Analytic infinite gaps

We completely determine the lq and C(K) spaces which are isomorphic to a subspace of lp⊗ˆπC(α), the projective tensor product of the classical lp space, 1≤p<∞, and the space C(α) of all scalar valued continuous functions defined on the interval of ordinal numbers [1,α], α<ω1. In order to do this, we extend a result of A. Tong concerning diagonal block matrices representing operators from lp to l1, 1≤p<∞. The first main theorem is an extension of a result of E. Oja and states that the only lq space which is isomorphic to a subspace of lp⊗ˆπC(α) with 1≤p≤q<∞ and ω≤α<ω1 is lp. The second main theorem concerning C(K) spaces improves a result of Bessaga and Pelczynski which allows us to classify, up to isomorphism, the separable spaces (X,Y) of nuclear operators, where X and Y are direct sums of lp and C(K) spaces. More precisely, we prove the following cancellation law for separable Banach spaces. Suppose that K1 and K3 are finite or countable compact metric spaces of the same cardinality and $1 (a) (lp⊕C(K1),lq⊕C(K2)) and (lp⊕C(K3),lq⊕C(K4)) are isomorphic.

In this paper we assume that all spaces are Tychonoff. For a space X, dim X denotes the Cech-Lebesgue dimension of X (see [3]). J. de Groot proved that a separable metrizable space X has a metrizable compactification aX with dim (aX^ X) <0 iff X is rim-compact (see [4]). A space X is rim-compact if each point of X has arbitrarily small neighborhoods with compact boundary. Modified the concept of rim-compactness, he defined the compactness degree of a space X, cmp X, inductively,as follows. A space X satisfies cmp X= ―1 iffX is compact. If n is a non-negative integer, then cmp X<n means that each point of X has arbitrarily small neighborhoods U with cmp Bd U<n ―1. We put cmp X=n if cmp X<n and cmp X^n ―l.If there is no integer n for which cmp X<n, then we put cmp X= oo. By the compactness deficiency of a Tychonoff space (resp. a separable metrizable space) Xwe mean the least integer n such that X has a compactification (resp. a metrizable compactification) aXwith dim (otXs≫X)=n. We denote this integer by def* X (resp. def X). We allow n to be oo. Thus, with this terminology, J. de Groot's result above asserts that cmp X<0 iffdef X <0 for every separable metrizable space X. The general problem whether cmp X<n iff def X< n for arbitrary separable metrizable space X has been known as J. de Groot's conjecture, and was unsolved for a long time. However, in 1982 R. Pol [7] constructed a separable metrizable space X such that cmp X=l and def X=2. In the class of separable metrizable spaces, another example -X with the property that cmp X^def X seems to be stillunknown but Pol's example above. On the other hand, in the class of TychonofF spaces, M. G. Charalambous [1] has already constructed a space X such that cmp X= 0 and def* X= n for each m = 1,2,・・・,<≫. J. van Mill [6] has constructed a Lindelof space X such that cmpX=l and def*Z=oo. In this paper we construct a countably compact space X such that cmpJ=m and

Extending the class of known Stone–Čech remainders for ψ-spaces

Strongly sequentially separable function spaces, via selection principles

In this dissertation, we show that the Central Limit Theorem and the Invariance Principle for Discrete Fourier Transforms discovered by Peligrad and Wu can be extended to the quenched setting. We show that the random normalization introduced to extend these results is necessary and we discuss its meaning. We also show the validity of the quenched Invariance Principle for fixed frequencies under some conditions of weak dependence. In particular, we show that this result holds in the martingale case. The discussion needed for the proofs allows us to show some general facts apparently not noticed before in the theory of convergence in distribution. In particular, we show that in the case of separable metric spaces the set of test functions in the Portmanteau theorem can be reduced to a countable one, which implies that the notion of quenched convergence, given in terms of convergence a.s. of conditional expectations, specializes in the right way in the regular case when the state space is metrizable and second-countable. We also collect and organize several disperse facts from the existing theory in a consistent manner towards the statistical spectral analysis of the Discrete Fourier Transforms, providing a comprehensive introduction to topics in this theory that apparently have not been systematically addressed in a self-contained way by previous references.

Ernest A. Michael has given a characterization of the regular quotient images of separable metric spaces. His result is generalized here to a characterization of the T 1 {T_1} quotient images of separable metric spaces (which are the same as the T 1 {T_1} quotient images of second countable spaces). This result is then used to characterize the Hausdorff pseudo-open images of separable metric spaces.

One of the standard goals of in nite-dimensional topology is the topological classi cation of linear metric spaces and their convex subsets. The origins of this problem go back to the questions of Fr echet [Fr] and Banach [Ba] whether all in nite-dimensional separable Fr echet (Banach) spaces are homeomorphic (we explain our terminology at the end of the introduction). These questions have been a rmatively answered by the combined results of Anderson and Kadec (see [BP]) which may be stated as follows: every separable Fr echet space is homeomorphic to a Hilbert space. This result was generalized by Toru nczyk [To] for the nonseparable case. Since then the main interest in the eld has turned to topological classi cation of incomplete or nonlocally convex linear metric spaces. Quite recently, Cauty [Ca2] has constructed an example of a separable linear complete metric space X which is not an AR. In particular, X is not homeomorphic to any convex subset of a locally convex linear metric space. The topological classi cation of incomplete locally convex metric linear spaces, or even incomplete pre-Hilbert spaces, is much more complex than for the complete ones. First, there exist pre-Hilbert spaces of an arbitrary Borel class. Moreover, the Borel type invariant does not classify such spaces, and in the class of -compact spaces there exist uncountably many pairwise nonhomeomorphic such spaces [BP], [Ca1]. Until recently, there was hope that, in spite of these di culties, the classi cation problem could be restricted to subspaces of Hilbert spaces. The result of Bessaga and Dobrowolski [BD] stating that every locally convex linear -compact metric space is homeomorphic to a pre-Hilbert space looked like a promising rst step. The main objective of this paper is to show that, in general, it is not possible to restrict ones attention to subspaces of Hilbert spaces only. We show that the classes of pre-Hilbert,

Dyadicity index and metrizability of compact continuous images of function spaces

The space of continuous maps from a topological spaceX to topological spaceY is denoted byC(X,Y) with the compact-open topology. In this paper we prove thatC(X,Y) is an absolute retract ifX is a locally compact separable metric space andY a convex set in a Banach space. From the above fact we know thatC(X,Y) is homomorphic to Hilbert spacel2 ifX is a locally compact separable metric space andY a separable Banach space; in particular,C(Rn,Rm) is homomorphic to Hilbert spacel2.

This thesis consists of an introduction and four independent chapters. In Chapter 1, we study homeomorphism groups of metrizable compactifications of the natural numbers. Those groups can be represented as almost zero-dimensional Polishable subgroups of the group S∞. We show that all Polish groups are continuous homomorphic images of almost zero-dimensional Polishable subgroups of S∞. We also find a sufficient condition for these groups to be one dimensional. In Chapter 2, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding shift action of Γ on MX, where M is a measure space. In particular, we show that the action of Γ on X is amenable iff the corresponding shift has almost invariant sets. This is joint work with Alexander Kechris. In Chapter 3, we prove that if the Koopman representation associated to a measure-preserving action of a countable group on a standard non-atomic probability space is non-amenable, then there does not exist a countable-to-one Borel homomorphism from its orbit equivalence relation to the orbit equivalence relation of any modular action (i.e., an action on the boundary of a countably splitting tree), generalizing previous results of Hjorth and Kechris. As an application, for certain groups, we connect antimodularity to mixing conditions. This is joint work with Inessa Epstein. In Chapter 4, we study full groups of countable, measure-preserving equivalence relations. Our main results include that they are all homeomorphic to the separable Hilbert space and that every homomorphism from an ergodic full group to a separable group is continuous. We also find bounds for the minimal number of generators of a dense subgroup of full groups allowing us to distinguish full groups of equivalence relations generated by free, ergodic actions of the free groups Fn and Fm if m and n are sufficiently far apart. We also show that an ergodic equivalence relation is generated by an action of a finitely generated group iff its full group has a finitely generated dense subgroup. This is joint work with John Kittrell.

We show that there are locally compact spaces that can be condensed on separable spaces, but not on compact separable spaces. We also show that for every cardinal $\kappa ,$ there is a locally compact topological group of cardinality $2^\kappa $ that can be condensed on a compact space but not on a compact topological group. These answer some questions of Arhangel’skii and Buzyakova.

Compact spaces of diversity two

In this paper we study linear into isometries of non-reflexive spaces (embeddings) that preserve finite dimensional structure of the range space. We consider this for various aspects of the finite dimensional structure, covering the recent notion of an almost isometric ideals introduced by Abrahamsen et.al., the well studied notions of a M M -ideal and that of an ideal. We show that if a separable non-reflexive Banach space X X , in all embeddings into its bidual X ∗ ∗ X^{\ast \ast } , is an almost isometric ideal and if X ∗ X^\ast is isometric to L 1 ( μ ) L^1(\mu ) , for some positive measure μ \mu , then X X is the Gurariy space. For a fixed infinite compact Hausdorff space K K , if every embedding of a separable space X X into C ( K ) C(K) is an almost isometric ideal and X ∗ X^\ast is a non-separable space, then again X X is the Gurariy space. We show that if a separable Banach space contains an isometric copy of c 0 c_0 and if it is a M M -ideal in its bidual in the canonical embedding, then there is another embedding of the space in its bidual, in which it is not a M M -ideal.

The main result of the first part of the paper is a generalization of the classical result of Menger-Urysohn : $\dim (A\cup B)\le \dim A+\dim B+1$. Theorem. Suppose $A,B$ are subsets of a metrizable space and $K$ and $L$ are CW complexes. If $K$ is an absolute extensor for $A$ and $L$ is an absolute extensor for $B$, then the join $K*L$ is an absolute extensor for $A\cup B$. As an application we prove the following analogue of the Menger-Urysohn Theorem for cohomological dimension: Theorem. Suppose $A$, $B$ are subsets of a metrizable space. Then \begin{equation*}\dim _{\mathbf {R} }(A\cup B)\le \dim _{\mathbf {R} }A+\dim _{\mathbf {R} }B+1 \end{equation*} for any ring $\mathbf {R}$ with unity and \begin{equation*}\dim _{G}(A\cup B)\le \dim _{G}A+\dim _{G}B+2\end{equation*} for any abelian group $G$. The second part of the paper is devoted to the question of existence of universal spaces: Suppose $\{K_{i}\}_{i\ge 1}$ is a sequence of CW complexes homotopy dominated by finite CW complexes. Then [a.] Given a separable, metrizable space $Y$ such that $K_{i}\in AE(Y)$, $i\ge 1$, there exists a metrizable compactification $c(Y)$ of $Y$ such that $K_{i}\in AE(c(Y))$, $i\ge 1$. [b.] There is a universal space of the class of all compact metrizable spaces $Y$ such that $K_{i}\in AE(Y)$ for all $i\ge 1$. [c.] There is a completely metrizable and separable space $Z$ such that $K_{i}\in AE(Z)$ for all $i\ge 1$ with the property that any completely metrizable and separable space $Z’$ with $K_{i}\in AE(Z’)$ for all $i\ge 1$ embeds in $Z$ as a closed subset.

We owe to E. H. Moore the introduction of a very general type of double sequence { Pi} which he called a development. With such sequences there may be associated certain abstract spaces investigated at length by Chittenden and Pitcher with noteworthy results particularly in connection with the problem of metrization. They dealt at considerable length with so-called regular developments of a space 9S. In a regular development the P's are neighborhoods of 9R and a has a finite range for every i. Our first object in the present paper is to investigate a type of development called normal whose sets are subjected to more stringent conditions of convergency than with the regular type. It turns out, however, that every separable metric space possesses normal developments. As a consequence they seem to be just what is needed for the treatment of many questions on separable metric. Making use in part of normal developments, we have put on a solid basis the theory of the order for separable spaces, and in particular, completely extended to them the fundamental order theorem (Lebesgue order theorem). It was then a simple matter to prove that every separable metric space can be mapped topologically on a compact metric space of the same dimension. As a consequence certain mapping theorems that we have obtained previously for compact spaces2 hold for separable spaces. By means of