Non-linear characterization of Jordan $*$-isomorphisms via maps on positive cones of $C^*$-algebras
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911793517953024 |
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| author | Hatori, Osamu Oi, Shiho |
| author_facet | Hatori, Osamu Oi, Shiho |
| contents | We study maps between positive definite or positive semidefinite cones of unital $C^*$-algebras. We describe surjective maps that preserve
(1) the norm of the quotient or multiplication of elements;
(2) the spectrum of the quotient or multiplication of elements;
(3) the spectral seminorm of the quotient or multiplication of elements. These maps relate to the Jordan $*$-isomorphisms between the specified $C^*$-algebras. While a surjection between positive definite cones that preserves the norm of the quotient of elements may not be extended to a linear map between the underlying $C^*$-algebras, the other types of surjections can be extended to a Jordan $*$-isomorphism or a Jordan $*$-isomorphism followed by the implementation by a positive invertible element. We also study conditions for the centrality of positive invertible elements. We generalize "the corollary" regarding surjections between positive semidefinite cones of unital $C^*$-algebras. Applying it, we provide positive solutions to the problem posed by Molnár for general unital $C^*$-algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07341 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-linear characterization of Jordan $*$-isomorphisms via maps on positive cones of $C^*$-algebras Hatori, Osamu Oi, Shiho Operator Algebras Functional Analysis We study maps between positive definite or positive semidefinite cones of unital $C^*$-algebras. We describe surjective maps that preserve (1) the norm of the quotient or multiplication of elements; (2) the spectrum of the quotient or multiplication of elements; (3) the spectral seminorm of the quotient or multiplication of elements. These maps relate to the Jordan $*$-isomorphisms between the specified $C^*$-algebras. While a surjection between positive definite cones that preserves the norm of the quotient of elements may not be extended to a linear map between the underlying $C^*$-algebras, the other types of surjections can be extended to a Jordan $*$-isomorphism or a Jordan $*$-isomorphism followed by the implementation by a positive invertible element. We also study conditions for the centrality of positive invertible elements. We generalize "the corollary" regarding surjections between positive semidefinite cones of unital $C^*$-algebras. Applying it, we provide positive solutions to the problem posed by Molnár for general unital $C^*$-algebras. |
| title | Non-linear characterization of Jordan $*$-isomorphisms via maps on positive cones of $C^*$-algebras |
| topic | Operator Algebras Functional Analysis |
| url | https://arxiv.org/abs/2403.07341 |