### Operator Algebras - 8EC - M2

Prerequisites

Necessary. You must have followed an introductory course on functional analysis and be familiar with notions as: Hilbert space, Banach space, bounded operator, operator norm, duality of Banach spaces (like duality between l_p-spaces), the Hahn-Banach theorem. As an indication you should have seen the larger part of Chapters II IV and VI of Hirzebruch and Scharlau:

http://hirzebruch.mpim-bonn.mpg.de/117/1/M5.pdf

Or chapters I, II.1, II.2 and III of John B. Conway's book "A course in functional analysis" (sections without a star). Note that in particular we assume that you have followed a course on topology and are familiar with notions like compactness and Hausdorff.

Recommended. In addition students who take this course may benefit from the mastermath course on Functional Analysis, where locally convex spaces, compact operators and spectral theory is treated. We will sometimes use (and explain) such results without proving them; the actual proofs are in the book by Murphy that we use, so the interested student can study them which keeps the course self-contained. The general idea is to focus on the operator algebras rather than single operator theory.

Aim of the course

In the late 1920's Murray and von Neumann initiated the study of their "rings of operators" in order to understand quantum physics and quantum probability. Their founding ideas lead to an entire mathematical area that studies algebras of bounded operators on a Hilbert space, which basically splits into two parts: C*-algebras and von Neumann algebras.

The first part of the course deals with the theory of C*-algebras which should be seen as "non-commutative topological spaces" in the sense that every commutative C*-algebra is isomorphic to the continuous functions on a topological space (Gelfand-Naimark theorem 1940's). We will prove this result in the course and treat continuous functional calculus. We will then study states and representation theory of C*-algebras including the GNS-construction. If time permits we treat some elements of K-theory which is an essential tool in the classification of C*-algebras.

The second part of the course deals with von Neumann algebras. These are considered as "non-commutative measure spaces" in the sense that every commutative/abelian von Neumann algebra is isomorphic to essentially bounded functions on a measure space. We will construct the predual of a von Neumann algebra and prove Kaplansky's density theorem together with some consequences. If time permits we will spend one lecture on the connection with discrete groups and give examples of non-isomorphic von Neumann algebras through amenability and Connes's fields medal awarded work on classification of injective factors.

C*- and von Neumann algebras have strong connections to and consequences for representation theory of groups, ergodic theory, geometric group theory, non-commutative geometry, quantum information theory and quantum physics. Some of these connections will be highlighted during the course.

Lecturers

Martijn Caspers (TU Delft)