Introduce the enveloping von Neumann algebra of a C*-algebra. Show that it is isometrically isomorphic to the double dual of the C*-algebra, and identify the weak-* topology at the level of the enveloping von Neumann algebra. Present a proof, using the Hahn-Banach separation theorem, that every ideal in a C*-algebra has quasi-central approximate units. One could also prove the equivariant version of this statement due to Kasparov. (A possible source for preparation can be Blackadar's book.)

Broadly speaking, cover the basics of the theory of completely positive maps between C*-algebras. Show Stinespring's theorem, which asserts that completely positive maps can be seen as compressions of *-homomorphisms. Further possibilities that can be cherry-picked: Explain injectivity for operator algebras. Discuss the minimal/maximal tensor product and study its compatibility with respect to completely positive maps. Get into a discussion about nuclearity/exactness, in particular the completely positive approximation property. Prove the Choi-Effros theorem. (The canonical source of preparation is Brown-Ozawa's book.)

Provide a detailed exposition on Connes' famous result in his Annals paper that every injective factor of type II_1 is hyperfinite.

Introduce Tomita-Takesaki theory, showing in reasonable detail why every von Neumann algebra with distinguished faithful normal state (or: semi-finite weight) carries a canonical time evolution. For two different choices of states, construct the Connes cocycle, culminating in the noncommutative Radon-Nikodym theorem. Introduce the finer subtype classification for factors of type III. Sketch why every (separable) factor of type III_lambda for 0 < lambda < 1 can be realized as a crossed product of a single (trace-scaling) automorphism on a type II_infty factor.

This talk can only be given after the previous two. From the last one, motivate that in order to get a uniqueness theorem for injective factors of type III_lambda for 0 < lambda < 1, it suffices to classify certain single automorphisms on injective factors of type II. Reproduce Connes' noncommutative Rokhlin-type theorem for outer automorphisms on factors, and sketch how this leads to the classification of trace-scaling automorphisms on the injective II_infty factor.

Present a proof of Brown's theorem, which asserts that two sigma-unital C*-algebras are stably isomorphic if one of them can be included into the other as a full hereditary subalgebra.

Present a proof of Voiculescu's famous theorem. This theorem states that any faithful essential representation of a separable C*-algebra on a Hilbert space approximately absorbs every other representation in a suitable sense. If time permits, hint at a few interesting applications. A detailed account of Voiculescu's theorem can be found near the beginning of Davidson's book.

Discuss (twisted) topological groupoids and their associated C*-algebras, giving some motivating examples along the way. Introduce Cartan subalgebras in the C*-algebraic context. Present Renault's theorem, asserting that every C*-Cartan pair comes from a twisted étale groupoid. Discuss the role of continuous orbit equivalence in this context. (There is a possibility to first present the analogous well-known theory in the context of von Neumann algebras.)

Develop the theory of extensions for C*-algebras, culminating in the Ext-group and (some of) its connections to K-theory. Cherrypick some nice applications: For example, present a computation for the K-theory of Cuntz algebras using extension theory. A possible (and more advanced) follow-up is to get into the theory of (twisted) extensions of C*-dynamical systems. In the long term, this topic could segway into the new approach to classification of finite C*-algebras.

Show the Akemann-Ostrand property for free groups of several generators. Use this to prove that the free group factors are solid, and deduce for example that they are tensorially prime, meaning that they cannot decompose into any non-trivial tensor product of factors.