An introductory course in functional analysis is required. You must be familiar with most notions like Hilbert spaces, projections and closed subspaces, bounded operators on a Banach space, the Hahn-Banach theorem and preferably its consequences for the separation of sets with functionals, Banach-Alaoglu theorem, spectrum of bounded operators and the fact that it is closed and non-empty (but not necessarily the spectral theorem), compact operators, basic examples of Banach spaces such as Lp-spaces (mostly for p=1, p=2 or p=infinity).

Aim of the course
Operator algebras originate from the late 1930's when John von Neumann proposed a new framework in terms of "rings of operators" to study quantum mechanics and which we call "von Neumann algebras" nowadays. Soon after, Israel Gelfand developed the more general theory of C*-algebras and laid further connections to representation theory. The theory of C*-algebras and von Neumann algebras form a single area that we call operator algebras. The theory saw a century of remarkable developments and is nowadays tightly connected to several mathematical areas such as: representation theory, geometric and combinatorial group theory, logic & descriptive set theory, ergodic theory, topology, differential geometry, knot theory, probability, quantum information theory and recently there have even been links to theoretical computer science.

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. One can think of all bounded operators on a Hilbert space or the compact operators as first examples of operator algebras. A fundamental theorem proved in the course is that every commutative C*-algebra can be identified as complex valued continuous functions on a (locally) compact Hausdorff space. Von Neumann algebras can always be identified with L-infinity functions on a good measure space or as a probability space. Starting from these cornerstone theorems the theory develops in many directions of which we will see some in the course such as representation theory. In the final part of the course we will focus on K-theory: this can be viewed as a group of homotopy classes of non-commutative vector bundles (though no knowledge of differential geometry is required for this course) and it connects the theory of operator algebras to topology and geometry via Fredholm index theory.

The aim of the first half of this course is to give an introduction to the theory of C*-algebras and von Neumann algebras. In the second part of the course we will develop the basic properties of K-theory, a homology theory for C*-algebras built from the non-commutative analogue of vector bundles, and explain its link to index theory of Fredholm operators .
The precise topics and learning goals of this course are:

Part I: Theory of operator algebras

  • Gelfand Naimark theorem characterising commutative C*-algebras, both unital and non-unital. Continuous functional calculus.
  • The cone of positive elements in a C*-algebra and its various characterisations.
  • States on C*-algebras and the GNS-construction.
  • Unitizations of C*-algebras: multiplier algebras, one-point unitisation, approximate units.
  • 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.

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

Martijn Caspers (TU Delft)
Bram Mesland (Universiteit Leiden)