Seyed Mohammad Sadegh Nabavi Sales - Department of Mathematics, Hakim Sabzevari University, P.O. Box ۳۹۷, Sabzevar, Iran.

We offer a new definition of $varepsilon$-orthogonality in normed spaces, and we try to explain some properties of which. Also we introduce some types of $varepsilon$-orthogonality in an arbitrary  $C^ast$-algebra $mathcal{A}$, as a Hilbert $C^ast$-module over itself, and investigate some of its properties in such spaces. We state some results relating range-kernel orthogonality in $C^*$-algebras.

Approximate orthogonality, Birkhoff--James orthogonality, Range-kernel orthogonality, $C^ast$-algebra, $ast$-orthogonality