  Renormalizable nonanticommutative SYM theories with chiral matter in the
adjoint representation of the gauge group have been recently constructed in
[arXiv:0901.3094]. In the present paper we focus on the U*(1) case with matter
interacting through a cubic superpotential. For a single flavor, in a
superspace setup and manifest background covariant approach we perform the
complete one-loop renormalization and compute the beta-functions for all
couplings appearing in the action. We then generalize the calculation to the
case of SU(3) flavor matter with a cubic superpotential viewed as a nontrivial
NAC generalization of the ordinary abelian N=4 SYM and its marginal
deformations. We find that, as in the ordinary commutative case, the NAC N=4
theory is one-loop finite. We provide general arguments in support of all-loop
finiteness. Instead, deforming the superpotential by marginal operators gives
rise to beta-functions which are in general non-vanishing. We study the
spectrum of fixed points and the RG flows. We find that nonanticommutativity
always makes the fixed points unstable.
