An operator algebra is an algebra of  continuous linear operators on a Hilbert space. Such algebras can be associated to a  variety of problems in mathematics and mathematical physics. The study of operator algebras is based on methods  from analysis, algebra, and algebraic topology. Topics of particular  interest are the theory of C*-algebras and their  classification, K-theory of operator algebras, von Neumann algebras, (C*)-dynamical systems, representation theory of locally compact groups, noncommutative  geometry, and mathematical physics.

The spectrum of the canonical operator in the noncommutative 2-torus T_θ depending on the parameter θ ∈ [0,1]

The specialization Operator Algebras has strong connections  to topology, but there are also links to other areas of mathematics like algebra  and  number theory, differential geometry, partial differential equations, probability, and geometric group theory. Some knowledge in any of these fields is therefore quite useful.

A course  on Functional Analysis  with the following content:

Hahn-Banach theorems, weak and weak-*-topologies, operators on Banach and Hilbert  spaces, spectral theorem for compact operators on Hilbert spaces.