Hausdorff spaceX Research Articles

Weighted composition operators on nonlocally convex weighted spaces of continuous functions

LetV be a system of weights on a completely regular Hausdorff spaceX and letB(E) be the topological vector space of all continuous linear operators on a general topological vector spaceE. LetCV0(X, E) andCVb(X, E) be the weighted spaces of vector-valued continuous functions (vanishing at infinity or bounded, respectively) which are not necessarily locally convex. In the present paper, we characterize in this general setting the weighted composition operatorsWπ,ϕ onCV0(X, E) (orCVb(X, E)) induced by the operator-valued mappings π:X→B(E) (or the vector-valued mappings π:X→E, whereE is a topological algebra) and the self-map ϕ ofX. Also, we characterize the mappings π:X→B(E) (or π:x→E) and ϕ:X→X which induce the compact weighted composition operators on these weighted spaces of continuous functions.

Crossed Products byC0(X)-Actions

Suppose thatGhas a representation groupH, thatGab≔G/[G,G]is compactly generated, and thatAis aC*-algebra for which the complete regularization of Prim(A) is a locally compact Hausdorff spaceX. In a previous article, we showed that there is a bijectionα↦(Zα,fα) between the collection of exterior equivalence classes of locally inner actionsα:G→Aut(A), and the collection of principalGab-bundlesZαtogether with continuous functionsfα:X→H2(G,T). In this paper, we compute the crossed productsA⋊αGin terms of the dataZα,fα, andC*(H).

Riesz product spaces and representation theory

Let {E i:i∈I} be a family of Archimedean Riesz spaces. The Riesz product space is denoted by ∏ i∈I Ei. The main result in this paper is the following conclusion: There exists a completely regular Hausdorff spaceX such that ∏ i∈I Ei is Riesz isomorphic toC(X) if and only if for everyi∈I there exists a completely regular Hausdorff spaceX i such thatE i is Riesz isomorphic toC(X i).

The Lyusternik-Shnirel'man category,¶vector bundles, and immersions of manifolds

One of our two main results exhibits for a vector bundle over a compact Hausdorff spaceX an interplay between its span, its possible splittings, and the Lyusternik-Shnirel’man category ofX. The other main result, also on vector bundles and the Lyusternik-Shnirel’man category, enables us to derive certain inequalities connecting the immersion codimension, the stable span, and the Lyusternik-Shnirel’man category of a smooth closed manifold which is not stably parallelizable. Our results are applicable in various situations of general interest.

Functional Calculus for Banach Function Algebras and Banach Function Spaces of Continuous Functions Vanishing at Infinity

We show among other things that ifBis a Banach function space of continuous real-valued functions vanishing at infinity on a locally compact Hausdorff spaceX, with the property that for some odd natural numberp>1,b1/p∈Bfor allb∈B, thenB=C0(X).

Spaces betweenX and its Freudenthal compactification

Let ϕX indicate the Freudenthal compactification of a rimcompact, completely regular Hausdorff spaceX. In this paper the spacesY which satisfyX⊑Y⊑ϕX are characterized. From this a characterization of whenX lies between its locally compact partL(X) and ϕ(L(X)) follows. Such spaces necessarily possess a compactification αX for whichCl αX (αX−X) is 0-dimensional. Conditions, including those internal toX, are provided which are necessary and sufficient for this property to hold.

Algebraic and essentially algebraic composition operators onC(X)

Given a compact Hausdorff spaceX, we may associate with every continuous mapa: X → X a composition operatorC a onC(X) by the rule(C a f)(x) = f(a(x)). We describe all self-mapsa for whichC a is an algebraic operator or an essentially algebraic operator (i.e. an operator algebraic modulo compact operators), determine the characteristic polynomialp a (z) and the essentially characteristic polynomialq a (z) in these cases and show how the connectivity ofX may be characterized in terms of the quotientsp a (z)/q a (z).

Essential extensions of Hausdorff spaces

In the category Haus of Hausdorff spaces the only injectives are the one-point spaces. Even though every Hausdorff spaceX has a maximal essential extension,X fails to have an injective hull, providedX has more than one point. A non-empty Hausdorff space has a proper essential extension if and only ifX is locally H-closed but not H-closed. In this case,X has (up to isomorphism) precisely one proper essential extension: the Obreanu-Porter extension (being simultaneously its maximal essential extension and its minimal H-closed extension). Completely parallel results hold for the categories SReg, Reg, and Tych of semi-regular, regular, and completely regular spaces respectively. In particular, the Alexandroff compactifications of locally compact, non-compact Hausdorff spaces are characterized categorically as the proper essential extensions of non-empty spaces in Tych (resp. Reg).

Spaces for which the first uncountable ordinal space is a remainder

A remainder of a locally compact, non-compact Hausdorff spaceX is anyαX − X whereαX is a Hausdorff compactification ofX. LetK(X) be the lattice of compactifications ofX. Conditions onK(X) and an internal condition are obtained which characterize when the first uncountable ordinal space is a remainder ofX.

Linear problems (with extended range) have linear optimal algorithms

LetF1 andF2 be normed linear spaces andS:F0 →F2 a linear operator on a balanced subsetF0 ofF1. IfN denotes a finite dimensional linear information operator onF0, it is known that there need not be alinear algorithmφ:N(F4) →F2 which is optimal in the sense that ‖φ(N(f)) −S(f‖ is minimized. We show that the linear problem defined byS andN can be regarded as having a linear optimal algorithm if we allow the range ofφ to be extended in a natural way. The result depends upon imbeddingF2 isometrically in the space of continuous functions on a compact Hausdorff spaceX. This is done by making use of a consequence of the classical Banach-Alaoglu theorem.

Weakly compact, unconditionally converging, and Dunford-Pettis operators on spaces of vector-valued continuous functions

Given a compact Hausdorff spaceX, EandFtwo Banach spaces, letT: C(X, E) → Fdenote a bounded linear operator (hereC(X, E)stands for the Banach space of all continuousE-valued functions defined onXunder supremum norm). It is well known [4] that any such operatorThas a finitely additive representing measureGthat is defined on the σ–field of Borel subsets ofXand thatGtakes its values in the space of all bounded linear operators fromEinto the second dual of F. The representing measure G enjoys a host of many important properties; we refer the reader to [4] and [5] for more on these properties. The question of whether properties of the operator T can be characterized in terms of properties of the representing measure has been considered by many authors, see for instance[1], [2], [3] and [6]. Most characterizations presented (see [3] concerning weakly compact operators or [3] and [6] concerning unconditionally converging operators) were given under additional assumptions on the Banach space E. The aim of this paper is to show that one cannot drop the assumptions on E, indeed as we shall soon show many of the operator characterizations characterize the Banach space E itself. More specifically, it is known [3] that ifE*andE**have the Radon-Nikodym property then a bounded linear operatorT: C(X, E) → Fis weakly compact if and only if the measureGis continuous at Ø (also called strongly bounded), i.e. limn||G|| (Bn) = 0 for every decreasing sequenceBn↘ Ø of Borel subsets ofX(here ||G|| (B) denotes the semivariation ofGatB), and if for every Borel setBthe operatorG(B)is a weakly compact operator fromEtoF. In this paper we shall show that if one wants to characterize weakly compact operators as those operators with the above mentioned properties thenE*andE**must both have the Radon-Nikodym property. This will constitute the first part of this paper and answers in the negative a question of [2]. In the second part we consider unconditionally converging operators onC(X, E). It is known [6] that ifT: C(X, E) → Fis an unconditionally converging operator, then its representing measureGis continuous at 0 and, for every Borel setB, G(B)is an unconditionally converging operator fromEtoF. The converse of the above result was shown to be untrue by a nice example (see [2]). Here again we show that if one wants to characterize unconditionally converging operators as above, then the Banach space E cannot contain a copy of c0. Finally, in the last section we characterize Banach spacesEwith the Schur property in terms of properties of Dunford-Pettis operators onC(X, E)spaces.

Formal Power Series Over Commutative N-Algebras

A Banach algebra P over C with identity element is called an N-algebra if any closed ideal in P is the intersection of maximal ideals. An example is given by the algebra of the continuous C-valued functions on a compact Hausdorff space X under the supremum norm; two others are discussed in § 3.

Resolutions of 𝐻-closed spaces

It is shown that every Hausdorff space X is the perfect, irreducible, continuous image of a Hausdorff space X ~ \tilde X which has a basis with open closures. Further, w ( X ~ ) ⩽ w ( X ) w(\tilde X) \leqslant w(X) , where w ( X ~ ) w(\tilde X) represents the weight of X, and if X is H-closed then X ~ \tilde X is also H-closed. A corollary of this result is that if f : X → Y f:X \to Y is a continuous map of the H-closed space X onto the semi-regular Hausdorff space Y, then w ( Y ) ⩽ w ( X ) w(Y) \leqslant w(X) .

A counterexample on extreme operators

We give examples of extreme operators in the unit ball ofL(C(X),C(Y)) which are not nice operators (i.e., their adjoints do not carry extreme points into extreme points in the corresponding unit balls.) Such counterexamples exist in fact for every non-dispersed compact Hausdorff spaceX when the scalars are complex.

Extension Function and Subcategories of Haus

AbstractFor each Hausdorff spaceX, letFXbe an Hausdorff extension ofX. The existence of the largest subcategory of HAUS on whichFs a functor and an epi-reflection is investigated.

Some ℵ0-bounded subsets of stone-čech compactifications

We characterize the maximalm-bounded extension of an arbitrary completely regular Hausdorff spaceX. The other principal results are:Theorem. LetX be a locally compact, σ-compact non-compact space with no more than 2ℵ0 zero-sets. Then assuming the continuum hypothesis,βX − X can be written as the union of 22ℵ0 pairwise disjoint, dense ℵ0-bounded subspaces.Theorem. LetX be a locally compact, σ-compact metric space without isolated points. Then both the set of remote points ofβX and the complement of this set inβX −X are ℵ0-bounded.

On Total Masses of Balayaged Measures

Beurling and Deny [1], [2] introduced the notion of Dirichlet spaces. They [2] showed the existence of balayaged measures and equilibrium measures in the theory of Dirichlet spaces. In this paper, we shall show that the following equivalence is valid for a Dirichlet space on a locally compact Hausdorff spaceX.