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.

This thesis presents a collection disjointness analysis to find disjointness relations between collections in Java. We define the three types of disjointness relations between collections: must-shared, may-shared and not-may-shared. The collection- disjointness analysis is implemented following the way of a forward data-flow analysis using Soot Java bytecode analysis framework. For method calls, which are usually difficult to analyze in static analysis, our analysis provide a way of generating and reading annotations of a method to best approximate the behavior of the calling methods. Finally, this thesis presents the experimental results of the collection-disjointness analysis on several tests.

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.

This thesis presents a collection disjointness analysis to find disjointness relations between collections in Java. We define the three types of disjointness relations between collections: must-shared, may-shared and not-may-shared. The collection- disjointness analysis is implemented following the way of a forward data-flow analysis using Soot Java bytecode analysis framework. For method calls, which are usually difficult to analyze in static analysis, our analysis provide a way of generating and reading annotations of a method to best approximate the behavior of the calling methods. Finally, this thesis presents the experimental results of the collection-disjointness analysis on several tests.

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 Banach-Stone Theorem (respectly, Kadison Theorem) says that two abelian (respectively, general) C*-algebras are isomorphic as C*-algebras (respectively, JB*-algebras) if and only if they are isomorphic as Banach spaces. We are interested in using different structures to determine C*-algebras. Here, we would like to study the disjointness structures of C*-algebras and investigate if it suffices to determine C*-algebras. There are at least four versions of disjointness structures: zero product, range orthogonality, domain orthogonality and doubly orthogonality. In this thesis, we first study these disjointness structures in the case of standard operator algebras. Then we extend these results to general C*-algebras, namely, C*-algebras with continuous trace.

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.

Banachverbände spielen sowohl in der Theorie als auch in der Anwendung von geordneten normierten Räume eine bedeutende Rolle. Einerseits erweisen sich viele in der Praxis relevanten Räume als Banachverbände, andererseits ermöglichen die Vektorverbandsstruktur und die enge Beziehung zwischen Ordnung und Norm ein tiefes Verständnis solcher normierter Räume. An dieser Stelle setzen folgende Überlegungen an: - Die genaue Untersuchung einiger Resultate der reichhaltigen Banachverbandstheorie ließ (zu Recht) vermuten, dass in manchen Fällen die Verbandsnormeigenschaft keine notwendige Voraussetzung ist. In der Literatur gibt es bereits einige interessante Untersuchungen allgemeiner geordneter normierter Räume mit qualifizierten positiven Kegeln und in dem Zusammenhang eine Reihe wertvoller Dualitätsaussagen. An dieser Stelle sind die Eigenschaften der Normalität, der Nichtabgeflachtheit und der Regularität eines Kegels erwähnt, welche selbst im Falle eines mit einer Norm versehenen Vektorverbandes eine schwächere Relation zwischen Ordnung und Norm ergeben als die Verbandsnormeigenschaft. - In einer neueren Arbeit wurde der aus der Theorie der Vektorverbände gut bekannte Begriff der Disjunktheit bereits auf beliebige geordnete Räume verallgemeinert, wobei viele Eigenschaften disjunkter Vektoren, des disjunkten Komplements einer Menge usw., welche aus der Verbandstheorie bekannt sind, erhalten bleiben. Auf entsprechende Weise, d.h. durch das Ersetzen exakter Infima und Suprema durch Mengen unterer bzw. oberer Schranken, können der Modul eines Vektors sowie der Begriff der Solidität einer Menge für geordnete (normierte) Räume eingeführt werden. An solchen Überlegungen knüpft die vorliegende Arbeit an. Im Kapitel m-Normen ======== werden verallgemeinerte Formen der M-Norm Eigenschaft eingeführt und untersucht. AM-Räume und (approximative) Ordnungseinheit-Räume sind Beispiele für geordnete normierte Räume mit m-Norm. Die Schwerpunkte dieses Kapitels sind zum Einen Kegel- und Normeigenschaften dieser Räume und deren Charakterisierung mit Hilfe solcher Eigenschaften und zum Anderen Dualitätsaussagen, wie sie zum Teil bereits aus der Theorie der AM- und AL-Räume bekannt sind. Minimal totale Mengen ===================== Ziel dieses Kapitels ist es, den oben erwähnten verallgemeinerten Disjunktheitsbegiff für geordnete normierte Räume zu untersuchen. Eine zentrale Rolle spielen dabei totale Mengen im Dualraum und insbesondere minimal totale Mengen sowie deren Zusammenhang mit der Disjunktheit von Elementen des Ausgangsraumes. Normierte pre-Riesz Räume ========================= Wie bereits bekannt, lässt sich jeder pre-Riesz Raum ordnungsdicht in einen (bis auf Isomorphie) eindeutigen minimalen Vektorverband einbetten, die so genannte Riesz Vervollständigung. Ist der pre-Riesz Raum normiert und sein positiver Kegel abgeschlossen, dann kann eine Verbandsnorm auf der Riesz Vervollständigung eingeführt werden, welche sich in vielen Fällen als äquivalent zur Ausgangsnorm auf dem pre-Riesz Raum erweist. Es ist allgemein bekannt, dass sich dann auch stetige lineare Funktionale fortsetzen lassen. In diesem Kapitel wird nun untersucht, inwiefern sich Ordnungsrelationen auf einer Menge stetiger linearer Funktionale beim Übergang zur Menge der Fortsetzungen erhalten lassen. Die gewonnenen Erkenntnisse kommen anschließend bei Untersuchungen zur schwachen bzw. schwach*-Topologie auf geordneten normierten Räumen zur Anwendung. Hierbei werden zwei Fragestellungen behandelt. Zum Einen gilt das Augenmerk disjunkten Folgen in geordneten normierten Räumen. Als Beispiel seien ordnungsbeschränkte disjunkte Folgen in geordneten normierten Räumen mit halbmonotoner mNorm genannt, welche stets schwach gegen Null konvergieren. Zum Anderen werden monoton fallende Folgen und Netze bzw. disjunkte Folgen von stetigen linearen Funktionalen auf einem geordneten normierten Raum betrachtet. / Banach lattices play an important role in the theory of ordered normed spaces. One reason is, that many ordered normed vector spaces, that are important in practice, turn out to be Banach lattices, on the other hand, the lattice structure and strong relations between order and norm allow a deep understanding of such ordered normed spaces. At this point the following is to be considered. - The analysis of some results in the rich Banach lattice theory leads to the conjecture, that sometimes the lattice norm property is no necessary supposition. General ordered normed spaces with a convenient positive cone were already examined, where some valuable duality properties could be achieved. We point out the properties of normality, non-flatness and regularity of a cone, which are a weaker relation between order and norm than the lattice norm property in normed vector lattices. - The notion of disjointness in vector lattices has already been generalized to arbitrary ordered vector spaces. Many properties of disjoint elements, the disjoint complement of a set etc., well known from the vector lattice theory, are preserved. The modulus of a vector as well as the concept of the solidness of a set can be introduced in a similar way, namely by replacing suprema and infima by sets of upper and lower bounds, respectively. We take such ideas up in the present thesis. A generalized version of the M-norm property is introduced and examined in section m-norms. ======= AM-spaces and approximate order unit spaces are examples of ordered normed spaces with m-norm. The main points of this section are the special properties of the positive cone and the norm of such spaces and the duality properties of spaces with m-norm. Minimal total sets ================== In this section we examine the mentioned generalized disjointness in ordered normed spaces. Total sets as well as minimal total sets and their relation to disjoint elements play an inportant at this. Normed pre-Riesz spaces ======================= As already known, every pre-Riesz space can be order densely embedded into an (up to isomorphism) unique vector lattice, the so called Riesz completion. If, in addition, the pre-Riesz space is normed and its positive cone is closed, then a lattice norm can be introduced on the Riesz completion, that turns out to be equivalent to the primary norm on the pre-Riesz space in many cases. Positive linear continuous functionals on the pre-Riesz space are extendable to positive linear continuous functionals in this setting. Here we investigate, how some order relations on a set of continuous functionals can be preserved to the set of the extension. In the last paragraph of this section the obtained results are applied for investigations of some questions concerning the weak and the weak* topology on ordered normed vector spaces. On the one hand, we focus on disjoint sequences in ordered normed spaces. On the other hand, we deal with decreasing sequences and nets and disjoint sequences of linear continuous functionals on ordered normed spaces.