On a Class of Completely Positive Mappings

A. K. Gutnova

North Ossetian State University after Kosta Levanovich Khetagurov, Vladikavkaz

Abstract: We prove a noncommutative analog of the Radon–Nikodým theorem for $n$-tuples of completely positive mappings on Hilbert $C^*$-modules covariant with respect to the action of a locally compact group. As an application, we describe the structure of the convex set of equivalence classes of completely positive tuples majorized by a given completely positive tuple. We also show that the classical Radon–Nikodým theorem follows from its noncommutative analog.

Keywords: Hilbert $C^*$-module, $C^*$-algebra, $n$-completely positive mapping, Radon–Nikodým theorem, $G$-communant, locally compact group.

UDC: 517

PACS: 02.30.Sa

MSC: 46L08

Received: 02.07.2024
Revised: 15.01.2025

DOI: 10.4213/mzm14585

 English version: Mathematical Notes, 2025, 118:1, 60–74

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