On a dilation of a some class of completely positive maps
Vestnik rossijskih universitetov. Matematika, Tome 24 (2019) no. 127, pp. 333-339
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In
this article we investigate sesquilinear forms defined on the
Cartesian product of Hilbert $C^*$-module $\mathcal{M}$ over
$C^*$-algebra $B$ and taking values in $B.$ The set of all such
defined sesquilinear forms is denoted by
$\mathcal{S}_{B}(\mathcal{M}).$ We consider completely positive maps
from locally $C^*$-algebra $A$ to $\mathcal{S}_{B}(\mathcal{M}).$
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.
@article{VTAMU_2019_24_127_a7,
author = {Ya. Elsaev},
title = {On a dilation of a some class of completely positive maps},
journal = {Vestnik rossijskih universitetov. Matematika},
pages = {333--339},
publisher = {mathdoc},
volume = {24},
number = {127},
year = {2019},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/VTAMU_2019_24_127_a7/}
}
Ya. Elsaev. On a dilation of a some class of completely positive maps. Vestnik rossijskih universitetov. Matematika, Tome 24 (2019) no. 127, pp. 333-339. http://geodesic.mathdoc.fr/item/VTAMU_2019_24_127_a7/