[Paper Review] "Varopoulos paradigm": Mackey property vs. metrizability in topological groups

Lydia Außenhofer, Daniel de la Barrera Mayoral+2 more|arXiv (Cornell University)|Mar 12, 2016

Advanced Topology and Set Theory6 references3 citations

TL;DR

This paper establishes that an abelian group satisfies the Varopoulos paradigm—where every metrizable locally quasi-convex topology is Mackey—if and only if the group is of finite exponent (i.e., bounded). The authors prove that metrizable locally quasi-convex topologies on unbounded abelian groups fail to be Mackey, using duality theory and character group analysis to show that non-Mackey topologies exist in all unbounded cases.

ABSTRACT

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.

Motivation & Objective

• To determine under what algebraic conditions a metrizable locally quasi-convex topology on an abelian group is the Mackey topology.
• To investigate whether the classical result from locally convex spaces—that metrizable spaces carry their Mackey topology—extends to the broader class of locally quasi-convex abelian groups.
• To characterize the class of abelian groups for which all metrizable locally quasi-convex topologies are Mackey, identifying this as the class of bounded (finite exponent) groups.
• To resolve the Mackey problem in the context of locally quasi-convex abelian groups by showing that the existence of a top element in the compatible topologies is equivalent to boundedness.

Proposed method

• Use of duality theory in abelian topological groups, particularly the character group $G^\wedge = \text{Hom}(G, \mathbb{T})$.
• Application of the notion of locally quasi-convex topologies and their compatibility with the dual group.
• Construction of non-Mackey metrizable topologies on unbounded abelian groups via direct sum decompositions into primary components.
• Leveraging known results on non-Mackey topologies on subgroups such as $\mathbb{Q}$, $\mathbb{Z}(p^\infty)$, and $\bigoplus_{n} \mathbb{Z}_{p^n}$.
• Use of the fact that a group is bounded if and only if all its elements have finite order, and that boundedness implies the existence of a top compatible topology.
• Proof by contradiction in the case of reduced, torsion groups: assuming a basic subgroup is bounded leads to a contradiction with the structure of the quotient group.

Experimental results

Research questions

• RQ1 Does the classical Mackey property in locally convex spaces extend to metrizable locally quasi-convex abelian groups?
• RQ2 What algebraic condition on an abelian group ensures that all its metrizable locally quasi-convex topologies are Mackey?
• RQ3 Can non-Mackey metrizable topologies be constructed on unbounded abelian groups?
• RQ4 Is the existence of a top compatible topology (i.e., the Mackey topology) in the class of locally quasi-convex groups equivalent to boundedness?
• RQ5 How does the structure of the character group $G^\wedge$ relate to the Mackey property in abelian topological groups?

Key findings

• An abelian group satisfies the Varopoulos paradigm—every metrizable locally quasi-convex topology is Mackey—if and only if the group is bounded (i.e., of finite exponent).
• For unbounded abelian groups, there always exists a metrizable locally quasi-convex topology that is not Mackey, as shown via subgroup embeddings into $\mathbb{Q}$, $\mathbb{Z}(p^\infty)$, or $\bigoplus_{i} \mathbb{Z}_{p^{n_i}}$.
• The character group of a metrizable locally quasi-convex group topology $\tau_{\mathbf{c}}$ on $\bigoplus_{\omega} \mathbb{Z}_{m_n}$ is isomorphic to $\bigoplus_{\omega} \mathbb{Z}_{m_n}^\wedge$, which is used to show that certain characters do not belong to the dual, implying non-Mackey status.
• The proof relies on constructing a character $\chi$ such that $\chi(x) \notin \mathbb{T}_+$, contradicting the assumption that $x$ is in a neighborhood of 0, thus showing $\chi \notin (G, \tau_{\mathbf{c}})^\wedge$.
• In the case of reduced, torsion groups with finitely many nontrivial primary components, the existence of an unbounded basic subgroup implies the existence of a non-Mackey metrizable topology.
• The equivalence between the Mackey property for all metrizable locally quasi-convex topologies and boundedness is established as Theorem A, with the forward direction proven in Theorem C and the converse in Theorem B.