Math 684 (Phillips)

This is the home page for N. C. Phillips' Math 684 at the University of Oregon, Fall quarter 2017. Right now, only administrative information is posted here. Any additional material will be posted here, not on Canvas. (I do not use Canvas.)

This is part of a three quarter sequence. The second and third quarters will be on C*-algebras, with emphasis on group actions on C*-algebras and their crossed products. They will be taught by Qingyun Wang.

Course information:

Course outline (not all of this might actually get done):

  1. A little about bounded operators.
  2. Definition and examples of Banach algebras.
  3. Bott periodicity.
  4. Spectrum, spectral radius.
  5. Weak* topology and Alaoglu's Theorem (with material on general topological vector spaces left as exercises).
  6. Maximal ideal space of a commutative Banach algebra and the Gelfand transform.
  7. Proof of the existence of Haar measure on a locally compact group, probably restricted to the second countable case so as to avoid measure theoretic technicalities.
  8. Careful identification of the maximal ideal space of L^1 (G) for a locally compact abelian group G.
  9. Pontryagin duality
  10. Generalized Fourier inversion.
  11. Start the basics of C*-algebras (positivity, continuous functional calculus, states, representations).
  12. Representations of groups from positive functions on them.

Some things that are likely to be omitted from the course for lack of time (but whose development would make good problem sets):

  1. Krein-Milman Theorem.
  2. Irreducible representations.
  3. Fredholm theory.
  4. Holomorphic functional calculus.

Problems:

The official final exam time for this course is 10:15 am--12:15 pm Monday 4 December 2017.

This page maintained by N. Christopher Phillips, email.

Last significant change 22 September 2017.