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A(Em A(Fr additive group Assume Hp Banach Space Bohr compactification bounded topology bounded weak topology Choose Closed Graph Theorem Commutative Banach Algebra compact abelian group constant contradicts Lemma 2.3 define the subset degree of Doctor direct sum Doctor of Philosophy dual group element equal exp(x feA(F finite set following theorem Fourier transform fully adequate Graduate Division greater than 1/100 groups G Haar measure hence idempotents isomorphic to A(F Let F linear functional locally compact abelian map cp norm obtain isomorphisms Pick point measure positive integer pseudomeasures quotient algebras r(k+l rationals read this thesis real numbers relatively prime restricted extension property restriction to E2 roots of unity scope and quality sequence of integers set F standard isomorphism Suppose symmetric set Theorem 2.2 theorem of Bernstein Tt(w Tw(t weak dense Yerk zero