aNon-Commutative LP-Spaces

Gilles Pisier*Equipe d’Analyse, Université Paris VI, Case 186, F-75252 Paris Cedex 05, France
Texas A&M University, College Station, TX 77843, USA
E-mail: gip@ccrjussieu.fr

Quanhua Xu
Laboratoire de Mathématiques, Université de Franche-Comté, UFR des Sciences et Techniques,
16, Route de Gray, 25030 Besangon Cedex, France

E-mail: gx@math.univ-fcomtefr

Contents
Introduction . . . . . LL
1.
2.
3.
4.
5.
6.
7.
8.
9.

LLL

ee

eee

eee.

1461

Preliminaries . . . . . «co. citeeeeeee ee ee ieee...
1463
Interpolation . . . Lo. LLLeee
ee eee.
1466
General von Neumann algebras, including type IIT . . . . .........................1470
From classic LP to non-commutative LP”: similarities and differences . .................1477
Uniform convexity (real and complex) and uniform smoothness. . ...................1479
Non-commutative
Khintchine inequalities . . . . ..............................14%4
Non-commutative martingale inequalities . . . . ..............................149
Non-commutative
Hardy spaces . . . .................
ue...
1495
Hankel operators and Schur multipliers . . .................................1500

10. Isomorphismandembedding. . . . . . . ..... LL...
eeeee... 1506
References . . . . . oo.
iii
ee
ei ee
eee
1510

*Partially supported by NSF and Texas Advanced Research Program 010366-163.

HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 2
Edited by William B. Johnson and Joram Lindenstrauss© 2003 Elsevier Science B.V.

All rights reserved.

Non-commutative LP -spaces

Introduction

1461

TT

This survey is devoted to the theory of non-commutative L?-spaces. This theory (in the
tracial case) was laid out in the early 50’s by Segal [181] and Dixmier [47] (see also [110,183]). Since then the theory has been extensively studied, extended and applied, and by
now the strong parallelism between non-commutative and classical Lebesgue integration
is well-known.
We will see that on the one hand, non-commutative L?-spaces share many properties
with the usual L”-spaces (to which we will refer as commutative LP-spaces), and on theother, they are very different from the latter. They provide interesting (often “pathological”) examples which cannot exist among the usual function or sequence spaces. They arealso used as fundamental tools in some other directions of mathematics (such as operator
algebra theory, non-commutative geometry and non-commutative probability), as well as
in mathematical physics.
Some tools in the study of the usual commutative L”-spaces still work in the noncommutative setting. However, most of the time, new techniques must be invented. To
illustrate the difficulties one may encounter when studying non-commutative L?”-spaces,
we mention here three well-known facts. Let H be a complex Hilbert space, and let B(H)denote the algebra of all bounded operators on H. The first fact states that the usual triangle inequality for the modulus of complex numbers is no longer valid for the modulus ofoperators, namely, in general, we do not have |x + y| < |x| + |y| for x, y € B(H), where|x| = (x*x)'/? is the modulus of x. However, there is a useful substitute, obtained in [1],which reads as follows. For any x, y € B(H) there are two isometries © and v in B(H)such thatIx + | <ulxlu® + vlylv*.
The second fact is about operator monotone functions. Let « be a positive real number.
In general, the condition that 0 <x < y (x,y € B(H)) does not imply x* < y®. Thisimplication holds only in the case of « < 1. The last fact concerns the convexity of themap x > x* on the positive part B(H).. of B(H). For o < 1 this map is concave (actually,
the function (x, y) — x® ® y'~% is concave on B(H)y x B(H)., [112]), but for & > 1
convexity holds only if 1 < « < 2. The reader can find more results of this nature in [17].

Some even worse phenomena may happen. It is well known that composed with theusual trace Tr on B(H), all the preceding maps have the usual desired properties. For
instance, the function x >Tr(x*) becomes convex for all « > 1, as one can expect. Now

consider the function(X1se.ns Xn) > Te[ (xf +... +x2)']
on B(H)?. In the commutative setting, the convexity of this function for all o > 1 andn >= 1 is extremely useful in many situations. Again, in the non-commutative case, thisconvexity is not guaranteed, at least for « > 2 (cf: [36]; see also [10] for some. related
results).

‘

1462

G. Pisier and Q. Xu

Despite the difficulty caused by the lack of these elementary properties, we feel the theory has now matured enough for us to be able to present the reader with a rather satisfactory
picture. Of course much remains to be done, as shown by the many open problems which
we will encounter.
We now briefly describe the organization of this survey. After a preliminary section,
we discuss the interpolation of non-commutative L”-spaces (associated with a trace) in
Section 2. This is one of the oldest subjects in the field. The main result there allows
to reduce all interpolation problems on non-commutative L?-spaces to the corresponding

ones on commutative LP-spaces.
Section 3 can be still considered as a preliminary
one. There we introduce the noncommutative L?-spaces associated with a state or weight. This section also contains two
useful results. The first one says that the non-commutative LP -spaces over the hyperfinite
II; factor are the smallest ones among all those over von Neumann algebras not of type I.
The second one is Haagerup’s approximation theorem.
In the short Section 4 we discuss very briefly some similarities and differences between
the commutative L?-spaces and their non-commutative counterparts. One remarkable result in the early stage of the non-commutative L”-space theory is the Gordon-Lewis theorem on local unconditional structureof the Schatten classes. This (negative) result showsthat compared with the usual function spaces, the Schatten classes (and so the general
non-commutative L?-spaces) are, in a certain sense, “very non-commutative”.

Section 5 discusses the uniform convexities and smoothness, and the related type andcotype properties. Although the problem on the uniform (real) convexity of the noncommutative LP-spaces goes back to the 50’s, the best constant for the modulus of convexity was found only at the beginning of the 90s. Two uniform complex convexities (theuniform PL-convexity and Hardy convexity) are also discussed in this section.
The central object in Section 6 is the non-commutative Khintchine inequalities, of paramount importance in this theory. Like in the commutative case, they are the key to a largepart of non-commutative analysis, including of course the type and cotype properties of
non-commutative LP” -spaces, and closely linked to the non-commutative Grothendieck the-orem.
Section 7 presents some very recent results on non-commutative martingale inequalities.
In view of its close relations with quantum (= non-commutative) probability, this direction,
which is still at an early stage of development, is likely to get more attention in the near

future.

:Section 8 deals with the non-commutative Hardy spaces. We present there some noncommutative analogues of the classical theorems on the Hardy spaces in the unit disc, suchas the boundedness of the Hilbert transformation, Szegé and Riesz factorizations.

The first result in Section 9 is Peller’s characterization of the membership of a Hankel
operator in a Schatten class. This result is related to Schur multipliers. The rest of thissection gives an outline of the recent works by Harcharras on Schur multipliers and noncommutative A(p)-sets.
:
The last section: concerns: the embedding and isomorphism - of non-commutative
L?P-spaces. Almost all results given there were obtained just in the last few years. Thisis still a very active direction.
iNon-commutative LP -spaces

1463

We end this introductory section by pointing out that we will freely use standard notationand notions from operator algebra theory, for which we refer to [48,104,178,184,185,190].

1. Preliminaries
In this section we give some necessary preliminaries on non-commutative LP-spaces associated with a trace. This requires that the underlying von Neumann algebra be semifinite(see below the definition). In Section 3, we will consider the non-tracial case.
M will always denote a von Neumann algebra, and M. its positive part. We recall thata trace on M is a map t: M. — [0, oc] satisfying@ tx +y)=1(x)+71(), Vx, y e My;(ii) T(Ax) = AT (x), VA €[0, 00), x € M;(ii) t(*u) =t@u™), Vu

e M.

7 is said to be normal if sup, t(xy) = t(sup, x) for any betinded increasing net (xy) in
M.., semifinite if for any non-zero x € M
there is a zero ye Mysuchthaty <xand t(y) < oo, and faithful if T(x) = 0 implies x —=0.1f 7(1) < oo (1 denoting the identityof M), t is said to be finite. If 7 is finite, we will assume almost systematically that ©
is normalized, that is, 7(1) = 1. We often think of © as a non-commutative (= quantum)
probability. .
A von Neumann algebra M is called semifinite if it admits a normal semifinite faithful(abbreviated as n.s.f.) trace 7, which we assume in the remainder of this section. Then let
8S. be the set of all x € M. such that T(suppx) < oo, where supp x denotes the supportof x (defined as the least projection p in M such that px = x or equivalently xp = x). Let
8 be the linear span of Sy. It is easy to check that S is a #-subalgebra of AM which is
w*-dense in M, moreover for any 0 < p < oo, x € § implies |x|? € S; (and so7(|x|P) < 00), where |x| = (x*x)!/? is the modulus ofx. Now we define_
py1Y/P
Ixllp, =[z(x1?)]"",xe&.

One can show that | - ||, is anorm on S if 1 < p < oo, and a quasi-norm (more precisely;:a p-norm) if 0 < p < 1. The completion of (S, | - ||») is denoted by L? (M, 7). This is the
non-commutative L?-space associated with (M, 7). For converence, we set L°(M, 7) =M equipped with the operator norm. The trace t can be extended to a linear functional
on &, which will be still denoted by 7. Then~~|r@| <x,Vxes.

Thus * extends to a continuous functional on L!(M, 7).
The elements in L? (M, 1) can be viewed as closed densely defined operators on H (Hbeing the Hilbert space on which M acts). We recall this briefly. A closed densely defined
operator x on H is said to be affiliated with M if xu = ux for any unitary u in the commu-tant M’ of M. An affiliated operator x is said to be t-measurable or simply measurable if

‘

1464

G. Pisier and Q. Xu

7(ex(|x])) < oo for some A > 0, where e; (|x|) denotes the spectral resolution of |x| (corresponding to the indicator function of (A, 00)). For any measurable operator x we define
the generalized singular numbers bype (x) =inf{r > 0: T(ex(Ix])) <t},¢>0.

It will be convenient to denote simply by w(x) the function t — i; (x).
Note that w(x) is a non-increasing function on (0, 00). This notion is the generalizationof the usual singular numbers for compact operators on a Hilbert space (see [72]). It was
first introduced in a Bourbaki seminar note by Grothendieck [77]. It was studied in details

in [132,62] and [64].
Let LO(M, 1) denote the space of all measurable operators in M. Then LO(M, 7) is ax-algebra, which can be made into a topological %-algebra as follows. LetV(e, 8) ={x € L%M, 1): pe(x) <8}.
Then {V (¢, 8): ¢,8 > 0} is a system of neighbourhoods at 0 for which LO(M, ©) becomesa metrizable topological *-algebra. The convergence with respect to this topology is called .
the convergence in measure. Then M is depse in LYM, ©). We refer to [131] and [191]for more information.
~~
The trace 7 is extended to a positive tracial functional on the positive part Lo (M, 1) ofLO(M, 7), still denoted by , satisfying

g

-

<

= [meanxem.
0:Then for 0 < p < oo,.

LPM, 7) = [x € L%WM, 7): t(|x|”) < 00}

and[x], = (z(1xI?))"/".

Also note that x € LP (M, 1) iff u(x) € LP(0, 00), and | x lp = ll)
Lr (0,00)- Recall thatw(x) = pu(x*) = u(lx|); sox € LP(M, 7) iffx* € LP (M, 1), and we have xl, = lx*p.
The usual Holder inequality extends to the non-commutative setting. Let 0 < 7, p,q <oo be such that 1/r=1/p + 1/q. Thenx€ LPM, 7),ye LIM, 7) => xy € L"(M, 7) and xy» < lixll p17 llq- )In particular, if r = 1,

[ee < lxylls < Ixllpllyly,x €LP(M,7),y€ LI(M, 7).

This defines a natural duality between L? (M, 7) and LY (M, 1): (x, y) = T(xy). Then forany
1 < p<

oo we have

(LPM, D) = LIM, 7) (isometrically).

:ETENon-commutative LP -spaces
Thus,

L'WM, 7) is the predual of M,

1465

and LP (M, 7) is reflexive for 1 < p < co. Notethat the classical theorem of Day on the dual of L? for 0 <:p < 1 was extended to thenon-commutative setting by Saito [176]: the dual of LP (M, 7), 0.< p <1, is trivial iff M
has no minimal projection.
REMARK.

[114] contains a different construction.of non-commutative

non-commutative upper integral.

L? -spaces via a:

Although we will concentrate ON non comutative. L=-spaceq in this survey, the moregeneral so-called “symmetric operator spaces” are worth mentioning: let E be a rearrangement invariant (in short #4.) function space on (0, co), the symmetric operator space associated with (M, t) and E is defined by

EM, 7) = {xe LOM, 7): ux) € E}

and|xlgomo = |r| g-

In particular, if E = LP (0, co), we recover L? (M, 7). These symmetric operator spaces
have been extensively studied, see, e.g., [51,53,54,133,134] and [203] for more information.

We end this section by some examples.
(i) Commutative LP-spaces. Let M be an Abelian von Neumann algebra. Then M =L*®(£2, u) for a measure space (£2, uu), integration with respect to the measure yu gives us
an n.s.f. trace, and L? (M, 7) is just the commutative L?-space LP (£2, un).
(it) Schatten classes. Let M = B(H), the algebra of all bounded operators on H, and
7 = Tr, the usual trace on B(H). Then the associated L?-space L?(M, 7) is the Schattenclass SP(H). If H is separable and dim H = oo (resp. dim H = n), we denote SP.(H)by SP (resp. SP). Note that in our notation $% (H) is not the ideal of all compact operatorson H but B(H) itself. [72,128] and [182] contain elementary properties of S? (H).
(iii) The hyperfinite II] factor. Let M, denote the full algebra of all complex rn xX nmatrices, equipped with the normalized trace o,,. Let(R, 7) = QR (An, Ta)(An, Ta) = (M2,02), neN,

nzl

Cn

be the von Neumann algebra tensor product. Then R-is the hyperfinite IT; factor and 7 is the(unique) normalized trace on R. There is another useful description of R. Let (&,),>1 bea sequence of self-adjoint unitaries on a Hilbert space, satisfying the following canonicalanticommutation relations
sigj+ &j&; = 2845,

i,jeN.

.

(CAR)

Let Ry be the C*-algebra generated by the &;’s. Then Ry admits a unique faithful tracialstate, denoted by 7, which is defined as follows. For any finite subset A = {i1,..., i}CNwith i; <--- <i, we put wy = g; ---¢&;,, and wg = 1. Then the trace 7 is uniquely determined by its action on the wa’s: T(wa) =1 (resp. =0) if A = 0 (resp. # ©). Consider
Ro as a C*-algebra acting on L?(t) by left multiplication. Then the von Neumann algebra

1466

G. Pisier and Q. Xu

generated by Rg in B(L?(t)) can be (isomorphically) regarded as the hyperfinite II; factor R. Note that the family of all linear combinations of the w4’s are w*-dense in R anddense in L?(R) for all 0 < p < 00; also note that {w4: A C N} is an orthonormal basisof L2(R) (= L%(1)). Finally, we mention that the von Neumann subalgebra generated by{e1,...,&2,} is isomorphic to Mo», and then the restriction of t to this subalgebra is justthe normalized trace of M»». We refer to [25] and [158] for more information.
(iv) Group algebras. Consider a discrete group I". Let uN (I") C B(£5(I™)) be the associated von Neumann algebra generated by the left translations. Let tr be the canonical trace
on vN(I"), defined as follows: 7, (x) = (x(8.), 8.) for any x € yN(I"), where (8g)ger
denotes the canonical basis of (J), and where e is the identity of I". This is a normal
faithful normalized finite trace on vN(I"). A particularly interesting case is when I" = F,,,the free group on n generators. We refer to [67] and [196] for more on this theme.

2. Interpolation
This section is devoted to the interpolation of non-commutative LP -spaces. It is well known
that the non-commutative L”-spaces associated with a semifinite von Neumann algebra =form an interpolation scale with respect to both the real and complex interpolation methods (see (2.1) and (2.2) below). This result not only is useful in applications but also canbe taken as a starting point to define non-commutative L? -spaces associated to a von Neumann algebra of type III (which admits no n.s.f. trace). This is indeed the viewpoint taken
by Kosaki [106] (see also [192]). We will discuss this point in the next section. Here we
restrict ourselves only to semifinite von Neumann algebras. Thus throughout this section,

M will always denote a semifinite von Neumann algebra equipped with a faithful normal
semifinite trace T. We refer to [15] for all notions and notation from interpolation theory
used below. Let 1 < pg, p1 < oo and 0 <6

< 1. It is well known that

LP(M, 7) = (LP°(M, 7), LP{(M, 7)),
LPM, 7) = (LPM, 7), LP'(M, Dg.»(with equal norms),(with equivalent norms),

2.
2.2)

where 1/p = (1—6)/po+6/p1, and where (-, -)g, (-, -)g, , denote respectively the complex
and real interpolation methods. It is not easy to retrace the origin of these interpolation
results. Some weaker or particular forms go back to the 50’s (cf, e.g., [47,110,172]). The
results in the full generality were achieved by Ovchinikov [133,134] (see also [135] for thereal interpolation, and [141] in the case of Schatten classes).

:(2.1) and (2.2) easily follow from the following result. Recall that u(x) denotes thegeneralized singular number of x (see Section 1) and that a map T : X — ¥ is called contractive (or a contraction) if | T|| < 1.
THEOREM 2.1. For any fixed x € L\(M, ©) + L™®(M, 1) there are linear maps T and S(which may depend on x) satisfying the following properties: GA) T:LY(M,1) + L®°M, 1) — L'(0, 00) + L*®(0, 00), Tis contractive from
L?(M,7) to LP(0, 00) for p-=1andp = 00, and Tx = u(x);~Non-commutative LP -spac;

1467(i) S:L1(0, 00) + L®(0, 00) — LYM, 7) + L®WM,x),
L?(0, 00) to LP(M, 7) for
p=1 and p = 00, and Spu(xS is contractive from=x.

Although not explicitly stated, Theorem 2.1 is implicit in the literature. It is essentiallycontained in [4] for Schatten classes, and in some different (weaker) form in [53] for thegeneral case. We will include a proof at the end of the section.
REMARK. In interpolation language, Theorem 2.1 implies that the pair (LYM, 1),
L®(M, 1)) is a (contractive) partial retract of (L1(0, 0), L°°(0, c0)). We shouldemphasize the usefulness of such a result: it reduces all interpolation problems on(LYM, 1), L°(M, 1)) to those on (L!(0, 00), L®(0, 00)). Recall that (L!(0, co),L°(0, c0)) is one of the best understood pairs in interpolation theory. We now illustrate
this by some examples. More applications can be found in [4,53,54] and [133,134].
First let us show how to get (2.1) and 3.2) from their commutative counterparts.

PROOF OF (2.1) AND (2.2). Let x € LP(M, t) (noting that L?(M, 1) Cc L'(M, 7) +L%®(M, 1)). Let S be the map associated to x given by Theorem 2. 1. Then by interpolation

S$: (L1(0, 00), L™(0, 00),

= (L'(M, 7), L®(M, T)),is a contraction. However, it is classical that

(Lt (0, 00), L=(0, 00), =L?(0, 00):(with equal norms):

Thus we deduce

Ixlle = |Su@)|, < [e®|, =x;
whenceLPM, 1) C (! WM, 1), L®(M, 2)

a contractive inclusion.

The inverse inclusion is proved similarly by means of the map
shown (2.1). In the same way, we get (2.2).

T'. Therefore, we have0

The above argument works in a more general setting as well.
COROLLARY 2.2. Let F be an interpolation functor. Then

FLY

WM, 1), LPM, 1) = F(L1(0, 00), L%(0, 00) (M, 1).

More generally, for any ri. function spaces Ey, E1 on (0, 00)

F(EoM, 1), Ex(M, 1)) = F(Eo, ED)(M, 7).

.

1468

G. Pisier and Q. Xu

This corollary is contained in [53] (and in [4,141] for Schatten classes).

REMARK. As a consequence of Corollary 2.2, we have that for any ri. function space E
the map T (resp. S) in Theorem 2.1 is contractive from E(M, 7) (resp. E) to E (resp.
E(M, 7)). In particular, T and S are contractions between the L?-spaces in consideration
forall 1 < p<oo.

The following particular case of Corollary 2.2 is worth being mentioned explicitly. HereK; denotes the usual K-functional from interpolation theory.
:
COROLLARY
anyt>0_—2.3. Let 1 < po, p1 < 00. Then for any x € LP*(M, ©) + LP1(M, 1) andKi (x; LP(M, 7), LPY(M, 1) = Ki (u(x); LPO(0, 00), LP1(0, 00).
In particular,t

Kos
REMARKS.

LOM, 2), L2(M, 0) = |0 Ws (x) ds.

~:(i) Using a factorization argument, one can easily extend Corollary 2.3 to thecase of quasi-Banach spaces, so that po, p1

are now

allowed to be in (0, oo]. Then theequality there has to be replaced by an equivalence with relevant constants depending only

on py, pi (see also [135]).

:(il) As a consequence of the preceding remark, the indices po, p; in (2.2) can vary in
0, oo].

(iii) (2.1) also extends to the quasi-Banach space case (cf. [201]).
PROOF OF THEOREM 2.1. Fix an x € L'(M, 7) + L®(M, 7). We may assume x > 0.
Indeed, by polar decomposition, it is easy to reduce the proof to this case.
First we suppose x is an elementary operator, i.e., of the form:n.

x= > ake,
k=1

|

)

where for all 1 < k <n,a; € (0,00),

and where)

the e;’s are disjoint projections with(ex) € (0, 00). Then we define

Py=) i=(yer)
—e,
Tle)

yeL'(M,7)+L®WM,1).

Note that P is the orthogonal projection of L*(M, 7) onto its subspace generated by{e1,..., en}. In particular, P is selfadjoint. Let y € L®(M, 7). Thenjr (vey)|
1Pyloo < sup —2—=<
1<k<n

Ter)

loot) =_, |yloo.
sup =

ir

Tler)

:

yg

|

|

Non-commutative LP -spaces

1469

Therefore, P is a contraction on L>°(M, t). By duality, P is a contraction on L!(M, 1)
as well.

}

Let \V be the subalgebra generated by {e,..., e,} (the identity of \ being e =e; +-+++ ep). Then NV is isomorphic to I and t induces a weighted counting measure v on
IZ, namely, v({k}) = t(e) for all 1 < k < rn. Itis clear that for any p

LPN, Tt |p) =1;(v)isometrically.

Thus we can identify (NV, T|xr) with (I, v). With this identification, u(x) is exactly theusual non-increasing rearrangement of x with respect to the measure v. On the other hand,
it is classical (and easy to prove in our special case) that there are linear maps R and Q
satisfying (cf. [33])@ R:I}W) + Iv) — L1(0, 00) + L°°(0,00), R is contractive from Lo
toL?(0, 00) for p = 1, 00, and Rx = u(x);(ii) Q:L(0, 00) + L*®(0, 0) — Iv) +15,(v), Q is contractive from L?(0, 00) toIw) for
p=1,00,and Qu(x) =x.
Then we set T = RP and § = iQ, where i is the natural inclusion of IT (v) + I3,(v)(= LYW, t|n) + LP WV, |p) into LY (M, ©) + L®(M, 7). One easily checks that T
and § satisfy all requirements of Theorem 2.1. Therefore, Theorem 2.1 is proved for elementary. operators.
}
Before passing to general (positive) operators, we note that 7" and S constructed above
are positive in the sense that y > 0 (resp. f > 0) implies Ty > 0 (resp. Sf = 0).

Now for a positive x € L'(M, t) + L*®(M, 7), using the spectral decomposition ofx, we may choose an increasing sequence {x,} of elementary positive operators suchthat x, < x for all n > 1, lim,»oo ps (xy) = p(x) for all £ >)0 and lim,_—00Xp = x in
the topology o(LY(M,7) + L®(M,©), LM, 7). LP (M7). See [64], pp. 277278. By the first part of the proof, for each rn there are T, ‘and S, associated with
xp as in Theorem 2.1. Thus (7}) is a bounded sequence in B(L*°(M, t), L%(0, c0)).
Since B(L*™(M, 7), L*(0, 00)) is a dual space with predual
LR(M, t) ®" L(0, co),passing to a subsequence if necessary, we may assume that T,\converges to T’ in

B(L*(M, 1), L*°(0, 00)) with respect to the w*-topology. Thus 7” is\acontraction fromL®(M, 1) to L*®(0, 00). To show that T’ also defines a contraction from L'(M, 7) toL'(0,00) let y € LY(M, 7) N L®°(M, 7) and f € L'(0, 00) N L°°(0, c0).\Then| ror= tim [nos
oo

,

le)

\

whence[o.¢]|0ro] <limsup [T)[ 1f loo < M111 loo,n—oo:which implies that T7’y € L1(0, 00) and | T’ yll1 < llyll1. Hence T” extends to a contraction

from L!(M, 1) into L'(0, 0).

:

oo

1470

G. Pisier and Q. Xu

On the other hand, by the positivity of T;,, we have p(x,) = Tx, < T,x. Taking limits,
we find p(x) < T'x. Hence p(x) = wT’x for some w € L*°(0, 00) with ||w|leo < 1. Then
one sees that the map 7" defined by Ty = wT"y has the property (i) of Theorem 2.1.
Similarly, from the sequence (S,) we get the desired map S.
O
REMARK 2.4. The result of interpolation applied to a compatible pair (Xo, X1) of Banach

spaces depends in general very much on the way in which we view this pair as compatible.
There is however an elementary “invariance” property which we will invoke in the sequel,as follows: let (Xp, X1) be a compatible pair of Banach spaces. Now let (¥p, ¥1) be anothercompatible pair of Banach spaces and let ug: Yy — Xo and u;:Y; — X; be isometricisomorphisms,

which coincide on Yp NY;(in that case it is customary

in interpolation

theory to think of uo and u; as the “same” map!). Equivalently, we have an isometricisomorphism u: Yo + ¥1 — Xo + X1 such that the restrictions ug = uy, and u; = uy, areisometric isomorphisms respectively from Yp to Xo and from ¥; to X1. Then u defines anisometric isomorphism from (¥p, ¥1)g to (Xo, X1)g, so that
Jo,

Te

~

Xo,

X1)e

(0<6

<1).

oo

This follows from the interpolation property applied separately to u and its inverse. If weassume that the pairs are made compatible with respect to continuous injections J : Xo —Xi and j:Yp — Yi. Then to say that ug and 1 are the “same” map means that uj = Jug.
In particular, if Xo = Yy and if ug is the identity on Xo = Yj, then this reduces to uj = J.

a:
3. General von Neumann algebras, including type ITI.
CoThe construction of non-commutative L?-spaces based on n.s.f. traces outlinedin Section 1
does not apply to von Neumann algebras of type III, which do not admit 7.s.f traces.
However, it is known that any von Neumann algebra has an n.s.f weight (a weight issimply an additive and positively homogeneous functional on the positive cone with values
in [0, oo]). This section is devoted to the non-commutative LP -spaces associated with a von
Neumann algebra equipped with an n.s.f. weight. There are several ways to construct thelatter spaces (cf., e.g., [3,78,87,106,113,192]). We will present two of them. The first one

is to reduce von Neumann algebras of type III to semifinite von Neumann algebras withthe help of crossed products, as proposed by Haagerup [78]. The second way is via thecomplex interpolation; so it can be considered as a continuation of the results establishedin the previous section for the semifinite case. This was developed by Kosaki [106] andTerp [192] (see also [88,89] for related results).
'
We begin with the construction via interpolation. Let M be a von Neumann algebra. We
know that M is a dual space with a unique predual, denoted by M,. We define, as usual,

LYM) = M, and L®(M) = M. Now we are confronted with the problem of defining
LP(M) for any 1 < p < co. For simplicity and clarity we will consider only the casewhere M is o finite, as in [106]. The reader is referred to [192] for the general case. Fix a|Non-commutative LP -spaces

1471

distinguished normal faithful state ¢ on M. Then we embed M into M,, by the following
left injection:

JiM—=> My,jx)=x¢(here xp(y)=¢(yx) Vy e M).

It is clear that j is a contractive injection with dense range. Thus we obtain a “compatibility” for the pair (M, M.) with respect to which we may consider interpolation spaces
betweenM

and M,.. Now

let 1 < p.< oo. Following

Kosaki,we introduce the corre-

sponding non-commutative LP-space asN

LPM, ¢) = (M, M:)1/pTo show the so-defined non-commutative L?-spaces possess all properties one can expect,
one should first note the important fact that L>(M, ¢) is a Hilbert space, more precisely,

L*(M, 9) = H,, where H, is the Hilbert space in the GNS construction induced by ¢(obtained after completion of M equipped with the inner product (x, y} = ¢(y*x)). The
proof of this fact given in [106] uses the modular theory. Here, we would like to point out
that it directly follows from a general result in interpolation theory, that we describe as
follows.
Let X be a complex Banach space. Let X denote the conjugate space of X, i.e., X is justX itself but equipped with the conjugate complex multiplication: A - x = Ax for any A € C
andxe

X. ForxeX,

x denotes the element x considered as an‘element in X. Given a

linear map v:X — ¥, we denote by #: X — Y the same map acting\on the “conjugates”.
Now suppose that there is a bounded linear map J : X* — X which is injective and of dense
range. This allows us to consider (X*, X) as a compatible pair of Banach
spaces. Suppose
further that J is positive, i.e., £(J(§)) = 0 for any & € X*. Then (£, n)=&(J(n)) definesa scalar product on X*. (Note: for & € X* and x € X we write £(x) = £(x)\) Let H be thecompletion of X* with respect to the above scalar product. Note that H contains X* as adense linear subspace. Thus we can define a bounded linear injection v: H — X by simply
setting (on an element of X™) v(§) = J(&), and extending by density to the whole of H.

Identifying H* with H as well as (X)* with X*, and denoting by ‘v: (X)* — H*\=H theadjoint of v (in the Banach space sense) we see that J = vv. These facts are well known(and easy to check). The general theorem referred to above is the following
THEOREM 3.1.

With the above assumptions, (X*, X); 72 = H with equal norms.

REMARK. This is well-known ([116]) with the additional assumption that X is reflexive.
The general form as above was observed in [153], p. 26 (see also [197] and [42] for related
results).
:

COROLLARY 3.2. L2(M, 9) = H, with equal norms.

NT

PROOF. We let X = M,, X* = M. Recall that the involution on M, is defined by
YH (x) = ¥ (x) ( € My, x € M). Let J : M — M, be the map takingx to (x)*j= px*
and let u;: M, — M, be the (linear) isometry taking ¢ to ¥*. We have (£,7) =1472

G. Pisier and Q. Xu

EJ)= E(pn®) = E(pn*) = (n*€), thus we find H = H,, with equal norms and wehave J = uy j, so that the result follows by invoking Remark 2.4 (here ug is simply theidentity of M).
a
Using this corollary and the reiteration theorem, we see that the dual space of L? (M, ¢)is (isometrically) equal to LP (M, ¢) forany 1 < p < 00 (1/p + 1/p’ = 1). The duality is
induced by the scalar product of Hy, that is, (x, y) = ¢(y*x), x, y € M. Corollary 3.2 also

yields the Clarkson inequalities in LP (M, ¢) forany 1 < p < oo (see Theorem 5.1 below).
Thus, LP (M, ¢) (1 < p < 00) is uniformly convex. We will see more precise results on
this in Section 5.
REMARK. Instead of the left injection considered previously, one could equally take theright injection of M into M,, i.e., x => ¢x (here px(y) = p(xy) Yy € M). Then the
resulting interpolation spaces are isometric to those obtained previously.
In view of the results in the last section, one is naturally led to consider the real interpo-

lation as well. Set, for 1 < p < ©

LPP (M, 9) = (M, Mi)1/p.pThe problem now is whether L?” (M, ¢) and LP(M, ¢) are isomorphic. For the special
case of p == 2, the answer is affirmative. Indeed, Theorem 3.1 admits a counterpart for thereal interpolation as well (see [116] in the case of reflexive spaces; [205], p. 519 for thegeneral case; see also [42] for more related results). Thus L%%(M, ¢) = L%(M, ¢) withequivalent norms. However, this is no longer true for all other values of p, as shown by thefollowing example, due to Junge and the second named author.
EXAMPLE 3.3. Let ¢ be the state of B(I%) given by a diagonal operator D of trace 1
and whose diagonal entries are all positive. Then, for any 1 < p #2 < oo, the two spaces(B®), BU*)+)1/p and (B(%), B(I%):)1,p, p do not coincide. Indeed, letR (resp. R,) bethe subspace of B(I%) (resp. B ®),) consisting of matrices whose all rows but the first are
zero. It is clear that R = 2 and R, =1%(d) isometrically, where d = (dn), is the sequenceof the diagonal entries of D, and where [2(d) is the weighted [2-space with the norms

1% x l 2a) = (= Xnd,
ndn| )

1/2
.

—

B

|

On the other hand, let P: BI?) — R be the natural projection. P is contractive on B(I%).
It is easy to check that under the left injection associated with ¢, P is also a contractive
projection from B(I%), onto R,. Now assume that for some 1 < p < oo the two interpola~._tion spaces (B(I?), B(1*:)1/p and (B12), BU) 1p, p have equivalent norms. Then we
deducethat (R, R)1/p

and (R, Ry)1/p, p have equivalent norms too. However, it is well-known that the-first space is still a weighted /2-space (and hence a Hilbert space), while thesecond one is isomorphic to a Hilbert space only when p = 2. Thus we have proved ourassertion.

NU

:

ca

Non-commutative LP -spaces

1473

This construction by interpolation has several disadvantages: there is no natural notion of

positive cone, no reasonably handy bimodule action by multiplication of M on LP (M, ¢),and finally the case p < 1 is excluded. However, these difficulties disappear in Haagerup’s
construction, to which we now turn. Our main reference for Haagerup’s L?-spaces is [191].
Let M be a von Neumann algebra equipped with a distinguished z.s.f weight ¢. Let oy =a), t €R, denote the one parameter modular automorphism group of R on M associatedwith ¢. We consider the crossed product R = M x, R. Recall briefly the definition of R. If
M acts on a Hilbert space H, R is a von Neumann algebra acting on L%(R, H), generated
by the operators w(x), x € M, and the operators A(s), s € R, defined by the following
conditions: for any § € LR, H)andt eRn(x)E)O)=o (x)§@)andA()E))=EF —5).

SE

Note that 7 is a normal faithful representation of AM on L*(R, H). Thus we may identify
M with w(M). Then the modular automorphism group {oc };cr is given by

xeM,

NN

teR.

There is a dual action {6:};cr of R on R. This is a one parameter automorphism groupof R on R, implemented by\the unitary representation {W (¢)};cr of R on L>(R, H):Gx) =WExWE),

teR, xeR,

where

WDE)(5) =e 8s),

Ee L’®R H), 1,5 €R.

Note that the dual action 6; is also uniquely determined by the following conditions

Gi(x)=x
Thus MandG(r(s))=e"A(s),is invariant under {6;};cr. In fact,

VxeM, s,t eR.

M is exactly the space of the fixed points of{6¢}ser, namely,
M={x eR: 6;(x) =x,Vt eR}.
Recall that the crossed product R is semifinite. Let 7 be its n.s.f. trace satisfying

Tob, =e,

YteR.

Also recall that any n.s.f weight ¥ on M induces a dual n.s.f. weight ¥ on R. Then ¥admits a Radon~Nikodym derivative with respect to t. In particular, the dual weight ¢ ofour distinguished weight ¢ has a Radon-Nikodym derivative D with respect to t. Then@(x)
= (Dx),

xR.

:

1474

G. Pisier and Q. Xu

Recall that D is an invertible positive selfadjoint operator on
L2(R, H), affiliated with R,
and that the regular representation A(z) above is given by

At) =D",teR.

.

Now we define the Haagerup non-commutative I” -spaces (0< p < 00) by
APM,

9) = {x e LOR, 7): Gr (x) =e H Py, VteR}.

(Recall that LO(R, 7) denotes the topological x-algebra of all operators
on L%(R, H) measurable with respect to (R, 7).) It is clear that A? (M, @) is avector subspace of LO(R, 7),

invariant under the *-operation. The algebraic structure of AP (M, p) is
inherited from thatof LOR, 7). Let x € AP(M, p) and x = u|x]| its polar decompos
ition. Then u €[x] € AP(M, 9). Recall that:

APM, 9)=M

and

M and

AM, ) = M,.

The latter equality is understood as follows. As mentioned
previously, for any » € MF,

the dual weight & has a Radon—Nikodym derivative, denotedby hg, with respect to 7:(x) =t(hyx),

xeRy.

Then

ho € LR, 7)

and& (hy) =e'h,,VicR.

Thus he, € AY(M, @)... This correspondence between Mj

and A'(M, 9), extends toa bijection between M, and Al (M, ©). Then for any weM,, if 0 = u|w| is its polar
decomposition, the corresponding f,, € A! (M, @) admits thepolar decomposition

ho =ulhy| = uh.
Thus we can define a norm on A! (M, gp) by

Itolli = lol) = lola,©

M,.

In this way, A(M, ¢) = M, isometrically. Now let 0 < p< 00. Since x € AP (Msp) iff[x]? € ALM, @), we define1
lp = 1x17],

xe 4PM,0).

:Thenif1< p < oo, | | » is a norm (cf. [78] and[191]),andi
f0< p < 1, | - Il» is a p-norm(cf. [108]). Equipped with | - | ,, AP (M, ¢) becomes a Banach
space or a quasi-Banach
space, according to 1 < p < 00 or 0 < p < 1! Clearly,
:

=]

=i.

Ei

FU

-

Non-commutative LP -spaces

1475

REMARKS. (i) Using [191] Lemma 115, one easily checks that AP (M, ¢) is isometricto a subspace of the non-commutative weak LP-space LP*®°(R, 1). Also note that inAP(M, p) the topology defined by | - ||, coincides with the topology induced by thatof LO(R, ©) (cf. [191]).

(ii) One weak point of the Haagerup non-commutative LP-spaces is the fact that for any
p # q the intersection of AP(M, ¢) and A7(M, ¢) is trivial. In particular, these spaces
do not form an interpolation scale. This causes some difficulties in applications (especiallywhen interpolation is used).

:

As usual, for 1 < p < oo the dual space of AP (M, ¢) is A” (M, ©), 1/p+1/p'=1.To
describe this duality, we need to introduce a distinguished linear functional on A'(M, ¢),called trace and denoted by tr, which is defined byr(x)= wx (1), xe A (M,p),where wy, € M,

is the nike

fication between M,

normal functional associated with x by the above identi-

and AMsatisfying|,¢). Then tr is a continuous functional on Al(M, ¢):|r| <tw(lxl)=xl}, x € ATM,9).

:

The usual Holder inequality also

holds for these non-commutativep.q,r1/q. Then<oosuchthat1/r =1/p +x € AP(M,p) and y € A7(M, 9)=

LP?-spaces. Let 0 <|ooxy e A"(M,9) and |lxy[l, <lxlipllylg-

In particular, for any 1 < p < oo we haveley) | < lxylls < xlpliyly,:x € AP(M, 9), y € AZ (M,9).

Thus, (x,y) +> tr(xy) defines a duality between A” (M, ¢) and A?’ (M, ¢), with respect
to which(APM, 9)" = AP (M, ¢)isometrically, 1 < p < 00.

This functional tr on A!(M, ¢) plays the role of a trace. Indeed, it satisfies the following
tracial propertyu(xy) =tr(yx),x € AP(M, 9p), y € A” (M, 9).

The reader is referred to [191] for more information.

)

:

1476

G. Pisier and Q. Xu

THEOREM 3.4. Let M be a von Neumann algebra.
(i) Let 0 < p < co. If T is an n.s.f. trace on M, then LP (M, 1) (the non-commutative
L? space described in Section 1) is isometric to AP (M, @).(ii) Let 0 < p < 00. Then AP (M, @) is independent of @, i.e., if ¢ and i are two n.s.f.
weights on M, then AP (M, ¢) and AP (M, ) are isometric.
(iii) Let ¢ be a normal faithful state on M and 1 < p < oo. Then LP (M, ¢) andAP (M, @) are isometric.
:
The first two parts of Theorem 3.4 are due to Haagerup [78] (see also [191]), and thethird one to Kosaki [106]. As can be expected, the proof of Theorem 3.4 heavily depends
on the modular theory.
The preceding statement allows a considerable simplification of the notation, as follows:
CONVENTION. From now on, given a von Neumann algebra M, LP (M) will denoteany one of the non-commutative LP” -spaces associated with M appearing in‘ Theorem 3.4.
(The latter shows that these spaces are all “the same”.) However, if M is semifinite, we will
always assume that LP (M) is the LP-space constructed from an n.s.f. trace as in Section 1.
The following basic result is very useful to reduce the failure of certain properties of

LP-spaces to the special case of the hyperfinite factor L? (R). Recall that R denotes thehyperfinite IT; factor (see Section 1).
THEOREM 3.5. Let M be a von Neumann algebra not of type 1. Theh forany 0 < p< oo(resp. 1 < p < 00) LP (R) is isometric to a (resp. 1-complemented) subspace of L? (M).

—The proof of Theorem 3.5 combines several more or less well-known facts. The key
point is that if M is not of type I, then R is isomorphic, as von Neumann algebra, to aw*-closed x-subalgebra of AM which is the range of a normal conditional expectation
on M. The reader is referred to [122] for more details and precise references.
We end this section with Haagerup’s approximation theorem of an L?(M) associatedwith an algebra M of type III by those associated with semifinite von Neumann algebras(cf. [79D).
THEOREM 3.6. Let M be a von Neumann algebra equipped with an n.s.f. weight ¢. Let
AP (M, 9) be the associated Haagerup LP-space (0 < p < 00). Then there are a Banach
space X (a p-Banach space if 0 < p < 1), a directed family {(M;, ©;))}icr of finite von
Neumann algebras M; (with normal faithful finite traces t;), and a family {j;};er of isometric embeddings j; : LP? (M;, tj) — X such that@)ji(LP(M;, 17) C ju (LP (My, Tp) forall i,i’ € I with i <i’;oC

(1) U;er(LPM, 1) is dense in X;:(iii) AP (M, @) is isometric to a (complemented for 1 < p < 00) subspace of X.

-

Non-commutative LP -spaces

1477

4. From classic L? to non-commutative L”: similarities and differences
A good part of the early theory consisted in extending commutative results over to thenon-commutative case; this usually required specific new methods, but without too many

surprises. For instance, we have already seen that a non-commutative L?-space LL” (M)is reflexive for any 1 < p < 00. Moreover, just like in the commutative case it is easy tocheck that L' (M) has the RNP iff M is atomic. Indeed, if M is not atomic, L1(M) contains a 1-complemented subspace isometric to L! (0, 1), hence fails the RNP. On the other

hand, L!(M) is weakly sequentially complete for any M. Moreover, there are characterizations of weakly compact subsets in L!(M), analogous to those in the commutative setting (cf. [190, II1.5] and the references given there; see also [140] for more recent results).
Moreover, we will see later in Section 5 that any non-commutative L?-space (0 < p <1)
has the analytic RNP.
oo
However, the differentiability of\the norms of non-commutative 7” -spaces has not been

well understood yet. This problem
was considered only for the Schatten classes in [194]. Tt
was announced there (with a sketch lof proof) that the norm of $7 had the same differentiability as that of I? (1 < p < oo). It seems unclear how to extend this to the general case (of,at least, to the semifinite case). In a different direction, the papers [55,56] are devoted tothe problem of characterizing the symmetric spaces of measurable operators for which theabsolute-value mapping x — |x| is Lipschitz continuous. In the case of non-commutative
L!-spaces, Kosaki proves in [108] the following useful inequality: for any ¢ and ¥ in sucha space, we have
1/2
lel = wil, <v2(le+ ¥lile — wi)".
The passage from the Schatten classes to von Neumann algebras with semifinite traces,ie, from the discrete-to.the continuous case, can sometimes be quite substantial. See, forinstafice, Brown's extension of-Weyl’s classical inequalities: (3 |A, (MPV?
Ts(here A, (T') are the eigenvalues of T repeated according to multiplicity). Brown [26] had
to invent a new kind of spectral measure (now called Brown’s measure) to extend this,
together with Lidskii’s trace theorem, to the semifinite case.
.
co
The study of non-commutative L?-spaces, or more generally, of symmetric operator

spaces, goes mainly in two closely related directions: lift topological or geometrical properties from the commutative setting to the non-commutative one, and reduce problems in
the non-commutative case to those in the commutative one. We have already seen several
examples in both directions.
To discuss more illustrations, it is better to place ourselves in the context of symmet-

ric operator spaces. Let M be a semifinite von Neumann algebra equipped with an n.s.f
trace 7, and let E be an ri. function space on (0, 0co).-One naturally expects that propertiesof E(M, 7) should be reflected by those of E. Works already done in this direction are
too numerous to enumerate. Here we content ourselves with only three examples. The firstone concerns the (uniform) Kadets—Klee properties. The lifting of these properties from E
to E(M,7) has been extensively studied (cf., e.g., [6,37,38,41,50,57]). The second exam-ple is about the reduction of weakly compact subsets in E(M, 1) to those in E. This was

1478

G. Pisier and Q. Xu

achieved in [52] and [58] (see also the references there for previous works on this problem). Finally, the geometry of the unit ball of E(M, t) was studied in [5] and [40]. More
examples and references of this kind will be given in appropriate places in the subsequent
sections (see also [49]).
Despite the strong analogy between the commutative and non-commutative settings,
non-commutative L?-spaces behave, in some aspects, very differently from their commutative counterparts.
One of the most spectacular differences concerns unconditional bases or “unconditional
structures”. Already in [111], it was proved that S! cannot be embedded into any spacewith an unconditional basis, in sharp contrast with £1. But the big surprise came when
Gordon and Lewis [74] proved that the Schatten class SP fails to have any unconditional
basis when p # 2 in sharp contrast with £7 or L?. More generally they proved that SP?
fails “local unconditional structure” in their sense (abbreviated as GL-Lu.st.; see [94] forthe precise definition). This was the first example of a reflexive Banach space which was
not isomorphic to any complemented subspace of a Banach lattice. More precisely, letlu(X) denote the GL-Lu.st. constant of a Banach space X (lu(X) is equal to the norm offactorization through a Banach lattice of the identity of X**). The following theorem wasprovedby Gordon and Lewis [74] using a criterion (necessary but not sufficient) for the:
GL-Lu.st. of a space X: any 1-absolutely summing operator on X must factor through L!
(this is now called the GL-property). More precisely, they obtained the first part of thenext statement (the second part comes from [143], see also [180] and [146], 8.d for related
results):
THEOREM 4.1.

There is a constant C > Q such that for any 1 <Cnll/P=1/21

p< ooandanyn > 1

lu (SP) <nll/p-172,—

Consequently, SP does not have the GL-Lu.st. for p # 2. More generally, let X be any
Banach lattice of finite cotype (resp. of type > 1), then there is a constant C > 0 suchthat, if E is any n*-dimensional subspace (resp. subspace of a quotient) of X, we haved(S?, E) > cnlt/P-1/72,

oo

:

:

Combining Theorem 3.5 and Theorem 4.1, we immediately obtain

COROLLARY 4.2. A non-commutative LP (M), 1 < p < 00 and p # 2, has the GL-Lu.st.
iff M is isomorphic, as Banach space, to L*° (82, i) for some measure space (£2, jv).
Moreover, this happens iff LP (M) is isomorphic to a subspace of a commutative LP -space.
Note that M is isomorphic, as Banach space, to a commutative L*® iff M is the directsum (£o sense) of finitely many algebras of the form L® (uw; B(H)) (= L® (1) ® B(H))
with dim(H) < oo.
iw:
Another striking divergence from the classical case is provided by the uniform approximation property (UAP in short): by an extremely complicated construction, Szankowski
proved that B(£>) fails the approximation property (AP in short), and moreover ([189])

Non-commutative LP -spaces

1479

that SP (or S”') fails the UAP for p > 80. It remains a challenging open problem to prove
this for any p # 2.
Wewill describe another striking difference in Section 7, that is, a non-commutative

LP-space LP(M), 0 < p <1, is never an analytic UMD space except when M is isomorphic, as Banach space, to a commutative L°°-space.
Surprisingly, by [122], “stability” provides us with one more sharp contrast. Recall thata Banach space X is stable (in Krivine-Maurey’s sense) if for any bounded sequences{xm}m>1, {¥n}nx1 in X and any ultrafilters U, V on N

Jim Hm [ly + ynll = Hm Hm fxn + yall.
It is well known that any commutative LP-space (1 < p = 00)is stable (cf. [109]). This is
no longer true in the non-commutative setting. In fact, we have the following characterization of stable non-commutative L? wn
THEOREM 4.3. Let 1 < p < 00, p 5 2. Then LP (M) is stable iffM is of type’ 1.
The “if” part of Theorem 4.3 was independently proved by Arazy [8] and Raynaud [166].
The “only if” part is dueto Marcolino [122]. Marcolino’s proof is divided into two steps.
The first one (the proof of which is relatively\easy) is that L?(R), p # 2, is not stable(recalling that R is the hyperfinite II; factor). The second step is the above Theorem 3.5.

5. Uniform convexity (real and complex) and uniform smoothness
The fact that

ZP (M),

1 < p < oo, is uniforinty.convex and smooth immediately follows

from the following Clarkson type inequalities.

THEOREM 5.1. Let 1 < p, p' <oco with 1/p+1/p'= 1. Then

0 #1<p<2

|

1

co

y

NEL

[50 +ylp +llx—ylp )]uv< (IB +1y15) 77, x,y€ LPM);

|

|
a.

(5.1

(i) f2< p< oo,
1

»

HC

»

1/p

+ylp+lx— o15)|<(Ixlp

a

+ ys)

/

1

/

Px,

yeLP(M).

(5.2)

Inequalities (5.1) and (5.2), of course, have their origin in the classical Clarkson inequalities for commutative L?-spaces. In the non-commutative setting, some partial or particular

;

1480

G. Pisier and Q. Xu

cases of (5.1) and (5.2) were obtained in [47,128] (see also [49] for additional references).
(5.1) and (5.2), as stated above were proved in [78] and [106] (see also [191] and [64]).
PROOF OF THEOREM 5.1. The proof is almost obvious via the complex interpolation.
Indeed, (5.1) (resp. (5.2)) is trivially true for p = 1,2 (resp. p = 2, oo). Then the complex
interpolation yields (5.1) and (5.2). We also note that (5.1) and (5.2) are dual to eachother.
aLet §x (resp. px) denote the modulus of convexity (resp. smoothness) of a Banach
space X. Theorem 5.1 implies the following
COROLLARY 5.2. Let 1 < p < 00. Then LP (M) is uniformly convex and smooth; more
precisely, we have@) fori<p<2S1(e) 2 PorSoa,

O0<e<2,

and

pron)

1,
< > ,

t>0;

1, ,

t>0.

i

(ii) for2<p<oo
Spe

(8) =

1

—gP,

p2r

O<e<2,

and

prem

(@) <

—tP

p_

The reader can find some applications of the uniform convexity of L?(M), eg.
in [107,108].
Let us comment on the estimate for the modulus of convexity given by Corollary 5.2 (thesame comment, of course, applies to the modulus of smoothness as well). This estimate isbest possible only in the case of 2 < p < 00. We should also point out that in this casethe relevant constant 1/(p2”) is optimal (for it is already so in the commutative case;see [115], p. 63). Keeping in mind the well-known result on the modulus of convexity-ofcommutative L”-spaces, one would expect that the order of §7r(a1)(e) for 1 < p <2 beO(g?). This is indeed the case (cf. [193]). In fact, we have a more precise result as follows.

THEOREM 5.3. Ler 1 < p < oo. Then(i) fori<pg2(1x12 + (= DIY) 1/2
:

1

< [50

»

+1541

507) |

1/p,

Vx,yeLP(M);(5.3)(i) for2<p<oo,[30 +05 +1 51)
1

:
1/p

< (112 + (2 = DIyI2) 172.7%,»

va, y e LPM).

Moreover, the constant p-— 1 is optimal in both (5.3) and (5.4).

;
BREE)
cio

:

Non-commutative LP -spaces

1481

This theorem was proved in [14] for Schatten classes. As pointed out by the authors, thearguments there work for semifinite von Neumann algebras as well. Then the general casefollows by Theorem 3.6. We should emphasize that the optimality of the constant p — 1
in (5.3) and (5.4) has important applications to hypercontractivity. We will later illustrate
this by discussing the Fermionic hypercontractivity.
Note that if one does not care about the best constants, one can deduce Theorem 5.3 fromthe optimal order of 8.,(A1y(¢) and ppp aq) (2) obtained in [193] (at least, for Schattenclasses). Note also that (5.3) and (5.4) are equivalent by duality. We will include a very
simple proof of (5.4) for p = 2" (n € N), and so by interpolation for all 2 <p < oo withsome constant C, instead of p — 1.
Theorem 5.3 gives the optimal estimates fox § Len(®
2 <p < 00).
\
COROLLARY 5.4.

REMARK.

We have, forany 0 <e <2 ny

Spray (e) 2 Poe,1<p<2,

pLrmy(®) < Pe,

2< p<oo.

The constants (p — 1)/8

(I< p <2) and prea)

0

.

and

and (p—=T1)72
in the above estimates are optimal(see [115], p. 63 for the-commutative case).
Corollaries 5.2 and 5.4 yield the type and cotype of L? (M) for 1 < p < co.
COROLLARY

5.5. Let1 < p < oo.

Then

LP(M)

max (2, p):

is of typemin(2, p)

oe

eT
and

cotype-

The type and cotype of L?(M) were determined in [193] for Schatten classes, andin [63] for the general case. We will see later that 7 (M) is of cotype 2 forO0 < p < 1.
Now we turn to the application of the optimality of the constant p — 1 in (5.3) and (5.4)
to the Fermionic hypercontractivity. Before starting our discussion, we should point out,
however, that in the scalar case (i.e., in the case where M= C) Theorem 5.3 is exactly Nel-son’s celebrated hypercontractivity inequality for the two point space (cf. [20] and [130]).
This two point hypercontractivity inequality easily yields the optimal hypercontractivity
for the classical Ornstein—Uhlenbeck semigroup. Carlen and Lieb used Theorem 5.3 (inthe case of Schatten classes) to obtain the optimal Fermionic hypercontractivity, thus solving a problem left open since Gross’ pioneer works in the domain (cf. [75]).
Let R be the hyperfinite II; factor. We recall that R is generated by a sequence(én)n>1 of self-adjoint unitaries satisfying (CAR) (see Section 1). We also recall that{wa: ACN, A finite} is an orthonormal basis of L?(R). We define the number operator N by
Nwy = |Alwa (|]A| denoting the cardinality of A). N is an unbounded positive
self-adjoint operator on L?(R). It generates the Fermionic Omstein—Uhlenbeck semigroup
PsP,

=e Nt

> 0. One can show that P; is a contraction on LP(R)The optimal Fermionic hypercontractivity is contained in the following

forall 1 € p < oc.

~

:

1482

G. Pisier and Q. Xu

THEOREM

5.6. Let 1 < p < gq < oo. Then P; is a contraction from LP (R) to LY(R) iff

e 2 <(p—1)/(q— 1).

Let us briefly comment on the proof of Theorem 5.6. First, since linear combinationsof the w4’s are dense in L?(R), it suffices to prove Theorem 5.6 in the finite-dimensionalcase, that is, when P; is restricted to the L”-spaces based on the von Neumann algebra
generated by {€1,...,&,} (n € N). Second, by standard arguments as for the classicalOrnstein—~Uhlenbeck semigroup, one can reduce Theorem 5.6 to the special case where2 = p < q. Assuming these reductions, one can use (5.4) to prove Theorem 5.6 by induction on x (noting that the case n = 1 corresponds to Nelson's two point hypercontractivity).
We refer to [35] for the details.
REMARKS. (i) Theorem 5.6 implies, and in fact, is equivalent to the optimal Fermioniclogarithmic Sobolev inequality, see [76] and [35].
(ii) Biane [18] obtained the analogue of Theorem 5.6 for the free Ornstein—Uhlenbeck
semigroup. Note that this latter semigroup is also ultracontractive (cf. [23,24]).
We end the discussion on the uniform convexity and smoothness by providing a simple
proof for Theorem 5.3 (except for the best constant). We need only to consider (5.4). We are:going to show that for 2 < p < oq there is a constant Cp, depending only on p; such that[301 p +0519)p 1p|aE<(Ix12+ Collyl2)'

1/2

Co
.

2, va, y e LPM).

(54)

To that end, by Theorem 3.6, we can assume that M is semifinite and equipped witha faithful normal semifinite trace t. The key step in the proof of (5.4,) is the implication “(5.4,) = (5.42). Let us show this. Assume (5.4,). Let x,y € L?*P(M), and set—"

a=x*x+y*y,b=x*y + y*x. Thena, b € L?(M) andapy _ 1pI?
51 (lx +ylh2p + lx = yliah)= 5 (la +bllp+ lla — pI?blip)< (lal? + Cpllp12)"? 2 (by (5.4):
co<[(1%13, + Iy13,)* 2 +4C, x3, Iy13,17% 2< (1x13, + @Cp + DIyI3,)

whence (5.42) with C2, < 2C), + 1. Therefore, starting with the trivial case p = 2 (notingthat Cp = 1), and by iteration, we get Co» < 2" — 1 (in fact, Con == 2" — 1). Thus for these
special values of p we obtain the best constant in (5.4). Then for any other value of p, say,2" < p <2" by complex interpolation, we deduce (5.4) withRR

cp< (2-1)

where 1/p = (1—8)/2" +6/2*+1.

— 1),

IE

ati

|

Oo

|

Non-commutative LP -spaces1483

Now we pass on to the uniform PL-convexity and Hardy convexity of non-commutative
LP-spaces. This time, we admit quasi-normed spaces (so p < 1 is allowed).
Let X be a (complex) quasi-Banach space. Let T be the unit circle equipped with nor-malized Lebesgue measure. For 0 < p < co we denote by L?(T, X) the usual L?-space ofBochner measurable functions with values in X. Note that LP (T, L? (M)) is just the non-commutative L?-space based on the von Neumann algebra tensor product L®(T) ® M.
Let P(X) denote the family of all complex polynomials with coefficients in X:

»
P00

={

;

Sond

mex

0<kn

nen

.

:

k=0

DEFINITION.

Let X be a quasi-Banach space. Let 0 < p <oe& > 0. We define

Hx(e) =inf{lix + 2ylipimpn — 1: Ix =1, yl 25, x, yeX}and\hs @) =mf{ll flier) — 1: [FO] =1, | f — FO) I Zeer. =e,

fePX}.

:|

X is said to be uniformly aalle > 0. Hx (g) (resp. nk (e)) is called

H?-convex)if Hx (g) > 0 (resp. nb (e) > 0) forthe modulus of PL-convexity (resp. H?-convexity)

of X.

ON

The uniform PL-convexity was introduced and studied Lin [45]. It was shown there thatin the definition of Hx(s) above, if the Ll-norm is replaced by an L?-norm, then the
resulting modulus is equivalent to Hy (¢). The uniform H?-convexity was explicitly introduced in [199]; however, it is already implicit in [80]. It was proved in [202] that if X is
uniformly H?-convex for one p € (0, 00), then so is it for all p € (0, 00). Thus we saythat X is uniformly H-convex if it is uniformly H?-convex for some p. The uniform PL-"
convexity (resp. H-convexity) is closely related to inequalities satisfied by analytic (resp.
Hardy) martingales with values in X. The Enflo~Pisier renorming theorem about the uniform (real) convexity admits analogues for these uniform complex convexities. We refer
to [45,199,201,202] and [149] for more information.
REMARKS. (i) For any given 0 < p < oo there is a constant «,, > 0 such that for all quasiBanach spaces X

Hx (e) 2 aphl(ape),0<e<l.

Consequently, the uniform H-convexity implies the uniform PL-convexity.
(ii) If a Banach space X is uniformly convex, it is uniformly H-convex.

|

1484

G. Pisier and Q. Xu

THEOREM 5.7. Assume 0 < p,q < 00. Let M

haan(© >a’,’

be a von Neumann algebra. Then

O0<e<l,where r = max(2, p,q) and a > 0 is a constant depending only on p,q.
REMARKS. (i) Inthe case g > 1, Theorem 5.7 easily follows from Corollaries 5.2 and 5.4.
Thus the non-trivial part of Theorem 5.7 lies in the case g < 1.
(ii) Theorem 5.7 implies, of course, that the same estimate holds for the modulus ofPL-convexity.
:(iii) In the case of ¢ = 1, Theorem 5.7 is contained in [80]. In fact, it is this result which
motivated the introduction of the umform H-convexity.
Theorem 5.7, as stated above, was proved in [201]. The ingredient of the proof is
the Riesz type factorization for Hardy spaces of analytic functions with values in Tioncommutative L?-spaces. In Section 8 below we will discuss such a factorization in a moregeneral context.
The following corollary completes Corollary 5.5. Thus the non-commutative L”-spaces
have the same type and cotype as the commutative L?-spaces.
COROLLARY 5.8.

L?(M)is of cotype 2 for any 0 < p < 1 and any von Neumann alge-

bra M.
This corollary was proved in [193] for
p=1 and in [201] for 0 < p < 1.
ol
We recall that a quasi-Banach space X has the analytic Radon-Nikodym property (abbreviated as analytic RNP) if any bounded analytic function ¥:D — X has a.e. radial
limits in X, where D denotes the unit disc (cf. [30,59], and also [32] for additional references). It is known that the uniform H-convexity implies the analytic RNP. Thus we get”

the
COROLLARY 5.9.
algebra M.
The

LP(M) has the analytic RNP for any 0 < p < 1 and any von Neumann

results discussed

in this section

have

all been

extended

to symmetric

operator

spaces. We refer to [70,195] for the cotype, uniform convexity, PL-convexity and smoothness in the unitary ideals, and in the general case, to [200] for the uniform convexity
and smoothness, to [203,204] for the uniform H-convexity, RNP and analytic RNP (see.
also [129]). Finally, we mention that [39] contains related results, especially thqse on thelocal uniform convexity for symmetric operator spaces.
)ce

6. Non-commutative Khintchine inequalities:

This section is devoted to the non-commutative Khintchine inequalities and the closely
related Grothendieck-type factorization theorems. Although all results in this section holdNon-commutative LP -spaces

1485

for the general non-commutative LP-spaces, we will restrict ourselves to the semifinite
ones, i.e., those constructed from an n.s.f. trace. Letters 4, Bp, ..., will denote positive

constants depending only on p.
Let (en)s>1 be a Rademacher (or Bernoulli) sequence, i.e., a sequence of independent random variables on some probability space (£2, F, P) such that P(g, = 1) =P(e, =—1)=1/2 forall n > 1. We first recall the classical Khintchine inequalities. Let
0 < p < oo. Then for all finite sequences (a) of complex numbers43!) > aren~

< > anonn>lLP(2,P)n=l< Bp | > anen
L2(£2,P)

(6.1)

n>1

LP(2,P)(Note that obviously | Dons 1anénllL2g, py = Qn lan |?)1/2.)
These inequalities remain
valid (suitably modified) when the eoefficients a,,’s are vectors froma
Banach space X. In
that case they are due to Kahane, and are usually called “Khintchine—Kahane inequalities”:
for all finite sequences (a,) in X

8 DICER BE D3
n>1

LP(2,P;X)

n>1< B, | >n>1

L%(2,P;X)

angen |:

(6.2)

LP(2,P;X)

In particular, if X is a commutative L?-space, say X = L? over (0, 1), (6.2) implies thatfor all finite sequences (a,) in LP (0, 1)

1/2Ser) 1, «| Zee]nzl

Lrwlge)],n>toe

L2(2,P;LP)

1/2

nzl

Le

PRES

It is (6.3) that we will extend to the non-commutative setting.

Now let M be a semifinite von Neumann algebra equipped with an n.s.f. trace 7. Let
a = (a) be a finite sequence in L?(M)(recalling that by our convention, L”(M)LP (M, 1)). Define

toms l(Z) |
:

N12

nz0

14

=| (5),

=

1/2

nz=0

p

This gives two norms (or quasi-norms if p < 1) on the family of all finite sequences in
LP(M). The corresponding completions (relative to the w*-topology for p = 00) are denoted by L?(M; 12) and LP (M; 1%), respectively.
The reader is referred to. [156] for a

discussion of these norms.

:

:

.

:

1486

G. Pisier and Q. Xu

Now we can state the non-commutative Khintchine inequalities.
THEOREM 6.1. Let 1 < p < oo, and let M be a semifinite von Neumann algebra. Let
a = (an)nz0 be a finite sequence in LP (M).
1) If2 < p < 00, there is a constant B, (depending only on p) such that
llall

2

12)

€

ErOnLICMit) = py

Endy,

Lo(2,P;LP(MY))< B, lall Lo (pts2ynee

msi):

(6.4)

(1) If 1 < p <2, there is an absolute constant A > 0 (independent of p and a) suchthat
Allall2

2

LPMGIE+LP (MIR)

S |

Sal

2

Ena

mr

|

LP(82,P;LP (MD)Lot 2yr Looms)

©5).

/

For the convenience of the reader we recall the norms in LP (M; 12) NLP(M; 12) andLP(M; 12) + LP(M; 13):

hall Lrrtznze vez) = max{lali ou), lal ou, )andNallLoatszy+romi2)= inf {151 Lp rt:i2) + elertz)},

where the infimum runs over all decompositions a = b + ¢ with b € LP? WM; 12) and c €LP (M; 13).

This result was first proved in [117] for 1 < p < oo in the case of the Schatten classes.
The general statement as above (including p = 1) is contained in [121]. Modulo the classical fact that in all preceding inequalities the sequence (¢,,) can be replaced by a lacunary
sequence, say, by @*H n3>1 on the unit circle T, the main ingredient of the proof in [121] isa Riesz type factorization theorem (see Theorem 8.3 below).
REMARKS. (i) Like in the classical Khintchine inequalities (6.1), the constant B, in (6.4)
is of order ,/p (the best possible) as p — oo (cf. [154, p. 106]).
.
(ii) We have already mentioned that in Theorem 6.1, the sequence (&,) can be replaced
by a lacunary sequence. It is also classical that (g,) can be replaced by a sequence ofindependent standard Gaussian variables.

(iii) More generally, Theorem 6.1 holds when (g,) is replaced by certain sequences in a
non-commutative L”-space L?(N) and ya, is replaced by ¢, ® a, in LP? WN ® M), forinstance, this holds for the generators of a free group, for a free semi-circular system. (inyaNon-commutative LP -spaces

1487

Voiculescu’s sense [196]) and for a sequence of CAR operators (as in Section 1). Note thatin the free cases, (6.4) even holds for p = oo! The reader is referred to [81,154] for more
information, and also to [27,29] for some related results and for the best constants in theseinequalities.
(iv) Theorem 6.1 also holds for non-commutative L”-spaces associated with a general
von Neumann algebra (cf. [100,101]). Note that [100,101] contains more inequalities re- .
lated to (6.4) and (6.5).
COROLLARY 6.2. Let (;;) be an independent collection (indexed by N x N) of mean zero
+1-valued random variables on (£2, F, P). For any 2 < p < 00, there is a constant Cp,
such that for any finitely supported function x : N? — C, we havexl, < > ex, hes)ij

<Cplixllp,

WLE($2,P,SP)

(6.6)

:

where
LN

2

xl» —max{ (3 ( Shei) )ip/2\1/p

)

. a2

(xz

j

|x, 7) )

j

p/2\

)

1/p

I

i
6.7)

A fortiori this implies| > eijx(i, jeij | LP(2. PP) < Cp om> ed, Hx(, Deij | oF

(6.8)

PROOF. Take LP(M) = SP. Let a;; = x(i, /)ej. Then (3; afa;j)'/* = 3; A jej; andCaai)'/? = 3 pies where hj = (3; |x, HIHY? and pi = (2; 1x, HI»). Thus
(6.6) is a special case of (6.4).

(|

REMARK 6.3. The preceding result remains valid with the same proof when 1 < p <2
provided one changes the definition of [|x], to the following one (dual to the other):lll = inf{ ly lle, ep) + 12], ep}
where the infimum runs over all possible decompositions of the form x = y +z.
REMARK. [101] contains more inequalities of type (6.6). Here we just mention one ofthem, which is an extension of (6.6). Let (f;;) be an independent collection of mean zerorandom variables in LP (£2, F, P) (2 < p < o0). Then

[52 ei]Pu pry~~

1 Pme] (002)

1/p

112(= (z 18)

ij)

pI2\1/p(Z(Zw))iwhere the equivalence constants depend only on p.

i
Do

7)

p/2\1/p)

1488

G. Pisier and Q. Xu

Inthe case of 0 < p < 1, itis easy to check that the second inequality of (6.5) still holds.
However, this is not clear for the first one.
:
PROBLEM 6.4. Does the first inequality of (6.5) hold for 0 < p < 1 (with some constantdepending on p)? Does Remark 6.3 extend to p < 1?
Likein thecommutative

setting,thenon-commutative

Khintchine

inequalities

areclosely related to non-commutative Grothendieck type factorization theorems. Indeed, it
was shown in [121] that (6.5) in the case of p = 1 is equivalent to the non-commutative
little Grothendieck theorem. To go further, we need one more definition.
DEFINITION.

Let

1 < p<

00,

0 <g

<r

< 00.

Let

Y

be

a Banachspace,

andletu:LP(M) — Y be an operator (M being a semifinite von Neumann algebra). u is said tobe (r, g)-concave if there is a constant C such that for all finite sequences (a,) in LP (M)(Duan)t/r<c] (rant)

1/q|

where |a|; = ((@*a+aa*)/2) 172 denotes the symmetric modulus of an operatora. If g =r,
u is simply said to be g-concave.

p

In the case of p = co (then M can be any C*-algebra), the above notion reduces to thatof (r, g)-C*-summing operators introduced in [144] and [147]. The following is an easy |consequence of the Hahn-Banach theorem (cf. [144] for a proof).

:

PROPOSITION 6.5. LetM be a semifinite von Neumann algebra. Let 1 < gq < p < 00 ands = p/q. Then for any operator u: LP (M) — Y the following assertions are equivalent -(1) u is g-concave;(ii) there are a constant C and f € (LS(M))*, f = 0, such thatluall < C(f (lal),Va e LPM).

|}

The following Grothendieck-type factorization theorem (when Y is a Hilbert space) is
equivalent to (6.5) with p’ in place of p.
THEOREM 6.6. Let2 < p < 00, and let Y be a Banach space of cotype 2. Then any operator u: LP (M) — Y is 2-concave, equivalently (via Proposition 6.5), there are a constant
C and f € (LP/?(M))* with f > 0 such thatCo

20\1/2

ual < Clull(f(lal2))’",»YaeLP(M).

Moreover, C can be chosen to depend. only on the cotype 2 constant of Y.
REMARK.

}vd

The basic case p = oo (=non-commutative Grothendieck theorem), is provedin [147] (see also [144,146]). In this case, M can actually be any C*-algebra. In the case of

Non-commutative LP -spaces1489

p < oo, Theorem 6.6 is essentially the main result in [118]. More generally, [118] proves
this for operators u:
E(M, 7) — H, where H is a Hilbert space and E is a 2-convex
ri. space with an additional mild condition. This, together with [121], implies that Theorem 6.1 can be extended to some symmetric operator spaces. The main difficulty in [118]is to obtain Theorem 6.6 with a constant C independent of p, or equivalently which remains bounded when p — co. If we ignore this important point, it is very easy to deduce
Theorem 6.6 from (6.4), as follows.
::
PROOF OF THEOREM 6.6 FOR p < 00 WITH C = C,. Since LP (M) is of type 2, by
Kwapieri’s theorem (cf. [146], Theorem 3.2), u factors through a Hilbert space. Thus we
may assume Y itself is a Hilbert space. Let (ay) be a finite sequence in L? (M). Then
a

(x

[tan]

)

=

2_u@n)en earn

< el] nen] | oi eocnny<G|(Zimk)2

1/2

7] byte)Therefore, u is 2-concave.

Te

Unfortunately, the preceding proof does not work for p = oo. The main difficulty inthis case is to show that an operator #4 from M into a space of cotype 2 factors througha Hilbert space. This was done in [147]. The proof given there relies on another result of
independent interest, that we state as follows.

.

THEOREM 6.7. Let 1 <q < 00. Let u: A — Y be an operator from a C*-algebra A intoa Banach space Y. Then the following assertions are equivalent@@) u is (g, 1)-C*-summing;::(ii) there are a constant C and a state f on A such thatiual < Cllull(£ (als)lal! =4, Vae 4;.

(iii) for any 1 <r < q there are a constant C and a state f on A such thatual < Clull(£(al})) 1 lal! _ =,Gv)

Vae a;

u is (gq, r)-C*-summing
for any 1 <r <q.

Thus Theorem 6.7 gives a characterization of (gq, r)-C*-summing operators defined on
a C*-algebra. (ii) and (iii) above can be reformulated as a Pietsch-type factorization of u
through a non-commutative Lorentz space L?'!, constructed from the state f via the realinterpolation in the spirit of Kosaki’s construction presented in Section 3. The resulting
spaces, denoted by L?-1(f), possess properties similar to the usual Lorentz spaces. The
reader is referred to [147] for more information.

|

1490

G. Pisier and Q. Xu.

REMARK. There does not seem to be a known characterization similar to that in Theorem 6.7 for (gq, r)-concave operators defined on LP (M) (p < 00).
Let us close this section by an application of Theorem 6.7.
THEOREM 6.8. Let M be any von Neumann algebra and X C M, a reflexive subspace.
Then there are a normal state f of M and p > 1 such that X embeds isomorphically into
LPP(f), where LPP (f) is the non-commutative LP -space referred to above.
This theorem, proved in [147], is a non-commutative version of a classical theorem due
to Rosenthal in the commutative setting. Its proof uses Theorem 6.7 and a previous resultin [92] that any reflexive subspace of M, is superreflexive. Note that the real interpolation
space LP-P(f) can be replaced by the corresponding complex interpolation space.
REMARK.

Let A be a C*-algebra, and let T : A — £; be absolutely summing (in the usual

sense). If A is commutative, it is well known that T factors as T = TT», where Th: A — V2)
is bounded and 7; € $2. In [161] it is shown that for a general C*-algebra A, one can geta factorization T = T1T», where T>: A — £; is bounded and 7}: £, — £5 belongs to the

Schatten class S4 (the exponent 4 is optimal). A fortiori T is compact. In particular, there
is no embedding of £; into a non-commutative L;-space with absolutely summing adjoint.
See [146, p. 68] for background on embeddings of this kind.
u
7. Non-commutative martingale inequalities|:

This section deals with non-commutative martingale inequalities. The reader is referredto [71] for the classical (= commutative) martingale inequalities. In what follows, M will
be a von Neumann algebra equipped with a normal faithful finite normalized trace ©. We ya
begin with some necessary definitions. Let A C M be a von Neumann subalgebra. The
non-commutative L?-space associated with (NV, t|xr) is naturally identified with a subspace of L”(M). There is a unique normal faithful conditional expectation £ : M — A
preserving the trace 7, i.e., T(€(x)) = v(x) for all x € M. For any 1 < p < 00, £ is extended to a contractive projection from L” (M) onto L? (A), still denoted by £.
Now let (Mp),0 be an increasing sequence of von Neumann subalgebras of M suchthat the union of all the M,,’s is w*-dense in M.

Let &£, be the conditional expectationfrom M onto M,,. Then as usual, we define a non-commutative martingale (with respect
to (Mn)n30) as a sequence x = (X,),>0 in LY(M) such thatEn xn)

=%0,

Yn20.

.

If additionally all x,’s are in L? (M), x is called an L? -martingale. Then we set
xl p = supnz0[xx] p-EAN

If Ix||p < 00, x is called 2 bounded L?-martingale. The difference sequence of x is defined
as dx = (dxp)n30 With dxg = xo and dx, = x, — x, _1 foralln > 1.
:.

Non-commutative LP -spaces

REMARK.

Let xo € LP(M).

1491

Set x, = E,(x00) for all n > 0. Then x = (x) is a bounded

LP-martingale and ||x||, = [|xco|l p; moreover, x, converges to Xoo in LP (M) (relative
to the w*-topology in the case p = 00). Conversely, if 1 < p < oo, every bounded
LP-martingale converges in L? (M), and so is given by some xo € L? (M) as previously.
Thus one can identify the space of all bounded LP-martingales with" L? (M) itself in thecase | < p <0.
The main result of [156] can be stated as follows. Recall that Ap, Bp, ..., denote constants depending only on p.
THEOREM 7.1. Let M and (M,)n>0 be as above. Let 1 < p < 00, and let x = (Xp)n0
be a finite LP -martingale with respect to (My)0. Then

ALS,(0) < lIxllp < BpSy (x),where for¢AY

2 < p < 00,Spx) = dx l Lo pui2ynLe

iid)

andforl <p <2,

Sp (0) =inf{dy lopez) + 1420 Loan
the infimum being taken over all decompositions

x =y +2 with LP -martingales y andz.

This is the non-commutative Burkholder-Gundy inequalities. Note that in the commutative case, Sp (x) is the LP-norm of the usual square function of x (so that the above
difference between the cases 2 < p < oo and 1 < p < 2 disappears). The proof of Theorem 7.1 in [156] is rather tortuous, due to the fact that the usual techniques from classicalmartingale theory, such as maximal functions, stopping times, etc., are no longer available
in the non-commutative setting. See [156] and [19] for applications to non-commutative
stochastic integrals. For Clifford martingales, some particular cases of Theorem 7.1 also
appear in [34].
REMARK 7.2.

The second inequality in (7.1) holds for p = 1 too. This follows from theduality between +! and BMO, proved in [156].
Like in the commutative case, Theorem 7.1 implies the unconditionality of noncommutative martingale differences. Let us record this explicitly as follows.
COROLLARY 7.3.

With the same assumptions as in Theorem 1.1, we have

] > endxy | <C, | > dx, | ,
nz=0

P

n=0

r

Ven, ==%1.

(7.2)

1492

G. Pisier and Q. Xu}

Some rather particular cases of (7.2) also appear in [65,66]. Note that in the case of 2 <p < 00, (7.2) is equivalent to (7.1), modulo the non-commutative Khintchine inequalities.
However, in the case of 1 < p < 2, to prove that (7.2) implies (7.1), one needs a noncommutative version of a classical inequality due to Stein. We refer to [156] for moredetails.
REMARK. If p is an even integer, the second inequality of (7.1) was extended in [155] tosequences more general than martingale difference sequences (the so-called p-orthogonalsequences); moreover, for these values of p, the method of [155] yields that the order ofthe constant B,, in (7.1) is O(p) (for even integers p), which is optimal as p — oo.
For the convenience of the reader, we recall the optimal order of the constants A p andB, in the commutative case (cf., €.g., [31]): B), is bounded as p — 1 and O(p) as p — oo;
Apis O((p — D1 as p— 1 and O(p'/?) as p — co. The constants Ap, and Bp in (7.1)
obtained in [156] are not satisfactory at all (they are of exponential type as p —> 0). Thus
finding the optimal order of A, and B, in Theorem 7.1 seemed a very interesting question.
Very recently, major progress on this was achieved by Randrianantoanina [165], as follows.
THEOREM 7.4 ([165]). There is a constant C such that for any finite non-commutative
martingale x in L' (M) and any sequence (g,) of signs[on], el Se,
n320

1,00

n320

on

1

:

By interpolation, this implies the optimal order of the constant Cp in (7.2), namely,
Cp =0(p)as p —> oo. This, in turn, combined with Theorem 6.1, yields better estimatesfor Ay, Bp in (7.1), namely A, is O((p — 1)~2) when p — 1 and both Ap and B, areO(p) when p — oo (which for Bj, is optimal). It was also shown in [102] that O(p) is
the optimal order of A, as p — co. Note that this order is the square of what it is in thecommutative case. On the other hand, it was proved in [100] that B,, remains bounded asp—> 1
We will now discuss two other inequalities: the Burkholder and Doob inequalities."
THEOREM 7.5.

With the same assumptions as in Theorem 7.1, we haveCAs @) < xl < Bpsp (a),
where

|

for 2 < p <.00,:

sp(x) = max] ( > Ia)

1/p(gear)

50)

n>=0

oo

, | ( > En (4x)

n=0

ER

3)

n>0

1/2

Co

hp

1/2

-

:
p

Ce

’

Non-commutative LP -spaces

1493

and for 1 <p <2,5px) = nt] (xz dw, 1?)

1/p+ (xz Eni (aus?)

n=0

|

n=0

1/2
4

<|(Zemtents) | }
1/2

nz0

:

P

‘

:

|

:

where the infimum runs over all decompositions x = w + u + v with LP -martingales w, u
and v.
This theorem comes from [100]. It is the non-commutative analogue of the classical

Burkholder inequality. Note that in the commutative case (3 &,—1(|dx,[?))'/2 is the conditioned square function of x. Like in the commutative case, Theorem 7.5 implies a noncommutative analogue of Rosenthal’s inequality concerning independent mean zero random variables; see [100,101] for more details and some applications.
THEOREM 7.6 ([97]). Let M and (M,,) be as in Theorem 7.1. Let 1 < p < 00. Let (ay)be a finite sequence of positive elements in LP (M).-Then

I
n>=0

P

n=0

FE —

r

:

Note that in the commutative case, (7.4) is the dual reformulation of Doob’s classicalmaximal inequality. Although it is clearly impossible to define the maximal function of anon-commutative martingale as in the commutative setting, Junge found in [97] a substitute, consistent with [154], which enables him to formulate a non-commutative analogueof Doob’s inequality itself, which is dual to (7.4). Note that the latter result immediately
implies the almost everywhere convergence of bounded non-commutative martingales in
LP(M) for all p > 1. Results of this kind on the almost everywhere convergence of noncommutative martingales go back to Cuculescu [43]. The reader is referred to [44] and[90,91] for more information.
REMARKS. (i) Like the constants in (7.1) the constants in (7.3) and (7.4) obtained in [100,97] are not satisfactory at all. In fact, they depend on those in (7.1) since the proofs of (7.3)
and (7.4) in [97] and [100] use (7.1). The more recent results of [165] imply better estimates
for these constants.
(ii) It was proved in [102] that the optimal order of the constant C), in (7.4) is O( p?) asp —> oo. This is in strong contrast with the commutative case for, in the commutative case,the optimal order of the corresponding constant is O(p) as p — oo. The same phenomenon
occurs for the optimal order of the best constant in the non-commutative Stein inequality
proved in [156], namely, this optimal order is O(p) as p — oo; again it is the square ofwhat it is in the commutative case. We refer to [102] for more information.
~ (iii) All the preceding results hold in the non-tracial case as well (cf. [100,101,97]).

1494

G. Pisier and Q. Xu

In the rest of this section, we briefly discuss the UMD property and the analytic UMD
property of non-commutative LP-spaces, a subject closely related to inequality (7.2). Applying Corollary 7.3 to commutative martingales with values in LP (M), 1 < p < 00, we

get the unconditionality of commutative martingale differences with values in LP” (M),that is, L? (M) is a UMD space in Burkholder’s sense (cf. [32] for information on UMD
spaces). This is a well-known fact, proved in [21] and [16]. In fact, these authors proved thatthe Hilbert transform extends to a bounded map on LP (T; LP (M)) for any 1 < p < 00;
but this property (called “HT” in short) is equivalent to UMD. We also refer to the next
section for discussions on Hilbert type transforms. Together with Theorem 3.6 we obtain
the
COROLLARY

7.7.

LP(M)

algebra M.

is a UMD space for any 1 < p < oo and any von Neumann

We mention an open problem circulated in the non-commutative world for almost two
decades on the UMD property for symmetric operator spaces.
Co
PROBLEM 7.8. Let

M be a semifinite von Neumann algebra equipped with an n.s.f. trace

7, and let E be a UMD

ri. space on (0, 00). Is E(M,7) a UMD

space?

We now turn to the analytic UMD property. Let TN be the infinite torus equipped withthe product measure, denoted by dm oo. Let £2, be the o-field generated by the coordinates(20, ..-, 2s), n 2 0. Let X be a quasi-Banach space. By a Hardy martingale in L? (TY; X)(0 < p < 00), we mean any sequence f = (f,) satisfying the following: for any n > 0,
fo € LP(TN, 2,: X) and Jn is analytic in the last variable z,, i.e., f, admits an expansion
as follows
~
Ja@0s

os

201,20)

=)

Oni

(Zo,

,

Zn-1)2E,

-

c k=l

:

where @, 1 € LP(TN, £2,_1; X) forn 2 0,k > 0. If in addition, on=0forallk >2, f iscalled an analytic martingale. Note that if X is a Banach space and 1 < Pp < 00, any Hardy
martingale in LP (TN; X) is a martingale in the usual sense.

DEFINITION. X is called an analytic UMD space if for some 0 < p < 00 (or equivaIently for all 0 < p < 00) there is a constant C such that all finite Hardy martingales +f in

LP (TN; X) satisfy> endfy’nz0< | > dfs
2

Tu>0

,
p

-

)

Vep,=+1.
}

This notion was introduced in [69]. The apparent weakening obtained by requiring theabove inequality be verified only for analytic martingales, is actually an equivalent definition of analytic UMD spaces (cf. [69]). Typical examples of Banach spaces which
are analytic UMD but not UMD are commutative L!-spaces. In fact, all commutative-Non-commutative LP -spaces

LP-spaces, 0 < p < 1, are analytic UMD

1495

spaces. We refer to [69] for more information(see also [32]).
However, this no longer holds in the non-commutative setting:

PROPOSITION 7.9. Let M be a von Neumann algebra and 0 < p < 1. Then LP (M) is ananalytic UMD space iff M is isomorphic, as Banach space, to a commutative L™-space.
It was proved in [80] that the trace class 1 is not an analytic UMD space. The ingredientof the proof there is the unboundedness of the triangular projection on § 1 (cf. [111]). (Thisprojection is in fact a non-commutative Riesz projection in the context of the next section.)
The same idea also shows that S? is not an analytic UMD space for 0 < p < 1. Noting thatthe analytic UMD property is “local”, we then deduce the general case from Theorem 3.5.

8. Non-commutative Hardy spaces
A classical theorem of Szegd says that if w is a positive function on the unit circle T suchthat log w € L!(T), there is an outer function ¢ such that |¢| = w a.e. on T. A lot of effort
has been made to extend this theorem to operator valued functions, not only for its intrinsic
interest, but also because it is the gateway to many useful applications (cf., e.g., [85,86,
46,179,198]). This problem makes sense in the broader context of subdiagonal algebras,
introduced by Arveson in the 60’s in order to unify several frequently used non-selfadjoint
algebras such as triangular matrices and bounded analytic operator valued functions. In thissection we will present the extension to this general context of someelassical results foranalytic functions in the unit disc, including Szeg6’s theorem, boundedness of the Hilberttransform and the Riesz factorization theorem.
Throughout this section, unless explicitly indicated otherwise, M will denote a finite
von Neumann algebra equipped with a normal fait
finite normalized trace 7. Let D
be a von Neumann subalgebra of M. Let £ be the (unique) normal faithful conditional
expectation of M with respect to D which leaves 7 invariant.
DEFINITION. A w*-closed subalgebra H* (M) of M
bra of M with respect to £ (or to Dy if@) {x +y* x,y€ H®(M)} is w*-densein M;

is called a finite subdiagonal alge::(1) E(xy) =EX)E®D), Yx,y € HOM);(iii) {x: x,x* € H®(M)} =D.
D is then called the diagonal of H®°(M).
This notion can be generalized further (see [12]). However, the theory we will give below
is, on one hand, satisfactory only for finite subdiagonal algebras as above, and on the other,interesting enough to cover many important cases.

REMARKS.

(i) IfH®(M) is a finite subdiagonal algebra of M, it is automatically max-

imal in the sense that it is contained in no proper subdiagonal algebra with respect to £other than itself (see [61]).

1496

G. Pisier and Q. Xu+ (ii) Consequently, H*°(M) admits the following useful characterization (cf. [12])H®(M)= {x e M: t(xy) =0, Vy € HP (M)},
where

HFM) = {x € HP (M): E(x) =0}.
Here are some examples (see [12] for more).
(1) Triangular matrices. Let M, be the full algebra of all complex n x n matrices
equipped with the normalized trace. Let 7, be the algebra of all upper triangular matrices in Mj. Then 7, is a finite subdiagonal algebra of M,,. In this case, the theory we will
give below is partly contained in [73].
(i1) Nest algebras. Let P be a totally ordered family of projections in M containing 0
and 1. LetooiN(P)={x € M: xe =exe, Yee P}.

’

Then NV (P) is a finite subdiagonal algebra of AM. The above example on triangular matrices is a special case of nest algebras.
(iii) Analytic operator valued functions. Let (M, 1) be a finite von Neumann algebra.’
Let (L°°(T), dm) ® (M, 7) be the von Neumann algebra tensor product (recalling that T
is the unit circle equipped with normalized Lebesgue measure dm). Let H™ (T, M) be thesubalgebra of (L*®(T), dm) ® (M, 1) consisting of all functions f such that

p=

[trop

dm(z) =0,

VxeL'(M), VneZ, n<0.

Then H*(T, M) is a finite subdiagonal algebra of (L®(T), dm) ® (M, t). This is thealgebra of “analytic” functions with values in M. More precisely, each element f in
H%(T, M) can be extended, using Poisson integrals, to an AM-valued function, analytic
and bounded in the unit disc admitting f as its (radial or non-tangential) weak-* boundary values. The particularly interesting case H°(T, M,,) or
H*®(T, B(l2)) was extensively
studied (cf., e.g., [85,461). Note that B(l2) does not fit into our setting; however, for almost

all problems we are concerned with, it can be recovered from M,, by approximation.
In the remainder of this section, unless specified otherwise, H%(M) will denote a finite subdiagonal algebra of AM with diagonal D. For 0 < p < co the corresponding Hardy
space HF (M) is defined as the closure of
H*°(AM) in LP (M). Many results on the clas-

sical Hardy spaces in the unit disc have been extended to the present setting. We refer,for instance, to [13,93,127,123,124,126,159,173,175] and [177] for more information-andreferences. We now give some of these extensions. The first one is the Szegd type theorem.

THEOREM 8.1. Suppose w € M and w=! € L2(M). Then there are a unitary u € M and© € H®(M)

with ole

HH(M) such thatw = up.

::inNon-commutative LP-spaces

1497

This theorem, proved by Saito [175], improves a previous factorization theorem

due

to Arveson [12], in which both w and w™! are supposed to belong to M. Saito’s proof

essentially follows the same fashion set out by Arveson, although some extra technicaldifficulties appear.
REMARKS. (i) The above theorem can be still improved as follows: let 0 <p, g < 00 andw € LP(M) with w™! € L9(M). Then there are a unitary u € M and ¢ € H? (M) with¢~! € H1(M) such that w = ug.
(ii) In the classical case of analytic functions in the unit disc, for a positive function
w on T, the condition logw € L!(T) is necessary and sufficient for the existence of afactorization w = ue, with # € L°°(T)unimodular and ¢ an outer function. It is an open

problem to extend this to the non-commutative setting. Some partial results can be foundin [46,86] and [198].
The following is an immediate consequence of Theorem 8.1 (and also of the remark (i)above).

COROLLARY

8.2. Let w € LY(M)such that w > 0 and w=" € LP(M) for some

0 < p < 00. Then there is ¢ € H*(M) such that w = ¢*¢.
By a rather standard argument, one can deduce from Theorem 8.1 the following Riesz
factorization theorem, which was proved in [124] (see also [177] for the case where p =q=2).

THEOREM 8.3. Let 1 < p,q, r<oo with 1/r =1/p + 1/q. Then any x € H" (M) can

be factored as x = yz with y € HP? (M) and z € HY (M); moreover,:Ix ll, =inf{
yl, lzllg: x =yz,y € HP(M), z€ HI(M)}.

:

REMARKS. (i) It seems unclear whether the infimum above is attained.
(ii) With the notations in Theorem 8.3, one has the following more ptecise statement:
for any ¢ > 0 there are y € HP (M) and z € H7(M) such that x = yz dnd.

we < (e@ +e)",pl) < (ux) +e)",

In particular, if x € H®(M), then y, z € H*(M)I¥llpllzllg= xl; +o(1)ase —0.

Ve >0.

|

and

BRE

This allows to partially extend Theorem 8.3 to the case of indices less than 1 (at least, forelements x €H®(M) CH" (M)). However, it is unknown whether Theorem 8.3, in its
full generality, still holds for indices less than 1.
~
The reader can find applications of Theorem 8.3 to Hankel operators in [179,123], toinvariant subspaces of L? (AM) in [175], and to the uniform H-convexity in [201] and [203]:

1498

G. Pisier and Q. Xu

We will now describe the Hilbert transform and Riesz projection. Let x € {a+b*: a,b €H>®(M)}. It is easy to see that x admits a unique decomposition
x=a+d+b*,

witha,be H°M),

deD.

Then we define the Hilbert transform H by

Hx =—i(a — b*).
Clearly, x +iHx € H®(M); moreover if x is self-adjoint, Hx is the unique self-adjoint
element in {a + b*: a,b € H*(M)}thatsuch that x + iHx € H®(M)

and £(Hx) = 0. Note

LAM) = Hy (M) @ L*(D) & (H*(M)) il
where

HEM)

= {x € H*(M):£(x) = 0}. One|
easilychecksthat

HZ (M)(resp...

(H*(M))1) is the closure of HS(M) (resp. {x*: x € H§O(M)}) in L?(M). This de-composition of L2(M) shows that H extends to a contraction on L*(M), still denoted .
by H.
Now let P be the orthogonal projection of L?(M) onto H2(M) (i.e., P is the “Riesz
projection”). Like in the classical case, H and P are linked together as follows
1.

1

Thus, as far as boundedness is concerned, it suffices to consider one of them.

-

THEOREM 8.4. (i) H extends to a bounded map on LP (M) for any 1 < p < 00; more
precisely, one has

I1Hxllp < Cplixllp,

Vx=a+b*a,be HM),

where Cp, < C p?/(p — 1) with C a universal constant.
(ii) H also extends to a bounded map from L' (M) into L1-%°(M) (the non-commutative
weak L!-space).
This result was proved in [160]. Of course, (i) above (for the case 1 < p < 2) follows by interpolation from (ii) and the L?-boundedness of H (and by duality for the case2 < p < 00). However, (i) admits a much simpler separate proof (see the discussion be-

low).

:

In the case of triangular matrices, (i) above is often referred to as Matsaev’s theorem(cf. [73]). In this case, the corresponding Riesz projection is the usual triangular projection (see [111] for more results on this projection; see also [208] for related results).
Let us discuss another particularly interesting case, that of analytic operator valued functions. Then Theorem 8.4(i) is equivalent to the UMD property of L? (M) that we alreadyNon-commutative LP -spaces1499

saw in the last section. Indeed, considering the finite subdiagonal algebra H°(T, M)
L>°(T)® M,

one sees that

H =H

® dpa Myofwhere H is the usual Hilbert transformon T. Thus the boundedness of H on L?(L*(T) ® M) is equivalent to the fact thatH ® id 2( rq) extends to a bounded map on LP (T; LP (M)) (noting that LP(L*°(T) ®
M) = LP(T; LP? (M))). In other words, L? (M) has the “HT property”, which is equivalent to the UMD property, as already mentioned in the last section. The main idea of theproof is an old trick due to Cotlar, which still works in the general setting as in Theorem 8.4. The ingredient is the following formula, whose proof is straightforward.

LEMMA 8.5. For any x =a + b* with a,b € Hi°(M)(Hx)*Hx =x"x + H(x*Hx + (Hx)*x).

:

Using Lemma 8.5, we easily check that the boundedness of H on L” (M) implies that on

L?P(M) (see also the proof of Theorem 5.3 above). Then starting from p = 2 and iterating,
we deduce that H is bounded on L?" (M) for all integers n > 1. Finally, interpolation andduality yield Theorem 8.4(i). We also point out that this argument gives the optimal orderof the constant C), as stated in Theorem 8.4.
REMARKS.

(i) It was shown in [150] that in the case of triangular matrices or analytic

operator valued functions, the non-commutative Hardy spaces form an interpolation scalewith respect to the real and complex methods. The same arguments work in the generalcase as well. Thus forany 0 < pp, p1 <o0and0(HP (M), HPY(M)),

<6

< 1

= (HP (M), HP'(M)), = HP(M),where 1/p=(1—6)/po+6/p1.
(ii) The Hilbert transform H enables us to identify the dual of H'(M) with the/noncommutative analog of the space BMO (for bounded mean oscillation) as in Fefferman’sclassical result, namely the space BMO(M) defined as follows:

BMO(M)= {x + Hy: x,y € L®(M)}

[I

-

[

equipped with the norm|[

lzll = inf{llxllco + ly loo:=x + Hy, x,y € L°(M)}.

]‘We refer to [124,125] for more information.

‘We end this section by an open problem. A famous theorem due to Bourgain states thatthe quotient space L!(T)/H(T) is a GT space of cotype 2 (cf., e.g., [146)). It is not clear
at all how to extend this theorem to the non-commutative case.
PROBLEM 8.6. Let H*(M) be a finite subdiagonal algebra in M. Is L1(M)/H(M)cotype 2? or merely of finite cotype?

of

1500

G. Pisier and Q. Xu

In the case of triangular matrices or vector-valued analytic functions, this problem has
been circulated in Banach space theory almost since Bourgain’s theorem. By the way,note that any quotient of L1(M) by a reflexive subspace is of cotype 2. This is the
non-commutative version of a theorem due to Kisliakov.and Pisier (cf. [146]). It follows:
from [92] and [145].

9. Hankel operators and Schur multipliers
In general it is not so easy to compute (up to equivalence) the S?-norm of an operator x in
SP, except when x = (x;;) is a column (or row) matrix and when it is a diagonal one, as
follows:

Snel,=| Seed, =

(ed)

1/2

|
PE

and

| Xmen], = (ber?)

1/p.

In view of their importance and ubiquity in Analysis, it was natural to wonder about thecase when x = (x;;) is a Hankel matrix, i.e., there is a (complex) sequence y in £7 suchthat

Xij =Yi+j,

Vi, j=0.

9.1,—

This case was solved in Peller’s remarkable paper [136] as follows.

THEOREM 9.1. Let x = (x;j) be given by (9.1) and let 1 < p < 00. Let2) =);50 2;

be “its symbol” and let

Aop(x) =%00,

Awp(@=Dy,

Ynz1@eD.
:

n=1 Lj<2n.

Then x € SP iffy, 2" Ang llh < oo (here | - ||, denotes the LP-norm on the unit circle .
.
with normalized Lebesgue measure). Moreover, |x | sr is equivalent to

(>
n0

an0l)

1/p

:

:

©2)

Actually, Peller also solved the cases p =1 and 0 < p < 1 but the solution is then a bit
more complicated to state (cf. [136] and [138]). The case 0 < p < 1 was obtained independently by Semmes (see [139] for this and for additional references). More generally,Non-commutative LP -spaces

1501

Peller [137] proved an extension of this result for Hankel matrices x = (x;;) with entries
xij = Yi+j in SP (H). In that case, Peller proved that |x [sre my) is equivalent to(x 2 1450 psy )

1/p

:

n=0

)

.

©.3)

N

Unfortunately, while (9.2) is very easy to use, in general the norm of A, in LP (SP)seems as untractable as that of x in S$? (£2(H)). However, when the spectrum of the symbolp=7 y; is restricted to be in a suitably “thin” set of integers A (meaning that theFourier coefficients y; € S” are zero when j ¢ A), then (9.3) can effectively be used, as
shown in the recent paper [83] (see (9.10) below).
To explain this, we will work in a (possibly non-commutative) discrete group I" but thecase of I' = Z is the most interesting one.
DEFINITION 9.2. Let p > 2 be an integer and let A C I" be a subset. Let ¢ = +1 if p iseven and & = —1 otherwise.
(i) A is called a B(p)-set if whenever two p-tuples (s;) and (#) in A satisfy
sit; Loot, .- ~spty! = e we have necessarily {s1,...,sp} = {t1, ..., 7p} with multiplicity (meaning that if an elementis repeated, it appears the same number of times
in both sets).
'(ii) For any ¢ in I", we denote by R(t, A) the number of p-tuples t;,1,...,7p in I"
with t; #¢; for all i # j such that tine! : tp = t; moreover we let Z(p, A) =sup{R,(t, A) | t #e}. We say that A is a Z(p)-setif Z(p, A) < co.
"The above is inspired by Zygmund’s study of the sets (called here Z(2)-sets) A C Z¢
such that Z(2, A) = sup, card{(n, m) € A? | n—m=t} < oo. As observed by Zygmund, the finite subsets Ay = {(n, m) € Z? | n* + m? = N} are uniformly Z(2)-sets, more
precisely we have.
sup Z(2, Ay) <2.
N21©4

Actually, the same is true if, replacing Z? by R?, we consider the circle of radius ~/Ninstead of Ay. A mere look at the picture of such a circle then establishes (9.4).
The paper [83] also shows that (generic) random

subsets of [1,..., N] with cardinal-

ity N1/2 are Z(2)-sets with constants uniformly bounded over N. On the other hand, as
pointed out in [83], B(p)-sets are a fortiori Z(p)-sets and this provides examples of adifferent kind: for instance free sets as well as any subset A C I" which does not satisfy any non-trivial relation of length < 2p. More generally, the generators of the free
Abelian groups such as Z¢ or ZN are B(p)-sets. On the other hand, because of torsion, theRademacher functions (= coordinateson {—1, 1}Y), identified to a subset R C {—1, 1M;
do not form a B(p)-set, but it is easy to see that they form a Z{p)-set for any p = 2 (withZ(p,R) =p).
i
g

1502

G. Pisier and Q. Xu

Thus, the following result in [83] can be viewed as an extension of Theorem 6.1 (=thenon-commutative Khintchine inequalities). Recall that vN (I™) is equipped with its normalized trace 7(see Section 1).

THEOREM 9.3. Let I' be any discrete group. Let p > 4 be an even integer and let A = {t, |n 20} CI" be a Z(p/2)-subset. Then there is a constant C such that for any semifinite
M and for any finite sequence a = (an)n30 in LP (M) we have

all Le ass2ynLe mz)< | >Mt) RayoY)< ClallLs

LP(uN(MHQM)

avizynreivzy-Moreover, the left side of (9.5) is actually valid for

9.5)

A=T.

To

REMARK 9.4. When I" is commutative and dim(M) = 1, the sets satisfying (9.5) areexactly the A(p)-sets in Rudin’s sense (see [171] and [22]). Because of this, the sets satisfying (9.5) when LP (M) = SP? are called A(p).p-sets and are studied in [83].

B.

DEFINITION 9.5. Consider two (commutative) LP-spaces LP(u), LP(v) (1 < p <o00) = and (closed) subspaces E C L? (i), F C L?(v). A linear mapping u: E — F is called
completely bounded (in short c.b.) if there is a constant C such that
YAR
<
(2)
v;
DITEOIORY
NNEC
DETR
sss

:
Pp
V3 . EE Vriese.

We denote by |u|.» the smallest C for which this holds.

:

This definition is coherent with the one used in the theory of operator spaces (cf. [154]).
REMARK 9.6. More generally if L”(u), LP (v) are non-commutative L?-spaces associated to semifinite traces u, v the preceding definition still makes sense using L p(n ® Tr)
and L? (v @ Tr) instead of L? (i; SP) and L? (v; SP).
te
COROLLARY 9.7. Let A and p be as in Theorem 9.3 (more generally, what follows is
valid for any p > 2 if A is assumed A(p)cp). Let (5) denote the Rademacher functions
on (£2, F, P), as in Section 6. Let Eg (resp. E 5) be the closed subspace of LP($2, F, P)(resp: LP(vN(I'))) generated by {e,, | n > 0} (resp. {A (t,) | n > 0}). Then the linear mappings u and u~! defined on the linear spans by u(en) = A(t) and u= (A (t,)) =e, ex-tend to c.b. maps u: Er — Ex and u™': Ey — Eg with ules < C and ues< Bj,

Moreover, the (orthogonal) projection P: LP (uN (I")) — E ,, defined by P(A(1)) = A(t)ifte A, and=0ift & A, is c.b. on LP (uN(I")).

Non-commutative LP -spaces

1503

The preceding results provide non-trivial new examples of c.b. Fourier multipliers on

LP(T). We now turn to Schur multipliers. A linear map 7": S? — SP (resp. T : B(£3) —B(£2)) is called a Schur multiplier if it is of the form

T(x) = [¢(, xij]
for some function ¢:N x N — C. In this case, we write T= M,,. The case p = 2 is ofcourse trivial: we have then ||M,|| = sup; ; lo@, 7)|. In the case p = o0, it is well known(due essentially to Grothendieck) that bounded Schur multipliers T' = Mj, : B(£2) — B(£2)
are all of the following special form: there are bounded sequences (x;) and (y;) in £; suchthat:0, J) = (xi, 3)(9.6)

Moreover, we have

1M] = inf{sup [i] sup I;1}
14

J

.

where the infimum runs over all possible (x;) and(y;) satisfying (9.6). This implies inparticular (due to Haagerup) that bounded Schur multipliers on B(£3) (or on § 1y are “automatically” c.b. (see [152]). However, the following remains open (we conjecture that theanswer is negative):
PROBLEM 9.8.

Is every bounded Schur multiplier on S? (1 < p #2 < oo) c.b.?

Note that it is rather easy to give examples of bounded Fourier multipliers on L?(G)which are not ¢.b. when G is any compact infinite commutative groupand 1 < p #2 < 0(see [83] or [152], p. 91).
PROBLEM 9.9. Is there a description of c.b. Schur multipliers on S$” extending (9.6) tol<ps#2<o00?
It is known ([83], see also [206]) that the space of bounded (or ¢.b.) Schur multipliers ofSP (2 < p < 00) does not coincide with any interpolation space between the cases p =2
and p = oo.
DEFINITION 9.10. A subset A C N x Nis called a o(p)-set (p = 2) if {e;j | (i, j) € A} isan unconditional basic sequence in S?.
A simple application of Corollary 6.2 shows that this holds iff there is a constant C suchthat for any finitely supported function x : A — C we havexl, < | > x6 fei |(i. )eA<Clixllp,
se

9.7)

1504

-

G. Pisier and Q. Xu

where [|x|| ;-is as defined in (6.7) above. It is easy to see by interpolation that, if 2 < p <oo, we have [|x |||, < [lx|lsr for all x in $7. Hence (9.7) implies that the idempotent Schurmultiplier corresponding to the indicator function of A is bounded on S? with norm < C.

For example, any set A for which either one of the two coordinate projections is one to
one when restricted to A, is obviously a o (p)-set. The following result provides much lesstrivial examples.
PROPOSITION 9.11([83]). Let p > 2. Let

A C Z be a Z(p/2)-set, or more generally aA(p)cp-set. Then the set Ay = {(i, j) e N* |i + j € A} is a o(p)-set.
PROOF. This follows from (6.4), applied to the series ¢(z) =},Indeed for any z we havele@]s, = | D>. x pei)(.))eAs
where a, = > jn

4 2" (3; 1 jn X05 Fei).

= > ay
sr

Wpea

NSP

x(, He; ’ . By the A(p)qp-property of A (see Remark 9.4) there is aconstant C such that:
1

"

ol

:

«

\1/2

N12

Zea], <mad| (Lean) | | (Cena) 7.)
nn< | DIE: Gn A.

©-8)

But as we just observed we have | } z"axl|Lr(spy = | 3_ anlls» and
NY)

RN)

Doanan=2_0 x Nley

Yanai=3"3 |x pl ei

ii

ij

Hence (9.8) implies (9.7) with
REMARK.

A = A 4.

O

In the situation of Proposition 9.11, the same argument shows that if

A = A 4then for any finitely supported function x : A — S$? we haveQ(x) < | >ej ®x(.))|(i, /)eA

<CO),

9.9)

SP (L245)

where

.

0x) — max] (|; (Zs
;
.

j

i

x.) g

(Zl(zeenen)ij1721p \ 1p
KYZ

1/29p

sp

)

;

7}
\1/p

co

i.

Non-commutative LP -spaces

1505

A set A C N? satisfying (9.9) for some constant C is called a o'(p)p-set. Equivalently, thismeans, by (6.4), that {e;; | (i, j) € A} is “completely unconditional” (sce Remark 9.6), i.e.,
for any choice of signs ¢;; = =1, the transformations>. x(, j)eij =>@i,j)eA@,j/)ehaxl, fei):

are c.b. on the closure in S? of {e;; | (i, j) € A}. Since the left side of (9.9) remains valid

for A =NZ, (9.9) implies that the indicator function of A is a c.b. Schur multiplier on S?.
In particular, if x = (x(i, j)) is Hankelian, i.e., x(i, j) = y (i + j) for some finitely
supported function y : A — S$”, then (9.9) implies

SRRCIARS PRICE
ned
where

> |
itj=n

< Ce),C910)

NS7(L®Db)|gy) = max (x(xz yoy)

2p

| )

jeg

\1/p

57

2p

\Upy

(ZI(Zremwer)0)§° 7}
FRY:

Thus we can “compute” (up to C) the norm of a Hanke! operator with “spectrum” in A:
COROLLARY 9.12 ([83]). Let p > 2 be an even integer.
(i)

There are §:> 0 and C such that, for any n, there is a subset A, C[1,..., nl? with

|A] = Sn t2/P such that {eij | GG, j) € An} is C-unconditional in §?; i.e., a o (p)-

set.

/

|

SH

(ii) There is an idempotent Schur multiplier T (idempotent means here T? = T) which
is bounded on SP but unbounded on S$? for any q > p.
a.
PROOF. The proof combines Theorem 9.3 with Rudin’s (combinatorial and number theoretic) construction of a B(p/2) set A C Z such that

limsup sup N~%?|AN[a,a+bN]| > 0.
N—oo

a,beN

0

REMARKS. (i) It is shown in [84] that, for any p > 2, n!+?/P is the maximal possibleorder of growth in the first part of Corollary 9.12.
(ii) The preceding corollary almost surely remains valid when p > 2 is not an eveninteger, but no proof is known at the time of this writing.
(iii) It is proved in [136] (see also [105] for related estimates on the case p = co) that theorthogonal projection from $2 onto the subspace of all Hankel matrices (i.e., the averaging

|
\
\

1506

G. Pisier and Q. Xu

projection) is bounded on S? iff 1 < p < 0, and for p = 1, it is bounded from S$! to §1:2
(a fortiori it is of “weak” type (1, 1)). See [2] for more recent results on (Hankel andToeplitz) Schur multipliers, in particular for the case SP with p < 1.

10. Isomorphism and embedding

In this section we discuss isomorphism and embedding of non-commutative L?-spaces.
Unless explicitly stated otherwise, we will assume all LP -spaces considered in this section are separable and infinite-dimensional, or equivalently, the underlying von Neumann
algebras are infinite-dimensional and act on separable Hilbert spaces.
Throughout this section, LP denotes the classical commutative LP-space on [0, 1].
The isomorphic classification of commutative LP-spaces is extremely simple, for there
are only two non-isomorphic commutative L”-spaces: [P and LP. However, in the noncommutative setting, the situation is far from simple. In fact, it is impossible to list all
non-commutative L?-spaces up to isomorphism. It even seems very hard to classify themaccording to the different types of the underlying von Neumann algebras. Despite these
difficulties, considerable progress has been achieved in the last few years.
Let K7 denote the direct sum in the [”-sense of the Ss, i.e.,k= (st)
nl

|
ro

:

Note that K'* is the non-commutative L”-space associated with the von Neumann algebra

M =D,>| Mn, the direct sum of the matrix algebras M,,, n > 1. We also recall that if X isa Banach space, L” (X) stands for the usual L?-space of Bochner measurable p-integrable
functions on [0, 1] with values in X. If X = L?(M), LP(X) is just the non-commutative
L?-space associated with L°°(0, 1) ® M. We should call the reader’s attention to the twodifferent notations for the Schatten classes, equally often used in the literature: S$? in ournotation is sometimes denoted by C?, and K? by SP! Recall that R denotes the hyperfinite
II; factor.

THEOREM 10.1. Let M be a hyperfinite semifinite von Neumann algebra. Let 1 < p < 00,
p # 2. Then LP (M) is isomorphic to precisely one of the following thirteen spaces:
17,

LP,

K?,

SP,

LP? ®

KP,

LP ® SP,LP(KP),SP ®

LP(KP),’

LP(SP), LP(R), S” ® LP(R), L?(S”) ® L”(R), L?(R ® B(1?)).
REMARKS. (i) The first nine spaces in the above list give precisely all non-commutative
LP-spaces, up to isomorphism, associated with von Neumann algebras of type I.
~~(ii) Theorem 10.1 is proved in [82]. Prior to that, the case of type I was studied in [187].
[82] also contains results on non-commutative L?-spaces associated with hyperfinite
factors of type III and free group von Neumann algebras. More precisely, it is shown thereNon-commutative LP-spaces

1507

that the non-commutative L?-spaces associated with hyperfinite factors of type III, for all

A € (0, 1] are isomorphic, and the non-commytative L?-space associated with a free groupvon Neumann algebra is independent, up to isomorphism, of the number
soon as this numberof generators as

is not less than 2. We refer the interested reader to [82] for more

information.

The proof of Theorem

10.1 can be reduced to the non-embedding of one non-commutative LP -space into another. The main general result on this is the following
THEOREM 10.2. Let 0 < p < 00, p # 2. Let M be a finite von Neumann algebra. Then
SP does not embed (isomorphically) into LP (M).
We get immediately the following corollary.
COROLLARY 10.3. LetO <p < 00, p + 2. Let M be a finite von Neumann algebra andN an infinite von Neumann algebra. Then LP(N) does not embed into LP{M).
Theorem 10.2 was proved in [186] for p > 2, in [82] for 1 £ p < 2, and in [188] forp < 1. Note that in the special case where M = L%°(0, 1), Theorem 10.2 was establishedby McCarthy in the pioneering paper [128]. His result was considerably improved in [74].

In particular, Theorem 4.1 implies that KX” does not embed into L?. In the converse direction, it was proved in [11] that L? does not embed into S7.

In the case of 0 < p < 2, we have the following result, much stronger than Theorem 10.2.
THEOREM 10.4. Let 0 < p <2. Let M be a finite von Neumann algebra. Let (u;,7)i, j>1
be an infinite matrix of elements in LP (M) such that sup; ; lu, l p < 00. Suppose that all
rows, columns and generalized diagonals of (u; ;)i,j>1 are unconditional. Then one of thefollowing three alternatives holds

RE(i) Some row or column has a subsequence equivalent to the canonical basis of 17;(ii) There is a constant A > 0 such that for every integer n some row and some column
contain n elements h-equivalent to the canonical basis of 11;(iii) There is a generalized diagonal (u;y, j, )k>1 such that

mE]
I

k=1

|
p

Here by a generalized diagonal of (u;,;);, j>>1 We mean a sequence (u;,, j )k>1 With i} <ip. <---andji < jy <---, Theorem

10.4 was proved in [82] for 1 < p <2 and in [188]for p < 1. Using Theorem 10.4, we can deduce the following refinement of Theorem 10.1,
which comes from [82] for I £ p < 2, and from [188] for p < 1.
::
THEOREM 10.5. Let M be as in Theorem 10.1, and let 0 < p < 2. If X #£ Y are listed in
the tree in the following figure, then X embeds into Y iff X can be joined to Y through a::descending branch

:

:

1508

G. Pisier and Q. Xu

yd

OK?

Ir

5°

KP, LP

SP @, LP

LP(K?)

SP @, LP(K?)

LP(R)

(57)

SP @, LP(R)

LP(SP) @, LP(R)

LP(RSB(£2))
Several non-embeddings in Theorem 10.5 are already contained in Corollary 10.3 andthe discussion just after it. On the other hand, the non-embedding of L? (KP) into L? & SP
was established in [187], and that of LP(R) into L?(S”) in [157]. The proof for the first
non-embedding in [187] uses-the classical result that L” contains a subspace isomorphic
to [7 forall:

0 < p < q < 2. This classical result admits a non-commutative version, which

is a remarkable result recently obtained by Junge (see Corollary 10.12 below), and whichis the main ingredient for the non-embedding of L?(R) into LP” (SP). The remaining nonembeddings in Theorem 10.5 can be reduced to the following
THEOREM 10.6. Let 0 < p <2, and let M and N be finite von Neumann algebras. Let
X C LP(M) be a closed subspace which contains no subspace isomorphic to IP, and let
Y be a quasi-Banach space which contains no subspace isomorphic to X. Then X ® , SPdoes not embed into Y & LP (N), where X ®,, SP denotes the closure of the algebraicitensor product X ® SP in LP (M @ B(?)).
Theorem 10.6 was proved in [188]. It extends some results in [82]. Like in [82], its proof
heavily relies upon Theorem 10.4. Using this theorem and Corollary 10.12 below (and its
commutative counterpart, cited above), we deduce that L? (S?) (resp. L? (R ® B(I?))) does
not embed in SP? @ LP(R) (resp. L? (SP) & LP (R)).
:
Subspaces of L?(M), which have no copy isomorphic to I”, can be characterized as.

follows.

.

THEOREM

10.7.

Let 0 < p <.00, p #2. Let M

be a finite von Neumann algebra and

X C LP(M) a closed subspace. Then the following assertions are equivalent:.

Non-commutative LP -spaces

1509(i) X contains a subspace isomorphic to IP.
(ii) Forany A > 1, X contains a subspace A-isomorphic to IP.

(iii) X contains Ij} ’s uniformly.

:(iv) Forany q suchthat0 <q < p the norms | - |; and | - ||, are not equivalent on X.
REMARK. If one of the preceding assertions holds, then X contains a perturbation of anormalized sequence formed of operators with disjoint support; consequently, if p > 1, X
contains, for any A > 1, a subspace A-isomorphic to [” and A-complemented in LP (M).
The

above theorem is the extension to the non-commutative

cal Kadets—Pelczyiniski results for commutative LP-spaces

setting of the classi-(cf. [103,170]). It was provedin [186] for p > 2, in [82] for 1 < p < 2, and in [163] and [188] for 0 < p < 1. In thecase p > 2, Theorem 10.7 yields the non-commutative analogue of the following striking
dichotomy:(
COROLLARY 10.8. Let M and X be as in Theorem 10.7 with 2 < p < 00. Then either X
is isomorphic to a Hilbert space or X contains a subspace isomorphic to 17.
REMARKS. (i) The above corollary is easier for subspaces of $7, and there it holds for all
0 < p < oo (cf. [68]).
(ii) More generally, Theorem 10.7 was extended in [169] to non-commutative L?-spaces
associated with any von Neumann algebra.
:(iil) [162,164] and [169] contain more results closely related to Theorem 10.7 and Corol-lary 10.8.
There are many open problems on the subject discussed above. Below we give twoof
them. Let M and NA be two von Neumann algebras of type A and u, respectively, whereA, we {1 II, Heo, II}. Combining Corollary 10.3, Theorem 10.5 and Theorem 3.5, we seethat if A < p and (A, pu) # (oo, ID), then LP? (M) and LP (N) are not isomorphic for all
0 < p < oo, p # 2. It is unknown whether this is still valid for (A, ©) = (Il, ITI).
PROBLEM 10.9.

Let

M and NV be two von Neumann algebras of type Il, and III, respec-tively. Are LP (M) and L? (AN) isomorphic for p # 2?
Theorem 10.5 solves the embedding problem for all spaces listed there in the case ofPp <2. On the other hand, Corollary 10.3 provides some partial solutions in the case ofp > 2. However, we do not know whether Theorem 10.5 holds in full generality for p > 2.
Below we state three of the most important cases left unsolved in Theorem 10.5.
PROBLEM

10.10.

Let p > 2, and let (X, Y) be one of the three couples (L?(K?), S? ®

LP), (LP(8P), SP @ LP(KP)) and (LP(R), L?(S?)). Does X embed into ¥?
All previous non-embedding results deal with a couple of non-commutative LP” -spaces
with the same index p. However, Junge’s theorem already mentioned above says that $9
does embed into L? (R) for p < gq < 2. In fact, Junge [96] proved the following striking
result, much stronger than the embedding of $7 into L”(R).
~

_.

1510

G. Pisier and Q. Xu

THEOREM
LP(R).

10.11.

Let 0 < p < q < 2. Then LY(R @ B(I?)) embeds isometrically into

As an immediate consequence, we get the

COROLLARY
into LP (R).

10.12.

Let 0 < p < g < 2. Then both S? and L1(R) embed isometricallyIn the commutative case, it is well-known that any IZ embeds (uniformly over rn) intosome / a. Junge also obtained the non-commutative version of this in [95].
THEOREM

10.13.

Let 0 <p<q<2,6>0,neN.

Then thereis

N = N(p,q,&,n) suchthat Sk contains a subspace (1 + &)-isomorphic to Sg.
Like in the commutative case, Junge’s arguments for the preceding results are probabilistic. They use non-commutative analogues of p-stable or Poisson processes. The reader isreferred to [95,96] for more details and more embedding results.
We conclude this section by a few words about the local theory of the non-commutative
LP-spaces, very recently developed in [99], in analogy with the classical £,-space theory. Actually, it is better (and more convenient in some sense) to develop this theory inthe operator space framework. Then the corresponding £-spaces are called OL ,-spacesin [60]. Many classical results concerning £,-spaces have been transferred to this noncommutative setting. In particular, any separable (JL ,-space (with an additional assumption) has a basis. It was also proved that L? (M) (1 < p < 00) is an OL ,-space when M
is injective or the von Neumann algebra of a free group (in the former case, p can be equal
to 1). Consequently, these non-commutative LP-spaces have bases. In the case of p = oo,
it was shown that any separable nuclear C*-algebra has a basis. The interested reader isreferred to [99] for more information.

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