On decompositions of Banach spaces of continuous functions on Mrowka's spaces

It is well known that if K is infinite compact Hausdor. and scattered (i.e., with no perfect subsets), then the Banach space C( K) of continuous functions on K has complemented copies of c(0), i.e., C( K) similar to c(0) + X similar to c(0) + c(0) + X similar to c(0) + C(K). We address the question if this could be the only type of decompositions of C(K) not similar to c(0) into infinite-dimensional summands for K infinite, scattered. Making a special set-theoretic assumption such as the continuum hypothesis or Martin's axiom we construct an example of Mrowka's space ( i. e., obtained from an almost disjoint family of sets of positive integers) which answers positively the above question.