Vol. 23, No. 10 (2000) 681–685 S0161171200002246 ©Hindawi Publishing Corp.
A NOTE ON
M
-IDEALS IN CERTAIN ALGEBRAS OF OPERATORS
CHONG-MAN CHO and WOO SUK ROH
(Received 7 October 1998)
Abstract.LetX=(∞n=1n
1)p, p >1. In this paper, we investigateM-ideals which are also ideals inL(X), the algebra of all bounded linear operators onX. We show thatK(X), the ideal of compact operators onXis the only proper closed ideal inL(X)which is both an ideal and anM-ideal inL(X).
Keywords and phrases. Compact operators, ideal,M-ideal.
2000 Mathematics Subject Classification. Primary 46A32.
1. Introduction. Since Alfsen and Effros [1, 2] introduced the notion of anM-ideal in a Banach space, many authors have studiedM-ideals in algebras of operators. An interesting problem has been characterizing and finding those Banach spacesX for whichK(X), the space of all compact linear operators onX, is anM-ideal inL(X), the space of all continuous linear operators onX[4, 8, 9, 11, 12].
It is known that ifXis a Hilbert space,p(1< p <∞)orc0, thenK(X)is anM-ideal in
L(X)[6, 8, 12] whileK(1)andK(∞)are notM-ideals in the corresponding spaces of
operators [12]. Smith and Ward [12] proved thatM-ideals in a complex Banach algebra with identity are subalgebras and that they are two-sided algebraic ideals if the algebra is commutative. They also proved thatM-ideals in a C∗-algebra are exactly the
two-sided ideals [12]. Later, Cho and Johnson [5] proved that ifXis a uniformly convex Banach space, then everyM-ideal inL(X)is a left ideal, and ifX∗ is also uniformly
convex, then everyM-ideal inL(X)is a two-sided ideal inL(X).
Flinn [7], and Smith and Ward [13] proved thatK(p)is the only nontrivialM-ideal in L(p)for 1< p <∞. Kalton and Werner [10] proved that if 1< p, q <∞, X=
(∞n=1nq)pwith complex scalars, thenK(X)is the only nontrivialM-ideal inL(X). In their proof of this fact, Kalton and Werner [13] used the uniform convexity ofXand
X∗. In this case,M-ideals inL(X)are two-sided closed ideals inL(X)[5].
In this paper, we investigateM-ideals which are also ideals inL(X)forX=(∞n=11n)p, 1< p <∞. In our case, neitherXnorX∗is uniformly convex. Therefore, no
relation-ships betweenM-ideals and algebraic ideals inL(X)seem to be known. But still we can use Kalton and Werner’s proof in [10] without using uniform convexity ofXand
X∗ to prove thatK(X)is the only nontrivialM-ideal in L(X)which is also a closed
ideal inL(X)(Theorem 3.3). By duality we have the same conclusion for the space
(∞n=1n∞)p, 1< p <∞.
2. Preliminaries. A closed subspace J of a Banach space X is said to be an
such that X is an algebraic direct sum X =J⊕J and satisfies a norm condition j+j = j+j(respectively,j+j =max{j,j}) for allj∈Jandj∈J.
In this case, we writeX=J⊕1J(respectively,X=J⊕∞J) and the projectionPonX
with rangJis called anL-projection (respectively, anM-projection). A closed subspace
Jof a Banach spaceXis anM-ideal inXif the annihilatorJ⊥ofJis anL-summand
inX∗.
LetAbe a complex Banach algebra with identitye. The state spaceSofAis defined to be{φ∈A∗:φ(e)= φ =1}. An elementh∈Ais said to be Hermitian ifeiλh =1 for all real numberλ. Equivalently,his Hermitian if and only ifφ(h)is real for every
φ∈S [3, page 46].
In what follows,Zalways denote a complex Banach space(∞n=1n1)p, thep-sum ofn
1’s for 1< p <∞. For eachn, let{enl}nl=1be the standard basis ofn1. Then these
bases string together to form the standard basis{en}∞n=1ofZand eachT∈L(Z)has
a matrix representation with respect to{en}∞n=1. If T ∈L(Z)has the matrix whose
(i,j)-entry istij, then we can writeT=i,j≥1tijej⊗ei, whereej⊗eiis the rank 1 map sendingejtoei. Observe thatT (ej)forms thejth column vector of the matrix ofT andT ej ≤ Tfor allj=1,2,.... If the matrix ofT has at most one nonzero entry in each row and column, thenTis thel∞-norm of the sequence of nonzero entries.
A number of facts which hold inL(p),1< p <∞, still hold inL(Z). If the matrix ofT∈L(Z)is a diagonal matrix(tij)with real diagonal entries, then for each realλ the matrix ofeiλT is also a diagonal matrix with diagonal matrix entrieseiλtii. Thus
T∈L(Z)is Hermitian if the matrixT is a diagonal matrix with real entries.
Flinn [7] proved that ifM is anM-ideal inL(p),1< p <∞and his a Hermitian element inL(p)withh2=I, thenhM ⊆Mand Mh⊆M. From this he proved that ifhis any diagonal matrix with real entries, thenhM⊆MandMh⊆M. His proof is valid forZin place ofp. Thus we have the following.
Lemma2.1. IfMis anM-ideal inL(Z)andh∈L(Z)is a diagonal matrix with real entries, thenhM⊆MandMh⊆M.
TheM-ideal structure of L(X) forX =(∞n=1nq)p,1< p,q <∞ was studied by Kalton and Werner [10]. Some of their proofs forXare still good forZ. One of them is the following.
Lemma2.2. There is a constantCsuch that, whenever(kn)is a sequence of positive
integers withlimsupkn= ∞, then(∞n=1kn1 )pisC-isomorphic to(∞n=1n1)p. Proof. See proof of Lemma 3.1 of [10].
We recall that a Banach spaceXisC-isomorphic to a Banach spaceY if there exists an isomorphismT formXontoY such that
1
Cx ≤ T x ≤Cx (2.1)
for everyx∈X. We use the following lemma which is due to Kalton and Werner [10].
Lemma2.3[10]. LetXbe a Banach space,᐀⊂L(X)be a two-sided ideal, andP a projection onto a complemented subspaceEofXwhich is isomorphic toX.
(b)IfEisC-isomorphic withXand᐀contains an operatorTwithT−P<(CP−1),
then᐀=L(X).
3. M-ideals inL((∞n=1n1)p). A matrix carpentry used by Flinn [7] to characterize theM-ideal structure inL(p)can be used to some extent in our caseZ=(∞n=11n)p. The proof of the following lemma is really a minor modification of Flinn’s proof in [7].
Lemma3.1. IfMis a nontrivialM-ideal inL(Z), thenK(Z)⊆M.
Sketch of the proof. Let us call two positive integersiandj are in the same block ifn(n+1)/2< i, j≤(n+1)(n+2)/2 for somen. Using Lemma 2.1, we can follow Flinn’s proof of the second corollary to Lemma 1 in [7]. The only modification is the following: to prove 21/q<|t
pl+tkl| ≤21/q, we consider two cases. Ifpandkare in a different block, Flinn’s proof just run through. Ifpandkare in the same block, then 21/q<|t
pl+tkl| ≤ T (el) ≤21/q.
The proof of the following lemma is contained in the proof of Theorem 3.3 in [10].
Lemma3.2. If ᐀is a closed ideal strictly containingK(Z)then᐀ contains all the operators which factor throughp.
The proof of the following theorem is a modification of that of Kalton and Werner [10]. Here we can go around the use of uniform convexity.
Theorem3.3. If᐀is a closed ideal and also anM-ideal inL(Z)strictly containing
K(Z), then᐀=L(Z).
Proof. We recall that the standard basis{enl}nl=1 ofn1 string together to form
the standard basis{en}∞n=1ofZ. If{en}∞n=1is the standard basis ofp, then the map
en→engives a contraction fromZtop. SinceE=span{enl}∞n=1is isometric top, there exists a norm one operatorA fromZ toE carryingen toen1 viaen. ThusA factors throughp. By Lemma 3.2,A∈᐀.
Since᐀ is also anM-ideal, by Proposition 2.3 in [14], there exists a net(Hα)⊆᐀ such that
limsup±A+(Id−Hα) =1. (3.1)
To simplify subsequent calculations, let us write the standard basis ofZas{enl:n∈ N, 1≤l≤n}and let {e∗
nl:n∈N, 1≤l≤n}be the corresponding biorthogonal functionals. ThenAenl=eml, wherem=(n−1)n/2+l.
Given 0< ε <1,
max
± ±A+(Id−Hα)<1+ε (3.2)
for infinitely manyα’s. For such anαand everyenl,
Putαkj=e∗kj(Id−Hα)enl. Then,
max± ±Aenl+(Id−Hα)enlp =max± ±em1−(Id−Hα)enlp
=max± |αm1±1|+|αm2|+···+|αmm| p
+ k≠m
k
j=1 |αkj|
p
< (1+ε)p.
(3.4)
Since max±|αm1±1| ≥1, it follows thatk≠m(kj=1|αkj|)p< (1+ε)p−1 and|αm2|+ ··· + |αmm|< ε. Since1+|αm1|2≤max±|αm1±1|<1+ε,|αm1|<√2ε+ε2<2√ε. Thus(Id−Hα)enl< ((3√ε)p+(1+ε)p−1)1/p→0 asε→0 uniformly innandl. It follows that, for anyn,
Pn(Id−Hα)Pn ≤ Pn(Id−Hα)jn ≤(3√ε)p+(1+ε)p−11/p, (3.5) wherePnis the projection onZwith rangen1⊆Zandjnis the canonical injection of
n 1 intoZ.
By Lemma 3.2 in [10], there exists a sequence(kn)such that, for the canonical pro-jectionPfromZonto(∞n=1kn1 )p,
P−PHαP = P(Id−Hα)P<3(3√ε)p+(1+ε)p−11/p. (3.6) SincePHαP∈᐀andε >0 is arbitrary small, by Lemmas 2.2 and 2.3,᐀=L(Z).
From Lemma 3.1 and Theorem 3.3, we have the following.
Corollary3.4. If᐀is a proper ideal and also anM-ideal inL(Z), then᐀=K(Z).
Remark. By duality, all the lemmas, Theorem 3.3 and Corollary 3.4 hold withZ∗=
(∞n=1n∞)p,1< p <∞, in place ofZ.
Acknowledgement. This research is supported by KOSEF No. 951-0102-003-2.
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Cho: Department of Mathematics, Hanyang University, Seoul133-791, Korea E-mail address:[email protected]