Fredholm module

Summary

A Fredholm module over a C*-algebra $A$ is a representation of $A$ on a Hilbert space with an extra operator $F$. The operator $F$ behaves like the phase of an elliptic operator. It is unitary and self-adjoint up to compact operators, and it commutes with $A$ up to compact operators. Fredholm modules are the cycles of K-homology, and they pair with K-theory through Fredholm indices. They are the bounded counterpart of spectral triples in Connes' noncommutative geometry.