For a finite group G, let $\operatorname { {AD}}(G)$ denote the Fourier norm of the antidiagonal in $G\times G$. The author showed recently in [‘An explicit minorant for the amenability constant of the Fourier algebra’, Int. Math. Res. Not. IMRN 2023 (2023), 19390–19430] that $\operatorname { {AD}}(G)$ coincides with the amenability constant of the Fourier algebra of G and is equal to the normalized sum of the cubes of the character degrees of G. Motivated by a gap result for amenability constants from Johnson [‘Non-amenability of the Fourier algebra of a compact group’, J. Lond. Math. Soc. (2) 50 (1994), 361–374], we determine exactly which numbers in the interval $[1,2]$ arise as values of $\operatorname { {AD}}(G)$. As a by-product, we show that the set of values of $\operatorname { {AD}}(G)$ does not contain all its limit points. Some other calculations or bounds for $\operatorname { {AD}}(G)$ are given for familiar classes of finite groups. We also indicate a connection between $\operatorname { {AD}}(G)$ and the commuting probability of G, and use this to show that every finite group G satisfying $\operatorname { {AD}}(G)< {61}/{15}$ must be solvable; here, the value ${61}/{15}$ is the best possible.

We consider a general twisted shift-invariant system, $V^{t}(\mathcal {A})$, consisting of twisted translates of countably many generators and study the problem of obtaining a characterization for the system $V^{t}(\mathcal {A})$ to form a frame sequence or a Riesz sequence. We illustrate our theory with some examples. In addition to these results, we study a dual twisted shift-invariant system and also obtain an orthonormal sequence of twisted translates from a given Riesz sequence of twisted translates.

Let 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 $A(H)^{\ast \ast }.$

Let $ H $ be a compact subgroup of a locally compact group $ G $. We first investigate some (operator) (co)homological properties of the Fourier algebra $A(G/H)$ of the homogeneous space $G/H$ such as (operator) approximate biprojectivity and pseudo-contractibility. In particular, we show that $ A(G/H) $ is operator approximately biprojective if and only if $ G/H $ is discrete. We also show that $A(G/H)^{**}$ is boundedly approximately amenable if and only if G is compact and H is open. Finally, we consider the question of existence of weakly compact multipliers on $A(G/H)$.

The purpose of this note is to construct an example of a discrete non-abelian group G and a subset E of G, not contained in any abelian subgroup, that is a completely bounded $\Lambda (p)$ set for all $p<\infty ,$ but is neither a Leinert set nor a weak Sidon set.

In this short note we first extend the validity of the spectral radius formula, obtained by M. Anoussis and G. Gatzouras, for Fourier–Stieltjes algebras. The second part is devoted to showing that, for the measure algebra on any locally compact non-discrete Abelian group, there are no non-trivial constraints among three quantities: the norm, the spectral radius, and the supremum of the Fourier–Stieltjes transform, even if we restrict our attention to measures with all convolution powers singular with respect to the Haar measure.

Motivated by the definition of a semigroup compactication of a locally compact group and a large collection of examples, we introduce the notion of an (operator) homogeneous left dual Banach algebra (HLDBA) over a (completely contractive) Banach algebra $A$. We prove a Gelfand-type representation theorem showing that every HLDBA over A has a concrete realization as an (operator) homogeneous left Arens product algebra: the dual of a subspace of $A^{\ast }$ with a compatible (matrix) norm and a type of left Arens product $\Box$. Examples include all left Arens product algebras over $A$, but also, when $A$ is the group algebra of a locally compact group, the dual of its Fourier algebra. Beginning with any (completely) contractive (operator) $A$-module action $Q$ on a space $X$, we introduce the (operator) Fourier space $({\mathcal{F}}_{Q}(A^{\ast }),\Vert \cdot \Vert _{Q})$ and prove that $({\mathcal{F}}_{Q}(A^{\ast })^{\ast },\Box )$ is the unique (operator) HLDBA over $A$ for which there is a weak$^{\ast }$-continuous completely isometric representation as completely bounded operators on $X^{\ast }$ extending the dual module representation. Applying our theory to several examples of (completely contractive) Banach algebras $A$ and module operations, we provide new characterizations of familiar HLDBAs over A and we recover, and often extend, some (completely) isometric representation theorems concerning these HLDBAs.

Let $G$ be a separable locally compact group with type $I$ left regular representation, $\widehat{G}$ its dual, $A(G)$ its Fourier algebra and $f\in A(G)$ with compact support. If $G=\mathbb{R}$ and the Fourier transform of $f$ is compactly supported, then, by a classical Paley–Wiener theorem, $f=0$. There are extensions of this theorem for abelian and some unimodular groups. In this paper, we prove that if $G$ has no (nonempty) open compact subsets, $\hat{f}$, the regularised Fourier cotransform of $f$, is compactly supported and $\text{Im}\,\hat{f}$ is finite dimensional, then $f=0$. In connection with this result, we characterise locally compact abelian groups whose identity components are noncompact.

Let $G$ be a locally compact amenable group and $A(G)$ and $B(G)$ be the Fourier and the Fourier–Stieltjes algebras of $G,$ respectively. For a power bounded element $u$ of $B(G)$ , let ${\mathcal{E}}_{u}:=\{g\in G:|u(g)|=1\}$ . We prove some convergence theorems for iterates of multipliers in Fourier algebras.

(a) If $\Vert u\Vert _{B(G)}\leq 1$ , then $\lim _{n\rightarrow \infty }\Vert u^{n}v\Vert _{A(G)}=\text{dist}(v,I_{{\mathcal{E}}_{u}})\text{ for }v\in A(G)$ , where $I_{{\mathcal{E}}_{u}}=\{v\in A(G):v({\mathcal{E}}_{u})=\{0\}\}$ .

(b) The sequence $\{u^{n}v\}_{n\in \mathbb{N}}$ converges for every $v\in A(G)$ if and only if ${\mathcal{E}}_{u}$ is clopen and $u({\mathcal{E}}_{u})=\{1\}.$

(c) If the sequence $\{u^{n}v\}_{n\in \mathbb{N}}$ converges weakly in $A(G)$ for some $v\in A(G)$ , then it converges strongly.

Let $G$ be a compact group. The aim of this note is to show that the only continuous *-homomorphism from $L^{1}(G)$ to $\ell ^{\infty }\text{-}\bigoplus _{[{\it\pi}]\in {\hat{G}}}{\mathcal{B}}_{2}({\mathcal{H}}_{{\it\pi}})$ that transforms a convolution product into a pointwise product is, essentially, a Fourier transform. A similar result is also deduced for maps from $L^{2}(G)$ to $\ell ^{2}\text{-}\bigoplus _{[{\it\pi}]\in {\hat{G}}}{\mathcal{B}}_{2}({\mathcal{H}}_{{\it\pi}})$.

In this paper we prove a new version of the Cowling-Price theorem for Fourier transforms on Rn. Using this we formulate and prove an uncertainty principle for operators. This leads to an analogue of the Cowling-Price theorem for nilpotent Lie groups. We also prove an exact analogue of the Cowling-Price theorem for the Heisenberg group.

We extend an uncertainty principle due to Cowling and Price to threadlike nilpotent Lie groups. This uncertainty principle is a generalization of a classical result due to Hardy. We are thus extending earlier work on Rn and Heisenberg groups.

We describe a generalization of the Hardy theorem on the motion group. We prove that for some weight functions νω growing very rapidly and a measurable function f, the finiteness of the Lp-norm of vf and the Lq-norm of ωf implies f=0 (almost everywhere).

Let F ⊂ G be closed and A(F) = A(G)/IF. If F is a Helson set then A(F)** is an amenable (semisimple) Banach algebra. Our main result implies the following theorem: Let G be a locally compact group, F ⊂ G closed, a ∈ G. Assume either (a) For some non-discrete closed subgroup H, the interior of F ∩ aH in aH is non-empty, or (b) R ⊂ G, S ⊂ R is a symmetric set and aS ⊂ F. Then A(F)** is a non-amenable non-semisimple Banach algebra. This raises the question: How ‘thin’ can F be for A(F)** to remain a non-amenable Banach algebra?

W. Rudin has proved that the union of the Riesz set N ⊆ R with a Λ(l)-subset of Z is again a Riesz set. In this note we generalize his result to compact groups whose contains a circle group, thereby extending an earlier F. and M. Riesz theorem for such groups by the author. We also investigate the possibility of constructing Λ(p)-sets for these groups, departing from Λ(p)-sets for the circle group in center.

Recently M. Benedicks showed that if a function f ∈ L2(Rd) and its Fourier transform both have supports of finite measure, then f = 0 almot everywhere. In this paper we give a version of this result for all noncompact semisimple connected Lie groups with finite centres.

Random Fourier series are studied for a class of compact abelian hypergroups. The randomizing factors are assumed to be independent and uniformly subgaussian. In the presence of a natural teachnical hypothesis, an entropy condition of Dudley is shown to be sufficient for almost sure continuity. The classical results on almost sure membership in Lp, where p < ∞, are generalized to this setting. As an application, it is shown that a simple condition on the dual object implies that the de Leeuw-Kahane-Katznelson phenomenon occurs. Another application is the analogue of a classical sufficient condition for almost sure continuity. Examples illustrating the general theory are given for the hypergroup of conjugacy classes of SU(2) and for a class of compact countable hypergroups.