# Roberts orthogonality and the Davis-Wielandt shell

## TL;DR

This paper characterizes Roberts orthogonality to the unit in a unital C*-algebra using the Davis-Wielandt shell and explores its relation to the symmetry of the numerical range.

## Contribution

It provides a new characterization of Roberts orthogonality via the Davis-Wielandt shell and links it to numerical range symmetry for specific algebra elements.

## Key findings

- Roberts orthogonality characterized by Davis-Wielandt shell
- Orthogonality equivalent to numerical range symmetry in certain cases
- New insights into the geometric structure of elements in C*-algebras

## Abstract

Let $\mathcal A$ be a unital $C^*$-algebra with the unit $e$. We consider the elements $a\in \mathcal A$ which are Roberts orthogonal to the unit $e$. We obtain a characterization of this orthogonality in terms of the Davis--Wielandt shell of $a$ and show that, for certain classes of elements of $\mathcal A$, the Roberts orthogonality of $a$ and $e$ is equivalent to the symmetry of the numerical range of $a$ with respect to the origin.