A remark on trace properties of K-cycles

In this paper we discuss trace properties of $d^+$-summable $K$-cycles considered by A.Connes in [\rfr(Conn4)]. More precisely we give a proof of a trace theorem on the algebra \A of a $K$--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 $K$-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 $K$-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 $K$. For suitable values of the parameters, we determine the dimensional spectrum and recover the Hausdorff measure of $K$ in terms of the residue of the volume functional $a\to$ tr$(a\,|D|^{-s})$ at its abscissa of convergence $d_D$, which coincides with the Hausdorff dimension $d_H$ of the fractal. We determine the associated Connes' distance showing that it is bi-Lipschitz equivalent to the distance on $K$ induced by the Euclidean metric of the plane, and show that the pairing of the associated Fredholm module with (odd) $K$-theory is non-trivial. When the parameters belong to a suitable range, the abscissa of convergence $\delta_D$ of the energy functional $a\to$ tr$(|D|^{-s/2}|[D,a]|^2\,|D|^{-s/2})$ takes the value $d_E=\frac{\log(12/5)}{\log 2}$, which we call energy dimension, and the corresponding residue gives the standard Dirichlet form on $K$.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