Jordan operator algebras revisited

TL;DR

This paper advances the theory of Jordan operator algebras by extending many results from associative operator algebra theory, including unitizations and positivity, and addresses an open problem from previous work.

Contribution

It completes the generalization of associative operator algebra results to Jordan operator algebras and solves an open problem from earlier research.

Findings

Generalization of associative operator algebra theory to Jordan algebras

Results on unitizations and real positivity for Jordan algebras

Resolution of an open problem from previous work

Abstract

Jordan operator algebras are norm-closed spaces of operators on a Hilbert space with a^2 in A for all a in A. In two recent papers by the authors and Neal, a theory for these spaces was developed. It was shown there that much of the theory of associative operator algebras, in particularly surprisingly much of the associative theory from several recent papers of the first author and coauthors, generalizes to Jordan operator algebras. In the present paper we complete this task, giving several results which generalize the associative case in these papers, relating to unitizations, real positivity, hereditary subalgebras, and a couple of other topics. We also solve one of the three open problems stated at the end of our earlier joint paper on Jordan operator algebras.