# MA8107 Operator Algebras

Lecturer: Franz Luef

# Lectures

**Monday: 15:15-17:00****Wednesday: 12:15-14:00**

Lectures are in room **922**, SBII.

On Mondays we are going to discuss one or two exercises at the beginning of the class, see here:Problem Set

## Covered Material

**Week 35:**Definition of normed algebras and Banach algebras. Basic properties of Banach algebras, involutive Banach algebras. Examples of Banach algebras: continuous functions on a compact and locally compact space, n-times differentiable functions on a compact topological space, Wiener's algebra. Wiener's lemma and its proof due to Newman combined with a method of Hulanicki.

**Week 36:**Convolution operators and a reformulation of Wiener's lemma. Weighted convolution algebras and Wiener algebras, submultiplicative weights. Spectrum, resolvent of elements in Banach algebras. Unitization of Banach algebras. Group of invertible elements is open in a Banach algebra. Neumann series.

- week 37: Gelfand theory for unital commutative Banach algebras: multiplicative linear functionals, maximal ideals, Gelfand-Mazur.

- week 38: Closed ideals in the space of continuous functions vanishing at infinity, see also Ideals in C(X), maximal ideals of the convolution algebra of absolutely convergent series, Gelfand transform, main theorem of Gelfand's theory for the case of unital commutative Banach algebras. Brief review of nets, lemma of Urysohn, theorem of Tychonov, weak*-topology and the theorem of Banach-Alaoglu.

- week 39: Semisimple Banach algebras, Gelfand theory for non-unital Banach algebras. C*-algebras, commutative C*-algebras and their characterization in terms of the Gelfand transform

- week 40: Consequences of the Gelfand transform, spectral mapping theorem, continuous functional calculus, square root of positive elements in C*-algebras, Hahn decomposition for selfadjoint elements, positivity in C*-algebras, approximate identities for C*-algebras, ideals in C*-algebras.

- week 41: Positivity in C*-algebras, hereditary C*-subalgebras, positive linear functionals.

- week 42: Representations of C*-algebras, non-degenerate, cyclic representations, traces, trace on finite-dimensional matrix algebras, first part of GNS-construction.

- week 43: GNS construction, Gelfand-Naimark Theorem, relation between pure states and irreducibility of representations of C*-algebras, Radon-Nikodym-type theorem for states, state space and the theorem of Krein-Milman.

- week 44: Compact operators, state space of compact operators, trace class and Hilbert-Schmidt operators.

- week 45: Group C*-algebras for discrete groups. Examples of group algebras: abelian groups and free group on two generators.

- week 46: Hilbert C*-modules, adjointable Hilbert C*-module operators, construction of Hilbert C*-modules over compact operators.

- week 47: Noncommutative tori, Morita equivalence for C*-algebras, strong and weak operator topologies, definition of von Neumann algebras, double commutant theorem.

# Prerequisites

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

# Contents of the course

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. Concretely, we are going to discuss the unexpected apperance of operator algebras in applied harmonic analysis and basic notions of noncommutative geometry.

In the final part of the course gives an introduction to C*-modules and in particular in the case of noncommutative tori. These investigations provide a link between operator algebras to wireless communication, pseudodifferential operators, partial differential equations, function spaces, harmonic analysis and time-frequency analysis, see http://www.sciencedirect.com/science/article/pii/S0022123609002468 and http://www.alainconnes.org/en/.

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 C*-modules, Morita equivalence, corners of C*-algebras, and their relevance for noncommutative geometry, in particular in the case of noncommutative tori.

# Projects

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
- Infinite-dimensional matrix algebras and GRS-condition with applications to pseudodifferential operators
- Wodzicki residue and pseudodifferential cacluli
- Spectral triples
- Complex structures in noncommutative geometry

# Textbooks

- “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.