Math 519 - Introduction to Operator Algebras, TR 2-3:20, CAB 563


Volker Runde

Office hours



The theory of operator algebras, i.e. of closed, self-adjoint algebras of bounded linear operators on Hilbert space, has its origins in the 1930s with the work of Gelfand, Neumark, von Neumann, and others. Even decades after its inception, it continues to be an extremely active area of research with numerous connections to other branches of mathematics such as ring theory, algebraic topology, differential geometry, and even computing (quantum computers).

This course is an introduction to the theory of operator algebras. I will certainly cover the following topics:

  1. Banach algebras: Definition and elementary properties of the spectrum; spectral radius formula; Gelfand theory; the commutative Gelfand-Naimark theorem.
  2. Operator theory: spectral theorem for normal bounded operators; Fredholm operators.
  3. C*- and von Neumann algebras: positivity; ideals and quotients; the GNS-construction; von Neumann's bicommutant theorem; Kaplansky's density theorem.

Further topics will be covered depending on the participants' background and the time avaialalbe. Possibilities are: holomorphic functional calculus; bounded approximate identities; Fredholm operators; type classification of von Neumann algebras; applications to harmonic analsysis.


Math 516 (Linear Analysis) or equivalent and some familiarity with algebraic concepts such as algebras, ideals, etc., are sufficient as prerequisites for the course.


None required, but the following books are useful supplementary reading:

  1. J. B. Conway, A Course in Functional Analysis. Springer Verlag, 1985.
  2. J. B. Conway, A Course in Operator Theory. American Mathematical Society, 2000.
  3. K. R. Davidson, C*-algebras by Example. American Mathematical Society, 1996.
  4. R. G. Douglas, Banach Algebra Techniques in Operator Theory. Springer Verlag, 1998.
  5. R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, I. American Mathematical Society, 1997.
  6. B.-R. Li, Introduction to Operator Algebras. World Scientific, 1992.
  7. G. J. Murphy, C*-Algebras and Operator Theory. Academic Press, 1990.


The grade will be based on regular homework assignments (50%) and a final project (50%).

Last update: 8/10/08.