Recommended textbooks:
[M] G.J. Murphy. C*-Algebras and Operator Theory.
[P] G.K. Pedersen. C*-Algebras and Their Automorphism Groups.
[D] K. Davidson. C*-Algebras by Example.
For the exam you will need a good understanding of the following topics:
- Different characterizations of positive elements.
- Functional calculus of selfadjoint elements.
- Homomorphisms, ideals, quotients, unitization, approximate units.
- GNS-construction.
- Irreducible representations and pure states.
- Structure of C*-algebras of compact operators.
- Three fundamental operator topologies.
- Von Neumann double commutant theorem and Kaplansky density theorem.
A reasonably good understanding of the following topics:
- Inductive limits of C*-algebras. AF-algebras. UHF-algebras.
- Bratteli diagrams.
- Different notions of equivalence of projections (Murray-von Neumann, unitary, homotopy). K0-group.
- Classification of AF-algebras by K-theory.
Some understanding (at the level that you can clearly convey the main definitions) of the following topics:
- Ultraweak and ultrastrong topologies.
- II-1 factors and normal traces.
- Trace preserving conditional expectations.
- Injectivity and amenability.
- Infinite tensor products and the hyperfinite II-1 factor.
- Stinespring dilation.
- Completely positive maps from and into matrix algebras.