spaces the local structure of non commut (1)
COL p spaces—the local structure of non-commutative L p spaces
M. Junge, c N.J. Nielsen, Zhong-Jin Ruan, and Q. Xu
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 of COL p spaces (a variation ofthe non-commutative analogue ofL p
spaces), we provide new tools to investigate the local structure ofnon-commutative L p spaces. Under mild assumptions on the underlying von Neumann algebras, non-commutative L p 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 L p spaces associated with hyperfinite von Neumann algebras with separable predual von Neumann algebras generated by free groups, and obtain a basis for separable nuclear C -algebras. r 2003 Elsevier Inc. All rights reserved.
MSC: 46L52; 46L07 Keywords: Non-commutative L p spaces; (cb-)basis; Non-commutative L p spaces
Corresponding author. E-mail addresses: [email protected] (M. Junge), [email protected] (N.J. Nielsen), ruan@
math.uiuc.edu (Z.-J. Ruan), [email protected] (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.
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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 L p spaces could be considered as non-commutative function spaces. In particular, the classical Banach spaces oftrace class operators, Hilbert– Schmidt operators and more generally Schatten p-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 ¼ 2 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-commutative L p 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 L p 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 f or some important spaces like nuclear (in particular type I) C -algebras, preduals of hyperfinite von Neumann algebras, and non-commutative L p spaces 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 of L p spaces. Following Lindenstrauss and Pe"czyn´ski [LP] a Banach space X is called an L p ;l space ifevery finite-dimensional
subspace ECX is contained in a finite-dimensional subspace ECF CX such that for n ¼ dimðFÞ the Banach–Mazur distance satisfies
ð0:1Þ Ifthis is true for some l ; X is called an L p space. Ifthis is true for all l41 ; then X is
d ðF; c n p Þpl:
isometrically isomorphic to L p ðO; 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 L p (The absence ofthe approximation property for general non-commutative L p spaces 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 an L p space
is an L p 0 space (p 0 ¼ p p 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 ofan L p space is an L p 0 space and that the copies of c n p in the definition of L p spaces may be assumed to be uniformly complemented. In order to underline the different notions in the non-commutative setting, we might call spaces satisfying the complemented condition CL p spaces and
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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 of CL 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 L p 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 of L 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 separable L p space admits a basis. Refining their techniques Nielsen and Wojtaszczyk showed that this basis locally looks like the basis of c n p : We use this approach as a guideline to discover the local structure ofa
separable COL p space and construct (very nice operator space) bases therein. Let us note that due to the work ofBourgain [Bo] , Bourgain et al. [BRS] many non- isomorphic L p spaces are known, and thus many ofthem are not isomorphic to standard examples L p
p or c p "c 2 : Therefore L p spaces have a very rich global structure. The right framework for the investigation of the local structure of non- commutative L p 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] . An operator space X is a norm closed subspace ofsome B ðHÞ equipped with the distinguished operator space matrix norm inherited from M n
2 ðHÞÞ: An abstract matrix norm characterization of operator spaces was given by Ruan (see e.g. [ER2] ). The morphisms in the category
ðXÞCBðc n
ofoperator spaces are completely bounded maps. Given operator spaces X and Y ;a linear map T : X -Y is completely bounded ifthe corresponding linear maps
T n :M n ðX Þ-M n ðYÞ defined by T n ð½x ij
ij
jjTjj cb ¼ sup jjT n jjoN:
nAN
A map T is a complete contraction (respectively, a complete isometry, or a complete quotient) if jjTjj cb p 1 (respectively, ifeach T n 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
d cb ðX ; YÞ ¼ inffjjTjj cb jjT jj cb : T a complete isomorphism from X onto Y g denote the completely bounded Banach–Mazur distance (in short cb-distance) of X
and Y (see [P4] ). Variations ofGrothendieck’s approximation property inspired crucial develop- ments in operator algebras and operator spaces. An operator space X CB ðHÞ has the operator space approximation property, in short OAP, ifthere exists a net offinite rank maps ðT i Þ such that id K # T i converges in the point-norm topology to the
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identity on K# min
X CB ðc 2 # H Þ; where K denotes the space ofcompact operators on c 2 : An operator space has the completely bounded approximation property (in short CBAP) ifthere exists a net ðT i Þ offinite rank maps converging in the point- norm topology to the identity on X and sup i jjT i jj cb oN : We say that an operator space X has a cb-basis if, X has a basis ðx n Þ and the natural projection maps
k ¼1
k ¼1
satisfy K ¼ sup n jjP n jj cb oN : In this case we call ðx n Þ a K-cb-basis.
For non-commutative L p spaces Pisier [P5] introduced a very natural operator space structure by interpolation (see [BL] for interpolation theory). Indeed, it is well- known that the Schatten p-classes S p can be obtained by complex interpolation
Here T ¼S 1 denotes the space oftrace class operators and K ¼S N the space of compact operators. Moreover, the natural (operator space structure preserving) duality between x ¼ ½x ij
ij
/ x ; yS ¼
ij y ij ¼ trðxy Þ:
ij
Pisier [P5] proved that
M n ðS p Þ ¼ ½M n ðKÞ; M n
define matrix norms on S p which satisfy Ruan’s abstract characterization for operator spaces. Therefore, there is an isometric embedding j p :S p -B ðc 2 Þ inducing these matrix norms and this is nowadays called the natural operator space structure of S p : We refer to [P5] for many nice features. Similarly, we may obtain a natural operator space structure on L p ðAÞ for every finite-dimensional C -algebra A :
Let us recall the operator space analogue of L p spaces. An operator space X is called an operator L p space (in short OL p ;l space) if X can be paved out by copies offinite-dimensional L p spaces, where the cb-distance is uniformly controlled by l : An operator space X is called an OL p ifit is an OL p ;l for some l41 : In this case, we use the parameter OL p ðXÞ ¼ inf l; where the infimum is taken over all l’s above. For a precise definition see Section 2.
During the last few years, OL 1 spaces have been intensively studied in [ER1,JOR,NO] . In particular, it was proved in [ER1] that the predual N ofa von Neumann algebra N is an OL 1 space ifand only ifN is hyperfinite. Moreover, a separable operator space X is an OL 1 space with OL 1 ðXÞ ¼ 1 ifand only ifit is the operator predual ofa hyperfinite von Neumann algebra (see [NO] ). Concerning OL N space, we recall that by Szankowski’s result (see [Sz1] ) the space
B ðHÞ does not have Grothendieck’s approximation property and hence is not an
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OL N space. In fact contrary to the commutative case, the OL N property for C - 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 a C -algebra
A is an OL N ;l space for some l ifand only ifA is nuclear. In Theorem 3.11, we will improve a recent result in [JOR] by showing OL N ðAÞp3: The microscopic index l even provides some additional information on the structure of the underlying C -
p ffiffi 1 algebra. For example, a C -algebra is stably finite if lp 1 ð þ 5 2 Þ 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 for OL p spaces in the range 1opoN : It is known that every OL p space is completely complemented in some non- commutative L p space. However, it can be derived from Szankowski’s work [Sz2] that there are finite von Neumann algebras with separable predual such that L p ðNÞ does not have the approximation property (see Theorem 2.19). Moreover, it is not known whether every OL p space has the CBAP. In order to use the concepts from Banach space theory, we will work with the analogue of CL p spaces. An operator space X is called a COL p ;l space ifit is paved by complemented copies ofL p ðAÞ’s where cb-distance and the cb-norm ofthe projections are uniformly controlled by l : Ifthis is true for some l ; X is called a COL p space. Ifwe can replace the L p ðAÞ’s by S n p ’s, we call this a COS p ;l ; COS p space, 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. Let 1opoN and X an operator space. X is a COS p space if and only if
X has the CBAP, id X admits a cb-factorization through an ultrapower of S p ; and X contains completely complemented S n p ’s uniformly.
Theorem 0.2. Let 1op ;p 0 oN with 1 p 1 þ p 0 ¼ 1 and X an operator space. Then X is a
COL p space if and only if X is a COL p 0 space.
The cases p ¼ 1; p ¼ N 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 space X such that X is COL 1 but X does not have the CBAP. In Section 4, we extend the results ofJohnson et al. [JRZ] , Nielsen and Wojtaszczyk [NW] to COL p spaces.
Theorem 0.3. Let 1pppN and X a separable COL p space (such that in addition X has the CBAP and X is locally reflexive for pA f1; Ng). Then X has a cb-basis.
Before we state our main application to non-commutative L p spaces, we have to clarify the ‘mild assumptions’ on the underlying von Neumann algebra N :AC - algebra A has the weak expectation property of Lance (in short WEP) iffor the
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universal representation ACA C B ðHÞ there is a contraction P : BðHÞ-A such that P j A ¼ id A :AC -algebra B is said to be QWEP ifthere exists a C -algebra A with the WEP and a closed two-sided ideal I such that B ¼ A=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 for 1opoN the following are equivalent
(i) L p ðNÞ has the OAP; (ii) L p ðNÞ has the CBAP; (iii) L p ðNÞ is a COL p space; (iv) L p ðNÞ has a cb-basis.
In particular, if one of the conditions above is satisfied, then L p ðNÞ is an OL p 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 for L p spaces associated to the von Neummann algebra VN ðF n Þ generated by the left regular representation ofthe free group F n : As so often in harmonic analysis, the spaces L p ðVNðF n ÞÞ behave much nicer for 1opoN than for the border cases pAf1; Ng:
Indeed, here L 1 ðVNðF n ÞÞ is not an OL 1 space and C red ðF n Þ is not an OL N space because F n is not amenable.
Theorem 0.5. Let 1opoN and F n the free group with n generators. Then L p ðVNðF n ÞÞ is a COL p space (hence an OL p space) and has a cb-basis.
We note that the existence ofa basis f or L 1 ðVNðF n ÞÞ or C red ðF n Þ 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 of A instead, we can obtain sufficiently many information about the local structure of A in the cases ofnuclear C -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, L p ðNÞ has Grothendieck’s approximation property iff it has a basis. A positive solution to the basis problem for non-commutative L p spaces has previously only been known for
the class oftype I von Neumann algebras and the hyperfinite II 1 and II N factors (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 techniques are very natural (and useful) in this setting. However, our project seems to be the first attempt to provide more specific information on the local structure of
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L p ðNÞ spaces 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 for L p spaces for all values 1pppN even in the commutative case (using the appropriate new notion ofcontaining c n 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 L p 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 space H ; we let B ðHÞ denote 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
P on an operator space X is a completely bounded projection if P 2 ¼ P: A subspace X 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 from Y onto X : If X is an operator space, then its dual space X is an operator space with matrix norms given by the isometric identifications
M n ðX Þ ¼ CBðX ; M n Þ
(see [BP,ER2] ). This operator space structure on X is called the operator (space) dual of
X : If X is an operator space, then the canonical embedding i : X -X is a completely isometric injection, i.e. id M n # i is isometric for all nAN : If T : X -Y is a completely bounded map, then its adjoint map T :Y -X is also completely bounded with
jjT jj cb ¼ jjTjj cb : Using the Arveson–Wittstock–Hahn–Banach theorem [ER2,Pa] , it is easy to show that if T is a complete isometry, then T is a complete quotient map, i.e. id M n # T maps the open unit ball onto the open unit ball for all nAN : Similarly, if T is
a complete quotient map, then T is a complete isometry. Given a von Neumann algebra N ; the canonical embedding i : N +N induces an operator space structure on N : With these matrix norms, we have the complete isometry
N ¼ ðN Þ :
In the following, we will use the notation S p (resp. S n p ) for the spaces of all compact operators on the Hilbert spaces c 2 ¼c 2 ðNÞ (resp. c n 2 ) such that
jjxjj p ¼ ½trððx x Þ 2 p oN :
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We will always work with the canonical duality between S p and S p 0 1 ð 1 p þ p 0 ¼ 1Þ given by
and obtain a complete isometry S p ¼S p 0 : Similarly, if A is a finite-dimensional C - algebra given by
A ¼M n 1 " N ? " N M n l ;
then we have
A n 1 " 1 ? " 1 S n ¼S l 1 1 ;
where " 1 is the operator space l 1 -direct sum. Let n ¼n 1 l : The canonical projection of S n onto A is completely contractive on S n
þ?þn
N and the same map is also completely contractive on S n 1 : Therefore, we may apply complex interpolation for the
compatible pair
ðA; A n ÞCðS N ;S 1 Þ and obtain the natural operator space structure on L p
p C S p : We refer to [BL] for the complex interpolation method. Note that by complementation,
ðAÞ ¼ ½A; A n 1 ¼S 1
we still have a complete isometry
L p ðAÞ ¼L p 0 ðAÞ
and L p ðAÞ is a completely contractively complemented subspace of S n p for 1pppN : In the sequel, we will also use an infinite-dimensional analogue ofthese spaces. Let m ¼ ðmðnÞÞ nAN
be a sequence ofnatural numbers and
b ðmÞ ¼ Y M m ðnÞ ;
the von Neumann algebra obtained as block diagonals in B ðc 2 Þ: In the Banach space literature one may also write b
P ðmÞ ¼ ð n " M m ðnÞ Þ N : Then the predual of b ðmÞ is
1 P ðmÞ ¼ ð ðnÞ n " S 1 Þ 1 ; i.e. the block diagonals in S 1 : Since the projection onto these
block diagonals is completely contractive in both cases, we see that
p ðmÞ ¼
p ¼L p n ðbðmÞÞ p
" S m p ðnÞ ¼ ½bðmÞ; s 1 1
is completely contractively complemented in S p : For p ¼ N; we use the notation s N ðmÞ for the c 0 sum. In the special case where m is given by m ðnÞ ¼ n for all nAN; we will simply use the notation s p : As in Banach space theory, ultraproducts turn out to be a useful tool in the study ofoperator spaces (see for example [P3] ). Let us recall that if U is a free ultrafilter on
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an infinite index set I and ðX i Þ iAI is a family of operator spaces, then we consider the ultraproduct
X i =J U ;
iAI
where Q iAI X i ¼ fðx i Þ j sup i jjx i jjoNg is the space ofall bounded families, and
is the norm closed subspace in Q iAI X i off amilies tending to 0 along U : An ultraproduct Q U X i ofoperator spaces carries canonical matrix norms given by
M n ðX i Þ:
For details see [ER2,P3,P5] . If Q ðA i Þ is a family of C -algebras, it is well-known that U A i is again a C -algebra. It is also known (see [Gr,Ra1,Ra2] ) that for 1ppoN ;
we have Q U S p ¼L p ðNÞ for some von Neumann algebra N: The following result is due to Junge [Ju2] and holds only for pA ð1; NÞ:
Theorem 1.1. Let E and F be finite-dimensional operator spaces and 1opoN : 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˜rjj cb jj˜sjj cb o jjrjj cb jjsjj cb þ e:
Approximation properties play an important role in operator algebras and operator spaces. Let X and Y be operator spaces. A linear map T : X -Y is said
to have the completely bounded approximation property (in short CBAP) ifthere exists a constant l and a net offinite rank maps T i : X -Y such that T i -T in the
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point-norm topology and sup i jjT i jj cb p l : In this case, we let L ðTÞ ¼ inf l
denote the infimum ofall l as above. If T ¼ id X we say that X has the CBAP and let
L ðXÞ ¼ Lðid X Þ:
Let 1pppN : A linear map T : X -Y is said to have the g p -approximation property (in short g p -AP) (i.e. can be approximately factored through S n p spaces) ifthere exist diagrams ofcompletely bounded maps
which converges in the point-norm topology to T and satisfies sup i jjr i jj cb jjs i jj cb p l for some constant loN : We let as above
g ap p ðTÞ ¼ inf l:
If T ¼ id X ; we say that X has the g p -approximation property and let
It is clear that if T has the g p -AP, then T has the CBAP with
ðTÞpg ap
p ðTÞ:
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 x 1 ; y; x n ;x 1 ; y; x n (i.e. jjx i jjp1; jjx j jjp1 and x i ðx j Þ¼d ij for all
i ;j ¼ 1; y; n). Let 0oeo1; and T : E-X a linear map such that
jjTðx i
i jjp
for all i ¼ 1; y; n: Then there exists a complete isomorphism W : X-X such that
WT ðxÞ ¼ x
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1 jjW jj cb p ð1 þ eÞ: 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 ¼ id F : If ECX is an n-dimensional subspace and 0oeo 1 2 such that
e jjxjj n jjrjj cb jjsjj cb
for all xAE : Then there exist maps ˜r : F -X and ˜s : X -F such that ˜s˜r ¼ id F ; ˜r˜s j E ¼ id E and
jj˜rjj cb jj˜sjj cb p
jjrjj cb jjsjj cb : Proof. Applying Lemma 1.2 to e 0 jjrjj e cb jjsjj cb p ¼ 1 2 and T ¼ rs; we may obtain a
e complete isomorphism W : X -X such that 0
cb p 1 0 and WT ðxÞ ¼ x for all xAE : Then we deduce
e 0 jjid F cb cb p jjrjj cb jjsjj cb 0 p 2e 1 :
Hence, for b ¼ ðsWrÞ we obtain the estimate
jjbjj cb p
We define ˜s ¼ bs and ˜r ¼ Wr: Clearly, ˜s˜r ¼ bsWr ¼ id F : For xAE; we observe that
sWrs ðxÞ ¼ sWTðxÞ ¼ sðxÞ:
Hence, sWr j s ðEÞ ¼ id s ðEÞ and therefore b j s ðEÞ ¼ id s ðEÞ : Thus, we get
˜r˜s ðxÞ ¼ WrbsðxÞ ¼ WrsðxÞ ¼ WTðxÞ ¼ x
for all xAE : Using the cb-norm estimates for b and W ; we obtain the assertions. & For Banach spaces (and nowadays also for operator spaces) it is well-known
that Lemma 1.2 implies that the ‘point-norm approximation’ can be improved to obtain finite rank maps which are the identity on a given finite-dimensional subspace.
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Lemma 1.4. Let T : X -Y be a completely bounded map. (i) T : X -Y has the CBAP with L ðTÞol if and only if for every finite-dimensional
subspace EDX ; there exists a finite rank map u : X -Y such that jjujj cb o l and u ðxÞ ¼ TðxÞ for all xAE: (ii) T : X -Y has the g -AP with g ap p p ðTÞol if and only if for every finite-dimensional subspace EDX
; there exist nAN and maps u : X -S n p ;v:S n p -Y such that jjujj cb jjvjj cb o l and vu ðxÞ ¼ TðxÞ for all xAE:
Proof. Obviously the second assertion in (i), (ii) implies the CBAP, g p -AP, respectively. Since the arguments are very similar, we will only show the missing implication in (i). If E is a finite-dimensional subspace of X ; then T ðEÞ is a finite- dimensional subspace of Y k : We can find vectors x
1 ; y; x k in E such that ðTðx i ÞÞ i ¼1 is part ofa biorthogonal system in T
ðEÞ: Choose 0odo1 such that ð1 þ dÞ 2 L ðTÞol: Since T has the CBAP, there exists a finite rank map T : X -Y such that ˜
k for all i ¼ 1; y; k: It follows from Lemma 1.2 that there exists a complete isomorphism W : Y -Y such that WT ðx i Þ¼
jj ˜ T jj cb o ð1 þ dÞLðTÞ and jj ˜ T
ðx d i i Þjjo
T ˜ ðx i Þ for ði ¼ 1; y; kÞ and jjW jj cb o ð1 þ dÞ: Hence, u ¼ W T : X -Y is a finite ˜ rank map which satisfies the requirement ofthe assertion. &
Using the uniform convexity of S p (see [TJ] ) it is easy to prove the following well- known fact. We refer to [ER1] for the details.
Lemma 1.5. Let 1opoN : Then U S p is reflexive for every ultrafilter U : Moreover, every OL p space is completely contractively complemented in some Q U S p and thus reflexive.
Proposition 1.6. Let 1opoN and X an operator space. Then X has the g p -AP if and only if X has the CBAP and there exists a free ultrafilter U on some index set I such Q that X is completely complemented in U S p ; i.e. there exists a commuting diagram of completely bounded maps
Proof. If X has the g p -AP, then X has the CBAP, and there exist diagrams of completely bounded maps
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which approximately commute in the point-norm topology and in addition satisfy
sup jjr i jj cb jjs i jj cb ¼ loN:
Ifwe let U be a free ultrafilter on the index set I ; then we obtain a commuting diagram ofcompletely bounded maps
where we let
s:X- U S n p i be the map given by s ðxÞ ¼ ðs i ðxÞÞ U and r: Q S n U p i -X the map given by
/ r ððz i Þ U Þ; x S ¼ lim U x ðr i ðz i ÞÞ
for all x A X : Since each S n p i is completely contractively complemented in S p ; Q
p i is completely contractively complemented in U S p ; and thus we can actually replace Q S n i in (1.4) by Q S
p : Hence X is isomorphic to s ðXÞC U S p and thus reflexive according to Lemma 1.5. Thus we obtain the commuting diagram
On the other hand, let us assume that X has the CBAP and satisfies diagram (1.3) with jjrjj cb jjsjj cb p C : It follows from Lemma 1.4 that for any finite-dimensional subspace EDX and e40 ; there exists a finite rank map u : X -X such that jjujj 2
cb o ð1 þ eÞLðX Þ and uj E ¼ id E for all xAE : In particular u j E ¼ id E and it suffices to show that u 2 factors through S m p : Let us consider the finite-dimensional
operator space G ¼ X=kerðuÞ with quotient map q G : X -G and the induced map ˆu : G-X such that u ¼ ˆuq G : Note that ˆu has the same cb-norm as u: Let F ¼
u ðXÞCX with inclusion map i F : F -X : Then u ˆu ¼ urs ˆu : G-F satisfies the assumption ofTheorem 1.1 and hence admits a f actorization u ˆu ¼ vw
; w : G-S m p ;v:S p -F such that
2 3 jjvjj 2
cb jjwjj cb p ð1 þ eÞjjujj cb jjrjj cb jjsjj cb p ð1 þ eÞ L ðX Þ C : Thus u 2 ¼i m F vwq G factors through S p and satisfies the corresponding cb-norm
estimate. Therefore, X has the g
p -AP with g p ðX ÞpCLðXÞ : &
ap
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Remark 1.7. The same result holds ifwe replace the cb-norm by the operator norm in all instances above. Indeed, in Theorem 1.1, this can be easily proved by using finite d-nets in the unit ball of E and F :
¼ 1: Indeed, if N is a von Neumann algebra, then g 1 ðN Þol ifand only ifN is l- semidiscrete, and thus injective by Pisier [P4] or Christensen and Sinclair [CS] . Let F n denote the free group of n generators. It is known that the von Neumann algebra VN ðF n Þ is not injective (for any l41), but satisfies LðVNðF n Þ Þ ¼ 1 (see [Ha3] ). Using an argument ofWassermann (or the fact that VN ðF n Þ is QWEP with the results in [EJR, Section 7] and [NO] ), we see that there are complete contractions r : VN
In the category ofoperator spaces Proposition 1.6 is no longer true for p
ap
ðF n Þ - U S 1 ;s: U S 1 -VN ðF n Þ such that
Hence VN ðF n Þ satisfies the assumptions ofProposition 1.6 without having the g 1 - AP. In Theorem 5.7, we will show that for 1opoN ;L p ðVNðF n ÞÞ has the g p -AP. This indicates that, as so often in harmonic analysis, the L p spaces in the range 1opoN behave much nicer than the extreme cases p ¼ 1 and p ¼ N:
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 space X is called an operator L p space (in short OL p space) ifthere exists a constant l41 and a family ðF i Þ iAI offinite-dimensional subspace such that
i F i is dense in X and for every index i there exists a finite-dimensional C -algebra
A i such that
ð2:1Þ In this case, we denote by OL p ðXÞ ¼ inf l; where the infimum is taken over all l as
d cb ðL p ðA i Þ; F i Þpl:
above. Moreover, we say that X is an OL p ;l -space, if OL p ðXÞpl: We call X an OS p space ifwe can replace the L p ðA i
;l n Þ’s in (2.1) by S
p i ’s.
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An operator space X is called a completely complemented OL p ;l space (in short COL p ;l space) for some constant l41 ifthere exist a family offinite-dimensional
C -algebras ðA i Þ and commuting diagrams ofcompletely bounded maps
such that r i s i -id X in the point-norm topology on X and jjr i jj cb jjs i jj cb p l : We call X
a completely complemented OS p ;l space (in short COS p ;l space) ifwe can replace L p i
ðA n Þ in (2.2) by S p i : We say that X is a COL p space (respectively, a COS p space) if it is a COL p ;l space (respectively, a COS p ;l space) for some lX1 : In this case, we denote by COL p ðX Þ ¼ inf l (respectively, COS p ðXÞ ¼ inf l), where the infimum is taken over all l such that X is a COL p ;l space (respectively, a COS 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- commutative L p spaces. 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 an OL p space with OL p ðX Þol if and only if there exists a l 0 o l such that for
every finite-dimensional subspace E of X there exists a finite-dimensional space ECF CX and a finite-dimensional C -algebra A such that
ðL 0 ðAÞ; FÞol :
d cb p
(ii) X is a COL p space satisfying COL p ðXÞol if and only if there exists a l 0 o l such that for every finite-dimensional subspace EDX ; there exist a finite-dimensional
C -algebra A and a commuting diagram of completely bounded maps
with jjrjj cb jjsjj cb p l 0 and rs ðxÞ ¼ x for all xAE:
A similar result holds for OS p spaces and COS p spaces. It follows from Lemma 2.1 that every COL p space is an OL p space. For p ¼ N;
the two notions are equivalent by the injectivity offinite-dimensional C -algebras. (However, ifwe consider the Banach space versions ofspaces paved out by Banach
M. Junge et al. / Advances in Mathematics 187 (2004) 257–319
space copies offinite-dimensional C -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 p ¼ 1 if l is sufficiently close to 1 (see [Oz] ). It is not known whether these notions are still equivalent for large l : 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 1pppN and X an operator space with the g p -AP. If X contains S n p ’s (respectively, complemented S n p ’s) then X is an OS p space (respectively, a COS p space).
Here X is said to contain S n p ’s ifthere exists a constant C such that for every nAN ; we can find G n C X such that
d cb n ðG n ;S p ÞpC:
We note that in the Banach space literature the term ‘X contains c n p ’s uniformly’ (respectively ‘X contains c n p ’s uniformly complemented’) is in use. If we want to specify the constant C we say X contains S n p ’s with constant C. Accordingly, we say that X contains complemented S n p ’s (with constant C) iffor every nAN there are r n :S n -X and s n
: X -S p such that
s n r n ¼ id S n p and jjr n jj cb jjs n jj cb p C :
As a technical (but important) modification we say that X contains complemented S n ’s with respect to Y if Y CX and s
n ðS p 0 ÞCY for all nAN: Although this clarifies the assumptions ofTheorem 2.2, the proofrequires ‘sufficiently many orthogonal’ copies of S n 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 n p ’s far out ifthere exists a constant C40 such that for every
finite-dimensional subspace ECX and for every nAN and e40 ; there exist r n :S n
p -X ; s n : X -S p such that s n r n ¼ id S p n ; jjr n s n j E jj cb p e and jjr n jj cb jjs n jj cb p C :
Again, we use ‘with constant C’ and ‘with respect to Y ’ as above. Similarly, we say that X contains S n p ’s far out (with constant C) iff or every finite rank map
T : X -X ; nAN and e40; there exists G n C X such that
d cb ðG n ;S n p ÞpC and jjTj G n jj cb p e :
Note that it suffices to have jjTj G n jjpe because G n is finite dimensional. Indeed, using a biorthogonal system one can easily prove that for a linear map T : E-X on
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a d-dimensional operator space E (see e.g. [EH] )
jjTjj cb p d jjTjj:
ð2:3Þ Similarly, the condition jjr n s n j E jj cb p e can be weakened to jjr n s n j E jjpe: 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 p ðs p Þ ¼ 1: s p contains complemented S n p ’s far out. Moreover, every operator space containing s p completely complemented contains complemented S n p ’s far out.
Proof. For nAN n : P ; we denote by r n
k ¼1 " S p Þ p -s p the natural completely
isometric inclusion map and by s n :s p
k ¼1 " S p Þ p the completely contractive projection. Then r n s n tends to the identity map in the point-norm topology and the
assertion follows from the fact that P n
k ¼1 " S p Þ¼L p ðA n Þ for the finite-dimensional
C -algebra A n ¼M 1 " N ? " N M n : In order to prove the second assertion, we consider the map v k :S k
p -s p which maps S p in the kth block and the natural projection u
k :s p -S p on the kth block. Clearly u k v k ¼ id S p k for all k and the sequence ofprojections ðP k Þ defined by P k ¼v k u k satisfies lim k P k ðxÞ ¼ 0 for all xAs p by the density ofelements with finitely many entries in s 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. Let 1pppN and X an operator space with the g p -AP and containing complemented S n p ’s far out with constant C : Then X is a COS p space satisfying COS p
ðX Þpð1 þ 2CÞð1 þ 2g ap
p ðXÞÞ:
Proof. Let E be a finite-dimensional subspace of X and 0oeo 1 2 : Since X has the g
-AP, we can apply Lemma 1.4 to obtain maps u : X -S n p and v : S p n p -X such that
E ¼ id E ; jjujj cb ¼ 1 and jjvjj cb p ð1 þ eÞg p ðX Þ: Let 0odoe
vu j ap
ð4ðC þ 1Þg 2 p ðX Þn Þ ; where C is the constant from the ‘far out’ definition. Let F
ap
¼ vðS n p Þ: According to the assumption, we may find r : S p -X and s : X -S n p such that
sr ¼ id S n p ; jjrjj cb ¼ 1; jjsjj cb p C and jjrsj F jj cb p d : We let P ¼ rs : X -X denote the completely bounded projection from X onto
the range of r ; and let ˜r : S n
p -X and ˜s : X -S p
be completely bounded maps
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given by ˜r ¼ v þ rðid S n p
However, in general ˜s˜r need not be the identity map on S n p : Since sr ¼ id S p n and ðid X
˜s˜r ¼ ½uðid X S n p
cb jjPj F jj cb jjvjj cb
ðC þ 1Þd2g ap
p ðXÞp 2 2n :
According to Lemma 1.2 we can find an isomorphism w : S n
p -S p such that w˜s˜r ¼ id S n p and
jjwjj cb p
e 1 p ð1 þ eÞ:
Ifwe define r E ;e ¼ ˜r and s E ;e ¼ w˜s; then we deduce jjr E ;e jj cb p jjvjj cb þ jjrjj cb ð1 þ jjujj cb jjvjj cb Þ
ð1 þ eÞg ap
ap
p ðX Þ þ ð1 þ ð1 þ eÞg p ðXÞÞ
ð1 þ eÞð1 þ 2g ap
p ðX ÞÞ
and jjs E ;e jj cb p
cb p ð1 þ eÞð1 þ C þ CÞ: Finally, we have to check that r E ;e s E ;e ðxÞ ¼ ˜rw˜sðxÞ ¼ x for all xAE: Ifwe let G ¼
u ðEÞ; then for y ¼ uðxÞAG; we have
uv ðyÞ ¼ uðvuðxÞÞ ¼ uðxÞ ¼ y;
hence
˜r ðyÞ ¼ vðyÞ ¼ vuðxÞ ¼ x:
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This implies that for all xAE
u ðxÞ ¼ y ¼ ðw˜s˜rÞðyÞ ¼ w˜sðxÞ:
In particular,
˜rw˜s ðxÞ ¼ ˜rðuðxÞÞ ¼ ˜rðyÞ ¼ x:
We have checked the conditions for COS p formulated in the introduction. Thus the assertion is proved. &
Remark 2.5. For a fixed subspace Y CX ; we may also define the g p -AP with respect to Y by requiring that the factorizations r i :S n
p -X and s i : X -S p with r i s i tending to id X satisfy the additional property s
i ðS p 0 ÞCY : The argument above shows that if
X has the g p -AP with respect to Y and contains complemented S n p ’s with respect to Y then X is a COS p space with respect to Y (defined as above). Let us point out that
these technical modifications are essential for the interesting applications in the cases p ¼ 1 or p ¼ N:
Remark 2.6. For a fixed nAN ; let us consider the following stronger version of the
g p -AP. We say that X has the g p ;n -AP ifthere exist diagrams ofcompletely bounded maps
which converges in the point-norm topology to id X and satisfies the inequalities sup
ap
i jjr i jj cb jjs i jj cb p g p ;n ðXÞoN: Similarly, we say that X contains complemented
c k n p ðS p Þ’s far out with constant C iffor every finite-dimensional subspace ECX; for every kAN and e40
; there exist r : c k n n
p ðS p Þ-X and s : X-c p ðS p Þ such that sr ¼ id c k p ðS p n Þ ; jjrsj E jj cb p e and jjrjj cb jjsjj cb p C :
The same proofas above shows that an operator space X with the g p ;n -AP and containing complemented c k n p ðS p Þ’s far out with constant C is a COL p space with constant
COL p
ðX Þpð1 þ 2CÞð1 þ 2g ap
p ;n ðX ÞÞ:
As an application ofProposition 2.4, we deduce that every operator space X with the g p -AP can be enlarged to provide an example ofa COS p space. This method provides many interesting examples of COS p spaces. We refer to Example 2.3 for the obvious fact that s p contains complemented S n p ’s far out.
M. Junge et al. / Advances in Mathematics 187 (2004) 257–319
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 of Q U S p for some ultrapower of S p ; (ii) X has the g p -AP.
(iii) X " p s p is a COS p space; (iv) X is completely complemented in a COS p space.
Proof. For the implication ðiÞ ) ðiiÞ; we note that X has the g p -AP according to Proposition 1.6. The implication ðiiÞ ) ðiiiÞ follows from Proposition 2.4 because X
and s p have the g p -AP and the space s p contains complemented S n p ’s far out, see Example 2.3. The implication ðiiiÞ ) ðivÞ is obvious because X is completely
contractively complemented in X " p s p : For the implication ðivÞ ) ðiÞ it suffices to note that every COL p space Y has the CBAP and according to Proposition 1.6 is completely complemented in some Q U S p : 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 of OL p spaces.
Proposition 2.8. Let 1pppN and X an operator space with the g p -AP. If X contains S n p ’s far out, then X is an OS p space.
Proof. Let assume that X contains S n p ’s far out with constant C : Let 0oeo
ð3CÞ ap ð1 þ 2g p ðXÞÞ and a finite-dimensional subspace ECX be given. Choose u : X -S n
p and v : S n p -X such that
cb p ð1 þ eÞg p ðXÞ and vuj E ¼ id E : Put F
jjujj cb p 1 ;
jjvjj ap
¼ vðS n p Þ and apply Lemma 1.4 (ii) to find a finite rank map T : X-X such that T
F ¼ id jjTjj cb ð1 þ eÞLðXÞpð1 þ eÞg p ðXÞ: By the assumptions there is finite-dimensional GCX such that
F and p j ap
d cb n ðG; S p ÞpC and jjT G jj cb p e :
C and jjw jj cb p 1 : We define R:S n p -X by R ¼ v þ wðid S
Let w : S n p -G be an isomorphism such that jjwjj cb p
p Þ as in the proofof Proposition 2.4. Thus it remains to show that R is an isomorphism from S n p onto its
range. To this end, fix an mAN and a unit vector xAM m n ðS jjTjj p Þ: Let d ¼
cb : Note that dX 1 3 : We consider two cases: jjðid M m # v Þxjj4d or jjðid M m # v ÞðxÞjjpd: Ifthe
1 cb þ2jjTjj
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former occurs, then by the choice of T (with id ¼ id M m ) jjTjj cb jjid#RðxÞjjX jjðid#TvÞx þ ðid#TÞðid#wðid S p n
X cb ð1 þ jjvjj cb jjujj cb Þe
X d ap p ðXÞÞe:
Ifwe are in the latter case, then jjðid#wÞðid#ðid S p n
X cb jjðid#vÞðxÞjj
Therefore, we get jjðid#RÞðxÞjjX jjðid#wÞðid#ðI S p n
1 þ 2jjTjj cb
This shows that
jjR jj cb p max 1 þ 2jjTjj cb ;
ap
jjTjj cb ðXÞÞ
ap
ð1 þ 2jjTjj cb Þ:
d p ðX ÞÞ
The assertion is proved and since e40 is arbitrary, we obtain OL
p ðX Þpð1 þ 2LðXÞÞðg p ðXÞ þ Cð1 þ g p ðXÞÞÞ: The assertion is proved. &
ap
ap
Apart from introducing the notion of containing S n 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 m ½ k
: Let F be a vector space and T : c p ðS p Þ-F a linear map with rkðTÞpl: Then there exists a subspace ECc m n
p ðS p Þ completely isometric to c p ðS p Þ and completely
kn
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contractively complemented such that
T j E ¼ 0:
Proof. We may assume that dim ðFÞ ¼ l and that f 1 ; y; f l are linear independent vectors in F
be the unit vector basis in c m p and ðe st Þ 1ps ;tpn denotes the matrix units in S n p : Consider the matrix
: We may assume m m 2 ¼ vk þ r; rok and v4lkn : Let ðh
i Þ i ¼1
a u ; ðw;s;t;jÞ ¼f j ðTðh uk þw # e st ÞÞ where 1pupv; 1pwpk;1ps; tpn and 1pjpl: Since v4lkn 2 ; there exists a non-trivial solution ða 1 ; y; a v Þ ofscalars such that
a u a u ; ðw;s;t;jÞ ¼0
u ¼1
for all 1ps ; tpn; 1pwpv; 1pjpl: We may assume jjða u Þjj p ¼ 1 and then (
a u b wst h uk þw # e st jb wst AC
1pupv ;1pwpk;1pstpn
is completely isometrically isomorphic to c k n p ðS p Þ and T vanishes on E: Using a
sequence ðb u Þ u ¼1 such that u a u b u ¼ 1 and jjðb u Þjj p 0 ¼ 1; we see that m
1pupv ;1pwpk
u 0 ¼1
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.
Lemma 2.10. Let 1pppN ; nAN fixed and X an operator space.
(i) If X contains c k n n p ðS p Þ’s for all k, then X contains c p ðS p Þ’s far out with e ¼ 0:
(ii) If X contains complemented c k n n p ðS p Þ’s, then X contains complemented c p ðS p Þ’s far out with e ¼ 0:
In particular, if X contains S k p ’s (complemented S k
p ’s), then it contains S p ’s far out, complemented S k p ’s far out, respectively.
Proof. (i) We assume that X contains c k n p ðS p Þ’s with constant C: Let T : X-X be a finite rank map, kAN : Choose m such that m
: Let G m C X such that
d cb ðG m ;c m n
p ðS p ÞÞpC: Let r : c p ðS p Þ-G m and s : G m -c p ðS p Þ such that sr ¼ id and
jjrjj n
cb jjsjj cb p C : According to Lemma 2.9, there exists a subspace ECc p ðS p Þ
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completely isometric to c k n p ðS p Þ such that Trj E ¼ 0: Hence, F ¼ rðEÞ is C-cb- isomorphic to c k n p ðS p Þ and Tj F ¼ 0: In order to prove (ii) we assume that X contains complemented c m n p ðS p Þ’s with constant C: Let FCX be a l-dimensional subspace. Let
: Let u : c m n ½ n
p ðS p Þ-X and v : c p ðS p Þ such that vu ¼ id c m p ðS p n Þ ; jjujj cb p 1 and jjvjj cb p C :
Let i
F : F -X : We apply Lemma 2.9 to ðvi F Þ :c p 0 ðS p 0 Þ-F and find a completely contractively complemented copy G of c k n p 0 ðS p 0 Þ such that ðvi F Þ j G ¼ 0: Using either the proofofLemma 2.9 or a simple duality argument, we find a completely
p ðS p Þ-c p ðS p Þ such that Qðc p ðS p ÞÞ is completely isometric to c k n p ðS p Þ and Qvj F ¼ 0: Then, we deduce that P ¼ uQv is a projection satisfying P j F ¼ 0 and id Q ðc m p ðS p n ÞÞ ¼ Qvu: This concludes the proofof(b). For the particular part, we only have to observe that c m n p ðS p Þ is completely contractively complemented in S nm p : Hence for all n the assumptions are satisfied. &
contractive projection Q : c m n
mn
Remark 2.11. In (a) and in (b), we may add ‘with respect to Y ’ 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 n p ’s. Then X is a COS p space. (ii) Let X be an operator space with the CBAP and containing S n 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 a COL p space has the g p -AP and thus Theorem 2.2 yields the assertion. In case (ii) again by Theorem 2.2, it remains to prove that X has the g p -AP. Let X CY such that Y is an OL p space. Let ECX be a finite-dimensional subspace and a finite rank map
T : X -X such that T j E ¼ id E according to Lemma 1.2. Then T ðXÞCX CY is a finite-dimensional subspace and we can find a finite-dimensional C -algebra A and T ðX ÞCFCY such that d cb ðF; L p ðAÞÞpC: Let v : L p ðAÞ-F and u : F-L p ðAÞ such
that u ¼v ; then we deduce for the inclusion map i X : X -Y that
i X T ¼i F vuT
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factors through L p ðAÞ; and thus factors through S m
for m large enough. Let P : Y -X be a completely bounded projection. Then
T ¼ Pi X ¼ Pi F vuT
factors through S m p and X has the g p -AP. & Resuming Lemma 1.5, Propositions 1.6, 2.4 and 2.8, we can formulate the
following result. Theorem 2.14. Let 1opoN and X an operator space with the CBAP. Then,
(i) X is a COS p space if and only if X is completely complemented in Q U S p and contains complemented S n p ’s. (ii) X is an OS p space if and only if X is completely complemented in Q U S p and contains S n p ’s.
As mentioned above our main motivation is the investigation ofnon-commutative L p 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 þ ¼ fx x j xANg; the cone of positive elements of N ; such that for all increasing nets ðx i Þ i with supremum in N and for all xAN þ
n. t ðsup i x i Þ ¼ sup i t ðx i Þ; s. For every 0ox there exists 0oyox such that t ðyÞoN;