In previous papers ([1, 2]) we defined the C*-algebra and the longitudinal pseudodifferential calculus of any singular foliation (M,). In the current paper we construct the analytic index of an elliptic operator as a KK-theory element, and prove that this element can be obtained from an “adiabatic foliation”  on M×ℝ, which we introduce here.

We investigate the uniform limits of the set of polynomials on the closed unit disc D¯"> with respect to the chordal metric χ. More generally, we examine analogous questions replacing the one-point compactification of CC∪{∞}"> by other metrizable compactifications.

In a previous paper ([1]), we associated a holonomy groupoid and a C*-algebra to any singular foliation (M,ℱ). Using these, we construct the associated pseudodifferential calculus. This calculus gives meaning to a Laplace operator of any singular foliation ℱ on a compact manifold M, and we show that it can be naturally understood as a positive, unbounded, self-adjoint operator on L2(M).