Groups of quasi-invariance and the Pontryagin duality

2008/12/09 by Saak Gabriyelyan, S. S. Gabriyelyan, Gabriyelyan, S. S. · 1 citation
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.GN #math.GT

paper · pdf · doi:10.48550/arxiv.0812.1671

openalex publication_date 2008/12/09 · arxiv created 2009/10/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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∈ X either g∈ G and the shift μ_g is equivalent to μ or g\not∈ 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 \mathbbT^H₂ of ℓ² 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 \mathbbT^H₂ 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.

Cited by

• A class of metrizable locally quasi-convex groups which are not Mackey

Related

• Pontryagin duality for Abelian s- and sb-groups
• Unitarizability, Maurey--Nikishin factorization, and Polish groups of\n finite type
• Locally compact abelian groups admitting non-trivial quasi-convex null sequences
• A locally quasi-convex abelian group without Mackey topology
• Quasi-Hermitian locally compact groups are amenable