Complex Vector Lattices Via Functional Completions

TL;DR

This paper investigates the properties of complex vector lattices, demonstrating that certain tensor products lack square mean completeness and introducing a new theory of complexification and tensor products for Archimedean vector lattices.

Contribution

It introduces the concept of complexification of Archimedean vector lattices and develops a theory of powers and tensor products for these structures.

Findings

Fremlin tensor product of C(X) and C(Y) is not square mean complete for uncountable spaces

Defined complexification of Archimedean vector lattices

Developed a theory of tensor powers for Archimedean complex vector lattices

Abstract

We show that the Fremlin tensor product $C(X)\bar{\otimes}C(Y)$ is not square mean complete when X and Y are uncountable metrizable compact spaces. This motivates the definition of complexification of Archimedean vector lattices, the Fremlin tensor product of Archimedean complex vector lattices, and a theory of powers of Archimedean complex vector lattices.