Semiprime ideals in C*-algebras
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866915034543685632 |
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| author | Gardella, Eusebio Kitamura, Kan Thiel, Hannes |
| author_facet | Gardella, Eusebio Kitamura, Kan Thiel, Hannes |
| contents | We show that a not necessarily closed ideal in a C*-algebra is semiprime if and only if it is idempotent, if and only if it is closed under square roots of positive elements. Among other things, it follows that prime and semiprime ideals in C*-algebras are automatically self-adjoint.
To prove the above, we isolate and study a particular class of ideals, which we call Dixmier ideals. As it turns out, there is a rich theory of powers and roots for Dixmier ideals. We show that every ideal in a C*-algebra is squeezed by Dixmier ideals from inside and outside tightly in a suitable sense, from which we are able to deduce information about the ideal in the middle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_17480 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Semiprime ideals in C*-algebras Gardella, Eusebio Kitamura, Kan Thiel, Hannes Operator Algebras Rings and Algebras Primary 46L05, Secondary 16N60, 16W10 We show that a not necessarily closed ideal in a C*-algebra is semiprime if and only if it is idempotent, if and only if it is closed under square roots of positive elements. Among other things, it follows that prime and semiprime ideals in C*-algebras are automatically self-adjoint. To prove the above, we isolate and study a particular class of ideals, which we call Dixmier ideals. As it turns out, there is a rich theory of powers and roots for Dixmier ideals. We show that every ideal in a C*-algebra is squeezed by Dixmier ideals from inside and outside tightly in a suitable sense, from which we are able to deduce information about the ideal in the middle. |
| title | Semiprime ideals in C*-algebras |
| topic | Operator Algebras Rings and Algebras Primary 46L05, Secondary 16N60, 16W10 |
| url | https://arxiv.org/abs/2311.17480 |