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.

In this paper we discuss what is known so far about diagonal acts of monoids. The first results that will be discussed comprise an overview of some work done on determining whether or not the diagonal act can be finitely generated or cyclic when looking at specific classes of monoids. This has been a topic of interest to a handful of semigroup theorists over the past seven years. We then move on to discuss some results pertaining to flatness properties of diagonal acts. The theory of flatness properties of acts over monoids has been of major interest over the past two decades, but so far there are no papers published on this subject that relate specifically to diagonal acts. We attempt to shed some light on this topic as well as present some new problems.

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.

Let X be a locally compact Hausdorff space along with n proper continuous maps σ = (σ1 , · · · , σn ). Then the pair (X, σ) is called a dynamical system. To each system one can associate a universal operator algebra called the tensor algebra A(X, σ). The central question in this theory is whether these algebras characterize dynamical systems up to some form of natural conjugacy. In the n = 1 case, when there is only one self-map, we will show how this question has been completely determined. For n ≥ 2, isomorphism of two tensor algebras implies that the two dynamical systems are piecewise conjugate. The converse was only established for n = 2 and 3. We introduce a new construction of the unitary group U (n) that allows us to prove the algebraic characterization question in n = 2, 3 and 4 as well as translating this conjecture into a conjecture purely about the structure of the unitary group.

This Master's thesis will develops a modern approach to complex interpolation problems studied by Carath\'odory, Nevanlinna, Pick, and Schur in the early $20^{th}$ century. The fundamental problem to solve is as follows: given complex numbers $z_1,z_2,...,z_N$ of modulus at most $1$ and $w_1,w_2,...,w_N$ additional complex numbers, what is a necessary and sufficiency condition for the existence of an analytic function $f: \mathbb{D} \rightarrow \mathbb$ satisfying $f(z_i) = w_i$ for $1 \leq i \leq N$ and $\vert f(z) \vert \leq 1$ for each $z \in \mathbb{D}$? The key idea is to realize bounded, analytic functions (the algebra $H^\infty$) as the \emph of the Hardy class of analytic functions, and apply dilation theory to this algebra. This operator theoretic approach may then be applied to a wider class of interpolation problems, as well as their matrix-valued equivalents. This also yields a fundamental distance formula for $H^\infty$, which provides motivation for the study of completely isometric representations of certain quotient algebras. Our attention is then turned to a related interpolation problem. Here we require the interpolating function $f$ to satisfy the additional property $f'(0) = 0$. When $z_i =0$ for some $i$, we arrive at a special case of a problem class studied previously. However, when $0$ is not in the interpolating set, a significant degree of complexity is inherited. The dilation theoretic approach employed previously is not effective in this case. A more function theoretic viewpoint is required, with the proof of the main interpolation theorem following from a factorization lemma for the Hardy class of analytic functions. We then apply the theory of completely isometric maps to show that matrix interpolation fails when one imposes this constraint.

The following thesis is divided into two main parts. In the first part we study the problem of characterizing algebras of functions living on analytic varieties. Specifically, we consider the restrictions M_V of the multiplier algebra M of Drury-Arveson space to a holomorphic subvariety V of the unit ball as well as the algebras A_V of continuous multipliers under the same restriction. We find that M_V is completely isometrically isomorphic to cM_W if and only if W is the image of V under a biholomorphic automorphism of the ball. In this case, the isomorphism is unitarily implemented. Furthermore, when V and W are homogeneous varieties then A_V is isometrically isomorphic to A_W if and only if the defining polynomial relations are the same up to a change of variables. The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. In the continuous homogeneous case, two algebras are isomorphic if and only if they are similar. However, in the multiplier algebra case the problem is much harder and several examples will be given where no such characterization is possible. In the second part we study the triangular subalgebras of UHF algebras which provide new examples of algebras with the Dirichlet property and the Ando property. This in turn allows us to describe the semicrossed product by an isometric automorphism. We also study the isometric automorphism group of these algebras and prove that it decomposes into the semidirect product of an abelian group by a torsion free group. Various other structure results are proven as well.

We consider the greatest prime factors of integers of certain form.

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.

In this paper we discuss what is known so far about diagonal acts of monoids. The first results that will be discussed comprise an overview of some work done on determining whether or not the diagonal act can be finitely generated or cyclic when looking at specific classes of monoids. This has been a topic of interest to a handful of semigroup theorists over the past seven years. We then move on to discuss some results pertaining to flatness properties of diagonal acts. The theory of flatness properties of acts over monoids has been of major interest over the past two decades, but so far there are no papers published on this subject that relate specifically to diagonal acts. We attempt to shed some light on this topic as well as present some new problems.

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.