Fredholm operators on C-Algebras

Abstract

The aim of this note is to generalize the notion of Fredholm operator to an arbitrary C-Algebra. Namely we define "finite type" elements in an axiomatic way, and also we define a Fredholm type element a as such an element of a given C-Algebra for which there are finite type elements p and q such that (1-q)a(1-p) is "invertible". We derive an index theorem for such operators. In Applications we show that many well-known operators are special cases of our theory. Those include: classical Fredholm operators on a Hilbert space, Fredholm operators in the sense of Breuer, Atiyah and Singer on a properly infinite von Neumann algebra, and Fredholm operators on Hilbert C-modules over a unital C-Algebra in the sense of Mishchenko and Fomenko.