Arens regularity of ideals of the group algebra of a compact Abelian group

Proceedings of the Royal Society of Edinburgh, page 1 of 17 DOI:10.1017/prm.2023.110 Arens regularity of ideals of the group algebra of a compact Abelian group Reza Esmailvandi Instituto Universitario de Matem´aticas y Aplicaciones (IMAC), Universidad Jaume I, E-12071 Castell´on, Spain (esmailv[email protected]) Mahmoud Filali Department of Mathematical Sciences, University of Oulu, Oulu, Finland (mfi[email protected].fi) Jorge Galindo Instituto Universitario de Matem´aticas y Aplicaciones (IMAC), Universidad Jaume I, E-12071 Castell´on, Spain ([email protected]) (Received 31 January 2023; accepted 18 September 2023) Let Gbe a compact Abelian group and Ea subset of the group  Gof continuous characters of G. We study Arens regularity-related properties of the ideals L1 E(G)of L1(G) that are made of functions whose Fourier transform is supported on E⊆ G. Arens regularity of L1 E(G), the centre of L1 E(G)∗∗ and the size of L1 E(G)∗/WAP(L1 E(G)) are studied. We establish general conditions for the regularity of L1 E(G) and deduce from them that L1 E(G) is not strongly Arens irregular if Eis a small-2 set (i.e. μ∗μ∈L1(G) for every μ∈M1 E(G)), which is not a Λ(1)-set, and it is extremely non-Arens regular if Eis not a small-2 set. We deduce also that L1 E(G) is not Arens regular when  G\Eis a Lust-Piquard set. Keywords: Arens product; Arens-regular algebra; centre; extremely non-Arens regular; Lust-Piquard set; Riesz set; strongly Arens irregular; small-2 set; Sidon set 2020 Mathematics Subject Classification: 22D15; 43A46; 43A60 1. Introduction It has long been known, since the work of Arens [1] in the fifties, that the bidual A∗∗ of a Banach algebra Acan be turned into a Banach algebra containing Aas a subalgebra. Two different multiplications can actually be introduced on A∗∗ to this effect. But, while both these multiplications are defined following completely symmetric and absolutely natural rules, they can be essentially different. The left multiplication operator defined by one of them is always weak∗-continuous but may ©The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh. This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives licence (https://creativecommons.org/licenses/ by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is unaltered and is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use or in order to create a derivative work. 1 https://doi.org/10.1017/prm.2023.110 Published online by Cambridge University Press 2R. Esmailvandi, M. Filali and J. Galindo fail to be so for the other, with the situation reversed for the right multiplication operator. The subset of A∗∗ made of those elements that produce weak∗-continuous multiplication operators from both sides is usually referred to as the topological centre of A∗∗, in symbols Z(A∗∗) and it always contains A. When the centre is as large as possible, i.e. when A∗∗ =Z(A∗∗), we say that Ais Arens regular, this is the case, for instance, of C∗-algebras. Following Dales and Lau [4], we say that Ais strongly Arens irregular (SAI for short) when Z(A∗∗) is as small as possible, i.e. when Z(A∗∗)=A. This is the case for the group algebra L1(G) discussed below. Facing the problem from a different point of view, Pym [22] considered the space WAP(A)ofweakly almost periodic functionals on A. This is the precise subspace of A∗on which the two Arens-multiplications agree. So, Ais Arens regular precisely when A∗=WAP(A), i.e. when the quotient A∗/WAP(A) is trivial. When the quotient A∗/WAP(A) contains a closed subspace isomorphic to A∗, and so it is as large as possible, we say that Ais extremely non-Arens regular (ENAR for short). Extreme non-Arens regularity was first studied in the context of Fourier algebras with a slightly different definition, see the papers by Granirer [11] and Hu [14]. I¸sik et al. [15] proved that the group algebra L1(G) of a compact group is always SAI. Shortly afterwards, Lau and Losert [17] proved the same fact for every locally compact group. Bouziad and Filali [3] proved that L1(G) is ENAR for locally compact groups whose compact covering number is not smaller than their local character (i.e. when G, topologically speaking, looks more discrete than compact) and compact metrizable groups. The group algebra L1(G)wasshowntobeENAR for every infinite locally compact group in [8]. In this paper, we work with ideals of L1(G) with Ga compact Abelian group. To describe these ideals, it is necessary to resort to duality. We denote by  Gthe group of all continuous homomorphisms into the multiplicative group of unimodular complex numbers, known as continuous characters. For μ∈M(G), the Fourier–Stieltjes transform of μis the bounded function μ: G→Cgiven by μ(γ)=G −x, γdμ(x). In terms of the duality between M(G)andC(G), for every γ∈ G, μ(γ)=ˇμ, γ, where for a measure μ∈M(G), we denote by ˇμthe measure in M(G) defined by ˇμ, φ=μ, ˇ φ=G φ(−x)dμ(x)(φ∈C(G)). If f∈L1(G) this definition produces the function ˇ f(x)=f(−x)(x∈G). https://doi.org/10.1017/prm.2023.110 Published online by Cambridge University Press Arens regularity of ideals of the group algebra 3 If Xis a linear subspace of M(G)andE⊂ G, we denote by XEthe subspace of X, XE={μ∈X:μ(γ)=0forγ∈ G\E}. Most prominent in our work will be the ideal ME(G)ofM(G) and its subspace the ideal L1 E(G)ofL1(G). 1.1. Summary of results In this paper, we address the Arens regularity properties of the ideals of L1(G) when Gis a compact Abelian group. These ideals are always of the form L1 E(G)for some subset Eof  G, see e.g. [13, Theorem 38.7]. We relate the Arens regularity of L1 E(G) with the size of the subspace of L1 E(G)∗made of restrictions to L1 E(G)of convolutions of the form ˇμ∗φwith μ∈ME(G)andφ∈L∞(G). As mentioned earlier, it is known that L1 E(G) is SAI and ENAR when E= G. On the contrary, if Eis finite, L1 E(G) has finite dimension and so is reflexive, and is thus Arens regular. One may therefore expect that the regularity properties of L1 E(G) improve as Edecreases in size. This is evidenced by the result of ¨ Ulger [26], the paper that inspired this work: if Eis a Riesz set, i.e. all measures on Gwith Fourier–Stieltjes transforms supported in Eare absolutely continuous, then L1 E(G) is Arens regular. As an example, L1 N(T) is Arens regular, Nbeing a Riesz subset of Zby the F. and M. Riesz theorem. The absolute continuity (with respect to Haar measure) of measures in ME(G)∗ ME(G) turns out to be important in this discussion. When ME(G)∗ME(G)⊆ L1 E(G) (such a set is said to be small-2), L1 E(G)∗∗ has a large centre and so L1 E(G) cannot be SAI, unless it is reflexive, see corollary 5.6. We do not know whether L1 E(G) can be Arens regular when Eis not Riesz (the main question in [26]). But, we are able to prove that regularity of L1 E(G) forces Eto be small-2 (see corollary 5.2). Another type of sets giving non-Arens regularity is provided by complements of Lust-Piquard sets (see below for the definition). For example, L1 Z\E(T) is not Arens regular when E⊆Zis the Lust-Piquard set consisting of the primes in the coset 5Z+ 2, see [20, Theorem 4]. Examples of L1 E(G) being SAI are provided by sets Ein the coset ring of  G.In particular all maximal ideals of L1(G)happentobeSAI. 2. Arens regularity In this section, we provide formal definitions for the concepts related to Arens regularity discussed in this paper. Let Abe a commutative Banach algebra and let A∗and A∗∗ be its first and second Banach duals, respectively. The multiplication of Acan be extended naturally to A∗∗ in two different ways. These multiplications arise as particular cases of the abstract approach of Arens [1,2] and can be formalized through the following three steps. For u, v in A,ϕin A∗and m, n ∈A∗∗,we define φ·u, u ·φ,m·φ, φ ·m∈A∗ https://doi.org/10.1017/prm.2023.110 Published online by Cambridge University Press 4R. Esmailvandi, M. Filali and J. Galindo and mn, 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 mn, φ=m,n·φ,m♦n, φ=n, φ ·m. When and ♦coincide on A∗∗,Ais said to be Arens regular. For any m∈A∗∗ the mapping n→ nmis weak∗–weak∗continuous on A∗∗. However, the mapping n→ mnneed not to be weak∗–weak∗continuous. The situation is reversed for ♦. The left topological centre of A∗∗ is then defined as Z(A∗∗)={m∈A∗∗ :n→ mnis weak∗--weak∗continuous on A∗∗}. Since we are assuming that Ais commutative, it is easy to see that Z(A∗∗)={m∈A∗∗ :mn=nm=m♦nfor all n∈A∗∗}. The algebra Ais therefore Arens regular if and only if Z(A∗∗)=A∗∗. Observe that Ais always contained in Z(A∗∗). Sometimes, the elements of the centre stop here. Definition 2.1. A commutative Banach algebra Ais strongly Arens irregular (SAI for short) when Z(A∗∗)=A. In [22], Pym considered the space WAP(A)ofweakly almost periodic functionals on A, this is the set of all ϕ∈A∗such that the linear map A→A∗:a→ a·ϕ is weakly compact. The functionals ϕ∈WAP(A) satisfy Grothendieck’s double limit criterion lim nlim mϕ, anbm= lim mlim nϕ, anbm for any pair of bounded sequences (an)n,(bm)min Afor which both the iterated limits exist. From this property, one may deduce that mn, φ=m♦n, φfor every m, n ∈A∗∗ if and only if φ∈WAP(A). So, Ais Arens 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 Ais extremely non-Arens regular (ENAR for short) when A∗/WAP(A)contains a closed subspace isomorphic to A∗. The term extreme non-Arens regularity was coined by Granirer [11] to characterize a slightly more general behaviour: Ais ENAR if the quotient space A∗/WAP(A) contains a closed linear subspace which has A∗as a quotient, i.e. as a continuous linear image. We have adopted here this simpler definition that is still enough to capture the extreme behaviour of many of the Banach algebras that harmonic analysis https://doi.org/10.1017/prm.2023.110 Published online by Cambridge University Press Arens regularity of ideals of the group algebra 5 associates with a locally compact group. Clearly, ENAR in the sense of definition 2.2 implies ENAR in the sense of Granirer, but we do not know whether the two definitions are actually the same. See [5–7]. 3. The structure of L1(G)∗∗ and L1 E(G)∗∗ We summarize here the structure of L1 E(G)∗∗ where Gis a compact Abelian group and E⊆ G. Notation will be additive and the identities of both Gand  Gwill be denoted by 0. All the facts mentioned here are well-known when E= G, see e.g. [15]. No new insight is needed for them to hold for arbitrary E⊆ Gbut having them stated beforehand will simplify our proofs. A good deal of the structure of L1(G)∗∗ is determined by the presence of right identities. These can be obtained as accumulation points in L1(G)∗∗ of bounded approximate identities of L1(G), which are always available (see e.g. [16,§1.3]). The first use of right identities is to bring measures on Ginto elements of L1(G)∗∗. For each μ∈ME(G), one considers the convolution operator: Cμ:L1(G)→L1 E(G) given by Cμ(u)=μ∗u, u ∈L1(G). Its double adjoint C∗∗ μthen maps L1(G)∗∗ into L1 E(G)∗∗. When necessary we will use i:L1 E(G)→L1(G) to denote the inclusion map, then i∗:L∞(G)→L1 E(G)∗ will be the restriction map and i∗∗ :L1 E(G)∗∗ →L1(G)∗∗ will be an embedding of Banach algebras. We will normally omit mentioning iand i∗∗ and see L1 E(G)∗∗ as an ideal of L1(G)∗∗. With these notations, if μ∈ME(G)andφ∈L∞(G) are given, a straightforward computation shows, that if eis a right identity in L1(G)∗∗, then C∗∗ μ(e)·i∗(φ)=i∗(ˇμ∗φ).(3.1) The lifting map Je:ME(G)→L1 E(G)∗∗ given by Je(μ)=C∗∗ μ(e), turns out to be an algebra isomorphism onto ei∗∗(L1 E(G)∗∗). The algebra L1 E(G) can be seen both as an ideal in ME(G) and as an ideal in L1 E(G)∗∗ and, in that sense, it is left invariant by Je, i.e. Je(f)=C∗∗ f(e)=ffor all f∈L1 E(G).(3.2) The canonical quotient map RE:L1 E(G)∗∗ →ME(G), defined, for each m∈ L1 E(G)∗∗,byRE(m)=mC(G)is then a left inverse for Jeand the composition Je◦REis a projection. Regardless of the right identity e,kerREcan always be identified with i∗(C(G))⊥, the annihilator of the subspace i∗(C(G)) in L1 E(G)∗∗. The projection Je◦REtherefore induces the decomposition L1 E(G)∗∗ =Je(ME(G)) ⊕i∗(C(G))⊥.(3.3) So, for a given right identity eof L1 E(G)∗∗, an element m∈L1 E(G)∗∗,maybe uniquely decomposed as m=C∗∗ μ(e)+r, (3.4) where μ∈ME(G)andr∈i∗(C(G))⊥. https://doi.org/10.1017/prm.2023.110 Published online by Cambridge University Press 6R. Esmailvandi, M. Filali and J. Galindo The above decomposition becomes handier if one observes that the elements of i∗(C(G))⊥are left annihilators of L1 E(G)∗∗. Indeed, for r∈i∗(C(G))⊥,ifm∈ L1 E(G)∗∗ is such that m=σL1 E(G)∗∗,L 1 E(G)∗−limαuα, with uα∈L1 E(G), and φ∈L∞(G), then mr, i∗(φ)=i∗∗ (mr),φ = lim αuα,i ∗∗(r)·φ = lim αr, i∗(ˇuα∗φ)=0,(3.5) where the last identity follows from ˇuα∗φ∈C(G). Next, as we see, left annihilators can actually be used to characterize Arens regularity. We first need a definition. Definition 3.1. Let Gbe a compact Abelian group, E⊆ Gand put S=i∗(C(G))⊥L1 E(G)∗∗. Note that for any fixed right identity e∈L1(G)∗∗, the set Sis given by S=rC∗∗ μ(e): r∈i∗(C(G))⊥and μ∈ME(G). Observe as well that, S∩L1 E(G)={0}. To see this, one can fix a right identity e∈ L1(G)∗∗ and a bounded approximate identity (uα)αin L1(G). Then, if rC∗∗ μ(e)∈ S∩L1 E(G), we have that rC∗∗ μ(e) = lim αuα∗rC∗∗ μ(e)=0. Theorem 3.2. Let Gbe a compact Abelian group and let E⊆ G.Then,L1 E(G)is Arens regular if and only if S={0}. Proof. Since S={0}immediately implies that L1 E(G)∗∗ is not commutative, by the preceding paragraph, we only need to show that S={0}implies that L1 E(G) is Arens regular. Assume S={0}and fix a right identity ein L1 E(G)∗∗ and let C∗∗ μ1(e)+s1and C∗∗ μ2(e)+s2be arbitrary in L1 E(G)∗∗, with s1,s 2∈i∗(C(G))⊥and μ1,μ 2∈ME(G). Then, C∗∗ μ1(e)+s1C∗∗ μ2(e)+s2=C∗∗ μ1∗μ2(e) =C∗∗ μ2∗μ1(e) =C∗∗ μ2(e)+s2C∗∗ μ1(e)+s1, and so L1 E(G)∗∗ is commutative, i.e. L1 E(G) is Arens regular.  We wish to record the following lemma, a restatement of Theorem 3.3(v) of [15], for later use. https://doi.org/10.1017/prm.2023.110 Published online by Cambridge University Press Arens regularity of ideals of the group algebra 7 Lemma 3.3. Let Gbe a compact Abelian group. Consider E⊆ Gand μ∈ME(G). If for every pair eand fof 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 ˇμ∗φis not continuous, see [13, Theorem 35.13]. By Lemma 2.3 of [15]we can find two different right identities f1,f 2∈L1(G)∗∗ such that f1,ˇμ∗φ>= f2,ˇμ∗φ. Since C∗∗ μ(fi),φ=fi,ˇμ∗φ,i=1,2, we deduce that C∗∗ μ(f1)=C∗∗ μ(f2), a contradiction with our hypotheses.  4. Special subsets of  G We describe here the sets E⊆ Gthat lead to the concrete ideals L1 E(G) that will appear later in the paper. We first recall that an invariant mean Mon L∞(G) is a linear functional on L∞(G) such that M,1=M= 1 and, for each φ∈L∞(G) and each x∈G, M,Lxφ=M,φwhere Lxis the translation operator by x. An invariant mean that is always available is the one produced by Haar measure: φ→ φ(x)dx.IfG is compact, L∞(G) always has other invariant means [23] but all them have the same effect on some functions. We say then that a function φ∈L∞(G)hasaunique invariant mean if M,φ=φ(x)dxfor every invariant mean Mon L∞(G). Definition 4.1. Let Gbe a compact Abelian group and E⊂ G. We say that E is a (i) Sidon set, if every f∈CE(G)has an absolutely convergent Fourier series. (ii) Λ(p)-set, p>0, if there are 0<q<pand C>0such that fp⩽Cfq,for every trigonometric polynomial, f=n k=1 ckχk,withχ1,...,χ n∈E. (iii) Rosenthal set, if L∞ E(G)=CE(G). (iv) 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 φis totally ergodic. (v) Riesz set, if ME(G)=L1 E(G). (vi) Small-2 set, if μ∗μ∈L1 E(G)for every μ∈ME(G). As pointed out to us by the referee, with the identity 2μ∗ν=(μ+ν)2−μ2−ν2, one quickly checks that μ∗ν∈L1 E(G) for every μ, ν ∈ME(G) if and only if μ∗μ∈ L1 E(G) for every μ∈ME(G) (i.e. if E⊆ Gis a small-2 set). Sidon sets are Rosenthal, see, e.g. [9, Corollary 6.2.5], and Rosenthal sets are Lust-Piquard (as continuous functions always have a unique invariant mean). LustPiquard sets are in turn always Riesz (see [18]) and Riesz sets are, obviously, small-2. https://doi.org/10.1017/prm.2023.110 Published online by Cambridge University Press 8R. Esmailvandi, M. Filali and J. Galindo Figure 1. Relations between properties of E⊂ G,Gcompact and Abelian. On the contrary, Sidon sets are Λ(p) for every p>0andΛ(p)setsareΛ(q)for every q<p[13, Section 37]. It is a result of Hare [12]thataΛ(p) is always a Λ(q) set for some q>p. The following is a consequence that is important in our context. Theorem 4.2 (Corollary in [12]). Let Gbe a compact Abelian group and let E⊂ G. The Banach space L1 E(G)is reflexive if and only if Eis a Λ(1) set. It follows from this Corollary that Λ(1)-sets are necessarily Riesz. For, if μ∈ ME(G)\L1 E(G) then Je(μ)∈eL1 E(G)∗∗ \L1 E(G), since Jeis an isomorphism that fixes L1 E(G). The preceding remarks are summarized in figure 1. To the authors’ knowledge, it is still unknown whether small-2 sets are Riesz. As already mentioned by ¨ Ulger in [26, p. 273], this is a long-standing open problem that goes back to Glicksberg [10]. It might therefore happen that the classes defined in items (v)–(vi) above are actually the same. Since L1 E(G) is Arens regular when EisRiesz(see[26] or corollary 6.3) we will not be interested in L1 E(G)forEin any class contained in that Riesz sets. However, sets Ewhose complement  G\Ebelongs to such a class will be of interest in §7.2, especially after one learns that the union of a Riesz set and Lust-Piquard set is Riesz [19], and hence that complements of Lust-Piquard sets are never Riesz. We turn now our attention to small-2 sets. 5. Small-2 sets We start with the following result of ¨ Ulger which reveals the relevance of non-small-2 sets in the analysis of Arens regularity. Theorem 5.1 (Theorem 2.2 of [25]). Let Abe a commutative, semisimple, weakly sequentially complete and completely continuous Banach algebra, then an element m∈A∗∗ is in the centre of Aif and only if mA∗∗ ⊆Aand A∗∗ m⊆A. Corollary 5.2. Let Gbe a compact Abelian group and assume that E⊆ Gis not a small-2 set. For every pair μ1,μ 2∈ME(G)such that μ1∗μ2/∈L1 E(G)and every right identity eof L1(G)∗∗, we have that neither C∗∗ μ1(e)nor C∗∗ μ2(e)is in Z(L1 E(G)∗∗). https://doi.org/10.1017/prm.2023.110 Published online by Cambridge University Press Arens regularity of ideals of the group algebra 9 Proof. Let μ1,μ 2∈M(G) such that μ1∗μ2/∈L1 E(G). Towards a contradiction, assume that C∗∗ μ1(e)∈Z(L1 E(G)∗∗). By theorem 5.1: C∗∗ μ1∗μ2(e)=C∗∗ μ1(e)C∗∗ μ2(e)∈L1 E(G). Since REisaleftinverseofJe, this is a contradiction.  Remark 5.3. With corollary 5.2, the last trivial implication Eis Riesz =⇒Eis small-2 in figure 1 may now be split into two non-trivial implications: Eis Riesz =⇒L1 E(G) is Arens regular =⇒Eis small-2. We shall further see in §7that L1 E(G) is even ENAR when Eis not a small-2 set. We proceed now to find non-trivial elements in the centre of L1 E(G)∗∗, when Eis a small-2 set. Recall that the set Swas defined in §3as S=i∗(C(G))⊥L1 E(G)∗∗. Theorem 5.4. Let Gbe a compact Abelian group and let E⊆ Gbe a small 2-set. Then, S⊆Z(L1 E(G)∗∗). Proof. We first fix 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∈ L1 E(G)∗∗, with s∈i∗(C(G))⊥and σ∈ME(G), then qp=0,asris a left annihilator, (3.5). Since sis also left annihilator and C∗∗ μ(e)C∗∗ σ(e)∈Z(L1 E(G)∗∗), for μ∗σ∈L1 E(G) since Eis a small-2 set, one gets: pq=rC∗∗ μ(e)C∗∗ σ(e)=rC∗∗ μ∗σ(e)=0. Hence, p∈Z(L1 E(G)∗∗), as needed.  We choose to express the main consequence of this theorem in two equivalent ways. Corollary 5.5. Let Gbe a compact Abelian group and let E⊆ Gbe a small-2 set. Then, L1 E(G)is SAI if and only if it is reflexive. Corollary 5.6. Let Gbe a compact Abelian group and let E⊆ G.IfEis a small 2-set that is not Λ(1), then L1 E(G)is not SAI. Proof. Suppose that Eis small-2 set with L1 E(G) SAI. Since S∩L1 E(G)={0} (see the remarks after definition 3.1), theorem 5.4 implies that Smust be trivial. Theorem 3.2 implies then that L1 E(G) is Arens regular, and so it must be reflexive, i.e. Emust be Λ(1) (theorem 4.2).  https://doi.org/10.1017/prm.2023.110 Published online by Cambridge University Press 16 R. Esmailvandi, M. Filali and J. Galindo As in (6.1), 0=s, i∗(ˇμ∗φ)>=sC∗∗ μ(e),φ>=sp, φ>, showing that p/∈Z(L1 E(G)∗∗) because ps=0.  The preceding proposition 7.9 yields the following result. Corollary 7.10. Let Gbe a compact metrizable Abelian group. If E⊆ Gis such that  G\Eis a Lust-Piquard set, then L1 E(G)is not Arens regular. We close the paper observing that, contrarily to what the previous corollary might suggest, the regularity properties of L1 E(G) do not determine those of L1  G\E(G). Example 7.11. L1 E(G)andL1  G\E(G) can be both SAI and regular. If E∈Ω G, then  G\E∈Ω G, then both L1 E(G)andL1  G\E(G) are SAI by corollary 7.1. If on the other hand we consider E=N⊆Zthe classical case of a Riesz set, then  G\E=−Nis also a Riesz set so that L1 E(G)andL1  G\E(G) are both Arens regular by corollary 6.3. Remark 7.12. In our forthcoming paper, we shall deal with more general Banach algebras of the same type dealt with in this paper. Our study will include the group algebra of a non-Abelian compact group and the Fourier algebra of an amenable discrete group. Acknowledgements We wish to thank the referee for the very careful reading of the paper, corrections and constructive recommendations and suggestions that have made the presentation of the paper much clearer and more compelling. The second author wishes to acknowledge the Department of Mathematics at the University of Jaume I in Castell´on. 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Approximate identities for ideals of Segal algebras on a compact group. J. Funct. Anal.191 (2002), 123–131. https://doi.org/10.1017/prm.2023.110 Published online by Cambridge University Press