Lecturer: Eduard Ortega

We will cover a variety of topics in the theory of operator algebras, in particular of C*-algebras and von Neumann algebras.

The goal is to expose students to the basic notions in the theory of operator algebras and to prepare them to understand fundamentals used in current areas of active research.

In order to gain a deeper understanding of the abstract concepts we examine a number of examples that illustrate different aspects of the theory, such as group algebras, convolution algebras and noncommutative tori.

Among the topics to be covered are the following:

• Basic facts about C*-algebras: homomorphisms, ideals, quotients, uniqueness of norm, characterization of general and commutative C*-algebras, representations and the GNS construction, unitizations and the multiplier algebra
• Main examples of C*-algebras: Group algebras, noncommutative tori
• Basic facts about Banach algebras: Convolution algebras, spectral invariance, symmetric involutive Banach algebras
• Basic facts about von Neumann algebras: factors, AF algebras, hyperfinite factor
• Basic facts about amenable groups and their C*-algebras.

• Mondays from 10:15 to 12:00 in room 734 (7th floor)
• Thursdays from 10:15 to 12:00 in room 734 (7th floor).

One will be able to follow the lectures on-line via Blackboard Ultra Collaborate.

The evaluation of the course will be done through an oral exam on the basic topics of the course, plus the presentation of one project.

There is also the option to write a project on a variety of topics and in various degrees of complexity and sizes. Potential topics are given below:

• K-theory of operator algebras
• Rieffel's version of Mackey's imprimitivity theorem
• Amenable groups
• Étale groupoids and their C*-algebras.
• Spectral triples
• Complex structures in noncommutative geometry

Functional analysis TMA4230 is the official prerequisite, but Linear Methods should suffice for an understanding of a large part of the material.

• “C*-algebras and operator theory” by Gerard J. Murphy. Boston : Academic Press, 1990.
• ”C*-algebras by example” av Kenneth R. Davidson. Fields Institute monographs; vol. 6 Providence, R.I. : American Mathematical Society, 1996.
• E. C. Lance. Hilbert C-modules — a toolkit for operator algebraists. London Math. Soc. Lecture Note Series 210. Cambridge Univ. Press, Cambridge, 1995.
• "Morita equivalence and continuous-trace C*-algebras" by Iain Raeburn, Dana P. Williams. Mathematical surveys and monographs ; no. 60. Providence, R.I. : American Mathematical Society, 1998.
• "An Introduction to Noncommutative Geometry" by Joseph C. Várilly. EMS series of lectures in mathematics European Mathematical Society, 2006.

• Graph C*-algebras
• Hermitian and symmetric Banach algebras
• Vector bundles and projective modules
• Hausdorff groupoids and their C*-algebras
• K-theory of C*-algebras
• Projections in noncommutative tori and Gabor frames
• Morita equivalence of C*-algebras
• C*-algebras associated to dynamical systems
• C*-algebras associated to irrational rotations