The local trajectories method establishes invertibility in algebras \(\mathcal {B}= \operatorname {alg}(\mathcal {A}, U_G)\) for a unital \(C^*\) -algebra \(\mathcal {A}\) with a non-trivial center and a unitary group \(U_g\) , \(g\in G\) , with G a discrete group, assuming that G is amenable and the action \(a\mapsto U_gaU_g^*\) is topologically free. It is applicable in particular to \(C^*\) -algebras associated with convolution type operators with amenable groups of shifts. We introduce here an M-local type condition that allows to establish an isomorphism between \({\mathcal {B}}\) and a \(C^*\) -crossed product, which is fundamental for the local trajectories method to work. We replace amenability of G by the more general condition that the action is amenable. The influence of the structure of the fixed points of the group action is analysed and a 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 essentially on \(U_G\) . It is shown that in \(\pi (\mathcal {B})= \operatorname {alg}(\pi (\mathcal {A}), \pi (U_G))\) , with \(\pi \) the local trajectories representation, the M-local type condition is satisfied, which allows establishing the isomorphism essential for the local trajectories method.