WEAK GROTHENDIECK’S THEOREM
LAHC `ENE MEZRAGReceived 14 June 2005; Revised 9 March 2006; Accepted 20 June 2006
LetEn⊂L2n1 be then-dimensional subspace which appeared in Kaˇsin’s theorem such that
L2n1 =En⊕E⊥n and theL12nandL2n2 norms are universally equivalent on bothEnandE⊥n.
In this paper, we introduce and study some properties concerning extension and weak Grothendieck’s theorem (WGT). We show that the Schatten spaceSp for all 0< p≤ ∞
does not verify the theorem of extension. We prove also thatSpfails GT for all 1≤p≤
∞and consequently by one result of Maurey does not satisfy WGT for 1≤p≤2. We conclude by giving a characterization for spaces verifying WGT.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
This work was inspired by the celebrated theorem of Kaˇsin [5]. We use his decomposition cited in the abstract and which states thatL2n
1 (this space is of dimension 2nand which
will be defined in the sequel) can be decomposed into two orthogonaln-dimensional sub-spaces “respecting” the inner product induced by the norm ofL2n2 and on each the norms
ofL2n
1 andL2n2 are universally equivalent on these subspaces. It is interesting to observe
that the constants of equivalence are independent ofn. Recently this was investigated by Anderson [1] and Schechtman [15]. We will say that a Banach spaceX verifies weak Grothendieck’s theorem ifπ2(X,l2)=B(X,l2). Let{ϕi}1≤i≤nbe a sequence of orthogonal
random variables inL2n
2 , which generatesEn. Consider 0< p≤ ∞. Letu:En→Snpbe a
lin-ear operator and letube any extension ofu. In this paper we show thatu ≥C√n, where Cis an absolute constant. We prove thatSpfails extension theorem for all 1≤p≤ ∞. We
also show thatSp does not verify GT for 1≤p≤ ∞and consequently fails WGT for all
1≤p≤2 by using one result of Maurey. We end this work by giving a characterization for operators satisfying WGT.
We start the first section by recalling some necessary notations and definitions such as the definition of cotypeq-Kaˇsin as studied in [9] and which is inspired by the Kaˇsin decomposition. We introduce also the property of weak Grothendieck’s theorem.
In section two, we recall the Schatten spacesSp which are the noncommutative
ana-logues of thelp-spaces and we give some properties concerning these spaces. After this,
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 43875, Pages1–12
we show that the spaceSpfails the property of extension for all 0< p≤ ∞and GT for all
1≤p≤ ∞. We deduce that the spaceSpdoes not verify WGT for allp, 1≤p≤2. We do
not know ifSpis of cotype 2-Kaˇsin for 1≤p≤2 like the classical cotype. We know that
the Schatten spaceSpis of cotype 2 for 1≤p≤2 as the usuallp-spaces; see [16]. By
an-other method which is not adjustable to our case we have proved in [10] thatLp([0, 1],dx)
andlpfor 0< p <1 fail the extension property.
InSection 4, we characterize the spaces which satisfy weak Grothendieck’s theorem.
2. Notation and preliminaries
Let 0< p≤+∞. We denote byLn
pthe spaceRn(orCn) equipped with the norm (and only
ap-norm if 0< p <1)
ai
Ln p=
1 n
n
i=1
aip 1/ p
, (2.1)
and ifp= ∞, we take max|ai|.
Recall that ap-norm on a vector spaceXis a functional · :X−→R+,
x−→ x (2.2)
such that
x =0⇐⇒x=0, λx = |λ|x ∀λinC,
x+y ≤xp+yp1/ p ∀x,yinX,
(2.3)
Xis called ap-normed space if its topology can be defined by ap-norm. Ln
p is isometric to Lnp(Ωn,ᏼ(Ωn),μn) where Ωn is the set {1, 2,. . .,n}, ᏼ(Ωn) the
σ- algebra of all subsetsA⊂Ωnandμnthe uniform probability onΩn(i.e.,μn(i)=1/n
for alliinΩn). Hence each element inLnpcan be considered as a random variable which
we denote in the sequel byϕand we have for 0< p≤q≤ ∞, ϕLn
p≤ ϕLnq≤n
1/ p−1/qϕ Ln
p. (2.4)
Moreover, we will denote byln
p(X) for any Banach spaceX(resp.,Lnp(X)), the spaceXn
equipped with the norm if 1≤p≤+∞and thep-norm if 0< p <1:
xiln p(X)=
n
i=1
xip X
1/ p
,
resp.,xiLn pt(X)=
1 n
n
i=1
xip X
1/ p (2.5)
We will use the following decomposition due to B. S. Kaˇsin (see also [13] and recently [1,15]), which is the principal inspiration of our idea.
Theorem 2.1 [5]. Considerpin{1, 2}andninN. There are three constantsAp,Bp, andC
(Cindependent ofpandn) and a sequence (ϕi)1≤i≤nof orthogonal random variables inL2n2
such that for all (ai)1≤i≤ninR, there exist
Ap
n
1
ai2 1/2
≤
n
1
aiϕi
L2n p
≤Bp
n
1
ai2 1/2
,
sup
1≤i≤n
ϕi
Ln∞≤C(logn)1/2.
(2.6)
Remark 2.2. It is well known that ifX is a finite dimensional space, then, all the norms are equivalent. But what is most remarkable inTheorem 2.1 is that the constants are independent of the dimensionn. It is also true for all pin ]0, 2]. We can and do choose theϕito be orthonormal, that is what we do in the sequel.
LetEnbe the subspace ofL2n1 spanned by the functions (ϕi)1≤i≤nand let en:En→L2n1
be the natural injection. By the above theorem,En is isomorphic to ln2, we denote by
βn:ln2→Enthe isomorphism which mapseiontoϕi, where (ei) the unit vector basis ofln2.
We have by (2.6) thatβn ≤B1andβn−1 ≤A−11.
Now we give the following definition which is introduced in [9].
Definition 2.3. LetXandY be Banach spaces and letu:X→Y be a linear operator. Say thatuis of cotypeq-Kaˇsin for 2≤q <+∞, if there is a positive constantK such that for all integernand for all finite sequence (xi)1≤i≤ninX, there exists
n
i=1
u xiq
1/q
≤K
n
i=1
ϕixi
L2n
1(X)
. (2.7)
Denote byKq(u) the smallest constant for which this holds.Xis of cotypeq-Kaˇsin if the
identity ofXis of cotypeq-Kaˇsin.
For exampleLp(1≤p≤2) is of cotype 2-Kaˇsin.
For being complete, we add (see [13, page 115]) that there is an orthonormal basis (ϕn)
ofL2([0, 1],ν) (νis the Lebesgue measure) such that theL1andL2norms are equivalent
on each of the spans of{ϕn,nodd}and{ϕn,neven}. LetE0be the space spanned by one
of these sequences inL1([0, 1],ν) and lete:E0→L1([0, 1],ν) be the isometric embedding.
We denote also byEn
0 the space spanned by thenfirstϕi.
Given two Banach spacesX andY, denote byX⊗ˆY their injective tensor product, that is, the completion ofX⊗Yunder the cross norm:
n
i=1
xi⊗yi
=sup
n
i=1
xi(ξ)yi(η)
:ξX∗≤1,ηY∗≤1
. (2.8)
the mappings
In⊗u:lnp⊗X−→lnp(Y), n
1
ei⊗xi−→
uxi
1≤i≤n
(2.9)
are uniformly bounded byC(i.e.,In⊗uln
p⊗X→lnp(Y)≤C).
We define thep-summing norm of an operatoruby
πp(u)=sup n
In⊗u
ln
p⊗X→lnp(Y). (2.10)
The following proposition is a characterization of spaces of cotype 2-Kaˇsin.
Proposition 2.4. LetCbe a positive constant. Then the following properties of a Banach spaceXare equivalent.
(i) The spaceX∗(X∗is the Banach space dual ofX) is of cotype 2-Kaˇsin andK2(X∗)≤
C.
(ii) For all integersnand for all finite sequences (xi)1≤i≤ninX, the operatoru:En→X
defined byu(ϕi)=xiadmits an extensionu:L2n1 →Xsuch thatu/E n=uandu ≤
C(ni=1xi2)1/2.
Proof. Letnbe a fixed integer. SinceX∗is of cotype 2-Kaˇsin, hence for all (ξi)1≤i≤n⊂X∗
we have
n
i=1
ξi2 X∗
1/2
≤C
n
i=1
ϕiξi
L2n
1(X∗)
. (2.11)
LetE= {n
i=1ϕiξi, (ξi)1≤i≤n⊂X∗}, which is a closed subspace ofL2n1 (X∗). We now define
the operators
T:E−→ln 2
X∗,
n
i=1
ϕiξi−→
ξi
1≤i≤n.
(2.12)
This definition is unambiguous (indeed,ni=1ϕiξi=in=1ϕiηiimplies thatξi=ηifor all
1≤i≤nbecause theϕiare orthogonal and consequently (ξi)1≤i≤n=(ηi)1≤i≤n).
Observe that
T ≤C. (2.13)
By duality we have
T∗:ln 2(X)−→
L2n
∞(X) E⊥ ,
xi
1≤i≤n−→ n
i=1
xiϕi+E⊥,
whereE⊥= {2n
i=n+1ϕixi, (xi)n+1≤i≤2n⊂X}is the subspace ofL2n∞(X) which is orthogonal toE.
SinceT = T∗(T∗is the adjoint operator ofT), hence we have
inf
R∈E⊥
n
i=1
xiϕi+R
L2∞n(X)
≤C n
i=1
xi2 1/2
. (2.15)
If nowu:L2n1 →Xis an extension ofu, by Riesz representation theorem then there isΨ
inL2n
∞ such that
∀ϕ∈L2n
1 , u(ϕ)=
1 2n
2n
i=1
ϕiΨi,
u = ΨL2∞n.
(2.16)
Sinceu(ϕi)=xi, we have
Ψ=
n
i=1
xiϕi+R. (2.17)
The correspondenceu→Ψis bijective and this implies that
infu = inf
R∈E⊥
n
i=1
xiϕi+R
L2∞n(X)
. (2.18)
This concludes the proof.
We say now that a Banach spaceX is of cotype strongly 2-Kaˇsin if there is a positive constantCsuch that, for all integersnand for all finite sequences (xi)1≤i≤ninX, we have
π2(v)≤C
n
i=1
ϕixi
L2n
1(X)
, (2.19)
wherev:l2n→Xis the operator defined byv(ei)=xifor all 1≤i≤n.
We denote by
K2strong(X)=inf
C: (2.19) holds∀xi
1≤i≤n,n≥1
. (2.20)
Corollary 2.5. LetX be a Banach space and letCbe a positive constant. The following assertions are equivalent.
(i) The spaceX∗is of cotype strongly 2-Kaˇsin andK2strong(X∗)≤C.
(ii) For all integersnand anyu:ln2→X,uadmits an extensionutoL2n1 such thatu/E n=
uβ−1
Proof. Fixed n in N, letE= {n
i=1ϕiξi, (ξi)1≤i≤n⊂X∗} which is a closed subspace of
L2n1 (X∗). We now define the operators
T:E−→π2
ln
2,X∗
,
n
i=1
ϕiξi−→v,
(2.21)
wherev:ln
2→X∗defined byv(ei)=ξi.
We have
T
n
i=1
ϕiξi =π2(v)≤C
n
i=1
ϕiξi
. (2.22)
By duality, we obtain
T∗:π2
X∗,ln
2
−→L2n∞E(⊥X),
w−→
n
i=1
xiϕi+E⊥,
(2.23)
wherew:X∗→l2nis a linear operator defined byw(ξ)= xi,ξ.
Letu(ei)=xi. We have
inf
R∈E⊥
n
i=1
xiϕi+R
L2∞n(X)
≤Cπ2
u∗. (2.24)
We conclude directly by using (2.18).
Remark 2.6. LetXbe a Banach space. IfXhas Gaussian (resp., Rademacher) cotype 2, then (2.19) holds with (gi) (resp., (ri)) and conversely. The spaceXis of cotype strongly
2-Kaˇsin implies thatXis of cotype 2-Kaˇsin. We do not know if the converse is true. Let us introduce the following definition.
Definition 2.7. LetXbe a Banach space. Say thatXsatisfies weak Grothendieck’s theorem if there is a positive constantCsuch that for allninNand any linear operatorufromX intol2n, there exists
π2(u)≤Cu. (2.25)
Remark 2.8. (1)Xsatisfies W.G.T. if and only ifX∗∗satisfies WGT.
(2)L1andL∞verify weak Grothendieck’s theorem. The spacesS1(see below) andB(l2)
(see [8, Corollary 4.2]) fail this.
(3) The classical definition is let X be a Banach space. We will say that X satisfies Grothendieck’s theorem if there is a constantCsuch that, for any linear operatorufrom Xinto a Hilbert spaceH, we have
(4) We can replaceHbyl2nfor any integern(i.e., there is a constantCsuch that for any
integernand anyu:X→ln2 we haveπ1(u)≤Cu). Also, this is equivalent to the dual
property (i.e., there is a constantCsuch that for every linear operator fromX∗into an L1-space, we haveπ2(u)≤Cu). GT implies WGT. IfXis of (classical) cotype 2, then
we have equivalence between GT and WGT becauseπp(X,Y)=π2(X,Y) for any Banach
spaceY and for allp≤2 (see [7]).
(5) The spaceL1verifies Grothendieck’s theorem. In [2] Bourgain proved thatL1/H1
is of cotype 2 and verifies Grothendieck’s theorem (L1is theL1-space relative to the circle
group andH1the subspace ofL1spanned by all functions{eint,n≥0}).
(6) Suppose thatX is a subspace ofC(K) and thatC(K)/X is reflexive. Then every operator with domainX and range a cotype 2 space is 2-summing [6,11]. As corollary, letXbe a reflexive subspace of anL1. Then, every operatoru:L1/X→l2is 1-summing.
(7) For any BanachEof cotype 2, Pisier has constructed in [12] a Banach spaceX which contains isometrically E such that, X and X∗ are both of cotype 2 and verify Grothendieck’s theorem.
3.Spfails WGT for all 1≤p≤2
We recall (see [14]) the noncommutative analogues oflpwhich is the Schatten classSp. Let
0< p <∞. We will denote byB(l2) the space of all bounded linear operatorsu:l2→l2and
bySpthe subspace of all compact operators such that tr|u|p<∞(where|u| =(uu∗)1/2).
We equip it with the norm if 1≤p <∞and thep-norm if 0< p <1:
up=tr|u|p1/ p (3.1)
for which it becomes a Banach space if 1≤p <∞and a quasi-Banach if 0< p <1. If p= ∞,S∞is the subspace of all compact operators onl2equipped with operator norm.
We have (Sp)∗=Sqfor 1< p≤ ∞and 1/ p+ 1/q=1, and alsoS∗1 =B(l2). We do not know
if the Schatten spacesSpare of the same cotype Kaˇsin as the usuallp-spaces for 1≤p≤2.
Finally, we denote bySn
p andB(ln2) the finite dimensional version ofSpandB(l2),
re-spectively.
Let 0< p≤q≤ ∞. We have foru∈B(ln2),
uq≤ up≤n1/ p−1/quq. (3.2)
LetRndenote the subspace ofSnpconsisting of alln×nmatricesusuch thatui,j=0 when
i=1 (first row matrices). Thena=uu∗is the matrix witha1,1=
n
j=1|u1,j|2= u22and
ai,j=0 when (i,j)=(1, 1). Hence|u|is the rank one operatoru2e1⊗e1. Its norm in
all spacesSn
p, 0< p≤ ∞is equal tou2. In particularRn equipped with theSnp-norm
is isometric toln
2. We denote by pnthe natural projection fromSnp intoRn defined by
pn(u)=vsuch thatv1j=u1jfor 1≤j≤n. We havepn ≤1.
The proposition to be proved now is the finite dimensional version of the theorem of extension.
Proposition 3.1. Suppose that for somep >0, there exits a constantCpsuch that for every
u ≤Cpu. Then
Cp≥C
√
n, (3.3)
whereCis an absolute constant.
Proof. Letunbe the operator sending thenvector basis ofEnto thenvector basis ofRn
(un(ϕi)=e1,i, 1≤i≤n). This operator is an isomorphism, by the above remark and (2.6).
We haveun ≤B1andu−n1 ≤A1. Letunbe an extension ofunto an operator fromL2n1
toSn
p, withun ≤Cpun. Consider now the following commutative diagram:
L2n1 un Sn
p
pn
En en
un
Rn u−1
n
En
(3.4)
Letqn=u−n1pnun. Thenqnis a projection fromL2n1 toEn. SinceEnisA1B1-isomorphic
tol2n(Theorem 2.1), we get by Grothendieck’s theorem [4] thatqn is 1-summing with
π1(qn)≤A1KGpnun. RestrictingqntoEnwe obtain for the identityinofEnthe
estima-tion √
n=π2
in
≤π2
qn
≤π1
qn
≤A1KGun≤A1KGCpun≤A1B1KGCp. (3.5)
This completes the proof.
Let nowᏮnbe theσ-algebra on [0, 1] generated by the Rademacher functions{r1,. . .,
rn}(rn(t)=sign(sin 2nπt)). The spaceLp([0, 1],Ꮾn,ν), whereνis the Lebesgue measure
in [0, 1], is isometric toL2n
p.
We denote byG (resp.,Gn) the closed linear subspace inL1([0, 1],ν) (resp.,L2
n
1 ) of
the Rademacher functions{rn}n∈N(resp.,{ri, 1≤i≤n}). Letg:G→L1([0, 1],ν) (resp.,
gn:Gn→L2
n
1) be the isometric embedding. By Khinchine’s inequalities, there are positive
constantsA1andB1such that for every (an) inl2we have
A1
n≥1
an2 1/2
≤
[0,1]
n≥1
anrn(t)
dν≤B1
n≥1
an2 1/2
. (3.6)
HenceG(resp.,Gn) is isomorphic tol2 (resp.,ln2). We will denote by α:l2→G(resp.,
αn:l2n→Gn) the isomorphism which mapsei ontori. We haveα ≤B1,α−1 ≤A1,
and also the same forαn.
Proposition 3.2. Suppose that for somep >0, there exits a constantCpsuch that for every
nand every linear operatorufromGntoSnpthere is an extensionu∈B(L2
n
1 ,Snp) ofuwith
u ≤Cpu. Then
Cp≥C
√
n, (3.7)
Proof. The same proof as inProposition 3.1. Theorem 3.3. Let 0< p≤ ∞. Letu:G→Spbe a compact linear operator. In general, there
is no continuous linear operatoruextendingutoL1([0, 1],ν).
Proof. Suppose that for any compact linear operatoru:G→Spthere is a bounded linear
operatoru:L1([0, 1],ν)→Sp extendingu. It follows from the open mapping theorem
that there is an absolute constantCpsuch that
u ≤Cpu (3.8)
for anyu. This implies byProposition 3.2thatCp≥C√nfor any integern. This is
im-possible whennis large enough.
Theorem 3.4. Let 0< p≤ ∞. Letu:E0→Spbe a compact linear operator. In general, there
is no continuous linear operatoruextendingu.
Proof. Using the same proof as inProposition 3.2(we takeEn
0instead ofGn) andTheorem
3.3, we show that the extension property concerning (L1([0, 1],ν),E0) fails for all 0< p≤
∞.
The following result shows that spaceSpfails GT.
Theorem 3.5. The spaceSpfails GT for all 1≤p≤ ∞and consequently WGT for 1≤p≤2.
Proof. Consider the following diagram:
Rn−−→in Snp pn
−−→Rn, (3.9)
whereinis the canonical injection. We have idRn=pn◦in. Since
√
n≤π1(idRn)≤π1(pn)
andpn ≤1, henceSpfails GT for all 1≤p≤ ∞. AsSpis of cotype 2 for 1≤p≤2 then,
by one result of Maurey, we haveπ1(pn)≤Cπ2(pn) for some constantC. This implies the
proof.
Remark 3.6. The spaceB(l2) fails weak Grothendieck’s theorem because by [8, Corollary
4.2] we haveπ2(B(l2),l2)=B(B(l2),l2).
4. Characterization of spaces which satisfy WGT
We start this section by recalling some notations and facts. We denote bylω
p(X) (resp.,
lnω
p (X)) the space of all sequences (xi) (resp., (xi)1≤i≤n) inXwith the norm
xi
lω
p(X)= sup
ξX∗=1
∞
1
xi,ξp 1/ p
<∞,
resp.,xilnω
p(X)= sup
ξX∗=1
n
1
xi,ξp 1/ p
.
We know (see [3]) thatlp(X)=lωp(X) for some 1≤p <∞if and only if dim(X) is finite. If
p= ∞, we havel∞(X)=l∞ω(X). We have also if 1< p≤ ∞,lωp(X)≡B(lp∗,X), andlω1(X)≡ B(cO,X) isometrically (where p∗ is the conjugate of p, i.e., 1/ p+ 1/ p=1). In other
words, letv:lp∗→Xbe a linear operator such thatv(ei)=xi(namely,v=∞1 ej⊗xj,ej denotes the unit vector basis oflp), then
v =xilω p(X)=
∞
1
ej⊗xj
lp⊗ˆ X
. (4.2)
We prove in the following theorem that the spaces which satisfy WGT and which hap-pen to be also of cotype strongly 2-Kaˇsin can be characterized by an extension property. Theorem 4.1. The following properties of a Banach spaceXare equivalent:
(i) the spaceX∗is of cotype strongly 2-Kaˇsin and verifies WGT;
(ii) there is a positive constantCsuch that for everyn∈Nand everyu:En→X, thenu
admits an extensionu:L2n
1 →Xsuch thatu/E n=uandu ≤Cu.
Proof. We prove that (ii)⇒(i). Letv:ln
2 →X be a linear operator. Consideru=vβ−n1:
En→X, thenuadmits an extensionu:L2n1 →Xsuch that
u ≤Cu ≤Cβn−1v ≤C/A1π2
v∗. (4.3)
FromCorollary 2.5, we obtain thatX∗is of cotype strongly 2-Kaˇsin andK2strong(X∗)≤
C/A1. Let nowu:X∗→ln2be an operator. First, we notice thatB(l2n,X∗∗)≡B(l2n,X)∗∗≡
B(X∗,l2n) isometrically. Sinceu:X∗→ln2 is inB(l2n,X)∗∗, then by Goldstine’s theorem,
there is a net of operatorsu∗i :X∗→ln
2which arew∗-continuous withui ≤ ufor all
iand{u∗i }converges touinw∗-topology ofB(ln2,X)∗∗. Asu∗i is 2-summing this implies
thatuis 2-summing andπ2(u)=lim i π2(u
∗
i ). Indeed,
π2(u)=sup
Tr(uv),v:l2n−→X∗∗∗π2(v)≤1}
=suplim
i Tr
u∗iv,v:ln
2−→X∗∗∗π2(v)≤1
=lim
i sup
Tru∗iv,v:ln
2−→X∗∗∗π2(v)≤1
=lim
i π2
u∗i
.
(4.4)
Let us consider the following commutative diagram:
L2n1
ui
En en
β−1
n
ln2 ui
X
by duality, we have
L2n
∞
e∗n
X∗
u∗i
u∗i
ln2 (β−1
n )∗
En∗
(4.6)
hence
π2
u∗i=π2
β∗nβ−1
n
∗u∗
i
≤βn∗π2
β−1
n
∗u∗
i
≤β∗nu∗i π2
e∗n≤β∗nβ−n1uiπ2
e∗n
≤A−1
1 B1uiπ2
e∗n
.
(4.7)
Thus
lim
i π2
u∗i ≤A−1
1 B1π2e∗nlim i
ui≤A−1
1 B1π2e∗nu. (4.8)
Consequently
π2(u)≤A−11B1π2
e∗nu. (4.9)
This shows thatXhas WGT because the numbersπ2(e∗n) are uniformly bounded by
Mau-rey’s theorem [7].
(i)⇒(ii). The spaceX∗is of cotype strongly 2-Kaˇsin which implies byCorollary 2.5 that for anyu:ln2→X,uadmits an extensionutoL2n1 such thatu/E n=uβ−n1andu ≤
K2strong(X∗)π2(u∗). AsX∗verifies WGT, thenπ2(u∗)≤Cuand hence
u ≤CK2
X∗u ≤CuC=CK2
X∗ (4.10)
which gives the extension.
We end this paper by the following remark.
Remark 4.2. We do not know ifSpfor 1≤p≤2 is of cotype 2-Kaˇsin.
Acknowledgment
The author is very grateful to the referees for pointing out some mistakes in the first version and for several valuable suggestions and comments which improved the paper.
References
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