Article

A Separation Property of Positive Definite Functions on Locally Compact Groups and Applications to Fourier Algebras

Authors:
Eberhard Kaniuth at Paderborn University
Eberhard Kaniuth
Anthony T. Lau
Anthony T. Lau
Anthony T. Lau
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Abstract

For a closed subgroup H of a locally compact group G consider the property that the continuous positive definite functions on G which are identically one on H separate points in G\H from points in H. We prove a structure theorem for almost connected groups having this separation property for every closed subgroup. Also, when a pair (G, H) has this separation property, there are interesting consequences in the ideal theory of the Fourier algebra of G.

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... The existence of invariant projections has been studied by several authors in connection with other separation properties and the existence of approximate identities for ideals in A (G) [1,9,10,15,17,24]. In particular, in [24] it is shown that if G has the H-separation property, then an invariant projection V N (G) → V N H (G) exists. ...
... The existence of invariant projections has been studied by several authors in connection with other separation properties and the existence of approximate identities for ideals in A (G) [1,9,10,15,17,24]. In particular, in [24] it is shown that if G has the H-separation property, then an invariant projection V N (G) → V N H (G) exists. We show in Section 4 that, in fact, the H-separation property is equivalent to the existence of a bounded approximate indicator for H consisting of positive definite functions that are identically one on H. ...
... In this section, we characterize the approximability of the characteristic function of a closed subgroup H of a locally compact group G in the spirit of the H-separation property of Kaniuth It is routine to verify that G has the H-separation property for any open, compact, or normal subgroup H, and it was shown by Forrest [16] that if G is a SIN group, then G has the H-separation property for every closed subgroup H. In [24], a fixed point argument is used to show that an invariant projection V N (G) → V N H (G) exists when the locally compact group G has the H-separation property (it is noted in Proposition 5.1 below that the projections arising this way are completely positive, in particular completely bounded). In fact, the following stronger result holds. ...
Weak separation properties for closed subgroups of locally compact groups
Preprint
  • Mar 2017
Three separation properties for a closed subgroup H of a locally compact group G are studied: (1) the existence of a bounded approximate indicator for H, (2) the existence of a completely bounded invariant projection of VN(G)VN\left(G\right) onto VNH(G)VN_{H}\left(G\right), and (3) the approximability of the characteristic function χH\chi_{H} by functions in McbA(G)M_{cb}A\left(G\right) with respect to the weak∗^{*} topology of McbA(Gd)M_{cb}A\left(G_{d}\right). We show that the H-separation property of Kaniuth and Lau is characterized by the existence of certain bounded approximate indicators for H and that a discretized analogue of the H-separation property is equivalent to (3). Moreover, we give a related characterization of amenability of H in terms of any group G containing H as a closed subgroup. The weak amenability of G or that GdG_{d} satisfies the approximation property, in combination with the existence of a natural projection (in the sense of Lau and \"Ulger), are shown to suffice to conclude (3). Several consequences of (2) involving the cb-multiplier completion of A(G)A\left(G\right) are given. Finally, a convolution technique for averaging over the closed subgroup H is developed and used to weaken a condition for the existence of a bounded approximate indicator for H.
View
... Slightly modifying the notation of [20] and [21], we call H a separating subgroup if for every x ∈ G \ H, there exists φ ∈ P H (G) such that φ(x) = 1. Also, G is said to have the separation property if each closed subgroup of G is separating. ...
... Also, G is said to have the separation property if each closed subgroup of G is separating. The interest in and importance of the separation properties arose from the fact that it turned out to be useful in studying the ideal theory of the Fourier algebra A(G) (see [20]). The extension and the separation properties have been studied by several authors (see [5], [7], [9], [13], [14], [19], [24] and [28] for the extension property and [20], [21] and [25] for the separation property). ...
... The interest in and importance of the separation properties arose from the fact that it turned out to be useful in studying the ideal theory of the Fourier algebra A(G) (see [20]). The extension and the separation properties have been studied by several authors (see [5], [7], [9], [13], [14], [19], [24] and [28] for the extension property and [20], [21] and [25] for the separation property). If H is a closed subgroup of G such that G has small H-conjugation invariant neighbourhoods of the identity (G ∈ [SIN ] H ), then H is extending as well as separating ( [7], [10] and [13]). ...
View
... Namely, that for every x ∈ G, x / ∈ H, there exists a continuous positive definite function ϕ such that ϕ(x) = 1 but ϕ(h) = 1 for each h ∈ H. This property has been studied extensively by Kaniuth and Lau in [9] and [10] who showed that it is of substantial interest on its own. In particular, they showed that while it is possible to establish the separation property for various types of subgroups, the general property fails outside of the class of [SIN ]−groups. ...
... In particular, they showed that while it is possible to establish the separation property for various types of subgroups, the general property fails outside of the class of [SIN ]−groups. However, in [9], they were also able to show that if G is an amenable locally compact group and if G satisfies the H-separation property, then there exists a norm-1 projection P from V N(G) onto V N H (G) = I(H) ⊥ that commutes with the module action of A(G) on V N(G). From this they were able to appeal to Proposition 1 of [2] to conclude that I(H) has a bounded approximate identity {u α } α∈Ω , where u α = 2 for each α. ...
... The final statement follows from[9, Theorem 3.3] or [3, Theorem 10], since if I(H) has a d−bounded approximate diagonal, then I(H) has a d−bounded approxiamte identity. At this point it does not seem obvious that we can always do better than 4−amenability for the ideal I(H). ...
Best bounds for approximate identities in ideals of the Fourier algebra vanishing on subgroups
Article
  • Aug 2005
In this paper we show that if G G is an amenable locally compact group and if H H is a closed subgroup, then the ideal I ( H ) I(H) has an approximate identity of norm 2. 2. If H H is not open, this bound is the best possible.
View
... The existence of invariant projections has been studied by several authors in connection with other separation properties and the existence of approximate identities for ideals in A (G) [1,9,10,15,17,24]. In particular, in [24] it is shown that if G has the H-separation property, then an invariant projection V N (G) → V N H (G) exists. ...
... The existence of invariant projections has been studied by several authors in connection with other separation properties and the existence of approximate identities for ideals in A (G) [1,9,10,15,17,24]. In particular, in [24] it is shown that if G has the H-separation property, then an invariant projection V N (G) → V N H (G) exists. We show in Section 4 that, in fact, the H-separation property is equivalent to the existence of a bounded approximate indicator for H consisting of positive definite functions that are identically one on H. ...
... In this section, we characterize the approximability of the characteristic function of a closed subgroup H of a locally compact group G in the spirit of the H-separation property of Kaniuth It is routine to verify that G has the H-separation property for any open, compact, or normal subgroup H, and it was shown by Forrest [16] that if G is a SIN group, then G has the H-separation property for every closed subgroup H. In [24], a fixed point argument is used to show that an invariant projection V N (G) → V N H (G) exists when the locally compact group G has the H-separation property (it is noted in Proposition 5.1 below that the projections arising this way are completely positive, in particular completely bounded). In fact, the following stronger result holds. ...
Weak separation properties for closed subgroups of locally compact groups
Article
  • Mar 2017
Three separation properties for a closed subgroup H of a locally compact group G are studied: (1) the existence of a bounded approximate indicator for H, (2) the existence of a completely bounded invariant projection of VN(G)VN\left(G\right) onto VNH(G)VN_{H}\left(G\right), and (3) the approximability of the characteristic function χH\chi_{H} by functions in McbA(G)M_{cb}A\left(G\right) with respect to the weak∗^{*} topology of McbA(Gd)M_{cb}A\left(G_{d}\right). We show that the H-separation property of Kaniuth and Lau is characterized by the existence of certain bounded approximate indicators for H and that a discretized analogue of the H-separation property is equivalent to (3). Moreover, we give a related characterization of amenability of H in terms of any group G containing H as a closed subgroup. The weak amenability of G or that GdG_{d} satisfies the approximation property, in combination with the existence of a natural projection (in the sense of Lau and \"Ulger), are shown to suffice to conclude (3). Several consequences of (2) involving the cb-multiplier completion of A(G)A\left(G\right) are given. Finally, a convolution technique for averaging over the closed subgroup H is developed and used to weaken a condition for the existence of a bounded approximate indicator for H.
View
... The separation property of locally compact groups with respect to closed subgroups was introduced by Lau and Losert [32] and Kaniuth and Lau [24], and was subsequently studied by several authors. A fundamental result is that the separation property is always satisfied with respect to compact subgroups. ...
... Definition 6.1 (Lau and Losert [32], Kaniuth and Lau [24]). Let G be a locally compact group and H be a closed subgroup of G. ...
... It was first observed in [32] that G has the H-separation property if H is either normal, compact or open. Generalizing a result of Forrest [15], it was proved that G has the Hseparation property provided that G has small H-invariant neighborhoods [24,Proposition 2.2]. The property was subsequently explored further in several papers, including [25,26]. ...
On Positive Definiteness Over Locally Compact Quantum Groups
Article
  • Oct 2014
  • CAN J MATH
The notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups to the quantum framework. Among these are theorems on "square roots" of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups.
View
... Slightly modifying the notation of [20] and [21], we call H a separating subgroup if for every x ∈ G \ H, there exists φ ∈ P H (G) such that φ(x) = 1. Also, G is said to have the separation property if each closed subgroup of G is separating. ...
... Also, G is said to have the separation property if each closed subgroup of G is separating. The interest in and importance of the separation properties arose from the fact that it turned out to be useful in studying the ideal theory of the Fourier algebra A(G) (see [20]). The extension and the separation properties have been studied by several authors (see [5], [7], [9], [13], [14], [19], [24] and [28] for the extension property and [20], [21] and [25] for the separation property). ...
... The interest in and importance of the separation properties arose from the fact that it turned out to be useful in studying the ideal theory of the Fourier algebra A(G) (see [20]). The extension and the separation properties have been studied by several authors (see [5], [7], [9], [13], [14], [19], [24] and [28] for the extension property and [20], [21] and [25] for the separation property). If H is a closed subgroup of G such that G has small H-conjugation invariant neighbourhoods of the identity (G ∈ [SIN ] H ), then H is extending as well as separating ( [7], [10] and [13]). ...
Extension and Separation Properties of Positive Definite Functions on Locally Compact Groups
Article
  • Jan 2007
Continuing earlier work, we investigate two related aspects of the set P(G) of continuous positive definite functions on a locally compact group G. The first one is the problem of when, for a closed subgroup H of G, every function in P(H) extends to some function in P(G). The second one is the question whether elements in G \ H can be separated from H by functions in P(G) which are identically one on H.
View
... It improved upon earlier work of the first author who showed in [5] that if G is an amenable [SIN ]−group and if H is a closed subgroup of G, then the ideal I(H) always admits a bounded approximate identity by showing that the pair (G, H) satisfy a strong separation property, namely that for every x ∈ G, x / ∈ H there exists a continuous positive definite function ϕ such that ϕ(x) = 1 but ϕ(h) = 1 for each h ∈ H. This property has been studied extensively by Kaniuth and Lau in [9] and [10] who showed that it is of substantial interest on its own. In particular, they showed that while it is possible to establish the separation property for various types of subgroups, the general property fails outside of the class of [SIN ]−groups. ...
... In particular, they showed that while it is possible to establish the separation property for various types of subgroups, the general property fails outside of the class of [SIN ]−groups. However, in [9], they were also able to show that if G is an amenable locally compact group and if G satisfies the H-separation property then there exists a norm-1 projection P from V N (G) onto V N H (G) = I(H) ⊥ that commutes with the module action of A(G) on V N (G). From this they were able to appeal to Proposition 1 of [2] to conclude that I(H) has a bounded approximate identity {u α } α∈Ω where u α = 2 for each α. ...
... The final statement follows from [9,Theorem 3.3] or [3, Theorem 10], since if I(H) has a d−bounded approximate diagonal, then I(H) has a d−bounded approxiamte identity. ...
Best bounds for approximate identities in ideals of the Fourier algebra vanishing on subgroups
Article
Full-text available
  • Jan 2006
In this paper we show that if G is an amenable locally compact group and if H is a closed subgroup then the ideal I(H) has an approximate identity of norm 2: If H is not open this bound is best posible.
View
... Using results from Delaporte and Derighetti [2], it can be shown that 2 is the best possible norm bound for an approximate identity of IðHÞ as long as H is not open in G (see [16]). In [16] it was shown that the norm bound 2 is achieved if the pair ðG; HÞ satisfies a certain separation property of positive definite functions. ...
... Using results from Delaporte and Derighetti [2], it can be shown that 2 is the best possible norm bound for an approximate identity of IðHÞ as long as H is not open in G (see [16]). In [16] it was shown that the norm bound 2 is achieved if the pair ðG; HÞ satisfies a certain separation property of positive definite functions. This property is satisfied in SIN-groups G for all closed subgroups H; but most likely fails in any other class of locally compact groups (see [16,18]). ...
... In [16] it was shown that the norm bound 2 is achieved if the pair ðG; HÞ satisfies a certain separation property of positive definite functions. This property is satisfied in SIN-groups G for all closed subgroups H; but most likely fails in any other class of locally compact groups (see [16,18]). We proceed to show that, nevertheless, if G is a locally nilpotent group or an almost connected group with a compact normal subgroup K such that G=K is nilpotent, then indeed IðHÞ has an approximate identity with norm bounded by 2 for every closed subgroup H of G (Theorems 3.3 and 3.7). ...
Ideals with bounded approximate identities in Fourier algebras
Article
  • Sep 2003
We make use of the operator space structure of the Fourier algebra A(G) of an amenable locally compact group to prove that if H is any closed subgroup of G, then the ideal I(H) consisting of all functions in A(G) vanishing on H has a bounded approximate identity. This result allows us to completely characterize the ideals of A(G) with bounded approximate identities. We also show that for several classes of locally compact groups, including all nilpotent groups, I(H) has an approximate identity with norm bounded by 2, the best possible norm bound.
View
... We now turn our attention to the operator projectivity of A(G) * * in A(G)-mod. In order to investigate this, we will need to make use of the so-called "separation property" for Fourier algebras introduced by Kaniuth and Lau in [26]. Let G be a locally compact group, and let H be a closed subgroup of G. ...
... Let G be a locally compact group, and let H be a closed subgroup of G. We recall from [26] (see also [11], [10]) that G has H-separation property if for every x / ∈ H, there is a continuous positive-definite function f ∈ P (G) such that f = 1 on H and f (x) = 1. It follows from [31] that G has H-separation property if H is either open, or compact, or normal in G. [30,Theorem 5.1.5], ...
... Note that the transference argument can be obtained by a similar diagram to (5.3). 26 Theorem 7.1. Let G be a locally compact group. ...
Projectivity of modules over Fourier algebras
Article
Full-text available
  • Jul 2008
  • P LOND MATH SOC
In this paper we study the homological properties of various natural modules associated to the Fourier algebra of a locally compact group. In particular, we focus on the question of identifying when such modules are projective in the category of operator spaces. We will show that projectivity often implies that the underlying group is discrete and give evidence to show that amenability also plays an important role.
View
... The existence of (completely) bounded A(G)-module projections P : V N (G) → V N (H) has been a recent topic of interest in harmonic analysis [8,9,12,15]. In particular, it was shown in [9,Theoerem 12] that if H is amenable, then a bounded A(G)-module projection always exists. ...
On hereditary properties of quantum group amenability
Preprint
  • Mar 2016
Given a locally compact quantum group G\mathbb{G} and a closed quantum subgroup H\mathbb{H}, we show that G\mathbb{G} is amenable if and only if H\mathbb{H} is amenable and G\mathbb{G} acts amenably on the quantum homogenous space G/H\mathbb{G}/\mathbb{H}. We also study the existence of L1(G^)L^1(\widehat{\mathbb{G}})-module projections from L∞(G^)L^{\infty}(\widehat{\mathbb{G}}) onto L∞(H^)L^{\infty}(\widehat{\mathbb{H}}).
View
... The existence of (completely) bounded A(G)-module projections P : V N (G) → V N (H) has been a recent topic of interest in harmonic analysis [8,9,12,15]. In particular, it was shown in [9,Theoerem 12] that if H is amenable, then a bounded A(G)-module projection always exists. ...
On hereditary properties of quantum group amenability
Article
  • Mar 2016
Given a locally compact quantum group G\mathbb{G} and a closed quantum subgroup H\mathbb{H}, we show that G\mathbb{G} is amenable if and only if H\mathbb{H} is amenable and G\mathbb{G} acts amenably on the quantum homogenous space G/H\mathbb{G}/\mathbb{H}. We also study the existence of L1(G^)L^1(\widehat{\mathbb{G}})-module projections from L∞(G^)L^{\infty}(\widehat{\mathbb{G}}) onto L∞(H^)L^{\infty}(\widehat{\mathbb{H}}).
View
... Remark. The existence of P projection of CV 2 (G) onto {T ∈ CV 2 (G) | supp T ⊂ H } with P(uT ) = uP(T ) for u ∈ A(G) has been recently investigated in several papers [2,11,15]. ...
Conditional expectations on CVp(G)CVp(G). Applications
Article
  • Jun 2007
We obtain for H, a closed amenable subgroup of a locally compact group G, the existence of an invariant conditional expectation of CVp(G) onto {T is an element of CVp(G) vertical bar supp T subset of H}. As a consequence we prove that H is locally p-Ditkin in G. We also establish relations between cv(p)(H), {T is an element of cv(p)(G) vertical bar supp T subset of H} and cv(p)(G) where cv(p) is the norm closure, in the set of all bounded operators of L-P, of the set all convolution operators with compact support.
View
... The dual and second dual spaces of A(G) are studied in [25,27,28,29,30,14,18,21]. On the ideal structure of Fourier algebras, see [13,16,82]. For other studies of A(G), see [19,22,26,36,37,38,39,40,43,44,54]. ...
The Fourier-Stieltjes and Fourier algebras for locally compact groupoids
Article
  • Nov 2003
The Fourier-Stieltjes and Fourier algebras B(G), A(G) for a general locally compact group G, first studied by P. Eymard, have played an important role in harmonic analysis and in the study of operator algebras generated by G. Recently, there has been interest in developing versions of these algebras for locally compact groupoids, justification being that, just as in the group case, the algebras should play a useful role in the study of groupoid operator algebras. Versions of these algebras for the locally compact groupoid case appear in three related theories: (1) a measured groupoid theory (J. Renault), (2) a Borel theory (A. Ramsay and M. Walter), and (3) a continuous theory (A. Paterson). The present paper is expository in character. For motivational reasons, it starts with a description of the theory of B(G), A(G) in the locally compact group case, before discussing these three realted theories. Some open questions are also raised.
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Operator biflatness of the Fourier algebra and approximate indicators for subgroups
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  • Mar 2002
We investigate if, for a locally compact group G, the Fourier algebra A(G) is biflat in the sense of quantized Banach homology. A central role in our investigation is played by the notion of an approximate indicator of a closed subgroup of G: The Fourier algebra is operator biflat whenever the diagonal in G×GG \times G has an approximate indicator. Although we have been unable to settle the question of whether A(G) is always operator biflat, we show that, for G=SL(3,C)G = SL(3,C), the diagonal in G×GG \times G fails to have an approximate indicator.
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Harmonic operators: the dual perspective
Preprint
  • Aug 2005
The study of harmonic functions on a locally compact group G has recently been transferred to a ``non-commutative'' setting in two different directions: C.-H. Chu and A. T.-M. Lau replaced the algebra L∞(G)L^\infty(G) by the group von Neumann algebra VN(G) and the convolution action of a probability measure μ\mu on L∞(G)L^\infty(G) by the canonical action of a positive definite function σ\sigma on \VN(G); on the other hand, W. Jaworski and the first-named author replaced L∞(G)L^\infty(G) by B(L2(G))B(L^2(G)) to which the convolution action by μ\mu can be extended in a natural way. We establish a link between both approaches. The action of σ\sigma on VN(G) can be extended to B(L2(G))B (L^2(G)). We study the corresponding space H~σ\tilde{H}_\sigma of ``σ\sigma-harmonic operators'', i.e., fixed points in B(L2(G))B(L^2(G)) under the action of σ\sigma. We show, under mild conditions on either σ\sigma or G, that H~σ\tilde{H}_\sigma is in fact a von Neumann subalgebra of B(L2(G))B (L^2(G)). Our investigation of H~σ\tilde{H}_\sigma relies, in particular, on a notion of support for an arbitrary operator in B(L2(G))B(L^2(G)) that extends Eymard's definition for elements of VN(G). Finally, we present an approach to H~σ\tilde{H}_\sigma via ideals in T(L2(G))T (L^2(G)) - where T(L2(G))T(L^2(G)) denotes the trace class operators on L2(G)L^2(G), but equipped with a product different from composition -, as it was pioneered for harmonic functions by G. A. Willis.
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Compact quantum subgroups and left invariant C*-subalgebras of locally compact quantum groups
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  • Apr 2010
We show that there is a one-to-one correspondence between compact quantum subgroups of a co-amenable locally compact quantum group G\mathbb{G} and certain left invariant C*-subalgebras of C0(G)C_0(\mathbb{G}). We also prove that every compact quantum subgroup of a co-amenable quantum group is co-amenable. Moreover, there is a one-to-one correspondence between open subgroups of an amenable locally compact group G and non-zero, invariant C*-subalgebras of the group C*-algebra C∗(G)C^*(G).
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We investigate the problem of when closed subgroups H H of a nilpotent locally compact group G G have the property that every continuous positive definite function on H H extends to some such function on G G .
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Amenability and Fan–Glicksberg theorem for set-valued mappings
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  • Jan 2018
In this paper, we begin by discussion of some well known results on the existence of left invariant means in the spaces: LUC(S), AP (S) and W AP (S) with Hahn-Banach extension theorem. We then give a new and precise proof of the well known Fan–Glicksberg fixed point theorem. This is then followed by a discussion on some related open problems.
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In this paper, we shall study contractive and pointwise contractive Banach function algebras, in which each maximal modular ideal has a contractive or pointwise contractive approximate identity, respectively, and we shall seek to characterize these algebras. We shall give many examples,including uniform algebras, that distinguish between contractive and pointwise contractive Banach function algebras, We shall describe a contractive Banach function algebra which is not equivalent to a uniform algebra. We shall also obtain results about Banach Sequence algebras and Banach function algebras that are ideals in their second duals.
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Power boundedness in Banach algebras associated with locally compact groups
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  • Jan 2014
Let G be a locally compact group and B(G) the Fourier-Stieltjes algebra of G. Pursuing our investigations of power bounded elements in B(G), we study the extension property for power bounded elements and discuss the structure of closed sets in the coset ring of G which appear as 1-sets of power bounded elements. We also show that L 1 -algebras of noncompact motion groups and of noncompact IN-groups with polynomial growth do not share the so-called power boundedness property. Finally, we give a characterization of power bounded elements in the reduced Fourier-Stieltjes algebra of a locally compact group containing an open subgroup which is amenable as a discrete group.
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We investigate the problem of when closed subgroups H of a nilpotent locally compact group G have the property that every continuous positive definite function on H extends to some such function on G.
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We obtain characterizations of left character amenable Banach algebras in terms of the existence of left φ-approximate diagonals and left φ-virtual diagonals. We introduce the left character amenability constant and find this constant for some Banach algebras. For all locally compact groups G, we show that the Fourier-Stieltjes algebra B(G) is C-character amenable with C ≤ 2 if and only if G is compact. We prove that if A is a character amenable, reflexive, commutative Banach algebra, then A≅Cn for some n ∈ N. We show that the left character amenability of the double dual of a Banach algebra A implies the left character amenability of A, but the converse statement is not true in general. In fact, we give characterizations of character amenability of L1G)** and A(G)**. We show that a natural uniform algebra on a compact space X is character amenable if and only if X is the Choquet boundary of the algebra. We also introduce and study character contractibility of Banach algebras.
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Let A and B be semisimple commutative Banach algebras with bounded approximate identities. We investigate the problem of extending a homomorphism φ: A → B to a homomorphism of the multiplier algebras M(A) and M(B) of A and B, respectively. Various sufficient conditions in terms of B (or B and φ) are given that allow the construction of such extensions. We exhibit a number of classes of Banach algebras to which these criteria apply. In addition, we prove a polar decomposition for homomorphisms from A into A with closed range. Our results are applied to Fourier algebras of locally compact groups.
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Let G be a locally compact group, K a compact subgroup of G and A(G/K) the Fourier algebra of the coset space G/K. Applying results from [E. Kaniuth, Weak spectral synthesis in commutative Banach algebras, J, Funct. Anal. 254 (2008), 987-1002], we establish injection and localization theorems relating weak spectral sets and weak Ditkin sets for A(G/K) to such sets for A(H/H∩K), where H is a closed subgroup of G. We also prove some results towards the analogue of Malliavin's theorem for weak spectral synthesis in A(G/K) and give illustrating examples.
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The aim of this paper is to investigate the cone of non-negative, radial, positive-definite functions in the set of continuous functions on ℝ d . Elements of this cone admit a Choquet integral representation in terms of the extremals. The main feature of this article is to characterize some large classes of such extremals. In particular, we show that there are many other extremals than the Gaussians, thus disproving a conjecture ofG. Choquet, and that no reasonable conjecture can be made on the full set of extremals. The last feature of this article is to show that many characterizations of positive definite functions available in the literature are actually particular cases of the Choquet integral representations we obtain.
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Harmonic operators: The dual perspective
Article
  • Mar 2007
The study of harmonic functions on a locally compact group G has recently been transferred to a “non-commutative” setting in two different directions: Chu and Lau replaced the algebra L ∞(G) by the group von Neumann algebra VN(G) and the convolution action of a probability measureμ on L ∞(G) by the canonical action of a positive definite function σ on VN(G); on the other hand, Jaworski and the first author replaced L ∞(G) by B (L2(G)){\mathcal B} (L^2(G)) to which the convolution action byμ can be extended in a natural way. We establish a link between both approaches. The action of σ on VN(G) can be extended to B (L2(G))\mathcal{B} (L^2(G)). We study the corresponding space [(H)\tilde]s\tilde{\mathcal H}_\sigma of “σ-harmonic operators”, i.e., fixed points in B (L2(G)){\mathcal B} (L^2(G)) under the action of σ. We show, under mild conditions on either σ or G, that [(H)\tilde]s\tilde{\mathcal H}_\sigma is in fact a von Neumann subalgebra of B (L2(G)){\mathcal B} (L^2(G)). Our investigation of [(H)\tilde]s\tilde{\mathcal H}_\sigma relies, in particular, on a notion of support for an arbitrary operator in B (L2(G)){\mathcal B} (L^2(G)) that extends Eymard’s definition for elements of VN(G). Finally, we present an approach to [(H)\tilde]s\tilde{\mathcal H}_\sigma via ideals in T (L2(G)){\mathcal T} (L^2(G)), where T(L2(G)){\mathcal T}(L^2(G)) denotes the trace class operators on L 2(G), but equipped with a product different from composition, as it was pioneered for harmonic functions by Willis.
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On topological centre problems and SIN quantum groups
Article
  • Jul 2009
Let A be a Banach algebra with a faithful multiplication and ∗〈A∗A〉 be the quotient Banach algebra of A∗∗ with the left Arens product. We introduce a natural Banach algebra, which is a closed subspace of ∗〈A∗A〉 but equipped with a distinct multiplication. With the help of this Banach algebra, new characterizations of the topological centre Zt(∗〈A∗A〉) of ∗〈A∗A〉 are obtained, and a characterization of Zt(∗〈A∗A〉) by Lau and Ülger for A having a bounded approximate identity is extended to all Banach algebras. The study of this Banach algebra motivates us to introduce the notion of SIN locally compact quantum groups and the concept of quotient strong Arens irregularity. We give characterizations of co-amenable SIN quantum groups, which are even new for locally compact groups. Our study shows that the SIN property is intrinsically related to topological centre problems. We also give characterizations of quotient strong Arens irregularity for all quantum group algebras. Within the class of Banach algebras introduced recently by the authors, we characterize the unital ones, generalizing the corresponding result of Lau and Ülger. We study the interrelationships between strong Arens irregularity and quotient strong Arens irregularity, revealing the complex nature of topological centre problems. Some open questions by Lau and Ülger on Zt(∗〈A∗A〉) are also answered.
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Operator biflatness of the Fourier algebra and approximate indicators for subgroups
Article
  • Apr 2004
We investigate if, for a locally compact group G, the Fourier algebra A(G) is biflat in the sense of quantized Banach homology. A central rôle in our investigation is played by the notion of an approximate indicator of a closed subgroup of G: The Fourier algebra is operator biflat whenever the diagonal in G×G has an approximate indicator. Although we have been unable to settle the question of whether A(G) is always operator biflat, we show that, for , the diagonal in G×G fails to have an approximate indicator.
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Compact quantum subgroups and left invariant C⁎-subalgebras of locally compact quantum groups
Article
  • Jul 2011
We show that there is a one-to-one correspondence between compact quantum subgroups of a co-amenable locally compact quantum group G and certain left invariant C⁎-subalgebras of C0(G). We also prove that every compact quantum subgroup of a co-amenable quantum group is co-amenable. Moreover, there is a one-to-one correspondence between open subgroups of an amenable locally compact group G and non-zero, invariant C⁎-subalgebras of the group C⁎-algebra .
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Operator Segal Algebras in Fourier Algebras
Article
  • Feb 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.
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Amenability and ideals in A(G)
Article
Full-text available
  • Oct 1992
  • J AUST MATH SOC
Closed ideals in A(G) with bounded approximate identities are characterized for amenable [SIN]-groups and arbitrary discrete groups. This extends a result of Liu, van Rooij and Wang for abelian groups.
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Almost Periodic Operators in VN(G)
Article
  • Jan 1990
Let G be a locally compact group, A(G) the Fourier algebra of G, B(G) the Fourier-Stieltjes algebra of G and VN(G) the von Neumann algebra generated by the left regular representationλ of G.Then A(G) is the predualof VN(G); VN(G) is a B(G)-module and A(G) is a closed ideal of B(G).Letis a compact operator from A(G) into VN(G)), the space of almost periodic operators inVN(G).Letbe the C*-algebra generated by (λ(x): x ∈ G). ThenFor a compact G, let E be the rank one operator on L2(G) that sends h∈ L2(G)to the constant function ∫ h(x) dx. We have the following results: (I) There exists a compact group G such that2) For a compact Lie group Ghas a unique left invariant mean ⇔G is semisimple. (3) If G is an extension of a locally compact abelian group by an amenable discrete group then(4) Let G = Fr, the free group with r generators, I <r < ∞. If T ∈ VN(G)and is a compact operator from B(G)into VN(G).
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Best bounds for the approximate units for certain ideals of L 1 (G) and of A p (G)
Article
  • Apr 1996
We compute the best bound for the approximate units of the augmentation ideal of the group algebra L1(G) of a locally compact amenable group G. More generally such a calculation is performed for the kernel of the canonical map from L1(G) onto L1(G/H), H being a closed amenable subgroup of G. Analogous results involving certain ideals of the Fourier algebra of an amenable group are also discussed.
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Duality and subgroups, II
Article
  • Oct 1972
Let (G) be the von Neumann algebra generated by the left regular representation λ of a locally compact group G, and H be the von Neumann subalgebra of (G) generated by the image λ(H) of a closed subgroup H. For an element (G) to fall in H it is necessary and sufficient that the support of x in the sense of Eymard, [4], is contained in H. This result yields that the correspondence of H and H is a lattice isomorphism.
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Spectral synthesis for A(G) and subspaces of VN(G)
Article
  • Jan 2001
Let G be a locally compact group, A(G) the Fourier algebra of G and VN(G) the von Neumann algebra generated by the left regular represen- tation of G. We introduce the notion of X-spectral set and X-Ditkin set when X is an A(G)-invariant linear subspace of VN(G), thus providing a unied approach to both spectral and Ditkin sets and their local variants. Among other things, we prove results on unions of X-spectral sets and X-Ditkin sets, and an injection theorem for X-spectral sets.
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Weakly Almost Periodic and Uniformly Continuous Functionals on the Fourier Algebra of any Locally Compact Group
Article
  • Mar 1974
We define for any locally compact group G , the space of bounded uniformly continuous functionals on Ĝ , U C B ( G ^ ) UCB(\hat G) , in the context of P. Eymard [Bull. Soc. Math. France 92 (1964), 181-236. MR 37 #4208] (for notations see next section). For u ∈ A ( G ) u \in A(G) let u ⊥ = { ϕ ∈ V N ( G ) ; ϕ [ A ( G ) u ] = 0 } {u^ \bot } = \{ \phi \in VN(G);\phi [A(G)u] = 0\} . Theorem . If for some norm separable subspace X ⊂ V N ( G ) X \subset VN(G) and some positive definite 0 ≠ u ∈ A ( G ) , U C B ( G ^ ) ⊂ 0 \ne u \in A(G),UCB(\hat G) \subset norm closure [ W ( G ^ ) + X + u ⊥ ] [W(\hat G) + X + {u^ \bot }] then G is discrete. If G is discrete then U C B ( G ^ ) ⊂ A P ( G ^ ) ⊂ W ( G ^ ) UCB(\hat G) \subset AP(\hat G) \subset W(\hat G) .
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Quelques observations concernant les ensembles de Ditkin d'un groupe localement compact
Article
  • Jun 1986
We prove that every closed normal subgroupH of a locally compact amenable groupG is a Ditkin set with respect to the Herz-Fig-Talamanca algebraA p (G) (p>1). Let be the canonical map ofG ontoG/H andF a closed subset ofG/H. We show thatF is a Ditkin set if and only if everyuA p (G), which vanishes on –1, lies on the norm closure of the subspace ofA p (G) generated by {u h |hH, vA p (G)C 00(G)} whereu h (x)=u(x h). As far as we know, this result seems to be new even forG abelian andp=2.
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Symmetry in Fourier-Stieltjes algebras
Article
  • Jan 1975
The concept of a commutative convolution measure algebra (c.c.m.a.), due to Taylor [24], has proved to be very important in the theory of harmonic analysis on locally compact abelian groups. Taylor, in effect, defines a c.c.m.a, to be the predual of a commutative W*-algebra d endowed with a cocommutative comultiplication c; i.e., a W*-morphism c: ~'-->~' ® d which obeys the diagrams dual to those characterizing a commutative associative multiplication. C.c.m.a.'s include LI(G) and M(G) for locally compact abelian groups G, and they share many of the important features of both algebras. If ~' is a commutative Hopf W*-algebra, i.e., if .~ is a W*-algebra together with a cocommutative comultiplication c, we shall call the predual a commutative predual algebra. Significant examples of such preduals are the Fourier algebra A(G) and the Fourier-Stieltjes algebra B(G) of a locally compact group G. These were introduced by Eymard [8], and have also been studied by Walter [26], among others. B(G) is by definition the linear combinations of (continuous) positive definite functions on G, and A(G) is a certain closed ideal in B(G). If G is abelian with dual group (~, then BIG ) is isomorphic to M((~) (Bochner's theorem), and A(G) is isomorphic to LI(G) under the inverse Fourier transform. Thus, for G non-abelian, B(G) and A(G) may be regarded as generalizations of M((J) and LI(~J), respectively. Our work here is motivated by the question of symmetry for M(G) when G is abelian. It is well-known (cf. [22], Chapter 5) that, unless G is discrete, M(G) is asymmetric; i.e., there is a complex homomorphism h of M(G) and a measure/~ in M(G) with h(/~*)4: h(#), where * is the involution in M(G). Furthermore, if G is not discrete, then the Hewitt-Kakutani phenomenon holds; namely, G contains a compact independent perfect set Q such that every bounded linear functional of norm less than or equal to 1 on the continuous measures supported on Qw - Q extends to a complex homomorphism of M(G). By analogy one might expect that B(G) is asymmetric if G is a noncompact locally compact group, and, that in fact there are always involution closed subpredual spaces L~ B(G) such that every element in L* of norm less than or equal to one extends to a complex homomorphism of B(G). However, if G is the group of Euclidean motions, then B(G) is symmetric (see Theorem 1.3), so such is not the case. The paper is organized as follows. In Section l we state our main theorem (Theorem 1.3), which characterizes those groups G for which B(G) is symmetric among those compactly generated or Lie groups which are a compact extension of a normal abelian subgroup. Examples are given to show why the hypothesis
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Extensions of positive-definite functions / John R. McMullen
Article
Thesis (Ph. D.)--University of Washington. Bibliography: l. [106]-109.
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Restrictions of certain function spaces to closed subgroups of locally compact groups
Article
  • Jan 1979
  • PAC J MATH
Let G be a locally compact group, and E(G) either the space Cu(G) of bounded left and right uniformly continuous functions on G, the space W(G) of weakly almost periodic functions on G, or the Fourier-Stieltjes algebra B(G) of G. Let E(G)|H be the space of restrictions of E(G)-functions to the closed subgroup H of G. A necessary and sufficient condition is given for an E(H)-function to belong to E(G)|H when H is a normal subgroup of G. It is also shown that E(G)|H is all of E(H) when H is any closed subgroup of a [SIN]-group. The techniques employed here can be used to deal with other function spaces.
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Weak∗^\ast-closed complemented invariant subspaces of L∞(G)L_\infty(G) and amenable locally compact groups.
Article
  • May 1986
  • PAC J MATH
One of the main results of this paper implies that a locally compact group G is amenable if and only ii whenever X is a weak*-closed left translation invariant complemented subspace of L∞(G), X is the range of a projection on L∞(G) commuting with left translations. We also prove that if G is a locally compact group and M is an invariant W*-subalgebra of the von Neumann algebra VN(G) generated by the left translation operators lg, g ∈ G, on L2(G), and Σ(M) = {g ∈ G; lg ∈ M} is a normal subgroup of G, then M is the range of a projection on VN(G) commuting with the action of the Fourier algebra A(G) on VN(G).
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Derighetti Best bounds for the approximate units of certain ideals of L1(G) and of Ap(G) Proc
  • J Delaporte
  • J Delaporte
Linear Operators, I,'' Interscience
  • Jan 1957
  • N Dunford
  • J T Schwartz
N. Dunford and J. T. Schwartz, ``Linear Operators, I,'' Interscience, New York, 1957.
Convoluteurs et projecteurs, in Harmonic Analysis
  • Jan 1988
  • 1422158
  • A Derighetti
A. Derighetti, Convoluteurs et projecteurs, in Harmonic Analysis,'' Lecture Notes in Math., Vol. 1359, pp. 1422158, Springer-Verlag, New YorkÂBerlin, 1988.
When G is abelian, then AP(G ) and WAP(G ) can be identified with the space of continuous almost periodic and continuous weakly almost periodic functions on G , respectively. For a discussion of these subspaces of VN(G ) see [2, 12, and 18]. If H is a closed subgroup of G
  • T Vn
The collection of operators T in VN(G ) for which the set [u } T : u # P(G ) & A(G ), u(e)=1] is relatively norm compact (weakly compact) is denoted AP(G ) (WAP(G )). Then both AP(G ) and WAP(G ) are closed A(G )-invariant subspaces of VN(G). When G is abelian, then AP(G ) and WAP(G ) can be identified with the space of continuous almost periodic and continuous weakly almost periodic functions on G, respectively. For a discussion of these subspaces of VN(G ) see [2, 12, and 18]. If H is a closed subgroup of G, then r*(UC(H ))=UC(G ) & VN H (G ), and similarly for AP(H ) and WAP(H ) [17, Lemma 3.2]. Note that if X is either UC c (G ) or UC(G ), then P(X ))X for any projec-tion P in K H. Indeed, if T # UC c (G ), then T=v } T for some v # A(G ) & C c (G ) and hence P(T)=P(v } T )=v } P(T ) # UC c (G ) because v has compact support. Since UC c (G ) is dense in UC(G ), it follows also that P(UC(G ))UC(G ). Also, if X=AP(G ) or WAP(G ), then P(X ))X. Indeed, this is straightforward from the fact that P(u } T )= u } P(T) for all T # VN(G ) and u # A(G ).
This proves that E is X-Ditkin. K In addition to UC c (G ), we now introduce some more subspaces of VN(G) UC(G ) was defined to be the closed linear span of
  • Since
Since v # J(E ) and =>0 was arbitrary, we conclude that (T, u) # (T, uJ(E )). This proves that E is X-Ditkin. K In addition to UC c (G ), we now introduce some more subspaces of VN(G) to which Theorem 3.5 applies. In [12], UC(G ) was defined to be the closed linear span of [u } T : u # A(G ), T # VN(G )]. Alternatively, UC(G ) can be defined as the norm closure of UC c (G ) in VN(G). When G is abelian, UC(G ) is the C*-algebra of bounded uniformly continuous func-tions on the dual group G of G (whence the notation in the general case).
UC(G ) is the C*-algebra of bounded uniformly continuous functions on the dual group G of G (whence the notation in the general case)
  • Abelian
abelian, UC(G ) is the C*-algebra of bounded uniformly continuous functions on the dual group G of G (whence the notation in the general case).
We have to show that, given T # X and u # I(H ), there exists v # A(G ) such that
  • Proof
Proof. We have to show that, given T # X and u # I(H ), there exists v # A(G ) such that (T, u) =(T, vu). If (T, u) =0, let v=0, and if (T, u) {0, then notice that since the function v Ä (T, vu) on A(G ) is REFERENCES
  • Jan 1992
  • 1433155
  • B Forrest
  • Amenability Ideals In
B. Forrest, Amenability and ideals in A(G), J. Austral. Math. Soc. Ser. A 53 (1992), 1433155.