An order-theoretic characterization of C*-algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908988977709056 |
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| author | Tiersma, Samuel |
| author_facet | Tiersma, Samuel |
| contents | We give an order-theoretic characterization of the essential image of the forgetful functor from the category of real/complex unital C*-algebras to the category of real/complex unital operator systems. It is based on the characterization of JB-algebras among the order unit spaces in terms of the existence of gauge-reversing bijections obtained by M. Roelands and the author in arXiv:2507.09526. To this end, we show that a unital operator system is completely order isomorphic to a C*-algebra if and only if each of its matrix spaces admits a compatible JB-algebra structure. As an application, we prove that for $n\ge 4$ the range of a unital n-positive projection on a unital real C*-algebra is unitally n-order isomorphic to a unital real C*-algebra, which is the analogue of a result proven for complex C*-algebras by Choi--Effros. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16904 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An order-theoretic characterization of C*-algebras Tiersma, Samuel Operator Algebras 46L07, 17C65, 46L05, 46L70 We give an order-theoretic characterization of the essential image of the forgetful functor from the category of real/complex unital C*-algebras to the category of real/complex unital operator systems. It is based on the characterization of JB-algebras among the order unit spaces in terms of the existence of gauge-reversing bijections obtained by M. Roelands and the author in arXiv:2507.09526. To this end, we show that a unital operator system is completely order isomorphic to a C*-algebra if and only if each of its matrix spaces admits a compatible JB-algebra structure. As an application, we prove that for $n\ge 4$ the range of a unital n-positive projection on a unital real C*-algebra is unitally n-order isomorphic to a unital real C*-algebra, which is the analogue of a result proven for complex C*-algebras by Choi--Effros. |
| title | An order-theoretic characterization of C*-algebras |
| topic | Operator Algebras 46L07, 17C65, 46L05, 46L70 |
| url | https://arxiv.org/abs/2604.16904 |