A STRONG VERSION OF THE BIRKHOFF-JAMES ORTHOGONALITY IN HILBERT C*-MODULES

In this paper we introduce a strong version of the Birkhoff-James orthogonality in Hilbert C*-modules. More precisely, we consider elements x and y of a Hilbert C*-module V over a C*-algebra A which satisfy parallel to x parallel to <= parallel to x + ya parallel to for all a is an element of A. We show that this relation can be described as the Birkhoff-James orthogonality of appropriate elements of V, and characterized in terms of states acting on the underlying C*-algebra A. Some analogous relations of this type are considered as well.