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Most of this article is an expanded version of our conference talk. It isessentially a survey, but some part, like most of the lengthy Section 5, iscomprised of new results whose proofs are unpublished elsewhere. We begin byreviewing the theory of real positivity of operator algebras initiated by theauthor and Charles Read. Then we present several new general results (mostlyjoint work with Matthew Neal) about real positive maps. The key point is thatreal positivity is often the right replacement in a general algebra $A$ forpositivity in C*-algebras. We then apply this to studying contractiveprojections (`conditional expectations') and isometries of operator algebras.For example we generalize and find variants of certain classical results onpositive projections on C*-algebras and JB algebras due to Choi, Effros,St{\o}rmer, Friedman and Russo, and others. In previous work with Neal we haddone the `completely contractive' case; we focus here on describing the realpositive contractive case from recent work with Neal. We also prove hereseveral new and complementary results on this topic due to the author, indeedthis new work constitutes most of Section 5. Finally, in the last section wedescribe a related part of some recent joint work with Labuschagne on what weconsider to be a good noncommutative generalization of the `characters' (i.e.homomorphisms into the scalars) on an algebra. Such characters are a specialcase of the projections mentioned above, and are shown to be intimately relatedto conditional expectations. The idea is to try to use these to generalizecertain classical function algebra results involving characters.

Real positive maps and conditional expectations on operator algebras