English

Lattice renormings of $C_0(X)$ spaces

Functional Analysis 2024-05-21 v1

Abstract

Suppose XX is a locally compact Polish space, and GG is a group of lattice isometries of C0(X)C_0(X) which satisfies certain conditions. Then we can equip C0(X)C_0(X) with an equivalent lattice norm  ⁣ ⁣ ⁣ ⁣| \! | \! | \cdot | \! | \! | so that GG is the group of lattice isometries of (C0(X), ⁣ ⁣ ⁣ ⁣)(C_0(X), | \! | \! | \cdot | \! | \! |). As an application, we show that for any locally compact Polish group GG there exists a locally compact Polish space XX, and an lattice norm  ⁣ ⁣ ⁣ ⁣| \! | \! | \cdot | \! | \! | on C0(X)C_0(X), so that GG is the group of lattice isometries of (C0(X), ⁣ ⁣ ⁣ ⁣)(C_0(X), | \! | \! | \cdot | \! | \! |).

Keywords

Cite

@article{arxiv.2405.11148,
  title  = {Lattice renormings of $C_0(X)$ spaces},
  author = {Timur Oikhberg and Mary Angelica Tursi},
  journal= {arXiv preprint arXiv:2405.11148},
  year   = {2024}
}