Curriculum

Prerequisite Notes

Please read the following notes before the GOALS mini-courses begin on Monday, June 29th: GOALS Prerequisite Notes.

Tentative curriculum for the mini-courses:

Operator Theory and Operator Algebras Basics
1. Introduction to operator theory and to important classes of operators: normal, self-adjoint, positive, projections, isometries, rank-1 operators, polar decomposition.
2. Introduction to operator algebras: Definitions of C*-algebras and von Neumann algebras as subalgebras of $B(\mathcal{H})$; first examples ($M_n(\mathbb{C}),C(X), L^\infty(X), K(\mathcal{H})$); ideals; functionals, states, and traces on operator algebras; functional calculus (Gelfand-Naimark Theorem).
C*-Algebras Von Neumann Algebras
1. Group C*-algebras; nuclearity for group C*-algebras. Amenable groups; some characterizations. 1. Strong/weak operator topologies and von Neumann’s bicommutant theorem.
2. Tensor products of C*-algebras, nuclearity for general C*-algebras, completely positive maps. 2. More examples of von Neumann algebras (type $\mathrm{I}$'s, hyperfinite $\mathrm{II}_1$, measure spaces, group von Neumann algebras, crossed products).
3. Examples of C*-algebras (Toeplitz algebra, Cuntz–Krieger algebras, AF algebras). 3. Abelian von Neumann algebras and the Borel functional calculus.
4. Introduction to crossed products; nuclearity for crossed products. 4. The predual and the ultra-weak/strong topologies.
5. Abstract C*-algebras and structural properties (C*-identity, unitizations). 5. The lattice of projections and construction of a trace for a $\mathrm{II}_1$ factor.
6. Gelfand-Naimark-Segal construction. 6. Conditional Expectations and the Basic Construction.
7. Stinespring’s dilation theorem; completely positive approximation property. 7. Bimodules for von Neumann algebras.