  In this paper we explicitly construct local $\nu$-Euler derivations $\mathsf
E_\alpha = \nu \partial_\nu + \Lie{\xi_\alpha} + \mathsf D_\alpha$, where the
$\xi_\alpha$ are local, conformally symplectic vector fields and the $\mathsf
D_\alpha$ are formal series of locally defined differential operators, for
Fedosov star products on a symplectic manifold $(M,\omega)$ by means of which
we are able to compute Deligne's characteristic class of these star products.
We show that this class is given by $1/\nu[\omega]+1/\nu[\Omega]$ where $\Omega
= \sum_{i=1}^\infty \nu^i \Omega_i$ is a formal series of closed two-forms on
$M$ the cohomology class of which coincides with the one introduced by Fedosov
to classify his star products. Moreover, our result implies that the
normalisation condition used by Fedosov does not have any effect on the
isomorphy class of the resulting star product. Finally we consider star
products that have additional algebraic structures and compute the effect of
these structures on the corresponding characteristic classes of these star
products. Specifying the constituents of Fedosov's construction we obtain star
products with these special properties. Finally we investigate equivalence
transformations between such special star products and prove existence of
equivalence transformations being compatible with the considered algebraic
structures.
