Algebraic orthogonality in $C^{\ast }$--algebras-II

Abstract

We prove the following: Let $A$ be a C${}^{\ast }$-algebra. Then for $a,b\in A^{+}\setminus \{0\}$, we have $ab=0$ if and only is $\Vert \Vert c{\Vert }^{-1}c+\Vert d{\Vert }^{-1}d\Vert =1$ whenever $0<c\le a$ and $0<d\le b$ in $A^{+}$.

Proof of Lemma is not convincing. That is, there is a gap in the proof