  The local trajectories method establishes invertibility in an algebra
$\mathcal{B}= \operatorname{alg}(\mathcal{A}, U_G)$, for a unital $C^*$-algebra
$\mathcal{A}$ and a unitary group action $U_g$, $g\in G$, of a discrete
amenable group $G$ on $\mathcal{A}$. We introduce here a local type condition
that allows to establish an isomorphism between $\mathcal{B}$ and a
$C^*$-crossed product, which is fundamental for the method to work. The
influence of the structure of the fixed points of the group action is analysed
and a new equivalent condition is introduced that applies when the action is
not topologically free. If $\mathcal{A}$ is commutative, the referred
conditions are related to the subalgebra $\operatorname{alg}(U_G)$ yielding, in
particular, a sufficient condition that depends only on the action. It is shown
that in $\pi(\mathcal{B})$, with $\pi$ the local trajectories representation,
the local type condition is verified, which allows establishing the isomorphism
on the local trajectories method. It is analysed whether the family of
representations of the local trajectories method is sufficient using the
notions of strictly norming and exhaustive families. It is shown that if
$\mathcal{A}$ is a commutative algebra or a matrix algebra of continuous
functions the family is sufficient.
