  For a long time, practitioners of the art of operator algebras always worked
over the complex numbers, and nobody paid much attention to real C*-algebras.
Over the last thirty years, that situation has changed, and it's become
apparent that real C*-algebras have a lot of extra structure not evident from
their complexifications. At the same time, interest in real C*-algebras has
been driven by a number of compelling applications, for example in the
classification of manifolds of positive scalar curvature, in representation
theory, and in the study of orientifold string theories. We will discuss a
number of interesting examples of these, and how the real Baum-Connes
conjecture plays an important role.
