Artykuły w formacie PDF dostępne są dla subskrybentów, którzy zapłacili za dostęp online, po podpisaniu licencji Licencja użytkownika instytucjonalnego. Czasopisma do 2009 są ogólnodostępne (bezpłatnie).

Locally compact groups and locally minimal group topologies

Tom 244 / 2019

Wenfei Xi, Dikran Dikranjan, Wei He, Zhiqiang Xiao Fundamenta Mathematicae 244 (2019), 109-124 MSC: Primary 22D05; Secondary 54H11. DOI: 10.4064/fm468-3-2018 Opublikowany online: 15 October 2018

Streszczenie

Minimal groups are Hausdorff topological groups $G$ satisfying the open mapping theorem with respect to continuous isomorphisms, i.e., every continuous isomorphism $G\to H$, with $H$ a Hausdorff topological group, is a topological isomorphism. A topological group $(G, \tau )$ is called locally minimal if there exists a neighbourhood $V$ of the identity such that for every Hausdorff group topology $\sigma \leq \tau $ with $V \in \sigma $ one has $\sigma = \tau $. Minimal groups, as well as locally compact groups, are locally minimal. According to a well known theorem of Prodanov, every subgroup of an infinite compact abelian group $K$ is minimal if and only if $K$ is isomorphic to the group $\mathbb {Z}_{p}$ of $p$-adic integers for some prime $p$.

We find a remarkable connection of local minimality to Lie groups and $p$-adic numbers by means of the following results extending Prodanov’s theorem: every subgroup of a locally compact abelian group $K$ is locally minimal if and only if either $K$ is a Lie group, or $K$ has an open subgroup isomorphic to $\mathbb {Z}_{p}$ for some prime $p$. In the nonabelian case we prove that all subgroups of a connected locally compact group are locally minimal if and only if $K$ is a Lie group, resolving Problem 7.49 from Dikranjan and Megrelishvili (2014) in the positive.

Autorzy

  • Wenfei XiInstitute of Mathematics
    Nanjing Normal University
    Nanjing 210046, China
    e-mail
  • Dikran DikranjanDipartimento di Matematica
    e Informatica
    Università di Udine
    33100 Udine, Italy
    e-mail
  • Wei HeInstitute of Mathematics
    Nanjing Normal University
    Nanjing 210046, China
    e-mail
  • Zhiqiang XiaoInstitute of Mathematics
    Nanjing Normal University
    Nanjing 210046, China
    e-mail

Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Przepisz kod z obrazka

Odśwież obrazek

Odśwież obrazek