Notes and Exercises

Friday, July 17th
  • Please refer to the posts in the Piazza for what notes to study.
  • Problem Set
Thursday, July 16th
Wednesday, July 15th
Tuesday, July 14th
Monday, July 13th
Friday, July 10th
Wednesday, July 8th
Monday, July 6th
Friday, July 3rd
Wednesday, July 1st
Monday, June 29th

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.