Noncommutative topology and Jordan operator algebras

Abstract

Jordan operator algebras are norm-closed spaces of operators on a Hilbert space with $a^2 \in A$ for all $a \in A$. We study noncommutative topology, noncommutative peak sets and peak interpolation, and hereditary subalgebras of Jordan operator algebras. We show that Jordan operator algebras present perhaps the most general setting for a `full' noncommutative topology in the C*-algebraic sense of Akemann, L. G. Brown, Pedersen, etc, and as modified for not necessarily selfadjoint algebras by the authors with Read, Hay and other coauthors. Our breakthrough relies in part on establishing several strong variants of C*-algebraic results of Brown relating to hereditary subalgebras, proximinality, deeper facts about $L+L^*$ for a left ideal $L$ in a C*-algebra, noncommutative Urysohn lemmas, etc. We also prove several other approximation results in $C^*$-algebras and various subspaces of $C^*$-algebras, related to open and closed projections, and technical $C^*$-algebraic results of Brown.