Welcome to Wasin So's Master Thesis Page
In 1986, I finished the master thesis entitled "Convexity properties of state spaces of uniform algebras"
under the supervision of Professor A. J. Ellis.
Following is the abstract of the thesis.
The theme of this thesis is two-folded. Some known results on the affine geometry of state-spaces of
uniform algebras are collected. On the other hand, we examine the relationship between members of the class
of state-spaces of uniform algebras. Altogether there are five chapters. Chapter 0 is a collection of definitions and results from the theory
of compact convex sets. Some properties of simplexes and a-polytopes are also included.
We introduce the state-space and the complex state-space of a uniform algebra in the beginning of chapter one.
Then the basic and facial properties of these compact convex sets are studied. In particular, we include the works of Briem and Ellis
which relate the generalized peak sets for a uniform algebra with the closed split
faces of its state-space and complex state-space. Section 2.1 consists of four examples of state-spaces of uniform algebras.
In �2.2 we examine the state-spaces of those uniform algebras A on X such that re A has finite codimension in
C_R(X). In chapter 3 we give a characterization of state-spaces of uniform algebras
in terms of certain mapping properties of compact convex sets. This result is due to Ellis.
The aim of the last chapter is to study the relationship between members of the class of state-spaces of uniform algebras. Firstly
we observe that the state-space of a uniform algebras is determined by its real affine space up to affine homeomorphisms. Then we prove
that every closed face of the state-space of a uniform algebra is again a member of the class.
Moreover it is proved that under certain conditions the operation of taking convex hulls is closed
in the class.
Following are published papers resulted from the thesis.