The survey article ``Set theory and C*-algebras'' by Nik Weaver might have some things along the lines you are looking for ( Bull. Symbolic Logic Volume 13, Issue 1 (2007), 1-20; see also math/0604198 on the arXiv). For a particular example of Weaver's recent work in this area, see Akemann & Weaver's paper
Consistency of a counterexample to Naimark's problem
We construct a C*-algebra that has only one irreducible representation up to unitary equivalence but is not isomorphic to the algebra of compact operators on any Hilbert space. This answers an old question of Naimark. Our construction uses a combinatorial statement called the diamond principle, which is known to be consistent with but not provable from the standard axioms of set theory (assuming those axioms are consistent). We prove that the statement ``there exists a counterexample to Naimark's problem which is generated by $\aleph_1$ elements'' is undecidable in standard set theory.
There has also been some work by I. Farah on applying set-theoretical technqiues to operator-algebraic problems.