Real positive maps and conditional expectations on operator algebras

Abstract

Most of this article is an expanded version of our conference talk. It is essentially a survey, but some part, like most of the lengthy Section 5, is comprised of new results whose proofs are unpublished elsewhere. We begin by reviewing the theory of real positivity of operator algebras initiated by the author and Charles Read. Then we present several new general results (mostly joint work with Matthew Neal) about real positive maps. The key point is that real positivity is often the right replacement in a general algebra $A$ for positivity in C*-algebras. We then apply this to studying contractive projections (`conditional expectations') and isometries of operator algebras. For example we generalize and find variants of certain classical results on positive 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 had done the `completely contractive' case; we focus here on describing the real positive contractive case from recent work with Neal. We also prove here several new and complementary results on this topic due to the author, indeed this new work constitutes most of Section 5. Finally, in the last section we describe a related part of some recent joint work with Labuschagne on what we consider to be a good noncommutative generalization of the `characters' (i.e. homomorphisms into the scalars) on an algebra. Such characters are a special case of the projections mentioned above, and are shown to be intimately related to conditional expectations. The idea is to try to use these to generalize certain classical function algebra results involving characters.