Paired $E_0$-Semigroups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915478113353728 |
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| author | Skeide, Michael |
| author_facet | Skeide, Michael |
| contents | In these notes we prove two main results:
1) It is well-known that two strongly continuous $E_0$-semigroups on $B(H)$ can be paired if and only if they have anti-isomorphic Arveson systems. For a new notion of pairing (which coincides only for $B(H)$ with the existing one), we show: For a von Neumann algebra $B$, a strongly continuous $E_0$-semigroup on $B$ and a strongly continuous $E_0$-semigroup on $B'$ can be paired if and only if their product systems are commutants of each other.
2) On the way to prove the former, en passant we have to fill in a long standing important gap in the theory of intertwiner product systems à la Arveson (known, so far, only for $B(H)$ in the separable case): Intertwiner product systems of faithful strongly continuous $E_0$-semigroups on von Neumann algebras have sufficiently many strongly continuous sections.
We explain why both results are entirely out of reach for Arveson's methods [Arv89,Arv90] and depend essentially on the alternative approach from Skeide [Ske16]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_05249 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Paired $E_0$-Semigroups Skeide, Michael Operator Algebras In these notes we prove two main results: 1) It is well-known that two strongly continuous $E_0$-semigroups on $B(H)$ can be paired if and only if they have anti-isomorphic Arveson systems. For a new notion of pairing (which coincides only for $B(H)$ with the existing one), we show: For a von Neumann algebra $B$, a strongly continuous $E_0$-semigroup on $B$ and a strongly continuous $E_0$-semigroup on $B'$ can be paired if and only if their product systems are commutants of each other. 2) On the way to prove the former, en passant we have to fill in a long standing important gap in the theory of intertwiner product systems à la Arveson (known, so far, only for $B(H)$ in the separable case): Intertwiner product systems of faithful strongly continuous $E_0$-semigroups on von Neumann algebras have sufficiently many strongly continuous sections. We explain why both results are entirely out of reach for Arveson's methods [Arv89,Arv90] and depend essentially on the alternative approach from Skeide [Ske16]. |
| title | Paired $E_0$-Semigroups |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2303.05249 |