J Fourier Anal ApplDOI 10.1007/s00041-013-9311-4

Spectral Synthesis in Fourier Algebrasof Ultrapherical Hypergroups

Sina Degenfeld-Schonburg · Eberhard Kaniuth ·Rupert Lasser

Received: 4 February 2013© Springer Science+Business Media New York 2013

Abstract Let H be an ultraspherical hypergroup associated to a locally compactgroup G and a spherical projector π (in Muruganandam, Math. Nachr. 281:1590–1603, 2008). We investigate spectral synthesis properties of the Fourier algebraA(H), partly building on results for A(G). Special emphasis is placed on doublecoset hypergroups. We also present several examples displaying the diverse behaviorof hypergroups in contrast to groups.

Keywords Hypergroup · Ultraspherical hypergroup · Double coset hypergroup ·Fourier algebra · Set of weak synthesis · Weak Ditkin set · Bounded approximateidentity · Injection theorem

Mathematics Subject Classification Primary 43A62 · 43A45 · 46J10 · Secondary43A30 · 43A46 · 46J10 · 46J20 · 20N20

1 Introduction

Let H be a locally compact hypergroup with (left) Haar measure and let C∗(H)

denote the enveloping C∗-algebra of the hypergroup algebra L1(H). In analogy to

Communicated by Hartmut Führ.

S. Degenfeld-Schonburg · R. LasserZentrum Mathematik, Technische Universität München, 85748 Garching, Germany

S. Degenfeld-Schonburge-mail: metzler@ma.tum.de

R. Lassere-mail: lasser@ma.tum.de

E. Kaniuth (B)Institut für Mathematik, Universität Paderborn, 33095 Paderborn, Germanye-mail: kaniuth@math.uni-paderborn.de

J Fourier Anal Appl

the group case, the Fourier-Stieltjes space B(H) of H is defined to be the Banachspace dual of C∗(H) and the Fourier space A(H) is the closed linear span of allconvolution products f ∗ g, where f,g ∈ Cc(H). However, in contrast to the groupcase, B(H) and A(H) need not be algebras in general because the product of twopositive definite functions on H may fail to be positive definite.

In recent years, some examples of (classes of) hypergroups have been discoveredfor which the Fourier space actually is a Banach algebra [1, 20, 27, 39]. Most no-tably, this concerns the class of ultraspherical hypergroups, which was introducedand studied in [28] and which includes in particular all double coset hypergroups andhence all orbit hypergroups.

Ultraspherical hypergroups arise from locally compact groups and certain averag-ing operators. Nevertheless, compared with locally compact groups they show a verydiverse behavior and are much less accessible than are groups and function algebrason groups. In the context of spectral synthesis, this is especially exemplified by thefollowing two facts. Let G be a locally compact group and K a compact subgroup ofG. Then

• Spectral synthesis holds for the Fourier algebra A(G) only if G is discrete, whereasthere are compact infinite double coset hypergroups G//K (in fact, orbit hyper-groups) such that every closed subset of G//K is a set of synthesis for A(G//K).

• Singletons in G are always sets of synthesis for A(G), whereas for many hyper-groups G//K almost all points in G//K fail to be sets of synthesis.

In this paper we investigate spectral synthesis properties for the Fourier algebrasA(H) of ultraspherical hypergroups H and the existence of bounded approximateidentities in ideals of A(H). Naturally, we try to exploit results existing for the lo-cally compact group G from which H is deduced. This method allows to establish,for a closed subset E of H , upper bounds for the characteristics ξ(E) and η(E) interms of the corresponding numbers for the inverse image of E in G (Theorem 3.1).Moreover, if G is amenable and K is a subhypergroup of H , it leads to the exis-tence of an approximate identity with norm bound 2 in the ideal I (K). In Sect. 3 wealso prove, applying results from [16], injection theorems for weak spectral sets andweak Ditkin sets (Theorem 3.11). Of course, special emphasis is placed on doublecoset hypergroups G//K and the question of when weak spectral synthesis holds forA(G//K). We show that this happens (if and) only if K is open in G provided that G

is nilpotent (Theorem 4.5). This result does not extend to solvable G. In fact, in [6] anexample is given of a 2-step solvable compact group G and a closed subgroup K suchthat G//K is infinite and nevertheless spectral synthesis holds for A(G//K). We alsoelaborate on that this phenomenon can occur only when G/K is totally disconnected.

In Sect. 5 we prove that the direct product of two ultraspherical hypergroups H1

and H2 is again ultraspherical and that A(H1 × H2) is isomorphic to the projectivetensor product A(H1) ⊗A(H2) provided that one of H1 and H2 is a so-called Pon-tryagin hypergroup. This allows to apply certain spectral synthesis results for pro-jective tensor products of general commutative Banach algebras. Finally, Sect. 6 isdevoted to present several illustrative examples.

J Fourier Anal Appl

2 Preliminaries

Let A be a regular and semisimple commutative Banach algebra with structure space�(A) and Gelfand transform a → a. For any subset M of A, the hull h(M) of M isdefined by h(M) = {ϕ ∈ �(A) : ϕ(M) = {0}}. Associated to each closed subset E of�(A) are two distinguished ideals of A with hull equal to E, namely

I (E) = {a ∈ A : a(ϕ) = 0 for all ϕ ∈ E}and

j (E) = {a ∈ A : a has compact support disjoint fromE}.Then I (E) is the largest ideal with hull E and j (E) is the smallest such ideal. Recallthat E is a spectral set (or set of synthesis) if I (E) = j (E) (equivalently, I (E) is theonly closed ideal with hull equal to E). One says that spectral synthesis holds for A ifevery closed subset of �(A) is a spectral set. Moreover, E is a Ditkin set if a ∈ aj (E)

for every a ∈ I (E).These concepts admit the following generalizations. A closed subset E of �(A)

is called a weak spectral set or set of weak synthesis if there exists n ∈ N such thatan ∈ j (E) for every a ∈ I (E). Adopting the notation of [16], we let ξ(E) denote thesmallest such number n and call it the characteristic of E. When this happens foreach E, we say that weak spectral synthesis holds for A. Generalizing the notion of aDitkin set, we call E a weak Ditkin set if there exists n ∈N such that an ∈ anj (E) forall a ∈ I (E), and η(E) will then stand for the minimal such n. So E is a spectral set(Ditkin set) if and only if ξ(E) = 1 (η(E) = 1). Weak spectral sets were introducedand first studied by Warner [40] in connection with the classical and still open unionproblem for sets of synthesis. However, they appeared implicitly earlier in the work ofVaropoulos and others. Subsequently, several authors have investigated weak spectralsets and the weak synthesis problem (see [16, 18, 19, 30, 32]).

We now recall from [28] the definition of the class of hypergroups the Fourier alge-bras of which are studied in this paper. Concerning the general theory of hypergroupswe refer the reader to [15] (and also [2]).

Definition 2.1 Let G be a locally compact group with left Haar measure. A linearmap π : Cc(G) → Cc(G) is called a spherical projector if it satisfies the followingconditions.

SH1 For all f,g ∈ Cc(G),

(1) π2 = π and π(f ) ≥ 0 whenever f ≥ 0;(2) π(π(f )g) = π(f )π(g);(3) 〈π(f ), g〉 = 〈f,π(g)〉;(4)

∫

Gπ(f )(x)dx = ∫

Gf (x)dx.

SH2 π(π(f ) ∗ π(g)) = π(f ) ∗ π(g).SH3 Let π∗ : M(G) → M(G) denote the adjoint map of π and let Ox = suppπ∗(δx)

for x ∈ G. Then, for all x, y ∈ G,

(1) either Ox ∩ Oy = ∅ or Ox = Oy ;

J Fourier Anal Appl

(2) if y ∈ Ox then y−1 ∈ Ox−1 and if Oxy = Oe then Oy = Ox−1 ;(3) the map x → Ox from G into the space K(G) of all closed subsets of G is

continuous with respect to the Michael topology on K(G).

The map π extends to a norm decreasing linear map on various spaces of functionson G, including Lp(G), 1 ≤ p ≤ ∞ (see [28, Remark 2.2] and [5], where a similarmap, called average operator, appears). A function f (respectively, a measure μ) iscalled π -radial if π(f ) = f (respectively, π∗(μ) = μ) holds.

Definition 2.2 Let H be the set consisting of all supports x = Ox , x ∈ G, equippedwith the quotient topology under the map p : G → H . Identifying M(H) with thesubspace of all π∗-radial measures in M(G), the product rule defined by δx ∗ δy =π∗(π∗(δx)∗π∗(δy)), x, y ∈ G, turns H into a locally compact hypergroup [28, Theo-rem 2.12]. Such an H is called a spherical hypergroup, and it is called ultrasphericalif in addition the modular function of G is π -radial.

The class of ultraspherical hypergroups contains all double coset hypergroups [28,Example 2.3]. In fact, let G be any locally compact group and K a compact subgroupof G with normalized Haar measure. Then the map π : Cc(G) → Cc(G), defined by

π(f )(x) =∫

K

∫

K

f (k1xk2) dk1dk2, f ∈ Cc(G), x ∈ G,

defines a spherical projector, and the resulting double coset hypergroup is ultras-pherical. Particularly interesting examples of double coset hypergroups are the or-bit hypergroups which are defined as follows. Let G be an arbitrary locally com-pact group and B a compact subgroup of the group of all topological automor-phisms of G [14, Sect. 26]. The orbit hypergroup GB is the space of all orbits B(x),x ∈ G, endowed with the quotient topology and the obvious hypergroup structure.Now, form the semidirect product G � B , the group defined by the product rule(x,α)(y,β) = (xα(y),αβ), x, y ∈ G, α,β ∈ B . Then the map B(x) → B(x,β)B isa topological hypergroup isomorphism between GB and the double coset hypergroup(G�B)//B (see [33] and [2]).

3 Spectral Synthesis Properties of Ultraspherical Hypergroups

In this section we study spectral synthesis problems for Fourier algebras of generalultraspherical hypergroups, whereas in Sect. 4 we confine ourselves to the class ofdouble coset hypergroups.

In what follows, G will always be a locally compact group and π : Cc(G) →Cc(G) a spherical projector such that the modular function of G is π -radial. The re-sulting ultraspherical hypergroup is denoted H and p : G → H,x → x is the quotientmap. A left Haar measure on H is given by

∫

Hf (x)dx = ∫

Gf (p(x))dx, f ∈ Cc(H).

The Fourier space A(H) is an algebra and is isometrically isomorphic to the subalge-bra Aπ(G) = {u ∈ A(G) : π(u) = u} of A(G) [28, Theorem 3.10]. Point evaluationsfurnish a homeomorphism between H and the Gelfand space �(A(H)). In this sense

J Fourier Anal Appl

we always identify A(H) with Aπ(G) and �(A(H)) with H . The Fourier algebraA(H) is semisimple, regular and Tauberian [28, Theorem 3.13] and ∅ is a Ditkin setfor A(H).

Theorem 3.1 Let E be a closed subset of H = �(A(H)).

(i) If p−1(E) is a set of weak synthesis for A(G), then E is a set of weak synthesisfor A(H) and ξ(E) ≤ ξ(p−1(E)).

(ii) If p−1(E) is a weak Ditkin set for A(G), then E is a weak Ditkin set for A(H)

and η(E) ≤ η(p−1(E)).

Proof (i) Let n = ξ(p−1(E)) and let u ∈ I (E) and ε > 0 be given. Then, identifyingA(H) with Aπ(G), u ∈ I (p−1(E)) and hence there exists v ∈ j (p−1(E)) such that‖un − v‖A(G) ≤ ε. Then

‖un − π(v)‖A(H) = ‖π(un − v)‖A(H) ≤ ‖un − v‖A(G) ≤ ε.

It therefore only remains to show that π(v) ∈ j (E). Let C = suppv, which is a com-pact subset of G disjoint from p−1(E). Then

E ∩ p(C) = p(p−1(E) ∩ C) = ∅ and p(C) = p(suppv) = suppπ(v).

Indeed, this latter equality is proved exactly as formula (2.7) of [28]. Thus π(v) hascompact support disjoint from E, as required.

(ii) Let n = η(p−1(E)) and u and ε as in (i). Then there exists v ∈ j (p−1(E))

such that ‖un − vun‖A(G) ≤ ε. Then, as seen above, π(v) ∈ j (E) and by (SH1),

π(v)π(u)n = π(π(u)nv) = π(unv) and π(un) = π(u)n,

since π(u) = u. Thus

‖un − π(v)un‖A(H) = ‖π(u)n − π(v)π(u)n‖A(H)

= ‖π(un) − π(vπ(un))‖A(H)

= ‖π(un − vun)‖A(H)

≤ ‖un − vun‖A(G) ≤ ε.

This shows that η(E) ≤ n. �

From Theorem 3.1 and results on Fourier algebras of locally compact groups wededuce the following

Corollary 3.2 Let K be a closed subhypergroup of H . Then K is a set of synthesisand a local Ditkin set for A(H).

Proof We first observe that the closed set p−1(K) is a subgroup of G. To see this, letx, y ∈ p−1(K). Then, by [28, (2.6)], supp(δx ∗ δy) = p(OxOy) and hence

xy−1 ∈ OxOy−1 ⊆ p−1(supp(δx ∗ δ ˙y−1)) ⊆ p−1(K)

J Fourier Anal Appl

since x, y ∈ K and K is a subhypergroup of H . Now, as shown in [36, Theorem3] and independently in [13], every closed subgroup of G is a set of synthesis forA(G). Moreover, every closed subgroup is a local Ditkin set [22, Corollary 5.5]. Thestatements of the corollary now follow from Theorem 3.1. �

Remark 3.3 As mentioned in [31, p. 298], the converse of Theorem 3.1 does not hold,even not for commutative double coset hypergroups. In fact, let G be a noncompact,connected, semisimple real Lie group with finite centre and fix a maximal compactsubgroup K of G. Suppose that G/K is a rank one Riemannian symmetric space ofdimension at least three, and let G = KA+K be a Cartan decomposition of G. Inthis situation Meaney [25] has shown that for every a ∈ A+, a �= e, the double cosetKaK fails to be a set of synthesis for A(G). On the other hand, G//K is topologicallyisomorphic to the locally compact Abelian group A+, which is a vector group, andhence singletons in A+ = A+ are sets of synthesis for L1(A) = A(G//K). Examplesof such groups G and maximal compact subgroups K are given in [25].

We now apply Theorem 3.1 and a result from [38] to deduce a lower bound, re-capitulating the one given in [3, p. 404], for the characteristic of SO(d − 1)-doublecosets in SO(d) = �(A(SO(d))). Here we identify SO(d − 1) with the subgroup ofall elements of SO(d) which fix the vector (1,0, . . . ,0)t ∈ R

d . Note that there is abijection

[−1,1] → SO(d)//SO(d − 1), x → SO(d − 1)a(x)SO(d − 1),

where

a(x) =⎛

⎝

(

x −√1 − x2√

1 − x2 x

)

02,d−2

0d−2,2 1d−2,d−2

⎞

⎠

and On,m and 1n,m denotes the zero and identity matrix of size n × m, respectively.

Corollary 3.4 Let d ≥ 3 and identify as above the double coset hypergroupSO(d)//SO(d − 1) as a set with the interval [−1,1]. Then, for each x ∈] − 1,1[,

ξ(SO(d − 1)a(x)SO(d − 1)) ≥⌊

d − 1

2

⌋

+ 1.

Proof As shown in [38, Theorem 3.11 and Sect. 4.1], for any x ∈]− 1,1[ the single-ton {x} is a weak spectral set with characteristic

ξ(x) =⌊

d − 2

2+ 1

2

⌋

=⌊

d + 1

2

⌋

.

The assertion now follows from Theorem 3.1. �

The hypergroups SO(d)//SO(d − 1), d ≥ 3, are examples of compact commuta-tive double coset hypergroups H for which almost all points in H fail to be sets of

J Fourier Anal Appl

synthesis for A(H). Another very large class of such hypergroups is provided by oneof the main results of [24], as is briefly outlined in the following example.

Example 3.5 Let G be a compact connected semisimple Lie group and let Inn(G) de-note the group of inner automorphisms of G, which is topologically isomorphic to thequotient group of G modulo its centre. Form the semidirect product ˜G = G� Inn(G)

and let H = ˜G//Inn(G). Then A(H) is isomorphic to the subalgebra Aci(G) consist-ing of all u ∈ A(G) which are conjugation-invariant, and �(Aci(G)) is homeomor-phic to the space of conjugacy classes Cx , x ∈ G. Now, denote the set of all so-calledregular elements of G by G′, a set the complement of which has measure zero in G.Then, as shown in [24, Theorem 3.3] (compare the proof), the singleton {Cx} fails tobe a set of synthesis for each x ∈ G′.

We continue to let (G,π) and H be as at the outset of this section. For locallycompact groups and closed subgroups, the next proposition is a standard.

Proposition 3.6 Let K be a closed subhypergroup of H .

(i) Given u ∈ A(K), there exists v ∈ A(H) such that v|K = u and ‖v‖ = ‖u‖.(ii) A(K) is isometrically isomorphic to the quotient algebra A(H)/I (K).

Proof (i) Let L = p−1(K), which is a closed subgroup of G. Then the pair (L,π |L)

defines the ultraspherical hypergroup K . Identify A(H) and A(K) with Aπ(G) andAπ |L(L) as Banach spaces, respectively [28, Theorem 3.10]. By [23], u ∈ Aπ |L(L) ⊆A(L) admits an extension w ∈ A(G) with the same norm. Then v = π(w) ∈ Aπ(G),v|L = π |L(w|L) = π |L(u) = u and

‖v‖A(H) = ‖v‖Aπ(G) ≤ ‖w‖A(G) = ‖u‖A(L) = ‖u‖A(K),

and hence v ∈ A(H) is an extension of u with the same norm.(ii) Since I (K) is the kernel of the restriction map A(H) → A(K), v → v|K , the

map v + I (K) → v|K defines an algebra isomorphism of A(H)/I (K) onto A(K). Itfollows from (i) that this map is isometric for the quotient norm on A(H)/I (K). �

In the sequel we shall several times require the group G to be amenable. For thedefinition and properties of amenable locally compact groups and the richness of thisclass of groups we refer to [11].

Lemma 3.7 If G is amenable, then A(H) has an approximate identity of norm bound1 and consisting of compactly supported positive definite functions.

Proof By Leptin’s theorem [21], A(G) has such an approximate identity, (eα)α say.Then, by SH1 and Theorems 3.3 and 3.4 of [28], for any u ∈ A(H) ∩ Cc(G) =Aπ(G) ∩ Cc(G),

‖π(eα)u − u‖ = ‖π(eα)π(u) − π(u)‖ = ‖π(eαπ(u) − u)‖≤ ‖eαπ(u) − u‖ = ‖eαπ(u) − π(u)‖,

which tends to zero. Since A(H) ∩ Cc(H) is dense in A(H), the lemma follows. �

J Fourier Anal Appl

In passing we remind the reader that the centre Z(H) of a hypergroup H is definedto be the set of all a ∈ H such that supp(δa ∗ δb) is a singleton for all b ∈ H . It isclear that Z(H) is a closed subgroup of H .

Lemma 3.8 As before, for every x ∈ G, let Ox = suppπ∗(δx) .

(i) Oe is a subgroup of G.(ii) An element x of H belongs to Z(H) if and only if OxOy = Oxy = OyOx for all

y ∈ G. In particular, Ox = xOe = Oex for each x ∈ Z(H).(iii) If x ∈ Z(H) and K is closed subhypergroup of H , then p−1(x ∗K) = xp−1(K).

Proof (i) is clear since Oe = p−1({e}) and {e} is a subgroup of H .(ii) First, let x ∈ Z(H) and for every y ∈ G, let zy ∈ G be such that δx ∗ δy = δzy .

Then xy ∈ OxOy = Ozy and hence Oxy = Ozy = OxOy and similarly Oxy = OxOy .Conversely, if Oxy = OxOy for all y ∈ G, then by [28, (2.6)],

supp(δx ∗ δy) = p(OxOy) = p(Oxy) = δxy,

and hence δx ∗ δy = δxy , so that x ∈ Z(H). Moreover, if x ∈ Z(H) then Ox = Oxe =OxOe ⊇ xOe and also x−1Ox ⊆ Ox−1Ox = Oe, so that Ox ⊆ xOe. Similarly, Oex =Ox .

(iii) Recall that the union of the sets supp(δx ∗ δk), k ∈ p−1(K), is dense in x ∗ K

and hence⋃

{p−1(supp(δx ∗ δk)) : k ∈ p−1(K)}is dense in p−1(x ∗K). For p−1(x ∗K) ⊆ xp−1(K) it therefore suffices to show thatif y ∈ G is such that y ∈ supp(δx ∗δk) for some k ∈ p−1(K), then y ∈ xp−1(K). Nowsupp(δx ∗ δk) = p(OxOk) and hence, since x ∈ Z(H),

y ∈ Oxk = OxOk = xOeOk = xOk = xp−1(k)

by (ii). Conversely, if k ∈ p−1(K) then

xk ∈ OxOk = supp(π∗(δx) ∗ π∗(δk)) =⋃

{Oy : y ∈ supp(δx ∗ δk)}

and hence xk ∈ p−1(x ∗ K). �

Proposition 3.9 Suppose that G is amenable. Let K be a closed subhypergroup ofH and x ∈ Z(H). Then the coset x ∗ K is a set of synthesis for A(H) and the idealI (x ∗ K) has an approximate identity with norm bounded by 2. In particular, x ∗ K

is a Ditkin set.

Proof Consider the closed subgroup p−1(K) of G. Since G is amenable, by [9, The-orem 2.6] the ideal I (p−1(K)) of A(G) has an approximate identity bounded by 2.Then the ideal I (xp−1K) also has an approximate identity bounded in norm by 2.Now observe that, by Lemma 3.8(iii) and the identification of A(H) with Aπ(G),

I (x ∗ K) = Aπ(G) ∩ I (p−1(x ∗ K)) = Aπ(G) ∩ I (xp−1(K)).

J Fourier Anal Appl

The proof of Lemma 3.7 then shows that I (x ∗ K) has an approximate identity withnorm bounded by 2. Since xp−1(K) is a set of synthesis for A(G), x ∗ K is one forA(H) by Corollary 3.2. Furthermore, since I (x ∗ K) has an approximate identity,x ∗ K is a Ditkin set for A(H) (see [17, Remark 5.1.8(1)]). �

The proof of the next lemma is an adaptation of the proof for the correspondingresult in the case of a group H .

Lemma 3.10 Suppose that G is amenable and let K be a closed subhypergroup ofH . Then, given u ∈ A(H) and ε > 0, there exists v ∈ A(H) such that v vanishes ona neighborhood of K and

‖u − vu‖A(H) ≤ 3‖u|K‖A(K) + ε.

Proof By Proposition 3.6(ii), we have

‖u|K‖A(K) = inf {‖u + w‖A(H) : w ∈ I (K)}.

So we can choose w ∈ I (K) such that

‖u − w‖A(H) ≤ ‖u|K‖A(K) + ε/4.

Since G is amenable and K is a set of synthesis (Corollary 3.2), by Proposition 3.9I (K) has an approximate identity bounded by 2 and contained in j (K). Thus thereexists v ∈ j (K) with ‖v‖A(H) ≤ 2 and ‖w − wv‖A(H) < ε/4. By standard estimatesit now follows that

‖u − uv‖A(H) ≤ 3‖u|K‖A(K) + ε

(see the proof of [18, Lemma 2.1]). �

Simply taking u ∈ I (K), Lemma 3.10 also shows that K is a Ditkin set providedthat G is amenable.

From the preceding results and Theorems 2.2 and 2.5 of [16] we can deduce thefollowing injection theorem for weak spectral sets and weak Ditkin sets in ultraspher-ical hypergroups.

Theorem 3.11 Let G be any locally compact group and K a closed subhypergroupof H . Let E be a closed subset of K and let i denote the embedding of K = �(A(K))

into H = �(A(H)). Then the following assertions hold.

(i) E is a weak spectral set for A(K) if and only if i(E) is a weak spectral set forA(H), and then ξ(E) = ξ(i(E)).

(ii) If i(E) is a weak Ditkin set for A(H), then E is a weak Ditkin set for A(K) andη(E) ≤ η(i(E)).

(iii) Suppose that G is amenable. If E is a weak Ditkin set for A(K), then i(E) is aweak Ditkin set for A(H) and η(i(E)) ≤ η(E)2.

J Fourier Anal Appl

Proof (i) Since A(H) is regular, h(I (K)) = K , which is a set of synthesis by Corol-lary 3.2. The statement now follows from Proposition 3.6(ii) and the general injectiontheorem for weak spectral sets [16, Theorem 2.2].

(ii) is a consequence of [16, Theorem 2.5(i)], and (iii) follows from Lemma 3.10and [16, Theorem 2.5(ii)]. �

Remark 3.12 The Fourier algebra A(Fn) of the free group on n generators, n ≥ 2,does not have a bounded approximate identity because Fn is not amenable [21]. How-ever, A(Fn) has an approximate identity which is bounded in the multiplier norm[12, Theorem 2.1]. Hence spectral synthesis holds for A(Fn). Consequently, by The-orem 3.1 spectral synthesis holds for the hypergroup discussed in [28, Example 2.4].

4 When Does Spectral Synthesis Hold for Double Coset Hypergroups?

Let G be a locally compact group and K a compact subgroup of G. It is known thatweak spectral synthesis holds for A(G) (if and) only if G is discrete [16, Theorem4.3]. This result was independently shown in [30] and will be used several times inthe sequel. In this section we mainly address the question of when (weak) spectralsynthesis holds for the Fourier algebra of the double coset hypergroup G//K .

Lemma 4.1 Let K and N be compact subgroups of G with N normal. If weakspectral synthesis holds for A(G//K), then weak spectral synthesis also holds forA((G/N)//(KN/N)).

Proof As N is normal in G, it is obvious that q : KxK → (KN/N)xN(KN/N) isa well defined map from G//K onto (G/N)//(KN/N), and q is clearly continuousfor the quotient topologies on both sides. For v ∈ A(G), let p(v) denote the functionon G/N defined by p(v)(xN) = ∫

Nv(xn)dn, where dn denotes normalized Haar

measure on N . Then p(v) ∈ A(G/N) and ‖p(v)‖ ≤ ‖v‖, and p maps A(G//K) intoA((G/N)//(KN/N)). Actually, p is onto since given u ∈ A((G/N)//(KN/N)).u ◦ q ∈ A(G//K) and p(u ◦ q) = u as is easily verified.

The proof now proceeds entirely as the proof of (i) of Theorem 3.1. If E is a closedsubset of (G/N)//(KN/N) and n = ξ(E) then, given u ∈ I (E) and ε > 0, we haveu ◦ q ∈ I (q−1(E)) and hence there exists v ∈ j (q−1(E)) with ‖(u ◦ q)n − v‖ ≤ ε.Since (u ◦ q)n = un ◦ q and p(un ◦ q) = un, we get

‖un − p(v)‖ = ‖p(un ◦ q)n) − p(v)‖ ≤ ‖(u ◦ q)n − v‖ ≤ ε.

The remaining bit, namely that p(v) ∈ j (E), follows as in the proof of Theo-rem 3.1. �

Lemma 4.2 Let K and L be compact subgroups of G such that K ⊆ L and letq : G//K → G//L, KxK → LxL, denote the quotient map. Then, for every closedsubset E of G//L, ξ(E) ≤ ξ(q−1(E)). In particular, if weak spectral synthesis holdsfor A(G//K), then it also holds for A(G//L).

J Fourier Anal Appl

Proof Suppose that m = ξ(q−1(E)) < ∞ and let u ∈ I (E) ⊆ A(G//L). Then u◦q ∈I (q−1(E)) ⊆ A(G//K) and hence there exists a sequence (vn)n in j (q−1(E)) suchthat ‖(u ◦ q)m − vn‖ → 0. Define wn on G by

wn(x) =∫

L

∫

L

vn(l1xl2)dl1dl2, x ∈ G,

where dl denotes normalized Haar measure on L. Then wn ∈ A(G//L) and, sinceu ◦ q is L-biinvariant,

‖(u ◦ q)m − wn‖ =∥

∥

∥

∥

∫

L

∫

L

[(u ◦ q)m(l1 · l2) − vn(l1 · l2)]dl1dl2

∥

∥

∥

∥

≤∫

L

∫

L

‖(u ◦ q)m(l1 · l2) − vn(l1 · l2)‖dl1dl2

= ‖(u ◦ q)m − vn‖.Moreover, wn ∈ j (q−1(E)) since supp wn ⊆ q−1(q(supp vn)), which is compact,and q−1(q(supp vn)) ∩ q−1(E) = ∅. �

In what follows, for any locally compact group G, G0 will denote the connectedcomponent of the identity e of G and, if G is understood, Z0 = {e} ⊆ Z1 ⊆ · · ·Zn ⊆· · · will denote the ascending central series of G. Recall that G is nilpotent if Zn = G

for some n ∈ N.The proof of the next lemma is analogous to the corresponding one [18] for the

Fourier algebra A(G/K) of the left coset space G/K , as defined and studied first in[8]. However, we include the short proof for the readers convenience.

Lemma 4.3 Let G be a nilpotent compact group and let K be a closed subgroup ofG. If weak spectral synthesis holds for A(G//K), then K has finite index in G.

Proof We prove by induction that Zj ∩ K has finite index [Zj : (Zj ∩ K)] in Zj .Assume that Zj−1 ∩ K has finite index in Zj−1 for some j ≥ 1. Since

[Zj : (Zj ∩ K)] = [Zj : (Zj ∩ Zj−1K] · [(Zj ∩ Zj−1K) : (Zj ∩ K)]≤ [Zj : (Zj ∩ Zj−1K] · [Zj−1K : K]= [Zj : (Zj ∩ Zj−1K] · [Zj−1 : (Zj−1 ∩ K)],

it remains to verify that Zj ∩ Zj−1K has finite index in Zj .Since weak spectral synthesis holds for A(G//K), it holds for A(ZjK//K) (The-

orem 3.11(i)), and hence also for A((ZjK//Zj−1K)) by Lemma 4.2. But Zj−1K isa normal subgroup of ZjK since Zj/Zj−1 is contained in the centre of G/Zj−1. Itfollows that ZjK/Zj−1K = Zj/Zj ∩ Zj−1K is finite, as required. �

It is worth pointing out (and has been mentioned in the introduction) thatLemma 4.3 does not hold for the larger class of solvable groups. In fact, as shown byDunkl and Ramirez [6, Theorem 10.6], there exist 2-step solvable compact groups G

J Fourier Anal Appl

with a nonopen closed subgroup K such that every closed subset of G//K is a set ofsynthesis for A(G//K). The simplest example is provided by the semidirect productG = �p �K , where p is a prime number, �p is the compact ring of p-adic integers,K is the multiplicative group of p-adic numbers of valuation 1 and K acts on �p bymultiplication. The hypergroup G//K , denoted Hw in [6], turns out to be the one-point compactification Z

∗+ = Z+ ∪ {∞} of Z+. We shall return to this hypergroup inSect. 6.

Proposition 4.4 Let G be a nilpotent locally compact group such that G0 has finiteindex in G. Suppose that there exists a compact subgroup K of G such that weakspectral synthesis holds for A(G//K). Then G is compact.

Proof We first suppose that G = G0. Note that then every member Zj of the ascend-ing central series of G is also connected. We show by induction on j that Zj ⊆ K .Assume that Zj−1 ⊆ K for some j ≥ 1. Then K is normal in ZjK since Zj/Zj−1

is contained in the centre of G/Zj−1 and Zj−1 ⊆ K . Since G is amenable and weaksynthesis holds for A(G//K), by Theorem 2.11(i) it also holds for A(ZjK//K).Because ZjK//K is actually a group, it follows that ZjK/K is discrete and hencetrivial as Zj is connected.

Now drop the hypothesis that G0 = G. Let Gc denote the set of all elements of G

which generate a relatively compact subgroup of G. Then Gc is a normal subgroupof G [10, Corollary 3.5.1 and Lemma 3.8] and Gc is compact because G is com-pactly generated [10, Theorem 9.7]. On the other hand, the discrete group G/G0G

c

is torsion-free [10, Theorem 8.3]. However, it is a finite group since G0 has finiteindex in G. So G = G0G

c and G/Gc is connected.Since K and Gc are both compact, so is KGc , and it therefore suffices to show

that KGc = G. Since weak spectral synthesis holds for A(G//K) and KGc is com-pact, weak spectral synthesis also holds for A(G//KGc) by Lemma 4.2. The normalsubgroup Gc of G being contained in KGc , A(G//KGc) is isometrically isomorphicto A((G/Gc)//(KGc/Gc)). Now G/Gc is connected and hence the first part of theproof applies to G/Gc and the compact subgroup KGc/Gc of G/Gc . It follows thatKGc/Gc = G/Gc and so KGc = G, as required. �

The next theorem is the main result of this section.

Theorem 4.5 Let G be a nilpotent locally compact group and K a compact subgroupof G. Then the following are equivalent.

(i) Spectral synthesis holds for A(G//K).(ii) Weak spectral synthesis holds for A(G//K).

(iii) K is open in G.

Proof (i) ⇒ (ii) being trivial, suppose that (iii) holds. Since K is open in G, G//K isdiscrete. Moreover, A(G//K) is Tauberian and, since G is amenable, has an approx-imate identity. It is standard that these facts together ensure that spectral synthesisholds for A(G//K) (compare [17, Corollary 5.2.15]).

J Fourier Anal Appl

It remains to show that (ii) implies (iii). Choose a relatively compact open subsetV of G containing K and let H denote the open subgroup of G generated by V . Sinceit suffices to show that K is open in H and since weak spectral synthesis holds forA(H//K), we can henceforth assume that G is compactly generated. Now, G beingcompactly generated, the subgroup G0G

c, which contains K , is compact modulo G0

and open in G (compare the proof of Proposition 4.4). Thus repeating the precedingreduction step and replacing G by G0G

c, we can even assume that G/G0 is com-pact. Then G is a projective limit of Lie groups, say G/Cα [26], and weak spectralsynthesis holds for each A((G/Cα)//(CαK/Cα)) by Lemmas 4.1 and 4.2.

It is enough to prove that G0 ⊆ K because then G is compact and Lemma 4.3yields that K is open in G. Now G0 ⊆ K once we have shown that G0 ⊆ CαK foreach α. Indeed, if x ∈ G0 and, for each α, x = yαzα with yα ∈ Cα and zα ∈ K , thenyα → e and hence x = limα zα ∈ K .

Fix α and let C = Cα . Then G0C/C = (G/C)0 since G0C/C is connected and(G/C)/(G0C/C) is totally disconnected. Recall that weak spectral synthesis holdsfor A((G/C)//(KC/C)). Therefore, passing to G/C, we can moreover assume thatG is a Lie group. Then G0 is open in G and hence has finite index in G. Now Propo-sition 4.4 applies and gives that G/C is compact. Finally, by Lemma 4.3, KC/C hasfinite index in G/C and hence contains (G/C)0 = G0C/C. From this we concludethat KC ⊇ G0, as was to be shown. �

The preceding theorem shows that nondiscrete double coset hypergroups G//K

such that weak spectral synthesis holds for A(G//K), cannot arise from nilpotentgroups G. In the remainder of this section we provide some further indication forwhen such examples can possibly occur.

Lemma 4.6 Let G be a nontrivial connected locally compact Abelian group, F afinite group of automorphisms of G and

AF (G) = {u ∈ A(G) : u ◦ α = u for all α ∈ F }.Then weak spectral synthesis does not hold for AF (G).

Proof We have to show the existence of a closed F -invariant subset E of G with thefollowing property. For each n ∈ N, there exists u ∈ I (E) ∩ AF (G) such that un /∈J (E), the closure of j (E) (equivalently, that 〈T ,un〉 �= 0 for some T ∈ J (E)⊥ ⊆A(G)∗). Recall that A(G)∗ = VN(G) and J (E)⊥ = {T ∈ VN(G) : suppT ⊆ E} [7].

Note first that there exists x0 ∈ G with |F(x0)| = |F |. Indeed, otherwise G =⋃

α∈F Gα , where Gα = {x ∈ G : α(x) = x}, so that at least one of the closed setsGα has nonempty interior and hence α(y) = y for all y ∈ G since G is connected. Itfollows that the open set U = {x ∈ G : |F(x)| = |F |} is nonempty.

By [32, Theorem 3.1] there exists a compact set E0 ⊆ U which fails to be ofweak synthesis for A(G). Now let E = ⋃

α∈F α(E0) and fix n ∈ N. Note that the setsα(E0), α ∈ F , are compact and pairwise disjoint since, for α,β ∈ F , α(U)∩β(U) �=∅ only if α = β . In particular, by regularity of A(G), we find w ∈ A(G) such thatw = 1 on a neighborhood V of E \ E0 and w = 0 on a neighborhood of E0.

J Fourier Anal Appl

Let v = v0 −v0w ∈ A(G) and u = ∑

α∈F v◦α ∈ AF (G). Then u ∈ I (E) since v ∈I (E) and E is F -invariant. We are going to show that 〈T ,un〉 �= 0, which will com-plete the proof since suppT ⊆ E0 ⊆ E. For α ∈ F , let α∗ : VN(G) → VN(G) denotethe adjoint of the map f → f ◦ α from A(G) into itself, i.e., 〈α∗(S), f 〉 = 〈S,f ◦ α〉for f ∈ A(G) and S ∈ VN(G). Then suppα∗(T ) = α−1(suppT ) ⊆ α−1(E0) and thisimplies, since v = 0 on V , 〈T , (v ◦ α)f 〉 = 〈α∗(T ), vf 〉 = 0 for all f ∈ A(G) when-ever α is not the identity id of F . Now

vn = (v0 − v0w)n = vn0 − w ·

n−1∑

j=0

(−1)n−j

(

n

j

)

vn0wn−1−j

and, since w = 0 on a neighborhood of E0,

⟨

T ,w ·n−1∑

j=0

(−1)n−j

(

n

j

)

vn0wn−1−j

⟩

= 0.

Thus 〈T ,vn〉 = 〈T ,vn0 〉 �= 0. It remains to observe that also 〈T ,un〉 = 〈T ,vn〉. Now

every summand in un = (∑

α∈F v ◦ α)n except for vn is of the form (v ◦ α)f , whereα �= id and f ∈ A(G). But then, as we have seen above, 〈T , (v◦α)f 〉 = 0. This showsthat 〈T ,un〉 = 〈T ,vn〉. �

Theorem 4.7 Let G be a solvable locally compact group and F a finite group ofautomorphisms of G. If weak spectral synthesis holds for A(G � F//F), then G

must be totally disconnected.

Proof Towards a contradiction, assume that G0 is nontrivial. Since all the groups inthe descending commutator series of G0 are connected and F -invariant, there existsa nontrivial connected, Abelian, F -invariant normal subgroup N of G. Recall thatN � F//F is a closed subhypergroup of G� F//F . Thus, since weak spectral syn-thesis holds for A(G� F//F), it also holds for A(N � F//F) by Theorem 3.10(i).However, this is impossible by Lemma 4.6. �

Corollary 4.8 Let G be a locally compact group such that G0 is Abelian, and letK be a compact subgroup of G. If weak spectral synthesis holds for A(G//K), thenK ⊇ G0.

Proof Arguing as in the proof of Theorem 4.5, we can reduce to the case whereG = KG0 and G0 has finite index in G.

Let L = K ∩ G0, which is a normal subgroup of G since G0 is Abelian and G =KG0. Towards a contradiction, assume that L is a proper subgroup of G0. ThenG0/L = R

p × Tq , where p + q ≥ 1. Using Lemma 4.6 again, we pass to G/L. As

[K : L] ≤ [G : G0] < ∞ and K/L ∩ G0/L is trivial, G/L is the semidirect productof Rp × T

q and the finite group F = K/L. Finally, since p + q ≥ 1, Theorem 4.7shows that weak spectral synthesis fails for A((G/L)//(K/L)). This contradictioncompletes the proof. �

J Fourier Anal Appl

5 Direct Products of Ultraspherical Hypergroups

In this section we show that the direct product of two ultraspherical hypergroups isagain ultraspherical and that the Fourier algebra of the direct product is isomorphicto the projective tensor product of the Fourier algebras of the factors provided thatone the two hypergroups is a Pontryagin hypergroup.

In proving the following proposition we shall use that if G is a locally compactgroup and π : Cc(G) → Cc(G) is a spherical projector, then for any compact subset C

of G, its π -saturation ˜C = ⋃

x∈C Ox is also compact. To see this, let (yα)α be a net inG such that yα ∈ Oxα for some xα ∈ C. Since C is compact, after passing to a subnetif necessary, we may assume that xα → x for some x ∈ C. Now recall that K(G),the collection of all nonempty compact subsets of G, carries Michael’s topology andthat the map t → Ot from G into K(G) is continuous. Let U be a neighborhoodbasis of the identity e of G. Then, since Oxα → Ox , for each U ∈ U , there existsan index α(U) such that Oxα ⊆ UOx for all α ≥ α(U). Thus yα(U) = sα(U)tα(U)

where sα(U) ∈ U and tα(U) ∈ Ox . Then sα(U) → e and, since Ox is compact, we canassume that tα(U) → t for some t ∈ Ox . Hence the subnet (yα(U))U of the originalnet converges in ˜C.

Proposition 5.1 The direct product of two spherical (ultraspherical) hypergroups isspherical (ultraspherical).

Proof Let G1 and G2 be locally compact groups with spherical projectors π1 andπ2, respectively, and let H1 and H2 denote the resulting spherical hypergroups. LetG = G1 × G2, f ∈ Cc(G) and choose compact subsets C1 and C2 of G1 and G2,respectively, such that suppf ⊆ C1 × C2. Then, for all s, t ∈ G1 and x2 ∈ G2,

|π2(f (s, ·))(x2) − π2(f (t, ·))(x2)| = |〈π2(δx2), f (s, ·) − f (t, ·)〉|≤ ‖π2(δx2)‖ · ‖f (s, ·) − f (t, ·)‖∞.

Since ‖π2(δx2)‖ ≤ 1, this estimate shows that the function gx2 , defined by gx2(s) =π2(f (s, ·))(x2), is continuous on G1. It is clear that the support of gx2 is contained inC1. Therefore we can associate to f the function π(f ) on G defined by

π(f )(x1, x2) = π1(gx2)(x1), xi ∈ Gi, i = 1,2.

We are showing next that π(f ) is continuous. Fix (x1, x2) ∈ G and let ε > 0 be given.Since the map μ → π1(μ) from M(G1) into itself is w∗-continuous, there exists aneighborhood V1 of x1 in G1 such that |〈π(δy1 − δx1), gx2〉| ≤ ε for all y1 ∈ V1. Then,for any y1 ∈ V1 and y2 ∈ G2,

|π(f )(y1, y2) − π(f )(x1, x2)| = |〈π1(δy1), gy2〉 − 〈π1(δx1), gx2〉|≤ |〈π1(δy1 − δx1), gx2〉| + |〈π1(δy1), gy2 − gx2〉|≤ ε + |〈π1(δy1), gy2 − gx2〉|≤ ε + ‖gy2 − gx2‖∞.

J Fourier Anal Appl

Now, since f ∈ Cc(G), f is uniformly continuous and hence for every t ∈ C1 thereexists an open neighborhood Ut in G1 such that ‖f (s, ·) − f (t, ·)‖∞ ≤ ε for alls ∈ Ut . Since C1 is compact, C1 ⊆ ⋃m

j=1 Utj for certain t1, . . . , tm ∈ C1. Because‖π2(δy2 − δx2)‖ ≤ 2, we then have, for all s ∈ Utj and 1 ≤ j ≤ m,

|gy2(s) − gx2(s)| = |π2(f (s, ·))(y2) − π2(f (s, ·))(x2)|= |〈π2(δy2 − δx2), f (s, ·)〉|≤ |〈π2(δy2 − δx2), f (s, ·) − f (tj , ·)〉|

+ |〈π2(δy2 − δx2), f (tj , ·)〉|≤ 2ε + |〈π2(δy2 − δx2), f (tj , ·)〉|.

Using that the map y2 → π2(δy2 − δx2) from G2 into M(G2) is continuous for thew∗-topology on M(G2), we find a neighborhood V2 of x2 in G2 such that

|〈π2(δy2 − δx2), f (tj , ·)〉 ≤ ε

for all y2 ∈ V2 and j = 1, . . . ,m. Since gy2(s) = 0 for s /∈ C1, we conclude that‖gy2 − gx2‖∞ ≤ ε for all y2 ∈ V2, and hence that

|π(f )(y1, y2) − π(f )(x1, x2)| ≤ 3ε

for all (y1, y2) ∈ V1 × V2. This shows that π(f ) is continuous at every point(x1, x2) ∈ G.

Moreover, π(f ) has compact support. To see this, let (x1, x2) ∈ G such thatπ(f )(x1, x2) �= 0. Then π1(gx2)(x1) �= 0 and hence, since suppgx2 ⊆ C1, suppπ(gx2)⊆ {Os : s ∈ C1}. Furthermore, since gx2 �= 0, π2(f (s, ·))(x2) �= 0 for some s ∈ G.This implies that x2 ∈ ⋃{Ot : t ∈ suppf (s, ·)} and hence x2 ∈ ⋃{Ot : t ∈ C2} assuppf ⊆ C1 × C2. Consequently,

suppπ(f ) ⊆⎛

⎝

⋃

s∈C1

Os

⎞

⎠ ×⎛

⎝

⋃

t∈C2

Ot

⎞

⎠ ,

which is a compact subset of G1 × G2, as we noticed at the outset of this section.Thus we have seen so far that π is a linear mapping from Cc(G) into itself. It

remains to show that π is a spherical projector, that is, satisfies conditions SH1, SH2and SH3 of Sect. 2. To that end, we first note that SH1(4) holds. In fact, retaining theabove notation,

∫

G

π(f )(x)dx =∫

G2

∫

G1

π(f )(x1, x2)dx2dx1

=∫

G2

∫

G1

π1(gx2)(x1)dx1dx2

=∫

G2

∫

G1

gx2(x1)dx1dx2

J Fourier Anal Appl

=∫

G1

∫

G2

π2(f (x1, ·))(x2)dx2dx1

=∫

G1

∫

G2

f (x1, ·)(x2)dx2dx1

=∫

G

f (x)dx.

This equation in particular shows that the map π : Cc(G) → Cc(G) is continuouswith respect to the L1-norms. This fact in turn implies that SH1(1), SH1(2) andSH1(3) and SH2 hold whenever they have been verified for functions f and g in somesubset of Cc(G), the linear span of which is dense in Cc(G) with respect to both the‖ · ‖∞-norm and the ‖ · ‖1-norm. Of course, the collection of product type functionshas this property. Thus, let fj ∈ Cc(Gj ), j = 1,2, and let f = f1 × f2 ∈ Cc(G).Then

gx2(s) = π2(f1(s)f2(·))(x2) = f1(s)π2(f2)(x2)

for s ∈ G1 and x2 ∈ G2, and therefore, for (x1, x2) ∈ G,

π(f )(x1, x2) = π1(gx2)(x1) = π1(f1)(x1)π2(f2)(x2).

Now, for such functions f (and g) it is straightforward to check (1), (2) and (3) ofSH1 and also SH2.

Thus we are left with checking SH3. Taking again for f a product type function,f = f1 × f2, we have for (x1, x2) ∈ G,

〈π(δ(x1,x2)), f 〉 = 〈δ(x1,x2), π1(f1) × π2(f2)〉 =2

∏

j=1

〈πj (δxj), fj 〉.

This forces π(δ(x1,x2)) = π1(δx1) × π2(δx2), and then (1), (2) and (3) of SH3 for π

follow from the corresponding properties of π1 and π2.Of course, from the mere definition of an ultraspherical hypergroup, it follows that

the direct product of two ultraspherical hypergroups is ultraspherical. This completesthe proof of the proposition. �

Let H1 and H2 be two ultraspherical hypergroups. For ui ∈ A(Hi), i = 1,2, letu on H1 × H2 be defined by u(x1, x2) = u1(x1)u2(x2), xi ∈ Hi , i = 1,2. Then(u1, u2) → u is a bilinear map from A(H1) × A(H2) into A(H1 × H2) satisfying‖u‖ ≤ ‖u1‖ · ‖u2‖ and hence extends uniquely to a norm decreasing linear map fromthe projective tensor product A(H1) ⊗A(H2) of A(H1) and A(H2) into A(H1 ×H2).We now exhibit a situation where this map turns out to be an isometric isomorphismfrom A(H1) ⊗A(H2) onto A(H1 × H2). For any hypergroup H , let VN(H) denotethe von Neumann subalgebra of bounded linear operators of L2(H) generated by theleft regular representation λH of H on L2(H). If H is ultraspherical, then VN(H) isisometrically isomorphic to the dual space of A(H) [28, Theorems 3.4, 3.8 and 3.9].The following result generalizes the corresponding one for locally compact groups(see [29]).

J Fourier Anal Appl

Proposition 5.2 Let H1 and H2 be ultraspherical hypergroups and suppose that oneof them is a Pontryagin hypergroup. Then A(H1 ×H2) is isometrically isomorphic tothe projective tensor product A(H1) ⊗A(H2).

Proof Suppose that H1 is Pontryagin and let Γ be the Abelian hypergroup withΓ = H1. Then A(H1) is isometrically isomorphic to L1(Γ ) via the Fourier transformand L1(Γ )⊗A(H2) is isometrically isomorphic to L1(Γ,A(H2)), the convolution al-gebra of A(H2)-valued Bochner integrable functions on Γ (compare [34, Sect. 2.3]).It therefore remains to show that also A(H1 × H2) is isometrically isomorphic toL1(Γ,A(H2)).

Now A(H1 × H2) is the predual of the von Neumann algebra VN(H1 × H2) and,by [35, Chap. IV, Theorem 7.17], L1(Γ,A(H2)) is the predual of the von Neumannalgebra tensor product L∞(Γ )⊗VN(H2). It thus suffices to verify that the von Neu-mann algebras VN(H1 × H2) and L∞(Γ )⊗VN(H2) are isomorphic. We have thefollowing sequence of isometric isomorphisms

L∞(Γ )⊗VN(H2) ∼= L1(Γ )∗ ⊗VN(H2) ∼= A(H1)∗ ⊗VN(H2)

∼= VN(H1)⊗VN(H2).

Consequently, we are left with observing that VN(H1 × H2) and VN(H1)⊗VN(H2)

are isometrically isomorphic. Now this follows as in the case of locally compactgroups simply taking into account that L2(H1 × H2) is isomorphic to the Hilbertspace tensor product of L2(H1) and L2(H2) and that the left regular representationλH1×H2 is unitarily equivalent to the (outer) tensor product of λH1 and λH2 . �

The results obtained in Sect. 5 of [19] for projective tensor products of commuta-tive Banach algebras now imply the following

Corollary 5.3 Let H1 and H2 be as in Proposition 4.2 and let Ej be a closed subsetof Hj , j = 1,2.

(i) For j = 1,2, ξ(Ej ) ≤ ξ(E1 × E2).(ii) For x1 ∈ H1, ξ({x1} × E2) = max{ξ(x1), ξ(E2)}, and similarly for x2.

(iii) For instance, if H1 is discrete, then (weak) spectral synthesis holds for theFourier algebra A(H1 × H2) if and only if it holds for A(H2).

6 Examples

We conclude the paper with a number of examples complementing and illustratingthe preceding sections. We first interpret Varopoulos’ result [37] in terms of the orbithypergroup (Rd)SO(d), d ≥ 2, where the special orthogonal group SO(d) acts on R

d

by rotation. Recall that (Rd)SO(d) is topologically isomorphic to the double cosethypergroup M(d)//SO(d), where M(d) = R

d� SO(d) is the motion group of Rd .

Note also that M(d)//SO(d) is nothing but the Bessel-Kingman hypergroup withparameter d

2 − 1 [2, (3.5.61) and (3.5.62)].

J Fourier Anal Appl

Example 6.1 Let Sd−1 denote the unit sphere in Rd and I = I (Sd−1) ⊆ A(Rd), and

let R denote the subalgebra of A(Rd) consisting of all rotation invariant functions.The Gelfand spectrum of R can be canonically identified with the half line [0,∞[ sothat u(r) = u(r) for u ∈ R and r ∈ [0,∞[. Let n = � d+1

2 �. Varopoulos [37, Lemma2(iv)] has shown the following.

(1) The ideals (I ∩ R)k , 1 ≤ k ≤ n, are the only closed ideals of R with hull equalto {1}. In particular, (I ∩ R)n = J ({1}).

(2) (I ∩ R)k+1 is properly contained in (I ∩ R)k for 1 ≤ k ≤ n − 1. In particular,(I ∩ R)n−1 �= (I ∩ R)n.

It follows from (1) and (2) that {1} is a weak spectral set for R and ξ({1}) =n. Now Aπ(M(d)) = A(M(d)//SO(d)) and thus {1} is a weak spectral set forA(M(d)//SO(d)) with ξ({1}) = � d+1

2 �. The same of course holds for any 0 < r <

∞. In contrast, the singleton {0} is a Ditkin set for M(d) and hence a Ditkin set forA(M(d)//SO(d)).

We now return to the example H = G//K of [6], mentioned in Sect. 4, for whichspectral synthesis holds although H is not discrete. We are going to show that for aclosed subset E of H , the ideal I (E) has a bounded approximate identity if and onlyif either E is finite or E is cofinite. This has also been accomplished in [4], usingdifferent and more complicated methods.

Example 6.2 We observe first that every closed subset of H = G//K is a Ditkinset. Indeed, since a closed countable union of Ditkin sets is a Ditkin set (see [17,Theorem 5.2.2]), it is enough to observe that singletons in H are Ditkin sets. Thisis clear for points of N, because A(H) is unital. Moreover, if p : G → H denotesthe projection, then p−1(0) = {0} × K . Since G is amenable, the ideal I ({0} × K)

of A(G) has a bounded approximate identity and hence so does the ideal I ({0}) ofA(H) (see Proposition 3.9). Thus we have seen that each closed ideal I of A(H) hasan approximate identity.

Note that A(H) = 1( H), where H = N0 with the hypergroup structure describedin [6, Theorem 3.8]. Moreover, by Theorem 7.8 of [6], a continuous function f on H

belongs to A(H) if and only if∑∞

n=1 |f (n) − f (n − 1)| < ∞.It is easy to see that if either E is finite or H \E is finite, then I (E) has a bounded

approximate identity. Thus, let E be a closed subset of H such that both E and H \E

are infinite. Using this hypothesis, it is straightforward to construct by induction asequence (kn)n∈N ⊆ N such that kn+1 > kn + 1 and kn /∈ E, but kn − 1 ∈ E. Then, foreach f ∈ I (E),

∞∑

n=1

|f (kn)| =∞∑

n=1

|f (kn) − f (kn − 1)| ≤∞∑

n=1

|f (n) − f (n − 1)|.

Towards a contradiction, assume now that I (E) has an approximate identity (fα)αwith norm bound c > 0, say. Since the norms f → ∑∞

n=1 |f (n) − f (n − 1)| andf → ‖f ‖A(H) are equivalent on A(H) [6, Remark 7.10], it follows that there existsd > 0 such that

∑∞n=1 |fα(kn)| ≤ cd for all α. However, since A(H) is regular and

J Fourier Anal Appl

the A(H)-norm dominates the ‖ · ‖∞-norm, given any finite subset F of N, thereexists α such that |fα(kn)| ≥ 1 − |F |−1 for all n ∈ F , and hence

cd ≥∑

n∈F

|fα(kn)| ≥ |F |(1 − |F |−1) = |F | − 1.

Since F can be chosen arbitrarily large, we have reached a contradiction. Conse-quently, I (E) cannot have a bounded approximate identity whenever E and H \ E

are both infinite.

We continue with a double coset hypergroup derived from a group which is usuallyreferred to as the reduced oscillator group.

Example 6.3 Let H denote the 3-dimensional Heisenberg group, realized as R3 withmultiplication

(t, x)(s, y) = (t + s + x1y2, x1 + x2, y1 + y2),

t, s ∈R, x = (x1, x2), y = (y1, y2) ∈R2. Let SO(2) act on H by A · (t, x) = (t,A(x))

and form the semidirect product G = H�SO(2). This group G is 3-step solvable andoften referred to as the reduced oscillator group.

Let K = {e} × SO(2), where e denotes the identity of H, and consider the doublecoset hypergroup G//K . Every double coset is of the form

K(t, x,A)K = {t} × SO(2)(x) × SO(2),

and hence the map K(t, x,A)K → (t,‖x‖) is a bijection between G//K and R ×[0,∞). It is easily verified that (t, r) ∗ (s,R) = (t + s, r ∗ R) for t, s ∈ R and r,R ∈[0,∞). Thus G//K is topologically isomorphic to the product of R and [0,∞), thehypergroup of SO(2)-orbits in R

2, and hence

�(A(G//K)) = �(A(R× [0,∞)) = R× [0,∞).

Now, by Proposition 4.2, A(R×[0,∞)) is isometrically isomorphic to the projectivetensor product A(R)⊗A([0,∞)). Since points in [0,∞) = M(2)//SO(2) are setsof synthesis, it follows from [16, Lemma 3.3] that points in R × [0,∞) are sets ofsynthesis for A(G//K).

Finally, we present an example of a double coset hypergroup H which arises froma p-adic Heisenberg type group and which allows to describe all the sets of weaksynthesis and the weak Ditkin sets for A(H) explicitly in terms of such sets in theadditive group of the p-adic number field. The main reason for this behavior is thatH decomposes in a feasible way into a countable union of subhypergroups.

Example 6.4 Let p be a prime number, Ωp the p-adic number field and, as before,�p the subring of p-adic integers. Equipped with the p-adic metric | · |p , Ωp isa locally compact totally disconnected field and �p is compact and open (see [14,

J Fourier Anal Appl

Sect. 10]). Let G denote the 2-step nilpotent totally disconnected locally compactgroup of all upper triangular matrices

⎛

⎝

1 x z

0 1 y

0 0 1

⎞

⎠ , x ∈ �p,y, z ∈ Ωp.

In what follows, we write such a matrix as a triple (x, y, z) and let K = {(x,0,0) :x ∈ �p} and H = G//K . It is easily verified that, for any (x, y, z) ∈ G,

K(x,y, z)K = (0, y, z)[�p × {0} × y�p] = [�p × {0} × y�p](0, y, z).

Notice next that K(x,y, z)K = K(x′, y′, z′)K if and only if y′ = y and z′ ∈ z+y�p .Thus, as a set

H =⋃

y∈Ωp

{y} × (Ωp/y�p).

Let S denote the unit sphere in Ωp , i.e. S = {x ∈ Ωp : |x|p = 1}. Since the valuation| · |p takes only the values pm, m ∈ Z, and 0, S is open in Ωp . Using this fact, it isstraightforward to determine the quotient topology on H . Let h = (y, z + y�p) ∈ H

and let (hα)α , hα = (yα, zα + yα�p), be a net in H . If y �= 0, then hα → h if andonly if eventually yα = yxα with xα ∈ S and xα → 1 and zα − z ∈ y�p . If y = 0,then hα → h = (0, z) if and only if yα → 0 and, since then yα�p → {0}, zα → z inΩp .

For k ∈ Z, let Hk = pk�p × Ωp/pk�p , which is open and closed in H andis a subhypergroup of H because, denoting by q : G → G//K the quotient map,q−1(Hk) = �p × pk�p × Ωp is a subgroup of G containing K .

Now let E be a weak spectral set with ξ(E) = n (respectively, a weak Ditkin setwith η(E) = m). Then, since E ∩Hk is open and closed in E, we have ξ(E ∩Hk) ≤ n

(respectively, η(E∩Hk) ≤ m). Fix k and let Ck be a system of coset representatives ofpk�p in Ωp . Then Ck is discrete and hence Hk decomposes as Hk = ⋃

z∈Ckpk�p ×

{z + pk�p}, a disjoint union of open and closed subsets of H homeomorphic topk�p . For z ∈ Ck , let

Ek,z = {y ∈ pk�p : (y, z + pk�k) ∈ E}.

Then ξ(Ek,z × {z + pk�p}) ≤ n (respectively, η(Ek,z × {z + pk�p}) ≤ m).In the following we let μC denote the normalized Haar measure of any compact

subgroup C of Ωp and π : Cc(G) → Cc(G) the spherical projector. Then, for y, z ∈Ωp and f ∈ Cc(G),

〈π(δ(0,y,z)), f 〉 =∫

�p

∫

�p

f (a, y, z + yb)dμ�p(a)dμ�p(b)

=⟨

∫

�p

∫

�p

δ(a,y,z+yb)dμ�p(a)dμ�p(b), f

⟩

,

J Fourier Anal Appl

and therefore, on the product space �p × Ωp × Ωp ,

π(δ(0,y,z)) = μ�p × δy × Tzμy�p ,

where Tz is translation by z. Let now hi = (yi, zi + pk�p) ∈ Hk , i = 1,2. Then, bythe definition of convolution on H ,

δh1 ∗ δh2 = π(

π(

δ(0,y1,z1)

) ∗ π(

δ(0,y2,z2)

))

= π((

μ�p × δy1 × Tz1μpk�p

)

∗(

μ�p × δy2 × Tz2μpk�p

))

= π

(

μ�p × δy1+y2 ×∫

�p

(

Tz1+z2+pka μpk�p

)

dμ�p

)

= π(

μ�p × δy1+y2 ×(

Tz1+z2μpk�p∗ μpk�p

))

= π(

μ�p × δy1+y2 × Tz1+z2μpk�p

)

= π(

δ(0,y1+y2,z1+z2)

)

= μ�p × δy1+y2 × Tz1+z2μpk�p.

This equation shows that each Ek,z is a weak spectral set for A(pk�p) with ξ(Ek,z) ≤n (respectively, weak Ditkin set with η(Ek,z) ≤ m). Let

H0 = H \⋃

k∈ZHk = {0} × Ωp.

Thus, since Ωp/pk�p is countable for each k, we have seen so far that E \ H0 isa countable union of pairwise disjoint open and closed subsets each of which is atranslate of a weak spectral set with ξ(·) ≤ n (respectively, weak Ditkin set withη(·) ≤ m). To deal with the case H0 ∩ E �= ∅, we distinguish two cases according towhether H0 ∩ E is open in E or not. If H0 ∩ E is open in E, then it is of course a setof weak synthesis. If H0 ∩ E fails to be open in E, then there exists a sequence (kj )jin Z such that kj → −∞ and Hkj

∩ E �= ∅ for all j . But then

H0 = {0} × Ωp ⊆∞⋃

j=1

Hkj∩ E

and

H0 ∩ E = H0.

Thus we have seen that every weak spectral set E of G with ξ(E) ≤ n (respec-tively, weak Ditkin set with η(E) ≤ m) can be written as a disjoint union of countablymany sets each of which (apart from possibly one, which equals the subgroup Ωp)is a translate in Ωp of some subset of pk�p , k ∈ Z, with ξ(·) ≤ n, (respectively,η(·) ≤ m).

J Fourier Anal Appl

Acknowledgements The authors are grateful to the referee for pointing out an inaccuracy in the state-ment of Theorem 3.11.

References

1. Amini, M., Medghalchi, A.: Fourier algebras on tensor hypergroups. Contemp. Math. 363, 1–14(2004)

2. Bloom, W.R., Heyer, H.: Harmonic Analysis of Probability Measures on Hypergroups. de GruyterStudies in Mathematics, vol. 20. Berlin (1995)

3. Cazzaniga, F., Meaney, C.: A local property of absolutely convergent Jacobi polynomial series. To-hoku Math. J. 34, 389–406 (1982)

4. Chilana, A.K., Kumar, A.: Ultra-strong Ditkin sets in hypergroups. Proc. Am. Math. Soc. 77, 353–358(1979)

5. Damek, E., Ricci, F.: Harmonic analysis on solvable extensions of H -type groups. J. Geom. Anal. 2,213–248 (1992)

6. Dunkl, C.F., Ramirez, D.E.: A family of countably compact P∗-hypergroups. Trans. Am. Math. Soc.202, 339–356 (1975)

7. Eymard, P.: L’algèbre de Fourier d’un groupe localement compact. Bull. Soc. Math. Fr. 92, 181–236(1964)

8. Forrest, B.E.: Fourier analysis on coset spaces. Rocky Mt. J. Math. 28, 173–190 (1998)9. Forrest, B.E., Spronk, N.: Best bounds for approximate identities in ideals of the Fourier algebra

vanishing on subgroups. Proc. Am. Math. Soc. 134, 111–116 (2006)10. Gluskov, V.M.: Locally nilpotent locally bicompact groups. Trudy Moscov. Obshch. 4, 291–332

(1955)11. Greenleaf, F.P.: Invariant means on topological groups. Van Nostrand, New York (1969)12. Haagerup, U.: An example of a non nuclear C∗-algebra, which has the metric approximation property.

Invent. Math. 50, 279–293 (1979)13. Herz, C.: Harmonic synthesis for subgroups. Ann. Inst. Fourier (Grenoble) 23, 91–123 (1973)14. Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. I. Springer, Berlin (1963)15. Jewett, R.I.: Spaces with an abstract convolution of measures. Adv. Math. 18, 1–101 (1975)16. Kaniuth, E.: Weak spectral synthesis in commutative Banach algebras. J. Funct. Anal. 254, 987–1002

(2008)17. Kaniuth, E.: A Course in Commutative Banach Algebras. Graduate Texts in Mathematics, vol. 246.

Springer, New York (2009)18. Kaniuth, E.: Weak spectral synthesis in Fourier algebras of coset spaces. Stud. Math. 197, 229–246

(2010)19. Kaniuth, E., Ülger, A.: Weak spectral synthesis in commutative Banach algebras. III. Preprint20. Lashkarizadeh Bami, M., Pourgholamhossein, M., Samea, H.: Fourier algebras on locally compact

hypergroups. Math. Nachr. 282, 16–25 (2009)21. Leptin, H.: Sur l’algèbre de Fourier d’un groupe localement compact. C. R. Math. Acad. Sci. Paris

266, 1180–1182 (1968)22. Ludwig, J., Turowska, L.: On the connection between sets of operator synthesis and sets of spectral

synthesis for locally compact groups. J. Funct. Anal. 233, 206–227 (2006)23. McMullen, J.R.: Extensions of Positive Definite Functions. Mem. Amer. Math. Soc., vol. 117. AMS,

Providence (1972)24. Meaney, C.: On the failure of spectral synthesis for compact semisimple Lie groups. J. Funct. Anal.

48, 43–57 (1982)25. Meaney, C.: Spherical functions and spectral synthesis. Compos. Math. 54, 311–329 (1985)26. Montgomery, D., Zippin, L.: Topological Transformation Groups. Wiley, New York (1966)27. Muruganandam, V.: Fourier algebras of hypergroups-I. J. Aust. Math. Soc. 82, 59–83 (2007)28. Muruganandam, V.: Fourier algebras of hypergroups-II. Spherical hypergroups. Math. Nachr. 281,

1590–1603 (2008)29. Nouyrigat, P.: Sur le predual de l’algèbre de von Neumann associée à une representation unitaire d’un

groupe localement compact. Publ. Dép. Math. (Lyon) 9, 31–59 (1973)30. Parthasarathy, K., Prakash, R.: Malliavin’s theorem for weak synthesis on nonabelian groups. Bull.

Sci. Math. 134, 561–578 (2010)

J Fourier Anal Appl

31. Parthasarathy, K., Shravan Kumar, N.: Ditkin sets in homogeneous spaces. Stud. Math. 203, 291–307(2011)

32. Parthasarathy, K., Varma, S.: On weak spectral synthesis. Bull. Aust. Math. Soc. 43, 279–282 (1991)33. Ross, K.A.: Hypergroups and centers of measure algebras. In: Symposia Math., vol. XXII, pp. 189–

203. Academic Press, London/New York (1977)34. Ryan, R.A.: Introduction to Tensor Products of Banach Spaces. Springer, London (2002)35. Takesaki, M.: Theory of Operator Algebras I. Springer, New York (1979)36. Takesaki, M., Tatsuuma, N.: Duality and subgroups. II. J. Funct. Anal. 11, 184–194 (1972)37. Varopoulos, N.T.: Spectral synthesis on spheres. Math. Proc. Camb. Philos. Soc. 62, 379–387 (1966)38. Vogel, M.: Spectral synthesis on algebras of orthogonal polynomial series. Math. Z. 194, 99–116

(1987)39. Vrem, R.C.: Harmonic analysis on compact hypergroups. Pac. J. Math. 85, 239–251 (1979)40. Warner, C.R.: Weak spectral synthesis. Proc. Am. Math. Soc. 99, 244–248 (1987)