Intermediate algebras in Archimedean semiprime f-algebras

TL;DR

This paper introduces bounded quasi-inversion closed semiprime f-algebras and shows that intermediate algebras within them are order ideals, extending previous results to a broader class of algebras with applications to function spaces.

Contribution

It extends the characterization of intermediate algebras as order ideals to non-unital, semiprime f-algebras, broadening the scope of prior work.

Findings

Intermediate algebras are order ideals in bounded quasi-inversion closed semiprime f-algebras.

The results apply to algebras of continuous and measurable functions.

Extension of previous unital case results to a more general 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.