On the Exel Crossed Product of Topological Covering Maps

For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product C ^*-algebras C(X)⋊ _α,ℒℕ introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical embedding of C(X) into C(X)⋊ _α,ℒℕ is a maximal abelian C ^*-subalgebra of C(X)⋊ _α,ℒℕ; any nontrivial two sided ideal of C(X)⋊ _α,ℒℕ has non-zero intersection with the embedded copy of C(X); a certain natural representation of C(X)⋊ _α,ℒℕ is faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product C ^*-algebras of homeomorphism dynamical systems.     

Web‐compact spaces,Fréchet‐Urysohn groups and a Suslin closed graph theorem

A topological space X is strongly web‐compact if X admits a family {A_α: α ∈ ?^?} of relatively countably compact sets covering X and such that A_α ? A_β for α ≤ β. The main result of this paper states the following: Theorem A Let X and Y be topological groups and f a homomorphism between X and Y with closed graph. If X is Fréchet‐Urysohn and Baire and Y is strongly web‐compact, then f is continuous. This extends a result of Valdivia. We provide an example showing that the property of being strongly web‐compact is not productive. This applies to show that there are quasi‐Suslin spaces X whose product X × X is not quasi‐Suslin (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem in complete random normed modules to stratification structure

Let (Ω,A,μ) be a probability space, K the scalar field R of real numbers or C of complex numbers,and (S,X) a random normed space over K with base (ω,A,μ). Denote the support of (S,X) by E, namely E is the essential supremum of the set {A ∈ A: there exists an element p in S such that X _p (ω) > 0 for almost all ω in A}. In this paper, Banach-Alaoglu theorem in a random normed space is first established as follows: The random closed unit ball S ^*(1) = {f ∈ S ^*: X ^* _f ⩽ 1} of the random conjugate space (S ^*,X ^*) of (S,X) is compact under the random weak star topology on (S ^*,X ^*) iff E∩A=: {E∩A | A ∈ A} is essentially purely μ-atomic (namely, there exists a disjoint family {A _n : n ∈ N} of at most countably many μ-atoms from E ∩ A such that E = ∪ _n=1^∞ A _n and for each element F in E ∩ A, there is an H in the σ-algebra generated by {A _n : n ∈ N} satisfying μ(FΔH) = 0), whose proof forces us to provide a key topological skill, and thus is much more involved than the corresponding classical case. Further, Banach-Bourbaki-Kakutani-Šmulian (briefly, BBKS) theorem in a complete random normed module is established as follows: If (S,X) is a complete random normed module, then the random closed unit ball S(1) = {p ∈ S: X _p ⩽ 1} of (S,X) is compact under the random weak topology on (S,X) iff both (S,X) is random reflexive and E ∩ A is essentially purely μ-atomic. Our recent work shows that the famous classical James theorem still holds for an arbitrary complete random normed module, namely a complete random normed module is random reflexive iff the random norm of an arbitrary almost surely bounded random linear functional on it is attainable on its random closed unit ball, but this paper shows that the classical Banach-Alaoglu theorem and BBKS theorem do not hold universally for complete random normed modules unless they possess extremely simple stratification structure, namely their supports are essentially purely μ-atomic. Combining the James theorem and BBKS theorem in complete random normed modules leads directly to an interesting phenomenum: there exist many famous classical propositions that are mutually equivalent in the case of Banach spaces, some of which remain to be mutually equivalent in the context of arbitrary complete random normed modules, whereas the other of which are no longer equivalent to another in the context of arbitrary complete random normed modules unless the random normed modules in question possess extremely simple stratification structure. Such a phenomenum is, for the first time, discovered in the course of the development of random metric theory.     

Multivalued perturbations ofm-accretive differential inclusions

Given anm-accretive operatorA in a Banach spaceX and an upper semicontinuous multivalued mapF: [0,a]×X→2 ^X , we consider the initial value problemu′∈−Au+F(t,u) on [0,a],u(0)=x ₀. We concentrate on the case when the semigroup generated by—A is only equicontinuous and obtain existence of integral solutions if, in particular,X* is uniformly convex andF satisfies β(F(t,B))≤k(t)β(B) for all boundedB⊂X wherek∈L ¹([0,a]) and β denotes the Hausdorff-measure of noncompactness. Moreover, we show that the set of all solutions is a compactR _δ-set in this situation. In general, the extra condition onX* is essential as we show by an example in whichX is not uniformly smooth and the set of all solutions is not compact, but it can be omited ifA is single-valued and continuous or—A generates aC _o-semigroup of bounded linear operators. In the simpler case when—A generates a compact semigroup, we give a short proof of existence of solutions, again ifX* is uniformly (or strictly) convex. In this situation we also provide a counter-example in ℝ⁴ in which no integral solution exists. The author gratefully acknowledges financial support by DAAD within the scope of the French-German project PROCOPE.     

Ergodicity of cylinder flows arising from irregularities of distribution

LetT be the mod 1 circle group, α∈T be irrational and 0<β<1. LetE be the closed subgroup ofR generated by β and 1. DefineX=T×E andT:X→X byT(x, t)=(x+α,t+1 _[0,β] (x)−β). Then we have the theorem:T is ergodic if and only if β is rational or 1, α and β are linearly independent over the rationals. This paper was prepared while I was very graciously hosted by the Centro de Investigacion y Estudios Avanzados, Mexico City.     

Entire functions having a concordant value sequence

A classic theorem of Pólya shows that 2 ^z is, in a strong sense, the “smallest” transcendental entire function that is integer valued on ℕ. An analogous result of Gel’fond concerns entire functions that are integer valued on the setX _a={a ⁿ:n ∈ ℕ}, wherea ∈ ℕ,|a|≥ 2. LetX=ℕ orX=X _a andκ ∈ ℕ orκ=∞. This paper pursues analogous results for entire functionsf having the following property: on any finite subsetD ofX with#D≤κ+1, the valuesf(z),z ∈D admit interpolation by an element of ℤ[z]. The results obtained assert that if the growth off is suitably restricted then the restriction off toX must be a polynomial. WhenX=X _a andκ<∞ a “smallest” transcendental entire function having the requisite property is constructed.     

Uniformly converging random variables for weakly converging laws

 Let {P _n , n ?ℕ} be a sequence of Borel probability measures on a compact and connected metric space X. We show that in case the measures P _n converge weakly to a fully supported limit measure P, there exist uniformly converging random variables X _n , n ?ℕ with these given laws. Connectivity and compactness are necessary conditions for our theorem to hold. We also present a decent generalization. We prove our theorem by means of a comparison of the Prokhorov and the so-called minimal L ^∞ metric. Then we only need to use the Strassen-Dudley theorem and Kellerer's measure extension theorem for decomposable families. Received: 2 November 2000 / Revised version: 5 January 2002/ Published online: 1 July 2002     

Symbolic representations of nonexpansive group automorphisms

Ifα is an irreducible nonexpansive ergodic automorphism of a compact abelian groupX (such as an irreducible nonhyperbolic ergodic toral automorphism), thenα has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts inX. In spite of this we are able to construct a symbolic spaceV and a class of shift-invariant probability measures onV each of which corresponds to anα-invariant probability measure onX. Moreover, everyα-invariant probability measure onX arises essentially in this way. The last part of the paper deals with the connection between the two-sided beta-shiftV _β arising from a Salem numberβ and the nonhyperbolic ergodic toral automorphismα arising from the companion matrix of the minimal polynomial ofβ, and establishes an entropy-preserving correspondence between a class of shift-invariant probability measures onV _β and certainα-invariant probability measures onX. This correspondence is much weaker than, but still quite closely modelled on, the connection between the two-sided beta-shifts defined by Pisot numbers and the corresponding hyperbolic ergodic toral automorphisms.     

On spaces in which countably compact sets are closed, and hereditary properties

A space X is called C-closed if every countably compact subset of X is closed in X. We study the properties of C-closed spaces. Among other results, it is shown that countably compact C-closed spaces have countable tightness and under Martin's Axiom or 2^ω0<2^ω1, C-closed is equivalent to sequential for compact Hausdorff spaces. Furthermore, every hereditarily quasi-k Hausdorff space is Fréchet-Urysohn, which generalizes a theorem of Arhangel'sk in [4]. Also every hereditarily q-space is hereditarily of pointwise countable type and contains an open dense first countable subspace.     

Closed ideals in topological algebras: A characterization of the topological Φ-algebra C k (x)

Let A be a uniformly closed and locally m-convex Φ-algebra. We obtain internal conditions on A stated in terms of its closed ideals for A to be isomorphic and homeomorphic to C _k (X), the Φ-algebra of all the real continuous functions on a normal topological space X endowed with the compact convergence topology.     

On a Banach space approximable by Jacobi polynomials

Let X represent either the space C[-1,1] L _p ^(α,β) (w), 1 ≦ p < ∞ on [-1, 1]. Then Xare Banach spaces under the sup or the p norms, respectively. We prove that there exists a normalized Banach subspace X ¹ _αβ of Xsuch that every f ∈ X ¹ _αβ can be represented by a linear combination of Jacobi polynomials to any degree of accuracy. Our method to prove such an approximation problem is Fourier–Jacobi analysis based on the convergence of Fourier–Jacobi expansions. This revised version was published online in June 2006 with corrections to the Cover Date.     

The structure of LC-continuous functions

A function f is LC-continuous if the inverse image of any open set is a locally closed set; i.e., an intersection of an open set and a closed set. The aim of this paper is to prove the following theorem: Let f: X→Y be an LC-continuous function onto a separable metric space Y. Then X can be covered by countably many subsets T _n ⊂X such that each restriction f∣T _n is continuous at all points of T _n .     

The extension of the operation in the Stone-Čech compactification of semigroups

For a large class of locally compact semitopological semigroups S, the Stone-Čech compactification β S is a semigroup compactification if and only if S is either discrete or countably compact. Furthermore, for this class of semigroups which are neither discrete nor countably compact, the quotient contains a linear isometric copy of ℓ _∞. These results improve theorems by Baker and Butcher and by Dzinotyiweyi.     

Classification of Ideals of Homeomorphism C*-Algebras and Quasidiagonality of Their Quotient Algebras

For the homeomorphism C*-algebra A(Σ) arising from a topological dynamical system Σ=(X,σ) where σ is a homeomorphism on an arbitrary compact Hausdorff space X, we first give detailed classification of its closed ideals into four classes. In case when X is a compact metric space, we then determine the conditions when the quotient algebras of A(Σ) become quasidiagonal. The case of A(Σ) itself was treated by M. Pimsner.     

Idealization of a decomposition theorem

In 1986, Tong [13] proved that a function f : (X,τ)→(Y,φ) is continuous if and only if it is α-continuous and A-continuous. We extend this decomposition of continuity in terms of ideals. First, we introduce the notions of regular-I-closed sets, A _I-sets and A _I -continuous functions in ideal topological spaces and investigate their properties. Then, we show that a function f : (X,τ,I)→(Y, φ) is continuous if and only if it is α-I-continuous and A _I-continuous. This revised version was published online in June 2006 with corrections to the Cover Date.     

Stable outer conjugacy and strong Morita equivalence of group actions on pro-C *-algebras

We show that two continuous inverse limit actions α and β of a locally compact group G on two pro-C ^*-algebras A and B are stably outer conjugate if and only if there is a full Hilbert A-module E and a continuous action u of G on E such that E and E ^*(the dual module of E) are countably generated in M(E)(the multiplier module of E), respectively M(E ^*) and the pair (E, u) implements a strong Morita equivalence between α and β. This is a generalization of a result of F. Combes [Proc. London Math. Soc. 49(1984), 289–306].      

On extensions of countable filterbases to ultrafilters and ultrafilter compactness

We show in the Zermelo-Fraenkel set theory ZF without the axiom of choice:

1. Given an infinite set X, the Stone space S(X) is ultrafilter compact.

2. For every infinite set X, every countable filterbase of X extends to an ultra-filter i? for every infinite set X, S(X) is countably compact.

3. ω has a free ultrafilter i? every countable, ultrafilter compact space is countably compact.

   We also show the following:

4. There are a permutation model 𝒩 and a set X ∈ 𝒩 such that X has no free ultrafilters and S(X) is not compact but S(X) is countably compact and every countable filterbase of X extends to an ultrafilter.

5. It is relatively consistent with ZF that every countable filterbase of ω extends to an ultrafilter but there exists a countable filterbase of ? which does not extend to an ultrafilter. Hence, it is relatively consistent with ZF that ? has free ultrafilters but there exists a countable filterbase of ? which does not extend to an ultrafilter.

     

Characterizations and Extensions of Lipschitz-α Operators

In this work, we prove that a map F from a compact metric space K into a Banach space X over F is a Lipschitz-α operator if and only if for each σ in X^＊ the map σoF is a Lipschitz-α function on K. In the case that K = [a, b], we show that a map f from [a, b] into X is a Lipschitz-1 operator if and only if it is absolutely continuous and the map σ→ （σ o f）＇ is a bounded linear operator from X^＊ into L^∞（[a, b]）. When K is a compact subset of a finite interval （a, b） and 0 〈 α ≤ 1, we show that every Lipschitz-α operator f from K into X can be extended as a Lipschitz-α operator F from [a, b] into X with Lα（f） ≤ Lα（F） ≤ 3^1-α Lα（f）. A similar extension theorem for a little Lipschitz-α operator is also obtained.     

Countably Compact Paratopological Groups

By means of the theory of bispaces we show that a countably compact T₀ paratopological group (G, τ) is a topological group if and only if (G, τ ∨ τ^-1) is ω-bounded (here τ^-1 is the conjugate topology of τ). Our approach is premised on the fact that every paratopological countably compact paratopological group is a Baire space and on the notion of a 2-pseudocompact space. We also prove that every ω-bounded (respectively, topologically periodic) Baire paratopological group admits a weaker Hausdorff group topology. In particular, ω-bounded (respectively, topologically periodic) 2-pseudocompact (so, also countably compact) paratopological groups enjoy this property. Some topological properties turning countably compact topological semigroups into topological groups are presented and some open questions are posed.     

Lattice-embedding scales ofL p spaces into orlicz spaces

We study the setP _X of scalarsp such thatL ^p is lattice-isomorphically embedded into a given rearrangement invariant (r.i.) function spaceX[0, 1]. Given 0<α≤β<∞, we construct a family of Orlicz function spacesX=L ^F [0, 1], with Boyd indicesα andβ, whose associated setsP _X are the closed intervals [γ, β], for everyγ withα≤γ≤β. In particular forα>2, this proves the existence of separable 2-convex r.i. function spaces on [0,1] containing isomorphically scales ofL ^p -spaces for different values ofp. We also show that, in general, the associated setP _X is not closed. Similar questions in the setting of Banach spaces with uncountable symmetric basis are also considered. Thus, we construct a family of Orlicz spaces ℓ ^F (I), with symmetric basis and indices fixed in advance, containing ℓ ^p (Γ-subspaces for differentp’s and uncountable Λ⊂I. In contrast with the behavior in the countable case (Lindenstrauss and Tzafriri [L-T₁]), we show that the set of scalarsp for which ℓ ^p (Γ) is isomorphic to a subspace of a given Orlicz space ℓ ^F (I) is not in general closed. Supported in part by DGICYT grant PB 94-0243.