Intermediate algebras in Archimedean semiprime f-algebras

TL;DR

This paper extends a known property of intermediate algebras being order ideals from unital Archimedean $f$-algebras to the nonunital setting by introducing bounded quasi-inversion closedness for Archimedean semiprime $f$-algebras. It establishes that if $A$ is bounded quasi-inversion closed, then every intermediate algebra in $A$ containing the bounded elements $A_b$ is an order ideal, with the unital case recovering bounded inversion closedness. The work connects to relative uniform completeness and provides applications to universal completions, showing that subalgebras of $L^{u}$ or $C^{\infty}$ with appropriate unit-containing ideals are Dedekind complete $f$-algebras, and illustrates these results in the contexts of $L^{0}(\mu)$ and $C^{\infty}(K)$. Overall, it offers structural insights into Archimedean semiprime function algebras and broadens the scope of order-ideal phenomena beyond the unital setting.

Abstract

We introduce the notion of bounded quasi-inversion closed semiprime f-algebras and we prove that, if A is such an algebra, then any intermediate algebra in A is an order ideal of A. This extends a recent result by Dominguez who has dealt with the unital case (the problem on C(X)-type spaces has been solved earlier by Dominguez, Gomez-Perez, and Mulero). Our results are illustrated by examples of algebras of continuous functions and algebras of measurable functions.

Theorems & Definitions (12)

• Lemma 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Lemma 3.4
• Theorem 3.5
• Corollary 3.6
• Corollary 3.7