Converging sequences of measures and complemented copies of c_0

If K is an infinite compact space then the Banach space C(K) contains natural copies of c_0. The question whether a given copy of c_0 is complemented in C(K) can be expressed in terms of converging sequences of measures on K.

We shall discuss some applications of such an approach; in particular, we consider compactifications K of natural numbers related to embeddings of the measure algebra into P(N)/fin.