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 | |
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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. |