Local type conditions for the $C^*$-crossed product and sufficient families of representations
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.