Modular sesquilinear forms and generalized Stinspring representation

A. V. Kalinichenko   ; I. N. Maliev   ; M. A. Pliev

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2018), pp. 50-59

Résumé

We consider completely positive maps defined on locally $C^{\ast}$-algebra and taking values in the space of sesquilinear forms on Hilbert $C^{\ast}$-module $\mathcal{M}$. We construct the Stinspring type representation for this type of maps and show that any two minimal Stinspring representations are unitarily equivalent.

Détail

Keywords: Hilbert $C^\ast$-module, locally $C^{\ast}$-algebra, sesquilinear form, completely positive map, positive definite kernel, Stinspring's representation.
Mots-clés : $\ast$-homomorphism

Bibliographie

[1] Asadi M. D., “Stinspring's theorem for Hilbert $C^{*}$-modules”, J. Operator Theory, 62:2 (2009), 235–238 | MR | Zbl

[2] Bhat R., Ramesh G., Sumesh K., “Stinspring's theorem for maps on Hilbert $C^{*}$-modules”, J. Operator Theory, 68:1 (2012), 173–178 | MR | Zbl

[3] Joita M., “Covariant version of the Stinespring type theorem for Hilbert $C^{*}$-modules”, Open Math., 9:4 (2011), 803–813 | MR | Zbl

[4] Masaev H. M., Pliev M. A., Elsaev Y. V., “The Radon–Nikodym type theorem for a covariant completely positive maps on Hilbert C*-modules”, Int. J. of Math. Anal., 9:35 (2015), 1723–1731 | MR

[5] Moslehian M. S., Kusraev A., Pliev M., “Matrix KSGNS construction and a Radon–Nikodym type theorem”, Indag. Math., 28:5 (2017), 938–952 | DOI | MR | Zbl

[6] Skeide M., Sumesh K., “$CP$-$H$-extendable maps between Hilbert modules and $CPH$-semigroups”, J. Math. Anal. Appl., 414 (2014), 886–913 | DOI | MR | Zbl

[7] Maliev I. N., Pliev M. A., “O predstavlenii tipa Stainspringa dlya operatorov v gilbertovykh modulyakh nad lokalnymi $C^{*}$-algebrami”, Izv. vuzov. Matem., 2012, no. 12, 51–58

[8] Pliev M. A., Tsopanov I. D., “O predstavlenii tipa Stainspringa dlya $n$-naborov vpolne polozhitelnykh otobrazhenii v gilbertovykh $C^*$-modulyakh”, Izv. vuzov. Matem., 2014, no. 11, 42–49

[9] Stinspring F., “Positive functions on $C^{\ast}$-algebras”, Proc. Amer. Math. Soc., 2 (1955), 211–216 | MR

[10] Hytonen T., Pellonpaa J. P., Ylinen K., “Positive sesquilinear form measures and generalized eigenvalue expansions”, J. Math. Anal. Appl., 336 (2007), 1287–1304 | DOI | MR | Zbl

[11] Pellonpaa J. P., Ylinen K., “Modules, completely positive maps, and a generalized KSGNS construction”, Positivity, 15:3 (2011), 509–525 | DOI | MR | Zbl

[12] Dubin D. A., Kiukas J., Pellonpaa J. P., Ylinen K., “Operator integrals and sesquilinear forms”, J. Math. Anal. Appl., 413 (2014), 250–268 | DOI | MR | Zbl

[13] Manuilov V. M., Troitskii E. V., $C^{*}$-gilbertovy moduli, Faktorial, M., 2001

[14] Merfi D., $C^{*}$-algebry i teoriya operatorov, Faktorial, M., 1997

[15] Lance E. C., Hilbert $C^*$-modules. A toolkit for operator algebraists, Cambridge Univ. Press, 1995 | MR

[16] Fragoulopoulou M., Topological algebras with involution, Elsevier, 2005 | MR | Zbl

[17] Joiţa M., Hilbert modules over locally $C^{*}$-algebras, Univ. of Bucharest Press, 2006 | MR

[18] Murphy G. J., “Positive definite kernels and Hilbert $C^*$-modules”, Proc. Edinburgh Math. Soc., 40 (1997), 367–374 | DOI | MR | Zbl