B-Fredholm Elements in Rings and Algebras

TL;DR

This paper characterizes B-Fredholm elements in rings and algebras, linking them to generalized Fredholm elements and Drazin invertibility, especially within unital primitive Banach algebras.

Contribution

It provides a new characterization of B-Fredholm elements in terms of generalized Fredholm elements and Drazin invertibility within certain Banach algebras.

Findings

B-Fredholm elements are characterized via generalized Fredholm elements.

A condition on the socle of a Banach algebra links B-Fredholm elements to Drazin invertibility.

In unital primitive Banach algebras, B-Fredholm elements of index 0 are sums of Drazin invertible elements and socle elements.

Abstract

In this paper, we study B-Fredholm elements in rings and algebras. After characterising these elements in terms of generalized Fredholm elements, we will give a condition on the socle of a unital primitive Banach algebra $A$, under which we prove that an element of $A$ is a B-Fredholm element of index $0$ if and only if it is the sum of a Drazin invertible element of $A$ and an element of the socle of $A$.