C*-algebras, number theory and classification

A C*-algebra is an algebra over the field of complex numbers which is isomorphic to an algebra of operators on a Hilbert space that is closed in the operator norm and closed under taking adjoints. Typical C*-algebras can for instance be associated naturally to objects from algebraic number theory such as algebraic number fields. A key tool for understanding such algebras is (topological) K-theory. This is a generalized cohomology theory. Amazing recent results show that a very large and natural class of C*-algebras (including the ones from number theory mentioned above) can be completely classified by K-theoretic invariants.