  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.
