The Birkhoff-James and Roberts orthogonality in C*-algebras

Orthogonality in normed linear spaces can be defined in different ways. We consider two types of orthogonality in C*-algebras. The Birkhoff-James orthogonality for arbitrary elements a and b of a C*-algebra A is characterized in terms of states acting on A. A characterization of a special case of the Roberts orthogonality for elements a and e, where e is the unit in A, is obtained in terms of the Davis-Wielandt shell of a.