Recent zbMATH articles in MSC 22https://zbmath.org/atom/cc/222021-01-08T12:24:00+00:00WerkzeugTopological loops with six-dimensional solvable multiplication groups having five-dimensional nilradical.https://zbmath.org/1449.220032021-01-08T12:24:00+00:00"Figula, Ágota"https://zbmath.org/authors/?q=ai:figula.agota"Ficzere, Kornélia"https://zbmath.org/authors/?q=ai:ficzere.kornelia"Al-Abayechi, Ameer"https://zbmath.org/authors/?q=ai:al-abayechi.ameerSummary: Using connected transversals we determine the six-dimensional indecomposable solvable Lie groups with five-dimensional nilradical and their subgroups which are the multiplication groups and the inner mapping groups of three-dimensional connected simply connected topological loops. Together with this result we obtain that every six-dimensional indecomposable solvable Lie group which is the multiplication group of a three-dimensional topological loop has one-dimensional centre and two- or three-dimensional commutator subgroup.Twisted topological graph algebras are twisted groupoid \(C^*\)-algebras.https://zbmath.org/1449.460462021-01-08T12:24:00+00:00"Kumjian, Alex"https://zbmath.org/authors/?q=ai:kumjian.alexander"Li, Hui"https://zbmath.org/authors/?q=ai:li.huiSummary: In \textit{H. Li} [Houston J. Math. 43, No. 2, 459--494 (2017; Zbl 1391.46067)], the second author showed how Katsura's construction of the \(C^*\)-algebra of a topological graph \(E\) may be twisted by a Hermitian line bundle \(L\) over the edge space \(E^1\). The correspondence defining the algebra is obtained as the completion of the compactly supported continuous sections of \(L\). We prove that the resulting \(C^*\)-algebra is isomorphic to a twisted groupoid \(C^*\)-algebra where the underlying groupoid is the Renault-Deaconu groupoid of the topological graph with Yeend's boundary path space as its unit space.Coset spaces and cardinal invariants.https://zbmath.org/1449.220012021-01-08T12:24:00+00:00"Fernández, M."https://zbmath.org/authors/?q=ai:fernandez.mauricio|fernandez.m-legua|fernandez.miguel-angel|fernandez.mario|fernandez.mirtha-lina|fernandez.maite|fernandez.moises|fernandez.m-p|fernandez.maximiliano|fernandez.maria-c|fernandez.max|fernandez.melchor|fernandez.manuel-jose|fernandez.m-v|fernandez.marcel|fernandez.michael|fernandez.mary-f|fernandez.m-angeles|fernandez.manel|fernandez.marcela|fernandez.manuel|fernandez.marcos|fernandez.maribel|fernandez.m-l-c|fernandez.marcelo-o|fernandez.margarita|fernandez.marisa|fernandez.mariela"Sánchez, I."https://zbmath.org/authors/?q=ai:sanchez.ivan"Tkachenko, M."https://zbmath.org/authors/?q=ai:tkachenko.mikhail-gThe authors extend various results about cardinal invariants of topological groups to homogeneous spaces \(G/H\) of topological groups. The notions used in the paper can be found for instance in \textit{A. Arhangel'skii} and \textit{M. Tkachenko} [Topological groups and related structures. Hackensack, NJ: World Scientific; Paris: Atlantis Press (2008; Zbl 1323.22001)].
A typical result: If \(H\) is a closed subgroup of a feathered topological group \(G\) then \(\pi\chi(G/H)=\chi(H)\) and \(\pi w(G/H)=w(G/H)\). (Here \(\pi\chi, \chi,\pi w\), and \(w\) are the \(\pi\)-character, character, \(\pi\)-weight and weight, respectively.)
The paper ends with a list of five open problems.
Reviewer: Mihail I. Ursul (Oradea)On soft ultrafilters.https://zbmath.org/1449.540362021-01-08T12:24:00+00:00"Salec, Alireza Bagheri"https://zbmath.org/authors/?q=ai:bagheri-salec.alirezaSummary: In this paper, the concept of soft ultrafilters is introduced and some of the related structures such as soft Stone-Čech compactification, principal soft ultrafilters and basis for its topology are studied.Countably compact group topologies on the free abelian group of size continuum (and a Wallace semigroup) from a selective ultrafilter.https://zbmath.org/1449.540432021-01-08T12:24:00+00:00"Boero, A. C."https://zbmath.org/authors/?q=ai:boero.ana-carolina"Pereira, I. C."https://zbmath.org/authors/?q=ai:pereira.i-castro"Tomita, A. H."https://zbmath.org/authors/?q=ai:tomita.artur-hideyukiIt is well known that a non-trivial free abelian group does not admit a compact Hausdorff group topology. On the other hand, it was shown that the free abelian group generated by \(\mathfrak{c}\) elements can be endowed with a countably compact Hausdorff group topology under some set-theoretic assumptions. \textit{R. E. Madariaga-Garcia} and \textit{A. H. Tomita} [Topology Appl. 154, No. 7, 1470--1480 (2007; Zbl 1116.54004)] also obtained such a group assuming the existence of \(\mathfrak{c}\) many pairwise incomparable selective ultrafilters and asked whether the existence of one selective ultrafilter implies the existence of a countably compact group topology on the free abelian group of size \(\mathfrak{c}\).
It was known that compact both-sided cancellative semigroups are topological groups. In the 1950's, Wallace asked whether every countably compact topological semigroup with both-sided cancellation is a topological group. A counterexample to Wallace's question has been called a Wallace semigroup. The main example in the above cited paper of \textit{R. E. Madariaga-Garcia} and \textit{A. H. Tomita} yields a Wallace semigroup from the existence of \(\mathfrak{c}\) selective ultrafilters.
In this paper, the authors prove that the existence of a selective ultrafilter implies the existence of a countably compact Hausdorff group topology on the free abelian group of size \(\mathfrak{c}\) answering the above first question, and the existence of a Wallace semigroup.
Reviewer: Kohzo Yamada (Shizuoka)On the continuity of the Lipsman mapping of semidirect products.https://zbmath.org/1449.220022021-01-08T12:24:00+00:00"Messaoud, Anis"https://zbmath.org/authors/?q=ai:messaoud.anis"Rahali, Aymen"https://zbmath.org/authors/?q=ai:rahali.aymenLet \(G= K\ltimes V\) be the semidirect product of a connected compact Lie group \(K\) acting by automorphisms on a finite dimensional vector space \(V\) equipped with an inner product. R. Lipsman established a bijection \(\Theta\) of the class of coadjoint orbits of \(G\) onto the unitary dual \(\widehat{G}\) of \(G\), that is, the set of all equivalence classes of irreducible unitary representations of \(G\). It is proved that the Lipsman mapping \(\Theta\) is continuous.
Reviewer: Yuri I. Karlovich (Cuernavaca)Essential spectrum and Fredholm properties for operators on locally compact groups.https://zbmath.org/1449.460582021-01-08T12:24:00+00:00"Măntoiu, Marius Laurenţiu"https://zbmath.org/authors/?q=ai:mantoiu.mariusSummary: We study the essential spectrum and Fredholm properties of certain integral and pseudo-differential operators associated to non-commutative locally compact groups~$G$. The techniques involve crossed product \(C^*\)-algebras. We extend previous results on the structure of the essential spectrum to self-adjoint operators belonging (or affiliated) to the Schrödinger representation of certain crossed products. When the group $G$ is unimodular and type~I, we cover a new class of global pseudo-differential differential operators with operator-valued symbols involving the unitary dual of~$G$. We use recent results of Nistor, Prudhon and Roch on the role of families of representations in spectral theory and the notion of quasi-regular dynamical system.An optimal system of one-dimensional subalgebras for the symmetry algebra of three-dimensional equations of the perfect plasticity.https://zbmath.org/1449.740512021-01-08T12:24:00+00:00"Kovalëv, Vladimir Aleksandrovich"https://zbmath.org/authors/?q=ai:kovalev.vladimir-aleksandrovich"Radaev, Yuriĭ Nikolaevich"https://zbmath.org/authors/?q=ai:radaev.yu-nSummary: The present paper is devoted to a study of a natural 12-dimensional symmetry algebra of the three-dimensional hyperbolic differential equations of the perfect plasticity, obtained by \textit{D. D. Ivlev} [Sov. Phys., Dokl. 4, 217--220 (1959; Zbl 0088.40801); translation from Dokl. Akad. Nauk SSSR 124, 546--549 (1959)] and formulated in isostatic coordinates. An optimal system of one-dimensional subalgebras constructing algorithm for the Lie algebra is proposed. The optimal system (total 187 elements) is shown consisting of of a 3-parametrical element, twelve 2-parametrical elements, sixty six 1-parametrical elements and one hundred and eight individual elements.Metaplectic transformations and finite group actions on noncommutative tori.https://zbmath.org/1449.460562021-01-08T12:24:00+00:00"Chakraborty, Sayan"https://zbmath.org/authors/?q=ai:chakraborty.sayan"Luef, Franz"https://zbmath.org/authors/?q=ai:luef.franzLet \(A_\Theta\) be the \(2n\)-dimensional noncommutative torus determined by a \(2n\)-dimensional real skew-symmetric matrix \(\Theta\), and suppose \(W\in\operatorname{SL}_{2n}(\mathbb{Z})\) has order \(k\) and satisfies \(W^T\Theta W=\Theta\). The authors define the ``metaplectic'' action of \(W\) on the set \(\mathcal{S}(\mathbb{R}^n)\) of Schwartz functions, and show that this action of \(\mathbb{Z}_k\) gives rise to a finitely generated projective module over the crossed product \(C^*\)-algebra \(A_\Theta\rtimes\mathbb{Z}_k\).
Reviewer: Vladimir M. Manuilov (Moskva)Infinitesimal aspects of idempotents in Banach algebras.https://zbmath.org/1449.460362021-01-08T12:24:00+00:00"Beltiţă, Daniel"https://zbmath.org/authors/?q=ai:beltita.daniel"Galé, José E."https://zbmath.org/authors/?q=ai:gale.jose-eThe authors introduce and study Stiefel bundles on flag manifolds, which are extensions of the well known Stiefel bundles on Grassmannians. The main ingredient of the investigation is the notion of connection on an infinite-dimensional bundle. This allows to investigate infinitesimal properties of sets of ordered \(n\)-tuples of idempotents in a symmetric Banach \(*\)-algebra.
For the entire collection see [Zbl 1404.42002].
Reviewer: Cătălin Badea (Villeneuve d'Ascq)Essential spectrum, quasi-orbits and compactifications: application to the Heisenberg group.https://zbmath.org/1449.470282021-01-08T12:24:00+00:00"Mougel, Jérémy"https://zbmath.org/authors/?q=ai:mougel.jeremyLet \(H\) be the Heisenberg group. Using the natural bijection between \(H\) and \(\mathbb{R}\sp{3}\), there is introduced a compactification \(\bar{H}\) of \(H\) induced by the spherical compactification of \(\mathbb{R}\sp{3}\). Let \(T=-\Delta + V\) be the Schrödinger-type operator on \(L\sp{2}(H)\), where \(V\) is a continuous function on \(\bar{H}\). The main result of the paper gives a representation of the essential spectrum of \(T\) as the union of spectra of some simpler operators. There are also obtained some similar results.
Reviewer: Vladimir S. Pilidi (Rostov-na-Donu)The Fourier transform on 2-step Lie groups.https://zbmath.org/1449.430052021-01-08T12:24:00+00:00"Lévy, Guillaume"https://zbmath.org/authors/?q=ai:levy.guillaumeThe author develops harmonic analysis on a nilpotent Lie group of step 2. The Fourier transform is expressed in terms of the canonical bilinear form and its matrix coefficients. The parameter space of these matrix coefficients and its completion with respect to a natural distance are computed explicitly, as well as the integral kernel of the matrix coefficients Fourier transform, the analogue for the above framework of the classical Fourier kernel \((x, \xi)\mapsto e^{i(x\cdot \xi)}\).
Reviewer: Anatoly N. Kochubei (Kyïv)