Ergodic decomposition in the space of unital completely positive maps

The classical decomposition theory for states on a C*-algebra that are invariant under a group action has been studied by using the theory of orthogonal measures on the state space. 4 In Ref. 3, we introduced the notion of generalized orthogonal measures on the space of unital completely positive (UCP) maps from a C*-algebra [Formula: see text] into [Formula: see text]. In this paper, we consider a group [Formula: see text] that acts on a C*-algebra [Formula: see text], and the collection of [Formula: see text]-invariant UCP maps from [Formula: see text] into [Formula: see text]. This paper examines a [Formula: see text]-invariant decomposition of UCP maps by using the theory of generalized orthogonal measures on the space of UCP maps, developed in Ref. 3. Further, the set of all [Formula: see text]-invariant UCP maps is a compact and convex subset of a topological vector space. Hence, by characterizing the extreme points of this set, we complete the picture of barycentric decomposition in the space of [Formula: see text]-invariant UCP maps. We establish this theory in Stinespring and Paschke dilations of completely positive maps. We end this note by mentioning some examples of UCP maps admitting a decomposition into [Formula: see text]-invariant UCP maps.