Recent zbMATH articles in MSC 54Dhttps://zbmath.org/atom/cc/54D2023-09-22T14:21:46.120933ZWerkzeugSome new results about left ideals of \(\beta S\)https://zbmath.org/1517.220012023-09-22T14:21:46.120933Z"Hindman, Neil"https://zbmath.org/authors/?q=ai:hindman.neil"Strauss, Dona"https://zbmath.org/authors/?q=ai:strauss.donaGiven a semigroup \(S\) with the discrete topology, there is a compact, right topological semigroup structure on its Stone-Čech compactification \(\beta S\). The minimal left ideals of \(\beta S\), which are also universal minimal \(S\)-flows with respect to the natural action of \(S\) on \(\beta S\), play a foundational part in the theory of \(\beta S\). In this paper the authors investigate (and answer) several natural questions concerning these minimal left ideals.
First is the question of when the minimal left ideals of \(\beta S\) are finite. If \(S\) already contains a finite left ideal, then this remains true (for relatively simple reasons) in \(\beta S\). The situation becomes more subtle, and more interesting, when all the minimal left ideals of \(\beta S\) are contained in \(\beta S \setminus S\). In this case the authors formulate some necessary conditions for the existence of finite minimal left ideals. They also show that these necessary conditions are not sufficient.
Next, the authors investigate how the taking of Cartesian products interacts with the taking of minimal left ideals. Are the minimal left ideals of a product of semigroups each equal (identifiable via a topological isomorphism) with the product of a minimal left ideal of each of the factors? Several results are proved in this vein.
Lastly, the authors show that, if \(S\) is any countably infinite cancellative semigroup, then every non-minimal semiprincipal left ideal in \(\beta S\) contains many semiprincipal left ideals defined by right cancelable elements of \(\beta S\). Some of the properties of these left ideals are explored.
Reviewer: Will Brian (Charlotte)Subspaces of Hilbert-generated Banach spaces and the quantification of super weak compactnesshttps://zbmath.org/1517.460052023-09-22T14:21:46.120933Z"Grelier, G."https://zbmath.org/authors/?q=ai:grelier.guillaume"Raja, M."https://zbmath.org/authors/?q=ai:raja.manakkulam-rohith|raja.m-mohan|raja.muhammad-saeed|raja.marimuthu-mohan|raja.matias|raya.maxim|raja.muhammad-asif-zahoorA Banach space \(X\) is said to be weakly compactly generated (WCG, for short) (respectively, Hilbert-generated (HG, for short)) whenever there exist a bounded linear operator from a reflexive (respectively, Hilbert) space into a dense subspace of \(X\). Due to the non-heredity of those notions, characterizing the class consisting of subspaces (i.e., SWCG, SHG, respectively) is relevant for the theory.
(i) By introducing the measure of weak noncompactness \(\gamma(A):=\inf\{\varepsilon>0: \overline{A}^{w^*} \subset X+ \varepsilon B_{X^{**}}\}\) for \(A\subset X\), elements in SWCG were characterized as those \(X\) that, for every \(\varepsilon>0\), \(B_X=\bigcup_{n\in\mathbb N}A^{\varepsilon}_{n}\) with \(\gamma(A^{\varepsilon}_{n})<\varepsilon\) for all \(n\in\mathbb N\) [\textit{M. Fabian} et al., J. Lond. Math. Soc., II. Ser. 69, No. 2, 457--464 (2004; Zbl 1059.46014)].
(ii) According to [\textit{M. Fabian} et al., Isr. J. Math. 124, 243--252 (2001; Zbl 1027.46012)], see also [\textit{M. Fabian} et al., J. Math. Anal. Appl. 297, No. 2, 419--455 (2004; Zbl 1063.46013)], \(X\) is SHG if, and only if, there exists a total set \(\Gamma\subset B_{X}\) such that, for every \(\varepsilon>0\), \(\Gamma=\bigcup_{n=1}^{\infty}\Gamma^\varepsilon_{n}\) where for every \(n\in\mathbb N\) and for every \(x^*\in B_{X^*}\), \(\sharp\{\gamma\in\Gamma^{\varepsilon}_{n}:\ \langle \gamma,x^*\rangle>\varepsilon\}<n\).
The main result here concerns (ii) above in the spirit of (i): \(X\) is SHG if, and only if, for every \(\varepsilon>0\), \(B_X=\bigcup_{n\in\mathbb N}A^{\varepsilon}_{n}\) with \(\Gamma(A^{\varepsilon}_{n})<\varepsilon\) for all \(n\in\mathbb N\), where \(\Gamma(\cdot)\) is a measure of super weak noncompactness, i.e., \(\Gamma(A):=\gamma(A^{\mathcal{U}})\), \(\mathcal{U}\) an (arbitrary) free ultrafilter on \(\mathbb N\). The notion of super weak compactness, due to
\textit{L.-X. Cheng} et al. [Stud. Math. 199, No.~2, 145--169 (2010; Zbl 1252.46009)]
-- equivalently, in the case of closed convex and bounded sets, the finite dentability, due to \textit{M.~Raja} [J. Convex Anal. 15, No.~2, 219--233 (2008; Zbl 1183.46018)]
-- lies strictly between norm and weak compactness, and it is crucial here. A~set \(K\subset X\) is super weakly compact whenever \(K^{\mathcal{U}}\) is weakly compact for any free ultrafilter \(\mathcal{U}\).
Related to the characterization in (ii) above, the relevant notion of uniformly weakly null sets is introduced, allowing for connections with several Banach-Saks type properties. A section is devoted to quantifying uniform convexity for operators. The paper ends by formulating an open problem for the class of the so-called super WCG spaces.
It is worth noticing (see [Zbl 1027.46012]) that SHG spaces are those that have an equivalent uniformly Gâteaux smooth norm. This is also mentioned in the paper.
Reviewer: Vicente Montesinos (Valencia)Menger and Menger-type star selection principles for hit-and-miss topologyhttps://zbmath.org/1517.540032023-09-22T14:21:46.120933Z"Cruz-Castillo, Ricardo"https://zbmath.org/authors/?q=ai:cruz-castillo.ricardo"Ramírez-Páramo, Alejandro"https://zbmath.org/authors/?q=ai:ramirez-paramo.alejandro"Tenorio, Jesús F."https://zbmath.org/authors/?q=ai:tenorio.jesus-fRecall that \({{\left[ A \right]}^{<\omega }}\) denotes the collection of all finite subsets of any set \(A\). \textit{Z. Li} [Topology Appl. 212, 90--104 (2016; Zbl 1355.54014)] defined the concepts of \({{\pi }_{F}}\)-network and \({{\pi }_{V}}\)-network for a space \(X\) to characterize the Rothberger and Menger properties for \((CL(X),{{\tau }_{F}})\) and \((CL(X),{{\tau }_{V}})\), respectively. The families of all \({{\pi }_{F}}\)-networks and all \({{\pi }_{V}}\)-networks are denoted by \({{\Pi }_{F}}\) and \({{\Pi }_{V}}\), respectively.
\textbf{Definition 2.1}. A family \(\zeta \) is called a \({{\pi }_{\Delta }}(\Gamma )\)-network of \(X\), if for each \(U\in {{\Gamma }^{c}}\), there exist \((B;{{V}_{1}},{{V}_{2}},\dots,{{V}_{n}})\in \zeta \) with \(B\subset U\) and \(F\in {{\left[ X \right]}^{<\omega }}\) such that \(F\cap U=\varnothing \) and for each \(i\in \{1,2,\dots,n\}\), \(F\cap {{V}_{i}}\ne \varnothing \). The family of all \({{\pi }_{\Delta }}(\Gamma )\)-networks is denoted by \({{\Pi }_{\Delta }}(\Gamma )\).
The authors define the following selection principle related with \({{\pi }_{\Delta }}(\Gamma )\)-networks.
\textbf{Definition 2.10.} Let \((X,\tau )\) be a topological space. We say that \((X,\tau )\) satisfies the principle \({{S}_{M}}({{\Pi }_{\Delta }}(\Gamma ),{{\Pi }_{\Delta }}(\Gamma ))\), if for any sequence \(\{{{J}_{n}}:\,\,n\in \mathbb{N}\}\subset {{\Pi }_{\Delta }}(\Gamma )\), there exists a sequence \(\{\mathcal{V}_{n}:\,\,n\in \mathbb{N}\}\subset {{\left[ {{\Gamma }^{c}} \right]}^{<\omega }}\), with \({{V}_{n}}\ne \varnothing \) for each \(n\in \mathbb N\), such that
\[
\mathcal{J}=\bigcup\limits_{n\in \mathbb N}{\{(B_{s}^{n};V_{1,s}^{n},V_{2,s}^{n},\dots ,V_{{{m}_{s}},s}^{n})\in {{J}_{n}}:\,\,\,\text{there exists }V\in {\mathcal{V}_{n}}\text{ with }B_{s}^{n}\subset V,\,\,V_{i,s}^{n}\not\subset V(1\le i\le {{m}_{s}})\}}
\]
is an element of \({{\Pi }_{\Delta }}(\Gamma )\).
The authors obtain the following results:
\textbf{Theorem 2.11}. Let \((X,\tau )\) be a topological space. The following conditions are equivalent:
\begin{itemize}
\item[1.] \((\Gamma ,\tau _{\Delta }^{+})\) is SSM;
\item[2.] \((X,\tau )\) satisfies \(S_{M}({{\Pi }_{\Delta }}(\Gamma ),{{\Pi }_{\Delta }}(\Gamma ))\).
\end{itemize}
\textbf{Theorem 2.16}. Let \((X,\tau )\) be a topological space. Suppose furthermore that for every \(x\in X\), \(\{x\}\in \Gamma \). The following conditions are equivalent:
\begin{itemize}
\item[1.] \((\Gamma ,\tau _{\Delta }^{+})\) is SM;
\item[2.] \(X\) satisfies \(S_{M}^{*}({{\Pi }_{\Delta }}(\Gamma ),{{\Pi }_{\Delta }}(\Gamma ))\).
\end{itemize}
\textbf{Corollary 2.17}. Let \((X,\tau )\) be a topological space. If \(\Gamma \) is any of the hyperspaces \(CL(X)\), \(\mathbb{K}(X)\), \(F(X)\) or \(\mathbb{CS}(X)\), then \((\Gamma ,\tau _{\Delta }^{+})\) is SM if and only if \(X\) satisfies the principle \(S_{M}^{*}({{\Pi }_{\Delta }}(\Gamma ),{{\Pi }_{\Delta }}(\Gamma ))\).
\textbf{Lemma 2.22}. Let \((X,\tau )\) be a topological space. A family \(\mathcal{U}\subset\Gamma^{c}\) is a \(c_{\Delta}(\Gamma)\)-cover of \(X\) if and only if the family \(\mathcal{U}^{c}\) is a dense subset of \((\Gamma ,\tau _{\Delta }^{+})\).
\textbf{Theorem 2.24}. Let \((X,\tau )\) be a topological space. The following conditions are equivalent:
\begin{itemize}
\item[1.] \((\Gamma ,\tau _{\Delta }^{+})\) satisfies \(S_{\mathrm{fin}}(\mathcal{D},\mathcal{D})\);
\item[2.] \((X,\tau )\) satisfies \(S_{\mathrm{fin}}(\mathbb{C}_{\Delta}(\Gamma), \mathbb{C}_{\Delta}(\Gamma))\).
\end{itemize}
Reviewer: Ruzinazar Beshimov (Tashkent)Johnstone-Gleason covers for partially ordered setshttps://zbmath.org/1517.540092023-09-22T14:21:46.120933Z"Abashidze, Vakhtang"https://zbmath.org/authors/?q=ai:abashidze.vakhtangThe classical Gleason cover \(\tilde{X}\) of a compact Hausdorff space \(X\) is the Stone dual of the complete Boolean algebra of its regular closed sets. \(\tilde{X}\) is an extremally disconnected compact Hausdorff space equipped with a continuous surjection \(p \colon \tilde{X} \to X\) with the property that every other continuous surjection from an extremally disconnected compact Hausdorff space onto \(X\) factors through \(p\) [\textit{A. M. Gleason}, Ill. J. Math. 2, 482--489 (1958; Zbl 0083.17401)].
Several authors have introduced various generalizations of this construction to many classes of spaces (usually referred to as an \textit{absolute} of the space). Further, in 1980, Johnstone introduced a construction of the Gleason cover for an arbitrary elementary topos which in particular provides a certain version of absolute for any topological space and, more generally, for locales.
In the paper under review, the author investigates properties of the Gleason cover in case of arbitrary partially ordered sets with the Alexandroff topology, that is, (not necessarily sober) \(T_0\) Alexandroff spaces. First, the notion of co-local homeomorphism for such spaces is introduced, and it is proved that for every finite \(T_0\) topological space \(X\) there exists a unique irreducible co-local homeomorphism \(p \colon \tilde{X}\to X\) from a finite extremally disconnected space \(\tilde{X}\) onto \(X\). Next, this approach is extended to Alexandroff topologies of arbitrary partially ordered sets. When this topology is sober, the generalization is straightforward, it merely repeats that of the finite case. However, for non-sober topologies, the Gleason cover will not necessarily produce a spatial locale, let alone an Alexandroff space of some other poset.
The paper ends with characterizations of posets for which the Gleason cover of the corresponding Alexandroff space is itself Alexandroff. Namely, the following theorem is proved:
Theorem 4.13. The Gleason cover of an Alexandroff topological space \(X\) is an Alexandroff space iff for any \(x\in X\) and for any infinite antichain \(S \subseteq\, \uparrow\! x\) there exist \(y_1, y_2 \in S\) such that \(\, \uparrow\! y_1\ \cap \uparrow\! y_2 \neq \emptyset\).
Reviewer: Jorge Picado (Coimbra)Dark energy star in gravity's rainbowhttps://zbmath.org/1517.830242023-09-22T14:21:46.120933Z"Bagheri Tudeshki, A."https://zbmath.org/authors/?q=ai:bagheri-tudeshki.a"Bordbar, G. H."https://zbmath.org/authors/?q=ai:bordbar.g-h"Eslam Panah, B."https://zbmath.org/authors/?q=ai:eslam-panah.bSummary: The concept of dark energy can be a candidate for preventing the gravitational collapse of compact objects to singularities. According to the usefulness of gravity's rainbow in UV completion of general relativity (by providing a new description of spacetime), it can be an excellent option to study the behavior of compact objects near phase transition regions. In this work, we obtain a modified Tolman-Openheimer-Volkof (TOV) equation for anisotropic dark energy as a fluid by solving the field equations in gravity's rainbow. Next, to compare the results with general relativity, we use a generalized Tolman-Matese-Whitman mass function to determine the physical quantities such as energy density, radial pressure, transverse pressure, gravity profile, and anisotropy factor of the dark energy star. We evaluate the junction condition and investigate the dynamical stability of dark energy star thin shell in gravity's rainbow. We also study the energy conditions for the interior region of this star. We show that the coefficients of gravity's rainbow can significantly affect this non-singular compact object and modify the model near the phase transition region.