  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
fom the scalar product of $\mathcal{H}$ and is multiplicative in the usual
sense of finite-dimensional Lie groupoid theory. We further show that the
groupoid
$(\widetilde{\mathcal{H}},\widetilde{\boldsymbol\omega})\rightrightarrows
\mathfrak{M}_*^+$ shares several other properties of finite-dimensional
presymplectic groupoids and we investigate the Poisson manifold structures of
its orbits as well as the leaf space the foliation defined by the degeneracy
kernel of the presymplectic form~$\widetilde{\boldsymbol\omega}$.
