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Статья
2018

Modular Sesquilinear Forms and Generalized Stinespring Representation


A. V. KalinichenkoA. V. Kalinichenko, I. N. MalievI. N. Maliev, M. A. PlievM. A. Pliev
Русская математика
https://doi.org/10.3103/S1066369X18120034
Abstract / Full Text

We consider completely positive maps defined on locally C*-algebra and taking values in the space of sesquilinear forms on Hilbert C*-module M. We construct the Stinespring type representation for this type of maps and show that any two minimal Stinespring representations are unitarily equivalent.

Author information
  • North-Caucasian Institute of Mining and Metallurgy named after K. L. Khetagurov (State Technological University), ul. Nikolaeva 44, Vladikavkaz, 362021, RussiaA. V. Kalinichenko
  • North-Ossetian State University, ul. Vatutina 44–46, Vladikavkaz, 362025, RussiaI. N. Maliev
  • Southern Mathematical Institute, Vladikavkaz Scientific Center of the Russian Academy of Sciences, ul. Markusa 22, Vladikavkaz, 362027, RussiaM. A. Pliev
References
  1. Asadi, M. D. Stinespring’s Theorem for Hilbert C*-Modules, J. Operator Theory 62, No. 2, 235–238 (2009).
  2. Bhat, R., Ramesh, G., Sumesh, K. Stinespring’s Theorem for Maps on Hilbert C*-Modules, J.Operator Theory 68, No. 1, 173–178 (2012).
  3. Joita, M. Covariant Version of the Stinespring Type Theorem for Hilbert C*-Modules, OpenMath. 9 (4), 803–813 (2011).
  4. Masaev, H.M., Pliev, M. A., Elsaev, Y.V. The Radon–Nikodym Type Theorem for a CovariantCompletely Positive Paps on Hilbert C*-Modules, Int. J. of Math. Anal. 9, No. 35, 1723–1731 (2015).
  5. Moslehian, M. S., Kusraev, A. and Pliev, M. Matrix KSGNS Construction and a Radon–Nikodym Type Theorem, Indag. Math. 28, No. 5, 938–952 (2017).
  6. Skeide, M., Sumesh, K. CP-H-Extendable Maps Between Hilbert Modules and CPH-Semigroups, J. Math. Anal. Appl. 414, 886–913 (2014).
  7. Maliev, I. N., Pliev, M. A. A Stinespring Type Representation for Operators in Hilbert Modules Over Local C-Algebras, RussianMathematics 56, No. 12, 43–49 (2012).
  8. Pliev, M. A., Tzopanov, I. D. On Representation of Stinespring’s Type for n-Tuples of Completely PositiveMaps in Hilbert C-Modules, RussianMathematics 58, No. 11, 36–42 (2014).
  9. Stinespring, F. Positive Functions on C*-Algebras, Proc. Amer.Math. Soc. 2, 211–216 (1955).
  10. Hytonen, T., Pellonpaa, J. P., Ylinen, K. Positive Sesquilinear Form Measures and Generalized Eigenvalue Expansions, J.Math. Anal. Appl. 336, 1287–1304 (2007).
  11. Pellonpaa, J. P., Ylinen, K. Modules, Completely Positive Maps, and a Generalized KSGNS Construction, Positivity 15, No. 3, 509–525 (2011).
  12. Dubin, D. A., Kiukas, J., Pellonpaa, J. P., Ylinen, K. Operator Integrals and Sesquilinear Forms, J. Math. Anal. Appl. 413, 250–268 (2014).
  13. Manuilov, V. M., Troitzkii, E. V. Hilbert C*-Modules (Factorial,Moscow, 2001) [in Russian].
  14. Murphy, G. J. C*-Algebras and Operators Theory (Factorial,Moscow, 1997) [Russian translation].
  15. Lance, E. C. Hilbert C*-Modules. A Toolkit for Operator Algebraists (Cambridge Univ. Press, 1995).
  16. Fragoulopoulou, M. Topological Algebras With Involution (Elsevier, 2005).
  17. Joitą, M. Hilbert Modules Over Locally C*-Algebras (Univ. of Bucharest Press, 2006).
  18. Murphy, G. J. Positive Definite Kernels and Hilbert C*-Modules, Proc. Edinburgh Math. Soc. 40, 367–374 (1997).