  A representation of general translation-invariant star products in the
algebra of M(C) = lim_N\to \infty M_N (C) is introduced which results in the
Moyal-Weyl-Wigner quantization. It provides a matrix model for general
translation-invariant noncommutative quantum field theories in terms of the
noncommutative calculus on differential graded algebras. Upon this machinery a
cohomology theory, the so called star-cohomology, with groups H^k_*(C), is
worked out which provides a cohomological framework to formulate general
translation-invariant noncommutative quantum field theories based on the
achievements for the commutative fields, and is comparable to the
Seiberg-Witten map for the Moyal case. Employing the Chern-Weil theory via the
integral classes of H^k_*(Z) a noncommutative version of the Chern character is
defined as an equivariant form which contains topological information about the
corresponding translation-invariant noncommutative Yang-Mills theory. Thereby
we study the mentioned Yang-Mills theories with three types of actions of the
gauge fields on the spinors, the ordinary, the inverse, and the adjoint action,
and then some exact solutions for their anomalous behaviors are worked out via
employing the homotopic correlation on the integral classes of *-cohomology.
Finally, the corresponding consistent anomalies are also derived from this
topological Chern character in the *-cohomology.
