Approximate ultrahomogeneity in $L_pL_q$ lattices

TL;DR

This paper investigates the approximate ultrahomogeneity of $L_pL_q$ Banach lattices, showing conditions under which they are AUH over finitely generated sublattices, with exceptions when $p/q$ is an integer.

Contribution

It establishes new conditions for approximate ultrahomogeneity in $L_pL_q$ lattices, including cases with band lattices, expanding understanding of their structural symmetry.

Findings

$L_pL_q$ is AUH over finitely generated sublattices when $p/q otin ats$

$L_pL_q$ is AUH over band lattices for $p eq q$

The property fails when $p/q$ is an integer

Abstract

We show that for $1\leq p, q<\infty$ with $p/q \notin \mathbb{N}$, the doubly atomless separable $L_pL_q$ Banach lattice $L_p(L_q)$ is approximately ultrahomogeneous (AUH) over the class of its finitely generated sublattices. The above is not true when $p/q \in \mathbb{N}$. However, for any $p\neq q$, $L_p(L_q)$ is AUH over the finitely generated lattices in the class $BL_pL_q$ of bands of $L_pL_q$ lattices.