Set theorists have started to seriously look at $C^*$-algebras and there have been several nice results in the last years. The most spectacular one is probably due to Farah, Phillips, and Weaver:
Whether all automorphism of the Calkin algebra (the quotient of the algebra of bounded operators on a separable Hilbert space by the ideal of compact operators) are inner automorphisms (i.e., conjugation by some unitary element) is independent over ZFC.
Philips and Weaver proved the existence of outer automorphisms assuming the continuum hypothesis (Duke Math J., 2009) and Farah showed the non-existence of outer automorphisms assuming Todorcevic's Open Coloring Axiom (Annals of Math, 2011). The interest of set theorists in this field was certainly increased by the Phillips-Weaver result.
There have been previous and clearly related results on automorphisms of the Boolean algebra $\mathcal P(\omega)/fin$, but the methods in the case of the Calkin algebra seem to be slightly different and also a bit more involved.
