  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, St{\o}rmer, 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 which are closedunder the
Jordan product. 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. We also 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.
