Characterization of a-Birkhoff–James orthogonality in C*-algebras and its applications

Abstract

In this paper, we aim to investigate the notion of Birkhoff–James orthogonality with respect to the a-norm in namely a-Birkhoff–James orthogonality. The characterization of a-Birkhoff–James orthogonality in by means of the elements of generalized state space is provided. As an application, a characterization for the best approximation to elements of in a subspace with respect to a-norm is obtained. Moreover, a formula for the distance of an element of to the subspace is given. We also study the strong version of a-Birkhoff–James orthogonality in The classes of -algebras in which these two types orthogonality relationships coincide are described. In particular, we prove that the condition of the equivalence between the strong a-Birkhoff–James orthogonality and -valued inner product orthogonality in implies that the center of is trivial. Finally, we show that if the (strong) a-Birkhoff–James orthogonality is right-additive (left-additive) in then the center of is trivial.