Autumn 2010

MAT4360: C*-algebras

MAT4360: C*-algebras

Recommended textbooks:

K.R. Davidson. C*-algebras by example.

G.J. Murphy. C*-algebras and operator theory.

G.K. Pedersen. C*-algebras and their automorphism groups.

Lecture 1 (26.08)

C*-algebras basics. Unitization.

Exercises

Lecture 2 (1.09)

Commutative C*-algebras. Functional calculus for normal elements.

Lecture 3 (8.09)

Gelfand duality. Positive elements.

Exercises

Lecture 4 (9.09)

States and representations.

Exercises

Lecture 5 (13.09)

Gelfand-Naimark theorem. Approximate units.

Lecture 6 (14.09)

Ideals. Tensor products. Minimal tensor product.

Exercises

Lecture 7 (22.09)

Von Neumann algebras.

Exercises

Lecture 8 (23.09)

The von Neumann double commutant theorem.

Exercises

Lecture 9 (29.09)

Kaplansky's density theorem. Irreducible representations.

Exercises

Lecture 10 (30.09)

Algebras of compact operators. Type I algebras.

Lecture 11 (13.10)

AF-algebras. Bratteli diagrams.

Lecture 12 (14.10)

Classification of AF-algebras.

Exercises

Lecture 13 (20.10)

Toeplitz algebra. Fredholm operators.

Exercises

Lecture 14 (21.10)

Toeplitz operators.

Lecture 15 (27.10)

Cuntz algebras.

Lecture 16 (28.10)

K-theory.

Lecture 17 (3.11)

K-theory of AF-algebras. Comparison of projections.

Lecture 18 (4.11)

K-theory of simple infinite algebras. Purely infinite algebras.

Lecture 19 (5.11)

Completely positive maps.

Exercises

Lecture 20 (10.11)

Tensor products of C*-algebras.

Exercises

Lecture 21 (11.11)

Nuclear C*-algebras.

Exercises

Lecture 22 (18.11)

Crossed products. Amenable groups.

Lecture 23 (24.11)

Amenable groups and nuclear C*-algebras.

Lecture 24 (25.11)

Crossed products of abelian C*-algebras.

Exercises

Lecture 25 (1.12)

Irrational rotation algebras.

Lecture 26 (2.12)

Pureliy infinite algebras arising from crossed products.

Exercises