Статья
2018
Modular Sesquilinear Forms and Generalized Stinespring Representation
A. V. Kalinichenko, I. N. Maliev, M. 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
- Asadi, M. D. Stinespring’s Theorem for Hilbert C*-Modules, J. Operator Theory 62, No. 2, 235–238 (2009).
- Bhat, R., Ramesh, G., Sumesh, K. Stinespring’s Theorem for Maps on Hilbert C*-Modules, J.Operator Theory 68, No. 1, 173–178 (2012).
- Joita, M. Covariant Version of the Stinespring Type Theorem for Hilbert C*-Modules, OpenMath. 9 (4), 803–813 (2011).
- 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).
- 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).
- Skeide, M., Sumesh, K. CP-H-Extendable Maps Between Hilbert Modules and CPH-Semigroups, J. Math. Anal. Appl. 414, 886–913 (2014).
- 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).
- 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).
- Stinespring, F. Positive Functions on C*-Algebras, Proc. Amer.Math. Soc. 2, 211–216 (1955).
- Hytonen, T., Pellonpaa, J. P., Ylinen, K. Positive Sesquilinear Form Measures and Generalized Eigenvalue Expansions, J.Math. Anal. Appl. 336, 1287–1304 (2007).
- Pellonpaa, J. P., Ylinen, K. Modules, Completely Positive Maps, and a Generalized KSGNS Construction, Positivity 15, No. 3, 509–525 (2011).
- Dubin, D. A., Kiukas, J., Pellonpaa, J. P., Ylinen, K. Operator Integrals and Sesquilinear Forms, J. Math. Anal. Appl. 413, 250–268 (2014).
- Manuilov, V. M., Troitzkii, E. V. Hilbert C*-Modules (Factorial,Moscow, 2001) [in Russian].
- Murphy, G. J. C*-Algebras and Operators Theory (Factorial,Moscow, 1997) [Russian translation].
- Lance, E. C. Hilbert C*-Modules. A Toolkit for Operator Algebraists (Cambridge Univ. Press, 1995).
- Fragoulopoulou, M. Topological Algebras With Involution (Elsevier, 2005).
- Joitą, M. Hilbert Modules Over Locally C*-Algebras (Univ. of Bucharest Press, 2006).
- Murphy, G. J. Positive Definite Kernels and Hilbert C*-Modules, Proc. Edinburgh Math. Soc. 40, 367–374 (1997).