TITLE: Nonseparable counterexamples in analysis
QUESTION [21 upvotes]: When asking for uncountable counterexamples in algebra I noted that in functional analysis there are many examples of things that “go wrong” in the nonseparable setting. But most of the examples I'm thinking of come from $\mathrm C^*$-algebras (equivalent formulations of being type I; anything having to do with direct integral decompositions; primeness versus primitivity, etc.).

Question. Are there also good examples from Banach space theory, or from harmonic analysis? Or from analysis in general? (Or even other
examples from $\mathrm C^*$-algebra that I might not know about?)

REPLY [3 votes]: 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.