Compact Metric Space Research Articles

Some aspects of dimension theory for topological groups

We discuss dimension theory in the class of all topological groups. For locally compact topological groups there are many classical results in the literature. Dimension theory for non-locally compact topological groups is mysterious. It is for example unknown whether every connected (hence at least 1-dimensional) Polish group contains a homeomorphic copy of [0,1]. And it is unknown whether there is a homogeneous metrizable compact space the homeomorphism group of which is 2-dimensional. Other classical open problems are the following ones. Let G be a topological group with a countable network. Does it follow that dimG=indG=IndG? The same question if X is a compact coset space. We also do not know whether the inequality dim(G×H)≤dimG+dimH holds for arbitrary topological groups G and H which are subgroups of σ-compact topological groups. The aim of this paper is to discuss such and related problems. But we do not attempt to survey the literature.

The disappearance of causality at small scale in almost-commutative manifolds

This paper continues the investigations of noncommutative ordered spaces put forward by one of the authors. These metaphoric spaces are defined dually by the so-called isocones which generalize to the noncommutative setting the convex cones of order-preserving functions. In this paper we will consider the case of isocones inside almost-commutative algebras of the form C(M)⊗Af, with M a compact metrizable space. We will give a family of isocones in such an algebra with the property that every possible isocone is contained in exactly one member of the family. We conjecture that this family is in fact a complete classification, a hypothesis related with the noncommutative Stone-Weierstrass conjecture. We also obtain that every isocone in C(M)⊗Af, with Af noncommutative, induces an order relation on M with the property that every point in M lies in a neighbourhood of incomparable points. Thus, if the causal order relation on spacetime is induced by an isocone in an almost-commutative (but not commutative) algebra, then causality must disappear at small scale. The usual Lorentzian causality would then only arise as an approximation of this noncommutative causality.

Real-linear surjective isometries between function spaces

We study surjective isometries between subspaces of continuous functions containing all constant functions and separating the points of the underlying spaces. In many contexts, every such isometry is represented by a combination of a weighted composition operator and its complex conjugate, called the canonical form, while there exists an isometry which does not take such a form ( [14] ). We seek a topological condition on compact Hausdorff spaces such that every surjective isometry on function spaces over the spaces has the canonical form. Also we extend the construction of [14] to show that, if a compact metrizable space X admits a semi-free action of the circle group with a global section, then there exists an isometry of a function space on X which does not take the canonical form.

Construction of minimal skew products of amenable minimal dynamical systems

For an amenable minimal topologically free dynamical system \alpha of a group on a compact metrizable space Z and for a compact metrizable space Y satisfying a mild condition, we construct a minimal skew product extension of \alpha on Z \times Y . This generalizes a result of Glasner and Weiss. We also study the pure infiniteness of the crossed products of minimal dynamical systems arising from this result. In particular, we give a generalization of a result of Rørdam and Sierakowski.

The structure of ideals, point derivations, amenability and weak amenability of extended Lipschitz algebras

Let $(X,d)$ be a compactmetric space and let $K$ be a nonempty compact subset of $X$. Let $alpha in (0, 1]$ and let ${rm Lip}(X,K,d^ alpha)$ denote the Banach algebra of all continuous complex-valued functions $f$ on$X$ for which$$p_{(K,d^alpha)}(f)=sup{frac{|f(x)-f(y)|}{d^alpha(x,y)} : x,yin K , xneq y}<infty$$when it is equipped with the algebra norm $||f||_{{rm Lip}(X, K, d^ {alpha})}= ||f||_X+ p_{(K,d^{alpha})}(f)$, where $||f||_X=sup{|f(x)|:~xin X }$. In this paper we first study the structure of certain ideals of ${rm Lip}(X,K,d^alpha)$. Next we show that if $K$ is infinite and ${rm int}(K)$ contains a limit point of $K$ then ${rm Lip}(X,K,d^alpha)$ has at least a nonzero continuous point derivation and applying this fact we prove that ${rm Lip}(X,K,d^alpha)$ is not weakly amenable and amenable.

Linear continuous surjections of Cp-spaces over compacta

Let X and Y be compact Hausdorff spaces and suppose that there exists a linear continuous surjection T:Cp(X)→Cp(Y), where Cp(X) denotes the space of all real-valued continuous functions on X endowed with the pointwise convergence topology. We prove that dim⁡X=0 implies dim⁡Y=0. This generalizes a previous theorem [7, Theorem 3.4] for compact metrizable spaces. Also we point out that the function space Cp(P) over the pseudo-arc P admits no densely defined linear continuous operator Cp(P)→Cp([0,1]) with a dense image.

On the Krein–Milman–Ky Fan theorem for convex compact metrizable sets

We extend the extension by Ky Fan of the Krein–Milman theorem. The $\Phi $-extreme points of a $\Phi $-convex compact metrizable space are replaced by the $\Phi $-exposed points in the statement of Ky Fan theorem. Our main results are based on new variational principles which are of independent interest. Several applications will be given.

Isometries of Spaces of Radon Measures

Let Ω and I denote a compact metrizable space with card(Ω)≥2 and the unit interval, respectively. We prove Milutin and Cantor-Bernstein type theorems for the spaces M(Ω) of Radon measures on compact Hausdorff spaces Ω. In particular, we obtain the following results: (1) for every infinite closed subset K of βN the spaces M(K), M(βN), and M(Ω2ℵ0) are order-isometric; (2) for every discrete space Γ with m≔card(Γ)&gt;ℵ0 the spaces M(βΓ) and M(I2m) are order-isometric, whereas there is no linear homeomorphic injection from C(βT) into C(I2m).

φ-contractibility of some classes of Banach function algebras

In this paper, we study ?-contractibility of natural Banach function algebras on a compact Hausdorff space. As a consequence, we characterize ?-contractibility of the Lipschitz algebra Lip(X,d?), for a compact metric space (X,d). We also characterize ?-contractibility of certain subalgebras of Lipschitz functions including rational Lipschitz algebras, analytic Lipschitz algebras and differentiable Lipschitz algebras.

Entropy approximation versus uniqueness of equilibrium for a dense affine space of continuous functions

We show that for a [Formula: see text]-action (or [Formula: see text]-action) on a non-empty compact metrizable space [Formula: see text], the existence of a affine space dense in the set of continuous functions on [Formula: see text] constituted by elements admitting a unique equilibrium state implies that each invariant measure can be approximated weakly[Formula: see text] and in entropy by a sequence of measures which are unique equilibrium states.

Minimum topological group topologies

A Hausdorff topological group topology on a group G is the minimum (Hausdorff) group topology if it is contained in every Hausdorff group topology on G. For every compact metrizable space X containing an open n-cell, n≥2, the homeomorphism group H(X) has no minimum group topology. The homeomorphism groups of the Cantor set and the Hilbert cube have no minimum group topology. For every compact metrizable space X containing a dense open one-manifold, H(X) has the minimum group topology. Some, but not all, oligomorphic groups have the minimum group topology.

Rokhlin Dimension for Flows

We introduce a notion of Rokhlin dimension for one parameter automorphism groups of C*-algebras. This generalizes Kishimoto's Rokhlin property for flows, and is analogous to the notion of Rokhlin dimension for actions of the integers and other discrete groups introduced by the authors and Zacharias in previous papers. We show that finite nuclear dimension and absorption of a strongly self-absorbing C*-algebra are preserved under forming crossed products by flows with finite Rokhlin dimension, and that these crossed products are stable. Furthermore, we show that a flow on a commutative C*-algebra arising from a free topological flow has finite Rokhlin dimension, whenever the spectrum is a locally compact metrizable space with finite covering dimension. For flows that are both free and minimal, this has strong consequences for the associated crossed product C*-algebras: Those containing a non-zero projection are classified by the Elliott invariant (for compact manifolds this consists of topological K-theory together with the space of invariant probability measures and a natural pairing given by the Ruelle-Sullivan map).

Reconstructing compact metrizable spaces

The deck, $\mathcal{D}(X)$, of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus \{x\}]\colon x \in X\}$, where $[Y]$ denotes the homeomorphism class of $Y$. A space $X$ is (topologically) reconstructible if whenever $\mathcal{D}(Z)=\mathcal{D}(X)$ then $Z$ is homeomorphic to $X$. It is known that every (metrizable) continuum is reconstructible, whereas the Cantor set is non-reconstructible. The main result of this paper characterises the non-reconstructible compact metrizable spaces as precisely those where for each point $x$ there is a sequence $\langle B_n^x \colon n \in \mathbb{N}\rangle$ of pairwise disjoint clopen subsets converging to $x$ such that $B_n^x$ and $B_n^y$ are homeomorphic for each $n$, and all $x$ and $y$. In a non-reconstructible compact metrizable space the set of $1$-point components forms a dense $G_\delta$. For $h$-homogeneous spaces, this condition is sufficient for non-reconstruction. A wide variety of spaces with a dense $G_\delta$ set of $1$-point components are presented, some reconstructible and others not reconstructible.

Fréchet–Urysohn subspaces of free topological groups

Let F(X) be the free topological group on a Tychonoff space X. For all natural numbers n we denote by Fn(X) the subset of F(X) consisting of all words of reduced length ≤n. In [10], the author found equivalent conditions on a metrizable space X for F3(X) to be Fréchet–Urysohn, and for Fn(X) to be Fréchet–Urysohn for n≥5. However, no equivalent condition on X for n=4 was found. In this paper, we prove that for a locally compact metrizable space X such that the set of all non-isolated points of X is compact, F4(X) is Fréchet–Urysohn if F4(X) is a k-space. We obtain as a corollary that if a metrizable space X is locally compact separable and the set of all non-isolated points of X is compact, then F4(X) is Fréchet–Urysohn. Consequently, for a metrizable space X=C⊕D such that C is an infinite compact set and D is a countably infinite discrete subset of X, F4(X) is Fréchet–Urysohn but F5(X) is not.

On the Krein–Milman–Ky Fan theorem for convex compact metrizable sets

We extend the extension by Ky Fan of the Krein–Milman theorem. The $\Phi$-extreme points of a $\Phi$-convex compact metrizable space are replaced by the $\Phi$-exposed points in the statement of Ky Fan theorem. Our main results are based on new variational principles which are of independent interest. Several applications will be given.

Invariant subsets under compact quantum group actions

We investigate compact quantum group actions on unital C^* -algebras by analyzing invariant subsets and invariant states. In particular, we come up with the concept of compact quantum group orbits and use it to show that countable compact metrizable spaces with infinitely many points are not quantum homogeneous spaces.

The Blum–Hanson property for 𝒞(K) spaces

We show that if K is a compact metrizable space, then the Banach space C.K/ has the so-called Blum‐Hanson property exactly when K has finitely many accumulation points. We also show that the space ‘1.N/ D C. N/ does not have the Blum‐Hanson property.

A Dixmier–Douady theory for strongly self-absorbing $C^\ast$-algebras II: the Brauer group

We have previously shown that the isomorphism classes of orientable locally trivial fields of C^\ast -algebras over a compact metrizable space X with fiber D \otimes \mathbb K , where D is a strongly self-absorbing C^\ast -algebra, form an abelian group under the operation of tensor product. Moreover this group is isomorphic to the first group \bar{E}^1_D (X) of the (reduced) generalized cohomology theory associated to the unit spectrum of topological K-theory with coefficients in D . Here we show that all the torsion elements of the group \bar{E}^1_D(X) arise from locally trivial fields with fiber D \otimes M_n (\mathbb C) , n\geq 1 , for all known examples of strongly self-absorbing C^\ast -algebras} D . Moreover the Brauer group generated by locally trivial fields with fiber {D\otimes M_n (\mathbb C)} , n\geq 1 is isomorphic to T or {\bar{E}^1_D(X)} .

Complex vector lattices via functional completions

We show that the Fremlin tensor product C(X)⊗¯C(Y) is not square mean complete when X and Y are uncountable metrizable compact spaces. This motivates the definition of complexification of Archimedean vector lattices, the Fremlin tensor product of Archimedean complex vector lattices, and a theory of powers of Archimedean complex vector lattices.

Classifying invariant $\sigma $-ideals with analytic base on good Cantor measure spaces

Let $X$ be a zero-dimensional compact metrizable space endowed with a strictly positive continuous Borel $\sigma$-additive measure $\mu$ which is good in the sense that for any clopen subsets $U,V\subset X$ with $\mu(U)<\mu(V)$ there is a clopen set $W\subset V$ with $\mu(W)=\mu(U)$. We study $\sigma$-ideals with Borel base on $X$ which are invariant under the action of the group $H_\mu(X)$ of measure-preserving homeomorphisms of $(X,\mu)$, and show that any such $\sigma$-ideal $\mathcal I$ is equal to one of seven $\sigma$-ideals: $\{\emptyset\}$, $[X]^{\le\omega}$, $\mathcal E$, $\mathcal M\cap\mathcal N$, $\mathcal M$, $\mathcal N$, or $[X]^{\le \mathfrak c}$. Here $[X]^{\le\kappa}$ is the ideal consisting of subsets of cardiality $\le\kappa$ in $X$, $\mathcal M$ is the ideal of meager subsets of $X$, $\mathcal N=\{A\subset X:\mu(A)=0\}$ is the ideal of null subsets of $(X,\mu)$, and $\mathcal E$ is the $\sigma$-ideal generated by closed null subsets of $(X,\mu)$.