Roberts orthogonality and Davis-Wielandt shell

We consider the elements a of a unital C*- algebra A which are Roberts orthogonal to the unit e of A. We present a characterization of this orthogonality in terms of the Davis- Wielandt shell of a and show that, for certain classes of elements of A, the Roberts orthogonality of a and e is equivalent to the symmetry of the numerical range of a with respect to the origin. When A is the C*-algebra of all 2x2 complex matrices, we give a complete description of the form of 2x2 matrices which are Roberts orthogonal.