Fourier Algebra Research Articles

Beurling-Fourier algebras and complexification

In this paper, we develop a new approach that allows to identify the Gelfand spectrum of weighted Fourier algebras as a subset of an abstract complexification of the corresponding group for a wide class of groups and weights. This generalizes recent related results of Ghandehari-Lee-Ludwig-Spronk-Turowska [11] about the spectrum of Beurling-Fourier algebras on some Lie groups. In the case of discrete groups we show that the spectrum of Beurling-Fourier algebra is homeomorphic to the group itself.

New results on spectral synthesis

In our former paper we introduced the concept of localization of ideals in the Fourier algebra of a locally compact Abelian group. It turns out that localizability of a closed ideal in the Fourier algebra is equivalent to the synthesizability of the annihilator of that closed ideal which corresponds to this ideal in the measure algebra. This equivalence provides an effective tool to prove synthesizability of varieties on locally compact Abelian groups. In this paper we utilize this tool to show that when investigating synthesizability of a variety, roughly speaking compact elements of the group can be neglected. Our main result is that spectral synthesis holds on a locally compact Abelian group G if and only if it holds on G/B, where B is the closed subgroup of all compact elements. In particular, spectral synthesis holds on compact Abelian groups. Also we obtain a simple proof for the characterization theorem of spectral synthesis on discrete Abelian groups.

On the growth of Fourier multipliers

We define a sequence of functions, namely, tame cuts, in the Fourier algebra A(G) of a locally compact group G , which satisfies certain convergence and growth conditions. This new consideration allows us to give a group admitting a Fourier multiplier that is not completely bounded. Furthermore, we show that the induction map MA(\Gamma)\rightarrow MA(G) is not always continuous. We also show how Liao’s property (T_{\mathrm{Schur}},G,K) opposes tame cuts. Some examples are provided.

Hyper-instability of Banach algebras

<abstract><p>In this paper, we introduce and study the concept of hyper-instability as a strong version of multiplicative instability. This concept provides a powerful tool to study the multiplicative instability of Banach algebras. It replaces the condition of the iterated limits in the definition of multiplicative instability with conditions that are easier to examine. In particular, special conditions are suggested for Banach algebras that admit bounded approximate identities. Moreover, these conditions are preserved under isomorphisms. This enlarges the class of studied Banach algebras. We prove that many interesting Banach algebras are hyper-unstable, such as $ C^* $-algebras, Fourier algebras, and the algebra of compact operators on Banach spaces, each under certain conditions.</p></abstract>

Group C*-Algebras of Locally Compact Groups Acting on Trees

Abstract It was proved by Samei and Wiersma that for every non-compact, closed subgroup $G$ of the automorphism group $\textrm {Aut}(T)$ of a (semi-)homogeneous tree $T$ acting transitively on the boundary $\partial T$ and every $2 \leq q < p \leq \infty $, the quotient map $C^{\ast }_{L^{p+}}(G) \twoheadrightarrow C^{\ast }_{L^{q+}}(G)$ is not injective. We prove that whenever $G$, moreover, acts transitively on the vertices of $T$ and has Tits’s independence property, the group $C^{\ast }$-algebras $C^{\ast }_{L^{p+}}(G)$ are the only group $C^{\ast }$-algebras of $G$ coming from ideals of the Fourier–Stieltjes algebra. We also show that given such a group $G$, every group $C^{\ast }$-algebra $C^{\ast }_{\mu }(G)$ that is distinguishable from $C^{\ast }(G)$ and whose dual space $C^{\ast }_{\mu }(G)^{\ast }$ is an ideal in $B(G)$ is abstractly $^{\ast}$-isomorphic to $C^{\ast }_{r}(G)$.

Time and band limiting for exceptional polynomials

The “time-and-band limiting” commutative property was found and exploited by D. Slepian, H. Landau and H. Pollak at Bell Labs in the 1960's, and independently by M. Mehta and later by C. Tracy and H. Widom in Random matrix theory. The property in question is the existence of local operators with simple spectrum that commute with naturally appearing global ones.Here we give a general result that insures the existence of a commuting differential operator for a given family of exceptional orthogonal polynomials satisfying the “bispectral property”. As a main tool we go beyond bispectrality and make use of the notion of Fourier Algebras associated to the given sequence of exceptional polynomials. We illustrate this result with two examples, of Hermite and Laguerre type, exhibiting also a nice Perline's form for the commuting differential operator.

Homomorphisms of L1 algebras and Fourier algebras

We investigate conditions for the extendibility of continuous algebra homomorphisms ϕ from the Fourier algebra A(F) of a locally compact group F to the Fourier-Stieltjes algebra B(G) of a locally compact group G to maps between the corresponding L∞ algebras which are weak* continuous. When ϕ is completely bounded and F is amenable, it is induced by a piecewise affine map α:Y→F where Y⊆G. We show that extendibility of ϕ is equivalent to α being an open map. We also study the dual problem for contractive homomorphisms ϕ:L1(F)→M(G). We show that ϕ induces a w* continuous homomorphism between the von Neumann algebras of the groups if and only if the naturally associated map θ (Greenleaf [1965], Stokke [2011]) is a proper map.

Projectivity of some Banach modules over semiprime Banach algebras

Let [Formula: see text] be a semiprime Banach algebra and let [Formula: see text] be a nonzero character on [Formula: see text]. In this paper, we discuss the projectivity of certain left Banach [Formula: see text]-modules in terms of left [Formula: see text]-contractibility of Banach algebras. We also apply our results for the category of quantum group algebras and eventually, we present illustrative examples of the projective left Banach modules over the group algebra [Formula: see text] and the Fourier algebra [Formula: see text] of a locally compact group [Formula: see text].

Electron Beams on the Brillouin Zone: A Cohomological Approach via Sheaves of Fourier Algebras

Topological states of matter can be classified only in terms of global topological invariants. These global topological invariants are encoded in terms of global observable topological phase factors in the state vectors of electrons. In condensed matter, the energy spectrum of the Hamiltonian operator has a band structure, meaning that it is piecewise continuous. The energy in each continuous piece depends on the quasi-momentum which varies in the Brillouin zone. Thus, the Brillouin zone of quasi-momentum variables constitutes the base localization space of the energy eigenstates of electrons. This is a continuous topological parameter space bearing the homotopy of a torus. Since the base localization space has the homotopy of a torus, if we vary the quasi-momentum in a direction, when the edge of the zone is reached, we obtain a closed path. Then, if we lift this loop from the base space to the sections of the sheaf-theoretic fibration induced by the localization of the energy eigenfunctions, we obtain a global topological phase factor which encodes the topological structure of the Brillouin zone. Because it is homotopically equivalent to a torus, the global phase factor turns out to be quantized, taking integer values. The experimental significance of this model stems from the recent discovery that there are observable global topological phase factors in fairly ordinary materials. In this communication, we show that it is the unitary representation theory of the discrete Heisenberg group in terms of commutative modular symplectic variables, giving rise to a joint commutative representation space endowed with an integral and Z2-invariant symplectic form that articulates the specific form of the topological conditions characterizing both the quantum Hall effect and the spin quantum Hall effect under a unified sheaf-theoretic cohomological framework.

Reflective prolate‐spheroidal operators and the adelic Grassmannian

AbstractBeginning with the work of Landau, Pollak and Slepian in the 1960s on time‐band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory, and integrable systems. Previously, such pairs were constructed by ad hoc methods, which essentially worked because a commuting operator of low order could be found by a direct calculation. We describe a general approach to these problems that proves that every point W of Wilson's infinite dimensional adelic Grassmannian gives rise to an integral operator , acting on for a contour , which reflects a differential operator with rational coefficients in the sense that on a dense subset of . By using analytic methods and methods from integrable systems, we show that the reflected differential operator can be constructed from the Fourier algebra of the associated bispectral function . The exact size of this algebra with respect to a bifiltration is in turn determined using algebro‐geometric methods. Intrinsic properties of four involutions of the adelic Grassmannian naturally lead us to consider the reflecting property above in place of plain commutativity. Furthermore, we prove that the time‐band limited operators of the generalized Laplace transforms with kernels given by the rank one bispectral functions always reflect a differential operator. A 90° rotation argument is used to prove that the time‐band limited operators of the generalized Fourier transforms with kernels admit a commuting differential operator. These methods produce vast collections of integral operators with prolate‐spheroidal properties, associated to the wave functions of all rational solutions of the KP hierarchy vanishing at infinity, introduced by Krichever in the late 1970s.

An Explicit Minorant for the Amenability Constant of the Fourier Algebra

Abstract We show that if a locally compact group $G$ is non-abelian, then the amenability constant of its Fourier algebra is $\geq 3/2$, extending a result of [9] who proved that this holds for finite non-abelian groups. Our lower bound, which is known to be best possible, improves on results by previous authors and answers a question raised by [16]. To do this, we study a minorant for the amenability constant, related to the anti-diagonal in $G\times G$, which was implicitly used in Runde’s work but hitherto not studied in depth. Our main novelty is an explicit formula for this minorant when $G$ is a countable virtually abelian group, in terms of the Plancherel measure for $G$. As further applications, we characterize those non-abelian groups where the minorant attains its minimal value and present some examples to support the conjecture that the minorant always coincides with the amenability constant.

On isomorphisms between ideals of Fourier algebras of finite abelian groups

On isomorphisms between ideals of Fourier algebras of finite abelian groups

On character amenability of the Fourier algebra

On character amenability of the Fourier algebra

On approximately biprojective and approximately biflat Banach algebras

In this paper, we study the approximate biprojectivity and the approximate biflatness of a Banach algebra A and find some relations between theses concepts with ?-amenability and ? -contractibility, where ? is a character on A. Among other things, we show that ?-Lau product algebra L1(G) ?? A(G) is approximately biprojective if and only if G is finite, where L1(G) and A(G) are the group algebra and the Fourier algebra of a locally compact group G, respectively. We also characterize approximately biprojective and approximately biflat semigroup algebras associated with the inverse semigroups.

Some tools to study sets of spectral synthesis for the Fourier algebra $A(G)$ of a LCA group $G$ and application

Some tools to study sets of spectral synthesis for the Fourier algebra $A(G)$ of a LCA group $G$ and application

Amenability, weak amenability and bounded approximate identities in multipliers of the Fourier algebra

Amenability, weak amenability and bounded approximate identities in multipliers of the Fourier algebra

Amenability constants of central Fourier algebras of finite groups

Amenability constants of central Fourier algebras of finite groups

ON THE ALGEBRAS AND OF AN ULTRASPHERICAL HYPERGROUP

AbstractLet H be an ultraspherical hypergroup and let $A(H)$ be the Fourier algebra associated with $H.$ In this paper, we study the dual and the double dual of $A(H).$ We prove among other things that the subspace of all uniformly continuous functionals on $A(H)$ forms a $C^*$ -algebra. We also prove that the double dual $A(H)^{\ast \ast }$ is neither commutative nor semisimple with respect to the Arens product, unless the underlying hypergroup H is finite. Finally, we study the unit elements of $A(H)^{\ast \ast }.$

On the Functorial Properties of the p-Analog of the Fourier–Stieltjes Algebras

In this paper, some known results about the functorial properties of the Fourier–Stieltjes algebra, B(G), will be generalized. First of all, the idempotent theorem on the Fourier–Stieltjes algebra will be promoted and linked to the p-analog one. Next, the p-analog of the \(\pi\)-Fourier space introduced by Arsac will be given, and by taking advantage of the theory of ultrafilters, the connection between the dual space of the algebra of p-pseudofunctions and the p-analog of the \(\pi\)-Fourier space will be fully investigated. As the main result, one of the significant and applicable functorial properties of the p-analog of the Fourier–Stieltjes algebras will be achieved.

Dimension-free bounds and structural results in communication complexity

The purpose of this article is to initiate a systematic study of dimension-free relations between basic communication and query complexity measures and various matrix norms. In other words, our goal is to obtain inequalities that bound a parameter solely as a function of another parameter. This is in contrast to perhaps the more common framework in communication complexity where poly-logarithmic dependencies on the number of input bits are tolerated.Dimension-free bounds are also closely related to structural results, where one seeks to describe the structure of Boolean matrices and functions that have low complexity. We prove such theorems for several communication and query complexity measures as well as various matrix and operator norms. In several other cases we show that such bounds do not exist.We propose several conjectures, and establish that, in addition to applications in complexity theory, these problems are central to characterization of the idempotents of the algebra of Schur multipliers, and could lead to new extensions of Cohen’s celebrated idempotent theorem regarding the Fourier algebra.