2025/10/20 by Christopher Schwanke, Schwanke, Christopher
Mathematics · #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical and Theoretical Analysis #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2510.17510
openalex publication_date 2025/10/20 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28
Given an Archimedean vector lattice E, we present one elementary property of E which is equivalent to the entire traditional list of axioms which makes E a Φ-algebra. We call a vector lattice with this property ``square closed". More generally, we then introduce the notion of a pseudo square closed vector lattice and prove that an Archimedean vector lattice is a semiprime f-algebra if and only if it is pseudo square closed. This theory serves as an efficient tool for determining whether or not an Archimedean vector lattice is a Φ-algebra (or a semiprime f-algebra). To illustrate this point, we generalize a well-known result for uniformly complete Archimedean vector lattices with a strong order unit by proving that every functionally complete Archimedean vector lattice with a strong order unit is a Φ-algebra.