Free Ultrafilters on the Natural Numbers

The notion of a filter was first introduced by Henri Cartan in 1937.  Filters play an important role in topology, order theory, lattice theory and functional analysis, as we will see. Primarily, the set of free ultrafilters on the set of natural numbers is in one-to-one correspondence with the corona of the natural numbers. This remarkable fact, which will be proven, allows us to make several constructions of C*-algebras; for instance, ultraproducts and ultrapowers. Afterwards, we will discuss the construction of the Murray-von Neumann hyperfinite II_1 factor and study its ultrapower.