Let $T$ be a bounded disjointness preserving linear operator from $C_0(X)$ into $C_0(Y)$, where $X$ and $Y$ are locally compact Hausdorff spaces. We give several equivalent conditions for $T$ to be compact; they are: $T$ is weakly compact; $T$ is completely continuous; $T= sum_n delta_{x_n} otimes h_n$ for a sequence of distinct points ${x_n}_n$ in $X$ and a norm null mutually disjoint sequence ${h_n}_n$ in $C_0(Y)$. The structure of a graph with countably many vertices associated to such a compact operator $T$ gives rise to a new complete description of the spectrum of $T$. In particular, we show that a nonzero complex number $la$ is an eigenvalue of $T$ if and only if $lambda^k= h_1(x_k) h_2(x_1) cdots h_k(x_{k-1})$ for some positive integer $k$. We also give a decomposition of compact disjointness preserving operators $T$ from $C_0(X,E)$ into $C_0(Y,F)$, where $X$ and $Y$ are locally compact Hausdorff spaces, $E$ and $F$ are Banach spaces. That is, $T= sum_n de_{x_n} otimes h_n$ for a sequence of distinct points ${x_n}_n$ in $X$ and a norm null mutually disjoint sequence ${h_n}_n$, where $h_n: Y o B(E,F)$ is continuous and vanishes at infinity in the uniform operator topology and $h_n(y)$ is compact for each $y$ in $Y$. For completely continuous disjointness preserving linear operators, we get a similar decomposition. More precisely, completely continuous disjointness preserving operators $T$ have a countable sum decomposition of completely continuous atoms $de_{x_n} otimes h_n$, where $h_n: Y o B(E,F)$ is continuous, vanishes at infinity in the strong operator topology and $h_n$ is uniformly completely continuous. In case of weakly compact disjointness preserving linear operators, $T$ have a countable sum decomposition of weakly compact atoms whenever the Banach space $E$ is separable. A counterexample is given whenever $E$ in nonseparable.

The thesis consists of four research papers. The first three deal with the Choquet theory of function spaces. In Chapter 1, a theory on products and projective limits of function spaces is developed. It is shown that the product of simplicial spaces is a simplicial space. The stability of the space of maximal measures under continuous affine mappings is studied in Chapter 2. The third chapter employs results from the previous chapters to construct an example of a function space where the abstract Dirichlet problem is not solvable for any class of Baire-n functions with $n\in N$. It is shown that such an example cannot be constructed via the space of harmonic functions. In the final chapter, the recently introduced class of sequentially Right Banach spaces is being investigated. Connections to other isomorphic properties of Banach spaces are established and several characterizations are given.

In this thesis we present an extensive characterization of weak* sequentially precompact subsets of the dual of a sequentially order complete M-space with an order unit. This central part of the thesis generalizes results due to H.H. Schaefer and X.D. Zhang showing that small weak* compact subsets of the dual of a space of bounded measurable real-valued functions (continuous real-valued functions on a compact quasi-Stonian space) are weakly compact. Moreover, while the proofs of Schaefer and Zhang use measure theoretical arguments, the arguments presented here are purely elementary and are based on the well-known result, that the space l1 has the Schur property. Finally some applications are given. For example, we investigate compact or sequentially precompact subsets, which consist of order-weakly compact operators, in the space of continuous linear operators defined on a sequentially order complete Riesz space with values in a Banach space provided with the strong operator topology: as an immediate consequence of the results, we can easily deduce extended versions of the Vitali-Hahn-Saks theorem for vector measures. For this we need a generalization of the Yosida-Hewitt decomposition theorem, which is proved here with other techniques like the factorization of an order-weakly compact operator through a Banach lattice with order continuous norm.