On a family of weighted convolution algebras

Transactions of the American Mathematical Society, 2015

Let G G be a compact connected Lie group. The question of when a weighted Fourier algebra on G G is completely isomorphic to an operator algebra will be investigated in this paper. We will demonstrate that the dimension of the group plays an important role in the question. More precisely, we will get a positive answer to the question when we consider a polynomial type weight coming from a length function on G G with the order of growth strictly bigger than half of the dimension of the group. The case of S U ( n ) SU(n) will be examined, focusing more on the details including negative results. The proof for the positive directions depends on a non-commutative version of the Littlewood multiplier theory, which we will develop in this paper, and the negative directions will be taken care of by restricting to a maximal torus.

Transactions of the American Mathematical Society, 1990

We give a direct transition from the existence of a bounded right approximate identity in the diagonal ideal for a weighted convolution algebra on a locally compact group to the existence of translation invariant means on an associated weighted L ∞ {L^\infty } -space, thus giving a characterization of amenability for such an algebra.

Advances in Mathematics

We investigate Beurling-Fourier algebras, a weighted version of Fourier algebras, on various Lie groups focusing on their spectral analysis. We will introduce a refined general definition of weights on the dual of locally compact groups and their associated Beurling-Fourier algebras. Constructions of nontrivial weights will be presented focusing on the cases of representative examples of Lie groups, namely SU (n), the Heisenberg group H, the reduced Heisenberg group Hr, the Euclidean motion group E(2) and its simply connected cover E(2). We will determine the spectrum of Beurling-Fourier algebras on each of the aforementioned groups emphasizing its connection to the complexification of underlying Lie groups. We also demonstrate "polynomially growing" weights does not change the spectrum and show the associated regularity of the resulting Beurling-Fourier algebras. CONTENTS 1. Introduction 2 1.1. Basic strategy 4 1.2. Organization 6 2. Preliminaries 6 2.1. Unbounded operators 6 2.1.1. Tensor products 8 2.1.2. Homomorphisms 8 2.1.3. Homomorphisms for non-commuting pairs 9 2.2. Lie groups, Lie algebras and related operators 9 2.3. Complexification of Lie groups 9 2.3.1. Operators associated to certain elements of the universal enveloping algebra and entire vectors 10 2.3.2. The choice of Fourier transforms 13 3. A refined definition for Beurling-Fourier algebras 13 3.1. Motivation: review of weights on abelian groups 13 3.2. Weights on the dual of G and Beurling-Fourier algebras 14 3.2.1. When W is bounded below 18 3.2.2. When G is separable and type I 19 3.3. Examples of weights 20 3.3.1. A list of weight functions on R k × Z n−k 20 3.3.2. Central weights 21 3.3.3. Extension from closed subgroups 23

Journal of Functional Analysis, 2012

For a compact group G we define the Beurling-Fourier algebra Aω(G) on G for weights ω : G → R >0. The classical Fourier algebra corresponds to the case ω is the constant weight 1. We study the Gelfand spectrum of the algebra realizing it as a subset of the complexification G C defined by McKennon and Cartwright and McMullen. In many cases, such as for polynomial weights, the spectrum is simply G. We discuss the questions when the algebra Aω(G) is symmetric and regular. We also obtain various results concerning spectral synthesis for Aω(G).

2010

We introduce the class of Beurling-Fourier algebras on locally compact groups and show that they are non-commutative analogs of classical Beurling algebras. We obtain various results with regard to the operator amenability, operator weak amenability and Arens regularity of Beurling-Fourier algebras on compact groups and show that they behave very similarly to the classical Beurling algebras of discrete groups. We then apply our results to study explicitly the Beurling-Fourier algebras on SU(2), the 2 × 2 unitary group. We demonstrate that how Beurling-Fourier algebras are closely connected to the amenability of the Fourier algebra of SU(2). Another major consequence of our results is that our investigation allows us to construct families of unital infinite-dimensional closed Arens regular subalgebras of the Fourier algebra of certain products of SU(2).

2006

Let G be a locally compact group, A(G) its Fourier algebra and L1(G) the space of Haar integrable functions on G. We study the Segal algebra SA(G)=A(G)\cap L1(G) in A(G). It admits an operator space structure which makes it a completely contractive Banach algebra. We compute the dual space of SA(G). We use it show that restriction operator u|->u|H:SA(G)->A(H), for some non-open closed subgroups H, is a surjective complete qutient map. We also show that if N is a non-compact closed subgroup, then the averaging operator tau_N:SA(G)->L1(G/N), tau_N u(sN)=\int_N u(sn)dn is a surjective complete quotient map. This puts an operator space perspective on the philosophy that SA(G) is ``locally A(G) while globally L1''. Also, using the operator space structure we can show that SA(G) is operator amenable exactly when when G is compact; and we can show that it is always operator weakly amenable. To obtain the latter fact, we use E. Samei's theory of hyper-Tauberian Banach algebras.

2015

Let G be a compact group. For 1≤ p≤∞ we introduce a class of Banach function algebras A^p(G) on G which are the Fourier algebras in the case p=1, and for p=2 are certain algebras discovered in forrestss1. In the case p=2 we find that A^p(G)A^p(H) if and only if G and H are isomorphic compact groups. These algebras admit natural operator space structures, and also weighted versions, which we call p-Beurling-Fourier algebras. We study various amenability and operator amenability properties, Arens regularity and representability as operator algebras. For a connected Lie G and p>1, our techniques of estimation of when certain p-Beurling-Fourier algebras are operator algebras rely more on the fine structure of G, than in the case p=1. We also study restrictions to subgroups. In the case that G=SU(2), restrict to a torus and obtain some exotic algebras of Laurent series. We study amenability properties of these new algebras, as well.

In this paper we introduce a new way of deforming convolution algebras and Fourier algebras on locally compact groups. We demonstrate that this new deformation allows us to reveal some informations of the underlying groups by examinining Banach algebra properties of deformed algebras. More precisely, we focus on representability as an operator algebra of deformed convolution algebras on compact connected Lie groups with connection to the real dimension of the underlying group. Similary, we investigate complete representability as an operator algebra of deformed Fourier algebras on some finitely generated discrete groups with connection to the growth rate of the group.

2010

We study multiplier algebras for a large class of Banach algebras which contains the group algebra L 1 (G), the Beurling algebras L 1 (G, ω), and the Fourier algebra A(G) of a locally compact group G. This study yields numerous new results and unifies some existing theorems on L 1 (G) and A(G) through an abstract Banach algebraic approach. Applications are obtained on representations of multipliers over locally compact quantum groups and on topological centre problems. In particular, five open problems in abstract harmonic analysis are solved.

Canadian Journal of Mathematics, 1974

We let G denote an infinite compact group, and Ĝ its dual. We use the notation of our book [3, Chapters 7 and 8]. Recall that A(G) denotes the Fourier algebra of G (an algebra of continuous functions on G), and denotes its dual space under the pairing 〈f, ϕ〉 , (f ∊ A (G), ϕ ∊ ). Further, note is identified with the C*-algebra of bounded operators on L 2(G) commuting with left translation. The module action of A (G) on is defined by the following: for f ∊ A (G), ϕ ∊ , f · ϕ ∊ by