On various notions of distance between subalgebras of operator algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908738665840640 |
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| author | Gupta, Ved Prakash Kumar, Sumit |
| author_facet | Gupta, Ved Prakash Kumar, Sumit |
| contents | Given any irreducible inclusion $\mB \subset \mA$ of unital $C^*$-algebras with a finite-index conditional expectation $E: \mA \to \mB$, we show that the set of $E$-compatible intermediate $C^*$-subalgebras is finite, thereby generalizing a finiteness result of Ino and Watatani (from \cite{IW}). A finiteness result for a certain collection of intermediate $C^*$-subalgebras of a non-irreducible inclusion of simple unital $C^*$-algebras is also obtained, which provides a $C^*$-version of a finiteness result of Khoshkam and Mashood (from \cite{KM}).
Apart from these finiteness results, comparisons between various notions of distance between subalgebras of operator algebras by Kadison-Kastler, Christensen and Mashood-Taylor are made. Further, these comparisons are used satisfactorily to provide some concrete calculations of distance between operator algebras associated to two distinct subgroups of a given discrete group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_05967 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On various notions of distance between subalgebras of operator algebras Gupta, Ved Prakash Kumar, Sumit Operator Algebras Given any irreducible inclusion $\mB \subset \mA$ of unital $C^*$-algebras with a finite-index conditional expectation $E: \mA \to \mB$, we show that the set of $E$-compatible intermediate $C^*$-subalgebras is finite, thereby generalizing a finiteness result of Ino and Watatani (from \cite{IW}). A finiteness result for a certain collection of intermediate $C^*$-subalgebras of a non-irreducible inclusion of simple unital $C^*$-algebras is also obtained, which provides a $C^*$-version of a finiteness result of Khoshkam and Mashood (from \cite{KM}). Apart from these finiteness results, comparisons between various notions of distance between subalgebras of operator algebras by Kadison-Kastler, Christensen and Mashood-Taylor are made. Further, these comparisons are used satisfactorily to provide some concrete calculations of distance between operator algebras associated to two distinct subgroups of a given discrete group. |
| title | On various notions of distance between subalgebras of operator algebras |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2403.05967 |