Non-Abelian Factors for Actions of $\mathbb{Z}$ and Other Non-$C^*$-Simple Groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916206148059136 |
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| author | Amrutam, Tattwamasi Glasner, Eli Glasner, Yair |
| author_facet | Amrutam, Tattwamasi Glasner, Eli Glasner, Yair |
| contents | Let $Γ$ be a countable group and $(X, Γ)$ a compact topological dynamical system. We study the question of the existence of an intermediate $C^*$-subalgebra $\mathcal{A}$ $$C^{*}_{r}(Γ)<\mathcal{A}<C(X)\rtimes_rΓ,$$ which is not of the form $\mathcal{A} = C(Y) \rtimes_r Γ$, corresponding to a factor map $(X,Γ) \to (Y,Γ)$. Here $ C^{*}_{r} (Γ)$ and $C(X) \rtimes_r Γ$ are the reduced $C^*$-algebras of $Γ$ and $(X,Γ)$ respectively. Our main results are (1) For $Γ$, which is not $C^*$-simple, if $(X,Γ)$ admits a $Γ$-invariant probability measure, then such a sub-algebra always exists. (2) For $Γ= \mathbb{Z}$ and $(X, Γ)$ an irrational rotation of the circle $X = S^1$, we give a full description of all these non-crossed-product subalgebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_14278 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-Abelian Factors for Actions of $\mathbb{Z}$ and Other Non-$C^*$-Simple Groups Amrutam, Tattwamasi Glasner, Eli Glasner, Yair Operator Algebras Dynamical Systems Functional Analysis 46L55, 37A55, 37B05, Let $Γ$ be a countable group and $(X, Γ)$ a compact topological dynamical system. We study the question of the existence of an intermediate $C^*$-subalgebra $\mathcal{A}$ $$C^{*}_{r}(Γ)<\mathcal{A}<C(X)\rtimes_rΓ,$$ which is not of the form $\mathcal{A} = C(Y) \rtimes_r Γ$, corresponding to a factor map $(X,Γ) \to (Y,Γ)$. Here $ C^{*}_{r} (Γ)$ and $C(X) \rtimes_r Γ$ are the reduced $C^*$-algebras of $Γ$ and $(X,Γ)$ respectively. Our main results are (1) For $Γ$, which is not $C^*$-simple, if $(X,Γ)$ admits a $Γ$-invariant probability measure, then such a sub-algebra always exists. (2) For $Γ= \mathbb{Z}$ and $(X, Γ)$ an irrational rotation of the circle $X = S^1$, we give a full description of all these non-crossed-product subalgebras. |
| title | Non-Abelian Factors for Actions of $\mathbb{Z}$ and Other Non-$C^*$-Simple Groups |
| topic | Operator Algebras Dynamical Systems Functional Analysis 46L55, 37A55, 37B05, |
| url | https://arxiv.org/abs/2306.14278 |