spaces the local structure of non commut (1)

http://www.elsevier.com/locate/aim Advances in Mathematics 187 (2004) 257–319

COL

_p

_{spaces—the local structure of}

non-commutative

L

p

spaces

M. Junge,

a,,1

N.J. Nielsen,

b,2

Zhong-Jin Ruan,

a,1

and Q. Xu

c

a_{Department of Mathematics, University of Illinois, 273 Altgeld Hall MC 382 1409, West Green Street,}

Urbana-Champaign, IL 61801, USA

b

Department of Mathematics & Computer Science, The University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark

c

Laboratoire de Mathe´matique, Universite´ de Franche-Comte´, Route de Gray, 25030 Besan@on, Cedex, France

Received 9 October 2001; accepted 14 August 2003 Communicated by Virgil Voiculescu

Abstract

Developing the theory ofCOL_{pspaces (a variation ofthe non-commutative analogue of}L_p spaces), we provide new tools to investigate the local structure ofnon-commutativeLpspaces. Under mild assumptions on the underlying von Neumann algebras, non-commutative Lp spaces with Grothendieck’s approximation property behave locally like the space ofmatrices equipped with the p-norm (ofthe sequences oftheir singular values). As applications, we obtain a basis for non-commutative Lp spaces associated with hyperfinite von Neumann algebras with separable predual von Neumann algebras generated by free groups, and obtain a basis for separable nuclearC_-algebras.

r2003 Elsevier Inc. All rights reserved.

MSC:46L52; 46L07

Keywords:Non-commutativeLpspaces; (cb-)basis; Non-commutativeLpspaces

_{Corresponding author.}

E-mail addresses: junge@math.uiuc.edu (M. Junge), njn@imada.sdu.dk (N.J. Nielsen), ruan@ math.uiuc.edu (Z.-J. Ruan), qx@math.univ-fcomte.fr (Q. Xu).

1_{Junge and Ruan were partially supported by the National Science Foundation, DMS 00-88928 and}

DMS 98-77157.

2_{Supported by the Danish Natural Science Research Council, Grant 9801867.}

0. Introduction

Non-commutative integration theory goes back to the work ofMurray and von Neumann and has been investigated in the context ofvon Neumann algebras by

Dixmier [D1], Segal[Se], Kunze [Ku], Nelson[Ne], Connes [C3], Haagerup[Ha2],

Kosaki[Ko]and many other researchers. Following the philosophy ofquantization,

non-commutative Lp spaces could be considered as non-commutative function

spaces. In particular, the classical Banach spaces oftrace class operators, Hilbert–

Schmidt operators and more generally Schattenp-classes share many properties with

their commutative counterparts, the classical c_p spaces (see [A1,A2,Fa,GK,TJ]).

Since these spaces are not compatible with the usual lattice structure ofclassical

function spaces (except for p_¼2 see [GL,P1]), their local structure has not been

investigated as thoroughly as for classical function spaces. The main intention of this

paper is to show that non-commutativeLp spaces with the bounded approximation

property (BAP) have very nice local properties, for instance, they can be paved out

by copies offinite-dimensional non-commutative Lp spaces. This can be achieved

under mild assumptions on the underlying von Neumann algebra by combining concepts from the local theory of Banach spaces with more recent tools from the theory ofoperator spaces. In contrast to the classical theory, these more abstract

techniques provide appropriate tools to prove the existence ofbases for some

important spaces like nuclear (in particular type I) C_{-algebras, preduals of}

hyperfinite von Neumann algebras, and non-commutativeLpspaces associated with

hyperfinite von Neumann algebras or the von Neumann algebra generated by the left regular representation ofa countable free group.

Let us first recall the classical notion ofL_{pspaces. Following Lindenstrauss and}

Pe"czyn´ski[LP]a Banach spaceX is called anL_p;l spaceifevery finite-dimensional

subspaceECXis contained in a finite-dimensional subspaceECFCXsuch that for

n¼dimðFÞthe Banach–Mazur distance satisfies

d_ðF;cn_pÞpl: ð0:1Þ

Ifthis is true for somel; Xis called anL_pspace. Ifthis is true for alll41;thenXis

isometrically isomorphic to LpðO;S;mÞ for some measure space ðO;S;mÞ and vice

versa. For 1opoN every separable L_{p space is isomorphic to a complemented}

subspace of Lp½0;1 and therefore inherits the bounded approximation property.

(The absence ofthe approximation property for general non-commutativeLpspaces

is a substantial but interesting drawback in the non-commutative setting.) The

‘paving’ definition (0.1) is not very practical for showing that the dual of anL_pspace

is an L_p0 space (p0¼_pp₁ the conjugate index). However, using the fundamental

Kadec–Pe"czyn´ski dichotomy and a ‘cut and paste’ technique, Lindenstrauss and

Rosenthal[LR]managed to prove that the dual ofanL_{pspace is an}L_p0 space and

that the copies ofcn_pin the definition ofL_p spaces may be assumed to be uniformly

complemented. In order to underline the different notions in the non-commutative

note by the remarks above that in the commutative setting L_{p spaces are indeed}

CL_p spaces.

In the non-commutative setting a Kadec–Pe"czyn´ski dichotomy (see[KP]) is not

available at the time ofthis writing. This forces us to introduce the class ofCOL_p

spaces, the non-commutative analogue ofCL_{p-space. In contrast to the}OL_p-spaces

defined by Effros and Ruan[ER1], we assume in addition that the finite-dimensional

copies ofnon-commutative Lp spaces are uniformly (completely) complemented.

This class of COL_{p spaces (for a precise definition see Section 2) seems to be the}

right substitute for the class ofL_p spaces in Banach space theory. We refer to the

end ofSection 2 for a discussion ofthese two notions. In the commutative theory

Johnson et al.[JRZ]showed that a separableL_{pspace admits a basis. Refining their}

techniques Nielsen and Wojtaszczyk showed that this basis locally looks like the

basis ofcn_p:We use this approach as a guideline to discover the local structure ofa

separableCOL_{pspace and construct (very nice operator space) bases therein. Let us}

note that due to the work ofBourgain [Bo], Bourgain et al. [BRS] many

non-isomorphicL_{p spaces are known, and thus many ofthem are not isomorphic to}

standard examplesLp½0;1;cporcp"c2:ThereforeLpspaces have a very rich global

structure.

The right framework for the investigation of the local structure of

non-commutative Lp spaces is the category ofoperator spaces. We will now indicate

some elementary operator space notations and in particular the notion of OL_p

-spaces, introduced by Effros and Ruan[ER1]. Anoperator space X is a norm closed

subspace ofsomeBðHÞequipped with the distinguished operator space matrix norm

inherited from MnðXÞC_Bðcn_2ðHÞÞ: An abstract matrix norm characterization of

operator spaces was given by Ruan (see e.g.[ER2]). The morphisms in the category

ofoperator spaces are completely bounded maps. Given operator spacesX andY;a

linear map T:X-Y is completely bounded ifthe corresponding linear maps

Tn:MnðXÞ-MnðYÞdefined byTnð½xijÞ ¼ ½TðxijÞare uniformly bounded, i.e.,

jjTjjcb¼sup nANjj

TnjjoN:

A mapT is a complete contraction(respectively, acomplete isometry, or acomplete

quotient) ifjjTjjcbp1 (respectively, ifeachTn is an isometry, or a quotient map). A

map T is said to be a complete isomorphism ifit is a completely bounded linear

isomorphism with a completely bounded inverse. In this case, we let

dcbðX;YÞ ¼inffjjTjjcbjjT1jjcb: T a complete isomorphism from X onto Yg

denote the completely bounded Banach–Mazur distance (in short cb-distance) of X

andY (see[P4]).

Variations ofGrothendieck’s approximation property inspired crucial

develop-ments in operator algebras and operator spaces. An operator spaceXCBðHÞhas the

operator space approximation property, in short OAP, ifthere exists a net offinite

identity onK#_minXCBðc₂#HÞ;whereKdenotes the space ofcompact operators

on c₂: An operator space has the completely bounded approximation property (in

short CBAP) ifthere exists a net ðTiÞoffinite rank maps converging in the

point-norm topology to the identity onX and supijjTijjcboN:We say that an operator

spaceX has acb-basisif,X has a basis_ðxn_Þand the natural projection maps

Pn

XN

k¼1

akxk

! ¼X

n

k¼1

akxk

satisfyK¼supnjjPnjjcboN:In this case we callðxnÞa K-cb-basis.

For non-commutative Lp spaces Pisier [P5] introduced a very natural operator

space structure by interpolation (see[BL]for interpolation theory). Indeed, it is

well-known that the Schattenp-classesSp can be obtained by complex interpolation

Sp¼ ½K;T1

p

:

HereT_{¼S1 denotes the space oftrace class operators and} K_{¼SNthe space of}

compact operators. Moreover, the natural (operator space structure preserving)

duality betweenx¼ ½xijAK _{andy¼ ½yij}AT_{is given by}

/x;yS_¼X

ij

xijyij¼trðxytÞ:

Pisier[P5]proved that

MnðSpÞ ¼ ½MnðKÞ;MnðTÞ1

p

define matrix norms on Sp which satisfy Ruan’s abstract characterization for

operator spaces. Therefore, there is an isometric embeddingjp:Sp-Bðc2Þinducing

these matrix norms and this is nowadays calledthe natural operator space structure of

Sp: We refer to [P5] for many nice features. Similarly, we may obtain a natural

operator space structure onLpðAÞfor every finite-dimensional C-algebraA:

Let us recall the operator space analogue ofL_{p spaces. An operator spaceX is}

called an operatorL_{p space (in short}OL_{p;l space) ifX can be paved out by copies}

offinite-dimensionalLp spaces, where thecb-distance is uniformly controlled byl:

An operator spaceXis called anOL_{pifit is an}OL_{p;lfor somel}41:In this case, we

use the parameterOL_{pðXÞ ¼infl; where the infimum is taken over alll’s above.}

For a precise definition see Section 2.

During the last few years, OL₁ spaces have been intensively studied in

[ER1,JOR,NO]. In particular, it was proved in [ER1]that the predualN ofa von

Neumann algebraN is anOL_{1 space ifand only ifN is hyperfinite. Moreover, a}

separable operator spaceX is anOL_{1space with}OL_{1ðXÞ ¼1 ifand only ifit is the}

operator predual ofa hyperfinite von Neumann algebra (see[NO]).

ConcerningOL_{Nspace, we recall that by Szankowski’s result (see[Sz1]) the space}

OL_{Nspace. In fact contrary to the commutative case, the} OL_{N property forC}

-algebras is very restrictive. More precisely, according to results by Pisier[P4], Effros

and Ruan[ER1], Kirchberg[Ki2], and Junge et al.[JOR]we know that aC-algebra

Ais anOL_{N;lspace for somelifand only ifAis nuclear. In Theorem 3.11, we will}

improve a recent result in[JOR]by showingOL_NðAÞp_3:The microscopic indexl

even provides some additional information on the structure of the underlyingC

-algebra. For example, aC-algebra is stably finite iflpð1þ

ffiffi 5

p

2 Þ 1

2 (see[JOR]). From

these results, we can see that the local operator space structure provides a very important tool for the investigation of operator algebras.

However, not much work has been done forOL_{pspaces in the range 1}opoN:It

is known that every OL_p space is completely complemented in some

non-commutative Lp space. However, it can be derived from Szankowski’s work[Sz2]

that there are finite von Neumann algebras with separable predual such thatLpðNÞ

does not have the approximation property (see Theorem 2.19). Moreover, it is not

known whether everyOL_{p space has the CBAP. In order to use the concepts from}

Banach space theory, we will work with the analogue ofCL_{p spaces. An operator}

spaceX is called a COL_p;l space ifit is paved by complemented copies ofLpðAÞ’s

wherecb-distance and thecb-norm ofthe projections are uniformly controlled byl:

Ifthis is true for somel;X is called aCOL_{pspace. Ifwe can replace theLpðAÞ’s by}

Sn

p’s, we call this aCOSp;l;COSpspace, respectively. Again we refer to Section 2 for

a precise definition. Combining Banach space techniques from [JRZ] with

applications ofthe Fubini Theorem from [Ju2], we obtain the following results on

COL_{p spaces.}

Theorem 0.1. Let1opoNand X an operator space.X is aCOS_{pspace if and only if}

X has the CBAP,idX admits a cb-factorization through an ultrapower of Sp;and X contains completely complemented Sn

p’s uniformly.

Theorem 0.2. Let1op; p0oNwith1

pþp10¼1and X an operator space. Then X is a

COL_{p space if and only if X is a}COL_p0 space.

The cases p¼1; p¼N remain true ifwe assume in addition that X _{has the}

CBAP and X is locally reflexive (in the operator space sense). Using an idea of

Kirchberg, we can construct an operator spaceXsuch thatXisCOL_{1butX does}

not have the CBAP. In Section 4, we extend the results ofJohnson et al. [JRZ],

Nielsen and Wojtaszczyk[NW]toCOL_{p spaces.}

Theorem 0.3. Let1pppNand X a separableCOL_pspace(such that in addition X

has the CBAP and X is locally reflexive for pAf1;Ng).Then X has a cb-basis.

Before we state our main application to non-commutativeLp spaces, we have to

clarify the ‘mild assumptions’ on the underlying von Neumann algebra N: AC

universal representation ACACBðHÞthere is a contraction P:BðHÞ-A such

thatPjA¼idA:AC-algebraB is said to beQWEP ifthere exists aC-algebra A

with the WEP and a closed two-sided idealIsuch thatB_¼A=I:It is a long standing

open problem whether every C-algebra is QWEP (see [Ki1] for many equivalent

formulations). Note that a hyperfinite von Neumann algebra is injective, hence has WEP and thus is QWEP.

Theorem 0.4. Let N be a QWEP von Neumann algebra with separable predual. Then

for1opoNthe following are equivalent

(i) LpðNÞhas the OAP;

(ii) LpðNÞhas the CBAP;

(iii) LpðNÞis aCOLp space;

(iv) LpðNÞhas a cb-basis.

In particular,if one of the conditions above is satisfied,then LpðNÞis anOLp space.

We apply Haagerup’s pioneering work [CH,Ha3] on approximation properties

and an interpolation argument (see e.g.[JR]) in order to obtain a result forLpspaces

associated to the von Neummann algebra VN_ðF_nÞ generated by the left regular

representation ofthe free group F_n: As so often in harmonic analysis, the spaces

LpðVNðFnÞÞbehave much nicer for 1opoNthan for the border casespAf1;Ng:

Indeed, here L1ðVNðFnÞÞis not an OL1 space and CredðFnÞis not an OLN space

becauseF_{n is not amenable.}

Theorem 0.5. Let1opoNandF_{nthe free group with n generators. Then LpðVNð}F_nÞÞ

is aCOL_p space(hence anOL_p space)and has a cb-basis.

We note that the existence ofa basis for L1ðVNðFnÞÞ or CredðFnÞ is an open

problem. In contrast to the commutative theory a non-commutative C-algebra A

might not have enough orthogonal finite-dimensional representations. Using the

operator space structure ofA instead, we can obtain sufficiently many information

about the local structure ofAin the cases ofnuclearC-algebras.

Theorem 0.6. Every separable nuclear C-algebra has a cb-basis.

For researchers interested only in Banach space theory, we should mention that all

the results hold in the Banach space sense. For example in Theorem 0.4,LpðNÞhas

Grothendieck’s approximation property iff it has a basis. A positive solution to the

basis problem for non-commutativeLp spaces has previously only been known for

the class oftypeIvon Neumann algebras and the hyperfiniteII1andIINfactors (see

[Su]). However, we note that passing to tensor products of COL_p spaces already

requires cb-norm estimates ofthe basis projections and thus operator space

LpðNÞspaces even on a purely Banach space level. We are indebted to W. B. Johnson

for stressing the fact that the existence of a basis in L_p spaces can be proved by

entirely local arguments. Indeed, this entirely local approach unifies the construction

ofbases forL_{p spaces for all values 1}pppNeven in the commutative case (using

the appropriate new notion ofcontainingcn_p’s ‘far out’).

In order to make this paper more accessible (for researchers with a Banach space background), we postpone arguments using modular theory ofvon Neumann

algebras to the end ofSection 5. In the subsequent paper[JRX], we will investigate

the isometric theory in the hyperfinite (non-semifinite) case. Further applications of

Lp spaces associated with discrete groups will be given in[JR].

1. Notation and preliminary results

We will use standard notation in operator algebras[D2,KR,Pe,Ta], and Banach

space theory[LT]. In particular, given a Hilbert spaceH; we letBðHÞdenote the

space ofall bounded linear operators on H: Our general references for operator

spaces are[ER2,P6]. Let us recall some basic notations. A completely bounded map

Pon an operator spaceXis acompletely bounded projectionifP2_{¼P:A subspaceX}

ofan operator space Y is called a completely complemented (respectively, a

completely contractively complemented) subspace in Y ifthere is a completely

bounded (respectively, completely contractive) projection fromY ontoX:IfX is an

operator space, then its dual spaceX is an operator space with matrix norms given

by the isometric identifications

MnðXÞ ¼CBðX;MnÞ

(see[BP,ER2]). This operator space structure onX_{is called theoperator(space)dualof}

X:IfX is an operator space, then the canonical embeddingi:X-X isa completely

isometric injection, i.e.idMn#i is isometric for allnAN: IfT:X-Y is a completely

bounded map, then its adjoint map T_{:Y-X is also completely bounded with}

jjT_jj

cb ¼ jjTjjcb:Using the Arveson–Wittstock–Hahn–Banach theorem[ER2,Pa], it is

easy to show that ifT is a complete isometry, thenT isa complete quotient map, i.e.

idMn#Tmaps the open unit ball onto the open unit ball for allnAN:Similarly, ifTis

a complete quotient map, then T is a complete isometry. Given a von Neumann

algebraN;the canonical embeddingi:N+Ninduces an operator space structure on

N:With these matrix norms, we have the complete isometry

N¼ ðNÞ:

In the following, we will use the notationSp (resp.Sn_p) for the spaces of all compact

operators on the Hilbert spacesc_2¼c_2ðN_Þ(resp.cn₂) such that

jjxjjp¼ ½trððxxÞ p

2_Þ 1

We will always work with the canonical duality betweenSpandSp0 ð_p1þ_p10¼1Þgiven by

/_½aij;½bijS_¼X

ij

aijbij ð1:1Þ

and obtain a complete isometry S_p¼Sp0: Similarly, if A is a finite-dimensional C

-algebra given by

N onto A is completely contractive onS

n

N and the same map is also

completely contractive onSn

1: Therefore, we may apply complex interpolation for the

compatible pair_ðA;A_ÞCðSn

N;S n

1Þand obtain the natural operator space structure on

LpðAÞ ¼ ½A;A1

p¼ Sn1

p "p?"pSpnlCSnp:

We refer to[BL]for the complex interpolation method. Note that by complementation,

we still have a complete isometry

LpðAÞ ¼Lp0ðAÞ

andLpðAÞis a completely contractively complemented subspace ofSnpfor 1pppN:In

the sequel, we will also use an infinite-dimensional analogue ofthese spaces. Letm_¼

ðmðnÞÞnAN be a sequence ofnatural numbers and

b_{ðmÞ ¼}Y

n

MmðnÞ;

the von Neumann algebra obtained as block diagonals inBðc_2Þ:In the Banach space

literature one may also write b_{ðmÞ ¼ ð}P

n"MmðnÞÞN: Then the predual of bðmÞ is s_{1ðmÞ ¼ ð}P

n"S mðnÞ

1 Þ1;i.e. the block diagonals inS1:Since the projection onto these

block diagonals is completely contractive in both cases, we see that

s_{pðmÞ ¼} X

is completely contractively complemented inSp:Forp¼N;we use the notations_NðmÞ

for thec0 sum. In the special case wheremis given bymðnÞ ¼nfor allnAN;we will

simply use the notations_p:

As in Banach space theory, ultraproducts turn out to be a useful tool in the study

an infinite index setIandðXiÞiAIis a family of operator spaces, then we consider the

ultraproduct

Y

U

Xi¼

Y

iAI

Xi=JU;

whereQ_iAI Xi¼ fðxiÞ jsupijjxijjoNgis the space ofall bounded families, and

JU¼ ðxiÞA Y

iAI Xi

limU jjxijjXi¼0

( )

is the norm closed subspace in Q_iAI Xi offamilies tending to 0 along U: An

ultraproductQU Xi ofoperator spaces carries canonical matrix norms given by

Mn

Y

U

Xi

! ¼Y

U

MnðXiÞ:

For details see[ER2,P3,P5]. IfðAiÞis a family ofC-algebras, it is well-known that

Q

U Aiis again aC-algebra. It is also known (see[Gr,Ra1,Ra2]) that for 1ppoN;

we haveQU Sp¼LpðNÞfor some von Neumann algebraN:The following result is

due to Junge[Ju2] and holds only forpAð1;NÞ:

Theorem 1.1. Let E and F be finite-dimensional operator spaces and1opoN:If we

have a commuting diagram of completely bounded maps

then for any e40; _{there exist an integer n and a commuting diagram of completely}

bounded maps

such that

jj˜r_jjcbjj˜s_jjcbojjrjj_cbjjsjj_cbþe:

Approximation properties play an important role in operator algebras and

operator spaces. LetX and Y be operator spaces. A linear map T:X-Y is said

to have the completely bounded approximation property (in short CBAP) ifthere

point-norm topology and sup_ijjTijjcbpl:In this case, we let

LðTÞ ¼infl

denote the infimum ofalllas above. IfT¼idX we say thatXhas the CBAP and let

LðXÞ ¼LðidXÞ:

Let 1pppN:A linear mapT:X-Y is said to have thegp-approximation property

(in shortgp-AP) (i.e. can be approximately factored throughSnpspaces) ifthere exist

diagrams ofcompletely bounded maps

which converges in the point-norm topology toT and satisfies sup_ijjrijjcbjjsijjcbpl

for some constantloN:We let as above

gap_{p ð}T_{Þ ¼}infl:

IfT _¼idX;we say thatX has theg_p-approximation propertyand let

gap_{p ð}X_{Þ ¼}gap_{p ð}idXÞ:

It is clear that ifT has theg_p-AP, thenT has the CBAP with

LðT_Þpgap_p ðT_Þ:

In the analysis ofapproximation properties, small perturbation arguments provide an essential technical tool. Let us recall the following operator space analogue of a

classical Banach space argument due to Pisier[P5].

Lemma 1.2. Let X be an operator space and ECX an n-dimensional subspace with a

biorthogonal system x1;y;xn;x1;y;xn(i.e.jjxijjp1;jjxjjjp1and xiðxjÞ ¼dijfor all i;j¼1;y;n).Let 0oeo1;and T:E-X a linear map such that

jjT_ðxiÞ xijjp

e

n

for all i_¼1;y;n:Then there exists a complete isomorphism W:X_-X such that

for all xAE and

jjWjjcbpð1eÞ

1_;

jjWidjjcbp

e

1e and jjW

1_jj

cbpð1þeÞ: ð1:2Þ

As in the category ofBanach spaces, we may obtain the following result.

Corollary 1.3. Let X and F be operator spaces with F finite-dimensional and let

r:F-X and s:X-F be maps such that sr_¼idF: If ECX is an n-dimensional subspace and0oeo1

2 such that

jjrsðxÞ xjjp ejjxjj

n_jjr_jjcbjjs_jjcb

for all xAE:Then there exist maps ˜r:F_-X and ˜s:X_-F such that ˜s˜r¼idF; ˜r˜sj E¼ idE and

jj˜r_jjcbjj˜s_jjcbpð1eÞ1ð12eÞ1jjrjj_cbjjsjj_cb:

Proof. Applying Lemma 1.2 to e0¼ e

jjrjjcbjjsjjcb p1

2 and T ¼rs; we may obtain a

complete isomorphismW:X-X such thatjjidWjjcbp e

0

1e0andWTðxÞ ¼xfor all

xAE:Then we deduce

jjidF sWrjjcb¼ jjsðidWÞrjjcbpjjrjjcbjjsjjcb

e0

1e0p2e:

Hence, forb¼ ðsWrÞ1 we obtain the estimate

jjbjjcbpð12eÞ

1_:

We define ˜s¼bs and ˜r¼Wr: Clearly, ˜s˜r¼bsWr¼idF: For xA_E; _{we observe}

that

sWrsðxÞ ¼sWTðxÞ ¼sðxÞ:

Hence,sWrjsðEÞ¼idsðEÞ and thereforebjsðEÞ¼idsðEÞ:Thus, we get

˜r˜s_ðx_{Þ ¼}Wrbs_ðx_{Þ ¼}Wrs_ðx_{Þ ¼}WT_ðx_{Þ ¼}x

for allxAE:Using thecb-norm estimates forbandW;we obtain the assertions. _&

Lemma 1.4. Let T:X-Y be a completely bounded map.

(i) T:X-Y has the CBAP withLðTÞolif and only if for every finite-dimensional subspace ED_X;_{there exists a finite rank map u:X-Y such that} jj_ujj

cboland u_ðx_{Þ ¼}T_ðx_Þfor all xAE:

(ii) T:X-Y has thegp-AP withgapp ðTÞolif and only if for every finite-dimensional subspace ED_X; _{there exist n}AN _{and maps u:X-S}n

p; v:Spn-Y such that

jju_jjcbjjv_jjcbol and vuðxÞ ¼TðxÞfor all xAE:

Proof. Obviously the second assertion in (i), (ii) implies the CBAP, gp-AP,

respectively. Since the arguments are very similar, we will only show the missing

implication in (i). IfE is a finite-dimensional subspace of X;then T_ðE_Þis a

finite-dimensional subspace ofY:We can find vectorsx1;y;xkinEsuch thatðTðxiÞÞki¼1is

part ofa biorthogonal system inTðEÞ:Choose 0odo1 such that_ð1_þdÞ2LðTÞol:

Since T has the CBAP, there exists a finite rank map T˜:X-Y such that

jjT˜_jjcbo_ð1_þd_ÞL_ðT_Þ and _jjT˜_ðxiÞ TðxiÞjjokd for all i¼1;y;k: It follows from

Lemma 1.2 that there exists a complete isomorphismW:Y-Ysuch thatWTðxiÞ ¼

˜

T_ðxiÞforði¼1;y;kÞandjjW1jjcboð1þdÞ:Hence,u¼W1T˜:X-Y is a finite

rank map which satisfies the requirement ofthe assertion. &

Using the uniform convexity ofSp(see[TJ]) it is easy to prove the following

well-known fact. We refer to[ER1]for the details.

Lemma 1.5. Let1opoN:ThenQUSpis reflexive for every ultrafilterU:Moreover,

everyOL_{p space is completely contractively complemented in some} Q_{U Sp and thus}

reflexive.

Proposition 1.6. Let1opoNand X an operator space. Then X has theg_p-AP if and

only if X has the CBAP and there exists a free ultrafilterU_{on some index set I such}

that X is completely complemented inQU Sp;i.e. there exists a commuting diagram of

completely bounded maps

ð1:3Þ

Proof. If X has the g_p-AP, then X has the CBAP, and there exist diagrams of

which approximately commute in the point-norm topology and in addition satisfy

sup_jjrijjcbjjsijjcb¼loN:

Ifwe let U _{be a free ultrafilter on the index set I; then we obtain a commuting}

diagram ofcompletely bounded maps

ð1:4Þ

where we let s:X-QU S_pni be the map given by sðxÞ ¼ ðsiðxÞÞU and

r: QU Sn_pi-X the map given by

/rððziÞUÞ;xS¼lim

U x

_ðriðziÞÞ

for all xAX: Since each Sni

p is completely contractively complemented in Sp;

Q

U Sn_pi is completely contractively complemented in

Q

U Sp; and thus we can

actually replaceQU Sn_pi in (1.4) by

Q

USp:HenceX is isomorphic tosðXÞCQU Sp

and thus reflexive according to Lemma 1.5. Thus we obtain the commuting diagram

On the other hand, let us assume thatX has the CBAP and satisfies diagram (1.3)

with _jjr_jjcbjjs_jjcbpC: It follows from Lemma 1.4 that for any finite-dimensional

subspace ED_{X and e}40; _{there exists a finite rank map u:X-X such that}

jjujjcboð1þeÞLðXÞ and ujE¼idE for all xAE: In particular u2jE ¼idE and it

suffices to show thatu2 _{factors throughS}m

p:Let us consider the finite-dimensional

operator spaceG_¼X=ker_ðu_Þ with quotient mapqG:X-G and the induced map

ˆu:G-X such that u¼ ˆuqG: Note that ˆu has the same cb-norm as u: Let F ¼

u_ðX_ÞCX with inclusion map i_F:F_-X: Then u ˆu¼urs ˆu:G_-F satisfies the

assumption ofTheorem 1.1 and hence admits a factorization u ˆu_¼

vw; w:G-Sm_p; v:S_pm-F such that

jjvjjcbjjwjjcbpð1þeÞjjujj

2

cbjjrjjcbjjsjjcbpð1þeÞ

3

LðXÞ2C:

Thus u2_¼i

FvwqG factors through Spm and satisfies the corresponding cb-norm

estimate. Therefore,X has theg_p-AP withgap

p ðXÞpCLðXÞ

Remark 1.7. The same result holds ifwe replace thecb-norm by the operator norm in all instances above. Indeed, in Theorem 1.1, this can be easily proved by using

finited-nets in the unit ball of EandF:

In the category ofoperator spaces Proposition 1.6 is no longer true for p¼1:

Indeed, if N is a von Neumann algebra, then gap_{1 ð}N_Þol ifand only ifN is l

-semidiscrete, and thus injective by Pisier[P4]or Christensen and Sinclair[CS]. LetF_n

denote the free group ofn generators. It is known that the von Neumann algebra

VNðF_{nÞ is not injective (for any l}41), but satisfies LðVNðF_nÞ

Þ ¼1 (see [Ha3]).

Using an argument ofWassermann (or the fact that VN_ðF_nÞ is QWEP with the

results in [EJR, Section 7]and [NO]), we see that there are complete contractions

r:VN_ðF_nÞ

-Q

U S1;s:

Q

U S1-VNðFnÞ such that

HenceVNðF_nÞ

satisfies the assumptions ofProposition 1.6 without having theg1

-AP. In Theorem 5.7, we will show that for 1opoN; L_pðVNðF_nÞÞ has theg_p-AP.

This indicates that, as so often in harmonic analysis, the Lp spaces in the range

1opoNbehave much nicer than the extreme casesp¼1 and p¼N:

2. COL_{p and} OL_{p spaces}

In this and the following sections (unless stated explicitly otherwise) we will work in the category ofoperator spaces. This means that all linear maps, inclusions, quotient maps and projections are to be understood as completely bounded maps, complete isomorphisms with values in the images, complete quotient maps and completely bounded projections, respectively. This convention will simplify our presentation but is by no means necessary. Let us point out that all the results (stated here in terms of operator spaces) hold true in the category ofBanach spaces. Some ofthe proofs are

slightly easier for Banach spaces or can be found in the literature, namely in[JRZ,NW].

Therefore, we decided to emphasize the modifications required for operator spaces.

An operator spaceX is called anoperatorL_{p space(in short}OL_{pspace) ifthere}

exists a constantl4_{1 and a family}ð_FiÞ

iAI offinite-dimensional subspace such that

S

iFi is dense inXand for every indexithere exists a finite-dimensionalC-algebra

Ai such that

dcbðLpðAiÞ;FiÞpl: ð2:1Þ

In this case, we denote byOL_{pðXÞ ¼infl;where the infimum is taken over alllas}

above. Moreover, we say that X is an OL_{p;l-space, if} OL_pðXÞpl: We call X an

OS_{p;l spaceifwe can replace theLpðAiÞ’s in (2.1) byS}ni

An operator space X is called acompletely complementedOL_{p;l space (in short} COL_p;l space) for some constant l41 ifthere exist a family offinite-dimensional

C_-algebrasðA

iÞand commuting diagrams ofcompletely bounded maps

ð2:2Þ

such thatrisi-idX in the point-norm topology onX andjjrijjcbjjsijjcbpl:We callX

a completely complemented OS_{p;l space (in short} COS_{p;l space) ifwe can replace}

LpðAiÞin (2.2) bySni

p:We say thatXis aCOLpspace (respectively, aCOSpspace) if

it is aCOL_{p;l space (respectively, a}COS_{p;l space) for somelX1:In this case, we}

denote byCOL_{pðXÞ ¼infl(respectively,}COS_{pðXÞ ¼infl), where the infimum is}

taken over alllsuch that X is a COL_p;l space (respectively, aCOS_p;l space). The

following perturbation result (Lemma 2.1) shows that these definitions of OL_p

(respectively,COL_{p spaces) are consistent with the idea ofpaving out the operator}

space X by copies (respectively, complemented copies) offinite-dimensional

non-commutativeLpspaces. Since the proofis very similar to the proofofLemma 1.4 we

will leave the details ofthe proofofLemma 2.1 to the reader.

Lemma 2.1. Let X be an operator space and l41:

(i) X is anOL_{pspace with}OL_pðXÞolif and only if there exists al0olsuch that for every finite-dimensional subspace E of X there exists a finite-dimensional space ECFCX and a finite-dimensional C-algebra A such that

dcbðLpðAÞ;FÞol0:

(ii) X is aCOL_{pspace satisfying}COL_pðXÞolif and only if there exists al0olsuch that for every finite-dimensional subspace ED_X; _{there exist a finite-dimensional} C-algebra A and a commuting diagram of completely bounded maps

with jjrjjcbjjsjjcbpl0 and rsðxÞ ¼x for all xAE: A similar result holds forOS_{p spaces and}COS_{p spaces.}

It follows from Lemma 2.1 that everyCOL_pspace is anOL_p space. Forp_¼N;

the two notions are equivalent by the injectivity offinite-dimensionalC-algebras.

space copies offinite-dimensionalC-algebras, it is not clear whether they might be assumed to be norm complemented.) In the context ofoperator space the two

notions are equivalent for p¼1 if l is sufficiently close to 1 (see [Oz]). It is not

known whether these notions are still equivalent for largel:We refer to the end of

this section for more open problems. Let us state the main result in this section.

Theorem 2.2. Let 1pppNand X an operator space with the g_p-AP. If X contains

Sn

p’s(respectively,complemented Snp’s)then X is anOSpspace(respectively,aCOSp space).

HereX is said to contain S_pn’sifthere exists a constantCsuch that for everynAN_;

we can findGnCX such that

dcbðGn;Sn_pÞpC:

We note that in the Banach space literature the term ‘X contains cn_p’s uniformly’

(respectively ‘X contains cn_p’s uniformly complemented’) is in use. If we want to

specify the constantCwe sayX contains S_pn’s with constant C. Accordingly, we say

that X contains complemented Sn

p’s (with constant C) iffor every nAN there are

rn:Sn_p-X andsn:X-S_pn such that

snrn¼idSn

p and jjrnjjcbjjsnjjcbpC:

As a technical (but important) modification we say that X contains complemented

Sn_p’s with respect to Y ifYC_{X ands}

nðSpn0ÞCY for allnAN:

Although this clarifies the assumptions ofTheorem 2.2, the proofrequires

‘sufficiently many orthogonal’ copies of Sn

p with respect to any finite-dimensional

subspace of X: Note that in the commutative setting this is an immediate

consequence ofthe Kadec–Pe"czyn´ski dichotomy. In our setting, we have to use a

formal definition of ‘sufficiently orthogonal’. We say that an operator space X

contains complemented S_pn’s far outifthere exists a constantC4_{0 such that for every}

finite-dimensional subspace ECX and for every nAN and e40; there exist

rn:Sn

p-X;sn:X-Spn such that

snrn¼idSn

p; jjrnsnjEjjcbpe and jjrnjjcb jjsnjjcbpC:

Again, we use ‘with constantC’ and ‘with respect toY’ as above. Similarly, we say

that X contains Sn

p’s far out (with constant C) iffor every finite rank map

T:X-X; nAN_ande40;there existsGnC_{X such that}

dcbðGn;Sn_pÞpC and jjTjGnjjcbpe:

Note that it suffices to have jjTjGnjjpe because Gn is finite dimensional. Indeed,

ad-dimensional operator spaceE (see e.g.[EH])

jjT_jjcbpdjjTjj: ð2:3Þ

Similarly, the condition_jjrnsnjEjjcbpecan be weakened to jjrnsnjEjjpe:Let us start

with the most natural class ofexamples (see the preliminaries for the definition ofs_p:)

Example 2.3. Let 1pppN: The space s_p is a COS_p-space with constant

COS_pðs_{pÞ ¼1:} s_{p contains complemented S}n

p’s far out. Moreover, every operator

space containings_{pcompletely complemented contains complementedS}n

p’sfar out.

Proof. For nAN_{; we denote by rn:ð}Pn

k¼1"SpkÞp-sp the natural completely

isometric inclusion map and by sn:s_p-ðPn

k¼1"SkpÞp the completely contractive

projection. Thenrnsn tends to the identity map in the point-norm topology and the

assertion follows from the fact that_ðPn_k¼1"Sk

pÞ ¼LpðAnÞfor the finite-dimensional

C-algebra An¼M1"N?"NMn: In order to prove the second assertion, we

consider the map vk:Skp-sp which maps Spk in the kth block and the natural

projectionuk:sp-Skpon thekth block. Clearlyukvk¼idSk

p for allkand the sequence

ofprojections_ðPkÞdefined byPk¼vkuksatisfies limkPkðxÞ ¼0 for allxAsp by the

density ofelements with finitely many entries ins_{p:The last assertion is an obvious}

consequence. &

In our context the techniques developed by Lindenstrauss and Rosenthal[LR]in

the commutative setting yield the following key result.

Proposition 2.4. Let1pppNand X an operator space with theg_p-AP and containing

complemented Sn

p’s far out with constant C: Then X is a COSp space satisfying

COS_pðX_Þpð1þ2CÞð1þ2gap_p ðXÞÞ:

Proof. Let E be a finite-dimensional subspace of X and 0oeo1

2: Since X has

the gp-AP, we can apply Lemma 1.4 to obtain maps u:X-Snp and v:Snp-X

such that

vujE¼idE; jjujjcb¼1 and jjvjjcbpð1þeÞgapp ðXÞ:

Let 0odoe_ð4_ðC_þ1_Þgap

p ðXÞn2Þ

1

; where C is the constant from the ‘far out’

definition. LetF¼vðS_pnÞ:According to the assumption, we may findr:S_pn-X and

s:X-Snp such that

sr_¼idSn

p; jjrjjcb¼1; jjsjjcbpC and jjrsjFjjcbpd:

We let P¼rs:X-X denote the completely bounded projection from X onto

the range of r; and let ˜r:Sn

given by

˜r¼vþrðidSn

puvÞ and ˜s¼uðidXPÞ þs:

However, in general ˜s˜r need not be the identity map on Sn

p: Since sr¼idSpn and

ðidXPÞr¼0;we have

˜s˜r¼ ½uðidXPÞ þs½vþrðidSn

puvÞ

¼uvuPv_þsv_{þ ð}idSn

puvÞ

¼idSn

puPvþsrsv¼idSnpþ ðsuÞPv:

Therefore,

jj˜s˜ridSn

pjjcbpjjðsuÞPvjjcbpjjsujjcbjjPjFjjcbjjvjjcb

pðCþ1Þd2gap_p ðXÞp e 2n2:

According to Lemma 1.2 we can find an isomorphismw:S_pn-Sn_p such thatw˜s˜r¼

idSn

p and

jjw_jjcbp 1

1e

2

pð1þeÞ:

Ifwe definerE;e¼˜randsE;e¼w˜s;then we deduce

jjrE;ejj_cbpjjvjj_cbþ jjrjj_cbð1þ jjujj_cbjjvjj_cbÞ

pð1þeÞgap_p ðXÞ þ ð1þ ð1þeÞgap_p ðXÞÞ

pð1þeÞð1þ2gap_p ðXÞÞ

and

jjsE;ejj_cbpð1þeÞjjuuPþsjj_cbpð1þeÞð1þCþCÞ:

Finally, we have to check thatrE;esE;eðxÞ ¼ ˜rw˜sðxÞ ¼x for all xA_E: _{Ifwe let G}¼

u_ðE_Þ;then for y_¼u_ðx_ÞAG;we have

uv_ðy_{Þ ¼}u_ðvu_ðx_{ÞÞ ¼}u_ðx_{Þ ¼}y;

hence

This implies that for allxAE

u_ðx_{Þ ¼}y_{¼ ð}w˜s˜r_Þðy_{Þ ¼}w˜s_ðx_Þ:

In particular,

˜rw˜s_ðx_{Þ ¼} ˜r_ðu_ðx_{ÞÞ ¼}˜r_ðy_{Þ ¼}x:

We have checked the conditions forCOS_p formulated in the introduction. Thus the

assertion is proved. &

Remark 2.5. For a fixed subspaceYC_X;_{we may also define theg}

p-AP with respect

to Y by requiring that the factorizationsri:S_pn-X andsi:X-Sn_p withrisi tending

toidX satisfy the additional propertysiðSnp0ÞCY:The argument above shows that if

Xhas thegp-AP with respect toYand contains complementedSnp’s with respect toY

thenX is aCOS_pspace with respect toY (defined as above). Let us point out that

these technical modifications are essential for the interesting applications in the cases

p¼1 orp¼N:

Remark 2.6. For a fixednAN_{;let us consider the following stronger version of the}

gp-AP. We say thatX has thegp;n-AP ifthere exist diagrams ofcompletely bounded

maps

which converges in the point-norm topology to idX and satisfies the inequalities

supijjrijjcbjjsijjcbpgapp;nðXÞoN: Similarly, we say that X contains complemented

ck_pðSn

pÞ’s far out with constantCiffor every finite-dimensional subspaceECX;for

everykANande40;there existr:ck_pðSn

pÞ-X ands:X-c k

pðSpnÞsuch that

sr_¼idck

pðSnpÞ; jjrsjEjjcbpe and jjrjjcbjjsjjcbpC:

The same proofas above shows that an operator space X with the gp;n-AP and

containing complemented ck_pðSn

pÞ’s far out with constant C is a COLp space with

constant

COL_pðXÞpð1þ2CÞð1þ2gap_p;nðXÞÞ:

As an application ofProposition 2.4, we deduce that every operator spaceX with

theg_p-AP can be enlarged to provide an example ofa COS_p space. This method

provides many interesting examples ofCOS_{pspaces. We refer to Example 2.3 for the}

obvious fact thats_{p contains complementedS}n

Corollary 2.7. Let 1opoN and X an operator space. Then the following are equivalent.

(i) X has the CBAP and X is completely isomorphic to a completely complemented subspace ofQU Spfor some ultrapower of Sp;

(ii) X has thegp-AP.

(iii) X"_ps_{p is a} COS_p space;

(iv) X is completely complemented in aCOS_{p space.}

Proof. For the implication _ði_{Þ ) ð}ii_Þ; we note that X has the g_p-AP according to

Proposition 1.6. The implicationðiiÞ ) ðiiiÞfollows from Proposition 2.4 becauseX

and s_{p have the gp-AP and the space} s_{p contains complemented S}n

p’s far out, see

Example 2.3. The implication _ðiii_{Þ ) ð}iv_Þ is obvious because X is completely

contractively complemented inX"_ps_{p:For the implicationðivÞ ) ðiÞit suffices to}

note that everyCOL_{p spaceY has the CBAP and according to Proposition 1.6 is}

completely complemented in some QU Sp: Both properties pass to completely

complemented subspaces. &

Similarly as for COL_{p spaces in Proposition 2.4, we can obtain a result in the}

context ofOL_{p spaces.}

Proposition 2.8. Let1pppNand X an operator space with theg_p-AP. If X contains

Sn

p’s far out,then X is anOSp space.

Proof. Let assume that X contains Sn_p’s far out with constant C: Let

0oeoð3CÞ1ð1þ2gap_p ðXÞÞ1 and a finite-dimensional subspace ECX be given.

Chooseu:X-Sn

p andv:Spn-X such that

jju_jjcbp1; jjv_jjcbpð1þeÞgap_p ðXÞ and vuj_E¼id_E:

PutF¼vðSn

pÞand apply Lemma 1.4 (ii) to find a finite rank mapT:X-Xsuch that

TjF ¼idF and jjTjjcbpð1þeÞLðXÞpð1þeÞgapp ðXÞ: By the assumptions there is

finite-dimensionalGCX such that

dcbðG;SnpÞpC and jjT Gjjcbpe

:

Let w:Sn

p-G be an isomorphism such thatjjwjjcbpC andjjw1jjcbp1: We define

R:Sn_p-X by R¼vþwðidSn

puvÞ: Then we have ECRðS

n

pÞ as in the proofof

Proposition 2.4. Thus it remains to show thatRis an isomorphism fromSn

p onto its

range. To this end, fix anmANand a unit vectorxAM_mðSn

pÞ:Letd¼

jjTjjcb

1þ2jjTjjcb:Note

that dX1

former occurs, then by the choice ofT (withid¼idMm)

jjTjjcbjjid#RðxÞjjXjjðid#TvÞxþ ðid#TÞðid#wðidSnpvuÞÞxjj

Xjjðid#v_Þx_{jj jj}w_jjcbð1_{þ jj}v_jjcbjju_jjcbÞe

XdCð1þ2gap_p ðXÞÞe:

Ifwe are in the latter case, then

jjðid#wÞðid#_ðidSn

puvÞÞxjjXjjðid#ðidSnpuvÞÞxjj

Xjjxjj jjujj_cbjjðid#v_Þðx_Þjj

X1d:

Therefore, we get

jjðid#RÞðxÞjjXjjðid#wÞðid#_ðISn

puvÞÞxjj jjðid#vÞxjj

X1dd¼12d¼ 1 1_þ2_jjTjjcb

:

This shows that

jjR1_jjcbpmax 1þ2jjTjj_cb; d

deCð1þ2gapp ðXÞÞ

d1_jjT_jjcb

p d

deC_ð1_þ2gapp ðXÞÞð

1þ2jjTjjcbÞ:

The assertion is proved and sincee40 is arbitrary, we obtain

OL_pðXÞpð1þ2LðXÞÞðgap_p ðXÞ þCð1þgap_p ðXÞÞÞ:

The assertion is proved. &

Apart from introducing the notion of containing Sn

p’s far out, the main new

ingredient in the proof of Theorem 2.2 is the fact that the ‘far out’ properties can be derived from more natural, weaker assumptions. After a first version of this paper circulated, E. Ricard considerably improved a technical lemma crucial for this kind ofresults. We want to thank him for the permission to publish his refinement ofour result which turned out to be crucial for the final version of Theorem 4.10.

Lemma 2.9. Let 1pppN and n;k;l;mAN such that the integer part _½m

k satisfies

½m

k4lkn2:Let F be a vector space and T:c m

pðSpnÞ-F a linear map with rkðTÞpl:Then there exists a subspace ECcm

contractively complemented such that

is completely isometrically isomorphic to ck_pðSn_pÞ and T vanishes on E: Using a

sequenceðbuÞvu¼1 such that

is a completely contractive projection. &

The following lemma can also be proved by using Ramsey-type arguments and

ultraproduct techniques (see [RX]), but our proofs based on Lemma 2.9 are

significantly more elementary.

In particular, if X contains Sk

completely isometric to ck_pðSn

pÞ such that TrjE ¼0: Hence, F ¼rðEÞ is C-cb

-isomorphic tock_pðSn

pÞandTjF ¼0:In order to prove (ii) we assume thatX contains

complementedcm_pðSn

pÞ’s with constantC:LetFCXbe al-dimensional subspace. Let

½m k4lkn

2_{:Let u:c}m

pðSnpÞ-X andv:cmpðSnpÞsuch that

vu_¼idcm_pðSn

pÞ; jjujjcbp1 and jjvjjcbpC:

Let iF:F-X:We apply Lemma 2.9 toðviFÞ:cmp0ðSnp0Þ-F and find a completely

contractively complemented copyG ofck_p0ðSn

p0Þsuch thatðviFÞjG¼0:Using either

the proofofLemma 2.9 or a simple duality argument, we find a completely

contractive projection Q:cm_pðSn

pÞ-c m

pðSnpÞ such that Qðc

m

pðSnpÞÞ is completely

isometric to ck_pðSn

pÞ and QvjF ¼0: Then, we deduce that P¼uQv is a projection

satisfying PjF ¼0 and idQðcm

pðSpnÞÞ¼Qvu: This concludes the proofof(b). For the

particular part, we only have to observe that cm_pðSn

pÞ is completely contractively

complemented inSnm_p :Hence for alln the assumptions are satisfied. &

Remark 2.11. In (a) and in (b), we may add ‘with respect toY’ in every place.

Proof of Theorem 2.2. Combine Proposition 2.4 and Lemma 2.10 in the

complemented case and Proposition 2.8 and Lemma 2.10 in the non-complemented

case. &

Remark 2.12. In the complemented case, we may again add ‘with respect to Y’

everywhere.

Corollary 2.13. Let 1pppN:

(i) Let X be a complemented subspace of a COL_{p space containing complemented}

S_pn’s. Then X is aCOS_{p space.}

(ii) Let X be an operator space with the CBAP and containing Sn

p’s. If X is a complemented subspace of an OL_{p space,then X is an}OS_{p space.}

Proof. In case (i), it suffices to note that a complemented subspace of aCOL_pspace

has the gp-AP and thus Theorem 2.2 yields the assertion. In case (ii) again by

Theorem 2.2, it remains to prove thatX has thegp-AP. LetXCY such thatY is an

OL_p space. Let ECX be a finite-dimensional subspace and a finite rank map

T:X-X such that TjE¼idE according to Lemma 1.2. Then TðXÞCXCY is a

finite-dimensional subspace and we can find a finite-dimensionalC_{-algebra Aand}

T_ðX_ÞCFCY such thatd_cbðF;L_pðAÞÞpC: Letv:L_pðAÞ_-F and u:F_-L_pðAÞsuch

thatu_¼v1_{;then we deduce for the inclusion map i}

X:X-Y that

factors through LpðAÞ; and thus factors through Sm

p for m large enough. Let

P:Y-X be a completely bounded projection. Then

T_¼PiX¼PiFvuT

factors throughSm

p andX has thegp-AP. &

Resuming Lemma 1.5, Propositions 1.6, 2.4 and 2.8, we can formulate the following result.

Theorem 2.14. Let 1opoNand X an operator space with the CBAP. Then,

(i) X is a COS_p space if and only if X is completely complemented in QUSp and contains complemented Sn

p’s.

(ii) X is an OS_p space if and only if X is completely complemented in QUSp and contains Sn

p’s.

As mentioned above our main motivation is the investigation ofnon-commutative

Lp spaces. Let us recall some definitions. A von Neumann algebra N is called

semifinite if there exists a normal semifinite faithful (in short n.s.f.) trace, i.e. a

positive homogeneous and additive function on N_{þ¼ f}xx_jxANg; the cone of

positive elements ofN;such that for all increasing netsðxiÞiwith supremum inNand

for allxAN

þ

n. t_ðsup_ixiÞ ¼supitðxiÞ;

s. For every 0oxthere exists 0oyox such thattðyÞoN;

f. t_ðx_{Þ ¼}0 implies x_¼0;

t. For all unitariesuAN:tðuxuÞ ¼tðxÞ:

A positive homogeneous and additive functionw: N_þ-½0;Nsatisfying n.s.f. but

not the last property t. is called an n.s.f. (normal semifinite faithful) weight. Iftis an

n.s.f. trace then

mðtÞ ¼ X

n

i¼1

yixijnAN;

Xn

i¼1

½tðy_iyiÞ þtðx_ixiÞoN

( )

is the definition ideal on which there exists a unique linear extension t:m_ðt_Þ-C

which satisfiest_ðxy_{Þ ¼}t_ðyx_Þ:TheLp-norm is defined forxAmðtÞand 1ppoNby

jjx_jjp¼t_ððxx_Þ p

2_Þ 1

p_:

Then LpðN;tÞ is the completion of mðtÞ with respect to the Lp-norm. For two

faithful tracest1andt2onN;we can find an elementdaffiliated with the center ofN

such that t1ðxÞ ¼t2ðdxÞ: Thus the space LpðN;t1Þ and LpðN;t2Þ are (completely)

isometrically isomorphic. Therefore, we will often use the notation LpðNÞ for this

[D1,FK,Ku,Ne,Se,Te] for more on this and for information on the topological

algebra oft-measurable operators affiliated withNin which all the spacesLpðN;tÞ

embed topologically. It is well-known that the complex interpolation method yields

LpðN;tÞ ¼ ½N;L1ðN;tÞ1

p

:

HereL1ðN;tÞinherits the natural operator space structure fromNopvia the map

b_ðx_Þðy_{Þ ¼}t_ðyx_Þ:

(Note thatNandNopcoincide as Banach spaces.) Then, we have

jj½b_ðxijÞjj_Sn

Here trn denotes the non-normalized trace on Mn: The complex interpolation (as

explained in the first section for the finite-dimensional case) defines the natural operator space structure

Note that these formulas slightly differ from[Fi]but are more consistent with[P5].

In particular, for every linear mapT:LpðN1;t1Þ-LpðN2;t2Þ;we deduce

jjTjjcb¼ jjid#T:LpðBðc2Þ#N1;tr#t1Þ-LpðBðc2Þ#N2;tr#t2Þjj: ð2:5Þ

This shows, as it should be, that thecb-norm can be obtained by replacing scalars

with matrix-valued coefficients. A corresponding formula also holds for maps