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We investigate some genuine Poisson geometric objects in the modular theoryof an arbitrary von Neumann algebra $\mathfrak{M}$. Specifically, for anystandard form realization $(\mathfrak{M},\mathcal{H},J,\mathcal{P})$, we find acanonical foliation of the Hilbert space $\mathcal{H}$, whose leaves are Banachmanifolds 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 thestructure of a Banach-Lie groupoid$\widetilde{\mathcal{H}}\rightrightarrows\mathfrak{M}_*^+$ which is isomorphicto 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 thepositive cone in the predual $\mathfrak{M}_*^+$, where$\mathcal{L}(\mathfrak{M})$ is the projection lattice of $\mathfrak{M}$. Thereis also a presymplectic form$\widetilde{\boldsymbol\omega}\in\Omega^2(\widetilde{\mathcal{H}})$ that comesfom the scalar product of $\mathcal{H}$ and is multiplicative in the usualsense of finite-dimensional Lie groupoid theory. We further show that thegroupoid$(\widetilde{\mathcal{H}},\widetilde{\boldsymbol\omega})\rightrightarrows\mathfrak{M}_*^+$ shares several other properties of finite-dimensionalpresymplectic groupoids and we investigate the Poisson manifold structures ofits orbits as well as the leaf space the foliation defined by the degeneracykernel of the presymplectic form~$\widetilde{\boldsymbol\omega}$.

Poisson geometrical aspects of the Tomita-Takesaki modular theory