  The main goal of this paper is to find operator algebra variants of certain
deep results of Stormer, Friedman and Russo, Choi and Effros, Effros and
Stormer, Robertson and Youngson, Youngson, and others, concerning projections
on C*-algebras and their ranges. (See papers of these authors referenced in the
bibliography.) In particular we investigate the `bicontractive projection
problem' and related questions in the category of operator algebras. To do
this, we will add the ingredient of `real positivity' from recent papers of the
first author with Read.
