Compact Hausdorff Research Articles

Beth definability and the Stone-Weierstrass Theorem

The Stone-Weierstrass Theorem for compact Hausdorff spaces is a basic result of functional analysis with far-reaching consequences. We introduce an equational logic ⊨Δ associated with an infinitary variety Δ and show that the Stone-Weierstrass Theorem is a consequence of the Beth definability property of ⊨Δ, stating that every implicit definition can be made explicit. Further, we define an infinitary propositional logic ⊢Δ by means of a Hilbert-style calculus and prove a strong completeness result whereby the semantic notion of consequence associated with ⊢Δ coincides with ⊨Δ.

Dynamic asymptotic dimension for actions of virtually cyclic groups

We show that the dynamic asymptotic dimension of an action of an infinite virtually cyclic group on a compact Hausdorff space is always one if the action has the marker property. This in particular covers a well-known result of Guentner, Willett, and Yu for minimal free actions of infinite cyclic groups. As a direct consequence, we substantially extend a famous result by Toms and Winter on the nuclear dimension of $C^{*}$-algebras arising from minimal free $\mathbb {Z}$-actions. Moreover, we also prove the marker property for all free actions of countable groups on finite-dimensional compact Hausdorff spaces, generalizing a result of Szabó in the metrisable setting.

Approximations in L^1 with convergent Fourier series

For a separable finite diffuse measure space {mathcal {M}} and an orthonormal basis {varphi _n} of L^2({mathcal {M}}) consisting of bounded functions varphi _nin L^infty ({mathcal {M}}), we find a measurable subset Esubset {mathcal {M}} of arbitrarily small complement |{mathcal {M}}{setminus } E|<epsilon , such that every measurable function fin L^1({mathcal {M}}) has an approximant gin L^1({mathcal {M}}) with g=f on E and the Fourier series of g converges to g, and a few further properties. The subset E is universal in the sense that it does not depend on the function f to be approximated. Further in the paper this result is adapted to the case of {mathcal {M}}=G/H being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of n-spheres with spherical harmonics is discussed. The construction of the subset E and approximant g is sketched briefly at the end of the paper.

Hausdorff compactifications

Previously, the authors used the insights of Robinson's nonstandard analysis as a powerful tool to extend and simplify the construction of some compactifications of regular spaces. They now show that any Hausdorff compactification is obtainable with their method.

Characterizations of self-adjointness, normality of pseudo-differential operators on homogeneous space of compact groups

Let G be a compact Hausdorff group and H be a closed subgroup of G. In this paper, we show that every bounded linear operator T on is a pseudo-differential operator with the symbol σ for . We present necessary and sufficient conditions on symbols to ensure that a bounded pseudo-differential operator on is self-adjoint, normal and present explicit formula for their symbols. A necessary and sufficient condition is also given such that the bounded linear operators on posses eigenvalues and eigenfunctions.

Forcing a classification of non-torsion Abelian groups of size at most [formula omitted] with non-trivial convergent sequences

We force a classification of all the Abelian groups of cardinality at most 2c that admit a countably compact group with a non-trivial convergent sequence. In particular, we answer (consistently) Question 24 of Dikranjan and Shakhmatov [10] for cardinality at most 2c, by showing that if a non-torsion Abelian group of size at most 2c admits a countably compact Hausdorff group topology, then it admits a countably compact Hausdorff group topology with non-trivial convergent sequences.

Nuclear dimension of crossed products associated to partial dynamical systems

We bound the nuclear dimension of crossed products associated to some partial actions of finite groups or Z on finite dimensional locally compact Hausdorff second countable spaces. Our results apply to globalizable partial actions, finite group partial actions, minimal partial automorphisms, and partial automorphisms acting on zero-dimensional spaces, or a class of one dimensional spaces, containing 1-dimensional CW complexes. This extends work on global systems by Hirshberg and Wu.

Category measures, the dual of $\mathrm {C}(K)^{\delta }$ and hyper-Stonean spaces

For a compact Hausdorff space $K$, we give descriptions of the dual of $C(K)^\delta$, the Dedekind completion of the Banach lattice $C(K)$ of continuous, real-valued functions on $K$. We characterize those functionals which are $\sigma$-order continuous and order continuous, respectively, in terms of Oxtoby's category measures. This leads to a purely topological characterization of hyper-Stonean spaces.

Groupoid algebras as covariance algebras

Suppose G is a second-countable locally compact Hausdorff \'{e}tale groupoid, G is a discrete group containing a unital subsemigroup P, and c:G→G is a continuous cocycle. We derive conditions on the cocycle such that the reduced groupoid C∗-algebra C∗r(G) may be realised as the covariance algebra of a product system over P with coefficient algebra C∗r(c−1(e)). When (G,P) is a quasi-lattice ordered group, we also derive conditions that allow C∗r(G) to be realised as the Cuntz--Nica--Pimsner algebra of a compactly aligned product system.

A remark on non-surjective composition operators

Assume that A, B are uniform algebras on compact Hausdorff spaces X and Y, respectively, and ∂A, ∂B are the Šilov boundaries of A, B. Let T : A −1 → B− 1 be a map with T1 = 1. We show that, if there exist constants α, β ≥ 1 such that β− 1‖f·g− 1‖ ≤ ‖Tf·(Tg) − 1∥ ≤ α∥f·g− 1∥ for all f, g ∈ A− 1, then there is a non-empty closed subset Y 0 of ∂B and a surjective continuous map τ : Y 0 → ∂A such that for all f ∈ A− 1 and all y ∈ Y 0. Moreover we give an example which shows that the multiple α 2 β in the above inequality is the best possible.

On Banach spaces of the form $$C_0(\alpha \times L)$$ with few operators

For an exotic locally compact Hausdorff space L, constructed under the assumption of Ostaszewski’s $$\clubsuit $$ -principle, and a countable ordinal space $$\alpha $$ , we prove that all operators defined on $$C_0(\alpha \times L)$$ have the simplest possible form. We also investigate the geometry of such space $$C_0(\alpha \times L)$$ and we classify up to isomorphisms all its complemented subspaces.

The Maximum Entropy of a Metric Space

AbstractWe define a one-parameter family of entropies, each assigning a real number to any probability measure on a compact metric space (or, more generally, a compact Hausdorff space with a notion of similarity between points). These generalize the Shannon and Rényi entropies of information theory. We prove that on any space X, there is a single probability measure maximizing all these entropies simultaneously. Moreover, all the entropies have the same maximum value: the maximum entropy of X. As X is scaled up, the maximum entropy grows, and its asymptotics determine geometric information about X, including the volume and dimension. And the large-scale limit of the maximizing measure itself provides an answer to the question: what is the canonical measure on a metric space? Primarily, we work not with entropy itself but its exponential, which in its finite form is already in use as a measure of biodiversity. Our main theorem was first proved in the finite case by Leinster and Meckes.

Characterizations of continuous operators on C_b(X) with the strict topology

Let X be a completely regular Hausdorff space and C_b(X) be the space of all bounded continuous functions on X, equipped with the strict topology beta . We study some important classes of (beta ,Vert cdot Vert _E)-continuous linear operators from C_b(X) to a Banach space (E,Vert cdot Vert _E): beta -absolutely summing operators, compact operators and beta -nuclear operators. We characterize compact operators and beta -nuclear operators in terms of their representing measures. It is shown that dominated operators and beta -absolutely summing operators T:C_b(X)rightarrow E coincide and if, in particular, E has the Radon–Nikodym property, then beta -absolutely summing operators and beta -nuclear operators coincide. We generalize the classical theorems of Pietsch, Tong and Uhl concerning the relationships between absolutely summing, dominated, nuclear and compact operators on the Banach space C(X), where X is a compact Hausdorff space.

A note on locally compact subsemigroups of compact groups

An elementary proof is given for the fact that every locally compact subsemigroup of a compact topological group is a closed subgroup. A sample consequence is that every commutative cancellative pseudocompact locally compact Hausdorff topological semigroup with open shifts is a compact topological group.

The Topological Entropy Conjecture

For a compact Hausdorff space X, let J be the ordered set associated with the set of all finite open covers of X such that there exists nJ, where nJ is the dimension of X associated with ∂. Therefore, we have Hˇp(X;Z), where 0≤p≤n=nJ. For a continuous self-map f on X, let α∈J be an open cover of X and Lf(α)={Lf(U)|U∈α}. Then, there exists an open fiber cover L˙f(α) of Xf induced by Lf(α). In this paper, we define a topological fiber entropy entL(f) as the supremum of ent(f,L˙f(α)) through all finite open covers of Xf={Lf(U);U⊂X}, where Lf(U) is the f-fiber of U, that is the set of images fn(U) and preimages f−n(U) for n∈N. Then, we prove the conjecture logρ≤entL(f) for f being a continuous self-map on a given compact Hausdorff space X, where ρ is the maximum absolute eigenvalue of f*, which is the linear transformation associated with f on the Čech homology group Hˇ*(X;Z)=⨁i=0nHˇi(X;Z).

Computing with continuous objects: a uniform co-inductive approach

AbstractA uniform approach to computing with infinite objects like real numbers, tuples of these, compacts sets and uniformly continuous maps is presented. In the work of Berger, it was shown how to extract certified algorithms working with the signed digit representation from constructive proofs. Berger and the present author generalised this approach to complete metric spaces and showed how to deal with compact sets. Here, we unify this work and lay the foundations for doing a similar thing for the much more comprehensive class of compact Hausdorff spaces occurring in applications. The approach is of the same computational power as Weihrauch’s Type-Two Theory of Effectivity.

Purely Infinite Locally Compact Hausdorff étale Groupoids and Their C*-algebras

AbstractIn this paper, we introduce properties including groupoid comparison, pure infiniteness, and paradoxical comparison as well as a new algebraic tool called groupoid semigroup for locally compact Hausdorff étale groupoids. We show these new tools help establishing pure infiniteness of reduced groupoid $C^*$-algebras. As an application, we show a dichotomy of stably finiteness against pure infiniteness for reduced groupoid $C^*$-algebras arising from locally compact Hausdorff étale minimal topological principal groupoids. This generalizes the dichotomy obtained by Bönicke–Li and Rainone–Sims. We also study the relation among our paradoxical comparison, $n$-filling property, and locally contracting property that appeared in the literature for locally compact Hausdorff étale groupoids.

A Banach space induced by an almost disjoint family, admitting only few operators and decompositions

We consider the closed subspace of ℓ∞ generated by c0 and the characteristic functions of elements of an uncountable, almost disjoint family A of infinite subsets of N. This Banach space has the form C0(KA) for a locally compact Hausdorff space KA that is known under many names, including Ψ-space and Isbell–Mrówka space.We construct an uncountable, almost disjoint family A such that the algebra of all bounded linear operators on C0(KA) is as small as possible in the precise sense that every bounded linear operator on C0(KA) is the sum of a scalar multiple of the identity and an operator that factors through c0 (which in this case is equivalent to having separable range). This implies that C0(KA) has the fewest possible decompositions: whenever C0(KA) is written as the direct sum of two infinite-dimensional Banach spaces X and Y, either X is isomorphic to C0(KA) and Y to c0, or vice versa. These results improve previous work of the first named author in which an extra set-theoretic hypothesis was required. We also discuss the consequences of these results for the algebra of all bounded linear operators on our Banach space C0(KA) concerning the lattice of closed ideals, characters and automatic continuity of homomorphisms.To exploit the perfect set property for Borel sets as in the classical construction of an almost disjoint family by Mrówka, we need to deal with N×N matrices rather than with the usual partitioners of an almost disjoint family. This noncommutative setting requires new ideas inspired by the theory of compact and weakly compact operators and the use of an extraction principle due to van Engelen, Kunen and Miller concerning Borel subsets of the square.

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.