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

P oceedings o he Royal Socie y o Edinbu gh, page 1 o 17 DOI:10.1017/p m.2023.110 A ens egula i y o ideals o he g oup algeb a o a compac Abelian g oup Reza Esmail andi Ins i u o Uni e si a io de Ma em´a icas y Aplicaciones (IMAC), Uni e sidad Jaume I, E-12071 Cas ell´on, Spain (esmail [email p o ec ed]) Mahmoud Filali Depa men o Ma hema ical Sciences, Uni e si y o Oulu, Oulu, Finland (mfi[email p o ec ed].fi) Jo ge Galindo Ins i u o Uni e si a io de Ma em´a icas y Aplicaciones (IMAC), Uni e sidad Jaume I, E-12071 Cas ell´on, Spain ([email p o ec ed]) (Recei ed 31 Janua y 2023; accep ed 18 Sep embe 2023) Le Gbe a compac Abelian g oup and Ea subse o he g oup  Go con inuous cha ac e s o G. We s udy A ens egula i y- ela ed p ope ies o he ideals L1 E(G)o L1(G) ha a e made o unc ions whose Fou ie ans o m is suppo ed on E⊆ G. A ens egula i y o L1 E(G), he cen e o L1 E(G)∗∗ and he size o L1 E(G)∗/WAP(L1 E(G)) a e s udied. We es ablish gene al condi ions o he egula i y o L1 E(G) and deduce om hem ha L1 E(G) is no s ongly A ens i egula i Eis a small-2 se (i.e. μ∗μ∈L1(G) o e e y μ∈M1 E(G)), which is no a Λ(1)-se , and i is ex emely non-A ens egula i Eis no a small-2 se . We deduce also ha L1 E(G) is no A ens egula when  G Eis a Lus -Piqua d se . Keywo ds: A ens p oduc ; A ens- egula algeb a; cen e; ex emely non-A ens egula ; Lus -Piqua d se ; Riesz se ; s ongly A ens i egula ; small-2 se ; Sidon se 2020 Ma hema ics Subjec Classi ica ion: 22D15; 43A46; 43A60 1. In oduc ion I has long been known, since he wo k o A ens [1] in he fi ies, ha he bidual A∗∗ o a Banach algeb a Acan be u ned in o a Banach algeb a con aining Aas a subalgeb a. Two diffe en mul iplica ions can ac ually be in oduced on A∗∗ o his effec . Bu , while bo h hese mul iplica ions a e defined ollowing comple ely symme ic and absolu ely na u al ules, hey can be essen ially diffe en . The le mul iplica ion ope a o defined by one o hem is always weak∗-con inuous bu may ©The Au ho (s), 2023. Published by Camb idge Uni e si y P ess on behal o The Royal Socie y o Edinbu gh. This is an Open Access a icle, dis ibu ed unde he e ms o he C ea i e Com- mons A ibu ion-NonComme cial-NoDe i a i es licence (h ps://c ea i ecommons.o g/licenses/ by-nc-nd/4.0/), which pe mi s non-comme cial e-use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is unal e ed and is p ope ly ci ed. The w i en pe mis- sion o Camb idge Uni e si y P ess mus be ob ained o comme cial e-use o in o de o c ea e a de i a i e wo k. 1 h ps://doi.o g/10.1017/p m.2023.110 Published online by Camb idge Uni e si y P ess 2R. Esmail andi, M. Filali and J. Galindo ail o be so o he o he , wi h he si ua ion e e sed o he igh mul iplica ion ope a o . The subse o A∗∗ made o hose elemen s ha p oduce weak∗-con inuous mul- iplica ion ope a o s om bo h sides is usually e e ed o as he opological cen e o A∗∗, in symbols Z(A∗∗) and i always con ains A. When he cen e is as la ge as possible, i.e. when A∗∗ =Z(A∗∗), we say ha Ais A ens egula , his is he case, o ins ance, o C∗-algeb as. Following Dales and Lau [4], we say ha Ais s ongly A ens i egula (SAI o sho ) when Z(A∗∗) is as small as possible, i.e. when Z(A∗∗)=A. This is he case o he g oup algeb a L1(G) discussed below. Facing he p oblem om a diffe en poin o iew, Pym [22] conside ed he space WAP(A)o weakly almos pe iodic unc ionals on A. This is he p ecise subspace o A∗on which he wo A ens-mul iplica ions ag ee. So, Ais A ens egula p ecisely when A∗=WAP(A), i.e. when he quo ien A∗/WAP(A) is i ial. When he quo ien A∗/WAP(A) con ains a closed subspace isomo phic o A∗, and so i is as la ge as possible, we say ha Ais ex emely non-A ens egula (ENAR o sho ). Ex eme non-A ens egula i y was fi s s udied in he con ex o Fou ie algeb as wi h a sligh ly diffe en defini ion, see he pape s by G ani e [11] and Hu [14]. I¸sik e al. [15] p o ed ha he g oup algeb a L1(G) o a compac g oup is always SAI. Sho ly a e wa ds, Lau and Lose [17] p o ed he same ac o e e y locally compac g oup. Bouziad and Filali [3] p o ed ha L1(G) is ENAR o locally compac g oups whose compac co e ing numbe is no smalle han hei local cha ac e (i.e. when G, opologically speaking, looks mo e disc e e han compac ) and compac me izable g oups. The g oup algeb a L1(G)wasshown obeENAR o e e y infini e locally compac g oup in [8]. In his pape , we wo k wi h ideals o L1(G) wi h Ga compac Abelian g oup. To desc ibe hese ideals, i is necessa y o eso o duali y. We deno e by  G he g oup o all con inuous homomo phisms in o he mul iplica i e g oup o unimodula com- plex numbe s, known as con inuous cha ac e s. Fo μ∈M(G), he Fou ie –S iel jes ans o m o μis he bounded unc ion μ: G→Cgi en by μ(γ)=G −x, γdμ(x). In e ms o he duali y be ween M(G)andC(G), o e e y γ∈ G, μ(γ)=ˇμ, γ, whe e o a measu e μ∈M(G), we deno e by ˇμ he measu e in M(G) defined by ˇμ, φ=μ, ˇ φ=G φ(−x)dμ(x)(φ∈C(G)). I ∈L1(G) his defini ion p oduces he unc ion ˇ (x)= (−x)(x∈G). h ps://doi.o g/10.1017/p m.2023.110 Published online by Camb idge Uni e si y P ess A ens egula i y o ideals o he g oup algeb a 3 I Xis a linea subspace o M(G)andE⊂ G, we deno e by XE he subspace o X, XE={μ∈X:μ(γ)=0 o γ∈ G E}. Mos p ominen in ou wo k will be he ideal ME(G)o M(G) and i s subspace he ideal L1 E(G)o L1(G). 1.1. Summa y o esul s In his pape , we add ess he A ens egula i y p ope ies o he ideals o L1(G) when Gis a compac Abelian g oup. These ideals a e always o he o m L1 E(G) o some subse Eo  G, see e.g. [13, Theo em 38.7]. We ela e he A ens egula i y o L1 E(G) wi h he size o he subspace o L1 E(G)∗made o es ic ions o L1 E(G)o con olu ions o he o m ˇμ∗φwi h μ∈ME(G)andφ∈L∞(G). As men ioned ea lie , i is known ha L1 E(G) is SAI and ENAR when E= G. On he con a y, i Eis fini e, L1 E(G) has fini e dimension and so is eflexi e, and is hus A ens egula . One may he e o e expec ha he egula i y p ope ies o L1 E(G) imp o e as Edec eases in size. This is e idenced by he esul o ¨ Ulge [26], he pape ha inspi ed his wo k: i Eis a Riesz se , i.e. all measu es on Gwi h Fou ie –S iel jes ans o ms suppo ed in Ea e absolu ely con inuous, hen L1 E(G) is A ens egula . As an example, L1 N(T) is A ens egula , Nbeing a Riesz subse o Zby he F. and M. Riesz heo em. The absolu e con inui y (wi h espec o Haa measu e) o measu es in ME(G)∗ ME(G) u ns ou o be impo an in his discussion. When ME(G)∗ME(G)⊆ L1 E(G) (such a se is said o be small-2), L1 E(G)∗∗ has a la ge cen e and so L1 E(G) canno be SAI, unless i is eflexi e, see co olla y 5.6. We do no know whe he L1 E(G) can be A ens egula when Eis no Riesz ( he main ques ion in [26]). Bu , we a e able o p o e ha egula i y o L1 E(G) o ces E o be small-2 (see co olla y 5.2). Ano he ype o se s gi ing non-A ens egula i y is p o ided by complemen s o Lus -Piqua d se s (see below o he defini ion). Fo example, L1 Z E(T) is no A ens egula when E⊆Zis he Lus -Piqua d se consis ing o he p imes in he cose 5Z+ 2, see [20, Theo em 4]. Examples o L1 E(G) being SAI a e p o ided by se s Ein he cose ing o  G.In pa icula all maximal ideals o L1(G)happen obeSAI. 2. A ens egula i y In his sec ion, we p o ide o mal defini ions o he concep s ela ed o A ens egula i y discussed in his pape . Le Abe a commu a i e Banach algeb a and le A∗and A∗∗ be i s fi s and second Banach duals, espec i ely. The mul iplica ion o Acan be ex ended na u ally o A∗∗ in wo diffe en ways. These mul iplica ions a ise as pa icula cases o he abs ac app oach o A ens [1,2] and can be o malized h ough he ollowing h ee s eps. Fo u, in A,ϕin A∗and m, n ∈A∗∗,we define φ·u, u ·φ,m·φ, φ ·m∈A∗ h ps://doi.o g/10.1017/p m.2023.110 Published online by Camb idge Uni e si y P ess 4R. Esmail andi, M. Filali and J. Galindo and mn, m♦n∈A∗∗ as ollows: φ·u, =φ, u ,u·φ, =φ, u 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 o be A ens egula . Fo any m∈A∗∗ he mapping n→ nmis weak∗–weak∗con inuous on A∗∗. Howe e , he mapping n→ mnneed no o be weak∗–weak∗con inuous. The si ua ion is e e sed o ♦. The le opological cen e o A∗∗ is hen defined as Z(A∗∗)={m∈A∗∗ :n→ mnis weak∗--weak∗con inuous on A∗∗}. Since we a e assuming ha Ais commu a i e, i is easy o see ha Z(A∗∗)={m∈A∗∗ :mn=nm=m♦n o all n∈A∗∗}. The algeb a Ais he e o e A ens egula i and only i Z(A∗∗)=A∗∗. Obse e ha Ais always con ained in Z(A∗∗). Some imes, he elemen s o he cen e s op he e. De ini ion 2.1. A commu a i e Banach algeb a Ais s ongly A ens i egula (SAI o sho ) when Z(A∗∗)=A. In [22], Pym conside ed he space WAP(A)o weakly almos pe iodic unc ionals on A, his is he se o all ϕ∈A∗such ha he linea map A→A∗:a→ a·ϕ is weakly compac . The unc ionals ϕ∈WAP(A) sa is y G o hendieck’s double limi c i e ion lim nlim mϕ, anbm= lim mlim nϕ, anbm o any pai o bounded sequences (an)n,(bm)min A o which bo h he i e a ed limi s exis . F om his p ope y, one may deduce ha mn, φ=m♦n, φ o e e y m, n ∈A∗∗ i and only i φ∈WAP(A). So, Ais A ens egula when A∗=WAP(A), i.e. when he quo ien A∗/WAP(A) is i ial. This is he mo i a ion o he ollowing defini ion. De ini ion 2.2. A Banach algeb a Ais ex emely non-A ens egula (ENAR o sho ) when A∗/WAP(A)con ains a closed subspace isomo phic o A∗. The e m ex eme non-A ens egula i y was coined by G ani e [11] o cha ac e - ize a sligh ly mo e gene al beha iou : Ais ENAR i he quo ien space A∗/WAP(A) con ains a closed linea subspace which has A∗as a quo ien , i.e. as a con inuous linea image. We ha e adop ed he e his simple defini ion ha is s ill enough o cap u e he ex eme beha iou o many o he Banach algeb as ha ha monic analysis h ps://doi.o g/10.1017/p m.2023.110 Published online by Camb idge Uni e si y P ess A ens egula i y o ideals o he g oup algeb a 5 associa es wi h a locally compac g oup. Clea ly, ENAR in he sense o defini ion 2.2 implies ENAR in he sense o G ani e , bu we do no know whe he he wo defini ions a e ac ually he same. See [5–7]. 3. The s uc u e o L1(G)∗∗ and L1 E(G)∗∗ We summa ize he e he s uc u e o L1 E(G)∗∗ whe e Gis a compac Abelian g oup and E⊆ G. No a ion will be addi i e and he iden i ies o bo h Gand  Gwill be deno ed by 0. All he ac s men ioned he e a e well-known when E= G, see e.g. [15]. No new insigh is needed o hem o hold o a bi a y E⊆ Gbu ha ing hem s a ed be o ehand will simpli y ou p oo s. A good deal o he s uc u e o L1(G)∗∗ is de e mined by he p esence o igh iden i ies. These can be ob ained as accumula ion poin s in L1(G)∗∗ o bounded app oxima e iden i ies o L1(G), which a e always a ailable (see e.g. [16,§1.3]). The fi s use o igh iden i ies is o b ing measu es on Gin o elemen s o L1(G)∗∗. Fo each μ∈ME(G), one conside s he con olu ion ope a o : Cμ:L1(G)→L1 E(G) gi en by Cμ(u)=μ∗u, u ∈L1(G). I s double adjoin C∗∗ μ hen maps L1(G)∗∗ in o L1 E(G)∗∗. When necessa y we will use i:L1 E(G)→L1(G) o deno e he inclusion map, hen i∗:L∞(G)→L1 E(G)∗ will be he es ic ion map and i∗∗ :L1 E(G)∗∗ →L1(G)∗∗ will be an embedding o Banach algeb as. We will no mally omi men ioning iand i∗∗ and see L1 E(G)∗∗ as an ideal o L1(G)∗∗. Wi h hese no a ions, i μ∈ME(G)andφ∈L∞(G) a e gi en, a s aigh o wa d compu a ion shows, ha i eis a igh iden i y in L1(G)∗∗, hen C∗∗ μ(e)·i∗(φ)=i∗(ˇμ∗φ).(3.1) The li ing map Je:ME(G)→L1 E(G)∗∗ gi en by Je(μ)=C∗∗ μ(e), u ns ou o be an algeb a isomo phism on o ei∗∗(L1 E(G)∗∗). The algeb a L1 E(G) can be seen bo h as an ideal in ME(G) and as an ideal in L1 E(G)∗∗ and, in ha sense, i is le in a ian by Je, i.e. Je( )=C∗∗ (e)= o all ∈L1 E(G).(3.2) The canonical quo ien map RE:L1 E(G)∗∗ →ME(G), defined, o each m∈ L1 E(G)∗∗,byRE(m)=mC(G)is hen a le in e se o Jeand he composi ion Je◦REis a p ojec ion. Rega dless o he igh iden i y e,ke REcan always be iden ified wi h i∗(C(G))⊥, he annihila o o he subspace i∗(C(G)) in L1 E(G)∗∗. The p ojec ion Je◦RE he e o e induces he decomposi ion L1 E(G)∗∗ =Je(ME(G)) ⊕i∗(C(G))⊥.(3.3) So, o a gi en igh iden i y eo L1 E(G)∗∗, an elemen m∈L1 E(G)∗∗,maybe uniquely decomposed as m=C∗∗ μ(e)+ , (3.4) whe e μ∈ME(G)and ∈i∗(C(G))⊥. h ps://doi.o g/10.1017/p m.2023.110 Published online by Camb idge Uni e si y P ess 6R. Esmail andi, M. Filali and J. Galindo The abo e decomposi ion becomes handie i one obse es ha he elemen s o i∗(C(G))⊥a e le annihila o s o L1 E(G)∗∗. Indeed, o ∈i∗(C(G))⊥,i m∈ L1 E(G)∗∗ is such ha m=σL1 E(G)∗∗,L 1 E(G)∗−limαuα, wi h uα∈L1 E(G), and φ∈L∞(G), hen m , i∗(φ)=i∗∗ (m ),φ = lim αuα,i ∗∗( )·φ = lim α , i∗(ˇuα∗φ)=0,(3.5) whe e he las iden i y ollows om ˇuα∗φ∈C(G). Nex , as we see, le annihila o s can ac ually be used o cha ac e ize A ens egula i y. We fi s need a defini ion. De ini ion 3.1. Le Gbe a compac Abelian g oup, E⊆ Gand pu S=i∗(C(G))⊥L1 E(G)∗∗. No e ha o any fixed igh iden i y e∈L1(G)∗∗, he se Sis gi en by S= C∗∗ μ(e): ∈i∗(C(G))⊥and μ∈ME(G). Obse e as well ha , S∩L1 E(G)={0}. To see his, one can fix a igh iden i y e∈ L1(G)∗∗ and a bounded app oxima e iden i y (uα)αin L1(G). Then, i C∗∗ μ(e)∈ S∩L1 E(G), we ha e ha C∗∗ μ(e) = lim αuα∗ C∗∗ μ(e)=0. Theo em 3.2. Le Gbe a compac Abelian g oup and le E⊆ G.Then,L1 E(G)is A ens egula i and only i S={0}. P oo . Since S={0}immedia ely implies ha L1 E(G)∗∗ is no commu a i e, by he p eceding pa ag aph, we only need o show ha S={0}implies ha L1 E(G) is A ens egula . Assume S={0}and fix a igh iden i y ein L1 E(G)∗∗ and le C∗∗ μ1(e)+s1and C∗∗ μ2(e)+s2be a bi a y in L1 E(G)∗∗, wi h 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 commu a i e, i.e. L1 E(G) is A ens egula .  We wish o eco d he ollowing lemma, a es a emen o Theo em 3.3( ) o [15], o la e use. h ps://doi.o g/10.1017/p m.2023.110 Published online by Camb idge Uni e si y P ess A ens egula i y o ideals o he g oup algeb a 7 Lemma 3.3. Le Gbe a compac Abelian g oup. Conside E⊆ Gand μ∈ME(G). I o e e y pai eand o igh iden i ies in L1(G)∗∗,C∗∗ μ(e)=C∗∗ μ( ), hen μ∈L1(G). P oo . Suppose ha μ∈ME(G) bu μ/∈L1(G), we can hen find φ∈L∞(G) such ha ˇμ∗φis no con inuous, see [13, Theo em 35.13]. By Lemma 2.3 o [15]we can find wo diffe en igh iden i ies 1, 2∈L1(G)∗∗ such ha  1,ˇμ∗φ>=  2,ˇμ∗φ. Since C∗∗ μ( i),φ= i,ˇμ∗φ,i=1,2, we deduce ha C∗∗ μ( 1)=C∗∗ μ( 2), a con adic ion wi h ou hypo heses.  4. Special subse s o  G We desc ibe he e he se s E⊆ G ha lead o he conc e e ideals L1 E(G) ha will appea la e in he pape . We fi s ecall ha an in a ian mean Mon L∞(G) is a linea unc ional on L∞(G) such ha M,1=M= 1 and, o each φ∈L∞(G) and each x∈G, M,Lxφ=M,φwhe e Lxis he ansla ion ope a o by x. An in a ian mean ha is always a ailable is he one p oduced by Haa measu e: φ→ φ(x)dx.I G is compac , L∞(G) always has o he in a ian means [23] bu all hem ha e he same effec on some unc ions. We say hen ha a unc ion φ∈L∞(G)hasaunique in a ian mean i M,φ=φ(x)dx o e e y in a ian mean Mon L∞(G). De ini ion 4.1. Le Gbe a compac Abelian g oup and E⊂ G. We say ha E is a (i) Sidon se , i e e y ∈CE(G)has an absolu ely con e gen Fou ie se ies. (ii) Λ(p)-se , p>0, i he e a e 0<q<pand C>0such ha  p⩽C q, o e e y igonome ic polynomial, =n k=1 ckχk,wi hχ1,...,χ n∈E. (iii) Rosen hal se , i L∞ E(G)=CE(G). (i ) Lus -Piqua d se , i γφ has a unique in a ian mean o e e y φ∈L∞ E(G)and e e y γ∈ G. We say in his case ha φis o ally e godic. ( ) Riesz se , i ME(G)=L1 E(G). ( i) Small-2 se , i μ∗μ∈L1 E(G) o e e y μ∈ME(G). As poin ed ou o us by he e e ee, wi h he iden i y 2μ∗ν=(μ+ν)2−μ2−ν2, one quickly checks ha μ∗ν∈L1 E(G) o e e y μ, ν ∈ME(G) i and only i μ∗μ∈ L1 E(G) o e e y μ∈ME(G) (i.e. i E⊆ Gis a small-2 se ). Sidon se s a e Rosen hal, see, e.g. [9, Co olla y 6.2.5], and Rosen hal se s a e Lus -Piqua d (as con inuous unc ions always ha e a unique in a ian mean). Lus - Piqua d se s a e in u n always Riesz (see [18]) and Riesz se s a e, ob iously, small-2. h ps://doi.o g/10.1017/p m.2023.110 Published online by Camb idge Uni e si y P ess 8R. Esmail andi, M. Filali and J. Galindo Figu e 1. Rela ions be ween p ope ies o E⊂ G,Gcompac and Abelian. On he con a y, Sidon se s a e Λ(p) o e e y p>0andΛ(p)se sa eΛ(q) o e e y q<p[13, Sec ion 37]. I is a esul o Ha e [12] ha aΛ(p) is always a Λ(q) se o some q>p. The ollowing is a consequence ha is impo an in ou con ex . Theo em 4.2 (Co olla y in [12]). Le Gbe a compac Abelian g oup and le E⊂ G. The Banach space L1 E(G)is eflexi e i and only i Eis a Λ(1) se . I ollows om his Co olla y ha Λ(1)-se s a e necessa ily Riesz. Fo , i μ∈ ME(G) L1 E(G) hen Je(μ)∈eL1 E(G)∗∗ L1 E(G), since Jeis an isomo phism ha fixes L1 E(G). The p eceding ema ks a e summa ized in figu e 1. To he au ho s’ knowledge, i is s ill unknown whe he small-2 se s a e Riesz. As al eady men ioned by ¨ Ulge in [26, p. 273], his is a long-s anding open p oblem ha goes back o Glicksbe g [10]. I migh he e o e happen ha he classes defined in i ems ( )–( i) abo e a e ac ually he same. Since L1 E(G) is A ens egula when EisRiesz(see[26] o co olla y 6.3) we will no be in e es ed in L1 E(G) o Ein any class con ained in ha Riesz se s. Howe e , se s Ewhose complemen  G Ebelongs o such a class will be o in e es in §7.2, especially a e one lea ns ha he union o a Riesz se and Lus -Piqua d se is Riesz [19], and hence ha complemen s o Lus -Piqua d se s a e ne e Riesz. We u n now ou a en ion o small-2 se s. 5. Small-2 se s We s a wi h he ollowing esul o ¨ Ulge which e eals he ele ance o non-small-2 se s in he analysis o A ens egula i y. Theo em 5.1 (Theo em 2.2 o [25]). Le Abe a commu a i e, semisimple, weakly sequen ially comple e and comple ely con inuous Banach algeb a, hen an elemen m∈A∗∗ is in he cen e o Ai and only i mA∗∗ ⊆Aand A∗∗ m⊆A. Co olla y 5.2. Le Gbe a compac Abelian g oup and assume ha E⊆ Gis no a small-2 se . Fo e e y pai μ1,μ 2∈ME(G)such ha μ1∗μ2/∈L1 E(G)and e e y igh iden i y eo L1(G)∗∗, we ha e ha nei he C∗∗ μ1(e)no C∗∗ μ2(e)is in Z(L1 E(G)∗∗). h ps://doi.o g/10.1017/p m.2023.110 Published online by Camb idge Uni e si y P ess A ens egula i y o ideals o he g oup algeb a 9 P oo . Le μ1,μ 2∈M(G) such ha μ1∗μ2/∈L1 E(G). Towa ds a con adic ion, assume ha C∗∗ μ1(e)∈Z(L1 E(G)∗∗). By heo em 5.1: C∗∗ μ1∗μ2(e)=C∗∗ μ1(e)C∗∗ μ2(e)∈L1 E(G). Since REisale in e seo Je, his is a con adic ion.  Rema k 5.3. Wi h co olla y 5.2, he las i ial implica ion Eis Riesz =⇒Eis small-2 in figu e 1 may now be spli in o wo non- i ial implica ions: Eis Riesz =⇒L1 E(G) is A ens egula =⇒Eis small-2. We shall u he see in §7 ha L1 E(G) is e en ENAR when Eis no a small-2 se . We p oceed now o find non- i ial elemen s in he cen e o L1 E(G)∗∗, when Eis a small-2 se . Recall ha he se Swas defined in §3as S=i∗(C(G))⊥L1 E(G)∗∗. Theo em 5.4. Le Gbe a compac Abelian g oup and le E⊆ Gbe a small 2-se . Then, S⊆Z(L1 E(G)∗∗). P oo . We fi s fix a igh iden i y e∈L1(G)∗∗. Le ∈i∗(C(G))⊥and μ∈ME(G). Pu p= C∗∗ μ(e). I q=C∗∗ σ(e)+s∈ L1 E(G)∗∗, wi h s∈i∗(C(G))⊥and σ∈ME(G), hen qp=0,as is a le annihi- la o , (3.5). Since sis also le annihila o and C∗∗ μ(e)C∗∗ σ(e)∈Z(L1 E(G)∗∗), o μ∗σ∈L1 E(G) since Eis a small-2 se , one ge s: pq= C∗∗ μ(e)C∗∗ σ(e)= C∗∗ μ∗σ(e)=0. Hence, p∈Z(L1 E(G)∗∗), as needed.  We choose o exp ess he main consequence o his heo em in wo equi alen ways. Co olla y 5.5. Le Gbe a compac Abelian g oup and le E⊆ Gbe a small-2 se . Then, L1 E(G)is SAI i and only i i is eflexi e. Co olla y 5.6. Le Gbe a compac Abelian g oup and le E⊆ G.I Eis a small 2-se ha is no Λ(1), hen L1 E(G)is no SAI. P oo . Suppose ha Eis small-2 se wi h L1 E(G) SAI. Since S∩L1 E(G)={0} (see he ema ks a e defini ion 3.1), heo em 5.4 implies ha Smus be i ial. Theo em 3.2 implies hen ha L1 E(G) is A ens egula , and so i mus be eflexi e, i.e. Emus be Λ(1) ( heo em 4.2).  h ps://doi.o g/10.1017/p m.2023.110 Published online by Camb idge Uni e si y P ess 16 R. Esmail andi, M. Filali and J. Galindo As in (6.1), 0=s, i∗(ˇμ∗φ)>=sC∗∗ μ(e),φ>=sp, φ>, showing ha p/∈Z(L1 E(G)∗∗) because ps=0.  The p eceding p oposi ion 7.9 yields he ollowing esul . Co olla y 7.10. Le Gbe a compac me izable Abelian g oup. I E⊆ Gis such ha  G Eis a Lus -Piqua d se , hen L1 E(G)is no A ens egula . We close he pape obse ing ha , con a ily o wha he p e ious co olla y migh sugges , he egula i y p ope ies o L1 E(G) do no de e mine hose o L1  G E(G). Example 7.11. L1 E(G)andL1  G E(G) can be bo h SAI and egula . I E∈Ω G, hen  G E∈Ω G, hen bo h L1 E(G)andL1  G E(G) a e SAI by co olla y 7.1. I on he o he hand we conside E=N⊆Z he classical case o a Riesz se , hen  G E=−Nis also a Riesz se so ha L1 E(G)andL1  G E(G) a e bo h A ens egula by co olla y 6.3. Rema k 7.12. In ou o hcoming pape , we shall deal wi h mo e gene al Banach algeb as o he same ype deal wi h in his pape . Ou s udy will include he g oup algeb a o a non-Abelian compac g oup and he Fou ie algeb a o an amenable disc e e g oup. Acknowledgemen s We wish o hank he e e ee o he e y ca e ul eading o he pape , co ec ions and cons uc i e ecommenda ions and sugges ions ha ha e made he p esen a ion o he pape much clea e and mo e compelling. 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