On Moore-Penrose ideals

TL;DR

This paper explores Moore-Penrose ideals in general C*-algebras, characterizing those that ensure invertibility modulo the ideal implies generalized invertibility, and examines their projection lifting properties.

Contribution

It characterizes Moore-Penrose ideals as dual ideals in C*-algebras and investigates their projection lifting properties.

Findings

Moore-Penrose ideals coincide with dual ideals.

Invertibility modulo Moore-Penrose ideals implies generalized invertibility.

Analysis of projection lifting properties of these ideals.

Abstract

Every bounded linear operator on a Hilbert space which is invertible modulo compact operators has a closed range and is, thus, generalized invertible. We consider the analogue question in general $C^*$-algebras and describe the closed ideals (called Moore-Penrose ideals in what follows) with the property that whenever an element is invertible modulo that ideal, then it is generalized invertible. In particular, we will see that the class of Moore-Penrose ideals coincides with the class of the dual ideals. Finally, we study some questions related with the projection lifting property of Moore-Penrose ideals.