Let X be a compact metric space, and Lip(X) is the space of all bounded real-valued Lipschitz functions on X. A linear map T:Lip(X)->Lip(Y) is called disjointness preserving if fg=0 in Lip(X) implies TfTg=0 in Lip(Y). We prove that a biseparating linear bijection T(i.e. T and T^-1 are separating) is a weighted composition operator Tf=hf¡³£p, f is Lipschitz space from X onto R, £p is a homeomorphism from Y onto X, and h(y) is a Lipschitz function in Y.

In this thesis, we shall study disjointness preserving linear functionals of the Wiener ring. It is clear that Wiener ring is a dense subalgebra of C(T)in the usual supremum norm .However, Wiener ring is also isomorphic to L1(Z). So it has an 1 norm . By studying the structure of ideals of the Wiener ring, we discover that disjointness preserving linear functionals are the same under different norms. Bounded disjointness preserving linear functionals of the Wiener ring is a multiple of the point mass in both cases. Finally, we establish the existence of unbounded disjointness preserving linear functionals of the Wiener ring.

Let X, Y be locally compact Hausdorff spaces. A linear operator T from C0(X,E) to C0(Y,F) is called disjointness preserving if coz(Tf)¡äcoz(Tg) = whenever coz(f)¡äcoz(g) = ∅. We discuss some cases on these compact disjointness preserving operators T and prove that if £f0 is a nonzero point of £m(T), then £f0 is an eigenvalue of T and we find a projection ∏: C0(X,E) ¡÷C0(X,E), such that for Y1 = ∏C0(X;E) and Y2 = (1-∏)C0(X;E), the operator T|Y1 -£f0 is a nilpotent and £f0-T|Y2 is invertible.

In this thesis, we study the question when a local automorphism of continuous functions, or in general, of an operator algebra, is an automorphism. We also study the question how to write an n-disjointness preserving operator as a finite sum of orthomorphisms locally.

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.

Peetre characterized local operators defined on the smooth section space over an open subset of an Euclidean space as ``linear differential operators'. We look for an extension to such maps of smooth vector sections of smooth Banach bundles. Since local operators are special disjointness preserving operators, it leads to the study of the disjointness structure of smooth Banach manifolds. In this thesis, we take an abstract approach to define the``smooth functions', via the so-called S-category. Especially, it covers the standard classes C^{n} and local Lipschitz functions, where 0≤n≤¡Û. We will study the structure of disjointness preserving linear maps between S-smooth functions defined on separable Banach manifolds. In particular, we will give an extension of Peetre's theorem to characterize disjointness preserving linear mappings between C^n or local Lipschitz functions defined on locally compact metric spaces.