Matrix KSGNS construction and a Radon–Nikodym type theorem

Abstract

In this paper, we introduce the concept of completely positive matrix of linear maps on Hilbert A-modules over locally C * -algebras and prove an analogue of Stinespring theorem for it. We show that any two minimal Stinespring representations for such matrices are unitarily equivalent. Finally, we prove an analogue of the Radon-Nikodym theorem for this type of completely positive n × n matrices.2010 Mathematics Subject Classification. Primary 46L08; Secondary 46L05.

Similarities and differences between real and complex Banach spaces: an overview and recent developments

“…One of the fundamental results on completely positive maps, essentially due to Stinespring [241], assures that if A and B are two C * -algebras and one of them is commutative, then every positive operator T : A → B is completely positive (see [201,Theorems 3.9 and 3.11] or [245, Corollary IV.3.5 and Proposition IV.3.9] as well as [185]). This conclusion does not hold in the real setting (see Example 5.13).…”

The triangle equality in Hilbert A-modules

“…Òåîðèÿ ãèëüáåðòîâûõ A-ìîäóëåé òåñíî ñâÿçàíà ñ ìíîãèìè ðàçäåëàìè ñîâðåìåííîé ìàòåìàòèêè ( [10]). Îïåðàòîðû, äåéñòâóþùèå â A-ìîäóëÿõ, èçó÷àëèñü â ðàáîòàõ [11] [14].…”