From Magnetic Schrödinger Operators to L²-Betti Numbers

This is a continuation of my talk on mathematical problems in solid state physics. I describe a non-commutative version of Bloch theory which allows to overcome the limitations of ordinary (commutative) Bloch theory. It applies not only to magnetic Schrödinger operators but to quite general elliptic gauge-invariant operators. One of the main tools is a generalized trace which is (in a special case) familiar from the theory of L²-Betti numbers, which we describe briefly.