Singular traces and their applications to geometry

Born as a mathematical curiosity when Dixmier showed their existence on B(H), singular traces turned out to play a central role in the integration on noncommutative manifolds in Connes' setting. Indeed Connes observed that the (logarithmic) Dixmier trace of a pseudo-differential operator of negative order coincides (up to a constant) with the Wodzicki residue of such an operator, and may be used to "redefine" the integral of functions on a compact spin manifold. This finally led Connes to the proposal of a noncommutative (compact) manifold as a triple (A,H,D) where A plays the role of the algebra of functions, H that of a Clifford bundle, and D of the Dirac operator. A non-commutative dimension d can be associated with this triple according to the Weyl asymptotics relation, namely to the order of growth of the eigenvalues of D. This can be restated by saying that d is characterized by the logarithmic divergence of the trace of D-d. Then the noncommutative dimension appears as the analogue of the Hausdorff dimension, namely as the unique number such that the corresponding d-integration, the singular trace on the ideal generated by D-d, is non trivial. In this paper we review how this idea can be further pursued, making use of different generalizations of singular traces to the von Neumann and C*-algebra settings. Indeed on the one hand the family of singularly traceable operators has been enlarged in order to contain also trace class elements, and on the other hand a new family of singular traces appeares, detecting the "rate of divergence" of some unbounded measurable operators affiliated to a continuous semifinite von Neumann algebra. This family may be defined on C*-algebras with a trace too, with the aid of noncommutative Riemann integration. The first phenomenon produced a criterion for singular traceability in terms of the infinitesimal order of an operator, irrespective of the trace class membership. Therefore such an order has a dimensional interpretation, because a singular trace (a non commutative integration) is associated with it. We then propose an interpretation of the second as associated with the asymptotic dimension of a manifold. This dimension is a large-scale analogue of the Kolmogorov dimension and may be attached to any metric space. On a suitable class of open manifolds, it may be computed in terms of the spectral behaviour of some geometrical operator, e.g. as the "order of infinite" of the inverse of the Laplace-Beltrami operator. In this case also, the asymptotic dimension is a noncommutative dimension, namely a non-trivial singular trace is defined on the bimodule generated by the corresponding power of the Laplacian. In the particular case of universal coverings, the asymptotic dimension coincides with one of the classical L2-invariants of the manifold, namely with the 0-th Novikov-Shubin invariant. Finally we notice the following property of the Novikov-Shubin invariants $a_p$: there exists a nontrivial singular trace on the bimodule generated by $D_p^{-a_p/2}$. This furnish a dimensional interpretation of the $a_p$, as the corresponding (non-commutative) integration is non trivial.