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On a dilation of a some class of completely positive maps

Annotation

In this article we investigate sesquilinear forms defined on the Cartesian product of Hilbert C^*-module M over C^*-algebra B and taking values in B. The set of all such defined sesquilinear forms is denoted by S_B (M). We consider completely positive maps from locally C^*-algebra A to S_B (M). Moreover we assume that these completely positive maps are covariant with respect to actions of a group symmetry. This allow us to view these maps as generalizations covariant quantum instruments which are very important for the modern quantum mechanic and the quantum field theory. We analyze the dilation problem for these class of maps. In order to solve this problem we construct the minimal Stinespring representation and prove that every two minimal representations are unitarily equivalent.

Keywords

locally C^*-algebra; Hilbert C^*-module; completely positive map; sesquilinear form; covariant Stinespring representation

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DOI

10.20310/2686-9667-2019-24-127-333-339

UDC

517.98; 519.46

Pages

333-339

References

[1] A. V. Kalinichenko, I. N. Maliev, M. A. Pliev, “Modular sesquilinear forms and generalized stinespring representation”, Russian Mathematics, 62:12 (2018), 42–49. [2] I. N. Maliev, M. A. Pliev, “A stinespring type representation for operators in Hilbert modules over local C∗ -algebras”, Russian Mathematics, 56:12 (2012), 43–49. [3] M. A. Pliev, I. D. Tsopanov, “On representation of Stinespring’s type for n -tuples of completely positive maps in Hilbert C∗ -modules”, Russian Mathematics, 58:11 (2014), 36–42. [4] J.P. Pellonpaa, K. Ylinen, “Modules, completely positive maps, and a generalized KSGNS construction”, Positivity, 15:3 (2011), 509–525. [5] M. S. Moslehian, A. G. Kusraev, M. A. Pliev, “Matrix KSGNS construction and a Radon-Nikodym type theorem”, Indagationes Mathematicae, 28:5 (2017), 938–952. [6] F. Stinspring, “Positive functions on C∗ -algebras”, Proc. Amer. Math. Soc., 6:2 (1955), 211–216. [7] D. A. Dubin, J. Kiukas, J.P. Pellonpaa, K. Ylinen, “Operator integrals and sesquilinear forms”, Journal of Mathematical Analysis and Applications, 413 (2014), 250–268. [8] V. Manuilov, E. Troitsky, Hilbert C∗ -modules, American Mathematical Society, Providence, 2005. [9] G. J. Murphy, C∗ -Algebras and Operator Theory, Academic Press, Inc., San Diego; Academic Press Limited, London, 1990. [10] M. Fragoulopoulou, Topological Algebras with Involution. V. 200, 1st ed., Elsevier, North Holland, 2005.

Received

2019-05-20

Section of issue

Scientific articles

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