  We present a class of interacting nonlocal quantum field theories, in which
the CPT invariance is violated while the Lorentz invariance is present. This
result rules out a previous claim in the literature that the CPT violation
implies the violation of Lorentz invariance. Furthermore, there exists the
reciprocal of this theorem, namely that the violation of Lorentz invariance
does not lead to the CPT violation, provided that the residual symmetry of
Lorentz invariance admits the proper representation theory for the particles.
The latter occurs in the case of quantum field theories on a noncommutative
space-time, which in place of the broken Lorentz symmetry possesses the twisted
Poincar\'e invariance. With such a CPT-violating interaction and the addition
of a C-violating (e.g., electroweak) interaction, the quantum corrections due
to the combined interactions could lead to different properties for the
particle and antiparticle, including their masses.
