  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 Fredholm type element $a$ as such 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 index theorem for such operators.
In applications we show that 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 an unital $C^*$-algebra in the sense of Mishchenko and Fomenko are special
cases of our theory.
