In this thesis we define even and odd p-summable almost Fredholm modules, which may be viewed as a generalization of even and odd p-summable Fredholm modules. To define a character of an odd almost Fredholm module, we use a duality map, which was developed by A. Gorokhovsky in his thesis. The duality map is defined by using the character of an almost representation which was developed by A. Connes. To do something similar in the even case, we extend the duality map to Z/2 graded algebras, and using this duality map we define a character for a pair of even almost Fredholm modules, and for a subset of the even almost Fredholm modules. Using this character, we are able to extend a result by J. Avron, R. Seiler, and B. Simon.

In particular, they show that if P is a projection and U is a unitary such that P − UPU⁻¹ is in the (2n + 1)st Schatten Class, then Trace((P − UPU⁻¹ ) ²ⁿ⁺¹) ∈ Z. One can recover this result by using Connes' character of an even almost Fredholm module. Our character allows us to give an extension of their result to almost projections, which are self adjoint maps such that P - P² is in the $p$th Schatten class.