.: Dilations of Irreversible Evolutions in Algebraic Quantum Theory. Communications of the Dublin Institute of Advanced Studies – Series A, no. 24, v\(+\)104 pp (1977)

  8. Ferreira, J.C., Menegatto, V.A.: Positive definiteness, reproducing kernel Hilbert spaces and beyond. Ann. Funct. Anal. 4(1), 64–88 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Frank, M.: Geometrical aspects of Hilbert \(C^*\)-modules. Positivity 3(3), 215–243 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lance, E.C.: Hilbert \(C^*\)-Modules. London Math. Soc. Lecture Note Series, vol. 210, Cambridge University Press, Cambridge (1995)

  11. Ma, Z., Jiang, L., Sun, H.: \(C^*\)-algebra-valued metric spaces and related fixed point theorems. Fixed Point Theory Appl. 2014, 11 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Manuilov, V.M., Troitsky, E.V.: Hilbert \(C^*\)-modules. Translated from the 2001 Russian original by the authors. Trans. Math. Monog. 226. AMS, Providence, RI (2005)

  13. Murphy, G.J.: Positive definite kernels and Hilbert \(C^*\)-modules. Proc. Edinb. Math. Soc. (2) 40(2), 367–374 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Micchelli, C.A.: Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx. 2(1), 11–22 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  15. Moslehian, M.S., Joita, M., Ji, U.C.: KSGNS type constructions for \(\alpha \)-completely positive maps on Krein \(C^*\)-modules. Complex Anal. Oper. Theory 10(3), 617–638 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pellonpää, J.-P.: Modules and extremal completely positive maps. Positivity 18(1), 61–79 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Schoenberg, I.J.: Metric spaces and positive definite functions. Trans. Am. Math. Soc. 44(3), 522–536 (1938)

    Article  MathSciNet  MATH  Google Scholar 

  18. Takesaki, M.: Theory of Operator Algebras. I. Reprint of the first (1979) edition. Encyclopaedia of Mathematical Sciences. 124. Operator Algebras and Non-commutative Geometry, 5. Springer-Verlag, Berlin (2002)

Download references

Acknowledgements

The author was supported by a grant from Ferdowsi University of Mashhad (No. 2/42346). He would like to thank the referee for several useful comments improving the paper.

Author information

Authors and Affiliations

  1. Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad, 91775, Iran

    Mohammad Sal Moslehian

Authors
  1. Mohammad Sal Moslehian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammad Sal Moslehian.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Moslehian, M.S. Conditionally positive definite kernels in Hilbert \(C^*\)-modules. Positivity 21, 1161–1172 (2017). https://doi.org/10.1007/s11117-016-0458-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-016-0458-5

Keywords

Mathematics Subject Classification

Search

Navigation