arXiv subjects

Lydia Außenhofer

Publications and source records attributed to Lydia Außenhofer.

3 recordsLinked to original sources

arXiv2025

The Protasov-Zelenyuk topology and ideal convergence

The so-called $T$-sequences $\mathbf u$ in a group $G$, and the related finest Hausdorff group topology $T_\mathbf u$ on $G$ that makes $\mathbf u$ a null sequence, were introduced by Protasov and Zelenyuk 35 years ago and since then they became a fundamental tool in the field of topological groups. More recently, in the abelian case, the subfamily of $T$-sequences called $TB$-sequences was introduced, as well as the finest precompact group topology $T_\mathbf{pu}$ that makes $\mathbf u$ a null sequence. Here we study the counterpart of all these notions with respect to ideal convergence in place of the classical notion of convergence of a sequence. Also, we study their relation to the already established field of $I$-characterized subgroups of compact abelian groups.

math.GR↗

arXiv2016

"Varopoulos paradigm": Mackey property vs. metrizability in topological groups

The class of all locally quasi-convex (lqc) abelian groups contains all locally convex vector spaces (lcs) considered as topological groups. Therefore it is natural to extend classical properties of locally convex spaces to this larger class of abelian topological groups. In the present paper we consider the following well known property of lcs: "A metrizable locally convex space carries its Mackey topology ". This claim cannot be extended to lqc-groups in the natural way, as we have recently proved with other coauthors (\cite{AD}, \cite{DMPT}). We say that an abelian group $G$ satisfies the \emph{Varopoulos paradigm} (VP) if any metrizable locally quasi-convex topology on $G$ is the Mackey topology. Surprisingly VP - which is a topological property of the group - is characterizes an algebraic feature, namely being of finite exponent.

math.AT↗

arXiv2009

Locally minimal topological groups

A Hausdorff topological group $(G,τ)$ is called locally minimal if there exists a neighborhood $U$ of 0 in $τ$ such that $U$ fails to be a neighborhood of zero in any Hausdorff group topology on $G$ which is strictly coarser than $τ.$ Examples of locally minimal groups are all subgroups of Banach-Lie groups, all locally compact groups and all minimal groups. Motivated by the fact that locally compact NSS groups are Lie groups, we study the connection between local minimality and the NSS property, establishing that under certain conditions, locally minimal NSS groups are metrizable. A symmetric subset of an abelian group containing zero is said to be a GTG set if it generates a group topology in an analogous way as convex and symmetric subsets are unit balls for pseudonorms on a vector space. We consider topological groups which have a neighborhood basis at zero consisting of GTG sets. Examples of these locally GTG groups are: locally pseudo--convex spaces, groups uniformly free from small subgroups (UFSS groups) and locally compact abelian groups. The precise relation between these classes of groups is obtained: a topological abelian group is UFSS if and only if it is locally minimal, locally GTG and NSS. We develop a universal construction of GTG sets in arbitrary non-discrete metric abelian groups, that generates a strictly finer non-discrete UFSS topology and we characterize the metrizable abelian groups admitting a strictly finer non-discrete UFSS group topology. We also prove that a bounded abelian group $G$ admits a non-discrete locally minimal and locally GTG group topology iff $|G|\geq \cont$.

math.GN↗