In this talk, we will present some recent study about the index
problem of longitudinal elliptic operators on open foliated manifolds. As
the operators under consideration are not elliptic on the whole (not
necessarily closed) manifold, they in general fail to be Fredholm. We will
introduce some operator algebra tools to study the index of such operators.
As an application, we will present a Lichnerowicz type vanishing result for
foliations on open manifolds.