We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not a tree (i.e. not simply connected). This result relates Fredholm modules and topology, refines and improves known results on p.c.f. fractals. We also discuss weakly summable Fredholm modules and the Dixmier trace in the cases of some finitely and infinitely ramified fractals (including nonself-similar fractals) if the so-called spectral dimension is less than 2. In the finitely ramified self-similar case we relate the p-summability question with estimates of the Lyapunov exponents for harmonic functions and the behavior of the pressure function.

The generalized Effros-Hahn conjecture for groupoid C * -algebras says that, if G is amenable, then every primitive ideal of the groupoid C *algebra C * (G) is induced from a stability group. We prove that the conjecture is valid for all second countable amenable locally compact Hausdorff groupoids. Our results are a sharpening of previous work of Jean Renault and depend significantly on his results.

Abstract. If G is a second countable locally compact Hausdorff groupoid with Haar system, we show that every representation induced from an irreducible representation of a stability group is irreducible.

For the Laplacian Δ \Delta defined on a p.c.f. self-similar fractal, we give an explicit formula for the resolvent kernel of the Laplacian with Dirichlet boundary conditions and also with Neumann boundary conditions. That is, we construct a symmetric function G ( λ ) G^{(\lambda )} which solves ( λ I − Δ ) − 1 f ( x ) = ∫ G ( λ ) ( x , y ) f ( y ) d μ ( y ) (\lambda \mathbb {I} - \Delta )^{-1} f(x) = \int G^{(\lambda )}(x,y) f(y) \, d\mu (y) . The method is similar to Kigami’s construction of the Green kernel and G ( λ ) G^{(\lambda )} is expressed as a sum of scaled and “translated” copies of a certain function ψ ( λ ) \psi ^{(\lambda )} which may be considered as a fundamental solution of the resolvent equation. Examples of the explicit resolvent kernel formula are given for the unit interval, standard Sierpinski gasket, and the level-3 Sierpinski gasket S G 3 SG_3 .