A1 Refereed original research article in a scientific journal

Kolmogorov Decompositions and Extremality Conditions for Positive Covariant Kernels




AuthorsErkka Haapasalo, Juha-Pekka Pellonpää

PublisherPERGAMON-ELSEVIER SCIENCE LTD

Publication year2019

JournalReports on Mathematical Physics

Journal name in sourceREPORTS ON MATHEMATICAL PHYSICS

Journal acronymREP MATH PHYS

Volume83

Issue2

First page 253

Last page271

Number of pages19

ISSN0034-4877

DOIhttps://doi.org/10.1016/S0034-4877(19)30042-4(external)


Abstract
We study positive kernels on X x X, where X is a set equipped with an action of a group, and taking values in the set of A-sesquilinear forms on a (not necessarily Hilbert) module over a C*-algebra A. These maps are assumed to be covariant with respect to the group action on X and a representation of the group in the set of invertible (A-linear) module maps. Such maps are generalizations of covariant instruments appearing in quantum theory of measurement. Minimal Kolmogorov decompositions for positive covariant kernels are given in this paper. We find necessary and sufficient conditions for extreme elements in certain convex subsets of positive covariant kernels and also study extreme rays of these sets.



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