arens regularity of ideals of the group algebra reza

ARENS REGULARITY OF IDEALS OF THE GROUP

ALGEBRA

REZA ESMAILVANDI1, MAHMOUD FILALI2, JORGE GALINDO3,

1,3 Instituto Universitario de Matematicas y Aplicaciones (IMAC),Universidad Jaume I, E-12071, Castellon, Spain;

2Department of Mathematical Sciences, University of Oulu, Oulu Finland;

Abstract. Let G be a compact Abelian group and E a subset of the

group G of continuous characters of G. We study Arens regularity-related properties of the ideals L1

E(G) of L1(G) that are made of func-

tions whose Fourier transform is supported on E ⊆ G. Arens regularityof L1

E(G), the center of L1E(G)∗∗ and the size of L1

E(G)∗/WAP(L1E) are

studied.We establish general conditions for the regularity of L1

E(G) and de-duce from them that L1

E(G) is not strongly Arens irregular if M1E(G) ∗

M1E(G) is contained in L1

E(G) or G \ E is a Lust-Piquard set and thatL1

E(G) is extremely non-Arens regular if M1E(G) ∗M1

E(G) is not con-tained in L1

E(G).

1. Introduction

It has long been known, at least since the work of Arens [Are51a] in thefifties, that the bidual A∗∗ of a Banach algebra A can be turned into aBanach algebra containing A as a subalgebra. Two different multiplicationscan actually be introduced on A∗∗ to this effect. But, while both thesemultiplications are defined following completely symmetric and absolutelynatural procedures, they can be essentially different. The left multiplicationoperator defined by one of them is always weak∗-continuous but may fail tobe so for the other, with the situation reversed for the right multiplicationoperator.

The subset of A∗∗ made of those elements that produce weak∗-continuousmultiplication operators from both sides is usually referred to as the topo-logical center of A∗∗, in symbols Z(A∗∗) and it always contains A. Whenthe center is as large as possible, i.e., when A∗∗ = Z(A∗∗), we say that A isArens regular, this is the case, for instance, of C∗-algebras. Following Dalesand Lau [DL05], we say that A is strongly Arens irregular (SAI for short)

E-mail address: [email protected], [email protected], [email protected]: December 12, 2021.2010 Mathematics Subject Classification. 22D15, 43A46, 43A60; Secondary: 54H11.Research of the third named author supported by Research supported by Spanish AEI

Project PID2019-106529GB-I00 / AEI / 10.13039/501100011033.

1

2 R. ESMAILVANDI, M. FILALI, J. GALINDO

when Z(A∗∗) is as small as possible, i.e., when Z(A∗∗) = A. This is the caseof the group algebra L1(G) discussed below.

Facing the problem from a different point of view, Pym [Pym65], consid-ered the space WAP(A) of weakly almost periodic functionals on A. This isthe precise subspace of A∗ on which the two Arens-multiplications agree. So,A is Arens regular precisely when A∗ = WAP(A), i.e., when the quotientA∗/WAP(A) is trivial.

When the quotient A∗/WAP(A) contains a closed subspace isomorphicto A∗, and so it is as large as possible, we say that A is extremely non-Arensregular (ENAR for short). Extreme non-Arens regularity was first studiedin the context of Fourier algebras with a slightly different definition, see thepapers by Granirer [Gra96] and Hu [Hu97].

Isik, Pym and Ulger [IkPU87] proved that the group algebra L1(G) of acompact group is always strongly Arens irregular. Shortly afterwards, Lauand Losert [LL88] proved the same fact for every locally compact group.Bouziad and Filali [BF11] proved that L1(G) is ENAR for locally com-pact groups whose compact covering number is not smaller than their localcharacter (i.e. when G, topologically speaking, looks more discrete thancompact) and compact metrizable groups. The group algebra L1(G) wasshown to be ENAR for every infinite locally compact group in [FG18].

In this paper we work with ideals of L1(G) with G a compact Abeliangroup. To describe these ideals, we need to resort to duality. We denote

by G is the group of all continuous homomorphisms into the multiplicativegroup of unimodular complex numbers, known as continuous characters.For µ ∈M(G), the Fourier-Stieltjes transform of µ is the bounded function

µ : G→ C given by

µ(γ) =

∫G〈−x, γ〉dµ(x).

In terms of the duality between M(G) and C(G), for every γ ∈ G,

µ(γ) = 〈µ, γ〉,

where for a measure µ ∈ M(G), we denote by µ the measure in M(G)defined by

〈µ, φ〉 = 〈µ, φ〉 =

∫Gφ(−x)dµ(x) (φ ∈ C(G)).

If f ∈ L1(G) this definition produces the function f(x) = f(−x) (x ∈ G).

If X is a linear subspace of M(G) and E ⊂ G, we denote by XE thesubspace of X ,

XE = µ ∈ X : µ(γ) = 0 for γ ∈ G \ E.

Most prominent in our work will be the ideal ME(G) of M(G) and itssubspace the ideal L1

E(G) of L1(G). The spaces L1E(G) will be in particular

1.1. Summary of results. In this paper we address the Arens regularityproperties of the ideals of L1(G) when G is a compact Abelian group. These

ideals are always of the form L1E(G) for some subset E of G, see e.g. [HR70,

Theorem 38.7].

ARENS REGULARITY OF IDEALS OF THE GROUP ALGEBRA 3

As mentioned earlier, it is known that L1E(G) is strongly Arens irregular

and ENAR when E = G. If E is finite, L1E(G) has finite dimension and so

is reflexive, and is thus Arens regular. One may therefore expect that theregularity properties of L1

E(G) improve as E decreases. This is evidenced by

the result of Ulger [U11], the paper that inspired this work: if E is a Rieszset, i.e., if E is a set that will not support transforms of measures that arenot in L1(G), then L1

E(G) is Arens regular.In this paper we relate the Arens regularity of L1

E(G) with the size of thesubspace of L1

E(G)∗ made of restrictions to L1E(G) of convolutions of the

form µ ∗ φ with µ ∈ME(G) and φ ∈ L∞(G).The absolute continuity (with respect to Haar measure) of measures in

ME(G) ∗ME(G) turns to be important in this discussion. When ME(G) ∗ME(G) ⊆ L1

E(G), L1E(G) has a large center and cannot be SAI. It has to

be mentioned that it is unknown whether this property implies that E is aRiesz set. If that was the case the L1

E(G) would actually be regular. If, onthe other hand, ME(G) ∗ME(G) is not contained in L1

E(G), then the centerof L1

E(G) is small and L1E(G) is ENAR.

We also discuss the impact of thinness properties on the Arens regularityof L1

E(G). As already stated, L1E(G) is Arens regular if E is a Riesz. We

prove in Section 7.3 below that L1E(G) cannot be Arens regular if G \ E is

a Lust-Piquard set. If we take into account that N is a Riesz subset of Z,by the F. and M. Riesz Theorem, and that the set E ⊆ Z consisting of theprimes in the coset 5Z + 2 is a Lust-Piquard set, by [LP89, Theorem 4],

then, we have that L(NT) is Arens regular, despite N not being too thin and

that L1Z\E(G) is not Arens regular, despite can offer to nontrivial ideals of

L1(G) one not too thin, the other not roo thick, such that finsee that Z \Enot being too thick.

Figure 1 summarizes what is known on the Arens regularity properties of

L1E(G), as E increases from a singleton to G, when the contributions of the

present paper are put together with what was previously known. In additionto the above results we can add that L1

E(G) will be SAI if E is the coset

ring of G, in particular all maximal ideals of L1(G) happen to be SAI.

2. Arens regularity

In this section we provide formal definitions for the concepts related toArens regularity discussed in this paper.

Let A be a commutative Banach algebra and let A∗ and A∗∗ be its firstand second Banach duals, respectively. The multiplication of A can beextended naturally to A∗∗ in two different ways. These multiplications ariseas particular cases of the abstract approach of R. Arens [Are51b, Are51a] andcan be formalized through the following three steps. For u, v in A, ϕ in A∗and m,n ∈ A∗∗, we define ϕ ·u, u ·ϕ, m ·ϕ,ϕ ·m ∈ A∗ and mn,m♦n ∈ A∗∗as follows:

〈ϕ · u, v〉 = 〈ϕ, uv〉, 〈u · ϕ, v〉 = 〈ϕ, vu〉〈m · ϕ, u〉 = 〈m,ϕ · u〉, 〈ϕ ·m,u〉 = 〈m,u · ϕ〉〈mn, ϕ〉 = 〈m ,n · ϕ〉, 〈m♦n, ϕ〉 = 〈n, ϕ ·m〉.


𝐸 finite

𝐿𝐸1 ሺ𝐺ሻ finite

dimensional

𝐸 ∈ Λሺ1ሻ ⊇ Λሺ𝑝)

𝐿𝐸1 ሺ𝐺ሻ reflexive

𝐸 Riesz

𝐿𝐸

1 ሺ𝐺ሻ regular

E small 1-1

𝐿𝐸1 ሺ𝐺ሻ not SAI

E not small-1-1

𝐿𝐸1 ሺ𝐺ሻ ENAR

𝐺\E Lust Piquard

𝐿𝐸1 ሺ𝐺ሻ not

Arens regular

𝐺\E finite

𝐿𝐸1 ሺ𝐺ሻ SAI

Figure 1. Several subsets E ⊆ G, G compact Abelian, and Arensregularity properties of the corresponding L1

E(G). The upper line con-tains sets that can be regarded as small, large sets are placed in the thelower line.

When and ♦ coincide on A∗∗, A is said Arens regular.For any m ∈ A∗∗ the mapping n 7→ nm is weak∗-weak∗ continuous

on A∗∗. However, the mapping n 7→ mn need not to be weak∗-weak∗

continuous. The situation is reversed for ♦. The left topological center ofA∗∗ is then defined as

Z(A∗∗) = m ∈ A∗∗ : n 7→ mn is weak∗-weak∗ continuous on A∗∗.Since we are assuming that A is commutative, it is easy to see that

Z(A∗∗) = m ∈ A∗∗ : mn = nm = m♦n for all n ∈ A∗∗ .The algebra A is therefore Arens regular if and only if Z(A∗∗) = A∗∗.Observe that A is always contained in Z(A∗∗). Sometimes, the elements ofthe center stop here:

Definition 2.1. A commutative Banach algebra A is strongly Arens irreg-ular (SAI for short) when Z(A∗∗) = A.

In [Pym65], J. Pym considered the space WAP(A) of weakly almost pe-riodic functionals on A, this is the set of all ϕ ∈ A∗ such that the linearmap

A→ A∗ : a 7→ a · ϕis weakly compact. The functionals ϕ ∈ WAP(A) satisfy Grothendick’sdouble limit criterion

limn

limm〈ϕ, anbm〉 = lim

mlimn〈ϕ, anbm〉

for any pair of bounded sequences (an)n, (bm)m in A for which both theiterated limits exist. From this property, one may deduce that 〈mn, φ〉 =〈m♦n, φ〉, for every m,n ∈ A∗∗, if and only if φ ∈ WAP(A). So, A isArens regular when A∗ = WAP(A), i.e., when the quotient A∗/WAP(A)is trivial. This is the motivation for the following definition.


Definition 2.2. A Banach algebra A is extremely non-Arens regular (ENARfor short) when A∗/WAP(A) contains a closed subspace isomorphic to A∗.

The term extreme non-Arens regularity was coined by Granirer [Gra96] tocharacterize a slightly more general behaviour. We have adopted here thissimpler definition that is still enough to capture the extreme behaviour ofmany of the Banach algebras that Harmonic Analysis associates to a locallycompact group, see [FGa, FGb].

3. The structure of L1(G)∗∗ and L1E(G)∗∗

We summarize here the structure of L1E(G)∗∗ where G is a compact

Abelian group and E ⊆ G. Notation will be additive and the identities

of both G and G will be denoted by 0. All the facts mentioned here are

well-known when E = G, see e.g. [IkPU87]. No new insight is needed for

them to hold for arbitrary E ⊆ G but having them stated beforehand willsimplify our proofs.

A good deal of the structure of L1(G)∗∗ is determined by the presence ofright identities. These can be obtained as accumulation points in L1(G)∗∗ ofbounded approximate identities of L1(G), which are always available (see e.g.[Kan09, Section 1.3]). The first use of right identities is to bring measureson G into elements of L1(G)∗∗.

For each µ ∈ME(G), one considers the convolution operator

Cµ : L1(G)→ L1E(G), given by Cµ(u) = µ ∗ u, u ∈ L1(G).

Its double adjoint C∗∗µ then maps L1(G)∗∗ into L1E(G)∗∗. When necessary we

will use i : L1E(G)→ L1(G) to denote the inclusion map, then i∗ : L∞(G)→

L1E(G)∗ will be the restriction map and i∗∗ : L1

E(G)∗∗ → L1(G)∗∗ will be anembedding of Banach algebras. We will normally omit mentioning i and i∗∗

and see L1E(G)∗∗ as an ideal of L1(G)∗∗.

With these notations, if µ ∈ME(G) and φ ∈ L∞(G) are given, a straight-forward computation shows that

C∗∗µ (e).i∗(φ) = i∗(µ ∗ φ

).(3.1)

If e is a right identity in L1(G)∗∗, the lifting map

Je : ME(G)→ L1E(G)∗∗ given by Je(µ) = C∗∗µ (e),

turns out to be an isomorphism onto e i∗∗(L1E(G)∗∗

).

The algebra L1E(G) can be seen both as an ideal in ME(G) and as an

ideal in L1E(G)∗∗ and, in that sense, it is left invariant by Je, i.e.

(3.2) Je(f) = C∗∗f (e) = f for all f ∈ L1E(G).

The canonical quotient map RE : L1E(G)∗∗ → M(G), defined, for each

m ∈ L1E(G)∗∗, by RE(m) = m

∣∣C(G)

is then a left inverse for Je and the com-

position Je RE is a projection. Regardless of the right identity e, kerREcan always be identified with i∗(C(G))⊥, the annihilator of the subspacei∗(C(G)

)in L1

E(G)∗∗. The projection Je therefore induces the decomposi-tion

L1E(G)∗∗ = Je(ME(G))⊕ i∗

(C(G)

)⊥(3.3)


So, for a given right identity e of L1E(G)∗∗, an element m ∈ L1

E(G)∗∗, maybe uniquely decomposed as

(3.4) m = C∗∗µ (e) + r,

where µ ∈ME(G) and r ∈ i∗(C(G))⊥.The above decomposition becomes handier if one observes that the el-

ements of i∗(C(G)

)⊥are left annihilators of L1

E(G)∗∗. Indeed, for r ∈i∗(C(G)

)⊥, if m ∈ L1

E(G)∗∗ is such that m = σ(L1E(G)∗∗, L1

E(G)∗)−limα uα,

with uα ∈ L1E(G), and φ ∈ L∞(G), then

〈m r, i∗(φ)〉 = 〈i∗∗(m r

), φ〉

= limα〈uα, i∗∗(r).φ〉

= limα〈r, i∗

(quα ∗ φ

)〉 = 0,(3.5)

where the last identity follows from quα ∗ φ ∈ C(G).While L1(G) always has bounded approximate identities (BAIs) whose

accumulation points in L1(G)∗∗ are right identities, L1E(G) may well fail to

have any. Approximate identities that are bounded in the multiplier norm(MBAIs) are however always available. This is proved in proved in [Zha02,Proposition 1], for convenience, we include a complete proof here.

Lemma 3.1. Let G be a compact Abelian group and E ⊂ G. Then L1E(G)

contains a net (eα)α∈Λ with the following properties:

(1) If u ∈ L1E(G), then limα‖u ∗ eα−u‖ = 0 ((eα)α∈Λ is an approximate

identity).(2) If u ∈ L1

E(G), then ‖u ∗ eα‖ ≤ ‖u‖ for every α ∈ Λ ((eα)α∈Λ is anMBAI).

(3) For every µ ∈ME(G), limα eα ∗ µ = µ in σ(M(G), C(G)).

Proof. Let (uα)α be an approximate identity that is made of continuousfunctions of norm 1 (see e.g [Kan09, Section 1.3]). By Plancherel’s theorem,

(uα) ⊆ `2(G). So (uα · 1E) ⊆ `2(G). Let now (eα) be a net in L2(G) suchthat

eα = uα · 1E (α ∈ I).

Then, for each µ ∈ME(G), we have

(eα ∗ µ)= eα · µ = uα · 1E · µ = (uα ∗ µ) .

Thus, by the uniqueness theorem, eα ∗ µ = uα ∗ µ for each α ∈ I. From thisall the assertions of the Lemma easily follow.

Lemma 3.2 ([IkPU87]). Let G be a compact Abelian group. Consider E ⊆G and µ ∈ME(G). If for every pair e and f of right identities in L1(G)∗∗,C∗∗µ (e) = C∗∗µ (f), then µ ∈ L1(G).

Proof. Suppose that µ ∈ ME(G) but µ /∈ L1(G), we can then find φ ∈L∞(G) such that µ∗φ is not continuous, see [HR70, Theorem 35.13]. Arguingas in Lemma 2.3 of [IkPU87] we can find then two different right identitiesf1, f2 ∈ L1(G)∗∗ such that 〈f1, µ ∗ φ〉 6= 〈f2, µ ∗ φ〉. Since

〈C∗∗µ (fi), φ〉 = 〈fi, µ ∗ φ〉, i = 1, 2,

we deduce that C∗∗µ (f1) 6= C∗∗µ (f2), a contradiction with our hypotheses.


Sidon +3

Λ(p) for every p +3 Λ(1) ks +3

L1E(G) is reflexive

Rosenthal +3 Lust Piquard +3 Riesz +3 small-1-1 +3 small-2

Figure 2. Relations between properties of E ⊂ G, G compact and Abelian.

4. Special subsets of G

We describe here the sets E ⊆ G that lead to the concrete ideals L1E(G)

that will appear later in the paper.We first recall than an invariant mean M on L∞(G) is a linear functional

on L∞(G) such that 〈M, 1〉 = ‖M‖ = 1 and, for each φ ∈ L∞(G) andeach x ∈ G, 〈M,Lxφ〉 = 〈M,φ〉 where Lx is the translation operator byx. An invariant mean that is always available is the one produced by Haarmeasure: φ 7→

∫φ(x)dx. If G is compact, L∞(G) always has other invariant

means [Rud72] but all them have the same effect on some functions. We saythen that a function φ ∈ L∞(G) has a unique invariant mean if 〈M,φ〉 =∫φ(x)dx for every invariant mean M on L∞(G).

Definition 4.1. Let G be a compact Abelian group and E ⊂ G. We saythat E is a

(1) Sidon set if every f ∈ CE(G) has an absolutely convergent Fourierseries.

(2) Λ(p)-set, with p > 0 if there are 0 < q < p and C > 0 such that‖f‖p ≤ C‖f‖q , for every trigonometric polynomial, f =

∑nk=1 ckχk,

with χ1, . . . , χn ∈ E.(3) Rosenthal set if L∞E (G) = CE(G).(4) Lust-Piquard set if γφ has a unique invariant mean for every φ ∈

L∞E (G) and every γ ∈ G. We say in this case that φ is totallyergodic.

(5) Riesz set if ME(G) = L1E(G).

(6) small-1-1 set if µ ∗ ν ∈ L1E(G) for every µ, ν ∈ME(G).

(7) small-2 set if µ ∗ µ ∈ L1E(G) for every µ ∈ME(G).

Sidon sets are Rosenthal, see, e.g. [GH13, Corollary 6.2.5], and Rosenthalsets are Lust-Piquard (as continuous functions always have a unique invari-ant mean). Lust-Piquard sets are on their turn always Riesz (see [Li93]) andRiesz sets are, obviously, small-1-1.

On the other hand, Sidon sets are Λ(p) for every p > 0 [Rud60, Theorem3.1] and Λ(p) sets are Λ(q) for every q < 1. It is a result of Hare [Har88] thata Λ(p) is always a Λ(q) set for some q > p. The following is a consequencethat is important in our context.

Theorem 4.2 (Corollary in [Har88]). Let G be a compact Abelian group

and let E ⊂ G. The Banach space L1E(G) is reflexive if and only if E is a

Λ(1) set.

The above remarks are summarized in Figure 2. One should remark that


it is still unknown, to these authors knowledge, whether every small-2 setis Riesz. So, it might happen that the classes defined in the items (5)-(7)above are all the same.

Since L1E(G) is Arens regular when E is Riesz (see [U11] and Section 6)

we will not be interested in L1E(G) for E in any class contained in that of

Riesz sets. However, sets E whose complement G \ E belongs to such aclass will at some points be of interest, especially after one learns that theunion of a Riesz set and Lust-Piquard set is Riesz [LRP06], and hence thatcomplements of Lust-Piquard sets are never Riesz.

5. The role of ME(G) ∗ L∞(G)

Many Arens regularity properties of L1E(G)∗∗ can de described through

the size of the subspace i∗(ME(G) ∗ L∞(G)

).

We begin with the following observation.

Lemma 5.1. Let G be a compact Abelian group and let E ⊆ G. Then

i∗(M−E(G)∗L∞(G)

)⊆ i∗(C(G)) if and only if i∗(C(G))⊥ ⊆ Z(L1

E(G)∗∗).

Proof. Assume first that i∗(M−E(G) ∗ L∞(G)

)⊆ i∗(C(G)) and let r ∈

i∗(C(G))⊥. Then, for each µ ∈ME(G) and φ ∈ L∞(G), i∗(µ∗φ

)∈ i∗(C(G))

and we have, by (3.1), that

(5.1) 〈r C∗∗µ (e), φ〉 = 〈r, i∗(µ ∗ φ)〉 = 0.

Thus if m ∈ L1E(G)∗∗ is decomposed as in (3.4) m = C∗∗µ (e) + s, with e

being some right identity in L1(G)P ∗∗, µ ∈ ME(G) and s ∈ i∗(C(G))⊥,we have, using that elements of i∗(C(G))⊥ are left annihilators of L1

E(G)∗∗,(3.5), that

r m = r C∗∗µ (e) = 0 = m r.

Hence r ∈ Z(L1E(G)∗∗

).

For the converse, assume that i∗(C(G))⊥ ⊆ Z(L1E(G)∗∗

)and let µ ∈

M−E(G) and φ ∈ L∞(G). Then for each r ∈ i∗(C(G))⊥, we have

〈r, i∗(µ ∗ φ)〉 = 〈r C∗∗µ (e), φ〉 = 〈C∗∗µ (e) r, φ〉 = 0,

where the last equalities follow from r being in the center of L1E(G)∗∗ (by hy-

pothesis) and a left annihilator in L1E(G)∗∗. Hence i∗(µ∗φ) ∈ i∗(C(G))⊥⊥ =

i∗(C(G)).

Theorem 5.2. Let G be a compact Abelian group and let E ⊆ G. Then:

(1) L1E(G) is Arens regular if and only if i∗

(M−E(G)∗L∞(G)

)⊆ i∗(C(G)).

(2) L1E(G) is SAI if and only if the convex hull of i∗

(M−E(G) ∗L∞(G)

)is dense in L1

E(G)∗.

Proof. We start with Statement (1). Assume that i∗(M−E(G) ∗ L∞(G)

)⊆

i∗(C(G)). Lemma 5.1 then shows that i∗(C(G))⊥

is contained in Z(L1E(G)∗∗

).

Let e be a right identity in L1E(G)∗∗. If m1 = C∗∗µ1

(e) + r1 and m2 =

C∗∗µ2(e) + r2 are two arbitrary elements of L1

E(G)∗∗, decomposed following(3.4), and we use that r1 and r2 are left annihilators, (3.5),

m1 m2 = C∗∗µ1∗µ2(e) = C∗∗µ2∗µ1

(e) = m2 m1,


proving that L1E(G) is Arens regular.

Lemma 5.1 proves the converse statement.We now prove Statement (2). Assume first that i∗

(M−E(G) ∗ L∞(G)

)is

dense in L1E(G)∗ and let m ∈ Z

(L1E(G)∗∗

). Pick a right identity e ∈ L1(G)∗∗

and let m = C∗∗µ (e) + r be a decomposition of m following (3.4).Take ν ∈ ME(G) and φ ∈ L∞(G), then 〈m, i∗ (ν ∗ φ)〉 = 〈mC∗∗ν (e), φ〉.

Since m is in the center, and r is a left annihilator, (3.5),

〈m, i∗ (ν ∗ φ)〉 = 〈C∗∗ν (e) (C∗∗µ (e) + r

), φ〉

= 〈C∗∗µ (e), i∗ (ν ∗ φ)〉.

Since the convex hull of i∗(M−E(G) ∗L∞(G)

)is dense in L1

E(G)∗, it followsthat m = C∗∗µ (e) and this for every right identity e. It follows from Lemma

3.2 that µ ∈ L1E(G) and we conclude that L1

E(G) is SAI.Assume now that L1

E(G) is SAI. Let r ∈ L1E(G)∗∗ be such that r ∈(

i∗(M−E(G) ∗ L∞(G)))⊥

. Then, if m = C∗∗µ (e) + s ∈ L1E(G)∗∗, with µ ∈

ME(G) and s ∈ i∗(C(G))⊥, and φ ∈ L∞(G)

〈r m, i∗(φ)〉 = 〈r, i∗(µ ∗ φ)〉 = 0.

This means that r m = 0 and, hence, that r ∈ Z(L1E(G)∗∗

). Since L1

E(G)

is SAI, L1E(G)∩ i∗(C(G))⊥ = 0 and i∗

(M−E(G) ∗L∞(G)

)⊥ ⊆ i∗(C(G))⊥,

we conclude that r = 0. Having shown that i∗(M−E(G) ∗ L∞(G)

)⊥= 0,

the denseness of the convex hull of i∗ (M−E(G) ∗ L∞(G)) in L1E(G)∗ is a

simple consequence of the Hahn-Banach theorem.

6. The regular side

Theorem 5.2 immediately implies Ulger’s theorem

Corollary 6.1 ([U11]). Let G be a compact Abelian group and let E ⊆ G.If E is a Riesz set, then L1

E(G) is Arens regular.

Proof. Simply apply (1) of Theorem 5.2, taking into account that L1(G) ∗L∞(G) ⊆ C(G).

We now turn our attention to small-1-1 sets. Recall that, as indicated inSection 4, we say that E is a small-1-1 set when ME(G) ∗ME(G) ⊆ L1

E(G).We see now that the center of L1

E(G)∗∗ is necessarily large when E is small-1-1.

In the next Theorem we consider, for any right identity e ∈ L1(G)∗∗, theset

Se =r C∗∗µ (e) : r ∈ i∗(C(G))⊥ and µ ∈ME(G)

.

Theorem 6.2. Let G be a compact Abelian group and let E ⊆ G be aninfinite small-1-1 set. Then, for any right identity e ∈ L1(G)∗∗, Se ⊆Z(L1

E(G)∗∗) and, either

• L1E(G) is not SAI, or

• Se = 0 and L1E(G) is Arens regular.

In particular, L1E(G) is not SAI unless it is reflexive.


Proof. We begin by fixing a right identity e ∈ L1(G)∗∗.Let r ∈ i∗(C(G))⊥ and µ ∈ ME(G). Put p = r C∗∗µ (e). If q = C∗∗σ (e) +

s ∈ L1E(G)∗∗, with s ∈ i∗(C(G))⊥ and σ ∈ ME(G), then q p = 0, as r is a

left annihilator, (3.5). Since s is also left annihilator and C∗∗µ (e) C∗∗σ (e) ∈Z(L1

E(G)∗∗) , for µ ∗ σ ∈ L1E(G), one gets

p q = r C∗∗µ (e) C∗∗σ (e) = r C∗∗µ∗σ(e) = 0.

Hence p ∈ Z(L1E(G)∗∗).

We next observe that Se∩L1E(G) = 0. To see that it suffices to consider

any approximate identity (uα)α in L1(G)∗∗, then, if r C∗∗µ (e) ∈ Se∩L1E(G),

we have that

r C∗∗µ (e) = limαuα ∗

(r C∗∗µ (e)

)= 0.

Since Se is central and intersects trivially L1E(G), the only way for L1

E(G)to be SAI is that Se = 0. In that case, we would have(

C∗∗µ1(e) + s1

)(C∗∗µ2

(e) + s2

)= C∗∗µ1∗µ2

(e)

= C∗∗µ2∗µ1(e)

=(C∗∗µ2

(e) + s2

)(C∗∗µ1

(e) + s1

),

for every s1, s2 ∈ i∗(C(G))⊥ and every µ1, µ2 ∈ ME(G) and, therefore,L1E(G) would be Arens regular.

7. The irregular side

The easy way to show that L1E(G) to be strongly Arens irregular is to

require the presence of a bounded approximate identity. By [Kan09, Corol-lary 5.6.2], this happens if and only E ∈ Ω

G, where Ω

Gdenotes the Boolean

ring generated by the left cosets of subgroups of G, known as the coset ring

of G. This is a consequence of P. J. Cohen’s theorem to the effect that for

a subset E of G, 1E is in B(G), the Fourier-Stieltjes algebra on G, if andonly if E ∈ Ω

G(see [Rud90, Theorem 3.1.3] for a exposition of this result).

With these facts in mind, it is an immediate consequence of statement(2) of Theorem 5.2 that L1

E(G) is strongly Arens irregular if E ∈ ΩG

. Itis enough to observe, that, for µ ∈ M(G) with µ = 1E , one has thati∗(µ ∗ φ) = i∗(φ).

Corollary 7.1. Let G be a compact Abelian group and let E ∈ ΩG

. Then

L1E(G) is SAI. In particular, L1

G\Fis SAI if F is finite.

7.1. L1E(G) is ENAR if E is not a small-1-1 set. The following result

of Ulger reveals the relevance of non-small-1-1 sets in the analysis of Arensregularity.

Theorem 7.2 (Theorem 2.2 of [U99]). Let A be a commutative, semisimple,weakly sequentially complete and completely continuous Banach algebra, thenan element m ∈ A∗∗ is in the center of A if and only if mA∗∗ ⊆ A andA∗∗ m ⊆ A.


Corollary 7.3. Let G be a compact Abelian group and assume that E ⊆ Gis not a small-1-1 set. For every pair µ1, µ2 ∈ ME(G) such that µ1 ∗ µ2 /∈L1E(G) and every right identity e of L1(G)∗∗, we have that neither C∗∗µ1

(e)

nor C∗∗µ2(e) are in Z(L1

E(G)∗∗).

Proof. Let µ1, µ2 ∈M(G) such that µ1 ∗ µ2 /∈ L1E(G). Towards a contradic-

tion, assume that C∗∗µ1(e) ∈ Z(L1

E(G)∗∗). By Theorem 7.2,

C∗∗µ1∗µ2(e) = C∗∗µ1

(e) C∗∗µ2(e) ∈ L1

E(G).

Since RE is a left inverse of Je, this is a contradiction.

Under conditions similar to those of Theorem 7.2 we can even find thatthe set WAP(A) is as small as possible. For this we need to produce ap-proximations of triangles in A that are `1-sets, as done in [FGa]. We recallhere the main concepts and results developed in that reference. To avoidfurther technicalities we will restrict ourselves to the countable case.

7.2. Approximating triangles.

Definition 7.4. Let A be a Banach algebra. Consider two sequences in A,A = an : n ∈ N and B = bn : n ∈ N. Then

(1) the sets

T uAB = anbm : n,m ∈ N, n ≤ m and T lAB = anbm : n,m ∈ N,m ≤ n

are called, respectively, the upper and lower triangles defined by Aand B.

(2) A set X ⊆ A is said to approximate segments in T uAB, if it can beenumerated as

X = xnm : n, m ∈ N, n ≤ m ,

and for each n ∈ N

limm‖xnm − anbm‖ = 0.

(3) A set X ⊆ A is said to approximate segments in T lAB, if it can beenumerated as

X = xnm : n, m ∈ N, m ≤ n ,

and for each m ∈ N,

limn‖xnm − anbm‖ = 0.

(4) a double indexed subset X = xnm : n, m ∈ N is vertically injectiveif the identity xnm = xn′m′ implies m = m′. If xnm = xn′m′ impliesn = n′ we say that X is horizontally injective.

Definition 7.5. Let E be a normed space. A bounded sequence B is an `1-base in E, with constant K > 0, when, for every choice of scalars, z1, . . . , zpand of elements a1, . . . , ap ∈ B, the following inequality holds.∥∥ p∑

n=1

znan∥∥ ≥ K p∑

n=1

|zn|.


Theorem 7.6 (Corollary 3.10 of [FGa]). Let A be a Banach algebra. Sup-pose that A contains two bounded sequences A and B and two disjoint setsX1 and X2 with the following properties:

(1) X = X1 ∪X2 is an `1-base in A.(2) X1 and X2 approximate segments in T uAB and T lAB, respectively.(3) X1 is vertically injective and X2 is horizontally injective.

Then there is a bounded linear map of A∗/WAP(A) onto `∞. If, in addition,A is separable, then A is ENAR.

Theorem 7.7. Let A be a commutative weakly sequentially complete Banachalgebra that is an ideal in A∗∗. If A contains a sequential MBAI (en)n andthere are p, q ∈ A∗∗ such that (en p q)n does not converge weakly, thenA∗/WAP(A) is not separable. If A is in addition separable, that A is ENAR.

Proof. We start by observing that the sequence (en p q)n cannot haveweakly Cauchy subsequences. If (en(k) p q)k was such, weak sequentialcompleteness of A would produce a ∈ A such that, in the σ(A,A∗)-topology,limk en(k) p q = a. Now, for any subnet (en(β) p q)β of the sequence(en p q)n, we have that, in the σ(A,A∗)-topology, which on A coincideswith the σ(A∗∗,A∗),

limβen(β) p q = lim

βlimken(k)

(en(β) p q

)(multiplication by en(β) is weak∗-continuous)

= limβen(β)

(limken(k) p q

)= lim

βen(β)a = a,

showing that a is the only accumulation point of (en p q) and, hencethat the sequence (en p q) is convergent. Since this goes against ourassumption, we can invoke Rosenthal’s theorem to deduce that there is asubsequence of (en p q)n that is an `1-base. We denote this `1-base againas (en p q)n.

Now put

A = e2n p : n ∈ N , and

B = e2n+1 q : n ∈ N

and define, for each m,n ∈ N, xnm = e2m+1 (p q), if m < n and xnm =e2n (p q), if n < m. If we let

X1 = xnm : m,n ∈ N, n < m and

X2 = xnm : m,n ∈ N,m < n.

Then we have

limm

∥∥xnm − (e2n p)(e2m+1 q)∥∥ = lim

m

∥∥e2n (p q)− (e2n p)(e2m+1 q)∥∥

= limm

∥∥e2n (p q)− e2m+1 (e2n (p q))∥∥ = 0.

for each n ∈ N. A symmetric computation yields, for each m ∈ N,

limn

∥∥xnm − (e2n p)(e2m+1 q)∥∥ = lim

n

∥∥e2m+1 (p q)− (e2n p)(e2m+1 q)∥∥ = 0.


The sets X1, X2 therefore approximate the segments in T uAB and T lAB, re-spectively.

Since (en (p q)) is an `1-base, we have that en (p q) 6= en′ (p q)when n 6= n′ and, hence, X1 is vertically injective and X2 is horizontallyinjective.

Theorem 7.6 can then be applied to deduce that A is ENAR.

Corollary 7.8. Let G be a metrizable compact Abelian group. If E ⊂ G isnot a small-1-1 set, then L1

E(G) is ENAR.

Proof. The algebra L1E(G) is commutative, weakly sequentially complete

and has a sequential MBAI (en), as shown in Lemma 3.1. By (3) of Lemma3.1, en C∗∗µ (e), µ ∈ME(G), can only converge (weakly) to C∗∗µ (e). By weaksequential completeness, this implies that the sequence (en C∗∗µ1

(e) C∗∗µ2(e))n

cannot be weakly convergent unless µ1 ∗ µ2 ∈ L1E(G).

7.3. Complements of Lust-Piquard sets. Certain thin subsets of G havea complement E that may not be very large, but is large enough to ensurethat L1

E(G) is not Arens regular. Complements of Lust-Piquard sets can beregarded as such.

Lemma 7.9. Let G be a compact metrizable Abelian group. If G \ E is aLust-Piquard set, then for each measure µ ∈ ME(G) \ L1(G), there exists

φ ∈ L∞(G) such that i∗(µ ∗ φ) /∈ i∗(C(G)).

Proof. Notice first that, by [Li93, Proposition 2], there exists φ ∈ L∞(G)such that µ ∗ φ is not totally ergodic. Fix such a φ ∈ L∞(G). Towards acontradiction, assume that there is a sequence (ψn) in C(G) such that

limni∗(ψn) = i∗(µ ∗ φ).

Then, since the restriction mapping i∗ is a quotient map, one can find (see,e.g., [Meg98, Theorem 1.7.7]) a sequence (ξn) in L∞(G) and ξ ∈ L∞(G)such that, for each n ∈ N, i∗(ψn) = i∗(ξn), and limn ξn = ξ. The equality

i∗(ψn) = i∗(ξn) entails that ψn− ξn ∈ L∞−G\E(G), so that (−G \E is a Lust-

Piquard set) ψn−ξn is totally ergodic. Since ψn is continuous, hence totallyergodic, we deduce that ξn is totally ergodic for each n ∈ N. Thus, ξ is

totally ergodic. Indeed, for γ ∈ G, limn ξn(−γ) = ξ(γ), so for each invariant

mean M , we have, using that, by total ergodicity, 〈M,γξn〉 = ξn(−γ),

〈M,γξ〉 = limn〈M,γξn〉 = lim

nξn(−γ) = ξ(−γ),

so ξ is totally ergodic. On the other hand, i∗(µ ∗ φ) = i∗(ξ) and, hence,µ ∗φ− ξ ∈ L∞

−G\E(G). So µ ∗φ− ξ is totally ergodic, and, therefore, µ ∗φ is

totally ergodic, a contradiction. We conclude that i∗(µ∗φ) /∈ i∗(C(G)

).

With the aid of Lemma 7.9, the elements of Z(L1E(G)∗∗) can be substan-

tially cornered when G \ E is a Lust-Piquard set.

Proposition 7.10. Let G be a compact metrizable Abelian group and let

E ⊂ G such that G \ E is a Lust-Piquard set. Fix a right identity e ∈


L1(G)∗∗. Then

Z(L1E(G)∗∗) ⊆

C∗∗u (e) + r : u ∈ L1

E(G), r ∈ i∗(C(G))⊥

Proof. Let p = C∗∗µ (e) + r ∈ L1E(G)∗∗ be an arbitrary element of L1

E(G)∗∗

with µ ∈ME(G) and r ∈ i∗(C(G))⊥.Assume that µ ∈ ME(G) \ L1

E(G). By Lemma 7.9, there exists φ ∈L∞(G) such that µ ∗φ /∈ i∗(C(G)). So there exists s ∈ i∗(C(G))

⊥such that

〈s, i∗(µ ∗ φ)〉 6= 0. As in (5.1),

0 6= 〈s, i∗(µ ∗ φ)〉 = 〈sC∗∗µ (e), φ〉 = 〈s p, φ〉,

showing that p /∈ Z(L1E(G)∗∗) because p s = 0.

The preceding Proposition 7.10 yields the following result.

Corollary 7.11. Let G be a compact metrizable Abelian group. If E ⊆ G

is such that G \ E is a Lust-Piquard set, then L1E(G) is not Arens regular.

8. Final Remarks

At some points we have hinted at a relation between Arens regularityproperties of L1

E(G) and the size or thinness of E. The actual cause behindthese properties is the presence (or absence) of algebraic properties in E.Thinness can then be regarded as the trigger for the absence of algebraicrelations.

That is why it is easy to construct the next examples.

Example 1. L1E(G) and L1

G\E(G) can be both SAI and both regular.

If E ∈ ΩG

, then G \ E ∈ ΩG

, then L1E(G) and L1

G\E(G) are both SAI by

Corollary 7.1.If on the other hand we consider E = N ⊆ Z the classical case of a Riesz

set, then G \ E = −N is also a Riesz set so that L1E(G) and L

G\E are both

Arens regular by Corollary 6.1.

Corollary 7.1 can be read as saying that L1E(G) is SAI if 1E ∈ B(G).

One could be tempted to conjecture that 1E ∈ B(G)‖·‖∞

could be anothersufficient condition for strong Arens irregularity. That is clearly not thecase.

Example 2. For every compact Abelian group G, there is E ⊂ G such that

1E is the uniform limit of a sequence of functions in B(G) but L1E(G) is

Arens regular.

It is enough to consider an infinite Sidon subset E ⊂ G. Then 1E ∈

B(G)‖·‖∞

, see [GH13, Corollary 6.3.2]. Since Sidon sets are Riesz, L1E(G) is

Arens regular,The obvious questions our work leaves unanswered are

Question 8.1. Let G be a compact Abelian group. Are there sets E ⊆ Gthat are not Riesz and, yet, L1

E(G) is Arens regular?


Should all small-2 sets turn to be Riesz sets, then the answer to theprevious question would be necessarily be in the negative.

Question 8.2. Let G be a compact Abelian group. Are there subsets E ⊆ Gwith E /∈ Ω

Gsuch that L1

E(G) is SAI?

References

[Are51a] Richard Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2(1951), 839–848. MR 0045941

[Are51b] , Operations induced in function classes, Monatsh. Math. 55 (1951), 1–19.MR 0044109

[BF11] Ahmed Bouziad and Mahmoud Filali, On the size of quotients of function spaceson a topological group, Studia Math. 202 (2011), no. 3, 243–259. MR 2771653(2012c:43004)

[DL05] H. G. Dales and A. T.-M. Lau, The second duals of Beurling algebras, Mem. Amer.Math. Soc. 177 (2005), no. 836, vi+191. MR 2155972 (2006k:43002)

[FGa] M. Filali and J. Galindo, On the extreme non-arens regularity of banach algebras,J. London Math. Soc, to appear. https://doi.org/10.1112/jlms.12485.

[FGb] , Orthogonal `1-sets and extreme non-arens regularity of preduals of vonneumann algebras, Preprint available at: https://arxiv.org/abs/2006.00851.

[FG18] Mahmoud Filali and Jorge Galindo, Extreme non-Arens regularity of the groupalgebra, Forum Math. 30 (2018), no. 5, 1193–1208. MR 3849641

[GH13] Colin C. Graham and Kathryn E. Hare, Interpolation and Sidon sets for com-pact groups, CMS Books in Mathematics/Ouvrages de Mathematiques de la SMC,Springer, New York, 2013. MR 3025283

[Gra96] Edmond E. Granirer, Day points for quotients of the Fourier algebra A(G), ex-treme nonergodicity of their duals and extreme non-Arens regularity, Illinois J.Math. 40 (1996), no. 3, 402–419. MR 1407625 (98c:43005)

[Har88] Kathryn E. Hare, An elementary proof of a result on Λ(p) sets, Proc. Amer. Math.Soc. 104 (1988), no. 3, 829–834. MR 964865

[HR70] Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Struc-ture and analysis for compact groups. Analysis on locally compact Abelian groups,Springer-Verlag, New York, 1970. MR 41 #7378

[Hu97] Zhiguo Hu, Extreme non-Arens regularity of quotients of the Fourier algebra A(G),Colloq. Math. 72 (1997), no. 2, 237–249. MR 1426699

[IkPU87] Nilgun Isı k, John Pym, and Ali Ulger, The second dual of the group algebra ofa compact group, J. London Math. Soc. (2) 35 (1987), no. 1, 135–148. MR 871771

[Kan09] Eberhard Kaniuth, A course in commutative Banach algebras, Graduate Textsin Mathematics, vol. 246, Springer, New York, 2009. MR 2458901

[LL88] Anthony To Ming Lau and Viktor Losert, On the second conjugate algebra ofL1(G) of a locally compact group, J. London Math. Soc. (2) 37 (1988), no. 3, 464–470. MR MR939122 (89e:43007)

[Li93] Daniel Li, A class of Riesz sets, Proc. Amer. Math. Soc. 119 (1993), no. 3, 889–892.MR 1176071

[LRP06] Pascal Lefevre and Luis Rodrıguez-Piazza, The union of a Riesz set and a Lust-Piquard set is a Riesz set, J. Funct. Anal. 233 (2006), no. 2, 545–560. MR 2214587

[LP89] Francoise Lust-Piquard, Bohr local properties of CΛ(T ), Colloq. Math. 58 (1989),no. 1, 29–38. MR 1028158

[Meg98] Robert E. Megginson, An introduction to Banach space theory, Graduate Textsin Mathematics, vol. 183, Springer-Verlag, New York, 1998. MR 1650235

[Pym65] John S. Pym, The convolution of functionals on spaces of bounded functions,Proc. London Math. Soc. (3) 15 (1965), 84–104. MR 0173152 (30 #3367)

[Rud60] Walter Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203–227.MR 0116177

[Rud72] , Invariant means on L∞, Studia Math. 44 (1972), 219–227. MR 304975


[Rud90] Walter Rudin, Fourier analysis on groups, John Wiley & Sons Inc., New York,1990, Reprint of the 1962 original, A Wiley-Interscience Publication. MR 91b:43002

[U99] A. Ulger, Central elements of A∗∗ for certain Banach algebras A without boundedapproximate identities, Glasg. Math. J. 41 (1999), no. 3, 369–377. MR 1720442

[U11] , Characterizations of Riesz sets, Math. Scand. 108 (2011), no. 2, 264–278.MR 2805606

[Zha02] Yong Zhang, Approximate identities for ideals of Segal algebras on a compactgroup, J. Funct. Anal. 191 (2002), no. 1, 123–131. MR 1909265

arens regularity of ideals of the group algebra reza

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