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.

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 thesis we study ideals in the Fourier algebra, A(G), of a locally compact group G. For a locally compact abelian group G, necessary conditions for a closed ideal in A(G) to be weakly complemented are given, and a complete characterization of the complemented ideals in A(G) is given when G is a discrete abelian group. The closed ideals in A(G) with bounded approximate identities are also characterized for any locally compact abelian group G. When G is an arbitrary locally compact group, we exploit the natural operator space structure that A(G) inherits as the predual of the group von Neumann algebra, VN(G), to study ideals in A(G). Using operator space techniques, necessary conditions for an ideal in A(G) to be weakly complemented by a completely bounded projection are given for amenable G, and the ideals in A(G) possessing bounded approximate identities are completely characterized for amenable G. Ideas from homological algebra are then used to study the biprojectivity of A(G) in the category of operator spaces. It is shown that A(G) is operator biprojective if and only if G is a discrete group. This result is then used to show that every completely complemented ideal in A(G) is invariantly completely complemented when G is discrete. We conclude by proving that for certain discrete groups G, there are complemented ideals in A(G) which fail to be complemented or weakly complemented by completely bounded projections.

In this thesis we study ideals in the Fourier algebra, A(G), of a locally compact group G. For a locally compact abelian group G, necessary conditions for a closed ideal in A(G) to be weakly complemented are given, and a complete characterization of the complemented ideals in A(G) is given when G is a discrete abelian group. The closed ideals in A(G) with bounded approximate identities are also characterized for any locally compact abelian group G. When G is an arbitrary locally compact group, we exploit the natural operator space structure that A(G) inherits as the predual of the group von Neumann algebra, VN(G), to study ideals in A(G). Using operator space techniques, necessary conditions for an ideal in A(G) to be weakly complemented by a completely bounded projection are given for amenable G, and the ideals in A(G) possessing bounded approximate identities are completely characterized for amenable G. Ideas from homological algebra are then used to study the biprojectivity of A(G) in the category of operator spaces. It is shown that A(G) is operator biprojective if and only if G is a discrete group. This result is then used to show that every completely complemented ideal in A(G) is invariantly completely complemented when G is discrete. We conclude by proving that for certain discrete groups G, there are complemented ideals in A(G) which fail to be complemented or weakly complemented by completely bounded projections.

The goal of this thesis is to define and study the Fourier algebra A(G) of a locally compact groupoid G. If G is a locally compact group, its Fourier-Stieltjes algebra B(G) and its Fourier algebra A(G) were defined by Eymard in 1964. Since then, a rich theory has been developed. For the groupoid case, the algebras B(G) and A(G) have been studied by Ramsay and Walter (borelian case, 1997), Renault (measurable case, 1997) and Paterson (locally compact case, 2004). In this work, we present a new definition of A(G) in the locally compact case, specially well suited for studying locally trivial groupoids. Let G be a locally compact proper groupoid. Following the group case, in order to define A(G), we consider the closure under certain norm of the span of the left regular G-Hilbert bundle coefficients. With the norm mentioned above, the space A(G) is a commutative Banach algebra of continuous functions of G vanishing at infinity. Moreover, A(G) separates points and it is also a B(G)-bimodule. If, in addition, G is compact, then B(G) and A(G) coincide. For a locally trivial groupoid G we present an easier to handle definition of A(G) that involves "trivializing" the left regular bundle. The main result of our work is a decomposition of A(G), valid for transitive, locally trivial groupoids with a "nice" Haar system. The condition we require the Haar system to satisfy is to be compatible with the Haar measure of the isotropy group at a fixed unit u. If the groupoid is transitive, locally trivial and unimodular, such a Haar system always can be constructed. For such groupoids, our theorem states that A(G) is isomorphic to the Haagerup tensor product of the space of continuous functions on Gu vanishing at infinity, times the Fourier algebra of the isotropy group at u, times space of continuous functions on Gu vanishing at infinity. Here Gu denotes the elements of the groupoid that have range u. This decomposition provides an operator space structure for A(G) and makes this space a completely contractive Banach algebra. If the locally trivial groupoid G has more than one transitive component, that we denote Gi, since these components are also topological components, there is a correspondence between G-Hilbert bundles and families of Gi-Hilbert bundles. Thanks to this correspondence, the Fourier-Stieltjes and Fourier algebra of G can be written as sums of the algebras of the Gi components. The theory of operator spaces is the main tool used in our work. In particular, the many properties of the Haagerup tensor product are of vital importance. Our decomposition can be applied to (trivially) locally trivial groupoids of the form X times X and X times H times X, for a locally compact space X and a locally compact group H. It can also be applied to transformation group groupoids X times H arising from the action of a Lie group H on a locally compact space X and to the fundamental groupoid of a path-connected manifold.

The goal of this thesis is to define and study the Fourier algebra A(G) of a locally compact groupoid G. If G is a locally compact group, its Fourier-Stieltjes algebra B(G) and its Fourier algebra A(G) were defined by Eymard in 1964. Since then, a rich theory has been developed. For the groupoid case, the algebras B(G) and A(G) have been studied by Ramsay and Walter (borelian case, 1997), Renault (measurable case, 1997) and Paterson (locally compact case, 2004). In this work, we present a new definition of A(G) in the locally compact case, specially well suited for studying locally trivial groupoids. Let G be a locally compact proper groupoid. Following the group case, in order to define A(G), we consider the closure under certain norm of the span of the left regular G-Hilbert bundle coefficients. With the norm mentioned above, the space A(G) is a commutative Banach algebra of continuous functions of G vanishing at infinity. Moreover, A(G) separates points and it is also a B(G)-bimodule. If, in addition, G is compact, then B(G) and A(G) coincide. For a locally trivial groupoid G we present an easier to handle definition of A(G) that involves "trivializing" the left regular bundle. The main result of our work is a decomposition of A(G), valid for transitive, locally trivial groupoids with a "nice" Haar system. The condition we require the Haar system to satisfy is to be compatible with the Haar measure of the isotropy group at a fixed unit u. If the groupoid is transitive, locally trivial and unimodular, such a Haar system always can be constructed. For such groupoids, our theorem states that A(G) is isomorphic to the Haagerup tensor product of the space of continuous functions on Gu vanishing at infinity, times the Fourier algebra of the isotropy group at u, times space of continuous functions on Gu vanishing at infinity. Here Gu denotes the elements of the groupoid that have range u. This decomposition provides an operator space structure for A(G) and makes this space a completely contractive Banach algebra. If the locally trivial groupoid G has more than one transitive component, that we denote Gi, since these components are also topological components, there is a correspondence between G-Hilbert bundles and families of Gi-Hilbert bundles. Thanks to this correspondence, the Fourier-Stieltjes and Fourier algebra of G can be written as sums of the algebras of the Gi components. The theory of operator spaces is the main tool used in our work. In particular, the many properties of the Haagerup tensor product are of vital importance. Our decomposition can be applied to (trivially) locally trivial groupoids of the form X times X and X times H times X, for a locally compact space X and a locally compact group H. It can also be applied to transformation group groupoids X times H arising from the action of a Lie group H on a locally compact space X and to the fundamental groupoid of a path-connected manifold.

This thesis develops the theory of operator algebras from the perspective of abstract harmonic analysis, and in particular, the theory of von Neumann algebras. Results from operator algebras are applied to the study of spaces of coefficient functions of unitary representations of locally compact groups, and in particular, the Fourier algebra of a locally compact group. The final result, which requires most of the material developed in earlier sections, is that the group von Neumann algebra of a locally compact group is in standard form.

This thesis develops the theory of operator algebras from the perspective of abstract harmonic analysis, and in particular, the theory of von Neumann algebras. Results from operator algebras are applied to the study of spaces of coefficient functions of unitary representations of locally compact groups, and in particular, the Fourier algebra of a locally compact group. The final result, which requires most of the material developed in earlier sections, is that the group von Neumann algebra of a locally compact group is in standard form.

Weighted group algebras have been studied extensively in Abstract Harmonic Analysis.Complete characterizations have been found for some important properties of weighted group algebras, namely, amenability and Arens regularity. Also studies on some other features of these algebras, say weak amenability and isomorphism to operator algebras, have attracted attention. Hypergroups are generalized versions of locally compact groups. When a discrete group has all its conjugacy classes finite, the set of all conjugacy classes forms a discrete commutative hypergroup. Also the set of equivalence classes of irreducible unitary representations of a compact group forms a discrete commutative hypergroup. Other examples of discrete commutative hypergroups come from families of orthogonal polynomials. The center of the group algebra of a discrete finite conjugacy (FC) group can be identified with a hypergroup algebra. For a specific class of discrete FC groups, the restricted direct products of finite groups (RDPF), we study some properties of the center of the group algebra including amenability, maximal ideal space, and existence of a bounded approximate identity of maximal ideals. One of the generalizations of weighted group algebras which may be considered is weighted hypergroup algebras. Defining weighted hypergroups, analogous to weighted groups, we study a variety of examples, features and applications of weighted hypergroup algebras. We investigate some properties of these algebras including: dual Banach algebra structure, Arens regularity, and isomorphism with operator algebras. We define and study Folner type conditions for hypergroups. We study the relation of the Folner type conditions with other amenability properties of hypergroups. We also demonstrate some results obtained from the Leptin condition for Fourier algebras of certain hypergroups. Highlighting these tools, we specially study the Leptin condition on duals of compact groups for some specific compact groups. An application is given to Segal algebras on compact groups.