For AF C*-algebras, there are a number of differences between separable and nonseparable algebras, and something weird happens at $\aleph_2$, too.

Recall that an AF (C*)-algebra is the norm-closure of a directed limit of finite-dimensional C*-algebras (the norm is uniquely determined), equivalently, for every $\epsilon > 0$, any finite set of elements can be approximated to within $\epsilon$ by elements of a finite-dimensional algebra.

George Elliott proved that for separable AF algebras (the usual kind), ordered pointed K$_0$ is a complete invariant (for isomorphism), and Effros Handelman and Shen determined exactly which partially ordered abelian groups can so arise; these are known as dimension groups (partially ordered abelian gps satisfying Riesz interpolation and unperforation)

When the locally semisimple subalgebra has first uncountable dimension (which corresponds to the smallest nonseparability condition for the C*-algebra), it was already known (from the 40s, in a Russian paper) that ordered pointed K$_0$ is not complete. However, it was shown (by Goodearl and Handelman) that every first uncountable dimension group could arise as the ordered K$_0$ of an AF algebra with first uncountable dimensional locally semisimple algebra.

However, when the dimension group is of cardinality $\aleph_2$, it was shown (by a well-known mathematician whose name escapes me at the moment), that there exist dimension groups of this cardinality which cannot be realized by a suitable direct limit, hence cannot be realized by corresponding AF algebras.