Lattice renormings of $C_0(X)$ spaces

Timur Oikhberg; Mary Angelica Tursi

arXiv:2405.11148·math.FA·May 21, 2024

Lattice renormings of $C_0(X)$ spaces

Timur Oikhberg, Mary Angelica Tursi

PDF

Open Access

TL;DR

This paper demonstrates how to construct equivalent lattice norms on $C_0(X)$ spaces to realize any locally compact Polish group as the group of lattice isometries, linking group actions with functional analysis.

Contribution

It introduces a method to renorm $C_0(X)$ spaces so that their lattice isometry groups match any given locally compact Polish group, extending the understanding of symmetry groups in Banach lattices.

Findings

01

Constructed equivalent lattice norms for $C_0(X)$ spaces.

02

Realized any locally compact Polish group as a lattice isometry group.

03

Established a link between group actions and Banach lattice symmetries.

Abstract

Suppose $X$ is a locally compact Polish space, and $G$ is a group of lattice isometries of $C_{0} (X)$ which satisfies certain conditions. Then we can equip $C_{0} (X)$ with an equivalent lattice norm $∣ ∣ ∣ \cdot ∣ ∣ ∣$ so that $G$ is the group of lattice isometries of $(C_{0} (X), ∣ ∣ ∣ \cdot ∣ ∣ ∣)$ . As an application, we show that for any locally compact Polish group $G$ there exists a locally compact Polish space $X$ , and an lattice norm $∣ ∣ ∣ \cdot ∣ ∣ ∣$ on $C_{0} (X)$ , so that $G$ is the group of lattice isometries of $(C_{0} (X), ∣ ∣ ∣ \cdot ∣ ∣ ∣)$ .

Peer Reviews

No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.

Videos

No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.

Taxonomy

TopicsAdvanced Banach Space Theory · Advanced Topology and Set Theory · Advanced Harmonic Analysis Research