On topological spaces of orders

We study the space of linear orders on a given set X, denoted by Op(X), endowed with the topology of pointwise convergence. We show, in particular, that if |X|=ω₁ or |X|=ω₀ then Op(X) is homeomorphic to ω₁² and ω₀², respectively.     

Transfinite Hausdorff dimension

Making extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or −1 or Ω) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorphisms), in fact like Hausdorff dimension, it does not increase under Lipschitz maps, and it also satisfies the intermediate dimension property (Theorem 2.7). The primary goal of transfinite Hausdorff dimension is to classify metric spaces with infinite Hausdorff dimension. Indeed, if tHD(X)?ω₀, then HD(X)=+∞. We prove that tHD(X)?ω₁ for every separable metric space X, and, as our main theorem, we show that for every ordinal number α<ω₁ there exists a compact metric space Xα (a subspace of the Hilbert space l₂) with tHD(Xα)=α and which is a topological Cantor set, thus of topological dimension 0. In our proof we develop a metric version of Smirnov topological spaces and we establish several properties of transfinite Hausdorff dimension, including its relations with classical Hausdorff dimension.     

Left inverses of matrices with polynomial decay

It is known that the algebra of Schur operators on ?² (namely operators bounded on both ?¹ and ?^∞) is not inverse-closed. When ?²=?²(X) where X is a metric space, one can consider elements of the Schur algebra with certain decay at infinity. For instance if X has the doubling property, then Q. Sun has proved that the weighted Schur algebra Aω(X) for a strictly polynomial weight ω is inverse-closed. In this paper, we prove a sharp result on left-invertibility of the these operators. Namely, if an operator A∈Aω(X) satisfies ‖Afp_‖?‖fp_‖, for some 1?p?∞, then it admits a left-inverse in Aω(X). The main difficulty here is to obtain the above inequality in ?². The author was both motivated and inspired by a previous work of Aldroubi, Baskarov and Krishtal (2008) [1], where similar results were obtained through different methods for X=Zd, under additional conditions on the decay.     

On the maximal resolvability of monotonically normal spaces

We continue the work started in [6] and show that all monotonically normal (in short: MN) spaces are maximally resolvable if and only if all uniform ultrafilters are maximally decomposable. As a consequence we get that the existence of an MN space which is not maximally resolvable is equi-consistent with the existence of a measurable cardinal. We also show that it is consistent (modulo the consistency of a measurable cardinal) that there is an MN space X with |X| = Δ(X) = ? _ω which is not ω ₁-resolvable. It follows from the results of [6] that this is best possible.     

When is a Pixley-Roy hyperspace CCC?

A space X is said to satisfy condition (C) if for every Y?X with |Y|=ω₁, any family $\text{G}$ of open subsets of Y with |$\text{G}$|=ω₁ has a countable network. It is easy to see that if X satisfies condition (C), then its Pixley-Roy hyperspace $\text{F}$[X] is CCC. We show that under MA_ω1 condition (C) is also necessary for $\text{F}$[X] to be CCC, but under CH it is not.     

Minimal hyperspace actions of homeomorphism groups of h-homogeneous spaces

Suppose that X is an h-homogeneous zero-dimensional compact Hausdorff space, i.e., X is a Stone dual of a homogeneous Boolean algebra. Using the dual Ramsey theorem and a detailed combinatorial analysis of what we call stable collections of subsets of a finite set, we obtain a complete list of the minimal sub-systems of the compact dynamical system (Exp(Exp(X)), Homeo(X)), where Exp(X) denotes the hyperspace comprising the closed subsets of X equipped with the Vietoris topology. The importance of this dynamical system stems from Uspenskij’s characterization of the universal ambit of G = Homeo(X). The results apply to the Cantor set, the generalized Cantor sets X = {0,1} ^κ for noncountable cardinals κ, and to several other spaces. A particular interesting case is X = ω* = βω \ ω, where βω denotes the Stone- ?ech compactification of the natural numbers. This space, called the corona or the remainder of ω, has been extensively studied in the fields of set theory and topology.     

Hyperspace of convex compacta of nonmetrizable compact convex subspaces of locally convex spaces

Our main result states that the hyperspace of convex compact subsets of a compact convex subset X in a locally convex space is an absolute retract if and only if X is an absolute retract of weight ?ω₁. It is also proved that the hyperspace of convex compact subsets of the Tychonov cube Iω₁ is homeomorphic to Iω₁. An analogous result is also proved for the cone over Iω₁. Our proofs are based on analysis of maps of hyperspaces of compact convex subsets, in particular, selection theorems for such maps are proved.     

Operator spaces with prescribed sets of completely bounded maps

Suppose A is a dual Banach algebra, and a representation π:A→B(?₂) is unital, weak* continuous, and contractive. We use a “Hilbert-Schmidt version” of Arveson distance formula to construct an operator space X, isometric to ?₂⊗?₂, such that the space of completely bounded maps on X consists of Hilbert-Schmidt perturbations of π(A)⊗I_?2. This allows us to establish the existence of operator spaces with various interesting properties. For instance, we construct an operator space X for which the group K₁(CB(X)) contains Z₂ as a subgroup, and a completely indecomposable operator space containing an infinite dimensional homogeneous Hilbertian subspace.     

A dual form of Ramsey's Theorem

Let k?ω, where ? is the set of all natural numbers. Ramsey's Theorem deals with colorings of the k-element subsets of ω. Our dual form deals with colorings of the k-element partitions of ω. Let (ω)^k (respectively (ω)^ω) be the set of all partitions of ω having exactly k (respectively infinitely many) blocks. Given X? (ω)^ω let (X)^k be the set of all Y? (ω)^k such that Y is coarser than X. Dual Ramsey Theorem. If (ω)^k = C₀ ∪ … ∪ C_t?1 where each C_i is Borel then there exists X? (ω)^ω such that (X)^k ? C_i for some i <l. Dual Galvin-Prikry Theorem. Same as before with k replaced by ω. We also obtain dual forms of theorems of Ellentuck and Mathias. Our results also provide an infinitary generalization of the Graham-Rothschild “parameter set” theorem [Trans. Amer. Math. Soc.159 (1971), 257–292] and a new proof of the Halpern-Läuchli Theorem [Trans. Amer. Math. Soc.124 (1966), 360–367].     

Asymptotic centers and best compact approximation of operators intoC(K)

The existence of best compact approximations for all bounded linear operators fromX intoC(K) is related to the behavior of asymptotic centers inX ^*. IfK is just one convergent sequence, the condition is that everyω ^*-convergent sequence inX ^* will have an asymptotic center. We first study this property, solving some open problems in the theory of asymptotic centers. IfK is more “complex,” the asymptotic centers should behave “continuously.” We use this observation to construct operators fromC[0,1] intoC(ω ²) and from ?₁ intoL ₁ without best compact approximation. We also construct spacesX ₁,X ₂, isomorphic to a Hilbert space, and operatorsT ₁,∶X ₁→C(ω ²),T ₂∶?₁→X ₂ without best compact approximations.     

A limit theorem for continuous selectors

Any (measurable) function K from Rⁿ to R defines an operator K acting on random variables X by K(X) = K(X₁,..., X_n), where the X_j are independent copies of X. The main result of this paper concerns continuous selectors H, continuous functions defined in Rn and such that H(x₁, x₂,..., x_n) ∈ {x₁, x₂,..., x_n}. For each such continuous selector H (except for projections onto a single coordinate) there is a unique point ω_H in the interval (0, 1) so that, for any random variable X, the iterates H^(N) acting on X converge in distribution as N → ∞ to the ωH-quantile of X.     

Similarity invariants of operators on a class of Gowers-Maurey spaces

This paper studies the similarity invariants of operators on a class of Gowers-Maurey spaces, Σ _dc spaces, where an infinite dimensional Banach space X is called a Σ _dc space if for every bounded linear operator on X the spectrum is disconnected unless it is a singleton. It shows that two strongly irreducible operators T ₁ and T ₂ on a Σ _dc space are similar if and only if the K ₀-group of the commutant algebra of the direct sum T ₁⊕T ₂ is isomorphic to the group of integers ?. On a Σ _dc space X, it uses the semigroups of the commutant algebras of operators to give a condition that an operator is similar to some operator in (ΣSI)(X), it further gives a necessary and sufficient condition that two operators in (ΣSI)(X) are similar by using the ordered K ₀-groups. It also proves that every operator in (ΣSI)(X) has a unique (SI) decomposition up to similarity on a Σ _dc space X, where (ΣSI)(X) denotes the class of operators which can be written as a direct sum of finitely many strongly irreducible operators.     

On box products

We prove two theorems about box products. The first theorem says that the box product of countable spaces is pseudonormal, i.e. any two disjoint closed sets one of which is countable can be separated by open sets. The second theorem says that assuming CH a certain uncountable box product is normal (i.e. <_ω1?□_α<ω1X_α where each Xα is a compact metric space).     

Spaces which contain a copy of Sω or S2 and their applications

Let S_ω and S₂ denote the sequential fan and Arens' space, respectively. In this paper, we shall prove the following main results. (1) If Π^∞_i=1 X_i contains a copy of S_ω (S₂), then some Πⁿ_i=1 X_i contains a copy of S_ω (S_ω or S₂, respectively). (2) Let f: X → Y be a closed map such that any f^-1(y) contains no closed copy of S_ω (S₂). If X contains a closed copy of S_ω (S₂), then Y contains a closed copy of S_ω (S_ω or S₂, respectively).As applications of (1) and (2), we shall consider the Fréchet or strongly Fréchetness, or sequentiality of products of finitely or countably many spaces.     

Banach spaces in various positions

We formulate a general theory of positions for subspaces of a Banach space: we define equivalent and isomorphic positions, study the automorphy index a(Y,X) that measures how many non-equivalent positions Y admits in X, and obtain estimates of a(Y,X) for X a classical Banach space such as ?p,Lp,L₁,C(ωω) or C[0,1]. Then, we study different aspects of the automorphic space problem posed by Lindenstrauss and Rosenthal; namely, does there exist a separable automorphic space different from c₀ or ?₂? Recall that a Banach space X is said to be automorphic if every subspace Y admits only one position in X; i.e., a(Y,X)=1 for every subspace Y of X. We study the notion of extensible space and uniformly finitely extensible space (UFO), which are relevant since every automorphic space is extensible and every extensible space is UFO. We obtain a dichotomy theorem: Every UFO must be either an L_∞-space or a weak type 2 near-Hilbert space with the Maurey projection property. We show that a Banach space all of whose subspaces are UFO (called hereditarily UFO spaces) must be asymptotically Hilbertian; while a Banach space for which both X and X^∗ are UFO must be weak Hilbert. We then refine the dichotomy theorem for Banach spaces with some additional structure. In particular, we show that an UFO with unconditional basis must be either c₀ or a superreflexive weak type 2 space; that a hereditarily UFO Köthe function space must be Hilbert; and that a rearrangement invariant space UFO must be either L_∞ or a superreflexive type 2 Banach lattice.     

Generalizing D-graphs

Let ω₀(G) denote the number of odd components of a graph G. The deficiency of G is defined as def(G)=max_X⊆V(G)(ω₀(G-X)-|X|), and this equals the number of vertices unmatched by any maximum matching of G. A subset X⊆V(G) is called a Tutte set (or barrier set) of G if def(G)=ω₀(G-X)-|X|, and an extreme set if def(G-X)=def(G)+|X|. Recently a graph operator, called the D-graph D(G), was defined that has proven very useful in examining Tutte sets and extreme sets of graphs which contain a perfect matching. In this paper we give two natural and related generalizations of the D-graph operator to all simple graphs, both of which have analogues for many of the interesting and useful properties of the original.     

On the square root of a positive B(X,X1)-valued function

Let (Ω, $\text{B}$, μ) be a measure space, $\text{X}$ a separable Banach space, and $\text{X}$¹ the space of all bounded conjugate linear functionals on $\text{X}$. Let f be a weak¹ summable positive B($\text{X, X}$¹)-valued function defined on Ω. The existence of a separable Hilbert space $\text{K}$, a weakly measurable B($\text{X, K}$)-valued function Q satisfying the relation $\text{Q}{}^1\text{(ω)Q(ω) = f(ω)}$ is proved. This result is used to define the Hilbert space L_2,f of square integrable operator-valued functions with respect to f. It is shown that for B⁺($\text{X, X}$¹)-valued measures, the concepts of weak¹, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained.     

Formal loops III: Additive functions and the Radon transform

To any algebraic variety X and closed 2-form ω on X, we associate the “symplectic action functional” T(ω) which is a function on the formal loop space LX introduced by the authors earlier. The correspondence ω→T(ω) can be seen as a version of the Radon transform. We give a characterization of the functions of the form T(ω) in terms of factorizability (infinitesimal analog of additivity in holomorphic pairs of pants) as well as in terms of vertex operator algebras.These results will be used in the subsequent paper which will relate the gerbe of chiral differential operators on X (whose lien is the sheaf of closed 2-forms) and the determinantal gerbe of the tangent bundle of LX (whose lien is the sheaf of invertible functions on LX). On the level of liens this relation associates to a closed 2-form ω the invertible function expT(ω).