Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

The operator algebras theory including the theory of von Neumann algebras gives the mathematical background for quantum mechanics. A similar role is played by Poisson geometry in classical mechanics. Such notion as: Lie groupoid and Lie algebroid, symplectic manifold, fibre-wise linear Poisson structures or Lie-Poisson space are important ingredients of the mathematical framework for the contemporary geometric mechanics. In this presentation I will show that these structures are also generated in a canonical way by the structure of W*-algebra (von Neumann algebra). Some constructions and related theorems describing this structures will be presented as well.