Introduction to C*-Algebra Theory

Timetable

2h lecture followed by 2h exercise classes - early hours preferably.

Module aim (aims)

C*-algebra theory is the language of modern analysis. It brings together specialists from various fields of analysis: abstract harmonic analysis and measure theory, operator algebras and group algebras, noncommutative geometry, free probability theory, dynamical systems among others. The aim of the course is to introduce this beautiful theory to wider audience and allow students for further study and research.

Pre-requisites in terms of knowledge, skills and social competences (where relevant)

The knowledge from Banach Space and Banach Algebra Theory is advised for better understanding of the course.

Syllabus

Week 1: C*-algebras: definitions, examples and basic properties.

Week 2: The Spectrum.

Week 3: Multiplicative functionals and maximal ideals.

Week 4: The Gelfand Transform.

Week 5: Continuous Functional Calculus.

Week 6: Positivity in C*-algebras.

Week 7: Bounded Approximate Identities.

Week 8: Weak topologies and density properties.

Week 9: Von Neumann algebras.

Week 10: Von Neumann double commutant theorem.

Week 11: Polar decomposition.

Week 12: Russo--Dye theorem.

Week 13: Kadison transitivity theorem.

Week 14: Pure states and irreducible representations.

Week 15: The GNS constructions.

Reading list

• K. R. Davidson, C*-Algebras by Example, AMS, 1996.
• R. V. Kadison, J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. 1 & 2, AMS, 1997.
• G. K. Pedersen, C*-Algebras and Their Automorphism Groups, Academic Press, 1979.
• N. P. Brown, N. Ozawa, C*-Algebras and Finite-Dimensional Approximations, AMS, 2008.