  For every formal power series $B=B_0 + \lambda B_1 + O(\lambda^2)$ of closed
two-forms on a manifold $Q$ and every value of an ordering parameter $\kappa\in
[0,1]$ we construct a concrete star product $\star^B_\kappa$ on the cotangent
bundle $\pi : T^*Q\to Q$. The star product $\star^B_\kappa$ is associated to
the formal symplectic form on $T^*Q$ given by the sum of the canonical
symplectic form $\omega$ and the pull-back of $B$ to $T^*Q$. Deligne's
characteristic class of $\star^B_\kappa$ is calculated and shown to coincide
with the formal de Rham cohomology class of $\pi^*B$ divided by $\im\lambda$.
Therefore, every star product on $T^*Q$ corresponding to the Poisson bracket
induced by the symplectic form $\omega + \pi^*B_0$ is equivalent to some
$\star^B_kappa$. It turns out that every $\star^B_kappa$ is strongly closed. In
this paper we also construct and classify explicitly formal representations of
the deformed algebra as well as operator representations given by a certain
global symbol calculus for pseudodifferential operators on $Q$. Moreover, we
show that the latter operator representations induce the formal representations
by a certain Taylor expansion. We thereby obtain a compact formula for the WKB
expansion.
