On the Boolean algebra tensor product via Caratheodory spaces of place functions

TL;DR

This paper establishes an isomorphism between Carathéodory spaces of place functions on free products of Boolean algebras and Fremlin's Riesz space tensor product, solving a problem on completeness in Boolean algebra free products.

Contribution

It demonstrates a new isomorphism linking Carathéodory spaces and Riesz space tensor products, and addresses a longstanding problem on completeness in Boolean algebra free products.

Findings

Proves Riesz isomorphism between Carathéodory spaces and tensor products.

Provides a solution to Fremlin's problem 315Y(f) on completeness.

Enhances understanding of tensor products in Boolean algebra contexts.

Abstract

We show that the Carath\'{e}odory space of place functions on the free product of two Boolean algebras is Riesz isomorphic with Fremlin's Archimedean Riesz space tensor product of their respective Carath\'{e}odory spaces of place functions. We provide a solution to Fremlin's problem 315Y(f) in \cite{fremlin_measure} concerning completeness in the free product of Boolean algebras by applying our results on the Archimedean Riesz space tensor product to Carath\'{e}odory spaces of place functions.