Kolmogorov Decompositions and Extremality Conditions for Positive Covariant Kernels

: Erkka Haapasalo, Juha-Pekka Pellonpää

Publisher: PERGAMON-ELSEVIER SCIENCE LTD

: 2019

: Reports on Mathematical Physics

: REPORTS ON MATHEMATICAL PHYSICS

: REP MATH PHYS

: 83

: 2

: 253

: 271

: 19

: 0034-4877

DOI: https://doi.org/10.1016/S0034-4877(19)30042-4

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.