On von Neumann regularity of algebras of quotients of Jordan algebras

TL;DR

This paper investigates whether Jordan algebras of quotients are von Neumann regular, showing that the property holds for algebras satisfying a polynomial identity but not in general.

Contribution

It extends Johnson's theorem to Jordan algebras, identifying conditions under which the algebra of quotients is von Neumann regular.

Findings

Von Neumann regularity does not hold universally for Jordan algebras of quotients.

The property holds for Jordan algebras satisfying a polynomial identity.

The paper provides conditions where the Jordan version of Johnson's theorem applies.

Abstract

We address a Jordan version of Johnson theorem on (associative) algebras of quotients, namely whether a strongly nonsingular (the Jordan version of nonsingularity) has a von Neumann regular algebra of quotients. Although the answer is negative in general, we prove that the result holds in the important case of algebras satisfying a polynomial identity.