Let A be a C*-algebra with unit. An unbounded Fredholm module over A is an object that abstracts the properties of a self-adjoint, first order elliptic pseudodifferential operator on a compact manifold. Alain Connes has shown that a C*-algebra equipped with an unbounded Fredholm module can be viewed as a sort of "noncommutative" metric space. In my talk, I will give some examples of unbounded Fredholm modules, and I will define a notion of an isometry of a noncommutative metric space. In many cases, the group of such isometries has properties similar to those enjoyed by the isometry group of a compact metric space or a compact Riemannian manifold. I will discuss these properties and give several examples.
The first half of the talk should be accessible to graduate students.