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Banach Lie-Poisson spaces and reduction. (English) Zbl 1044.53057

The goal of this paper is to introduce Banach Poisson manifolds (as the infinite-dimensional analogs of Poisson manifolds) and to study the linear structures (the Banach Lie-Poisson spaces) in this setting in more details.

Recall that in the finite-dimensional case, a linear Poisson structure on a vector space determines a Lie algebra structure on the dual vector space. Conversely, every Lie algebra structure gives rise to the linear Lie-Poisson structure on the dual space. Because of possible non-reflexivity of Banach spaces, a Lie algebra structure corresponds to a linear Poisson structure iff it admits a predual invariant under the coadjoint representation. Motivated by this observation, the authors define a Poisson structure on a Banach manifold as a Lie bracket on the space of smooth functions which satisfies the Leibniz identity and an extra condition, which allows to associate to each function a Hamiltonian vector field. This generalizes the notion of previously introduced strong symplectic manifolds.

The authors derive the basic consequences of this definition and describe the compatibility conditions of Poisson structures on Banach manifolds with almost complex and holomorphic structures. The classical Poisson reduction procedure is generalized to the Banach manifolds setting. Lie-Poisson Banach spaces and linear continuous Poisson maps between them are studied in more details. If a Lie algebra of a Banach Lie group is in this category, the coadjoint orbits in the predual are shown to be weak symplectic manifolds.

The main examples of Banach Lie-Poisson spaces are the preduals to ${W}^{*}$-algebras and duals to ${C}^{*}$-algebras. The authors also give some indications that the definition of Banach Poisson manifolds might have to be modified to include some of the important examples.

##### MSC:
 53D17 Poisson manifolds; Poisson groupoids and algebroids 46L99 Selfadjoint operator algebras (${C}^{*}$-algebras, ${W}^{*}$-algebras, etc.) 37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics 58B25 Group structures and generalizations on infinite-dimensional manifolds