On the Covariance of Moore-Penrose Inverses in Rings with Involution

Abstract

We consider the so-called covariance set of Moore-Penrose inverses in rings with an involution. We deduce some new results concerning covariance set. We will show that if $a$ is a regular element in a $C^{\ast }$-algebra, then the covariance set of $a$ is closed in the set of invertible elements (with relative topology) of $C^{\ast }$-algebra and is a cone in the $C^{\ast }$-algebra.