Abelian groups of continuous functions and their duals

Let A^* = Hom (A, Z) for an Abelian group A, were Z is the group of integers. A^* is endowed with the topology as a subspace of Z^A. Then, for a 0-dimensional space X and an infinite cardinal κ the following are equivalent. (1) There exists a free summand of C(X, Z) of rank κ; (2) there exists a subgroup of C(X, Z)^* isomorphic to Z^κ; (3) there exists a compact subset K of β_NX with w(K)≥κ; (4) there exists a compact subset K of C(X, Z)^* with w(K)≥κ. There exist groups A such that A^* is a subgroup of Z^N and A^* is not isomorphic to A^***.