Note on the Covariance Coset of the Moore-Penrose Inverses in C-Algebras

We will introduce and study several algebraic properties of the covariance cosets in C-algebras. Indeed, we will characterize the covariance coset in terms of commutators. Also, we will show that for an invertible element b, the covariance coset of b?1 coincides with the covariance coset of b. Moreover, if b is normal, then the covariance coset of b coincides with the covariance coset of b. In addition, we will prove that the covariance coset is a cone.