Ideal-quasi-Cauchy sequences

An ideal $I$ is a family of subsets of positive integers $\textbf{N}$ which is closed under taking finite unions and subsets of its elements. A sequence $(x_n)$ of real numbers is said to be $I$-convergent to a real number $L$, if for each \;$\varepsilon> 0$ the set $\{n:|x_{n}-L|\geq \varepsilon\}$ belongs to $I$. We introduce $I$-ward compactness of a subset of $\textbf{R}$, the set of real numbers, and $I$-ward continuity of a real function in the senses that a subset $E$ of $\textbf{R}$ is $I$-ward compact if any sequence $(x_{n})$ of points in $E$ has an $I$-quasi-Cauchy subsequence, and a real function is $I$-ward continuous if it preserves $I$-quasi-Cauchy sequences where a sequence $(x_{n})$ is called to be $I$-quasi-Cauchy when $(\Delta x_{n})$ is $I$-convergent to 0. We obtain results related to $I$-ward continuity, $I$-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, $\delta$-ward continuity, and slowly oscillating continuity.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1005.494

K-homological finiteness and hyperbolic groups

Motivated by classical facts concerning closed manifolds, we introduce a strong finiteness property in K-homology. We say that a C*-algebra has uniformly summable K-homology if all its K-homology classes can be represented by Fredholm modules which are finitely summable over the same dense subalgebra, and with the same degree of summability. We show that two types of C*-algebras associated to hyperbolic groups - the C*-crossed product for the boundary action, and the reduced group C*-algebra - have uniformly summable K-homology. We provide explicit summability degrees, as well as explicit finitely summable representatives for the K-homology classes.Comment: v1: 34 pages, expands and supersedes our preprint `Finitely summable Fredholm modules for boundary actions of hyperbolic groups', arXiv:1208.0856; v2: final version, to appear in Crell

Compact κ\kappa-deformation and spectral triples

We construct discrete versions of $\kappa$-Minkowski space related to a certain compactness of the time coordinate. We show that these models fit into the framework of noncommutative geometry in the sense of spectral triples. The dynamical system of the underlying discrete groups (which include some Baumslag--Solitar groups) is heavily used in order to construct \emph{finitely summable} spectral triples. This allows to bypass an obstruction to finite-summability appearing when using the common regular representation. The dimension of these spectral triples is unrelated to the number of coordinates defining the $\kappa$-deformed Minkowski spaces.Comment: 30 page

Derivations and Dirichlet forms on fractals

We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not a tree (i.e. not simply connected). This result relates Fredholm modules and topology, and refines and improves known results on p.c.f. fractals. We also discuss weakly summable Fredholm modules and the Dixmier trace in the cases of some finitely and infinitely ramified fractals (including non-self-similar fractals) if the so-called spectral dimension is less than 2. In the finitely ramified self-similar case we relate the p-summability question with estimates of the Lyapunov exponents for harmonic functions and the behavior of the pressure function.Comment: to appear in the Journal of Functional Analysis 201

The Method of almost convergence with operator of the form fractional order and applications

The purpose of this paper is twofold. First, basic concepts such as Gamma function, almost convergence, fractional order difference operator and sequence spaces are given as a survey character. Thus, the current knowledge about those concepts are presented. Second, we construct the almost convergent spaces with fractional order difference operator and compute dual spaces which are help us in the characterization of matrix mappings. After we characterize to the matrix transformations, we give some examples.Comment: 20 pages, 4 table