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.

DOI: 10.14232/actasm-015-526-5

AMS Subject Classification (1991): 47A53, 46L08, 46L80

Keyword(s): $C^*$-algebra, Fredholm operators, $K$ group, index

Received April 8, 2015, and in final form July 26, 2017. (Registered under 26/2015.)