A Radon-Nikodým theorem for completely multi-positive linear maps and its applications

Abstract

The order relation on the set of completely n-positive linear maps from a pro-C * -algebra A to L(H), the C * -algebra of bounded linear operators on a Hilbert space H, is characterized in terms of the representation associated with each completely n-positive linear map. Also, the pure elements in the set of all completely n-positive linear maps from A to L(H) and the extreme points in the set of unital completely n-positive linear maps from A to L(H) are characterized in terms of the representation induced by each completely n-positive linear map. 2000 Mathematical Subject Classification: 46L05

Matrix KSGNS construction and a Radon–Nikodym type theorem

“…Remark 3.6. In [10], Joiţa proved a Radon-Nikodym type theorem for completely positive n × n matrices of maps from locally C ⋆ -algebra A to the C ⋆ -algebra L (H) of all bounded linear bounded operators on a Hilbert space H. If we regard a completely positive n × n matrix as a completely positive map [ϕ] : M n (A ) → M n (L (H)), then Theorem3.4 can be considered as a Radon-Nikodym theorem for a module [ϕ]-map in the sense of recent works of ϕ-maps (see for instance [2,3,12,17]).…”

On representations associated with completely n-positive linear maps on pro-C *-algebras

“…Let i ∈ {1, · · · , n}. Since 1 n ρ ii ≤ ρ, by Radon Nikodym type theorem for completely positive linear maps (see [9,Theorem 3.5…”

“…Then ρ is pure and for each i, j ∈ {1, 2, · · · , n} there is λ ij ∈ C such that T ρ ij = λ ij id Hρ (see [9,Corollary 3.6]). Moreover, [λ ij ] n i,j=1 is a positive matrix in M n (C) with n k=1 λ kk = n, and since…”

“…Conversely, if ρ = [λ ij θ] n i,j=1 and n k=1 λ kk = n, then ρ = θ and since θ is pure, the representation of A associated with ρ is irreducible (see [9,Corollary 3.6]). Therefore the representation (Φ ρ , H ρ ) of A associated with ρ is irreducible.…”