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For a locally compact group G and p ∈ (1,8), let Bp(G) is the multiplier algebra of the Figà-Talamanca-Herz algebra Ap(G). For p = 2 and G amenable, the algebra B(G) := B2(G) is the usual Fourier-Stieltjes algebra. In this paper, we show that Ap(G) is a Bochner-Schoenberg-Eberlin (BSE) algebra and every clopen subset of G is a synthetic set for Ap(G). Furthermore, we characterize idempotent elements of the Banach algebra Bp(G). This result generalizes the Cohen-Host idempotent theorems for the case of Figà-Talamanca-Herz algebras. Characterization of idempotent elements of Bp(G) is of paramount importance to study homomorphisms in Figà-Talamanca-Herz algebras. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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.

Studies of various reproducing function systems emphasized the role of translations and the Fourier periodization function. These influenced the development of the concept of dual integrable representations, a large and important class of unitary representations on LCA groups. The key ingredient is the bracket function that enables the explicit description of corresponding cyclic spaces. Since its introduction, the notion was extended to some specific classes of non-abelian groups, and a natural problem emerged, i.e., whether it can be extended to the entire class of (including non-abelian) locally compact groups. In this paper we solve this problem. Interestingly enough, the bracket function (or rather an operator in this case) keeps all of its useful properties, except the Cauchy-Schwarz inequality. Nevertheless, we show that cyclic spaces still can be represented in terms of bracket-weighted spaces. [ABSTRACT FROM AUTHOR]

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 Z 2 -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. [ABSTRACT FROM AUTHOR]

In this paper, we study the Radon-Nikodým Property for the Fourier space of a commutative compact hypergroup and that of a compact (non necessarily commutative) hypergroup. We prove the coincidence of the weak-* topology and the norm topology on the unit sphere of the subset AK (H) of the Fourier space A(H) of a commutative hypergroup H consisting of elements that have support in a fixed compact subset K of the hypergroup H. Finally, we derive the fact that AK (H) has the Radon-Nikodým property. [ABSTRACT FROM AUTHOR]

In this paper, for a locally compact group and a fixed number , we give some sufficient conditions for the set to be spaceable in . Also, by some special Segal algebras which recently have been introduced, we find spaceable subsets of the Fourier algebra . Finally, we give some necessary and sufficient conditions for a locally compact group to be compact or discrete.

In this paper, we present a general version of the algebra AM(G) which was introduced by B. Forrest. Indeed, for a faithful commutative Banach algebra A, we embed it in ℳ(A), the multiplier algebra of A, and obtain Banach algebra AM. Then, we study the spaceability of AM− A and AM (G) ℒ ℒA(G). These results give some characterizations of compactness and discreteness of locally compact groups. Also, we show that AM(G) is an ideal in its second dual if and only if G is discrete. Finally, we study the BSE-property of AM(G). [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

Let A be a Banach algebra and let φ be a non-zero character on A . Suppose that A M is the closure of the faithful Banach algebra A in the multiplier norm. In this paper, topologically left invariant φ -means on A M ∗ are defined and studied. Under some conditions on A , we will show that the set of topologically left invariant φ -means on A ∗ and on A M ∗ have the same cardinality. The main applications are concerned with the quantum group algebra L 1 (G) of a locally compact quantum group G . In particular, we obtain some characterizations of compactness of G in terms of the existence of a non-zero (weakly) compact left or right multiplier on L M 1 (G) or on its bidual in some senses. [ABSTRACT FROM AUTHOR]

The Fourier algebra of the affine group of the real line has a natural identification, as a Banach space, with the space of trace-class operators on L 2 ( R × , d t / | t |) . In this paper we study the “dual convolution product” of trace-class operators that corresponds to pointwise product in the Fourier algebra. Answering a question raised in work of Eymard and Terp, we provide an intrinsic description of this operation which does not rely on the identification with the Fourier algebra, and obtain a similar result for the connected component of this affine group. In both cases we construct explicit derivations on the corresponding Banach algebras, verifying the derivation identity directly without requiring the inverse Fourier transform. We also initiate the study of the analogous Banach algebra structure for trace-class operators on L p ( R × , d t / | t |) for p ∈ (1 , 2) ∪ (2 , ∞) . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For an inverse semigroup S with the set of idempotents E, we find necessary and sufficient conditions for the Fourier algebra A(S) to be module amenable, module character amenable, module (operator) biflat and module (operator) biprojective (as l 1 (E) -module). Some examples show that when S is either a bicyclic inverse semigroup or a Brandt inverse semigroup, A(S) is module amenable, module biprojective and module biflat as well. [ABSTRACT FROM AUTHOR]

A locally compact group G is discrete if and only if the Fourier algebra A(G) has a non-zero (weakly) compact multiplier. We partially extend this result to the setting of ultraspherical hypergroups. Let H be an ultraspherical hypergroup and let A(H) denote the corresponding Fourier algebra. We will give several characterizations of discreteness of H in the terms of the algebraic properties of A(H). We also study Arens regularity of closed ideals of A(H). [ABSTRACT FROM AUTHOR]

In this paper we give an example of a Banach algebra A and a closed ideal I of A such that the multiplier algebra of I is equal to A but I does not have any bounded approximate identity. In the case that I has an approximate identity, we give a necessary condition on I for which A = M (I) , where M (I) denotes the multiplier algebra of I. Finally, as a corollary of our results, we show that the Fourier algebra of an amenable group is strictly dense in the Fourier–Stieltjes algebra. [ABSTRACT FROM AUTHOR]

In this paper, we investigate key assumptions on some Banach algebras to have quasi-weak normal structure (resp. quasi-weak ⋆ normal structure) and to prove in particular that fixed point property for Kannan mappings is satisfied on weakly compact convex subsets in this setting. We establish some conditions on a locally compact group G for which the Fourier and Fourier–Stieltjes–Banach algebras A(G) and B(G) have quasi-normal structure. In addition, we give some examples of Banach spaces X such that L (X) (the Banach algebra of bounded linear operators on X) and some of its closed two-sided ideals associated with Fredholm perturbations have fixed point properties. Finally, we give some comments and interesting questions related to this area. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for fractional Schrödinger equation with cubic convolution nonlinearity (i ∂ t u − (− Δ) α 2 u ± (K ⁎ | u | 2) u = 0) with Cauchy data in the modulation spaces M p , q (R d). For K (x) = | x | − γ (0 < γ < min { α , d / 2 }) , we establish global well-posedness results in M p , q (R d) (1 ≤ p ≤ 2 , 1 ≤ q < 2 d / (d + γ)) when α = 2 , d ≥ 1 , and with radial Cauchy data when d ≥ 2 , 2 d 2 d − 1 < α < 2. Similar results are proven in Fourier algebra F L 1 (R d) ∩ L 2 (R d). [ABSTRACT FROM AUTHOR]

We address two errors made in our paper [7]. The most significant error is in Theorem 1.1. We repair this error, and show that the main result, Theorem 2.5 of [7] , is true. The second error is in one of our examples, Remark 2.4 (iv), and we partially resolve it. [ABSTRACT FROM AUTHOR]

Let A be a complex commutative semisimple Banach algebra. In this paper, we study some ergodic properties of Cesàro bounded multipliers on A. The results are linked to the sets of synthesis and the main applications are concerned with Fourier and Fourier-Stieltjes algebras on locally compact groups. We study also the structure of ideals associated with multipliers of A and A-invariant projections of the dual space of A. Some related problems are also discussed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For an amenable inverse semigroup S with the set of idempotents E and a minimal idempotent, we explicitly construct a contractive and positive module operator virtual diagonal on the Fourier algebra A(S), as a completely contractive Banach algebra and operator module over ℓ1(E). This generalizes a well known result of Zhong-Jin Ruan on operator amenability of the Fourier algebra of a (discrete) group Ruan (Am J Math 117:1449-1474, 1995). [ABSTRACT FROM AUTHOR]

Let G be a locally compact group and let A(G) be its Fourier algebra. A closed subset H of G is said to be a Helson set if the restriction homomorphism ø : A(G) → C0(H), ø(a) = a|H, is surjective. In this paper, under the hypothesis that G is amenable, we prove that every Helson subset H of G that is also a set of synthesis is a Ditkin set. This result is new even for G = R. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Suppose that f is the characteristic function of a probability measure on the real line $$\mathbb R$$ . In this paper, we deal with the following problem posed by N.G. Ushakov: Is it true that f is never determined by its imaginary part $$\mathfrak {I}f$$ ? In other words, is it true that for any characteristic function f there exists a characteristic function g such that $$\mathfrak {I}f\equiv \mathfrak {I}g$$ but $$ f\not \equiv g$$ ? We study this question in the more general case of the characteristic function defined on an arbitrary locally compact abelian group. A characterization of what characteristic functions are uniquely determined by their imaginary parts are given. As a consequence of this characterization, we obtain that several frequently used characteristic functions on the classical locally compact abelian groups are uniquely determined by their imaginary parts. [ABSTRACT FROM AUTHOR]

Suppose that f is the characteristic function of a probability measure on the real line R. We deal with the following open problem posed by N.G. Ushakov: Is it true that f is never determined by its imaginary part Im f? In other words, is it true that for any characteristic function f, there exists a characteristic function g such that Im f = Im g, but f ≠ g? The answer to this question is no. We give a characterization of those characteristic functions, which are uniquely determined by their imaginary parts. Also, several examples of characteristic functions, which are uniquely determined by their imaginary parts, are given.

We study locally compact groups for which the Fourier algebra coincides with the Rajchman algebra. In particular, we show that there exist uncountably many non-compact groups with this property. Generalizing a result of Hewitt and Zuckerman, we show that no non-compact nilpotent group has this property, whereas non-compact solvable groups with this property are known to exist. We provide several structural results on groups whose Fourier and Rajchman algebras coincide as well as new criteria for establishing this property. Finally, we study the relation between groups with completely reducible regular representation and groups whose Fourier and Rajchman algebras coincide. For unimodular groups with completely reducible regular representation, we show that the Fourier algebra may in general be strictly smaller than the Rajchman algebra. [ABSTRACT FROM AUTHOR]

Different notions of amenability on hypergroups and their relations are studied. Developing Leptin's theorem for discrete hypergroups, we characterize the existence of a bounded approximate identity for hypergroup Fourier algebras. We study the Leptin condition for discrete hypergroups derived from the representation theory of some classes of compact groups. Studying amenability of the hypergroup algebras for discrete commutative hypergroups, we obtain some results on amenability properties of some central Banach algebras on compact and discrete groups. [ABSTRACT FROM AUTHOR]

We study various operator homological properties of the Fourier algebra A ( G ) of a locally compact group G . Establishing the converse of two results of Ruan and Xu [35] , we show that A ( G ) is relatively operator 1-projective if and only if G is IN, and that A ( G ) is relatively operator 1-flat if and only if G is inner amenable. We also exhibit the first known class of groups for which A ( G ) is not relatively operator C -flat for any C ≥ 1 . As applications of our techniques, we establish a hereditary property of inner amenability, answer an open question of Lau and Paterson [24] , and answer an open question of Anantharaman-Delaroche [1] on the equivalence of inner amenability and Property (W). In the bimodule setting, we show that relative operator 1-biflatness of A ( G ) is equivalent to the existence of a contractive approximate indicator for the diagonal G Δ in the Fourier–Stieltjes algebra B ( G × G ) , thereby establishing the converse to a result of Aristov, Runde, and Spronk [3] . We conjecture that relative 1-biflatness of A ( G ) is equivalent to the existence of a quasi-central bounded approximate identity in L 1 ( G ) , that is, G is QSIN, and verify the conjecture in many special cases. We finish with an application to the operator homology of A c b ( G ) , giving examples of weakly amenable groups for which A c b ( G ) is not operator amenable. [ABSTRACT FROM AUTHOR]

We initiate the study of the completely bounded multipliers of the Haagerup tensor product of two copies of the Fourier algebra of a locally compact group . If is a closed subset of we let and show that if is a set of spectral synthesis for then is a set of local spectral synthesis for . Conversely, we prove that if is a set of spectral synthesis for and is a Moore group then is a set of spectral synthesis for . Using the natural identification of the space of all completely bounded weak* continuous -bimodule maps with the dual of , we show that, in the case is weakly amenable, such a map leaves the multiplication algebra of invariant if and only if its support is contained in the antidiagonal of . [ABSTRACT FROM AUTHOR]

We investigate involutions and trivolutions in the second dual of algebras related to a locally compact topological semigroup and the Fourier algebra of a locally compact group. We prove, among the other things, that for a large class of topological semigroups namely, compactly cancellative foundation $$*$$ -semigroup S when it is infinite non-discrete cancellative, $$M_a(S)^{**}$$ does not admit an involution, and $$M_a(S)^{**}$$ has a trivolution with range $$M_a(S)$$ if and only if S is discrete. We also show that when G is an amenable group, the second dual of the Fourier algebra of G admits an involution extending one of the natural involutions of A( G) if and only if G is finite. However, $$A(G)^{**}$$ always admits trivolution. [ABSTRACT FROM AUTHOR]

The notion of reduced synthesis in the context of harmonic analysis on general locally compact groups is introduced; in the classical situation of commutative groups, this notion means that a function f in the Fourier algebra is annihilated by any pseudofunction supported on f (0). A relationship between reduced synthesis and compact synthesis (i.e., the possibility of approximating compact operators by pseudointegral ones without increasing the support) is determined, which makes it possible to obtain new results both in operator theory and in harmonic analysis. Applications to the theory of linear operator equations are also given. [ABSTRACT FROM AUTHOR]

For a Banach function algebra A , we consider the problem of representing a continuous d -homogeneous polynomial P : A → X , where X is an arbitrary Banach space, that satisfies the property P ( f + g ) = P ( f ) + P ( g ) whenever f , g ∈ A are such that supp ( f ) ∩ supp ( g ) = ∅ . We show that such a polynomial can be represented as P ( f ) = T ( f d ) ( f ∈ A ) for some continuous linear map T : A → X for a variety of Banach function algebras such as the algebra of continuous functions C 0 ( Ω ) for any locally compact Hausdorff space Ω, the algebra of Lipschitz functions lip α ( K ) for any compact metric space K and α ∈ ] 0 , 1 [ , the Figà–Talamanca–Herz algebra A p ( G ) for some locally compact groups G and p ∈ ] 1 , + ∞ [ , the algebras A C ( [ a , b ] ) and B V C ( [ a , b ] ) of absolutely continuous functions and of continuous functions of bounded variation on the interval [ a , b ] . In the case where A = C n ( [ a , b ] ) , P can be represented as P ( f ) = ∑ T ( n 1 , … , n d ) ( f ( n 1 ) ⋯ f ( n d ) ) , where the sum is taken over ( n 1 , … , n d ) ∈ Z d with 0 ≤ n 1 ≤ … ≤ n d ≤ n , for appropriate continuous linear maps T ( n 1 , … , n d ) : C n − n d ( [ a , b ] ) → X . [ABSTRACT FROM AUTHOR]

Motivated by and extending a theorem of Reiter on sets of synthesis in $$\mathbb {R}^N$$ , we establish a general result for Fourier algebras of locally compact groups, even in the wider context of regular, semisimple and Tauberian commutative Banach algebras, which contains Reiter's theorem as a special case and explains why it holds. In addition, we give a number of examples and several further results on weak spectral sets. [ABSTRACT FROM AUTHOR]

By analogy with the classical construction due to Forrest, Samei and Spronk, we associate to every compact quantum group , a completely contractive Banach algebra , which can be viewed as a deformed Fourier algebra of . To motivate the construction, we first analyze in detail the quantum version of the integration over the diagonal subgroup, showing that although the quantum diagonal subgroups in fact never exist, as noted earlier by Kasprzak and Sołtan, the corresponding integration represented by a certain idempotent state on makes sense as long as is of Kac type. Finally, we analyze as an explicit example the algebras , , associated to Wang's free orthogonal groups, and show that they are not operator weakly amenable. [ABSTRACT FROM AUTHOR]

We introduce a certain property of commutative Banach alge- bras which we call property OB. We prove that every bounded disjointness-preserving linear map from a commutative Banach algebra with the aforesaid property to any semisimple, commutative Banach algebra is a weighted composition map. Further, it is shown that a variety of important Banach algebras in harmonic analysis have the property OB. [ABSTRACT FROM AUTHOR]

We show that for a locally compact group G , amongst a class which contains amenable and small invariant neighbourhood groups, its Fourier algebra A ( G ) satisfies a completely bounded version Pisier's similarity property with similarity degree at most 2. Specifically, any completely bounded homomorphism π : A ( G ) → B ( H ) admits an invertible S in B ( H ) for which ‖ S ‖ ‖ S − 1 ‖ ≤ ‖ π ‖ cb 2 and S − 1 π ( ⋅ ) S extends to a ⁎-representation of the C*-algebra C 0 ( G ) . This significantly improves some results due to Brannan and Samei (2010) [5] and Brannan, Daws and Samei (2013) [4] . We also note that A ( G ) has completely bounded similarity degree 1 if and only if it is completely isomorphic to an operator algebra if and only if G is finite. [ABSTRACT FROM AUTHOR]

Let Γ be a locally compact group. We answer two questions left open in [8] and [10] : (i) For abelian Γ, we prove that if χ S ∈ B ( Γ ) is an idempotent with norm ‖ χ S ‖ < 4 3 , then S is the union of two cosets of an open subgroup of Γ. (ii) For general Γ, we prove that if χ S ∈ M c b A ( Γ ) is an idempotent with norm ‖ χ S ‖ c b < 1 + 2 2 , then S is an open coset in Γ. [ABSTRACT FROM AUTHOR]