Functional Completions of Archimedean Vector Lattices

TL;DR

This paper explores various methods to complete Archimedean vector lattices using positively-homogeneous functions, leading to new concepts like complexification and tensor products in the lattice framework.

Contribution

It introduces a general approach to functional completions of Archimedean vector lattices and develops a universal theory of complexification and tensor products.

Findings

Examples include square mean closed and geometric closed vector lattices.

Provides a universal definition of complexification for Archimedean vector lattices.

Lays groundwork for tensor products and powers of complex vector lattices.

Abstract

We study completions of Archimedean vector lattices relative to any nonempty set of positively-homogeneous functions on finite-dimensional real vector spaces. Examples of such completions include square mean closed and geometric closed vector lattices, amongst others. These functional completions also lead to a universal definition of the complexification of any Archimedean vector lattice and a theory of tensor products and powers of complex vector lattices in a companion paper.