Decomposable Weak Expectations

In this article we define a special class of weak expectations for a representation of a separable unital C*-algebra, called decomposable weak expectation. We give necessary and sufficient conditions for such kind of weak expectations to exist for a given representation. Then we define decomposable measures on the state space of a C*-algebra and show that the GNS representation of a state admits a decomposable weak expectation if and only if there is a decomposable measure on the state space. Further we give an example of a decomposable weak expectation.