Results concerning some convolution algebras



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Texas Tech University


This dissertation presents results concerning the properties of two classes of Banach algebras of integrable functions. The theory of algebras of integrable functions on a topological group is a point of nexus between two of the main branches of mathematlcal analysis, abstract harmonic analysis and the theory of Banach algebras.

Abstract harmonlc analysis developed from the classical theory of Fourier series and Fourier Integrals on the circle, the integers, and the real line (28). The process of abstraction began in the 1920's with the explicit definition of the notion of a topological group. The work of Haar (8), von Neumann (22), and Weil (26) established the existence of Haar measure and extended the realm of Fourier analysis to certain classes of topological groups. The main thrust of abstract harmonic analysis has been an effort to interpret and generalize the results of classical Fourier analysis In this abstract setting. Results in this direction are collected in various texts and reference works (11) (12) (17) (24).



Multipliers, Banach algebras, Spectral synthesis