Operator algebras is a deep and fascinating subject, but it requires a sophisticated understanding of functional analysis including, but not limited to, bounded linear operators on Hilbert space and theirfunctional calculus. Math 103 and 113 are a good start, but you will also want to look at my lecture notes on the spectral theorem. In addition, I have written two books designed for graduate students embarking on the study of operator algebras. The first, written with Iain Raeburn, is entitled Morita Equivalence and Continuous-Trace C*-Algebras. The second is entitled Crossed Product C*-Algebras. Both have extensive appendices that should be useful.

As for choosing an advisor, or being chosen, that is a personal matter. But a successful partnership -- one that leads to a Ph.D. -- can be a lifetime affair. I certainly have depended on my advisor for advice, and not to mention countless letters of recommendation, throughout my career.