A new characterization of (pre)liminary C*-algebras
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914340672634880 |
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| author | Lupini, Martino |
| author_facet | Lupini, Martino |
| contents | Given an arbitrary countable ordinal $α$, we introduce the notion of type $I_{α}$ C*-algebra and $α$-subhomogeneous C*-algebra. When $α=0$, these recover the notions of Fell C*-algebra and of commutative C*-algebra, respectively. When $α=n<ω$, these recover the notions of type $I_{n}$ C*-algebra and of $n$-subhomogeneous C*-algebra, respectively. We prove that a separable C*-algebra is liminary if and only if it is type $I_{α}$ for some $α<ω_{1}$, and it is preliminary (i.e., has no infinite-dimensional irreducible representation) if and only if it is $α$-subhomogeneous for some $α<ω_{1}$. We also prove that for any countable ordinal $α$ there exists a separable C*-algebra that is type $I_{α}$ and not type $I_{β}$ for $β<α$, and a separable C*-algebra that is $α$-subhomogeneous and not $β$-subhomogeneous for any $β<α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20980 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new characterization of (pre)liminary C*-algebras Lupini, Martino Operator Algebras Functional Analysis General Topology Logic 46L40, 46L57 (Primary), 03E15, 54H05 (Secondary) Given an arbitrary countable ordinal $α$, we introduce the notion of type $I_{α}$ C*-algebra and $α$-subhomogeneous C*-algebra. When $α=0$, these recover the notions of Fell C*-algebra and of commutative C*-algebra, respectively. When $α=n<ω$, these recover the notions of type $I_{n}$ C*-algebra and of $n$-subhomogeneous C*-algebra, respectively. We prove that a separable C*-algebra is liminary if and only if it is type $I_{α}$ for some $α<ω_{1}$, and it is preliminary (i.e., has no infinite-dimensional irreducible representation) if and only if it is $α$-subhomogeneous for some $α<ω_{1}$. We also prove that for any countable ordinal $α$ there exists a separable C*-algebra that is type $I_{α}$ and not type $I_{β}$ for $β<α$, and a separable C*-algebra that is $α$-subhomogeneous and not $β$-subhomogeneous for any $β<α$. |
| title | A new characterization of (pre)liminary C*-algebras |
| topic | Operator Algebras Functional Analysis General Topology Logic 46L40, 46L57 (Primary), 03E15, 54H05 (Secondary) |
| url | https://arxiv.org/abs/2509.20980 |