The Daugavet Property of C*-Algebras and Non-commutative Lp-Spaces

We prove that a C^*-algebra A or a predual N_* of a von Neumann algebra N has the Daugavet property if and only if A (or N) is non-atomic. We also prove a similar (although somewhat weaker) result for non-commutative L_p-spaces corresponding to non-atomic von Neumann algebras.