Order type relations on the set of tripotents in a JB$^*$-triple
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| Format: | Preprint |
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2021
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| _version_ | 1866917114601799680 |
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| author | Hamhalter, Jan Kalenda, Ondřej F. K. Peralta, Antonio M. |
| author_facet | Hamhalter, Jan Kalenda, Ondřej F. K. Peralta, Antonio M. |
| contents | We introduce, investigate and compare several order type relations on the set of tripotents in a JB$^*$-triple. The main two relations we address are $\le_h$ and $\le_n$. We say that $u\le_h e$ (or $u\le_n e$) if $u$ is a self-adjoint (or normal) element of the Peirce-2 subspace associated to $e$ considered as a unital JB$^*$-algebra with unit $e$. It turns out that these relations need not be transitive, so we consider their transitive hulls as well. Properties of these transitive hulls appear to be closely connected with types of von Neumann algebras, with the results on products of symmetries, with determinants in finite-dimensional Cartan factors, with finiteness and other structural properties of JBW$^*$-triples. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_03155 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Order type relations on the set of tripotents in a JB$^*$-triple Hamhalter, Jan Kalenda, Ondřej F. K. Peralta, Antonio M. Operator Algebras 17C65, 17C27, 17C10, 06A99, 46L10 We introduce, investigate and compare several order type relations on the set of tripotents in a JB$^*$-triple. The main two relations we address are $\le_h$ and $\le_n$. We say that $u\le_h e$ (or $u\le_n e$) if $u$ is a self-adjoint (or normal) element of the Peirce-2 subspace associated to $e$ considered as a unital JB$^*$-algebra with unit $e$. It turns out that these relations need not be transitive, so we consider their transitive hulls as well. Properties of these transitive hulls appear to be closely connected with types of von Neumann algebras, with the results on products of symmetries, with determinants in finite-dimensional Cartan factors, with finiteness and other structural properties of JBW$^*$-triples. |
| title | Order type relations on the set of tripotents in a JB$^*$-triple |
| topic | Operator Algebras 17C65, 17C27, 17C10, 06A99, 46L10 |
| url | https://arxiv.org/abs/2112.03155 |