Course Identification

Basic courses in functional analysis and measure theory

What is this course all about?

An operator algebra on a Hilbert space H is a closed subalgebra of the algebra B(H) of bounded operators on H. These include all closed algebras of continuous functions on a compact space, matrix algebras, and many more. My intention is to divide the course into two halves: the first will focus on operator algebras, with the first few weeks dedicated to self-adjoint ones, namely C*-algebras, and the second will be devoted to the fruitful interaction between operator algebras and groups

The very flexible plan is:

• C*-algebras (3–5 lectures)
• Operator algebras and operator systems (3–4 lectures)
• Groups, dynamics, and related operator algebras (5–7 lectures) 

Upon successful completion of this course students should be able to
demonstrate an understanding of the concepts of operator algebras as well as their relations to dynamics.

1. D. P. Blecher and C. L. Merdy. Operator Algebras and Their Modules — An Operator Space Approach. The Clarendon Press, Oxford University Press, Oxford, 2004.
2. E. Breuillard, M. Kalantar, M. Kennedy, and N. Ozawa. C*-simplicity and the unique trace property for discrete groups. Publ. Math. Inst. Hautes Études Sci., 126:35–71, 2017.
3. N. P. Brown and N. Ozawa. C*-algebras and Finite-Dimensional Approximations. American Mathematical Society, Providence, 2008.
4. K. R. Davidson. C*-algebras by example, volume 6 of Fields Institute Monographs. American Mathematical Society, Providence, RI, 1996.
5. M. Kalantar and M. Kennedy. Boundaries of reduced C ∗ -algebras of discrete groups. J. Reine Angew. Math., 727:247–267, 2017.
6. V. Paulsen. Completely Bounded Maps and Operator Algebras, volume 78 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2002.