Autumn 2006
MAT4360: C*-algebras

Main book:
K.R. Davidson. C*-algebras by example.

Additional literature:
G.J. Murphy. C*-algebras and operator theory.
G.K. Pedersen. C*-algebras and their automorphism groups.
M. Rørdam, F. Larsen, N. Laustsen. An introduction to K-theory for C*-algebras.



Lecture 1 (21.08)
C*-algebras basics. States.
Sections: 1.1-1.2, 1.4, 1.9.

Lecture 2 (30.08)
Representations. Gelfand-Naimark theorem. Abelian C*-algebras.
Sections: 1.3, 1.9.
Exercises: these few.

Lecture 3 (6.09)
Approximate units. Quotients of C*-algebras.
Sections: 1.4, 1.5.

Lecture 4 (11.09)
Operator topologies. Functional calculus for Borel functions. Von Neumann algebras. Polar decomposition.
Sections: 1.6, 1.8.
Exercises: these six.

Lecture 5 (27.09)
The density theorems.
Sections: 1.7.
Exercises: these two.

Lecture 6 (2.10)
Irreducible representations. Pure states.
Sections: 1.9.

Lecture 7 (4.10)
Algebras of compact operators. Type I algebras.
Sections: 1.10.
Exercises: these four.

Lecture 8 (11.10)
AF-algebras.
Sections: 3.1-3.3.

Lecture 9 (18.10)
Classification of AF-algebras. UHF-algebras.
Sections: 3.3, 3.5.
Exercises: 1, 2 and 6 from Chapter 3 and these four.

Lecture 10 (25.10)
Ideals of AF-algebras. K-theory.
Sections: 3.4, 4.1, 4.2.
Exercises: these two.

Lecture 11 (1.11)
Equivalence of projections. Direct limits.
Sections: 4.1-4.3.

Lecture 12 (8.11)
Stability of K-theory. K-theory of AF-algebras.
Sections: 4.3-4.5.
Exercises: 14, 15 from Chapter 4.

Lecture 13 (13.11)
Dimension groups, examples. Fredholm operators. Toeplitz algebra.
Sections: 4.6, 4.3, 5.1, 5.2.

Lecture 14 (15.11)
Toeplitz operators. Cuntz algebras.
Sections: 5.1, 5.2, 5.4.
Exercises: these two.

Lecture 15 (20.11)
Cuntz algebras. Purely infinite algebras. Cuntz-Toeplitz algebras.
Sections: 5.4, 5.5.
Exercises: 5.11.

Lecture 16 (27.11)
Faithfulness of the Fock representation. Long exact sequence of K-theory. K-theory of Cuntz algebras.
Exercises: these four.

Lecture 17 (29.11)
K-theory of Cuntz-Toeplitz algebras. Bott periodicity.
Exercises: 5.15 plus compute the K-theory of that algebra.