86 research outputs found

### 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

### 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

### 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

### 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

### 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

### 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

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