ABSTRACT
In Chapter 3 we discussed the problem of embedding an analytic structure in the carrier space of a real function algebra. Even when such an analytic structure cannot be embedded, we can still ask whether the (Gelfand transforms of) functions in a real function algebra exhibit some properties of analytic functions. By the maximum modulus principle, we know that the functions in the real disk algebra attain their maximum absolute value on the unit circle. The aim of the present chapter is to study subsets on which every function attains its maximum absolute value. Such a subset is called a boundary. If a boundary is closed, we can regard the given algebra as a function algebra on this boundary. Hence we look for a minimal closed boundary. Our search for a minimal closed boundary proceeds via the study of subsets on which the real parts of functions assume their maximum value. Such a subset is called a Choquet set.
