Bases in banach spaces of smooth functions on cantor-type sets

Ankara : The Department of Mathematics and the Graduate School of Engineering and Science of Bilkent University, 2013.Thesis (Ph. D.) -- Bilkent Universty, 2013.Includes bibliographical references leaves 77-79.We construct Schauder bases in the spaces of continuous functions C p (K) and in the Whitney spaces E p (K) where K is a Cantor-type set. Here different Cantortype sets are considered. In the construction, local Taylor expansions of functions are used. Also we show that the Schauder basis which we constructed in the space Cp(K), is conditional.Özfidan, NecipPh.D

Prescribed Szlenk index of separable Banch spaces

In a previous work, the first named author described the set $\cal P$ of all values of the Szlenk indices of separable Banach spaces. We complete this result by showing that for any integer $n$ and any ordinal $\alpha$ in $\cal P$, there exists a separable Banach space $X$ such that the Szlenk of the dual of order $k$ of $X$ is equal to the first infinite ordinal $\omega$ for all $k$ in $\{0,..,n-1\}$ and equal to $\alpha$ for $k=n$. One of the ingredients is to show that the Lindenstrauss space and its dual both have a Szlenk index equal to $\omega$. We also show that any element of $\cal P$ can be realized as a Szlenk index of a reflexive Banach space with an unconditional basis.Comment: 17 pages. It is a revised version of the previous preprint "Prescribed Szlenk index of iterated duals": arXiv:1710.01638. The paper has been reorganized and the title has been changed. To appear in Studia Mat

On the complexity of the uniform homeomorphism relation between separable Banach spaces

We consider the problem of determining the complexity of the uniform homeomorphism relation between separable Banach spaces in the Borel reducibility hierarchy of analytic equivalence relations. We prove that the complete $K_{\sigma}$ equivalence relation is Borel reducible to the uniform homeomorphism relation, and we also determine the possible complexities of the relation when restricted to some small classes of Banach spaces. Moreover, we determine the exact complexity of the local equivalence relation between Banach spaces, namely that it is bireducible with $K_{\sigma}$. Finally, we construct a class of mutually uniformly homeomorphic Banach spaces such that the equality relation of countable sets of real numbers is Borel reducible to the isomorphism relation on the class

Compactifications of ω\omega and the Banach space c0c_0

We investigate for which compactifications $\gamma\omega$ of the discrete space of natural numbers $\omega$, the natural copy of the Banach space $c_0$ is complemented in $C(\gamma\omega)$. We show, in particular, that the separability of the remainder of $\gamma\omega$ is neither sufficient nor necessary for $c_0$ being complemented in $C(\gamma\omega)$ (for the latter our result is proved under the continuum hypothesis). We analyse, in this context, compactifications of $\omega$ related to embeddings of the measure algebra into $P(\omega)/fin$. We also prove that a Banach space $C(K)$ contains a rich family of complemented copies of $c_0$ whenever the compact space $K$ admits only measures of countable Maharam type.Comment: 20 pages, version of Jan 23, 201

Where the Buffalo Roam: Infinite Processes and Infinite Complexity

These informal notes, initially prepared a few years ago, look at various questions related to infinite processes in several parts of mathematics, with emphasis on examples.Comment: latex-2e. 110 pages with inde

A metric interpretation of reflexivity for Banach spaces

We define two metrics $d_{1,\alpha}$ and $d_{\infty,\alpha}$ on each Schreier family $\mathcal{S}_\alpha$, $\alpha<\omega_1$, with which we prove the following metric characterization of reflexivity of a Banach space $X$: $X$ is reflexive if and only if there is an $\alpha<\omega_1$, so that there is no mapping $\Phi:\mathcal{S}_\alpha\to X$ for which $$cd_{\infty,\alpha}(A,B)\le \|\Phi(A)-\Phi(B)\|\le C d_{1,\alpha}(A,B) \text{ for all $A,B\in\mathcal{S}_\alpha$.}$$ Secondly, we prove for separable and reflexive Banach spaces $X$, and certain countable ordinals $\alpha$ that $\max(\text{ Sz}(X),\text{ Sz}(X^*))\le \alpha$ if and only if $({\mathcal S}_\alpha, d_{1,\alpha})$ does not bi-Lipschitzly embed into $X$. Here $\text{Sz}(Y)$ denotes the Szlenk index of a Banach space $Y$

Norming Sets and Related Remez-type Inequalities

The classical Remez inequality bounds the maximum of the absolute value of a real polynomial $P$ of degree $d$ on $[-1,1]$ through the maximum of its absolute value on any subset $Z\subset [-1,1]$ of positive Lebesgue measure. Extensions to several variables and to certain sets of Lebesgue measure zero, massive in a much weaker sense, are available. Still, given a subset $Z\subset [-1,1]^n\subset {\mathbb R}^n$ it is not easy to determine whether it is ${\mathcal P}_d({\mathbb R}^n)$-norming (here ${\mathcal P}_d({\mathbb R}^n)$ is the space of real polynomials of degree at most $d$ on ${\mathbb R}^n$), i.e. satisfies a Remez-type inequality: $\sup_{[-1,1]^n}|P|\le C\sup_{Z}|P|$ for all $P\in {\mathcal P}_d({\mathbb R}^n)$ with $C$ independent of $P$. (Although ${\mathcal P}_d({\mathbb R}^n)$-norming sets are exactly those not contained in any algebraic hypersurface of degree $d$ in ${\mathbb R}^n$, there are many apparently unrelated reasons for $Z \subset [-1,1]^n$ to have this property.) In the present paper we study norming sets and related Remez-type inequalities in a general setting of finite-dimensional linear spaces $V$ of continuous functions on $[-1,1]^n$, remaining in most of the examples in the classical framework. First, we discuss some sufficient conditions for $Z$ to be $V$-norming, partly known, partly new, restricting ourselves to the simplest non-trivial examples. Next, we extend the Turan-Nazarov inequality for exponential polynomials to several variables, and on this base prove a new fewnomial Remez-type inequality. Finally, we study the family of optimal constants $N_{V}(Z)$ in the Remez-type inequalities for $V$, as the function of the set $Z$, showing that it is Lipschitz in the Hausdorff metric

Representations of \'etale groupoids on LpL^p-spaces

For $p\in (1,\infty)$, we study representations of \'etale groupoids on $L^{p}$-spaces. Our main result is a generalization of Renault's disintegration theorem for representations of \'etale groupoids on Hilbert spaces. We establish a correspondence between $L^{p}$-representations of an \'etale groupoid $G$, contractive $L^{p}$-representations of $C_{c}(G)$, and tight regular $L^{p}$-representations of any countable inverse semigroup of open slices of $G$ that is a basis for the topology of $G$. We define analogs $F^{p}(G)$ and $F_{\mathrm{red}}^{p}(G)$ of the full and reduced groupoid C*-algebras using representations on $L^{p}$-spaces. As a consequence of our main result, we deduce that every contractive representation of $F^{p}(G)$ or $F_{\mathrm{red}}^{p}(G)$ is automatically completely contractive. Examples of our construction include the following natural families of Banach algebras: discrete group $L^{p}$-operator algebras, the analogs of Cuntz algebras on $L^{p}$-spaces, and the analogs of AF-algebras on $L^{p}$-spaces. Our results yield new information about these objects: their matricially normed structure is uniquely determined. More generally, groupoid $L^{p}$-operator algebras provide analogs of several families of classical C*-algebras, such as Cuntz-Krieger C*-algebras, tiling C*-algebras, and graph C*-algebras.Comment: 33 pages. v2: minor changes. v3: more minor changes. To appear in Advances in Mat

Weil-Petersson geometry and determinant bundles on inductive limits of moduli spaces

In an earlier paper [Acta Mathematica, v. 176, 1996, 145-169, alg-geom/9505024 ] the present authors and Dennis Sullivan constructed the universal direct system of the classical Teichm\"uller spaces of Riemann surfaces of varying genus. The direct limit, which we called the universal commensurability Teichm\"uller space, $T_{infty}$, was shown to carry on it a natural action of the universal commensurability mapping class group, $MC_{\infty}$. In this paper we identify an interesting cofinal sub-system corresponding to the tower of finite-sheeted characteristic coverings over any fixed base surface. Utilizing a certain subgroup inside $MC_{\infty}$, (associated intimately to the inverse system of characteristic coverings), we can now descend to an inductive system of moduli spaces, and construct the direct limit ind-variety $M_{\infty}$. Invoking curvature properties of Quillen metrics on determinant bundles, and naturality under finite coverings of Weil-Petersson forms, we are able to construct on $M_{\infty}$ the natural sequence of determinant of cohomology line bundles, as well as the Mumford isomorphisms connecting these.Comment: 26 pages, Latex. To appear in Contemporary Math. Series (American Math. Soc.), ``Proceedings of the first Bers Colloquium'', New York, 1995; eds. J.Dodziuk, L.Kee

On the number of permutatively inequivalent basic sequences in a Banach space

AbstractLet X be a Banach space with a Schauder basis (en)n∈N. The relation E0 is Borel reducible to permutative equivalence between normalized block-sequences of (en)n∈N or X is c0 or ℓp saturated for some 1⩽p<+∞. If (en)n∈N is shrinking unconditional then either it is equivalent to the canonical basis of c0 or ℓp, 1<p<+∞, or the relation E0 is Borel reducible to permutative equivalence between sequences of normalized disjoint blocks of X or of X∗. If (en)n∈N is unconditional, then either X is isomorphic to ℓ2, or X contains 2ω subspaces or 2ω quotients which are spanned by pairwise permutatively inequivalent normalized unconditional bases