On Bounded Approximate Identities and Existence of Dense Ideals in Real Locally C*- and Locally JB-Algebras

TL;DR

This paper extends known results about dense ideals and bounded approximate identities from complex locally C*-algebras to real locally C*-algebras and locally JB-algebras, broadening their theoretical understanding.

Contribution

It provides the first analogues of Inoue and Fritzsche's results for real locally C*-algebras and locally JB-algebras, establishing conditions for dense ideals and approximate identities.

Findings

Real locally C*-algebras with dense ideals have bounded approximate identities.

Unbounded elements in unital real locally C*-algebras imply the existence of dense one-sided ideals.

Analogues of complex algebra results are established for real and JB-algebras.

Abstract

It has been established by Inoue that a complex locally C*-algebra with a dense ideal posesses a bounded approximate identity which belonges to that ideal. It has been shown by Fritzsche that if a unital complex locally C*-algebra has an unbounded element then it also has a dense one-sided ideal. In the present paper we obtain analogues of the aforementioned results of Inoue and Fritzsche for real locally C*-algebras (projective limits of projective families of real C*-algebras), and for locally JB-algebras (projective limits of projective families of JB-algebras).