# MA8107 Operator Algebras

Lecturer: Magnus B. Landstad

# Lectures

This will be given as a regular course.

** NOTE: Change: ** We have reserved the following rooms:

**Wednesday: 12:15-14:00 in 822****Friday: 10:15-12:00 in 734,**

First time 24.8 was 12:15-14:00 in R81.

# Latest news

Curriculum:

**Chap. 1.1-3.****Chap. 2.1-3. Chap 2.4 is supposed to be known from Functional analysis.****Chap. 3.1-4.****Chap. 5.1.****From Rørdam's book: Chap. 1.1, 1.3, 2.2, 2.3, 3.1.****Finite dimensional C*-algebras.****Individual project.****Relevant exercises.**

**1.11** Eksamen:
Friday 25.november.

**24.10** Todays seminar:

26th Oct. 14.15-15.00 in 734

Speaker: Petter Kjeverud Nyland (NTNU)

Title: An introduction to AF-algebras

Abstract: The Approximately Finite dimensional C*-algebras (AF-algebras) is a well known class of C*-algebras that were introduced by the Norwegian mathematician Ola Bratteli. They form a rich class of C*-algebras, yet at the same time they are (relatively) easily analysed using combinatorial graphs known as Bratteli-diagrams. The AF-algebras provide a bunch of examples that were well suited to test conjectures in they early days of C*-theory. The AF-algebras was also the first nontrivial class of C*-algebras to be classified solely by their K-theory. Because of this, the AF-algebras have played an important part in the development of C*-theory. In this talk I will introduce the AF-algebras and their Bratteli-diagrams, some of their interesting properties and also give a few concrete examples. The talk is based on parts of my master thesis; “Bratteli Diagrams: Modeling AF-algebras and Cantor Minimal Systems Using Infinite Graphs”.

PS: The talk will be given in Norwegian

**24.10** Here is Petter Nyland's notes on K-theory. Notatene er basert på delkaptlene 1.1, 1.3, 2.2, 2.3, 3.1 og 3.3 i boka til Rørdam.

**4.10** Friday Oct 14, Eduard Ortega and Christian Skau will give information about possible projects. For projects given by Trond Digernes or me, take contact by email.

**4.10** Wednesday Oct 12 and 19, Petter Nyland will give an introduction to K-theory.

**26.9** Here is a short version of Chapter 2.1 in Murphy's book.

# Projects

There will be no lectures from Oct 7-21. Instead you should work on a project. Here are some suggestions of topics in various degrees of complexity and sizes:

- Graph C*-algebras (Ortega)
- C*-algebras and symbolic dynamics (Ortega)
- K-theory of operator algebras (Ortega)
- Spectral triples (Ortega)
- Elliott's Theorem: Classification of AF-algebras (Ortega)
- Dimension groups (Skau)
- Compact and Fredholm operators (See Murphy's book)
- Toeplitz Operators (See Murphy's book)
- Group algebras (See Davidson's book)
- The irrational rotation algebra (See Davidson's book)

- 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

# Exam

** **
Oral.
Friday 25.november.

# Covered Material in 2016

**Week 34:**From Murphy's book: Chap. 1.1.**Week 35:**Chap. 1.2.**Week 36:**Chap. 1.3.**Week 37:**Chap. 2.1.**Week 38:**Chap. 2.1.**Week 39:**Chap. 2.2.**Week 40:**Chap. 2.3.**Week 41-42:**From Rørdam's book: Chap. 1.1, 1.3, 2.2, 2.3, 3.1. +Work on project.**Week 43:**Chap. 3.1.**Week 44:**Chap. 3.2.**Week 45:**Chap. 3.3-4.**Week 46:**Finite dim C*-algebras, Chap. 5.1.**Week 47:**Exam.

## Covered Material in 2014

This can be thought of as a preliminary list of topics we will cover this year.

**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.

In 2014 one or two exercises from Problem Set were discussed during the lectures.

# Prerequisites

Functional analysis TMA4230 is the official prerequisite, but Linear Methods TMA4145 should suffice for an understanding of a large part of the material. Note that the course is given only every second year, so for some students it is an option to follow the course now, but take the exam later.

# 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.

# 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.
- "Crossed products of C*-algebras" by Dana P. Williams. Mathematical surveys and monographs vol. 134 (2007)

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