Autumn 2010
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