The Fourier algebra A(G) can be viewed as a dual object for the group G and, in turn, for the group algebra L1(G). It is a commutative Banach algebra constructed using the representation theory of the group, and from which the group G may be recovered as its spectrum. When G is abelian, A(G) coincides with L1(G^); for non-abelian groups, it is viewed as a generalization of this object. B. Johnson has shown that G is amenable as a group if and only if L1(G) is amenable as a Banach algebra. Hence, it is natural to expect that the cohomology of A(G) will reflect the amenability of G. The initial hypothesis to this effect is that G is amenable if and only if A(G) is amenable as a Banach algebra. Interestingly, it turns out that A(G) is amenable only when G has an abelian group of finite index, leaving a large class of amenable groups with non-amenable Fourier algebras. The dual of A(G) is a von Neumann algebra (denoted VN(G)); as such, A(G) inherits a natural operator space structure. With this operator space structure, A(G) is a completely contractive Banach algebra, which is the natural operator space analogue of a Banach algebra. By taking this additional structure into account, one recovers the intuition behind the first conjecture: Z.-J. Ruan showed that G is amenable if and only if A(G) is operator amenable. This thesis concerns both the non-amenability of the Fourier algebra in the category of Banach spaces and why Ruan's Theorem is actually the proper analogue of Johnson's Theorem for A(G). We will see that the operator space projective tensor product behaves well with respect to the Fourier algebra, while the Banach space projective tensor product generally does not. This is crucial to explaining why operator amenability is the right sort of amenability in this context, and more generally, why A(G) should be viewed as a completely contractive Banach algebra and not merely a Banach algebra.

Surprisingly, Fourier series on certain fractals can have better convergence properties than classical Fourier series. This is a result of the existence of gaps in the spectrum of the Laplacian. In this work we prove a general criterion for the existence of gaps. Most of the known examples on which the Laplacians admit spectral decimation satisfy the criterion. Then we analyze the infinite family of Vicsek sets, finding an explicit formula for the spectral decimation functions in terms of Chebyshev polynomials. The Laplacians on this infinite family of fractals are also shown to satisfy our criterion and thus have gaps in their spectrum.

The Fourier algebra A(G) can be viewed as a dual object for the group G and, in turn, for the group algebra L1(G). It is a commutative Banach algebra constructed using the representation theory of the group, and from which the group G may be recovered as its spectrum. When G is abelian, A(G) coincides with L1(G^); for non-abelian groups, it is viewed as a generalization of this object. B. Johnson has shown that G is amenable as a group if and only if L1(G) is amenable as a Banach algebra. Hence, it is natural to expect that the cohomology of A(G) will reflect the amenability of G. The initial hypothesis to this effect is that G is amenable if and only if A(G) is amenable as a Banach algebra. Interestingly, it turns out that A(G) is amenable only when G has an abelian group of finite index, leaving a large class of amenable groups with non-amenable Fourier algebras. The dual of A(G) is a von Neumann algebra (denoted VN(G)); as such, A(G) inherits a natural operator space structure. With this operator space structure, A(G) is a completely contractive Banach algebra, which is the natural operator space analogue of a Banach algebra. By taking this additional structure into account, one recovers the intuition behind the first conjecture: Z.-J. Ruan showed that G is amenable if and only if A(G) is operator amenable. This thesis concerns both the non-amenability of the Fourier algebra in the category of Banach spaces and why Ruan's Theorem is actually the proper analogue of Johnson's Theorem for A(G). We will see that the operator space projective tensor product behaves well with respect to the Fourier algebra, while the Banach space projective tensor product generally does not. This is crucial to explaining why operator amenability is the right sort of amenability in this context, and more generally, why A(G) should be viewed as a completely contractive Banach algebra and not merely a Banach algebra.

Surprisingly, Fourier series on certain fractals can have better convergence properties than classical Fourier series. This is a result of the existence of gaps in the spectrum of the Laplacian. In this work we prove a general criterion for the existence of gaps. Most of the known examples on which the Laplacians admit spectral decimation satisfy the criterion. Then we analyze the infinite family of Vicsek sets, finding an explicit formula for the spectral decimation functions in terms of Chebyshev polynomials. The Laplacians on this infinite family of fractals are also shown to satisfy our criterion and thus have gaps in their spectrum.

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Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (M.S.)--University of Wisconsin--Madison, 1967. / eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--University of Chicago, Department of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, August 2000. / Includes bibliographical references. Also available on the Internet.