Abstract. Summarizing the notion of regular Dirichlet forms (E,F) on locally compact measured spaces (X,m), we then recall the construction and the main properties of the associated differential calculus represented by a closable derivation ∂ on F ∩ C0(X) taking values in a Hilbert tangent bimodule. As an application, the construction of a Hodge theory from self-similar Dirichlet forms on the Sierpinski gasket is illustrated. Subsequently we show how to use the derivation to construct the Dirac operator D on X proving that, if the associated Lipschitz algebra L(F) ⊂ F is dense in C0(X), this gives rise to a spectral triple in the sense of A. Connes, i.e. a framework of metric geometry on X, from which topological invariants (cyclic cocycle) can be extracted.
To take care of situations where the Lipschitz algebra trivializes, as on fractals, we introduce a central subject of potential theory, namely the algebra M(F) of finite energy multipliers of the Dirichlet space F.
Using the Deny’s inequality or embedding, we prove that i) M(F) is a form core for (E,F) and ii) the commutators [F,a], between multipliers a ∈ M(F) and the symmetry F with respect to the graph of the derivation, are compact operators. Whenever the algebra M(F) is dense in C0(X), this gives rise to a Fredholm module in the sense of M. Atiyah, i.e. a framework of a conformal geometry on X, from which cyclic cocycle and index formulas can be derived. Asymptotic of the eigenvalues distribution of commutators [F, a] and Dixmier traces allows to construct nonlinear energy forms analog to the d-Dirichlet integral of a d-dimensional Riemannian manifold.
In the final part of the seminar, we will illustrate how Dirichlet forms, their potential theory and the constructions above, can be considered on noncommutative C∗-algebras endowed with a trace (A,τ).
This part will be based on examples including dual of discrete groups as the free groups Fn, smooth dynamical systems on Riemannian manifolds as the Kronecker foliation of a torus, the Brillouin zone of a quasi-crystal introduced by J. Bellissard, the free entropy introduced by D.V. Voiculescu.
Works in collaboration with J.-L. Sauvageot, D. Guido, T. Isola, A. Kula, U. Franz.