  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.
