Compact Hausdorff Research Articles

Scott functions, their representations on domains, and applications to random sets

Choquet theorems (1954) on integral representation for capacities are fundamental to probability theory. They inspired a growing body of research into different approaches and generalizations of Choquet's results by many other researchers. Notably Mathéron's work (1975) on distributions over the space of closed subsets has led to further advancements in the theory of random sets. This paper was inspired by the work of Norberg (1989) who generalized Choquet's results to distributions over domains. While Choquet's original theorems were obtained for locally compact Hausdorff (LCH) spaces, both Mathéron's and Norberg's depend on the assumption of separability in their application of the Carathéodory's method. Our Radon measure approach differs from the work of Mathéron and Norberg, in that it does not require separability. This investigation naturally leads to the introduction of finite and locally finite valuations, which allows us to characterize finite and locally finite random sets in terms of capacities on the class of compact subsets. Finally, the treatment of Lévy exponent by Mathéron and Norberg is revisited, and the notion of exponential valuation is proposed for the representation of general Poisson processes.

On Vector-Valued Banach Function Algebras

We consider vector-valued Banach function algebras on a compact Hausdorff space. Then, we define the subalgebras generated by vector-valued polynomials and rational functions, and determine their maximal ideal spaces and Silov boundaries. We finally make use the results for a certain category of these algebras such as vector-valued Lipschitz algebras, vector-valued Dales–Davie algebras (algebras of vector-valued differentiable functions) and the algebras of vector-valued differentiable Lipschitz functions.

A topological invariant for continuous fields of Cuntz algebras

We wish to investigate continuous fields of the Cuntz algebras. The Cuntz algebras \({\mathcal {O}}_{n+1}, n\ge 1\) play an important role in the theory of operator algebras, and they are characterized by their K-groups \(K_0({\mathcal {O}}_{n+1})={\mathbb {Z}}_n\), the cyclic groups of order \(n\ge 1\). Since the mod n K-group for a compact Hausdorff space can be realized by the K-group of the trivial continuous field of \({\mathcal {O}}_{n+1}\) over the space, one can regard \({\mathcal {O}}_{n+1}\) as a noncommutative analogue of the Moore space of \({\mathbb {Z}}_n\), and classifying continuous fields of the Cuntz algebras is an interesting problem. M. Dadarlat classifies these fields which are constructed from the vector bundles of rank \(n+1\), and he also showed that not every continuous field comes from the vector bundle. For a continuous field of \({\mathcal {O}}_{n+1}\) over a finite CW complex, we introduce a topological invariant, which is an element in Dadarlat–Pennig’s generalized cohomology group, and prove that the invariant is trivial if and only if the field comes from a vector bundle via Pimsner’s construction.

Rota's Fubini lectures: The first problem

In his 1998 Fubini Lectures, Rota discusses twelve problems in probability that “no one likes to bring up”. The first problem calls for a revision of the notion of a sample space, guided by the belief that mention of sample points in a probabilistic argument is bad form and that a “pointless” foundation of probability should be provided by algebras of random variables.In 1958 Chang introduced MV-algebras to prove the completeness theorem of Łukasiewicz logic Ł∞. The aim of this paper is to show that MV-algebras provide a solution of Rota's first problem.The adjunction between MV-algebras and unital commutative C*-algebras equips every MV-algebra A with a natural ring structure, as advocated by Nelson for algebras of random variables. The closed compact set S(A)⊆[0,1]A of finitely additive probability measures on A (the states of A) coincides with the set of [0,1]-valued functions on A whose finite restrictions are consistent in de Finetti's sense. MV-algebras and Ł∞ thus provide the framework for a generalization (known as ŁIPSAT) of Boole's probabilistic inference problem, and its modern reformulation known as probabilistic satisfiability, PSAT. We construct an affine homeomorphism γA of S(A) onto the weakly compact space of regular Borel probability measures on the maximal spectral space μ(A). The latter is the most general compact Hausdorff space. As a consequence, for every Kolmogorov probability space (Ω,FΩ,P), with FΩ the sigma-algebra of Borel sets of a compact Hausdorff space Ω, and P a regular probability measure on FΩ, there is an MV-algebra A and a state σ of A such that (Ω,FΩ,P)≅(μ(A),Fμ(A),γA(σ)).

Two extensions of the stone duality to the category of zero-dimensional Hausdorff spaces

Extending the Stone Duality Theorem, we prove two duality theorems for the category ZHaus of zero-dimensional Hausdorff spaces and continuous maps. They extend also the Tarski Duality Theorem; the latter is even derived from one of them. We prove as well two new duality theorems for the category EDTych of extremally disconnected Tychonoff spaces and continuous maps. Also, we describe two categories which are dually equivalent to the category ZComp of zero-dimensional Hausdorff compactifications of zero-dimensional Hausdorff spaces and obtain as a corollary the Dwinger Theorem about zero-dimensional compactifications of a zero-dimensional Hausdorff space.

An Expository on Some Nonstandard Compactifications

Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The general procedure is as follows: A suitable equivalence relation is defined on an enlargement *X of the space X which is a completely regular space or a locally compact Hausdorff space or a locally compact Abelian group. Accordingly, every f in C(X,R) (the space of bounded continuous real valued functions on X) or Cc(X,R) (the space of continuous real valued functions on X with compact support) or the dual group Γ of the locally compact Abelian group G is extended to the set of the above mentioned equivalence classes. A compact topology on is obtained as the weak topology generated by these extensions of f. Then X is naturally imbedded densely in .

Proximity inductive dimension and Brouwer dimension agree on compact Hausdorff spaces

We show that the proximity inductive dimension defined by Isbell agrees with the Brouwer dimension originally described by Brouwer (for Polish spaces without isolated points) on the class of compact Hausdorff spaces. This shows that Fedorchuk?s example of a compact Hausdorff space whose Brouwer dimension exceeds its Lebesgue covering dimension is an example of a space whose proximity inductive dimension exceeds its proximity dimension as defined by Smirnov. This answers Isbell?s question of whether or not proximity inductive dimension and proximity dimension coincide.

Additive local multiplications and zero-preserving maps on C(X)

Suppose $X$ is a compact Hausdorff space. In terms of topolocical properties of $X$, we find topological conditions on $X$ that are equivalent to each of the following: 1. every additive local multiplication on $C\left( X\right) $ is a multiplication, 2. every additive local multiplication on $C_{R}\left( X\right) $ is a multiplication, and 3. every additive map on $C\left( X\right) $ that is zero-preserving (i.e., $f\left( x\right) =0$ implies $\left( Tf\right) \left( x\right) =0$) has the form $T\left( f\right) =T\left( 1\right) \operatorname{Re}f+T\left( i\right) \operatorname{Im}f$.

On spaces of *-measures on ultrametric spaces

The notion of * -measure on a compact Hausdorff space is introduced and investigated in a previous publication of the first named author. In the present note we consider the set of all * -measures of compact support on an ultrametric space. An ultrametrization of this set is provided, which determines a functor in the category of ultrametric spaces and non-expanding maps. We prove that this functor is locally non-expanding and preserves the class of complete ultrametric spaces.

A study on D-spaces and the metrizability of compact spaces with property (σ-A)

In this article, we introduce two notions which are called property (σ-A) and property (σ-B). They are generalizations of property (A) and property (B), respectively. Every space with a point-countable base (or σ-NSR pair-base) satisfies property (σ-A). Every space with the Collins-Roscoe property satisfies property (σ-B). We show that every compact Hausdorff space with property (σ-A) is metrizable. Thus some known conclusions can be generalized. This shows that property (σ-A) plays a key role in the metrizability of compact Hausdorff spaces.We show that the properties of property (σ-A) and property (σ-B) are closed under finite products. Every finite product of T1-spaces which satisfy property (σ-B) (property (σ-A), σ-sheltering (F), σ-well-ordered (F)) is hereditarily a D-space. If (X,T) satisfies ω1-sheltering (F), then (X,Tω) is hereditarily a D-space. We show that if a space X satisfies ω1-sheltering (F) and every countable discrete subspace of X is closed, then X is hereditarily a D-space. This gives a partial answer to a question posed by Z.Q. Feng and J.E. Porter in 2015.We finally give examples to show that there exists a space which has property (σ-A) but it does not have a point-countable base and there exists a space which has property (C) but it does not have property (σ-A).

On Sjölin–Soria–Antonov type extrapolation for locally compact groups and a.e. convergence of Vilenkin–Fourier series

A Sjolin–Soria–Antonov type extrapolation theorem for locally compact $$\sigma$$ -compact non-discrete Hausdorff groups is proved. Applying this result it is shown that the Fourier series with respect to the Vilenkin orthonormal systems on the Vilenkin groups of bounded type converge almost everywhere for functions from the class $$L {\rm log}^{+} L {\rm log}^{+} {\rm log}^{+} {\rm log}^{+} L$$ .

On the irreducibility of induced representations of some groupoid $$C^*$$-algebras

Let $${\mathcal {G}}$$ be a topological locally compact Hausdorff and second countable groupoid with a Haar system and $${\mathcal {K}}$$ a proper subgroupoid of $${\mathcal {G}}$$ with a Haar system too. $$({\mathcal {G}},{\mathcal {K}})$$ is an internally Gelfand pair if for any u in the unit space, the pair of isotropy groups $$({\mathcal {G}}(u), {\mathcal {K}}(u))$$ is a Gelfand pair. In this work, we prove, when $$({\mathcal {G}},{\mathcal {K}})$$ is an internally Gelfand pair that any representation of $${\mathcal {C}}^{*} ({\mathcal {G}}, {\mathcal {K}},\lambda , \alpha )$$ induced from a positive definite spherical function on an isotropy group is irreducible. We introduce also the notion of spherical bundle and after identifying it with the spectrum of some $$C^*$$ -algebra, we obtain some properties of its topology.

On a variant of Tingley's problem for some function spaces

Let (Ω,A,μ) and (Γ,B,ν) be two arbitrary measure spaces, and p∈[1,∞]. SetS(Lp(μ))+:={f∈Lp(μ):‖f‖p=1;f≥0μ-a.e.}, that is, the positive part of the unit sphere of Lp(μ). We show that every surjective isometry Φ:S(Lp(μ))+→S(Lp(ν))+ can be extended (necessarily uniquely) to an isometric order isomorphism from Lp(μ) onto Lp(ν). A Lamperti form, i.e., a weighted composition like form, of Φ is provided, when (Γ,B,ν) is localizable (in particular, when it is σ-finite). On the other hand, we show that for compact Hausdorff spaces X and Y, if Φ is a surjective isometry from the positive part of the unit sphere of C(X) to that of C(Y), then there is a homeomorphism τ:Y→X satisfying Φ(f)(y)=f(τ(y)) for f∈S(C(X))+ and y∈Y.

Compact groups in which all elements have countable right Engel sinks

A right Engel sink of an element g of a group G is a set ${\mathscr R}(g)$ such that for every x ∈ G all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$. (Thus, g is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$.) It is proved that if every element of a compact (Hausdorff) group G has a countable right Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent.

Stone–Weierstrass theorems for Riesz ideals of continuous functions

Notions of convergence and continuity specifically adapted to Riesz ideals I of the space of continuous real-valued functions on a Lindel\"of locally compact Hausdorff space are given, and used to prove Stone-Weierstra{\ss}-type theorems for I. As applications, sufficient conditions are discussed that guarantee that various types of positive linear maps on I are uniquely determined by their restriction to various point-separating subsets of I. A very special case of this is the characterization of the strong determinacy of moment problems, which is rederived here in a rather general setting and without making use of spectral theory.

Non-linear modular Birkhoff-James orthogonality preservers between spaces of continuous functions

A new notion of non-linear isomorphisms between Banach modules is introduced based on modular Birkhoff-James orthogonality. It is shown that a (possibly non-linear) bijective modular Birkhoff-James orthogonality preserver in both direction between spaces of continuous functions (vanishing at infinity) induces a homeomorphism between underlying locally compact Hausdorff spaces. Moreover, characterizations of linear and additive modular Birkhoff-James orthogonality preservers between spaces of continuous functions are also given in terms of the induced homeomorphisms and continuous functions having constant absolute values.

On 2-local diameter-preserving maps between C(X)-spaces

The 2-locality problem of diameter-preserving maps between C(X)-spaces is addressed in this paper. For any compact Hausdorff space X with at least three points, we give an example of a 2-local diameter-preserving map on C(X) which is not linear. However, we show that for first countable compact Hausdorff spaces X and Y, every 2-local diameter-preserving map from C(X) to C(Y) is linear and surjective up to constants in some sense. This fact yields the 2-algebraic reflexivity of isometries with respect to the diameter norms on the quotient spaces.

Corson reflections

A reflection principle for Corson compacta holds in the forcing extension obtained by Levy-collapsing a supercompact cardinal to ℵ2. In this model, a compact Hausdorff space is Corson if and only if all of its continuous images of weight ℵ1 are Corson compact. We use the Gelfand–Naimark duality, and our results are stated in terms of unital abelian C⁎-algebras.

Range preserving maps between the spaces of continuous functions with values in a locally convex space

Let C ( X , E ) C(X,E) be the linear space of all continuous functions on a compact Hausdorff space X X with values in a locally convex space E E . We characterize maps T : C ( X , E ) → C ( Y , E ) T:C(X,E)\to C(Y,E) which satisfy R a n ( T F − T G ) ⊂ R a n ( F − G ) \mathrm {Ran}(TF-TG)\subset \mathrm {Ran}(F-G) for all F , G ∈ C ( X , E ) F,G\in C(X,E) .

Causal variational principles in the σ-locally compact setting: Existence of minimizers

Abstract We prove the existence of minimizers of causal variational principles on second countable, locally compact Hausdorff spaces. Moreover, the corresponding Euler–Lagrange equations are derived. The method is to first prove the existence of minimizers of the causal variational principle restricted to compact subsets for a lower semi-continuous Lagrangian. Exhausting the underlying topological space by compact subsets and rescaling the corresponding minimizers, we obtain a sequence which converges vaguely to a regular Borel measure of possibly infinite total volume. It is shown that, for continuous Lagrangians of compact range, this measure solves the Euler–Lagrange equations. Furthermore, we prove that the constructed measure is a minimizer under variations of compact support. Under additional assumptions, it is proven that this measure is a minimizer under variations of finite volume. We finally extend our results to continuous Lagrangians decaying in entropy.