Higher K-groups for operator systems
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917829016551424 |
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| author | van Suijlekom, Walter D. |
| author_facet | van Suijlekom, Walter D. |
| contents | We extend our previous definition of K-theoretic invariants for operator systems based on hermitian forms to higher K-theoretical invariants. We realize the need for a positive parameter $δ$ as a measure for the spectral gap of the representatives for the K-theory classes. For each $δ$ and integer $p \geq 0$ this gives operator system invariants $\mathcal V_p^δ(-,n)$, indexed by the corresponding matrix size.
The corresponding direct system of these invariants has a direct limit that possesses a semigroup structure, and we define the $K_p^δ$-groups as the corresponding Grothendieck groups. This is an invariant of unital operator systems, and, more generally, an invariant up to Morita equivalence of operator systems. Moreover, there is a formal periodicity that reduces all these groups to either $K_0^δ$ or $K_1^δ$. We illustrate our invariants by means of the spectral localizer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02981 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher K-groups for operator systems van Suijlekom, Walter D. Operator Algebras Functional Analysis K-Theory and Homology We extend our previous definition of K-theoretic invariants for operator systems based on hermitian forms to higher K-theoretical invariants. We realize the need for a positive parameter $δ$ as a measure for the spectral gap of the representatives for the K-theory classes. For each $δ$ and integer $p \geq 0$ this gives operator system invariants $\mathcal V_p^δ(-,n)$, indexed by the corresponding matrix size. The corresponding direct system of these invariants has a direct limit that possesses a semigroup structure, and we define the $K_p^δ$-groups as the corresponding Grothendieck groups. This is an invariant of unital operator systems, and, more generally, an invariant up to Morita equivalence of operator systems. Moreover, there is a formal periodicity that reduces all these groups to either $K_0^δ$ or $K_1^δ$. We illustrate our invariants by means of the spectral localizer. |
| title | Higher K-groups for operator systems |
| topic | Operator Algebras Functional Analysis K-Theory and Homology |
| url | https://arxiv.org/abs/2411.02981 |