Poisson geometrical aspects of the Tomita-Takesaki modular theory
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2019
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917543455752192 |
|---|---|
| author | Beltita, Daniel Odzijewicz, Anatol |
| author_facet | Beltita, Daniel Odzijewicz, Anatol |
| contents | We investigate some genuine Poisson geometric objects in the modular theory of an arbitrary von Neumann algebra $\mathfrak{M}$. Specifically, for any standard form realization $(\mathfrak{M},\mathcal{H},J,\mathcal{P})$, we find a canonical foliation of the Hilbert space $\mathcal{H}$, whose leaves are Banach manifolds that are weakly immersed into~$\mathcal{H}$, thereby endowing $\mathcal{H}$ with a richer Banach manifold structure to be denoted by~$\widetilde{\mathcal{H}}$. We also find that $\widetilde{\mathcal{H}}$ has the structure of a Banach-Lie groupoid $\widetilde{\mathcal{H}}\rightrightarrows\mathfrak{M}_*^+$ which is isomorphic to the action groupoid $\mathcal{U}(\mathfrak{M})\ast\mathfrak{M}_*^+\rightrightarrows\mathfrak{M}_*^+$ defined by the natural action of the Banach-Lie groupoid of partial isometries $\mathcal{U}(\mathfrak{M})\rightrightarrows\mathcal{L}(\mathfrak{M})$ on the positive cone in the predual $\mathfrak{M}_*^+$, where $\mathcal{L}(\mathfrak{M})$ is the projection lattice of $\mathfrak{M}$. There is also a presymplectic form $\widetilde{\boldsymbolω}\inΩ^2(\widetilde{\mathcal{H}})$ that comes from the scalar product of $\mathcal{H}$ and is multiplicative in the usual sense of finite-dimensional Lie groupoid theory. We further explore some aspects of reduction theory for the groupoid endowed with the multiplicative presymplectic form $(\widetilde{\mathcal{H}},\widetilde{\boldsymbolω})\rightrightarrows \mathfrak{M}_*^+$, including the Poisson manifold structures of its orbits and the foliation defined by the degeneracy kernel of the presymplectic form~$\widetilde{\boldsymbolω}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_14466 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Poisson geometrical aspects of the Tomita-Takesaki modular theory Beltita, Daniel Odzijewicz, Anatol Operator Algebras Symplectic Geometry Primary 53D17, Secondary 22A22, 22E65, 46L10, 46L60, 58B25 We investigate some genuine Poisson geometric objects in the modular theory of an arbitrary von Neumann algebra $\mathfrak{M}$. Specifically, for any standard form realization $(\mathfrak{M},\mathcal{H},J,\mathcal{P})$, we find a canonical foliation of the Hilbert space $\mathcal{H}$, whose leaves are Banach manifolds that are weakly immersed into~$\mathcal{H}$, thereby endowing $\mathcal{H}$ with a richer Banach manifold structure to be denoted by~$\widetilde{\mathcal{H}}$. We also find that $\widetilde{\mathcal{H}}$ has the structure of a Banach-Lie groupoid $\widetilde{\mathcal{H}}\rightrightarrows\mathfrak{M}_*^+$ which is isomorphic to the action groupoid $\mathcal{U}(\mathfrak{M})\ast\mathfrak{M}_*^+\rightrightarrows\mathfrak{M}_*^+$ defined by the natural action of the Banach-Lie groupoid of partial isometries $\mathcal{U}(\mathfrak{M})\rightrightarrows\mathcal{L}(\mathfrak{M})$ on the positive cone in the predual $\mathfrak{M}_*^+$, where $\mathcal{L}(\mathfrak{M})$ is the projection lattice of $\mathfrak{M}$. There is also a presymplectic form $\widetilde{\boldsymbolω}\inΩ^2(\widetilde{\mathcal{H}})$ that comes from the scalar product of $\mathcal{H}$ and is multiplicative in the usual sense of finite-dimensional Lie groupoid theory. We further explore some aspects of reduction theory for the groupoid endowed with the multiplicative presymplectic form $(\widetilde{\mathcal{H}},\widetilde{\boldsymbolω})\rightrightarrows \mathfrak{M}_*^+$, including the Poisson manifold structures of its orbits and the foliation defined by the degeneracy kernel of the presymplectic form~$\widetilde{\boldsymbolω}$. |
| title | Poisson geometrical aspects of the Tomita-Takesaki modular theory |
| topic | Operator Algebras Symplectic Geometry Primary 53D17, Secondary 22A22, 22E65, 46L10, 46L60, 58B25 |
| url | https://arxiv.org/abs/1910.14466 |