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

TL;DR

This paper characterizes orthogonality in C*-algebras through a norm condition involving positive elements, providing a new algebraic criterion for orthogonality.

Contribution

It establishes a novel algebraic characterization of orthogonality in C*-algebras using norm relations among positive elements.

Findings

Orthogonality $ab=0$ is equivalent to a specific norm condition.

Provides a new criterion for orthogonality in C*-algebras.

Enhances understanding of algebraic structure of positive elements.

Abstract

We prove the following: Let $A$ be a C$^{\ast}$-algebra. Then for $a, b \in A^+ \setminus\{ 0 \}$, we have $a b = 0$ 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 $A^+$.