C*-algebras and K-theory 2003-04

Lecture Notes

Exercises

Part I: [pdf] [ps] (exam included)
Part II: [pdf] [ps]

Take-home exam for Part II

Give a complete proof of Cuntz' theorem: Any functor E from the category of C*-algebras to the category of abelian groups that is i) stable; ii) half-exact; iii) homotopy invariant, satisfies Bott periodicity (where the higher E-groups are defined via suspension, as for E=K).
Literature:
J. Cuntz, K-theory and C*-algebras, Lecture Notes in Mathematics 1046 (1984) 55-79
N.E. Wegge-Olsen, K-Theory and C*-Algebras (Oxford, 1993)
N. Higson and J. Roe, Analytic K-Homology (Oxford, 2000).

Answers to the take-home exam (Part I)

[pdf] [ps]

Answers to the exercises (Part I by C. Carvalho, Part II by F. Behrouzi, B. Norouzizadeh, O. Maldonado Molina, A. Stanulescu, with corrections and additions by C. Carvalho)

Part I: [pdf] [ps] Part II: [pdf] [ps]

Lecturer:

prof.dr. N.P. Landsman npl@science.uva.nl

Exercise class:

dr. C. Carvalho ccarvalh@science.uva.nl

Audience:

MRI Masterclass students, M.Sc. students in mathematics or theoretical physics.

Prerequisites:

Elementary topology and differential geometry, basic theory of Hilbert and Banach spaces (definitions, basic theorems, and simple examples). Mathematics students should have acquired this background in Analysis 1B, 2A, Real analysis, and Differental geometry. Theoretical physics students should know this material from Quantum Mechanics, Geometric Methods in Physcs, and general education.

Dates:

Semester I (2003): September 25, October 2, 9, 16, 23, November 6, 13, 20, 27, December 4, 11
Semester II (2004): February 5, 12, 26, March 4, 11, 18, 25, April 1, 8, 22, 29.

Place and time:

Thursdays: 13:15-16:00, room P.014 (2003), P.017 (2004), Euclides, Plantage Muidergracht 24, 1018 TV AMSTERDAM
directions

Goal:

Familiarity with two key tools in noncommutative geometry.

The philosophy of noncommutative geometry
Banach algebras and C*-algebras
Theory of commutative C*-algebras
Gelfand-Naimark duality theorem
Categories and functors
Natural transformations and equivalence of categories
Categorical version of the Gelfand-Naimark duality theorem
Gelfand-Naimark representation theorem
Ideals and compact operators
Tensor products and Hopf C*-algebras

Part II: K-theory for C*-algebras

Overview of K-theory
Projections in C*-algebras
Definition of K_0
Topological K-theory
K_0 for nonunital C*-algebras
Some explicit computations
Properties of C*-algebraic K-theory (half-exactness, excision, etc.)
Higher K-groups
The index map
Bott periodicity

Literature:

A. Connes, Noncommutative Geometry (Academic Press, San Diego, 1994).
This book forms the basis of the subject, written by its founder. To some, it is the most inspiring mathematics book ever written. However, even pros find the going tough.

J.M. Gracia-Bondia, J.C. Varilly, and H. Figueroa, Elements of Noncommutative Geometry (Birkhauser, Boston, 2001).
This is more reader-friendly, but still advanced.

G. Landi, An Introduction to Noncommutative Spaces and their Geometries (Springer, Heidelberg, 1997). This is really an introduction.

M. Roerdam, F. Larsen, N. Laustsen, An Introduction to K-theory for C*-algebras (Cambridge University Press, Cambridge, 2000). This is the text we'll use for Part II.

J. Brodzki, An Introduction to K-theory and Cyclic Cohomology ps-file
This is a useful starting point for the study of cyclic cohomology.

Exam:

to be announced

Reward:

Part I: 3 ``studiepunten'' (brownie points). Part I and II: 7 studiepunten. It is not possible to take Part II without Part I, but Part I can be done by itself.

Lecture Notes:

In preparation. For Part I my Lecture Notes on C*-algebras and Quantum Mechanics will be helpful, though they by no means cover all material in this new course.