K-theory

K-theory assigns to each C*-algebra A two abelian groups K_0(A) and K_1(A). The K-theory of a C*-algebra contains deep information about the algebra A and there are strong tools that allow you to compute the K-theory. K-theory is probably the most important invariants in operator algebras, non-commutative geometry and in topology with a host of applications in mathematics and in physics. For commutative unital C*-algebras, aka continuous functions on compact spaces, there are two equivalent descriptions of the K-groups, each with its own advantages. In one description K_0 classifies (stable equivalence of) projections and in the other description it classifies (stable equivalence of) vector bundles over the compact space (the spectrum) associated to the algebra.

The course will contain the following specific elements:

• Projections and unitaries in C*-algebras
• Definition, standard picture and basic properties of the K-groups: K_0 and K_1.
• Classification of AF-algebras
• Exact sequences and calculation of K-groups.
• Bott periodicity.
• The six term exact sequence in K-theory.