An Introduction to Index Theory and Noncommutative Geometry

In this chapter we introduce some of the basic ideas of Index Theory and of the Noncommutative Geometry of Alain Connes. Our main specific objective is to state and prove the Toeplitz Index Theorem, a nice relationship between a certain analytic invariant of a non-vanishing continuous, complex-valued function on the circle, and the topology of the function (actually the winding number of the function ’around the origin.)