Compact Hausdorff Space Research Articles

Witnessing the lack of the Grothendieck property in C(K)-spaces via convergent sequences

Let K be a compact Hausdorff space and let C(K) be the space of all scalar-valued, continuous functions on K. We show that C(K) is an $$\ell _1(K)$$ -Grothendieck space but not a Grothendieck space exactly when the spaces $$C_p(K)$$ and $$C_p(K \oplus {\mathbb {N}}^{\#})$$ are not linearly isomorphic, where $${\mathbb {N}}^{\#}$$ is the one-point compactificiation of the discrete space of natural numbers. (That is, if C(K) contains a complemented copy of $$c_0$$ , then C(K) fails to be $$\ell _1(K)$$ -Grothendieck if and only if the topologies of pointwise convergence in $$C_p(K)$$ and $$C_p(K \oplus {\mathbb {N}}^{\#})$$ are linearly isomorphic.) Moreover, for infinite compact spaces K and L, there exists a compact space G that has a non-trivial convergent sequence and such that $$C_{p}(K\times L)$$ and $$C_{p}(G)$$ are linearly isomorphic. This extends a remarkable theorem of Cembranos and Freniche. Some examples illustrating the above results are provided.

Orthograph related to mutual strong Birkhoff–James orthogonality in $$C^*$$-algebras

We study the relation of mutual strong Birkhoff–James orthogonality in two classical $$C^*$$ -algebras: the $$C^*$$ -algebra $${\mathbb {B}}(H)$$ of all bounded linear operators on a complex Hilbert space H and the commutative, possibly nonunital, $$C^*$$ -algebra. With the help of the induced graph it is shown that this relation alone can characterize right invertible elements. Moreover, in the case of commutative unital $$C^*$$ -algebras, it can detect the existence of a point with a countable local basis in the corresponding compact Hausdorff space.

Conjugations on Banach $$*$$-algebras

The notion of conjugation is extended to Banach $$*$$ -algebras. The aim of this paper is to characterize conjugations on the Banach algebra of all bounded linear operators on a complex Hilbert space, the algebra of J-symmetric operators on a complex Hilbert space with given conjugation J and the algebra of all complex valued continuous functions, defined on a connected locally compact Hausdorff space, which vanish at infinity.

Additive jointly separating maps and ring homomorphisms

Let X and Y be compact Hausdorff spaces, E and F be real or complex normed spaces and be a subspace of . For a function , let be the cozero set of f. A pair of additive maps is said to be jointly separating if whenever . In this paper, first, we give a partial description of additive jointly separating maps between certain spaces of vector-valued continuous functions (including spaces of vector-valued Lipschitz functions, absolutely continuous functions and continuously differentiable functions). Then we apply the results to characterize continuous ring homomorphisms between certain Banach algebras of vector-valued continuous functions. In particular, the results provide some generalizations of the recent results on unital homomorphisms between vector-valued Lipschitz algebras, with a different approach.

Gelfand theory for real Banach algebras

We are concerned with the development of the more general real case of the classical theorem of Gelfand on representation of a complex commutative unital Banach algebra as an algebra of continuous functions defined on a compact Hausdorff space. To that end, we use only intrinsic methods which do not depend on the complexification of the algebra, and obtain two representation theorems for commutative unital real Banach algebras as algebras of continuous real (respectively, complex) functions on the compact space of real-valued (respectively, complex-valued) $$\mathbb R$$ -algebra homomorphisms.

A note on “Posets having continuous interval”

In this note we present a counterexample for a result in [2], where the concept of continuous interval posets (CI-posets for short) was introduced. A CI-poset is a poset in which every nonempty interval [x,y] is a continuous poset (in the restricted order) and the above mentioned result is about continuous functions spaces [X⟶P], where X is a core compact topological space and P is a CI-poset (with its Scott topology).

Relatively Weakly Open Convex Combinations of Slices and Scattered C$$^*$$-Algebras

We prove that given a locally compact Hausdorff space, $K$, and a compact C$^*$-algebra, $\mathcal{A}$, the C$^*$-algebra $C(K, \mathcal{A})$ satisfies that every convex combination of slices of the closed unit ball is relatively weakly open subset of the closed unit ball if and only if $K$ is scattered and $\mathcal{A}$ is the $c_0$-sum of finite-dimensional C$^*$-algebras. We introduce and study Banach spaces which have property $(\overline{\hbox{P1}})$, i. e. For every convex combination of slices $C$ of the unit ball of a Banach space $X$ and $x\in C$ there exists $W$ relatively weakly open set containing $x$, such that $W\subseteq \overline{C}$. In the setting of general C$^*$-algebras we obtain a characterization of this property. Indeed, a C$^*$-algebra has property $(\overline{\hbox{P1}})$ if and only if is scattered with finite dimensional irreducible representations. Some stability results for Banach spaces satisfying property $(\overline{\hbox{P1}})$ are also given. As a consequence of these results we prove that a real $L_1$-predual Banach space contains no isomorphic copy of $\ell_1$ if and only if it has property $(\overline{\hbox{P1}})$.

On Loeb and sequential spaces in ZF

A topological space is called Loeb if the collection of all its non-empty closed sets has a choice function. Loeb and sequential spaces are investigated in the absence of the axiom of choice. Among other results, the following theorems are proved in ZF: (i) if X is a Cantor completely metrizable second-countable space, then Xω is Loeb; (ii) if a sequential, locally sequentially compact space X has the property that every infinite countable collection of non-empty closed subsets of X has a choice function, then the Cartesian product X×Y of X with any sequential space Y is sequential; (iii) if R is sequential, then every second-countable compact Hausdorff space is sequential.Several statements on Loeb and sequential spaces are shown to be independent of ZF. Open problems are posed.

Jointly separating maps between vector-valued function spaces

AbstractLetXandYbe compact Hausdorff spaces,Ebe a real or complex Banach space andFbe a real or complex locally convex topological vector space. In this paper we study a pair of linear operatorsS,T:A(X,E) →C(Y,F) from a subspaceA(X,E) ofC(X,E) toC(Y,F), which are jointly separating, in the sense thatTfandSghave disjoint cozeros wheneverfandghave disjoint cozeros. We characterize the general form of such maps between certain classes of vector-valued (as well as scalar-valued) spaces of continuous functions including spaces of vector-valued Lipschitz functions, absolutely continuous functions and continuously differentiable functions. The results can be applied to a pairT:A(X) →C(Y) andS:A(X,E) →C(Y,F) of linear operators, whereA(X) is a regular Banach function algebra onX, such thatf⋅g= 0 impliesTf⋅Sg= 0, for allf∈A(X) andg∈A(X,E). IfTandSare jointly separating bijections between Banach algebras of scalar-valued functions of this class, then they induce a homeomorphism betweenXandYand, furthermore,T−1andS−1are also jointly separating maps.

When Is a Continuous Bijection a Homeomorphism?

Homeomorphisms are one of the main objects of study in topology and, hence, it is interesting to provide conditions that ensure that a map between topological spaces is a homeomorphism. Some examples of these kinds of results are the invariance of domain theorem (IDT) or the well-known lemma that states that a continuous bijection from a compact topological space to a Hausdorff topological space is a homeomorphism. In this note, a similar result is provided. More specifically, we show that any continuous bijection from a path-connected topological space to a set endowed with the order topology is a homeomorphism. In particular, we show how this generalizes the easiest case of the IDT for dimension n = 1. Furthermore, we apply the result in the context of special relativity and Lorentz transformations.

Gelfand-Naimark-Stone duality for normal spaces and insertion theorems

Gelfand-Naimark-Stone duality provides an algebraic counterpart of compact Hausdorff spaces in the form of uniformly complete bounded archimedean ℓ-algebras. In [5] we extended this duality to completely regular spaces. In this article we use this extension to characterize normal, Lindelöf, and locally compact Hausdorff spaces. Our approach gives a different perspective on the classical theorems of Katětov-Tong and Stone-Weierstrass.

On closed non-vanishing ideals in CB(X) I; Connectedness properties

Let X be a completely regular topological space. We study closed ideals H of CB(X), the normed algebra of bounded continuous scalar-valued mappings on X equipped with pointwise addition and multiplication and the supremum norm, which are non-vanishing, in the sense that, there is no point of X at which every element of H vanishes. This is done by studying the (unique) locally compact Hausdorff space Y associated to H in such a way that H and C0(Y) are isometrically isomorphic. We are interested in various connectedness properties of Y. In particular, we present necessary and sufficient (algebraic) conditions for H such that Y satisfies (topological) properties such as locally connectedness, total disconnectedness, zero-dimensionality, strong zero-dimensionality, total separatedness or extremal disconnectedness.

Abstract topological dynamics involving set-valued functions

Continuous functions over compact Hausdorff spaces have been completely characterised. We consider the more general problem: given a set-valued function T from an arbitrary set X to itself, does there exist a compact Hausdorff topology on X with respect to which T is upper semicontinuous? We give conditions that are necessary for T to be upper semicontinuous and point-closed if X is a compact Hausdorff space. We show that it is always possible to provide X with a compact T1 topology with respect to which T is lower semicontinuous, and consequently, if T:X→X is a function, then it is always possible to provide X with a compact T1 topology with respect to which T is continuous.

Ergodic algebras beyond the weak almost periodic functions

Ergodic algebras is a concept introduced in a paper by Zhikov and Krivenko in the 1980’s which generalizes the concept of almost periodic functions and capture the main properties satisfied by generic realizations of a continuous function in a compact topological space endowed with an ergodic dynamical system and an invariant measure. We address the question whether there exist ergodic algebras beyond the weak almost periodic functions, that is, functions whose set of translates is relatively compact in the weak topology of the bounded continuous functions in $${\mathbb {R}}^n$$ . This question is important since, so far, all known ergodic algebras are subalgebras of the weak almost periodic functions. Also, it is known that almost all realizations of a continuous functions in a compact space endowed with a dynamical system and an invariant measure, belongs to an ergodic algebra. In this paper we construct examples of ergodic algebras that are not contained in the weak almost periodic functions. Actually, our examples are not contained in the greater algebra formed by the sum of almost periodic functions with uniformly continuous functions with null mean value in $${\mathbb {R}}^n$$ .

Three Proofs of Tychonoff’s Theorem

We give three proofs of Tychonoff’s theorem on the compactness of a product of compact topological spaces. The first one proceeds “from scratch.” The second one relies on the characterization of compactness in terms of convergent subnets. The third one has a “categorical” flavor.

A generalized type semigroup and dynamical comparison

In this paper, we construct and study a semigroup associated to an action of a countable discrete group on a compact Hausdorff space that can be regarded as a higher dimensional generalization of the type semigroup. We study when this semigroup is almost unperforated. This leads to a new characterization of dynamical comparison and thus answers a question of Kerr and Schafhauser. In addition, this paper suggests a definition of comparison for dynamical systems in which neither the acting group is necessarily amenable nor the action is minimal.

Extensions of dualities and a new approach to the Fedorchuk duality

Applying a general categorical construction for the extension of dualities, we present a new proof of the Fedorchuk duality between the category of compact Hausdorff spaces with their quasi-open mappings and the category of complete normal contact algebras with suprema-preserving Boolean homomorphisms which reflect the contact relation.

Expansive Actions with Specification on Uniform Spaces, Topological Entropy, and the Myhill Property

We prove that every expansive continuous action with the weak specification property of an amenable group G on a compact Hausdorff space X has the Myhill property, i.e., every pre-injective continuous self-mapping of X commuting with the action of G on X is surjective. This extends a result previously obtained by Hanfeng Li in the case when X is metrizable.

Функтор IS в категории компактных Хаусдорфовых пространств

We construct a space of normed, homogeneous and max-plus-semiadditive functionals and we give its description. Further we establish that the construction of taking of a space of normed, homogeneous and max-plus-semiadditive functionals, forms a normal functor acting in the category of Hausdorff compact spaces and their continuous maps.

Chebyshev-Type Integral Inequalities for Continuous Fields of Operators Concerning Khatri–Rao Products and Synchronous Properties

We consider bounded continuous fields of self-adjoint operators which are parametrized by a locally compact Hausdorff space Ω equipped with a finite Radon measure μ . Under certain assumptions on synchronous Khatri–Rao property of the fields of operators, we obtain Chebyshev-type inequalities concerning Khatri–Rao products. We also establish Chebyshev-type inequalities involving Khatri–Rao products and weighted Pythagorean means under certain assumptions of synchronous monotone property of the fields of operators. The Pythagorean means considered here are three classical symmetric means: the geometric mean, the arithmetic mean, and the harmonic mean. Moreover, we derive the Chebyshev–Grüss integral inequality via oscillations when μ is a probability Radon measure. These integral inequalities can be reduced to discrete inequalities by setting Ω to be a finite space equipped with the counting measure. Our results provide analog results for matrices and integrable functions. Furthermore, our results include the results for tensor products of operators, and Khatri–Rao/Kronecker/Hadamard products of matrices, which have been not investigated in the literature.