The von Neumann algebra of a foliation

Abstract

Every smooth foliation 3 of a manifold V gives rise very naturally to a von Neumann algebra M = L~(V/3). The weights on M correspond exactly to operator valued forms on the "manifold" of leaves of 3. We compute their modular automor-phism group, this yields the continuous decomposition of M in terms of another foliation 3 ° of V and a one parameter group of automorphisms of 3 °. We then illustrate this decomposition with a few examples. Let V be a (smooth, finite dimensional real) manifold and 3 a smooth fo-liation of V. We assume for simplicity that the union of the leaves of 3 which have non-trivial holonomy is negligible (for the smooth measures on V). This is always the case if 3 is analytic. Definition 1-A random operator T = (Tf) a measurable family (Tf)f E £ for each leaf f E £ of 3 , Tf is an operator in L2(f) (~) where As an example, let Y E Co(V,T(3)) be a section (smooth with compact support) of the tangent bundle of 3. Then the flow F = exp tY induces on each leaf f t U t t of L2(f) For each t = (U) f E £ a one parameter group of unitaries Uf. , is a random operator. The random operators are added and composed as follows : (TI+T2) f = Tlf+T2f Vf E £ , (TIT2) f = TlfT2f gf E £ The natural norm is ]]TII ~ = Ess Sup IITfll , defined as the smallest % ~ 0 such that IITfll ~ % holds almost everywhere (i.e. the union of leaves where this fails is negligible). (~) Given a manifold X vector space Co(X, iAI I/2) c a n o n i c a l s c a l a r p r o d u c t : , we let L2(X) be the hilbert space completion of the of smooth half densities with compact support, with the < >