Generalized Orthogonal Measures on the Space of Unital Completely Positive Maps

TL;DR

This paper extends Effros' classical results by introducing generalized orthogonal measures for unital completely positive maps, linking barycentric decomposition with Stinespring dilation dis-integration in a non-commutative setting.

Contribution

It generalizes Effros' barycentric decomposition framework to unital completely positive maps, introducing generalized orthogonal measures in a non-commutative context.

Findings

Established a connection between barycentric decomposition and Stinespring dilation dis-integration.

Introduced the concept of generalized orthogonal measures.

Provided examples illustrating these new measures.

Abstract

A classical result by Effros connects the barycentric decomposition of a state on a C*-algebra to the disintegration of the GNS representation of the state with respect to an orthogonal measure on the state space of the C*-algebra. In this note, we take this approach to the space of unital completely positive maps on a C*-algebra with values in B(H), connecting the barycentric decomposition of the unital completely positive map and the dis-integration of the minimal Stinespring dilation of the same. This generalizes Effros' work in the non-commutative setting. We do this by introducing a special class of barycentric measures which we call generalized orthogonal measures. We end this note by mentioning some examples of generalized orthogonal measures.