  We present a natural imbedding of the crossed product $\mathcal A \rtimes_\xi
\mathbb Z^d$ into the $C^\ast$-algebra of adjointable operators over the
standard Hilbert $\mathcal A$-module $\mathcal H_{\mathcal A}$. By replacing
the representations on Hilbert spaces with this canonical imbedding, we define
Fredholm modules and corresponding Chern-Connes characters that are intrinsic
to the $C^\ast$-dynamical system $(\mathcal A,\xi,\mathbb Z^d)$. The
compression of the Dirac operator against projectors from $\mathcal A
\rtimes_\xi \mathbb Z^d$ produces generalized Fredholm operators over $\mathcal
H_{\mathcal A}$ and Mingo's index defines a $KK$-map from $K_0(\mathcal A
\rtimes_\xi \mathbb Z^d)$ to $K(\mathcal A)$. Using a generalized Fedosov
principle and a generalized Fedosov formula, we prove an index formula for the
pairing of the intrinsic Chern-Connes characters and $K_0(\mathcal A
\rtimes_\xi \mathbb Z^d)$. This pairing takes values in the image of
$K_0(\mathcal A)$ in $\mathbb R$ under a canonical trace. A local index formula
enables new applications in condensed matter physics to the so called weak
topological invariants.
