Course plan

Welcome to the C*-algebra course!

Our plan is to cover the following topics, roughly in this order:

• First examples of C*-algebras
• Commutative case
  • Gelfand transform and functional calculus
  • Gelfand-Naimark theorem
• Basic structure theory
  • Unitization Positive elements
  • Approximate units
  • Ideals and homomorphisms
  • Tensor products
• von Neumann algebras
  • Operator topologies; von Neumann algebras
  • von Neumann's double commutant theorem
  • Kaplansky's density theorem
• Representation theory of operator algebras
  • States; Gelfand-Naimark-Segal construction
  • Irreducible representations and pure states
  • Representations of algebras of compact operators; Voiculescu¡¯s theorem
• K-groups
• AF-algebras
  • Bratteli diagrams
  • Classification of AF-algebras
• Advanced topics
  • Completely positive maps; Conditional expectations
  • Crossed products
  • Amenability

Our recommended references are:

• Kenneth R. Davidson, C*-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, 1996.
• Karen R. Strung, An introduction to C*-algebras and the classification program, Advanced Courses in Mathematics. CRM Barcelona, Birkh?user/Springer, 2021.