$M$-Local type conditions for the $C^*$-crossed product and local trajectories

Abstract

The local trajectories method establishes invertibility in algebras $\mathcal{B}= \alg(\mathcal{A}, U_G)$, 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 $\cB$ 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 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 $\alg(U_G)$ yielding, in particular, a sufficient condition that depends essentially on $U_G$. It is shown that in $π(\mathcal{B})= \alg(π(\mathcal{A}), π(U_G))$, with $π$ the local trajectories representation, the $M$-local type condition is verified, which allows establishing the isomorphism essential for the local trajectories method.