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 M. Specifically, for any standard form realization (M, H, J, P), we find a canonical foliation of the Hilbert space H, whose leaves are Banach manifolds that are weakly immersed into H, thereby endowing H with a richer Banach manifold structure to be denoted by H. We also find that H has the structure of a Banach-Lie groupoid H ⇒ M + * which is isomorphic to the action groupoid U(M) * M + * ⇒ M + * defined by the natural action of the Banach-Lie groupoid of partial isometries U(M) ⇒ L(M) on the positive cone in the predual M + * , where L(M) is the projection lattice of M. There is also a presymplectic form ω ∈ Ω 2 ( H) that comes fom the scalar product of H and is multiplicative in the usual sense of finite-dimensional Lie groupoid theory. We further show that the groupoid ( H, ω) ⇒ 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 ω.

Banach Poisson–Lie Groups and Bruhat–Poisson Structure of the Restricted Grassmannian

“…In this way local sections of F will be regarded as local 1-forms on M . Such a bundle F will play the role of co-characteristic distribution in the sense of [4]. However in general it may not be a Banach subbundle of T * M .…”

“…Related work. The notion of Poisson manifold in the context of Banach manifolds was introduced in [19] and generalized in various directions in [2,4,6,8,18,20,22]. The notion of Poisson-Lie group in the finite-dimensional setting goes back to [9,15,17,21].…”