Thesis--Ph. D., University of Hong Kong. / Mimeographed.

Thesis--M. Phil., University of Hong Kong, 1980.

published_or_final_version / Mathematics / Master / Master of Philosophy

No description available.

The objective of this dissertation is to expand on the proofs and concepts of Degree Theory, dealt with in chapters 1 and 2 of Deimling [28], to make it more readable and accessible to anyone who is interested in the field. Chapter 1 is an introduction and contains the basic requirements for the subsequent chapters. The remaining chapters aim at defining a ll-valued map D (the degree) on the set M = {(F, Ω, y) / Ω C X open, F : Ὠ → X, y ɇ F(∂Ω)} (each time, the elements of M satisfying extra conditions) that satisfies : (D1) D(I, Ω, y) = 1 if y Є Ω. (D2) D(F, Ω, y) = D(F, Ω1 , y) + D(F, Ω2, y) if Ω1 and Ω2 are disjoint open subsets of Ω o such that y ɇ F(Ὠ \ Ω1 U Ω2 ). (D3) D(I - H(t, .), Ω, y(t)) is independent of t if H : J x Ὠ →X and y : J → X. An important property that follows from these three properties is (D4) F-1(y) ≠ Ø if D(F, Ω, y) ≠ 0. This property ensures that equations of the form Fx = y have solutions if D(F, Ω, y) ≠ 0. Another property that features in these chapters is the Borsuk property which gives us conditions under which the degree is odd and hence nonzero. / Thesis (M.Sc.)-University of Durban-Westville, 1989.

In the first half of the thesis a necessary and sufficient condition is given for a separable C*-algebra to be *-isomorphic to a maximal full algebra of cross-sections over a base-space such that the fibre algebras are primitive throughout a dense subset. The condition is that the relation of inseparability for pairs of points in the primitive ideal space should be an open equivalence relation. In the second half of the thesis a characterisation is given of those C*- algebras A for which each self-adjoint inner derivation D(α, A) satisfies ∥D(α, A)∥ = 2 inf {∥α-z∥ : z ∈Z(A), the centre of A}. This time the characterisation is that A should be quasicentral and the relation of inseparability for pairs of points in the primitive ideal space should be an equivalence relation. Those C*-algebras for which every inner derivation satisfies the equation are characterised in a similar way.

The notion of asymptotic structure of an infinite dimensional Banach space was introduced by Maurey, Milman and Tomczak-Jaegermann. The asymptotic structure consists of those finite dimensional spaces which can be found everywhere `at infinity'. These are defined as the spaces for which there is a winning strategy in a certain vector game. The above authors introduced the class of asymptotic $\ell_p$ spaces, which are the spaces having simplest possible asymptotic structure. Key examples of such spaces are Tsirelson's space and James' space. We prove some new properties of general asymptotic $\ell_p$ spaces and also compare the notion of asymptotic $\ell_2$ with other notions of asymptotic Hilbert space behaviour such as weak Hilbert and asymptotically Hilbertian. We study some properties of smooth functions defined on subsets of asymptotic $\ell_\infty$ spaces. Using these results we show that that an asymptotic $\ell_\infty$ space which has a suitably smooth norm is isomorphically polyhedral, and therefore admits an equivalent analytic norm. We give a sufficient condition for a generalized Orlicz space to be a stabilized asymptotic $\ell_\infty$ space, and hence obtain some new examples of asymptotic $\ell_\infty$ spaces. We also show that every generalized Orlicz space which is stabilized asymptotic $\ell_\infty$ is isomorphically polyhedral. In 1991 Gowers and Maurey constructed the first example of a space which did not contain an unconditional basic sequence. In fact their example had a stronger property, namely that it was hereditarily indecomposable. The space they constructed was `$\ell_1$-like' in the sense that for any $n$ successive vectors $x_1 < \ldots < x_n$, $\frac{1}{f(n)} \sum_{i=1}^n \| x_i \| \leq \| \sum_{i=1}^n x_i \| \leq \sum_{i=1}^n \| x_i \|,$ where $ f(n) = \log_2 (n+1) $. We present an adaptation of this construction to obtain, for each $ p \in (1, \infty)$, an hereditarily indecomposable Banach space, which is `$\ell_p$-like' in the sense described above. We give some sufficient conditions on the set of types, $\mathscr{T}(X)$, for a Banach space $X$ to contain almost isometric copies of $\ell_p$ (for some $p \in [1, \infty)$) or of $c_0$. These conditions involve compactness of certain subsets of $\mathscr{T}(X)$ in the strong topology. The proof of these results relies heavily on spreading model techniques. We give two examples of classes of spaces which satisfy these conditions. The first class of examples were introduced by Kalton, and have a structural property known as Property (M). The second class of examples are certain generalized Tsirelson spaces. We introduce the class of stopping time Banach spaces which generalize a space introduced by Rosenthal and first studied by Bang and Odell. We look at subspaces of these spaces which are generated by sequences of independent random variables and we show that they are isomorphic to (generalized) Orlicz spaces. We deduce also that every Orlicz space, $h_\phi$, embeds isomorphically in the stopping time Banach space of Rosenthal. We show also, by using a suitable independence condition, that stopping time Banach spaces also contain subspaces isomorphic to mixtures of Orlicz spaces.

Thesis (Ph.D.)--University of Adelaide, Dept. of Mathematics, 1965. / Typescript. Includes bibliographical references.

Thesis (Ph. D.)--Virginia Polytechnic Institute and State University, 1990. / Vita. Abstract. Includes bibliographical references (leaves 188-200). Also available via the Internet.

Thesis (Ph. D.)--University of Hong Kong, 1973. / Also available in print.