The von Neumann algebra of a foliation

Abstract

Every smooth foliation

of a manifold V gives rise very naturally to a von Neumann algebra

. The weights on M correspond exactly to operator valued forms on the “manifold” of leaves of

. We compute their modular automorphism group, this yields the continuous decomposition of M in terms of another foliation

of V and a one parameter group of automorphisms of

. We then illustrate this decomposition with a few examples.