On the Moore-Penrose inverse in $C^*$-algebras

TL;DR

This paper explores properties of the Moore-Penrose inverse within $C^*$-algebras, including a reverse order law characterization and the study of Moore-Penrose hermitian elements, linking them to normality and partial isometries.

Contribution

It provides a new characterization of the reverse order law and introduces a comprehensive study of Moore-Penrose hermitian elements in $C^*$-algebras.

Findings

Characterization of the reverse order law for Moore-Penrose inverse.

Full characterization of Moore-Penrose hermitian elements.

Equivalence of normal and Moore-Penrose hermitian elements with hermitian partial isometries.

Abstract

In this article, two results regarding the Moore-Penrose inverse in the frame of $C^*$-algebras are considered. In first place, a characterization of the so-called reverse order law is given, which provides a solution of a problem posed by M. Mbekhta. On the other hand, Moore-Penrose hermitian elements, that is $C^*$-algebra elements which coincide with their Moore-Penrose inverse, are introduced and studied. In fact,these elements will be fully characterized both in the Hilbert space and in the $C^*$-algebra setting. Furthermore, it will be proved that an element is normal and Moore-Penrose hermitian if and only if it is a hermitian partial isometry.