Contractive projections and real positive maps on operator algebras
We study contractive projections, isometries, and real positive maps on algebras of operators on a Hilbert space. For example we find generalizations and variants of certain classical results on contractive projections on $C^*$-algebras and JB-algebras due to Choi, Effros, Størmer, Friedman and Russo, and others. In fact most of our arguments generalize to contractive ‘real positive’ projections on Jordan operator algebras, that is, on a norm closed space $A$ of operators on a Hilbert space with $a^2 \in A$ for all $a \in A$. We also prove many new general results on real positive maps which are foundational to the study of such maps, and of interest in their own right. Moreover, we prove a new Banach–Stone type theorem for isometries between operator algebras or Jordan operator algebras. An application of this is given to the characterization of symmetric real positive projections.