A strong version of the Birkhoff--James orthogonality in Hilbert $C^*$-modules

Abstract

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 $\|x\|\le \|x+ya\|$ for all $a\in A.$ 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.