1,629 research outputs found
A remark on trace properties of K-cycles
In this paper we discuss trace properties of -summable -cycles
considered by A.Connes in [\rfr(Conn4)]. More precisely we give a proof of a
trace theorem on the algebra \A of a --cycle stated in [\rfr(Conn4)],
namely we show that a natural functional on \A is a trace functional. Then we
discuss whether this functional gives a trace on the whole universal graded
differential algebra \Q(\A). On the one hand we prove that the regularity
conditions on -cycles considered in [\rfr(Conn4)] imply the trace property
on \Q(\A). On the other hand, by constructing an explicit counterexample, we
remark that the sole -cycle assumption is not sufficient for such a property
to hold.Comment: 11 pages, plain Te
Spectral triples for the Sierpinski Gasket
We construct a family of spectral triples for the Sierpinski Gasket . For
suitable values of the parameters, we determine the dimensional spectrum and
recover the Hausdorff measure of in terms of the residue of the volume
functional tr at its abscissa of convergence , which
coincides with the Hausdorff dimension of the fractal. We determine the
associated Connes' distance showing that it is bi-Lipschitz equivalent to the
distance on induced by the Euclidean metric of the plane, and show that the
pairing of the associated Fredholm module with (odd) -theory is non-trivial.
When the parameters belong to a suitable range, the abscissa of convergence
of the energy functional
tr takes the value
, which we call energy dimension, and the
corresponding residue gives the standard Dirichlet form on .Comment: 48 pages, 9 figures. Final version, to appear in J.Funct.Ana
Integrals and Potentials of Differential 1-forms on the Sierpinski Gasket
We provide a definition of integral, along paths in the Sierpinski gasket K,
for differential smooth 1-forms associated to the standard Dirichlet form K. We
show how this tool can be used to study the potential theory on K. In
particular, we prove: i) a de Rham reconstruction of a 1-form from its periods
around lacunas in K; ii) a Hodge decomposition of 1-forms with respect to the
Hilbertian energy norm; iii) the existence of potentials of smooth 1-forms on a
suitable covering space of K. We finally show that this framework provides
versions of the de Rham duality theorem for the fractal K.Comment: Some proofs have been clarified, reference to previous literature is
now more accurate, 33 pages, 6 figure
A C*-algebra of geometric operators on self-similar CW-complexes. Novikov-Shubin and L^2-Betti numbers
A class of CW-complexes, called self-similar complexes, is introduced,
together with C*-algebras A_j of operators, endowed with a finite trace, acting
on square-summable cellular j-chains. Since the Laplacian Delta_j belongs to
A_j, L^2-Betti numbers and Novikov-Shubin numbers are defined for such
complexes in terms of the trace. In particular a relation involving the
Euler-Poincare' characteristic is proved. L^2-Betti and Novikov-Shubin numbers
are computed for some self-similar complexes arising from self-similar
fractals.Comment: 30 pages, 7 figure
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
A noncommutative Sierpinski gasket
A quantized version of the Sierpinski gasket is proposed, on purely topological grounds, as a C*-algebra A infinity with a suitable form of self-similarity. Several properties of A infinity are studied, in particular its nuclearity, the structure of ideals as well as the description of irreducible representations and extremal traces. A harmonic structure is introduced, giving rise to a self-similar Dirichlet form E. A spectral triple is also constructed, extending the one already known for the classical gasket, from which E can be reconstructed. Moreover we show that A infinity is a compact quantum metric space
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