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
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1
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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∗
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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
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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))⊥.
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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.
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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.
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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.
The second au ho wishes o acknowledge he Depa men o Ma hema ics
a he Uni e si y o Jaume I in Cas ell´on. All he suppo , including he pa -
ial financial suppo , by he Uni e si y o Jaume is g a e ully acknowledged;
he would ne e ha e been in his boa wi hou such suppo . Resea ch o he
fi s and hi d au ho s was suppo ed by g an PID2019-106529GB-I00 unded by
MCIN/AEI/10.13039/501100011033.
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