More on cyclic amenability of the Lau product of Banach algebras defined by a Banach algebra morphism

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by  Mohammad Ramezanpour

MORE ON CYCLIC AMENABILITY OF THE LAU PRODUCT OF BANACH
ALGEBRAS DEFINED BY A BANACH ALGEBRA MORPHISM
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M.RAMEZANPOUR
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n
a
J Abstract. FortwoBanachalgebrasAand B, the T-LauproductA×T B,wasre-
1 centlyintroducedandstudiedforsomeboundedhomomorphismT:B→Awith
2 kTk≤1.Here,wegivegeneralnessesaryandsufficentconditionsforA×TBtobe

(approximately)cyclicamenable. Inparticular,weextendsomerecentresultson
]
A (approximate)cyclicamenabilityofdirectproductA⊕BandT-LauproductA×TB
F andansweraquestiononcyclicamenabilityofA×TB.
.
h
t 1. IntroductionandsomePreliminaries
a
m
The notion of weak amenability for commutative Banach algebras was intro-
[
duced and studied for the first time by Bade, Curtis and Dales [3]. Johnson [11]

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extended this concept to the non commutative case and showed that group al-
v
6 gebras of all locally compact groups are weakly amenable. A Banach algebra A
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0 is called weakly amenable if every continuous derivation D : A→ A∗ is inner. It
6 is often useful to restrict one’s attention to derivations D : A → A∗ satisfying the
0
. property D(a)(c)+D(c)(a) = 0for all a,c ∈ A. Suchderivationsarecalled cyclic.
1
0 Clearly inner derivations are cyclic. A Banach algebra is called cyclic amenable
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if every continuous cyclic derivations D : A → A∗ is inner. This notion was pre-
1
: sentedbyGronbaek[9]. Heinvestigatedthehereditarypropertiesofthisconcept,
v
i foundsomerelationsbetweencyclicamenabilityofaBanachalgebraandthetrace
X
extensionpropertyofitsideals.
r
a
GhahramaniandLoy[7]introducedseveralapproximatenotionsofamenabil-
itybyrequiringthatallbounded derivationsfromagiven BanachalgebraAinto
certain Banach A-bimodules to be approximately inner. In the same paper and
thesubsequentone[8],theauthorsshowedthedistinctionbetweeneachofthese
concepts and the corresponding classical notions and investigated properties of
algebrasineachofthesenewclasses. Motivatedbythisnotions,Esslamzadehand
Shojaee[6]definedtheconceptofapproximatecyclicamenabilityforBanachalge-
brasand investigated the hereditarypropertiesfor this newnotion. Shojaee and
1991MathematicsSubjectClassification. Primary46H05;Secondary46H99.
Keywordsandphrases. Banachalgebra,cyclicamenability,T-Lauproduct.
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2 M.RAMEZANPOUR
Bodaghi in [14, Theorem 2.3] showed that for Banach algebras Aand B, if direct
productA⊕Bwithℓ1-normisapproximatelycyclicamenable,thensoareAandB.
TheyalsoshowedthattheconverseforthecasewhereA2isdenseinA.
Ontheotherhand,fortwoBanachalgebrasAandBandaboundedhomomor-
phism T : B → Awith kTk≤ 1, the T-LauproductA× B isdefined asthe space
T
A×Bequippedwiththenormk(a,b)k=kak+kbkandthemultiplication
(a,b)(a′,b′)=(aa′+aT(b′)+T(b)a′,bb′),
for all a,a′ ∈ Aand b,b′ ∈ B. This product was first introduced and studied by
BhattandDabhiin[4]forthecasewhereAiscommutative. JavanshiriandNemati
in [10] extended this product to the general Banach algebras and studied Arens
regularity,amenabilityandn-weakamenabilityof A× B inthegeneralcase; see
T
also[2]and[5]. WhenT =0,thismultiplicationistheusualcoordinatewiseprod-
uctand so A× B is in factthe directproductA⊕B. Furthermore, let Abe unital
T
with the identity element e and letθ : B → C be a non-zeromultiplicative linear
functional. Define Tθ : B → Aas Tθ(b) = θ(b)e, for each b ∈ B. Then the above
productwithrespecttoTθ coincideswiththeproductinvestigatedin[12].
Bhattand Dabhiin [4] showed thatcyclic amenability of A× B is stable with
T
respecttoT,forthecasewhereAiscommutative,buttheproofcontainsagap. In
[1]AbtahiandGhafarpanahfixedthisgapandextendedthisresulttoanarbitrary
BanachalgebraA. IndeedtheyprovedthatifA× BiscyclicamenablethenbothA
T
andBarecyclicamenable. Theyalsoprovedtheconverseforthecasewhereboth
AandB havefaithfuldualspaces. ButtheyleftitopenforallBanachalgebras;[1,
Question3.5].
In the present paper, we give general nessesary and sufficent conditions for
A× B to be (approximate) cyclic amenable. In particular we extend the recent
T
resultson (approximate)cyclic amenabilityof the directproductA⊕B, [14], and
the T-LauproductA× B andanswerQuestion3.5in[1]oncyclicamenabilityof
T
A× B.
T
2. Cyclicamenability
LetAbeaBanachalgebra,andX beaBanachA-bimodule. Thenthedualspace
X∗ of X becomes a dual Banach A-bimodule with the module actions defined by
(fa)(x)= f(ax)and(af)(x)= f(xa),foralla∈A,x ∈X and f ∈X∗. Aderivation
fromAintoX isalinearmappingD:A→X satisfying
D(ac)=D(a)c+aD(c) (a,c∈A).
MOREONCYCLICAMENABILITYOFTHELAUPRODUCTOFBANACHALGEBRASDEFINEDBYABANACHALGEBRAMORPHISM3
If x ∈X thend :A→X definedbyd (a)=ax−xaisaderivationwhichiscalled
x x
aninnerderivation.
AderivationD:A→A∗issaidtobecyclicifD(a)(c)+D(c)(a)=0foralla,c∈A.
If every continuous cyclic derivation D : A → A∗ is inner then A is called cyclic
amenable.
Asremarkedin[4],thedualspace(A× B)∗ canbeidentifiedwiththeBanach
T
spaceA∗×B∗equippedwiththenormk(f,g)k=max{kfk,kgk}via
(f,g)(a,b)= f(a)+g(b),
where a ∈A,f ∈A∗,b ∈ B and g ∈ B∗. Moreover,adirectverificationrevealsthat
(A× B)-moduleoperationsof(A× B)∗areasfollows.
T T
(f,g)(a,b)=(cid:0)fa+ fT(b),T∗(fa)+gb(cid:1),
(a,b)(f,g)=(cid:0)af +T(b)f,T∗(af)+bg(cid:1)
fora∈A,b∈B,f ∈A∗and g∈B∗.
To clarify the relation between cyclic amenability of A× B and that of Aand
T
B,weneedthenextresultwhichcharacterizethecontinuouscyclicderivationson
A× B.
T
Lemma2.1. SupposethatAand B areBanachalgebrasand T :B →Aisabounded(by
one) homomorphism. Then every continuous cyclic derivation D : A× B → (A× B)∗
T T
enjoysthepresentation
D(a,b)=(D (a)−S∗(b),D (b)+S(a))
1 2
foralla∈Aand b∈B,where
(a) D :A→A∗andD :B→B∗ arecontinuouscyclicderivations.
1 2
(b) S : A → B∗ is a bounded linear operator such that S(ac) = (T∗◦D )(ac) and
1
S(a)b=(T∗◦D )(a)bforalla,c∈Aand b∈B.
1
Moreover,DisinnerifandonlyifS=T∗◦D andbothD andD areinnerderivations.
1 1 2
Abtahi and Ghafarpanah in [1], proved that if A× B is cyclic amenable then
T
both A and B are cyclic amenable. They also proved the converse for the case
where both Aand B have faithfuldual spaces, but left it as an open question for
allBanachalgebras;[1,Question3.5].Animprovementofthisresulthasbeenalso
obtained in [13]. Indeed, the converse has been proved for the case where A2 is
denseinA; [13, Theorem2.6]. Weshould remarkthataBanachalgebraAhasleft
(right)faithfuldualspacejustwhenA2isdenseinA.
4 M.RAMEZANPOUR
HerewegivesgeneralnecessaryandsufficientconditionsforA× Btobecyclic
T
amenable. Thisresultimproves[13, Theorem2.6]andanswersalsoQuestion 3.5
in[1].
Theorem2.2. SupposethatAandBareBanachalgebrasandT :B→Aisabounded(by
one)homomorphism.ThenA× Biscyclicamenableifandonlyifthefollowingstatements
T
hold.
(1) AandBarecyclicamenable.
(2) A2isdenseinAorB2isdenseinB.
3. Approximatecyclicamenability
Recall from [7] that a derivation D : A → X is called approximately inner if
thereexistsanet{xα}⊆X suchthat D=limαdxα inthestrongoperatortopology.
ABanachalgebraAiscalled approximatelycyclic amenable, if everycontinuous
cyclicderivation D:A→A∗ isapproximatelyinner. Theconceptsofapproximate
cyclicamenabilitywasintroducedandstudiedin[6];seealso[14].
ThenextresultcharacterizesapproximatelyinnerderivationsonA× B.
T
Proposition3.1. SupposethatAandB areBanachalgebrasand T :B→Aisabounded
(byone) homomorphism. Let D : A× B → (A× B)∗ be a continuous cyclicderivation
T T
given by D(a,b) = (D (a)−S∗(b),D (b)+S(a)) for all a ∈ Aand b ∈ B. Then D is
1 2
approximately inner if and only if S = T∗◦D and both D and D are approximately
1 1 2
inner.
Applying Proposition 3.1 and using a similar argument used in the proof of
Theorem 2.2 we can prove the next theorem. This result improves [13, Theorem
2.3].
Theorem 3.2. Suppose that Aand B are Banach algebras and T : B → Ais a bounded
(byone)homomorphism. ThenA× B isapproximatelycyclicamenableifandonlyifthe
T
followingstatementshold.
(1) AandBareapproximatelycyclicamenable.
(2) A2isdenseinAorB2isdenseinB.
Shojaee and Bodaghi in [14, Theorem 2.3] showed that for Banach algebras A
andB,ifdirectproductA⊕Bisapproximatelycyclicamenable,thensoareAandB.
TheyalsoshowedthattheconverseforthecasewhereA2 isdenseinA. Applying
Theorems2.2and3.2forT =0,wegetthenextresultwhichextends[14,Theorem
2.3].
MOREONCYCLICAMENABILITYOFTHELAUPRODUCTOFBANACHALGEBRASDEFINEDBYABANACHALGEBRAMORPHISM5
Corollary3.3. LetAandBbeBanachalgebras. ThenthedirectproductA⊕Bis(approx-
imately)cyclicamenableifandonlyifthefollowingstatementshold.
(1) AandBare(approximately)cyclicamenable.
(2) A2isdenseinAorB2isdenseinB.
Let Abe unital and θ : B → C be a non-zero multiplicative linear functional.
Define Tθ(b):=θ(b)e. ThenA×Tθ Bistheθ-LauproductA×θ B,[12]. Asaconse-
quenceofTheorems2.2and3.2,wehavethenextresult.
Corollary3.4. LetAbeunitalandθ beanon-zeromultiplicativelinearfunctionalonB.
ThenA×θ B is(approximately)cyclicamenableifandonlyifbothAand B are(approxi-
mately)cyclicamenable.
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6 M.RAMEZANPOUR
SchoolofMathematicsandComputerScience, DamghanUniversity,P. O.Box36716, Damghan
41167,Iran.
E-mailaddress:[email protected]