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.