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

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*-algebra, then the covariance set of a is closed in the set of invertible elements (with relative topology) of C*-algebra and is a cone in the C*-algebra.