Introduction to $C^*$-algebras

1. Background material in functional analysis and Hilbert space theory; basics of Banach algebras, and Gelfand theory.
2. Basic results: commutative $C^*$-algebras, positivity, ideals and quotients, states and representations, the GNS constuction, the algebra of compact operators.
3. A discussion of some basic examples in the theory (e.g. AF algebras, Toeplitz and Cuntz algebras, irrational rotation algebras, as time permits). Introduction to K-theory.

Further topics as time permits.