[Paper Review] A Radon-Nikodym theorem for completely n-positive linear maps on pro-C*-algebras and its applications

Maria Joiţa|ArXiv.org|Jan 9, 2007

Advanced Operator Algebra Research10 references3 citations

TL;DR

This paper establishes a Radon-Nikodym-type theorem for completely $n$-positive linear maps on pro-$C^*$-algebras, characterizing the order structure, purity, and extremality of such maps via their associated Stinespring-type representations. The key contribution is a characterization of extreme points in the set of unital completely $n$-positive maps using the injectivity of a projection map on the commutant of the representation, generalizing Arveson's results to the pro-$C^*$-algebra setting.

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.

Motivation & Objective

• To extend Arveson's Radon-Nikodym theorem for completely positive maps to the setting of completely $n$-positive maps on pro-$C^*$-algebras.
• To characterize the order relation on the set of completely $n$-positive linear maps using their associated representations.
• To identify pure and extreme elements in the set of unital completely $n$-positive maps via representation-theoretic conditions.
• To generalize Kaplansky's multi-positive functional theory to the context of pro-$C^*$-algebras and $n$-positivity.
• To provide a characterization of extreme points in the space of unital completely $n$-positive maps using the injectivity of a projection map on the commutant of the representation.

Proposed method

• Establish a correspondence between the matricial order structure on $\mathcal{B}(A, L(H))$ and the comparability of nondegenerate representations of the pro-$C^*$-algebra $A$.
• Define $n$-positive and completely $n$-positive linear maps as $n \times n$ matrices of continuous linear maps from $A$ to $L(H)$, inducing a completely positive map on $M_n(A)$ to $M_n(L(H))$.
• Utilize the generalized Stinespring construction to associate each completely $n$-positive map with a representation of $A$ on a Hilbert space $H_\rho$, extending the GNS and KSGNS constructions.
• Define the map $\theta_T$ associated with a positive operator $T$ in the commutant $\Phi_\rho(A)'$, and use it to represent elements in the set of completely $n$-positive maps.
• Prove that the map $T \mapsto P_{H_0} T P_{H_0}$ from $\Phi_\rho(A)'$ to $L(H_\rho)$ is injective if and only if the map $\rho$ is an extreme point in the set of unital completely $n$-positive maps.
• Use the representation-theoretic framework to derive criteria for purity and extremality, generalizing results from Arveson and Kaplan.

Experimental results

Research questions

• RQ1 How can the order relation on the set of completely $n$-positive linear maps from a pro-$C^*$-algebra $A$ to $L(H)$ be characterized in terms of their associated representations?
• RQ2 What conditions on the representation associated with a completely $n$-positive map ensure that the map is pure?
• RQ3 Under what conditions is a unital completely $n$-positive map an extreme point in the convex set of such maps?
• RQ4 How does the injectivity of the projection map $T \mapsto P_{H_0} T P_{H_0}$ on the commutant relate to the extremality of the map $\rho$?
• RQ5 Can the characterization of extreme points in the unital completely positive case be extended to the $n$-positive setting on pro-$C^*$-algebras?

Key findings

• The order relation on the set of completely $n$-positive linear maps is characterized via the comparability of their associated nondegenerate representations on a Hilbert space.
• A completely $n$-positive map $\rho$ is pure if and only if its associated representation satisfies a specific irreducibility condition derived from the Stinespring construction.
• A unital completely $n$-positive map $\rho$ is an extreme point in the set of such maps if and only if the map $T \mapsto P_{H_0} T P_{H_0}$ from $\Phi_\rho(A)'$ to $L(H_\rho)$ is injective.
• If the diagonal components $\rho_{ii}$ of $\rho$ are pure and satisfy a unitary equivalence condition with $\rho_{ij}(u_{ij}) = I_H$ for $i \neq j$, then the map $\varphi(a) = [\rho_{ij}(a)]$ is an extreme point in the set of unital completely positive maps from $A$ to $M_n(L(H))$.
• The characterization of extreme points in $CP^n_\infty(A, L(H), I)$ generalizes Arveson's result for $C^*$-algebras to the pro-$C^*$-algebra setting.
• The proof relies on constructing perturbations $\rho_T$ via positive operators in the commutant and showing that extremality forces the perturbation operator to be a scalar multiple of identity.