Continuous actions on primitive ideal spaces lift to $\mathrm{C}^{\ast}$-actions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912112694001664 |
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| author | Pagliero, Matteo |
| author_facet | Pagliero, Matteo |
| contents | We prove that for any second-countable, locally compact group $G$, any continuous $G$-action on the primitive ideal space of a separable, nuclear $\mathrm{C}^{\ast}$-algebra $B$ such that $B \cong B\otimes\mathcal{K}\otimes\mathcal{O}_2$ is induced by an action on $B$. As a direct consequence, we establish that every continuous action on the primitive ideal space of a separable, nuclear $\mathrm{C}^{\ast}$-algebra is induced by an action on a $\mathrm{C}^{\ast}$-algebra with the same primitive ideal space. Moreover, we discuss an application to the classification of equivariantly $\mathcal{O}_2$-stable actions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04297 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Continuous actions on primitive ideal spaces lift to $\mathrm{C}^{\ast}$-actions Pagliero, Matteo Operator Algebras 46L55, 46L35 We prove that for any second-countable, locally compact group $G$, any continuous $G$-action on the primitive ideal space of a separable, nuclear $\mathrm{C}^{\ast}$-algebra $B$ such that $B \cong B\otimes\mathcal{K}\otimes\mathcal{O}_2$ is induced by an action on $B$. As a direct consequence, we establish that every continuous action on the primitive ideal space of a separable, nuclear $\mathrm{C}^{\ast}$-algebra is induced by an action on a $\mathrm{C}^{\ast}$-algebra with the same primitive ideal space. Moreover, we discuss an application to the classification of equivariantly $\mathcal{O}_2$-stable actions. |
| title | Continuous actions on primitive ideal spaces lift to $\mathrm{C}^{\ast}$-actions |
| topic | Operator Algebras 46L55, 46L35 |
| url | https://arxiv.org/abs/2406.04297 |