Noncommutative Riemann integration and and Novikov-Shubin invariants for open manifolds

Given a C*-algebra A with a semicontinuous semifinite trace tau acting on the Hilbert space H, we define the family R of bounded Riemann measurable elements w.r.t. tau as a suitable closure, a la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions, and show that R is a C*-algebra, and tau extends to a semicontinuous semifinite trace on R. Then, unbounded Riemann measurable operators are defined as the closed operators on H which are affiliated to A'' and can be approximated in measure by operators in R, in analogy with unbounded Riemann integration. Unbounded Riemann measurable operators form a tau-a.e. bimodule on R, denoted by R^, and such bimodule contains the functional calculi of selfadjoint elements of R under unbounded Riemann measurable functions. Besides, tau extends to a bimodule trace on R^. Type II_1 singular traces for C*-algebras can be defined on the bimodule of unbounded Riemann-measurable operators. Noncommutative Riemann integration, and singular traces for C*-algebras, are then used to define Novikov-Shubin numbers for amenable open manifolds, show their invariance under quasi-isometries, and prove that they are (noncommutative) asymptotic dimensions.Comment: 34 pages, LaTeX, a new section has been added, concerning an application to Novikov-Shubin invariants, the title changed accordingl

Dimensions and singular traces for spectral triples, with applications to fractals

Given a spectral triple (A,D,H), the functionals on A of the form a -> tau_omega(a|D|^(-t)) are studied, where tau_omega is a singular trace, and omega is a generalised limit. When tau_omega is the Dixmier trace, the unique exponent d giving rise possibly to a non-trivial functional is called Hausdorff dimension, and the corresponding functional the (d-dimensional) Hausdorff functional. It is shown that the Hausdorff dimension d coincides with the abscissa of convergence of the zeta function of |D|^(-1), and that the set of t's for which there exists a singular trace tau_omega giving rise to a non-trivial functional is an interval containing d. Moreover, the endpoints of such traceability interval have a dimensional interpretation. The corresponding functionals are called Hausdorff-Besicovitch functionals. These definitions are tested on fractals in R, by computing the mentioned quantities and showing in many cases their correspondence with classical objects. In particular, for self-similar fractals the traceability interval consists only of the Hausdorff dimension, and the corresponding Hausdorff-Besicovitch functional gives rise to the Hausdorff measure. More generally, for any limit fractal, the described functionals do not depend on the generalized limit omega.Comment: latex, 36 pages, no figures, to appear on Journ. Funct. Analysi

The problem of completeness for Gromov-Hausdorff metrics on C*-algebras

It is proved that the family of equivalence classes of Lip-normed C*-algebras introduced by M. Rieffel, up to isomorphisms preserving the Lip-seminorm, is not complete w.r.t. the matricial quantum Gromov-Hausdorff distance introduced by D. Kerr. This is shown by exhibiting a Cauchy sequence whose limit, which always exists as an operator system, is not completely order isomorphic to any C*-algebra. Conditions ensuring the existence of a C*-structure on the limit are considered, making use of the notion of ultraproduct. More precisely, a necessary and sufficient condition is given for the existence, on the limiting operator system, of a C*-product structure inherited from the approximating C*-algebra. Such condition can be considered as a generalisation of the f-Leibniz conditions introduced by Kerr and Li. Furthermore, it is shown that our condition is not necessary for the existence of a C*-structure tout court, namely there are cases in which the limit is a C*-algebra, but the C*-structure is not inherited.Comment: 31 pages. Accepted for publication in Journal of Functional Analysi

An asymptotic dimension for metric spaces, and the 0-th Novikov-Shubin invariant

A nonnegative number d_infinity, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and is invariant under rough isometries. It is then shown that for a class of open manifolds with bounded geometry the asymptotic dimension coincides with the 0-th Novikov-Shubin number alpha_0 defined previously (math.OA/9802015, cf. also math.DG/0110294). Thus the dimensional interpretation of alpha_0 given in the mentioned paper in the framework of noncommutative geometry is established on metrics grounds. Since the asymptotic dimension of a covering manifold coincides with the polynomial growth of its covering group, the stated equality generalises to open manifolds a result by Varopoulos.Comment: 17 pages, to appear on the Pacific Journal of Mathematics. This paper roughly corresponds to the third section of the unpublished math.DG/980904

Tangential dimensions I. Metric spaces

Pointwise tangential dimensions are introduced for metric spaces. Under regularity conditions, the upper, resp. lower, tangential dimensions of X at x can be defined as the supremum, resp. infimum, of box dimensions of the tangent sets, a la Gromov, of X at x. Our main purpose is that of introducing a tool which is very sensitive to the "multifractal behaviour at a point" of a set, namely which is able to detect the "oscillations" of the dimension at a given point. In particular we exhibit examples where upper and lower tangential dimensions differ, even when the local upper and lower box dimensions coincide. Tangential dimensions can be considered as the classical analogue of the tangential dimensions for spectral triples introduced in math.OA/0202108 and math.OA/0404295, in the framework of Alain Connes' noncommutative geometry.Comment: 18 pages, 4 figures. This version corresponds to the first part of v1, the second part being now included in math.FA/040517

Dimensions and spectral triples for fractals in R^N

Two spectral triples are introduced for a class of fractals in R^n. The definitions of noncommutative Hausdorff dimension and noncommutative tangential dimensions, as well as the corresponding Hausdorff and Hausdorff-Besicovitch functionals considered in math.OA/0202108, are studied for the mentioned fractals endowed with these spectral triples, showing in many cases their correspondence with classical objects. In particular, for any limit fractal, the Hausdorff-Besicovitch functionals do not depend on the generalized limit procedure.Comment: 24 pages, 4 figures. To appear in Proceedings of the Conference "Operator Algebras and Mathematical Physics" held in Sinaia, Romania, June 2003, O. Bratteli, R. Longo H. Siedentop Eds., Theta Foundation, Bucares

Fractals in Noncommutative Geometry

To any spectral triple (A,D,H) a dimension d is associated, in analogy with the Hausdorff dimension for metric spaces. Indeed d is the unique number, if any, such that |D|^-d has non trivial logarithmic Dixmier trace. Moreover, when d is finite non-zero, there always exists a singular trace which is finite nonzero on |D|^-d, giving rise to a noncommutative integration on A. Such results are applied to fractals in R, using Connes' spectral triple, and to limit fractals in R^n, a class which generalises self-similar fractals, using a new spectral triple. The noncommutative dimension or measure can be computed in some cases. They are shown to coincide with the (classical) Hausdorff dimension and measure in the case of self-similar fractals.Comment: 15 pages, LaTeX with fic-l.cls at ftp://ftp.ams.org/pub/author-info/packages/fic/amslatex/fic-l.cls To appear in the proceedings of the conference "Mathematical Physics in Mathematics and Physics", Siena 200

Zeta functions for infinite graphs and functional equations

The definitions and main properties of the Ihara and Bartholdi zeta functions for infinite graphs are reviewed. The general question of the validity of a functional equation is discussed, and various possible solutions are proposed.Comment: 23 pages, 3 figures. Accepted for publication in "Fractals in Applied Mathematics", Contemporary Mathematics, Editors Carfi, Lapidus, Pearse, van Frankenhuijse

Novikov-Shubin invariants and asymptotic dimensions for open manifolds

The Novikov-Shubin numbers are defined for open manifolds with bounded geometry, the Gamma-trace of Atiyah being replaced by a semicontinuous semifinite trace on the C*-algebra of almost local operators. It is proved that they are invariant under quasi-isometries and, making use of the theory of singular traces for C*-algebras developed in math/9802015, they are interpreted as asymptotic dimensions since, in analogy with what happens in Connes' noncommutative geometry, they indicate which power of the Laplacian gives rise to a singular trace. Therefore, as in geometric measure theory, these numbers furnish the order of infinitesimal giving rise to a non trivial measure. The dimensional interpretation is strenghtened in the case of the 0-th Novikov-Shubin invariant, which is shown to coincide, under suitable geometric conditions, with the asymptotic counterpart of the box dimension of a metric space. Since this asymptotic dimension coincides with the polynomial growth of a discrete group, the previous equality generalises a result by Varopoulos for covering manifolds. This paper subsumes dg-ga/9612015. In particular, in the previous version only the 0th Novikov-Shubin number was considered, while here Novikov-Shubin numbers for all p are defined and studied.Comment: 43 pages, LaTex2

A Spectral Triple for a Solenoid Based on the Sierpinski Gasket

The Sierpinski gasket admits a locally isometric ramified self-covering. A semifinite spectral triple is constructed on the resulting solenoidal space, and its main geometrical features are discussed