$M$-Local type conditions for the $C^{\ast }$-crossed product and local trajectories

Abstract

The local trajectories method establishes invertibility in algebras $\mathcal{B}= \alg(\mathcal{A}, U_G)$, for a unital $C^{\ast }$-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 $a\mapsto U_{g}a{U}_g^{\ast }$ is topologically free. It is applicable in particular to $C^{\ast }$-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 $C^{\ast }$-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 $\pi$ the local trajectories representation, the $M$-local type condition is verified, which allows establishing the isomorphism essential for the local trajectories method.

Revised version. Added examples, references, new structure of sections. Previous section on sufficient families not included. To appear in A. Böttcher et al. (eds.), Achievements and Challenges in the Field of Convolution Operators, Operator Theory: Advances and Applications 306, Birkhäuser