Two Applications of Boolean Valued Analysis

TL;DR

This paper applies Boolean valued analysis to decompose certain vector lattices and characterizes cyclic Banach lattices, providing new structural insights and transfer principles in functional analysis.

Contribution

It introduces a novel decomposition of universally complete vector lattices and extends the Ando Theorem to cyclic Banach lattices via Boolean valued transfer.

Findings

Decomposition of vector lattices into invariant sublattices

Extension of Ando Theorem to cyclic Banach lattices

Application of Boolean valued analysis to Banach lattice theory

Abstract

The paper contains two main results that are obtained by Boolean valued analysis. The first asserts that a universally complete vector lattice without locally one-dimensional bands can be decomposed into a direct sum of two vector sublattices that are laterally complete and invariant under all band projections and there exists a band preserving linear isomorphism of each of these sublattices to the original lattice. The second result establishes a counterpart of the Ando Theorem on the joint characterization of $A\!L^p$ and $c_0$ for the class of cyclic Banach lattices, using the Boolean valued transfer for injective Banach lattices.