published_or_final_version / abstract / Mathematics / Doctoral / Doctor of Philosophy

A research report submitted to the Faculty of Science, University of the Witwatersrand, in partial fulfillment of the degree of Master of Science. / The Toeplitz form is a most useful and important teo! in many areas of applied. mathematics today including signal processing, time-series analysis and prediction theory. It is even used in quantum mechanics in Ising model correlation functions. (Abbreviation abstract) / AC 2018

Thesis (Ph. D.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

Let H be a separable Hilbert space and {e_n : n belong to Z} be an orthonormal basis in H. A bounded operator T is called the slant Toeplitz operator if <T ej , ei> =c2i−j , where c_n is the n-th Fourier series of a bounded Lebesgue measurable function on the unit circle T = {z belong to C : |z| = 1}. It has been shown [7] that T* is an isometry if and only if |fi(z)|^2 +|fi(−z)|^2 = 2 a.e. on T and if this is the case and fi belong to C(T), then either T is unitarily equivalent to a shift or to the direct sum of a shift and a rank one unitary, with infinite multiplicity (for the shift part, that is). Moreover, with some additional assumption on the smoothness and the zeros of fi, T* is similar to either the constant multiple of a shift or to the constant multiple of the direct sum of a shift and a rank one unitary, with infinite multiplicity. On the other hand, according to the terminologies in [10], an operator A that is constant with respect to a shift S if AS = SA and A S = SA . Therefore, in this article, we will study the operators that is constant with respect to T , i.e., bounded operator A satisfying AT = T A and A T = T A .

published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 1994. / Includes bibliographical references (leaves 40-42).

Thesis (M. Phil.)--University of Hong Kong, 1993. / Photocopy typescript.

We consider the class of generalised Toeplitz algebras; those C*-algebras that can be expressed as an extension of C(X) by the compact operators K, for some compact metrizable space X. We show that one can generalise the result of Brake and Winter, that the nuclear dimension of the Toeplitz algebra is 1, to show that for any generalised Toeplitz algebra its nuclear dimension must be equal to dim_nuc C(X). This shows that Brake and Winter's dimension reduction phenomenon is applicable to a much wider class of algebras. We also introduce our definition for the relative nuclear dimension of a C*-algebra. This is a modification to the definition of nuclear dimension that requires us to factor through algebras of the form F \otimes B for F finite dimensional and B some fixed algebra we are working relative to. We explore various properties satisfied by the relative nuclear dimension with a particular eye to its being a modification of nuclear dimension.

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