Autumn 2006

MAT4360: C*-algebras

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.