Groundwork for Operator Algebras Lecture Series

July 15 - 27, 2024

Mini-courses

W*-Minicourse (Rolando de Santiago and Brent Nelson)

This course will provide an introduction to the theory of von Neumann algebras beginning with von Neumann’s Bicommutant Theorem. Topics to be covered include the Borel functional calculus, the structure of abelian von Neumann algebras, the predual, equivalence of projections, type decomposition, tracial states, notable examples and constructions. Time permitting we may also discuss modular theory or the basic construction. The lectures will provide a broad overview of the material, but the complete details can be found in the following notes.

Notes: W_-Notes – GOALS

 

C*-Minicourse (Elizabeth Gillaspy and Lara Ismert)

Through the C*-minicourse, you will learn the proofs of some of the most fundamental theorems in C*-algebra theory, as well as common techniques and examples that underpin modern research in the field. We will begin by classifying all commutative C*-algebras via the Gelfand-Naimark Theorem, which establishes the notion that the study of (noncommutative) C*-algebras is “noncommutative topology.” Other marquee theorems are the GNS construction (which makes precise the assertion that every C*-algebra is an infinite-dimensional generalization of the algebra of $n \times n$ matrices) and the Choi–Effros—Kirchberg Theorem, establishing the equivalence of the different characterizations of nuclear C*-algebras.

In this course, you will also develop your facility with key tools of the C*-algebra trade such as the continuous functional calculus, inductive limits, approximate identities, and Hilbert C*-modules. We will also study a variety of examples of C*-algebras, such as group C*-algebras, AF algebras, graph C*-algebras, and Cuntz–Pimsner algebras.

Notes: https://www.ipam.ucla.edu/wp-content/uploads/2024/07/C_-Notes-GOALS.pdf