Semi-Fredholm theory in $$C^{*}$$-algebras

Abstract

Kečkić and Lazović introduced an axiomatic approach to Fredholm theory by considering Fredholm type elements in a unital $$C^{*}$$ -algebra as a generalization of $$C^{*}$$ -Fredholm operators on the standard Hilbert $$C^{*}$$ -module introduced by Mishchenko and Fomenko and of Fredholm operators on a properly infinite von Neumann algebra introduced by Breuer. In this paper, we establish semi-Fredholm theory in unital $$C^{*}$$ -algebras as a continuation of the approach by Kečkić and Lazović. We introduce the notion of a semi-Fredholm type element and a semi-Weyl type element in a unital $$C^{*}$$ -algebra. We prove that the difference between the set of semi-Fredholm elements and the set of semi-Weyl elements is open in the norm topology, that the set of semi-Weyl elements is invariant under perturbations by finite type elements, and several other results generalizing their classical counterparts. Also, we illustrate applications of our results to the special case of properly infinite von Neumann algebras and we obtain a generalization of the punctured neighbourhood theorem in this setting.