  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 [2], we introduced the notion of
generalized orthogonal measures on the space of unital completely positive
(UCP) maps from a C*-algebra A into B(H). In this article, we consider a group
G that acts on a C*-algebra A, and the collection of G-invariant UCP maps from
A into B(H). This article examines a G-invariant decomposition of UCP maps by
using the theory of generalized orthogonal measures on the space of UCP maps,
developed in [2]. Further, the set of all G-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 G-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
G-invariant UCP maps.
