  There are three natural ways to define UHF (uniformly hyperfinite)
C*-algebras, and all three definitions are equivalent for separable algebras.
In 1967 Dixmier asked whether the three definitions remain equivalent for not
necessarily separable algebras. We give a complete answer to this question.
More precisely, we show that in small cardinality two definitions remain
equivalent, and give counterexamples in other cases. Our results do not use any
additional set-theoretic axioms beyond the usual axioms, namely ZFC.
