In this series of fourlectures I will define and investigate in the framework of Banach differentialgeometry the geometric structures related in a canonical way to the structureof a W*-algebra m
(vonNeumann algebra).These are the following structures:
(1) The Banach Liegroupoid G(m)of partial invertible elements of m whosemanifold of units is the lattice oforthogonal projections L(m).
(2) TheBanach Lie algebroid A(m)àL(m)of the groupoid G(m).
(3) Thevarious fiberwise linear Poisson structures and the Poisson VB-groupoidsnaturally related to m.
The obtainedresults show that the methods of Poisson geometry can be applied to thedescription of the classical as well as quantum physical systems.