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:
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.