MA8107 Operator Algebras, fall 2022

Lecturer: Eduard Ortega

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

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 approximately finite algebras, group algebras, convolution algebras and noncommutative tori.

Among the topics to be covered are the following:

• Banach Algebra basics, Commutative C*-algebras, positive elements, representations of C*-algebras.
• C*-algebra of compact operators.
• AF C*-algebras
• K-theory for C*-algebras.
• C*-algebras of isometries: Toeplitz algebras, Cuntz algebra, purely infinity C*-algebras.
• Group C*-algebras, amenable groups, the free group.

First day: Tuesday 23.08

* Tuesday 10:15 - 12:00 always in B22

* Thursday 10:15 - 12:00 in 25/8 in B2, from 1/9 to 29/9 in the seminar room 734 (7th floor in math department), and from 6/10 in KJL5.

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 will be announced during the course.

Project 1 CAR algebras: Define the algebra defined by Canonical Anti-commutative Relations. See that the CAR algebra is an AF-algebra. You can look for example this notes of Christopher Hawthorne.

Project 2 Classification of UHF-algebras. A UHF-algebra is a special class of AF-algebra. You should see that UHF-algebras are simple and that can be classified by its associated supernatural number. You can look for example this notes of Christopher Hawthorne.

Project 3 The non-commutative torus embeds into an AF-algebra. There are two papers one can follows: Paper 1 or/and Paper 2

Project 4 The graph C*-algebra. In this projects one should learn how to given a directed graph to define a C*-algebra. Follow the notes from Iain Raeburn.

Project 5 The group algebras of the free group. See that that reduced group algebra of the free group with 2 generators is not isomorphic to its full group C*-algebra. You can see it for example in Ian Putnam's notes

Project 6 Amenable groups. Amenable groups are those such that the reduced and the full group C*-algebras coincide. Study the basic properties and different charactarizations of amenable groups. You can follow the first chapters of Juschenko book

Project 7 Swan Theorem. Swan theorem relates the equivalence classes of vector bundles of a topological space X with the projective modules of C(X). Swan Theorem

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.

You can follow the books in the bibliography or this notes from Ian Putnam.

• Lecture 1: Basics on Banach algebras and C*-algebras. Examples (Commutative C*-algebras and Compact operators). Unitization.
• Lecture 2: Spectrum of an element and its properties.
• Lecture 3: Spectral radius and properties. Examples.
• Lecture 4: Commutative C*-algebras. The space of characters. The Gelfand transform. The Gelfand theorem, every commutative C*-algebras is isometrically isomorphic to the C*-algebra of continuous functions over its space of characters.
• Consequences of the Gelfand theorem: Spectral Calculus, *-homomorphisms between C*-algebras, spectral invariance.
• Lecture 5; Positive elements. An element is positive if it is of the from xx* for some x. Approximate units.
• Lecture 6: Ideals and quotients. Representations. Irreducible representations.
• Lecture 7: Given a state construct a representation (GNS-representation associated to the state).
• Lecture 8: The GNS-representation of a state is irreducible if and only if the state is pure.
• Lecture 9: Every C*-algebra can be isometrically identified with a sub-C*-algebra of B(H).
• Lecture 10: Introduction to AF-algebras and examples Notes on AF-algberas (by P. Nyland)
• Lecture 11: Description of AF-algebras by Bratteli diagrams.
• Lecture 12: Perturbation of finite dimensional algebras.
• Lecture 13: Characterisation of AF-algebras by approximations to finite dimensional algebras.
• Lecture 14: Introduction to the non-commutative torus. Universal property. Automorphisms group.
• Lecture 15: Definition of a Trace in the non-commutative torus. This trace is unique. The Non-commutative torus is simple for irrational angles.
• Lecture 16: Introduction of the Cuntz-algebras. Definition of an expectation on the Cuntz-Algebras.
• Lecture 17: Simplicity of Cuntz algebras.
• Lecture 18: Locally compact groups and the Haar measure. The convolution algebra.
• Lecture 19: Group representations and integrated representation to the convolution algebra.
• Lecture 20: Group algebras of abelian rings and finite groups.
• Lecture 21: Primitive ideal space of C*-algebras.
• Lecture 22: Primitive ideal space of the Toeplitz algebra.