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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.07019 |
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| _version_ | 1866915076381868032 |
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| author | Bastos, M. Amélia Carvalho, Catarina C. Dias, Manuel G. |
| author_facet | Bastos, M. Amélia Carvalho, Catarina C. Dias, Manuel G. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_07019 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $M$-Local type conditions for the $C^*$-crossed product and local trajectories Bastos, M. Amélia Carvalho, Catarina C. Dias, Manuel G. Operator Algebras 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. |
| title | $M$-Local type conditions for the $C^*$-crossed product and local trajectories |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2307.07019 |