Compact Hausdorff Research Articles

Cardinal-indexed classifying spaces for families of subgroups of any topological group

For G a topological group, existence theorems by Milnor (1956), Gelfand–Fuks (1968), and Segal (1975) of classifying spaces for principal G-bundles are generalized to G-spaces with torsion. Namely, any G-space approximately covered by tubes (a generalization of local trivialization) is the pullback of a universal space indexed by the orbit types of tubes and cardinality of the cover. For G a Lie group, via a metric model we generalize the corresponding uniqueness theorem by Palais (1960) and Bredon (1972) for compact G. Namely, the G-homeomorphism types of proper G-spaces over a metric space correspond to stratified-homotopy classes of orbit classifying maps.The former existence result is enabled by Segal's clever but esoteric use of non-Hausdorff spaces. The latter uniqueness result is enabled by our own development of equivariant ANR theory for noncompact Lie G. Applications include the existence part of classification for unstructured fiber bundles with locally compact Hausdorff fiber and with locally connected base or fiber, as well as for equivariant principal bundles which in certain cases via other models is due to Lashof–May (1986) and to Lück–Uribe (2014). From a categorical perspective, our general model EFκG is a final object inspired by the formulation of the Baum–Connes conjecture (1994).

Schatten class and nuclear pseudo-differential operators on homogeneous spaces of compact groups

Given a compact (Hausdorff) group G and a closed subgroup H of G, in this paper we present symbolic criteria for pseudo-differential operators on the compact homogeneous space G/H characterizing the Schatten–von Neumann classes \(S_r(L^2(G/H))\) for all \(0<r \le \infty .\) We go on to provide a symbolic characterization for r-nuclear, \(0< r \le 1,\) pseudo-differential operators on \(L^{p}(G/H)\) with applications to adjoint, product and trace formulae. The criteria here are given in terms of matrix-valued symbols defined on noncommutative analogue of phase space \(G/H \times \widehat{G/H}.\) Finally, we present an application of aforementioned results in the context of the heat kernels.

Free noncommutative hereditary kernels: Jordan decomposition, Arveson extension, kernel domination

We discuss (i) a quantized version of the Jordan decomposition theorem for a complex Borel measure on a compact Hausdorff space, namely, the more general problem of decomposing a general noncommutative kernel (a quantization of the standard notion of kernel function) as a linear combination of completely positive noncommutative kernels (a quantization of the standard notion of positive definite kernel). Other special cases of (i) include: the problem of decomposing a general operator-valued kernel function as a linear combination of positive kernels (not always possible), of decomposing a general bounded linear Hilbert-space operator as a linear combination of positive linear operators (always possible), of decomposing a completely bounded linear map from a C^* -algebra \mathcal{A} to an injective C^* -algebra \mathcal{L}(\mathcal{Y}) as a linear combination of completely positive maps from \mathcal{A} to \mathcal{L}(\mathcal{Y}) (always possible). We also discuss (ii) a noncommutative kernel generalization of the Arveson extension theorem (any completely positive map \phi from an operator system \mathbb{S} to an injective C^* -algebra \mathcal{L}(\mathcal{Y}) can be extended to a completely positive map \phi_e from a C^* -algebra containing \mathbb{S} to \mathcal{L}(\mathcal{Y})) , and (iii) a noncommutative kernel version of a Positivstellensatz (i.e., finding a certificate to explain why one kernel is positive at points where another given kernel is strictly positive).

Inverse systems of compact Hausdorff spaces and (m, n)-dimension

Inverse systems of compact Hausdorff spaces and (m, n)-dimension

Disjoint hypercyclic powers of weighted translations on locally compact Hausdorff spaces

Disjoint hypercyclic powers of weighted translations on locally compact Hausdorff spaces

On recurrence in zero-dimensional locally compact flow with compactly generated phase group

We define recurrence for a compactly generated para-topological group G acting continuously on a locally compact Hausdorff space X with dimX=0, and then we show that if Gx‾ is compact for all x∈X, the conditions are pairwise equivalent: (i) this dynamic is pointwise recurrent, (ii) X is a union of G-minimal sets, (iii) the G-orbit closure relation is closed in X×X, and (iv) X∋x↦Gx‾∈2X is continuous. Consequently, if this dynamic is pointwise product recurrent, then it is pointwise regularly almost periodic and equicontinuous; moreover, a distal, compact, and nonconnected G-flow has a nontrivial equicontinuous pointwise regularly almost periodic factor.

Browder’s theorem with general parameter space

It follows from Browder (Summa Bras Math 4:183–191, 1960) that for every continuous function $$F : (X \times Y) \rightarrow Y$$ , where X is the unit interval and Y is a nonempty, convex, and compact subset of a locally convex linear vector space, the set of fixed points of F, defined by $$C_F := \{ (x,y) \in X \times Y :F(x,y)=y\}$$ , has a connected component whose projection to the first coordinate is X. We extend Browder’s result to the case that X is a connected and compact Hausdorff space.

A generalized Powers averaging property for commutative crossed products

We prove a generalized version of Powers’ averaging property that characterizes simplicity of reduced crossed products C ( X ) ⋊ λ G C(X) \rtimes _\lambda G , where G G is a countable discrete group, and X X is a compact Hausdorff space which G G acts on minimally by homeomorphisms. As a consequence, we generalize results of Hartman and Kalantar on unique stationarity to the state space of C ( X ) ⋊ λ G C(X) \rtimes _\lambda G and to Kawabe’s generalized space of amenable subgroups Sub a ⁡ ( X , G ) \operatorname {Sub}_a(X,G) . This further lets us generalize a result of the first named author and Kalantar on simplicity of intermediate C*-algebras. We prove that if C ( Y ) ⊆ C ( X ) C(Y) \subseteq C(X) is an inclusion of unital commutative G G -C*-algebras with X X minimal and C ( Y ) ⋊ λ G C(Y) \rtimes _\lambda G simple, then any intermediate C*-algebra A A satisfying C ( Y ) ⋊ λ G ⊆ A ⊆ C ( X ) ⋊ λ G C(Y) \rtimes _\lambda G \subseteq A \subseteq C(X) \rtimes _\lambda G is simple.

On the continuity of lattice isomorphisms on C(X, I)

Abstract We investigate topological conditions on a compact Hausdorff space Y, such that any lattice isomorphism φ : C(X, I) → C(Y, I), where X is a compact Hausdorff space and I is the unit interval [0, 1], is continuous. It is shown that in either of cases that the set of G δ points of Y has a dense pseudocompact subset or Y does not contain the Stone-Čech compactification of ℕ, such a lattice isomorphism is a homeomorphism.

New Directions in Duality Theory for Modal Logic

AbstractIn this work we present some new contributions towards two different directions in the study of modal logic. First we employ tense logics to provide a temporal interpretation of intuitionistic quantifiers as “always in the future” and “sometime in the past.” This is achieved by modifying the Gödel translation and resolves an asymmetry between the standard interpretation of intuitionistic quantifiers.Then we generalize the classic Gelfand–Naimark–Stone duality between compact Hausdorff spaces and uniformly complete bounded archimedean $\ell $ -algebras to a duality encompassing compact Hausdorff spaces with continuous relations. This leads to the notion of modal operators on bounded archimedean $\ell $ -algebras and in particular on rings of continuous real-valued functions on compact Hausdorff spaces. This new duality is also a generalization of the classic Jónsson-Tarski duality in modal logic.Abstract taken directly from the thesis.E-mail: lcarai@unisa.itURL: https://www.proquest.com/openview/5d284dbfb954383da9364149fa312b6f/1?pq-origsite=gscholar&amp;cbl=18750&amp;diss=y

Composition in Modulus Maps on Semigroups of Continuous Functions

For locally compact Hausdorff spaces $X$ and $Y$, and function algebras $A$ and $B$ on $X$ and $Y$, respectively, surjections $T:A \longrightarrow B$ satisfying norm multiplicative condition $\|Tf\, Tg\|_Y =\|fg\|_X$, $f,g\in A$, with respect to the supremum norms, and those satisfying $\||Tf|+|Tg|\|_Y=\||f|+|g|\|_X$ have been extensively studied. Motivated by this, we consider certain (multiplicative or additive) subsemigroups $A$ and $B$ of $C_0(X)$ and $C_0(Y)$, respectively, and study surjections $T$ from $A$ onto $B$ satisfying the norm condition $\rho(Tf, Tg)=\rho(f,g)$, $f,g \in A$, for some classes of two variable positive functions $\rho$. It is shown that such a map $T$ is also a composition in modulus map.

A New Approach to the Katětov–Tong Theorem

We give a new proof of the Katětov–Tong theorem. Our strategy is to first prove the theorem for compact Hausdorff spaces, and then extend it to all normal spaces by showing how to extend upper and lower semicontinuous real-valued functions to the Stone–Čech compactification so that the less than or equal relation between the functions is preserved. In this way, the main step of the proof is the compact case, and our approach to handling this case also leads to a new proof of a version of the Stone–Weierstrass theorem.

Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

We describe homomorphisms $$\varphi :H\rightarrow G$$ for which G is acylindrically hyperbolic and H is a topological group which is either completely metrizable or locally countably compact Hausdorff. It is shown that, in a certain sense, either the image of $$\varphi $$ is small or $$\varphi $$ is almost continuous. We also describe homomorphisms from the Hawaiian earring group to G as above. We prove a more precise result for homomorphisms $$\varphi :H\rightarrow {\text {Mod}}(\Sigma )$$ , where H is as above and $${\text {Mod}}(\Sigma )$$ is the mapping class group of a connected compact surface $$\Sigma $$ . In this case there exists an open normal subgroup $$V\leqslant H$$ such that $$\varphi (V)$$ is finite. We also prove the analogous statement for homomorphisms $$\varphi :H\rightarrow {\text {Out}}(G)$$ , where G is a one-ended hyperbolic group. Some automatic continuity results for relatively hyperbolic groups and fundamental groups of graphs of groups are also deduced. As a by-product, we prove that the Hawaiian earring group is acylindrically hyperbolic, but does not admit any universal acylindrical action on a hyperbolic space.

Commutativity of the Haagerup tensor product and base change for operator modules

By computing the completely bounded norm of the flip map on the Haagerup tensor product [Formula: see text] associated to a pair of continuous mappings of locally compact Hausdorff spaces [Formula: see text], we establish a simple characterization of the Beck-Chevalley condition for base change of operator modules over commutative [Formula: see text]-algebras, and a descent theorem for continuous fields of Hilbert spaces.

Extension of Haar’s theorem

Haar’s theorem ensures a unique nontrivial regular Borel measure on a locally compact Hausdorff topological group, up to multiplication by a positive constant. In this article, we extend Haar’s theorem to the case of locally compact Hausdorff strongly topological gyrogroups. We simultaneously prove the existence and uniqueness of a Haar measure on a locally compact Hausdorff strongly topological gyrogroup, using the method of Steinlage. We then find a natural relationship between Haar measures on gyrogroups and on their related groups. As an application of this result, we study some properties of a convolution-like operation on the space of Haar integrable functions defined on a locally compact Hausdorff strongly topological gyrogroup

Diameter two properties in some vector-valued function spaces

We introduce a vector-valued version of a uniform algebra, called the vector-valued function space over a uniform algebra. The diameter two properties of the vector-valued function space over a uniform algebra on an infinite compact Hausdorff space are investigated. Every nonempty relatively weakly open subset of the unit ball of a vector-valued function space $$A(K, (X, \tau ))$$ over an infinite dimensional uniform algebra has diameter two, where $$\tau $$ is a locally convex Hausdorff topology on a Banach space X compatible to a dual pair. Under the assumption of X equipped with the norm topology being uniformly convex and the additional condition that $$A\otimes X\subset A(K, X)$$ , it is shown that Daugavet points and $$\Delta $$ -points on A(K, X) over a uniform algebra A are the same, and they are characterized by the norm-attainment at a limit point of the Shilov boundary of A. In addition, a sufficient condition for the convex diametral local diameter two property of A(K, X) is also provided. Similar results also hold for an infinite dimensional uniform algebra.

MODAL OPERATORS ON RINGS OF CONTINUOUS FUNCTIONS

AbstractIt is a classic result in modal logic, often referred to as Jónsson-Tarski duality, that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality for boolean algebras. Our goal is to generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead of working with the boolean algebra of clopen subsets of a Stone space, we work with the ring of continuous real-valued functions on a compact Hausdorff space. The main novelty is to define a modal operator on such a ring utilizing a continuous relation on a compact Hausdorff space.Our starting point is the well-known Gelfand duality between the category ${\sf KHaus}$ of compact Hausdorff spaces and the category $\boldsymbol {\mathit {uba}\ell }$ of uniformly complete bounded archimedean $\ell $ -algebras. We endow a bounded archimedean $\ell $ -algebra with a modal operator, which results in the category $\boldsymbol {\mathit {mba}\ell }$ of modal bounded archimedean $\ell $ -algebras. Our main result establishes a dual adjunction between $\boldsymbol {\mathit {mba}\ell }$ and the category ${\sf KHF}$ of what we call compact Hausdorff frames; that is, Kripke frames equipped with a compact Hausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between ${\sf KHF}$ and the reflective subcategory $\boldsymbol {\mathit {muba}\ell }$ of $\boldsymbol {\mathit {mba}\ell }$ consisting of uniformly complete objects of $\boldsymbol {\mathit {mba}\ell }$ . This generalizes both Gelfand duality and Jónsson-Tarski duality.

Lipschitz integral operators represented by vector measures

&lt;p&gt;In this paper we introduce the concept of Lipschitz Pietsch-p-integral &lt;br /&gt;mappings, (1≤p≤∞), between a metric space and a Banach space. We represent these mappings by an integral with respect to a vector&lt;br /&gt;measure defined on a suitable compact Hausdorff space, obtaining in this way a rich factorization theory through the classical Banach spaces C(K), L_p(μ,K) and L_∞(μ,K). Also we show that this type of operators fits in the theory of composition Banach Lipschitz operator ideals. For p=∞, we characterize the Lipschitz Pietsch-∞-integral mappings by a factorization schema through a weakly compact operator. Finally, the relationship between these mappings and some well known Lipschitz operators is given.&lt;/p&gt;

Geometrical properties of the space of idempotent probability measures

Although traditional and idempotent mathematics are "parallel'', by an application of the category theory we show that objects obtained the similar rules over traditional and idempotent mathematics must not be "parallel''. At first we establish for a compact metric space X the spaces P(X) of probability measures and I(X) idempotent probability measures are homeomorphic ("parallelism''). Then we construct an example which shows that the constructions P and I form distinguished functors from each other ("parallelism'' negation). Further for a compact Hausdorff space X we establish that the hereditary normality of I&lt;sub&gt;3&lt;/sub&gt;(X)\ X implies the metrizability of X.

On maximal compact frames

Every closed subset of a compact topological space is compact. Also every compact subset of a Hausdorff topological space is closed. It follows that compact subsets are precisely the closed subsets in a compact Hausdorff space. It is also proved that a topological space is maximal compact if and only if its compact subsets are precisely the closed subsets. A locale is a categorical extension of topological spaces and a frame is an object in its opposite category. We investigate to find whether the closed sublocales are exactly the compact sublocales of a compact Hausdorff frame. We also try to investigate whether the closed sublocales are exactly the compact sublocales of a maximal compact frame.