  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.
