Poisson geometrical aspects of the Tomita-Takesaki modular theory

Abstract

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 $\widetilde{\boldsymbol\omega}\in\Omega^2(\widetilde{\mathcal{H}})$ 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 $(\widetilde{\mathcal{H}},\widetilde{\boldsymbol\omega})\rightrightarrows \mathfrak{M}_*^+$, including the Poisson manifold structures of its orbits and the foliation defined by the degeneracy kernel of the presymplectic form~$\widetilde{\boldsymbol\omega}$.