[Paper Review] Groups of quasi-invariance and the Pontryagin duality

Saak Gabriyelyan|ArXiv.org|Dec 9, 2008

Advanced Topology and Set Theory12 references4 citations

TL;DR

This paper introduces groups of quasi-invariance (QI-groups), a class of Polish groups defined via measures on locally compact groups where shifts preserve or orthogonalize the measure. It proves that QI-groups are σ-compact and constructs a monothetic, non-locally quasi-convex QI-group $\mathbb{T}^{H}_{2}$ whose bidual is not a QI-group, demonstrating that the Pontryagin duality fails to preserve the QI property under biduality.

ABSTRACT

A Polish group $G$ is called a group of quasi-invariance or a QI-group, if there exist a locally compact group $X$ and a probability measure $μ$ on $X$ such that 1) there exists a continuous monomorphism of $G$ to $X$, and 2) for each $g\in X$ either $g\in G$ and the shift $μ_g$ is equivalent to $μ$ or $g ot\in G$ and $μ_g$ is orthogonal to $μ$. It is proved that $G$ is a $σ$-compact subset of $X$. We show that there exists a quotient group $\mathbb{T}^H_2$ of $\ell^2$ modulo a discrete subgroup which is a Polish monothetic non locally quasi-convex (and hence nonreflexive) pathwise connected QI-group, and such that the bidual of $\mathbb{T}^H_2$ is not a QI-group. It is proved also that the bidual group of a QI-group may be not a saturated subgroup of $X$.

Motivation & Objective

• To define and study groups of quasi-invariance (QI-groups) as Polish subgroups of locally compact groups with specific measure-theoretic invariance properties.
• To investigate the interplay between quasi-convexity, reflexivity, and Pontryagin duality in non-locally compact topological groups.
• To construct explicit examples of QI-groups whose biduals fail to be QI-groups, thereby challenging the preservation of the QI property under duality.
• To clarify the role of T-sequences and metric structures in defining Polish group topologies on dual and bidual groups.

Proposed method

• Define a QI-group as a Polish group $G$ embedded in a locally compact group $X$ such that for each $g \in X$, the measure shift $\mu_g$ is either equivalent to $\mu$ (if $g \in G$) or orthogonal (if $g \notin G$).
• Use the Mackey-Weil theorem to relate the structure of $E(\mu)$, the set of measure quasi-invariants, to local compactness and Polish group topologies.
• Construct a quotient group $\mathbb{T}^{H}_{2} = \ell^2 / H$ for a discrete subgroup $H$, showing it is a Polish, monothetic, pathwise connected, and non-locally quasi-convex QI-group.
• Employ $T$-sequences and metric structures on $\mathbb{T}^\infty$ and $G_p$ to define Polish group topologies and analyze convergence and duality.
• Analyze the dual and bidual groups using the compact-open topology and the canonical homomorphism $\alpha_G: G \to G^{\wedge\wedge}$.
• Prove that $G_p^\wedge = G_0^\wedge$ and $G_p^{\wedge\wedge} = G_0$, but $G_0$ is not a QI-group, showing bidual may fail to inherit the QI property.

Experimental results

Research questions

• RQ1 Does the Pontryagin duality theorem extend to non-locally compact groups, particularly QI-groups?
• RQ2 Can a QI-group have a bidual that is not a QI-group?
• RQ3 Is the bidual of a QI-group necessarily a saturated subgroup of the ambient locally compact group?
• RQ4 What is the relationship between local quasi-convexity and the QI property in dual and bidual groups?
• RQ5 Can one construct a monothetic, pathwise connected, non-locally quasi-convex QI-group?

Key findings

• The group $\mathbb{T}^{H}_{2} = \ell^2 / H$ is a Polish, monothetic, pathwise connected, non-locally quasi-convex QI-group.
• The bidual of $\mathbb{T}^{H}_{2}$ is not a QI-group, showing that the QI property is not preserved under biduality.
• The bidual of a QI-group may fail to be a saturated subgroup of the ambient locally compact group $X$.
• The dual group $G_p^\wedge$ is isomorphic to $G_0^\wedge$, and $G_p^{\wedge\wedge} = G_0$, but $G_0$ is not a QI-group.
• For each $\delta > 0$, the polar set $U_\delta^\triangleright$ is contained in $\phi_p(A(k,0))$ for some $k$, implying a uniform bound on coefficients in the group representation.
• The canonical map $\alpha_p: G_p \to G_p^{\wedge\wedge} = G_0$ has dense image, and $G_p^\wedge$, $G_0^\wedge$, and $G_0$ are reflexive, though $G_p$ is not locally quasi-convex.