  We investigate the breakdown of Lorentz symmetry in QED by a CPT violating
interaction term consisting of the coupling of an axial fermion current with a
constant vector field $b$, in the framework of algebraic renormalization -- a
regularization-independent method. We show, to all orders in perturbation
theory, that a CPT-odd and Lorentz violating Chern-Simons-like term,
definitively, is not radiatively induced by the axial coupling of the fermions
with the constant vector $b$.
