Operator Algebras - M2 - 8EC

1) Prerequisites

The M1 course "Functional Analysis" is a required prerequisite. We further assume familiarity with basics of tensor products of vector spaces.

 

2) Aim of the course

Operator algebras originate from the late 1930's when John von Neumann proposed a new framework to study quantum mechanics in terms of "rings of operators", which we now call von Neumann algebras. Soon after, Israel Gelfand developed the general theory of C*-algebras. 

 

The theory of Operator algebras is concerned with the study of C*- and von Neumann algebras. From its inception, the field has been tightly connected to other  areas of mathematics, such as: representation theory and harmonic analysis, geometric and combinatorial group theory, logic & descriptive set theory, ergodic theory, topology, differential geometry, knot theory, probability, and quantum information theory.

 

In a nutshell operator algebras concerns the theory of algebras of bounded operators on a Hilbert space. Thus instead of studying a single operator one rather studies algebras of operators as a whole. A fundamental theorem , proved in the course "Functional Analysis", is that every commutative C*-algebra can be identified as complex valued continuous functions on a (locally) compact Hausdorff space. Similarly, commutative von Neumann algebras can be identified with L-infinity functions on a measure space.

 

Starting from these cornerstone theorems the theory develops in many directions of which we will see some in the course. In the first part will focus on von Neumann algebras, representation theory and tensor products of C*-algebras. 

 

In the second part of the course we will focus on K-theory, which can be viewed as the group of homotopy classes of non-commutative vector bundles (though no knowledge of differential geometry is required for this course). It connects the theory of operator algebras to topology and geometry via Fredholm index theory.

 

The precise topics and learning goals of this course are:

Part I: Theory of operator algebras

  • Non-unital C*-algebras: one point compactification, approximate units, multiplier algebras.

  • von Neumann's double commutant theorem for von Neumann algebras.

  • Characterisations of von Neumann algebra's in terms of the strong and weak operator topology. Predual of a von Neumann algebra.

  • Existence of suprema and projections in a von Neumann algebra.

  • Trace class and Hilbert-Schmidt operators.

  • Characterisation of commutative von Neumann algebras in terms of measure spaces. The notion of maximal abelian subalgebras.

  • Kaplansky's density theorem.

  • Tensor products of C*-algebras and Takesaki's theorem on minimal tensor products.

Part II: K-theory

  • Serre-Swan theorem: equivalence of vector bundles and projective modules

  • Definition and first properties of K-theory: projections, unitaries, homotopies

  • Functoriality of K-theory, exactness properties

  • Index map and long exact sequence

  • Fredholm operators in topology and geometry

  • Toeplitz index theorem

  • Bott periodicity

 

3) Rules about homework / exam

  • We will post weekly exercises on the Mastermath website and provide some sketches and hints for the solutions to help you. We encourage people to discuss with other students. The ELO also has a forum which can be used to discuss with other students but which will not be moderated by us in principle. Every week we will use the 1st lecture hour as an exercise session. Depending on the needs we discuss questions of the previous week or make an exercise together.

  • During the semester, there will be 2 to 3 hand-in exercises that will be graded and can count towards the final grade as explained below.

  • At the end of the course there will be a written exam which is open book. Allowed sources are the books by Murphy and Roerdam, and the lecture notes by Mesland. No other materials are allowed.

    The final grade for the course is the exam grade rounded off to integers if the grade is strictly below 5.0. If your exam grade is a 5.0 or higher, then your final grade is the number max( 25% takehome exercises + 75% exam grade, exam grade). The grade will be rounded of to a half integer, except that grades between a 5 and 6 will be rounded off to either a 5 or a 6; so it is not possible to get a 5.5. 

  • The retake is a written open book exam (same rules as for the exam). If you take the retake your grade is the grade for the retake (so no bonus for the take home exercises). Again the grade will be rounded off to the nearest half integer, except for a 5.5. 

 

4) Lecture notes/ literature

We will make use of the books:
for Part I: G.J. Murphy, C*-algebras and operator theory (Chapters 1, 2,3, 4 and 6).
for Part II: Lecture notes by B.Mesland, Chapter 7 of Murphy. Although the lecture notes are self-contained, for Part II we also recommend the book
M. Roerdam, F. Larsen, N. Laustsen: An introduction to K-theory for C*-algebras (Chapters 2,3,4, 8 and 9).