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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.

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, 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.

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 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]

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]

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 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]

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]

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]

We show that for a connected Lie group G , its Fourier algebra A ( G ) is weakly amenable only if G is abelian. Our main new idea is to show that weak amenability of A ( G ) implies that the anti-diagonal, Δ ˇ G = { ( g , g − 1 ) : g ∈ G } , is a set of local synthesis for A ( G × G ) . We then show that this cannot happen if G is non-abelian. We conclude for a locally compact group G , that A ( G ) can be weakly amenable only if it contains no closed connected non-abelian Lie subgroups. In particular, for a Lie group G , A ( G ) is weakly amenable if and only if its connected component of the identity G e is abelian. [ABSTRACT FROM AUTHOR]

Motivated by the beautiful work of M.A. Rieffel (1965) and of M.E. Walter (1974), we obtain characterisations of the Fourier algebra A ( G ) of a locally compact group G in terms of the class of F -algebras (i.e. a Banach algebra A such that its dual A ′ is a W ⁎ -algebra whose identity is multiplicative on A ). For example, we show that the Fourier algebras are precisely those commutative semisimple F -algebras that are Tauberian, contain a nonzero real element, and possess a dual semigroup that acts transitively on their spectrums. Our characterisations fall into three flavours, where the first one will be the basis of the other two. The first flavour also implies a simple characterisation of when the predual of a Hopf–von Neumann algebra is the Fourier algebra of a locally compact group. We also obtain similar characterisations of the Fourier–Stieltjes algebras of G . En route, we prove some new results on the problem of when a subalgebra of A ( G ) is the whole algebra and on representations of discrete groups. [ABSTRACT FROM AUTHOR]

Properties of spectral synthesis are exploited to show that, for a large class of commutative hypergroups and for every compact hypergroup, every closed, re exive, left-translation-invariant subspace of L∝(K) is finite- dimensional. Also, we show that, for a class of hypergroups which includes many commutative hypergroups and all Z-hypergroups, every derivation of L1(K) into an arbitrary Banach L1-bimodule is continuous. [ABSTRACT FROM AUTHOR]

Let G be a locally compact group and 1 ≤ p < ∞ . A continuous unitary representation π : G → B ( H ) of G is an L p -representation if the matrix coefficient functions s ↦ 〈 π ( s ) x , x 〉 lie in L p ( G ) for sufficiently many x ∈ H . The L p -Fourier algebra A L p ( G ) is defined to be the set of matrix coefficient functions of L p -representations. Similarly, the L p -Fourier–Stieltjes algebra B L p ( G ) is defined to be the weak*-closure of A L p ( G ) in the Fourier–Stieltjes algebra B ( G ) . These are always ideals in the Fourier–Stieltjes algebra containing the Fourier algebra A ( G ) . In this paper we investigate how these spaces reflect properties of the underlying group and study the structural properties of these algebras. As an application of this theory, we characterize the Fourier–Stieltjes ideals of SL ( 2 , R ) . [ABSTRACT FROM AUTHOR]

We show that an n -homogeneous polynomial P on the Fourier algebra A ( G ) of a locally compact group G can be represented in the form P ( f ) = 〈 T , f n 〉 ( f ∈ A ( G ) ) for some T in the group von Neumann algebra VN ( G ) of G if and only if it is orthogonally additive and completely bounded. [ABSTRACT FROM AUTHOR]

Let A be a commutative Banach algebra with a BAI (=bounded approximate identity). We equip A** with the (first) Arens multiplication. To each idempotent element u of A** we associate the closed ideal Iu = {a ∊ A : au = 0} in A. In this paper we present a characterization of the closed ideals of A with BAI's in terms of idempotent elements of A**. The main results are: a) A closed ideal I of A has a BAI iff there is an idempotent u ∊ A** such that I = Iu and the subalgebra Au is norm closed in A**. b) For any closed ideal I of A with a BAI, the quotient algebra A/I is isomorphic to a subalgebra of A**. We also show that a weak* closed invariant subspace X of the group von Neumann algebra VN(G) of an amenable group G is naturally complemented in VN(G) iff the spectrum of X belongs to the closed coset ring Rc(Gd) of Gd, the discrete version of G. This paper contains several applications of these results. [ABSTRACT FROM AUTHOR]

Abstract: Using techniques of non-abelian harmonic analysis, we construct an explicit, non-zero cyclic derivation on the Fourier algebra of the real group. In particular this provides the first proof that this algebra is not weakly amenable. Using the structure theory of Lie groups, we deduce that the Fourier algebras of connected, semisimple Lie groups also support non-zero, cyclic derivations and are likewise not weakly amenable. Our results complement earlier work of Johnson (1994) [15], Plymen (2001) [18] and Forrest, Samei, and Spronk (2009) [9]. As an additional illustration of our techniques, we construct an explicit, non-zero cyclic derivation on the Fourier algebra of the reduced Heisenberg group, providing the first example of a connected nilpotent group whose Fourier algebra is not weakly amenable. [Copyright &y& Elsevier]

Abstract: This paper is concerned with weak⁎ closed masa-bimodules generated by -invariant subspaces of . An annihilator formula is established, which is used to characterise the weak⁎ closed subspaces of which are invariant under both Schur multipliers and a canonical action of on via completely bounded maps. We study the special cases of extremal ideals with a given null set and, for a large class of groups, we establish a link between relative spectral synthesis and relative operator synthesis. [Copyright &y& Elsevier]

Abstract: Let be a Lau algebra and let be the semigroup of all positive functionals in with norm one. We obtain some equivalent conditions for left amenability of in terms of a common fixed point property and the Hahn–Banach property. We then apply these results for certain Lau algebras on a locally compact group to give characterizations for amenability of . [Copyright &y& Elsevier]

Abstract: In this paper, the closed ideals with bounded approximate identities in the Fourier algebra of an amenable homogeneous space is characterized in terms of the coset ring of the corresponding group. Some of these results are also extended to the case of Figà–Talamanca–Herz algebra. The closure of the Fourier algebra in the -multiplier norm is also considered and we prove some results on spectral synthesis. We also derive some results on the ideals with bounded approximate identity. [Copyright &y& Elsevier]

Abstract: Let G be an arbitrary locally compact group and its Fourier–Stieltjes algebra. An element u of is called power bounded if . We present a detailed analysis of the structure of power bounded elements of and characterize them in terms of sets in the coset ring of G and -convergence of sequences , . [Copyright &y& Elsevier]

Abstract: This paper is an invitation to the study of the Fourier–Stieltjes algebra , the linear span of the continuous positive definite complex-valued functions on a topological group G. We study , when G has a host algebra or a group -algebra, the analogue of the group -algebra of a locally compact group. Our main challenge is that a topological group G cannot have a positive regular Borel measure which is left translation invariant unless G is locally compact. [Copyright &y& Elsevier]

Abstract: In this paper, we characterize surjective completely bounded disjointness preserving linear operators on Fourier algebras of locally compact amenable groups. We show that such operators are given by weighted homomorphisms induced by piecewise affine proper maps. [Copyright &y& Elsevier]

Abstract: We define the Haagerup property for C⁎-algebras and extend this to a notion of relative Haagerup property for the inclusion , where is a C⁎-subalgebra of . Let Γ be a discrete group and Λ a normal subgroup of Γ, we show that the inclusion has the relative Haagerup property if and only if the quotient group has the Haagerup property. In particular, the inclusion has the relative Haagerup property if and only if has the Haagerup property; has the Haagerup property if and only if Γ has the Haagerup property. We also characterize the Haagerup property for Γ in terms of its Fourier algebra . [Copyright &y& Elsevier]