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NONCOMMUTATIVE BURKHOLDER/ROSENTHALINEQUALITIES II: APPLICATIONS

M. JUNGE∗ AND Q. XU

Abstract. We show norm estimates for the sum of independent random variables innoncommutative Lp spaces for 1<p<∞ following previous work by the authors. Theseestimates generalize Rosenthal’s inequalities in the commutative case. Among other appli-cations, we derive a formula for p-norm of the eigenvalues for matrices with independententries, and characterize those symmetric subspaces and unitary ideal spaces which canbe realized as subspaces of noncommutative Lp for 2 < p < ∞.

0. Introduction and Notation

Martingale inequalities have a long tradition in probability. The applications of the

work of Burkholder and his collaborators [B73, ?, BDG72, B71a, B71b, BGS71, BG70, B66]

ranges from classical harmonic analysis to stochastical differential equations and the geom-

etry of Banach spaces. When proving the estimates for the ‘little square function’ Burk-

holder was aware of Rosenthal’s result [Ros] on sums of independent random variables.

Here we proceed differently and prove the noncommutative Rosenthal inequality along the

same line as the noncommutative Burkholder inequality from [JX1]. This slight modi-

fied prove yields a better constant. The main intention of this paper is to illustrate the

usefulness of the ‘little square function’ in several examples. For many applications it is

important to consider generalized notations of independence. This will allow us to explore

applications towards random matrices and symmetric subspaces of noncommutative Lpspaces.

Our estimates on random matrices are motivated by the noncommutative Khintchine

inequality. In [LP] Lust-Piquard showed that for 2 ≤ p < ∞ and scalar coefficients (aij)

one has

E‖∑ij

εijaijeij‖p ∼c(p)

∑i

(∑j

|aij|2) p

2

1p

+

∑j

(∑i

|aij|2) p

2

1p

.(0.1)

∗ The first author is partially supported by the National Science Foundation Foundation DMS-0301116.1

2 M. JUNGE AND Q. XU

We use the notation a ∼c b if a ≤ c1b, b ≤ c2a and c1c2 ≤ c. Let N be a von Neumann

algebra with a normal faithful trace τ . Then the Lp-norm of an operator x affiliated to N

is given by

‖x‖p = [τ(|x|p)]1p .

In particular for N = B(`2) and τ = tr, the p-norm of a matrix is the p-norm of its

singular values

‖∑ij

aijeij‖p =

(∑k

λk(|a|)p) 1

p

,

i.e. the eigenvalues λk(|a|) of |a|. In our first result we replace the coefficients aijεij in

(0.1) by arbitrary random variables:

Theorem 0.1. Let (fij) ⊂ Lp(Ω, µ) be a matrix of independent mean 0 random variables

defined on a probability space.

i) If 2 ≤ p <∞, then

‖∑ij

fijeij‖p

∼Cp max

(∑

ij

‖fij‖pp

) 1p

,

(∑j

(∑i

‖fij‖22

) p2

) 1p

,

(∑i

(∑j

‖fij‖22

) p2

) 1p

.

ii) If 1 < p ≤ 2, then

‖∑ij

eij ⊗ fij‖p ∼Cp′ inffij=gij+hij+dij(∑

j

‖(∑

i

E(g∗ijgij)

) 12

‖pp

) 1p

+

(∑i

‖(∑

j

E(hijh∗ij)

) 12

‖pp

) 1p

+

(∑ij

‖dij‖pp

) 1p

.

Here the infimum is taken of gij, hij dij with mean 0 and measurable with respect

to σ-algebra generated by fij.

The estimates in Theorem 0.1 for p ≥ 2 is a direct application of our main result

‖∑k

xk‖p ∼cp

(∑k

‖xk‖pp) 1

p + ‖(∑

k

E(x∗kxk + xkx∗k))1/2‖p(0.2)

which holds for independent mean 0 variables. Here E is allowed to be operator val-

ued. This allows us to replace the fij’s by operator valued (or matrix valued) coefficients

provided they satisfy appropriate independence conditions, for example if they are freely

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 3

independent (in the sense of Voiculescu [VDN]). We also extend our result to the non-

tracial setting and even non-faithful setting. The dual version of Rosenthal’s inequality

(in the non-tracial setting) provides Khintchine type inequalities for the generators of the

CAR algebra. As in the case of Khintchine inequalities [LPP] one has to replace the

maximum by an infimum, see section 4. In section 3 we provide a version of Rosenthal’s

inequality using the recent concept of maximal functions in the noncommutative setting.

Formally these formulations are much closer to Burkholder’s original version [B73] and we

are providing a stopping time free proof of these results for p > 2.

Symmetric subspaces of Lp spaces are motivated by probabilistic notions of exchange-

able random variables. In the commutative situation. The memoir of Johnson, Maurey,

Schechtman and Tzafriri [JMST] contains an impressive amount of information and many

sophisticated applications of probabilistic techniques. In this paper we extend one of the

more elementary inequalities from [JMST]:

Theorem 0.2. Let N and M be von Neumann algebras and 2 ≤ p <∞. Let ek ∈ Lp(N)

and C > 0 such that

(0.3) ‖∑k

εkaπ(k) ⊗ xk‖p ≤ C‖∑k

ak ⊗ xk‖p

holds for all εk = ±1, all permutations π on 1, .., n and coefficients ak ∈ Lp(M). Then

there are constants a, c, r such that

‖∑k

ak ⊗ ek‖p ∼C2cp a

(n∑k=1

‖ak‖pp

) 1p

+ b‖(n∑k=1

a∗kak)12‖p + c‖(

n∑k=1

aka∗k)

12‖p .(0.4)

holds for all ak ∈ Lp(M).

In Banach space theory a sequence (ek) satisfying (0.3) for scalar valued coefficients is

called a symmetric basis for spanek : k ∈ N. In analogy with the commutative case, we

obtain a characterization of symmetric subspaces for 2 < p <∞.

Corollary 0.3. Let 2 ≤ p < ∞ and X ⊂ Lp(N) be a subspace with a symmetric basis,

then X is isomorphic to `p or to `2.

We also explore the natural operator space version (operator-valued version) of this

result. A further application is a characterization of unitary ideals (and noncommutative

functions spaces) in noncommutative Lp spaces for p > 2. The non-faithful version of

Rosenthal’s allows us to show that Lp spaces are asymptotically symmetric (see [JR]).

The results in [JR] show that Corollary 0.3 still holds for spaces with a subsymmetric

basis (see section 5 for a precise definition).

4 M. JUNGE AND Q. XU

The paper is organized as follows. Section 1 contains basic information on noncommu-

tative Lp spaces and the abstract version of the noncommutative Rosenthal inequality for

2 ≤ p < ∞. In section 2 we extend these results to the non-faithful situation. Section

3 develops an improvement of the noncommutative Rosenthal inequality using the recent

theory of maximal functions and vector-valued Lp spaces. In section 4 we investigate the

case 1 < p ≤ 2. We refer to section 5 for more information on symmetric subspace of

noncommutative Lp for p ≥ 2. Applications to unitary ideals are contained in section 6.

We use standard notation from (noncommutative) probability and the theory of von

Neumann algebras (see e.g. [KRI, KRII], [SZ, Str] and [TakI, TakII, TakIII]). For more

information and definitions on operator space theory we refer to [Pis2] and [ER]. We refer

to [Pis1, JNRX] for the natural operator space structure on Lp spaces.

1. Rosenthal’s inequality for sums of independent random variables

In this section, we shall prove a noncommutative version of Rosenthal’s inequality. We

will work under the standard assumptions from [JX1], i.e. we fix a faithful normal state

φ on a von Neummann algebra N and we denote by D the density of φ in the Haagerup’s

space L1(N) such that

φ(x) = tr(xD) .

We refer to [JX1] for details on the Haagerup Lp-spaces needed in this paper. In our

situation, we may work with N oσφt

R, where σφt is the modular group associated to φ.

NoR is semifinite and posseses a unique trace τ such that τ(θs(x)) = e−sτ(x) is satisfied

for the dual action. Then T (x) =∫

R θs(x)ds is a positive operator valued weight and

D is the density of the weight φ T in the space L0(N o R, τ) of measurable operators

affiliated to N o R. Let us recall that

Lp(N) = x ∈ L0(N, τ) : θs(x) = e−spx .

It turns out that for x ∈ Lp(N) with polar decomposition x = u|x|, we have u ∈ N and

|x| ∈ Lp(N). The norm in Lp(N) is given by

‖x‖p = (tr(|x|p))1p .

Here tr : L1(N) → C, tr(Dφ) = φ(1) is the Haagerup trace. We will frequently use

Holder’s inequality

‖xy‖p ≤ ‖x‖r‖y‖qwhenever 1

p= 1

r+ 1

q. We say that M ⊂ N is a φ-invariant subalgebra, if σφt (M) ⊂ M .

According to Takesaki’s Theorem ([Tak72]), there exists a unique conditional expectation

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 5

E : N →M such that φ|M E = φ. This implies in particular

(1.1) σφt E = E σφt

for all t ∈ R. For 1 ≤ p ≤ ∞, the conditional expectation E extends to a contraction

Ep : Lp(N) → Lp(M) densely defined as follows

Ep(xD1p

φ ) = E(x)D1p

φ|M .

HereDφ|M is the density of φ|M in L1(M). However, equation (1.1) implies thatMoφ|Mσφ

t

R ⊂N oσφ

tR and that the restriction of τ to M o R is the unique trace for M o R. We obtain

an isometric embedding ιp : Lp(M) → Lp(N) such that ιp(D1/pφ|M ) = D

1/pφ (see [JX1] for

more details). Therefore, we will simple use D for the density of φ. Let us recall that the

conditional expectation E is characterized by

(1.2) φ(E(x)y) = φ(xy)

for all x ∈ N and y ∈ M . If we have several φ-invariant subalgebras (Mi), we write

(EMi) for the corresponding sequence of conditional expectations. We refer to [JX1] for

the following frequently used equality

(1.3) Ep(D1−θ

p xDθp ) = D

1−θp E(x)D

θp

which holds for all x ∈ N , all 0 ≤ θ ≤ 1 and 1 ≤ p ≤ ∞. It is convenient to omit the

index p. This is justified using Kosaki’s embedding I : Lp(N) → L1(N), I(xD1p ) = xD

since then E1(I(y)) = I(Ep(y)). In this sense all the maps Ep are ‘induced’ by the same

map E1.

For our formulation of Rosenthal’s inequality (and its natural matricial valued general-

izations) we will assume that M ⊂ N is φ- invariant and that there are further φ-invariant

von Neumann subalgebras (Ak) ⊂ N containing M . We will say that (Ak) are (faithfully)

independent over M if

I) EM(xy) = EM(x)EM(y) holds for all x ∈ Ak and y in the algebra generated by

A1, ..., Ak−1, Ak+1, ..., An.

The following notion has been introduced by Schurmann: The algebras (Ak) are increas-

ingly independent over M if

II) EA1∪···∪Ak−1(x) = EM(x) holds for all x ∈ Ak and 1 ≤ k ≤ n

6 M. JUNGE AND Q. XU

Lemma 1.1. Condition I) implies condition II). Moreover, if I) is satisfied and xk ∈Lp(Ak) satisfy EM(xk) = 0, then

‖n∑k=1

εkxk‖p ≤ 2‖n∑k=1

xk‖p

holds for all εk = ±1 and 1 ≤ p ≤ ∞.

Proof. Let us assume that condition I) is satisfied and let S ⊂ 1, ..., n be a subset

of 1, ..., n such that k /∈ S. Since all the Ak are φ-invariant so is the von Neumann

subalgebra BS generated by the Ak|k ∈ S. Let us denote by ES the unique conditional

expectation characterized by

φ(ES(x)y) = φ(xy)

for all y = a1 · · · am, aj ∈ Aij , ij ∈ S. Now, we assume that x ∈ Ak and EM(x) = 0. Let

y = a1 · · · am as above. Then we deduce from condition I) that

φ(xy) = φ(EM(xy)) = φ(EM(x)EM(y)) = 0 .

Thus ES(x) = 0. Hence for an arbitrary element x ∈ Ak, we get

ES(x) = ES((x− EM(x)) + ES(EM(x)) = EM(x)

because M ⊂ BS. If we apply this to the set S = 1, ..., k − 1 we obtain condition

II). To prove the second assertion consider εk = ±1 and define S = k : εk = 1. By

approximation with elements of the form xk = akD1p , ak ∈ Ak and EM(ak) = 0, we see

that

ES(n∑k=1

xk) =∑k∈S

xk +∑k/∈S

ES(xk) =∑k∈S

xk .

Since ES is a contraction, we deduce

‖n∑k=1

εkxk‖p ≤ ‖∑k∈S

xk‖p + ‖∑k∈Sc

xk‖p = ‖ES(n∑k=1

xk)‖p + ‖ESc(n∑k=1

xk)‖p ≤ 2‖n∑k=1

xk‖p .

Let us formulate our version of Rosenthal’s inequality. Following [JX1], we introduce

the notations

‖(xk)‖`p(Lp) =

(∑k

‖xk‖pp

) 1p

‖(xk)‖Lp(M,E;`c2) = ‖∑k

E(x∗kxk)‖12

p/2 and ‖(xk)‖Lp(M,E;`r2) = ‖∑k

E(xkx∗k)‖

12

p/2 .

Here E is a conditional expectation. For many applications it turns out to be quite useful

to work with these simplified ’little square functions’.

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 7

Theorem 1.2. Let 2 ≤ p < ∞, n ∈ N ∪ ∞. Let M ⊂ N , (Ak) ⊂ N von Neumann

subalgebras and xk ∈ Lp(Ak) with EM(xk) = 0.

I) If the (Ak)1≤k≤n are independent over M , then

1

Cp‖

n∑k=1

xk‖p ≤ max‖(xk)‖`p(Lp), ‖(xk)‖Lp(M,E;`c2), ‖(xk)‖Lp(M,E;`r2) ≤ 2‖n∑k=1

xk‖p .

II) If the (Ak)1≤k≤n are increasingly independent over M , then

1

Cp2‖

n∑k=1

xk‖p ≤ max‖(xk)‖`p(Lp), ‖(xk)‖Lp(M,E;`c2), ‖(xk)‖Lp(M,E;`r2) ≤ 2‖n∑k=1

xk‖p .

Here C is an absolute constant.

Proof. Since the algebra Nn generated by A1, ..., An is φ-invariant, we may assume that

N = Nn. The space Lp(N) = (1− EM)Lp(N) is 2-complemented in Lp(N) and therefore

the lower estimate (n∑k=1

‖xk‖p

) 1p

≤ 2‖n∑k=1

xk‖p

follows by interpolation (using the inclusion map x → xD12 ∈ L2(N), see [Kos] for more

details on these interpolation spaces). Moreover, for x =∑n

k=1 xk, we have

‖n∑k=1

EM(x∗kxk)‖ p2

= ‖EM(x∗x)‖ p2≤ ‖x∗x‖ p

2= ‖x‖2

p .

Therefore the lower estimates for the norm of the sum are proved. The main part is

the proof of the upper estimate. We will now consider the case I). Let us note that

for 1 ≤ p ≤ 2, we deduce from interpolation (and the fact that the space of mean 0 is

2-complemented in Lp(Ak) see above) that EM(xk) = 0 implies

‖n∑k=1

xk‖p ≤ 2

(n∑k=1

‖xk‖pp

) 1p

.

We will use the standard iteration procedure. Indeed, we may assume that for q = p2, we

have the estimate

‖n∑k=1

xk‖q ≤ C(q) max‖(xk)‖`q(Lq), ‖(xk)‖Lq(M,E;`c2), ‖(xk)‖Lq(M,E;`r2)

8 M. JUNGE AND Q. XU

for all xk ∈ Lq(Ak) with mean 0. Our aim is to prove the estimate for p. Of course we

may assume p > 2. Let xk ∈ Lp(Ak) and EM(xk) = 0. First we apply the Khintchine

inequality (see [PS] for the right order of constants) and deduce from Lemma 1.1 that

(1.4) ‖n∑k=1

xk‖p ≤ 2 E‖n∑k=1

εkxk‖p ≤ 2c1√p max‖

n∑k=1

x∗kxk‖12p2, ‖

n∑k=1

xkx∗k‖

12p2 .

Let us consider the first one of these two square functions. We define the selfadjoint mean

0 elements yk = x∗kxk − EM(x∗kxk). By hypotheses we have

‖n∑k=1

x∗kxk‖ p2≤ ‖

n∑k=1

EM(x∗kxk)‖ p2

+ ‖n∑k=1

yk‖ p2

≤ ‖n∑k=1

EM(x∗kxk)‖ p2

+ C(q) max‖(yk)‖`q(Lq), ‖(yk)‖Lq(M,E;`c2), ‖(yk)‖Lq(M,E;`r2)

Moreover, if 2 ≤ p ≤ 4, we have 1 ≤ q = p/2 ≤ 2 and we can disregard the second term.

Since EM is a contraction, we have(n∑k=1

‖yk‖qq

) 1q

=

(n∑k=1

‖x∗kxk − EM(x∗kxk)‖qq

) 1q

≤ 2

(n∑k=1

‖xk‖pp

) 2q

.

Hence, for 2 ≤ p ≤ 4, we find

‖n∑k=1

xk‖p ≤ 2c1√p(1 + 4)

12 max‖(xk)‖`p(Lp), ‖(xk)‖Lp(M,E;`c2), ‖(xk)‖Lp(M,E;`r2) .

For 4 < p <∞, we first note that

EM(y∗kyk) = EM((x∗kxk − EM(x∗kxk))∗(x∗kxk − EM(x∗kxk)))

= EM(x∗kxkx∗kxk)− EM(x∗kxk)EM(x∗kxk) ≤ EM(|xk|4) .

Now, we may use the interpolation result of [JX1, Lemma 5.2] (for En−1 = EM) and find

‖n∑k=1

EM(|xk|4)‖ q2≤ ‖

n∑k=1

EM(|xk|2)‖q−2q−1q

(n∑k=1

‖xk‖pp

) 1q−1

.

By homogeneity this implies

‖n∑k=1

EM(|xk|4)‖12q2≤ max

(

n∑k=1

‖xk‖pp

) 2p

, ‖n∑k=1

EM(|xk|2)‖q

.

Therefore we have proved that

‖(yk)‖Lq(M,E;`c2) ≤ max‖(xk)‖`p(Lp), ‖(xk)‖Lp(M,E;`c2)2 .

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 9

Applying the same argument for xkx∗k, we find

‖n∑k=1

xk‖p ≤ 2c1√p(1 + 2C(q))

12 max‖(xk)‖`p(Lp), ‖(xk)‖Lp(M,E;`c2), ‖(xk)‖Lp(M,E;`r2) .

This shows C(p) ≤ 2c1√p(1 + 2C(q))

12 for p > 4. We have seen that for 2 ≤ p ≤ 4, we

may assume C(q) ≤ C1q. Thus by induction hypothesis we may assume C(q) ≤ Cq for

some constant C ≥ C1 ≥ 1 and find

C(p) ≤ 2c1√p(1 + 2Cp/2)

12 ≤ 2

√2c1C

12p .

Thus for C = maxC1, 8c21 the induction argument works and we obtain the assertion I).

For increasingly independent algebras, Lemma 1.1 is no longer at our disposal. We have

to replace (1.4) by the Burgholder-Gundy inequality from [PX1, JX1] (see also [JX2]) for

constants):

‖n∑k=1

xk‖p ≤ cp max‖n∑k=1

x∗kxk‖12p2, ‖

n∑k=1

xkx∗k‖

12p2 .

Here the underlying martingale structure is given by the von Neumann algebras Bk−1

generated by A1, ..., Ak−1. Let us denote by Ek−1 the conditional on Bk−1. By assumption

(and approximation) we have

Ek−1(xk) = EM(xk)

Assuming EM(xk) = 0 for all k we deduce that∑

k xk is a martingale and the Burgkholder-

Gundy inequalities apply. With the iteration procedure from above this yields a constant

Cp2 for the upper estimate.

Remark 1.3. In the commutative case the best constant in the Burkholder Rosenthal

inequality is of the order p/1 + ln p. In view of this result the constant of order p seems

reasonable. At the time of this writing we don’t know whether this order of the constant

is optimal. It seems that the best order of growth for Burkholder’s inequality in the

noncommutative case is unknown. Our estimate works with Cp2.

Remark 1.4. a) As stated our result holds for σ-finite von Neumann algebras. It can

easily be extended to arbitrary von Neumann algebras provided E : N → M is a faithful

conditional expectation and EAkare normal conditional expectations satisfying the com-

mutation relation EAkE = EAk

E = E. Indeed, let ψ be a strictly semifinite weight on

M , i.e. for a weight of the form ψ =∑

i siφisi such that the si’s are mutually orthogonal,∑i si = 1 and the states φi are faithful on siMsi. Such a weight can always be constructed

by choosing a maximal orthogonal family of states. For a finite subset J ⊂ I, we define

sJ =∑

i∈I si and obtain an increasing family of projections such that limj sJ = 1. Then

10 M. JUNGE AND Q. XU

we may consider the state φJ = 1/|J |∑

i∈J φi E on sJNsJ and see that the assump-

tions of Theorem 1.2 are satisfied for Ak,J = sJAksJ . Moreover, for x ∈ Lp(N) we find

x = limJ sJx = limJ xsJ = limJ sJxsJ . For p < ∞ this is a norm limit. Thus the result

extends to Lp(N) by density.

b) There is also a standard procedure to reduce problems for arbitrary von Neumann

algebras to von Neumann algebras with separable dual (see [GGMS, Appendix]). Indeed,

let φ be a faithful normal state. Let B ⊂ N countable subset of N . Then, we may consider

the ∗-invariant subalgebra B generated by σφt (EAk(A))k∈N,t∈Q. The von Neumann algebra

MA ⊂ N generated by B satisfies Takesaki’s criterion that σφt (MB) ⊂MB and hence there

is a φ-invariant conditional expectation. Thus there exists a net (MB)B⊂Ncountable of von

Neumann subalgebras of N which admit a unique φ-invariant conditional expectations

EMBsatisfying EAk

(MB) ⊂ MB. By uniqueness we have EMB2|MB1

= EMB1whenever

B1 ⊂ B2. Therefore Lp(N) is the direct limit of the complemented subspaces Lp(MB) and

the assumptions of Theorem ?? are still satisfied for (MB ∩ Ak)k∈N.

As an illustration we prove our main result on random matrices.

Corollary 1.5. Let 2 ≤ p < ∞ and (fij) a matrix of independent random matrices in

Lp(N) defined on a probability space (Ω, µ) such that E(fij) =∫fijdµ = 0. Then

1

Cp‖∑ij

fij ⊗ eij‖Lp(Ω;Lp(N⊗B(`2))) ≤ max

(∑

ij

‖fij‖pLp(N)

) 1p

,

(∑j

‖(∑

i

E(f ∗ijfij)

) 12

‖pp

) 1p

,

(∑i

‖(∑

j

E(fijf∗ij)

) 12

‖pp

) 1p

≤ 2 ‖

∑ij

fij ⊗ eij‖Lp(Ω;Lp(N⊗B(`2))) .

If the fij are independent scalar-valued mean 0 random variables, then

‖∑ij

fijeij‖p

∼Cp max

(∑

ij

‖fij‖pp

) 1p

,

(∑j

(∑i

‖fij‖22

) p2

) 1p

,

(∑i

(∑j

‖fij‖22

) p2

) 1p

.

Proof. According to Remark 1.4 we may assume N is σ-finite N . Let φN be a normal

faithful state φN on N and let ψ be a faithful normal state on B(`2). We deduce that

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 11

φ(x) =∫

(φN ⊗ ψ)(x(ω))dµ(ω) is a faithful normal state on L∞(Ω, µ;N⊗B(`2)). Since

the (fij)’s are independent, we may assume that Ω =∏

ij Ωij such that fij is Ωij measur-

able. We define the algebras Aij = L∞(Ωij, µ;N⊗B(`2)). Obviously, these algebras are

independent over 1⊗N⊗B(`2) ⊂ L∞(Ω, N⊗B(`2)). We use the lexicographical order on

N × N. Since fij ⊗ eij ∈ Lp(Aij) = Lp(Ωij;Lp(N⊗B(`2))) the assumptions of Theorem

1.2 are satisfied. Let us calculate the little square functions

‖∑ij

E((fij ⊗ eij)∗fij ⊗ eij)‖ p

2= ‖

∑ij

E(f ∗ijfij)⊗ ejieij)‖ p2

=

(∑j

‖∑i

E(f ∗ijfij)‖p2p2

) 2p

.

The same calculation applies for the second square function. The second chain of inequal-

ities corresponds to the particular case N = C.

2. The non-faithful case for 2 ≤ p <∞

Non-faithful filtrations occur very naturally in operator algebras. The easiest example

is the filtration (Mn) in B(`2) given by matrix algebras Mn of matrices (aij) such that

aij = aji = 0 for i > n. The notion of non-faithful copies of von Neumann algebras is

important in the context of iterated ultraproducts of von Neumann algebras.

Let us fix the relevant notation. We will assume that M ⊂ N is a weakly closed ∗-

invariant subalgebra with unit e and a conditional expectation E : N → M with support

e. In addition we assume that φ is a normal state on N , faithful on eNe such that φ =

φ|M E = φ. Now, we consider a sequence (Ak) of weakly closed ∗-invariant subalgebras

of N containing M . Let us denote by rk the units of Ak. We will say that the algebras Akare independent over M if

i) The projections sk = rk − e are mutual orthogonal.

ii) There are conditional expectations Fk : skNsk → skAksk with support sk.

iii) EM(xy) = EM(x)EM(y) holds for all xk ∈ Ak and y in the ∗-algebra generated by

A1, ..., Ak−1, Ak+1, ..., An.

Let us observe that the sequence of von Neumann algebras (eAke)k are independent

over M in the sense of the previous section. In the following we will say that (eAke)k is

faithfully independent over E. This allows us to distinguish the notation from the previous

section from the on defined above. For our exposition we may assume that n is finite and

that the weakly closed subalgebras Bk = skAksk are σ-finite (see 3.5). Let us fix normal

faithful states ψ1, ..., ψk on Bk. Then the state

12 M. JUNGE AND Q. XU

(2.1) Φ(x) =1

n+ 1(φ(x) +

n∑k=1

ψk(Fk(x)))

is normal and faithful on N . It is easily checked that the support of ψk Fk is sk. We

obtain the new projection f = e+∑

k sk and Φ is a normal faithful state on fNf . Thanks

to our assumptions on the conditional expectations Fk, we may and will assume in the

following that f = 1 because all the algebras Ak are contained in fNf . Moreover, by

construction, the modular group of Φ may be calculated componentwise

(2.2) σΦt (x) = eσφt (exe)e+

n∑k=1

skσψkFkt (skxsk)sk .

By orthogonality of the sk, we deduce that the von Neumann algebras Ak are Φ-invariant.

Once this is established, we obtain Φ-invariant conditional expectations Ek : N → Ak such

that Ek(e) = e. We refer to the beginning of section 2 on how to extend these conditional

expectations to Lp(N). The new ingredient for the non-faithful version of Rosenthal’s

inequality is a separate treatment of the corners.

Lemma 2.1. Let 2 ≤ p <∞ and xk ∈ Lp(Ak). Then

‖∑k

skxke‖p ≤ (1 + Cp)12 max‖(skxke)‖`p(Lp), ‖(skxk)‖Lp(M,EM ,`c2) .

Proof. Let x =∑

k skxke. By the orthogonality of the sk, we obtain

‖x‖2p = ‖

∑k

ex∗kskxke‖ p2≤ ‖

∑k

EM(x∗kskxk)‖ p2

+ ‖∑k

ex∗kskxke− EM(x∗kskxk)‖ p2.

We observe that yk = ex∗kskxke − EM(x∗kskxk) ∈ eAke and satisfies EM(yk) = 0. As

observed before, the sequence (eAke) is faithfully independent over M . Now, we follow

the proof of Theorem 1.2. If 2 ≤ p ≤ 4, we deduce from Lemma 1.1 that

‖∑k

yk‖ p2≤ 2 E ‖

∑k

εkyk‖ ≤ 2

(n∑k=1

‖yk‖p2p2

) 2p

≤ 2

(n∑k=1

‖skxke‖pp

) 2p

.

For 4 < p <∞, we deduce from the fact that the yk’s are selfadjoint, Theorem 1.2 applied

to p/2 and Lemma [JX1, Lemma 5.2] that

‖∑k

yk‖ ≤Cp

2max

(∑k

‖yk‖p2p2

) 2p

, ‖(∑

k

EM(y∗kyk)) 1

2‖ p2

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 13

≤ Cp

2max

2

(∑k

‖skxke‖pp

) 2p

, ‖∑k

EM(|skxke|4)‖12p4

≤ Cp

2max

2

(∑k

‖skxke‖pp

) 2p

,

(∑k

‖skxke‖pp

) 1p−2

‖∑k

EM(|skxke|2)‖2(p/4−1)

p−2p2

.

The assertion follows by homogeneity.

The non-faithful version of Rosenthal’s inequality for p ≥ 2 reads as follows.

Theorem 2.2. Let 2 ≤ p < ∞. Let (Ak) be independent over M with units (rk). Let

sk = rk − e above. Let xk ∈ Lp(Ak) and yk = xk − EM(xk). Then

1

Cp‖∑k

xk‖p ≤ max‖∑k

EM(xk)‖p, ‖(yk)‖`p(Lp), ‖(yk)‖Lp(N,EM ,`c2), ‖(yk)‖Lp(N,EM ,`r2)

≤ 7‖∑k

xk‖p .

Proof. We recall that the algebras eAke ⊂ eNe are independent over EM (in the sense

of section 2). We need some easy facts. First skyk = sk(xk − EM(xk)) = skxk because

EM(xk) ∈ eNe. Similarly xksk = yksk. Theorem (1.2) implies in combination with Lemma

2.1 that

‖∑k

xk‖p

≤ ‖∑k

EM(xk)‖p + ‖∑k

skxksk‖p + ‖∑k

skxke‖p + ‖∑k

exksk‖p + ‖∑k

eyke‖p

≤ ‖n∑k=1

EM(xk)‖p +

(∑k

‖skxksk‖pp

) 1p

+ Cp

(∑k

‖exksk‖pp

) 1p

+ Cp

(∑k

‖skxke‖pp

) 1p

+ Cp

(∑k

‖eyke‖pp

) 1p

+ Cp‖(∑k

EM(x∗kskxk))12‖p + Cp‖(

∑k

EM(y∗keyk))12‖p

+ Cp‖(∑k

EM(xkskx∗k))

12‖p + Cp‖(

∑k

EM(ykey∗k))

12‖p

≤ ‖n∑k=1

EM(xk)‖p+ 4Cp

(∑k

‖yk‖pp

) 1p

14 M. JUNGE AND Q. XU

+ 3Cp‖(∑k

EM(y∗kyk))12‖p+ 3Cp‖(

∑k

EM(yky∗k))

12‖p.

This proves the upper estimate. EM being a contraction implies ‖∑k

EM(xk)‖p ≤ ‖∑k

xk‖p.

The estimate (∑k

‖yk‖pp

) 1p

≤ 2‖∑k

xk‖p

holds by interpolation. By orthogonality we have

‖∑k

EM(y∗kyk)‖12p2

= ‖∑kj

EM(y∗kyj)‖ p2

= ‖EM(y∗y)‖12p2≤ ‖y‖p

≤ ‖∑k

xk‖p + ‖∑k

EM(xk)‖p ≤ 2‖∑k

xk‖p .

This completes the proof of the lower estimate.

Example 2.3. Non-faithful independence occurs natural in the context of conditional

expectation with respect to corners. Let N be a von Neumann algebra, e a projection and

r1, ..., rn be a family of orthogonal projections such that e ≤ rj and

rjrk = rkrj = e .

We also assume ∨rj = 1. Let us consider

M = eNe , Ak = rkNrk .

The conditional expectation is given by E(x) = exe. Here Fk is the identity. We define

rk = ∨j 6=krj and observe that rkrk = e. The algebra Bk is given by rkNrk. Thus for

x ∈ Ak and y ∈ Bk we have

E(xy) = exye = exrkekye = exeye = E(xy) .

This situation occurs for example on a tensor product N = B⊗n where e = f1 ⊗ · · · ⊗ fnand rk = f1 ⊗ · · · fk−1 ⊗ 1⊗ fk+1 ⊗ · · · ⊗ fn.

3. A variation of Rosenthal’s inequality using maximal functions

We will now discuss a version of Rosenthal’s inequality where the term in `p(Lp(N)) is

replaced by Lp(N ; `∞). This is in perfect analogy with Burkholder’s original inequality in

the commutative case. Our argument is based on interpolation and, unfortunately, has a

singularity for p→ 2. We need some facts on vector-valued Lp spaces with values in `nq .

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 15

Let us recall the definition of the spaces Lp(N ; `n∞) and Lp(N ; `n1 ) defined for finite

sequences (xk)1≤k≤n, n ∈ N, by

‖x‖Lp(N ;`n∞) = infxk = aykb

‖a‖2p supk‖yk‖‖b‖2p

and

‖x‖Lp(N ;`n1 ) = infxk =

P

ja∗kjbkj

‖∑kj

a∗kjakj‖1pp ‖∑kj

b∗kjbkj‖1pp .

For more information (see [Jun1]) and [JRX2] where the connection to decomposable maps

is explained). We may then define

Lp(N ; `nq ) = [Lp(N ; `n∞), Lp(N ; `n1 )] 1q.

We will write Lp(N ; `nq ) if we want to emphasize the underlying von Neumann algebra.

Some preliminary facts about these spaces are needed in the following.

Lemma 3.1. Let N be von Neumann algebra and x ∈ L∞(N ; `np ) and a, b ∈ L2p. Then

‖axb‖`np (Lp(N)) ≤ ‖a‖2p‖x‖L∞(`np )‖b‖2p .

Proof. By trilinear interpolation it suffices to check this for p = ∞ (obvious) and for p = 1.

Given

xk =∑kj

v∗jkwjk

and a, b ∈ L2(N), we deduce from Holder’s inequality thatn∑k=1

‖axkb‖L1(N) =n∑k=1

‖∑j

av∗jkwjkb‖L1(N)

≤

(n∑k=1

‖∑j

av∗jkvjka∗‖1

) 12(

n∑k=1

‖∑j

b∗w∗jkwjkb‖1

) 12

=

(n∑k=1

tr(∑j

v∗jkvjka∗a)

) 12(

n∑k=1

tr(∑j

w∗jkwjkbb∗)

) 12

≤ ‖∑kj

v∗jkvjk‖12‖a‖2‖

∑kj

w∗jkwjk‖12‖b‖2 .

Before we prove a stability result with respect to conditional expectations, we have to

recall some facts about the Stinespring dilation theorem (see also [Jun1][JRX1]). Following

[Ru] there exists a normal representation π : M → N⊗B(`2) such that

E(x) = (1⊗ e11)π(x)(1⊗ e11) .

16 M. JUNGE AND Q. XU

(In the case of non-separable predual, we have to use a larger index set). Therefore, we

may define the map u : M → N⊗C by u(x) = π(x)(1⊗ e11) which satisfies

(3.1) E(y∗x) = u(y)∗u(x) .

Following [Jun1] this map has a natural extension up : Lp(N) → Lp(M ; `c2) given by

up(xD1p ) =

∑k

(1⊗ ekk)u(x)D1p ⊗ e1k .

We clearly have

up(yD1p )∗up(xD

1p ) = D

1pE(y∗x)D

1p = E p

2((yD

1p )∗xD

1p ) .

This shows that for 2 ≤ p ≤ ∞ the map up extends to an isometry Lcp(M,E) such that

up(y)∗up(x) = E(y∗x). As usual we drop the index p.

Lemma 3.2. Let (yk)nk=1 ⊂ L1(N). Then

‖(yk)‖L1(N ;`n∞) = ‖(yk)‖L∞(N ;`n1 )∗ .

Proof. Let (yk) ⊂ L1(N) and let us assume that the corresponding functional l : L∞(N ; `n1 )

given by

l((xk)) =∑k

tr(ykxk)

has norm ≤ 1. Then, we apply the Hahn-Banach argument in [Jun1, Proposition 3.6] and

find states φ and ψ such that

|tr(ykab)| ≤ φ(a∗a)12ψ(b∗b)

12 .

holds for all 1 ≤ k ≤ n. By approximation we deduce for all a, b ∈ N∗∗ that

|φyk(ab)| ≤ φ(a∗a)

12ψ(b∗b)

12 .

where φykis the unique extension of the linear functional φyk

(n) = tr(ykn) to N∗∗. Let z

be the central projection in N∗∗ such that N = zN∗∗. Then z.φyk.z = φyk

. This yields

|φyk(ab)| ≤ φ(za∗az)

12ψ(zb∗bz)

12 .

However, z.φ (z.ψ) is the normal part of φ (ψ, respectively). This implies that there are

d1 and d2 ∈ L1(N) such that

|tr(ykab)| ≤ tr(d1a∗a)

12 tr(d2b

∗a)12 .

Now, the proof in [Jun1, Propositon 3.6] may be completed as stated and yields

‖(yk)‖L∞(N ;`n1 )∗ ≥ ‖(yk)‖L1(N ;`n∞) .

The converse inequality is elementary.

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 17

Lemma 3.3. Let M ⊂ N , φ be a faithful normal state on N and E : N → M be a

conditional expectation such that φ = φ|M E. Then

Lp(N ; `nq ) ⊂ Lp(M ; `nq )

holds isometrically and E extends to a contraction on Lp(M ; `nq ).

Proof. We first observe that by definition the inclusion maps

Lp(M ; `n∞) ⊂ Lp(N ; `n∞) and Lp(M ; `n1 ) ⊂ Lp(N ; `n1 )

are contractive. Hence by interpolation the inclusion

Lp(M ; `nq ) ⊂ Lp(N ; `nq )

is contractive. The inclusion will certainly be isometric if we can show E⊗id : Lp(N ; `nq ) →Lp(M ; `nq ) is contractive. By interpolation it suffices to shows this for q = ∞ and q = 1.

We start with the latter case and consider

xk =∑j

a∗kjbkj .

Then we deduce from (3.1) that

E(xk) =∑j

E(a∗kjbkj) =∑j

u(akj)∗u(bkj) .

Since E is a contraction Lp(N) we deduce that

‖(E(xk))‖Lp(M ;`n1 ) ≤ ‖∑kj

u(akj)∗u(akj)‖

12p ‖∑kj

u(bkj)∗u(bkj)‖

12p

= ‖∑kj

E(a∗kjakj)‖12p ‖∑kj

E(b∗kjbkj)‖12p

≤ ‖∑kj

a∗kjakj‖12p ‖∑kj

b∗kjbkj‖12p .

Taking the infimum over all decompositions for (xk) yields the assertion. By duality (see

[Jun1, Proposition 3.6]) we deduce the assertion for Lp′(N ; `n∞) in the range 1 < p′ ≤ ∞.

For p′ = ∞ we apply Lemma 3.2. Then E∗ ⊗ id is contraction on L∞(N ; `n1 )∗. The

restriction to L1(N ; `n∞) is exactly E ⊗ id.

Proposition 3.4. Let 1 ≤ p ≤ ∞. Then

Lp(N ; `np ) = `np (Lp(N))

holds isometrically.

18 M. JUNGE AND Q. XU

Proof. We will first prove that the inclusion map

(3.2) Lp(N ; `np ) ⊂ `np (Lp(N))

is contractive for finite von Neumann algebras N . Let xk = fk(1/p) where (fk)nk=1 is a

family of analytic functions in Lp(N) such that

supt‖(fk(it)‖Lp(N ;`n∞) ≤ 1− ε and sup

t‖(fk(1 + it)‖Lp(N ;`n1 ) ≤ 1− ε .

By the continuous selection theorem, we may find continuous functions a, b and y defined

on iR satisfying

fk(it) = a(it)yk(it)b(it)

such that ‖a(it)‖2p ≤ 1, supk ‖yk(it)‖ ≤ 1 and ‖b(it)‖2p ≤ 1. We note that for z = 1 + it

we may write

fk(z) =∑j

a∗jkbjk

such that

‖∑kj

a∗jkajk‖p ≤ 1 and ‖∑kj

b∗jkbjk‖p ≤ 1 .

Using a = (∑kj

a∗kjakj)12 and vkj = a−

12akj, we see that

fk(z) = agkb = a(∑kj

v∗kjwkj)b

where (gk) ∈ L∞(N ; `n1 ). Applying the continuous selection theorem again we find contin-

uous maps a, b and y on 1 + iR such that

fk(z) = a(z)yk(z)b(z) .

We apply [PX2] and obtain an analytic invertible function α, β : Ω → L2p(N) such that

α(z)α(z)∗ = a(z)a(z)∗ + δ1 and β(z)∗β(z) = b(z)∗b(z) + δ1

for all z ∈ ∂Ω. Then we may define

yk(z) = α(z)−1fk(z)β(z)−1 .

For z ∈ ∂Ω we deduce from a(z)∗a(z) ≤ α(z)∗α(z) and b(z)∗b(z) ≤ β(z)∗β(z) that there

are contractions v(z) and w(z) satisfying a(z) = α(z)v(z), b(z) = w(z)β(z). This implies

that

‖(yk(1/p))‖L∞(N ;`np ) ≤ 1 .

By the definition of α and β is also clear that

‖α(1/p)‖p ≤ (1 + δ)12 and ‖β(1/p)‖p ≤ (1 + δ)

12 .

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 19

According to Lemma 3.1 we deduce

‖(xk)‖`np (Lp(N)) ≤ ‖α(1/p)‖2p‖(yk(1/p))‖L∞(`np )‖β(1/p)‖2p ≤ (1 + δ) .

Since ε > 0 and δ > 0 are arbitrary, the assertion is proved in the finite case. For

the general σ-finite case we use Haagerup’s crossed product construction and consider

N = N oσφtG where G =

⋃n 2−nZ is the group of duadic rationals. There is a conditional

expectation F : N → N given by the identity element. Then φ = φF is a normal faithful

state. We refer for [JX3] for the fact that there is an increasing sequence Nm of φ invariant

finite von Neumann algebras such that the corresponding conditional expectations Em :

N → Nm converge to the identity in the strong operator topology. Then Em(x) converges

to x in norm for every element x ∈ Lp(N ). According to Lemma 3.3 and our previous

argument, we deduce that

(3.3)

(n∑k=1

‖FEm(xk)‖pLp(N )

) 1p

≤ ‖(xk)‖Lp(`np ) .

Thus passing to the limit shows that the inclusion (3.2) is contractive. Applying the same

argument for p′ we also have

Lp′(N ; `np′) ⊂ `np′(Lp′(N)) .

By duality

`np (Lp(N)) ⊂ Lp′(N ; `np′)∗ .

By interpolation and [Jun1, Proposition 3.6] we have for 1 < p ≤ ∞ that

Lp′(N ; `np′)∗ = [Lp′(N ; `n∞), Lp′(N ; `n1 )]∗1

p′= [Lp′(N ; `n∞)∗, Lp′(N ; `n1 )∗] 1

p′

= [Lp(N ; `n1 ), Lp(N ; `n∞)]1− 1p

= [Lp(N ; `n∞), Lp(N ; `n1 )] 1p

= Lp(N ; `np ) .

Here we have used the equality [Lp′(N ; `n∞)∗, Lp′(N ; `n1 )∗]1p′ = [Lp′(N ; `n∞)∗, Lp′(N ; `n1 )∗] 1

p′

which follows immediately from the fact that topological all these space coincide with

(Lp′(N)∗)n and Bergh’s theorem [Be]. For p = 1 the inclusion `n1 (L1) ⊂ L1(`n1 ) follows

from the fact that the space L1(`n1 ) is normed (see [Pis1] for a similar argument).

Remark 3.5. Motivated from Pisier’s theory of vector valued Lp spaces we introduce

|||(xk) |||pq = infxk=aykb

‖a‖2p‖(yk)‖L∞(N ;`nq )‖b‖2p = ‖(xk)‖Lp(N ;`nq ) .

Then we have

|||(xk) |||pq = ‖(xk)‖Lp(`nq ) .

20 M. JUNGE AND Q. XU

Proof. We refer to [Pis1] for the argument showing showing that |||(xk) |||pq is a norm. The

proof above shows that for a finite von Neumann algebra N , we have

|||(xk) |||pq ≤ ‖(xk)‖Lp(N,`nq ) .

In the general case, we deduce from (3.3) that for every γ > 0, we may find xk,γ = FEm(xk)

such that ‖xk,γ − xk‖ ≤ γ and

|||(xk,δ) |||pq ≤ (1 + ε)‖(xk)‖Lp(N ;`nq ) .

Let δ > 0. Define zk,0 = xk,δ and zk,j = xk,δ2−(j+1) − xk,δ2−j for j ≥ 1. Then we have

xk = xk,δ + xk − xk,δ =∑

j≥0 zk,j. This implies with the triangle inequality

|||(xk) |||pq = |||(∑j≥0

zk,j) |||pq = |||(xk,δ) +∑j≥1

(zk,j) |||pq ≤ |||(xk,δ) |||p,q + |||∑j≥1

(zk,j) |||pq

≤ |||(xk,δ) |||p,q +n∑k=1

‖∑j

zk,j‖Lp ≤ (1 + ε)‖(xk)‖Lp(`nq ) +∑j≥0

nδ(2−(j+1) + 2−(j))

≤ (1 + ε)‖(xk)‖Lp(`nq ) + 3nδ .

Since δ > 0 and ε > 0 we deduce

|||(xk) |||pq ≤ ‖(xk)‖Lp(`nq ) .

The converse inequality follows from the contractive inclusion

L2p(N)L∞(N ; `nq )L2p(N) ⊂ Lp(N ; `nq ) .

Indeed, this is obvious for q = 1 and q = ∞ and hence valid for all q by (trilinear)

interpolation.

Lemma 3.6. Let 1 ≤ q ≤ p, r ≤ ∞ with 1q

= 1p

+ 1r

and (xk) ⊂ Lp(N). Then

‖(xk)‖Lp(N ;`nq ) = sup‖a‖2r‖b‖2r≤1

(n∑k=1

‖axkb‖qq

) 1q

.

Proof. Given an element (xk) we apply Remark 3.5 and write

xk = a1ykb1

such that

‖a1‖2p‖(yk)‖L∞(N ;`nq )‖b1‖2p ≤ (1 + ε)‖(xk)‖Lp(N ;`nq ) .

Thus for a ∈ L2r and b ∈ L2r, we deduce from Lemma 3.1 that

‖(axkb)‖`nq (Lq(N)) = ‖(aa1ykb1b)‖`nq (Lq(N)) ≤ ‖aa1‖2q‖yk‖L∞(N ;`nq )‖b1b‖2q

≤ ‖a‖2r‖a1‖2p‖yk‖L∞(N ;`nq )‖b1‖2p‖b‖2r ≤ (1 + ε)‖(xk)‖Lp(`nq )‖a‖2r‖b‖2r .

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 21

In order to prove the converse inequality we consider an element (yk) ∈ Lp′(N ; `nq′) of norm

less than one. According to Remark 3.5 we may write

yk = azkb

such that ‖a‖2p′ ≤ 1, ‖(zk)‖L∞(N ;`nq′ )≤ 1 and ‖b‖2p′ ≤ 1. We may decompose a = a1a2,

b = b2b1 such that

max‖a1‖2q′ , ‖a2‖2r ≤ 1 and max‖b2‖2q′‖b1‖2r ≤ 1 .

According to Proposition 3.4, we deduce that

‖(a1zkb1)‖`nq′ (Lq′ (N)) ≤ 1 .

This implies with Holder’s inequality that

|n∑k=1

tr(y∗kxk)| = |n∑k=1

tr((a2zkb2)∗xk)| = |

n∑k=1

tr((z∗ka∗2xkb2)|

≤

(n∑k=1

‖a∗2xkb∗2‖qq

) 1q

‖(zk)‖`nq′ (Lq′ (N)) ≤

(n∑k=1

‖a∗2xkb∗2‖qq

) 1q

.

Therefore, we have

‖(xk)‖Lp(`nq ) = sup‖(yk)‖Lp′ (`

nq′

)≤1

|n∑k=1

tr(y∗kxk)| ≤ sup‖a2‖2r‖b2‖2r≤1

(n∑k=1

‖a2xkb2‖qq

) 1q

and the assertion is proved.

Corollary 3.7. Let 2 ≤ p ≤ ∞. Then

[Lp(N, (`n2 )c), Lp(N, (`

n2 )r)] 1

2⊂ Lp(N ; `n2 ) .

Proof. Let 12

= 1p

+ 1r. We use trilinear interpolation and consider the map

T (a, (xk), b) = (axkb) .

First, we note that

T : L∞(N)× Lp(N, (`n2 )c)× Lr(N) → `n2 (L2(N))

is a contraction becausen∑k=1

‖axkb‖22 ≤

n∑k=1

tr(b∗x∗kxkb) = tr((n∑k=1

x∗kxk)bb∗)

≤ ‖n∑k=1

x∗kxk‖ p2‖bb∗‖r = ‖(xk)‖2

Lp(N ;`c2)‖b‖22r .

22 M. JUNGE AND Q. XU

Similarly, we see that

T : Lr(N)× Lp(N, (`n2 )r)× L∞(N) → `n2 (L2(N))

is a contraction. Thus by interpolation

T : L2r(N)× [Lp(N, (`n2 )c), Lp(N, (`

n2 )r)] 1

2× L2r(N) → `n2 (L2(N)) .

Lemma 3.6 implies the assertion.

Theorem 3.8. Let 2 < p <∞, n ∈ N ∪ ∞ and (Ak)1≤k≤n be independent over M . Let

xk ∈ Lp(Ak) with EM(xk) = 0. Then

‖n∑k=1

xk‖p ≤ max(2Cp)p

p−2‖(xk)‖Lp(N ;`∞), Cp‖(xk)‖Lp(N,E;`c2), Cp‖(xk)‖Lp(N,E;`r2) .

Here C is the constant from Theorem (1.2). For p ≥ 4

‖n∑k=1

xk‖p ≤ Cpmax‖(xk)‖Lp(`∞), ‖(xk)‖Lp(N,E;`c2), ‖(xk)‖Lp(N,E;`r2)

holds for an absolute constant C.

Proof. Let us denote by C1 the constant from Theorem (1.2). Let us assume that(n∑k=1

‖xk‖pp

) 1p

< max‖(xk)‖Lp(N,E;`c2), ‖(xk)‖Lp(N,E;`r2) .

Then Theorem (1.2) implies

‖∑k

xk‖p ≤ C1p max‖(xk)‖Lp(N,E;`c2), ‖(xk)‖Lp(N,E;`r2) .

Let us consider the second case

max‖(xk)‖Lp(N,E;`c2), ‖(xk)‖Lp(N,E;`r2) ≤

(n∑k=1

‖xk‖pp

) 1p

.

Then Theorem 1.2 implies

‖n∑k=1

xk‖p ≤ C1p

(n∑k=1

‖xk‖pp

) 1p

.

Let θ such that 1p

= 1−θ∞ + θ

2. From the reiteration theorem [BL], we deduce that

Lp(N ; `np ) = [Lp(N ; `n∞), Lp(N ; `n2 )]θ .

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 23

This implies (n∑k=1

‖xk‖pp

) 1p

≤ ‖(xk)‖1−θLp(N ;`n∞)‖(xk)‖

θLp(N ;`n2 ) .

We recall from the proof of Theorem 1.2 that

max‖(n∑k=1

x∗kxk)12‖p, ‖(

n∑k=1

xkx∗k)

12‖p ≤ 2‖

n∑k=1

xk‖p .

According to Corollary 3.7 this implies

‖(xk)‖Lp(N ;`n2 ) ≤ 2‖n∑k=1

xk‖p .

Combining these estimates we find (after cancellation) that

(3.4) ‖n∑k=1

xk‖p ≤ (2θC1p)1

1−θ ‖(xk)‖Lp(`n∞) .

Since θ = 2p≤ 1 we deduce the first assertion. If moreover p ≥ 4, we have

pp

p−2 = p1+ 2p−2 ≤ p1+ 4

p ≤ e4ep

and the assertion is follows.

Remark 3.9. The same improvement is also true in the context of Burkholder’s inequality

for martingales:

‖∑k

dk‖p ≤ max(Cp2)p

p−2‖(dk)‖Lp(N ;`∞), Cp2‖x‖hc

p, Cp2‖x‖hr

p

where ‖x‖hcp

= ‖(∑

k Ek−1(d∗kdk))

1/2‖p and ‖x‖hrp

= ‖x∗‖hcp

are the ‘little square functions’.

Proof. The only difference here is that we have to use the lower estimate in the square

function inequality (see [JX2] for constant:)

‖(dk)‖Lp(`2) ≤ ‖(dk)‖12

Lp(N ;`c2)‖(dk)‖12

Lp(N ;`r2) ≤ cp‖∑k

dk‖p .

We may assume again that

max‖(∑

k

Ek−1(d∗kdk)

) 12‖p, ‖

(∑k

Ek−1(dkd∗k)) 1

2‖p ≤

(∑k

‖dk‖pp

) 1p

.

24 M. JUNGE AND Q. XU

In this situation (3.4) has to be replaced by

‖n∑k=1

dk‖p ≤ ((cp)θCp2)1

1−θ ‖(xk)‖Lp(`n∞) ≤ (Cce2/ep2)p

p−2‖(xk)‖Lp(`n∞) .

4. Rosenthal’s inequality for 1 < p ≤ 2

We will now investigate the dual version of Rosenthal’s inequality for 1 < p ≤ 2. In the

following we will assume as before that M ⊂ N is a subalgebra, invariant under modular

group σφt of a normal faithful state φ.

Lemma 4.1. Let 2 ≤ p < ∞ and 2 ≤ r ≤ ∞ such that 12

= 1p

+ 1r. The dual space of

Lp(N,EM , `c2) consists of all sequences (xk) ⊂ Lp′(M) such that there is an a ∈ Lr(M)

and (yk) ⊂ `2(L2(M)) and

xk = yka .

Moreover,

‖(xk)‖Lp(N,EM ,`c2)∗ = inf

xk=yka

(∞∑k=1

‖yk‖22

) 12

‖a‖r .

Let D be the density of φ. The space of finite sequences (zkD1p′ ) is dense in Lp(N,EM , `c2)

∗

and for those elements

‖(zkD1p′ )‖Lp(N,EM ,`c2)

∗ = ‖∑k

D1p′EM(z∗kzk)D

1p′ ‖

12p′2

.

Proof. According to Remark (1.4) we may assume that M is σ-finite and even that N has

separable dual. Let us recall the map up : MD1p → Lp(M, `c2) constructed in (3.1) which

satisfies

up(xD1p )∗u(yD

1p ) = D

1pEM(x∗y)D

1p .

This means that u : Lp(N,EM) → Lp(N, `c2) is an isometric isomorphism when restricted

to MD1p . For 2 ≤ p <∞ this space is dense in Lp(M). We consider M⊗C as a M -valued

Hilbert C∗-module with the M -valued scalar product 〈(xk), (yk)〉 =∑

k x∗kyk. The map

constructed in [Jun1] using Kasparov’s dilation technique (see [Lan]) has the additional

advantage that there exists a right M -module map P : M⊗C →M⊗C with P 2 = P such

that u(N) is strongly dense in P (M⊗C) and

(4.1) 〈P ((xk)), (xk)〉 = 〈(xk), P (xk)〉 = 〈P (xk), P (xk)〉 ≤ 〈(xk), (xk)〉

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 25

Indeed, P is a the projection onto a submodule (see [Lan] for more details, see also [JS] for

a general treatment without assuming N∗ separable based on [Pas]). Thus for 2 ≤ p <∞the map up extends to an isometric isomorphism and P extends to a contractive projection

from Lp(M, `c2) to the image up. In [Jun1] we defined Lp(N,EM) as the closure of ND1p

with respect to the norm ‖D1pE(x∗x)D

1p‖1/2

p/2 for all 1 ≤ p ≤ ∞. We may also consider

up ⊗ id : Lp(M,EM ; `c2) → Lp(M, `c2(N2)) and see that

(up ⊗ id)((xj))∗(up ⊗ id)((xj)) =

∑j

up(xj)∗up(xj) =

∑j

Ep(x∗jxj) .

As above P ⊗ id : Lp(M, `c2(N2)) → Lp(M, `2(N2)) extends to a projection, because we

have

(P ⊗ id)((xjk))∗(P ⊗ id)((xjk)) =

∑j

P (xj,k)∗P (xj,k) ≤

∑j,k

x∗jkxjk .

We may hence define Lp(M,EM ; `c2) as the closure of the space of finite sequence (wjD1p )

with respect to the norm ‖∑

j D1pE(w∗jwj)D

1p‖1/2

p/2 for all 1 ≤ p < ∞. By density this is

consistent with the the definition given above for p ≥ 2. Then up⊗ id defines a isometric

isomorphism onto a complemented subspace of Lp(M, `2(N2)) for all values 1 ≤ p ≤ ∞.

Let 1 ≤ p <∞. Then we have Lp(M, `c2(N2))∗ = Lp′(M, `c2(N2). Therefore every functional

ψ : Lp(M,EM ; `c2) → C is given by

ψy((xj)) =∑jk

tr(y∗jkup(xj)k)

and ‖∑

jk y∗jkyjk‖

1/2p′/2 = ‖ψ‖. However, finite sequences of the form yjk = zjkD

1p′ are dense

in Lp′(M, `c2(N2)). For such an element we find (using (4.1))

ψy((vjD1p )) =

∑j

∑k

tr(D1p′ z∗jku(xj)kD

1p ) =

∑j

∑k

φ(z∗jkP (u(vj))k)

=∑j

φ(〈(zjk)k, Pu(vj)〉) =∑j

φ(〈P (zjk), u(vj)〉) .

This means that P ((zjk)k) is in the range of u and we find an element wj ∈M such that

ψy((vjD1p )) =

∑j

φ(〈P ((zj,k)k), u(vj)〉) =∑j

φ(u(wj)∗u(yj))

=∑j

φ(E(w∗jyj)) =∑j

tr(D1p′w∗jyjD

1p ) .

Recall that only finitely many wj’s are non-zero and hence ψy has a unique extension as

a linear functional (see also [Jun1, Proposition 2.15]). This shows that Lp(N,EM , `c2)∗

is

26 M. JUNGE AND Q. XU

the norm closure of finite sequences (wjD1/p′) and the norm of such a sequence satisfies

‖(wjD1p′ )‖2

Lp′ (N,EM ,`c2) = ‖∑j

D1p′E(w∗jwj)D

1p′ ‖p′/2 = ‖

∑j

D1p′ 〈P (zj), P (zj)〉D

1p′ ‖p′/2

≤ ‖∑j

D1p′∑k

z∗j,kzj,kD1p′ ‖p′/2 = ‖(yj,k)‖2

Lp′ (M,`2(N2)) .

Thus we have identified the dual of Lp(N,EM , `c2) as the closure of finite sequences (wjD

1p′ )

in Lp′(N,EM , `c2) with respect to the antilinear duality given by the trace. Let 1 ≤ p′ ≤

2 ≤ p and (wj) ⊂ N be a finite sequence. We define

a =n∑j=1

D1p′EM(w∗jwj)D

1p′ .

We may assume that tr(ap′/2) = 1. Let δ > 0 and define d = a + δD2/p′ . Note that

‖d‖p′/2p′/2 ≤ 1 + δ. Since D2/p′ ≤ δ−1d, we find v ∈ M such that D1/p′ = vd1/2. Then, we

define

dj = wjD1p′ d−

p′2r = wjvd

12− p′

2r = wjvdp′4 .

Note that wjD1/p′ = djd

p′/2r and ‖dp′/2r‖rr = ‖dp′/2‖p′/2p′/2 ≤ (1 + δ). On the other hand

we have

d12v∗∑j

EM(w∗jwj)vd12 = D

1p′∑j

EM(w∗jwj)D1p′ ≤ d .

Since d has full support we deduce∑

j v∗EM(w∗jwj)v ≤ 1 and hence

dp′4 v∗

∑j

EM(w∗jwj)vdp′4 ≤ d

p′2 .

Therefore, we have∑j

‖dj‖22 =

∑j

tr(EM(d∗jdj)) = tr(∑j

dp′4 v∗EM(w∗jwj)vd

p′4 ) ≤ tr(d

p′2 ) ≤ 1 + δ .

This yields

‖(wjD1p′ )‖`2(L2(M))Lr(N) ≤ (1 + δ)

1p′ ‖∑j

D1p′EM(w∗jwj)D

1p′ ‖

12p′2

.

This means that on a dense subset of Lp′(N,EM , `c2) we have

|||(xj) ||| = infxj=yja

(∑k

‖yj‖22

) 12

‖a‖r ≤ ‖(xj)‖Lp′ (N,EM ,`c2) .

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 27

Following the arguments in [Pis1] it is easily checked that the space `2(L2(M))Lr(N)

equipped with this homogenous expression ||| ||| is indeed a normed space. By density we

obtain the contractive inclusion

Lp′(N,EM , `c2) ⊂ `2(L2(M))Lr(N) .

For the converse we consider tj = djd such that d ∈ L2r(N) and (dj) ⊂ `2(L2(M)). Let

(xj) ⊂ Lp(N,EM ; `c2). Then, we deduce from the Cauchy-Schwarz inequality (see [Jun1,

Propostion 2.15]) that

|∑j

tr(x∗j tj)| = |∑j

tr(x∗jdjd)| = |tr(∑j

EM(x∗jdjd))|

≤ ‖∑j

EM(x∗jyj)‖r′‖d‖r ≤ ‖(∑j

EM(x∗jxj))12‖p‖(

∑j

EM(d∗jdj))12‖2‖d‖r

= ‖(xj)‖Lp(N,EM ;`c2) ‖(dj)‖`2(L2(M))‖d‖r .

In the following we will keep the notation Lp(N,EM ; `c2) as the closure of finite elements

(wjD1p ) with respect to the norm

‖(wkD1p )‖Lp(N,EM ,`c2) = ‖

∑j

D1pEM(w∗kwk)D

1p‖1/2

p/2

for all 1 ≤ p ≤ ∞. The row version Lp(N,EM ; `r2) is given by

‖(xk)‖Lp(N,EM ;`r2) = ‖(x∗k)‖Lp(N,EM ;`c2) .

We will also use the free probability notation Lp(A) for elements in Lp(A) with EM(x) = 0.

In the following we assume that M ⊂ Ak ⊂ N are von Neumann subalgebras with a

conditional expectation EAk: N → Ak satisfying EMEAk

= EAkEM = EM .

Lemma 4.2. Let 1 ≤ p ≤ ∞ and Xp(c) ⊂ Lp(N,EM , `c2) the closure of finite sequences

(xkD1p ) with xk ∈ A. Then Xp(c) is 2-complemented in Lp(N,EM , `

c2).

Proof. Let us consider a finite sequence (xkD1p ). Then we deduce from Kadison inequality

that

‖∑k

D1pEM(EAk

(xk)∗EAk

(xk))D1p‖p/2 ≤ ‖

∑k

D1pEM(EAk

(x∗kxk))D1p‖p/2

≤ ‖∑k

D1pEM(EAk

(x∗kxk))D1p‖p/2 = ‖

∑k

D1pEM(x∗kxk)D

1p‖p/2 .

This shows that the map F ((yk)) = (EAk(yk)) extends to a contraction on Lp(M,EM , `

c2).

The same argument shows that EM((yk)) = (EM(yk)) extends to a contraction. The

projection onto Xp(c) is given by (id− E)F and has norm ≤ 2.

28 M. JUNGE AND Q. XU

Theorem 4.3. Let 1 < p ≤ 2, 2 ≤ s < ∞ with 1p

= 12

+ 1s. Let (Ak) be σφt -invariant

algebras, faithfully independent over M . Let xk ∈ Lp(Ak). Then

1

2‖∑k

xk‖p ≤ infxk = yk(c)a+byk(r)+xk(d)

+

(∑k

‖xk(d)‖pp

) 1p

+

(∑k

‖xk(c)‖22

) 12

‖a‖s + ‖b‖s

(∑k

‖xk(r)‖22

) 12

≤ Cp′‖∑k

xk‖p

where the infimum is taken over yk(c), yk(r) ∈ L2(Ak), a, b ∈ Ls(M) and xk(d) ∈ Lp(Ak).

Proof. Let 1 ≤ p ≤ 2 and (xk) ∈ Lp(Ak). By Lemma 1.1 and the fact that Lp(N) has type

p, we have

‖∑k

xk‖p ≤ 2 E‖∑k

εkxk‖p ≤ 2

(n∑k=1

‖xk‖pp

) 1p

.

We denote by Xp(d) ⊂ `p(Lp) the subspace of sequences (xk) with xk ∈ Lp(Ak). For

xk = zkD1/p′ with EM(zk) = 0 and x =

∑k zkD

1/p, we deduce from [Jun1, section 2](see

also [JX1, section 7]) that

‖x‖2p = ‖x∗x‖ p

2≤ ‖EM(x∗x)‖ p

2= ‖

n∑k,j=1

D1pEM(z∗kzj)D

1p‖ p

2= ‖

n∑k=1

D1pEM(z∗kzk)D

1p‖ p

2.

By density this implies that

‖∑k

xk‖p ≤ ‖(xk)‖Lp(N,EM ;`c2) .

whenever (xk) ∈ Xp(c). We define Xp(r) = (x∗k) : (xk) ∈ Xp(c) with the canonical norm.

By convexity we find

‖∑k

xk‖p ≤ infxk=xk(d)+xk(c)+xk(r)

‖(xk(d))‖Xp(d) + ‖(xk(c))‖Xp(c) + ‖(xk(r))‖Xp(r) .

According to (the very last and easy part) in the proof of Lemma 4.1, we obtain the first

estimate. For the converse we consider x =∑

k xk ∈ Lp(N) such that xk ∈ Lp(Ak). We

define the functional Φ : Xp′(d) ∩Xp′(c) ∩Xp′(r) → C by

Φ(yk) =∑k

tr(x∗kyk) .

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 29

The symbol ∩ refers to the fact that the norm in this space is given by the maximum of

the three expressions. Theorem (1.2) implies that

|Φ(yk)| = |∑k

tr(y∗kxk)| = |∑kj

tr(y∗kxj)|

≤ ‖∑k

xk‖ ‖∑j

yj‖ ≤ Cp′ ‖∑k

xk‖ ‖(yk)‖Xp′ (d)∩Xp′ (c)∩Xp′ (r).

Here we used the orthogonality tr(y∗kxj) = tr(Fj(y∗kxj)) = tr(EM(yk)

∗xj) = 0. By the

Hahn-Banach theorem, we may extend Φ to a functional Φ on the space

Xp(d)⊕∞ Xp(c)⊕∞ Xp(r)

with the same norm. With the help of the complementation from Lemma 4.2 and Lemma

4.1, we deduce that Xp′(c)∗ = Xp(c), Xp′(r)

∗ = Xp(r) and Xp′(d)∗ = Xp(d) holds with

constant 2. Therefore Φ has three components (xk(c)) ∈ Xp′(c), (xk(r)) ∈ Xp′(r) (xk(d)) ∈Xp′(d) such that

‖(xk(d))‖Xp(d) + ‖(xk(c))‖Xp(c) + ‖(xk(r))‖Xp(r) ≤ 2‖Φ‖ ≤ 2Cp′‖∑k

xk‖p .

Moreover, since Φ extends Φ, we know that

(4.2) tr(x∗kyk) = Φ((0, 0, · · · , 0, yk, 0, · · · )) = tr(xk(c)∗yk)+ tr(xk(r)

∗yk)+ tr(xk(d)∗yk)

holds for all yk ∈ Lp(Ak). Since EM(xk) = 0 we deduce for all yk ∈ Lp(Ak) that

tr(x∗kyk) = tr(x∗k(yk − EM(yk)) + tr(x∗kEM(yk))

= tr(x∗k(yk − EM(yk)) + tr(EM(xk)∗EM(yk)) = tr(x∗k(yk − EM(yk))) .

Therefore (4.2) implies indeed xk = xk(d) + xk(c) + xk(r). Finally, we have to apply the

second part of Lemma 4.3 and observe that we find xk(c) = yk(c)a where ar ∈ L1(M) is

the density of a faithful state. Then we find yk(c)a = EAk(xk(c)) = EAk

(yk(c))a and hence

yk(c) = EAk(yk(c)). The same argument shows that 0 = EM(xk(c)) = EM(yk(c))a and

hence EM(yk(c)) = 0.

Remark 4.4. The maximal inequality of section 3 also implies an improvement for the

lower estimate. Under the assumptions above, we have

infxk=xk(c)+xk(r)+xk(d)

‖(xk(d))‖Lp(`1) + ‖(xk(c))‖Lp(N,EM ,`c2) + ‖(xk(c))‖Lp(N,EM ,`c2)

≤ (Cp′)p

2−p ‖∑k

xk‖p .

30 M. JUNGE AND Q. XU

Indeed, the modified spaces Xp′(d) ⊂ Lp′(`∞) given by sequences (xk) with xk ∈ Lp′(Ak)

is complemented to due Doob’s inequality. Then the same Hahn-Banach argument and

Theorem 3.8 provide the decomposition.

Proof of Theorem 0.1 for 1 < p ≤ 2. We can not formally apply Theorem (4.3). However,

we note that the right hand side of Corollary (1.5) defines space Xp(c), Xp(r), Xp(d)

which are complemented in `p(Lp(N,EM ; `c2)), `p(Lp(N,EM ; `r2)) and `p(N2;Lp(N)) (re-

specting the correct switch of the indices). The Hahn-Banach theorem then provides the

corresponding decomposition in combination with Lemma 4.1.

Due to the recent work of Pisier/Shlyahktenko and [Jun2, Jun4], it is clear conditional

expectations with respect states are very important for the theory of operators spaces and

closely connected to the classical Araki-Woods factors. Let us describe an application in

this direction. We consider the generators of the CAR algebra

ak = 1⊗ · · · ⊗ 1⊗ e12︸︷︷︸k-th position

⊗1⊗ · · · ⊗ 1 .

Let (µk) ⊂ (0, 1) and φµk= (1− µk)e11 + µke22. Here eij ∈ M2 are the standard matrix

units. Then the tensor product state φ = ⊗k∈Nφµkis a quasi-free state satisfying

φµ(a∗i1 · · · a∗iraj1 · · · ⊗ ajs) = δrs

∏s

l=1δil,jlµil

for all increasing sequences i1 < ... < ir and j1 < ... < jr. We denote by N (µ) the von

Neumann algebra generated by the ak’s in the GNS contruction with respect to φµ.

Theorem 4.5. Let Dµ be the density of φµ in L1(N (µ)). Let 1 < p ≤ 2, N be a von

Neumann algebra and xk ∈ Lp(N ). Then

‖∑k

xk ⊗D12pµ akD

12pµ ‖p ∼cp′

infxk=ck+dk

‖(∑k

(1− µk)1pµ

1p′

k c∗kck)

12‖p + ‖(

∑k

(1− µk)1p′ µ

1p

k dkd∗k)

12‖p

Here the infimum is taken over all (ck), (dk) ⊂ Lp(N ).

Proof. We define q = p′ ≥ 2. The natural conditional expectation E : N ⊗N (µ) → N ⊗ 1

given by E(x ⊗ y) = φµ(y)(x ⊗ 1). We may assume that N is σ-finite. We define the

algebras

Ak = 1⊗ · · · ⊗ 1⊗ M2︸︷︷︸k-th position

⊗ 1⊗ · · ·

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 31

and Ak = N ⊗ Ak. It is obvious that the Ak’s are (faithfully) independent over N ⊗ 1.

For q ≥ 2, we deduce from Theorem 1.2 that

‖∑k

y∗k ⊗D12qµ a

∗kD

12qµ ‖q ≤ Cqmax

(∑k

‖y∗k ⊗D12qµ a

∗kD

12qµ ‖qq

) 1q

,

‖∑k

yky∗k ⊗ E(D

12qµ akD

12qµ D

12qµ a

∗kD

12qµ )‖

q2q2, ‖∑k

y∗kyk ⊗ E(D12qµ a

∗kD

12qµ D

12qµ akD

12qµ )‖

q2q2

.

By approximation we may assume that only a finite number of the yk’s is non-zero. Thus

we may consider Dµ = ⊗mk=1((1− µk)e11 + µke22) ∈ S2m

1 = (S21)⊗m . This implies

‖D12qµ a

∗kD

12qµ ‖q = ‖((1− µk)µk)

12q e21‖q = ((1− µk)µk)

12q .

It is also easy to calculate the conditional expectation

E(D12qµ a

∗kD

12qµ D

12qµ akD

12qµ ) = (1− µk)

1qµ

− 1q

k D1qµE(a∗kak)D

1qµ = (1− µk)

1qµ

1q′

k D2qµ .

Similarly, we obtain

E(D12qµ akD

12qµ D

12qµ a

∗kD

12qµ ) = (1− µk)

1q′ µ

1qD

2qµ .

Using(∑

k ‖zk‖qq) 1

q ≤ ‖(∑

k z∗kzk)

1/2‖q, we may split the `q term according to µk ≥ 12

and

1− µk ≥ 12

and obtain(∑k

‖(1− µk)1/2qµ

1/2qk yk‖qq

) 1q

≤ 21

2q′−12q(‖(∑

k

(1− µk)1q′ µ

1q

k yky∗k

) 12‖q + ‖

(∑k

(1− µk)1qµ

1q′

k y∗kyk) 1

2‖q).

Therefore the Rosenthal inequality in this case simplifies to

‖∑k

y∗k ⊗D12qµ a

∗kD

12qµ ‖q

≤ 4Cqmax‖(∑

k

(1− µk)1q′ µ

1q

k yky∗k

) 12‖q, ‖

(∑k

(1− µk)1qµ

1q′

k y∗kyk) 1

2‖q.

Now, we observe that

tr(D1

2q′µ a∗jD

12q′µ D

12qµ akD

12qµ ) = δk,j(1− µk)

12µ

− 12

k φµ(a∗kak) = δkj√

(1− µk)µk .

Given xk ∈ Lp(N ) we deduce∣∣∣∣∣∑k

trN (y∗kxk)√

(1− µk)µk

∣∣∣∣∣ =

∣∣∣∣∣trN⊗N (µ)

((∑j

y∗j ⊗D12qµ a

∗jD

12qµ )(

∑k

xk ⊗D12pµ akD

12pµ )

)∣∣∣∣∣

32 M. JUNGE AND Q. XU

≤ ‖∑j

y∗j ⊗D12qµ a

∗jD

12qµ ‖q ‖

∑k

xk ⊗D12pµ akD

12pµ ‖p ≤ 4Cq‖

∑k

xk ⊗D12pµ akD

12pµ ‖p

max‖(∑k

(1− µk)1pµ

1q

k yky∗k)

12‖q, ‖(

∑k

(1− µk)1qµ

1p

k y∗kyk)

12‖q .

By duality this implies the decomposition. The converse inequality follows as in Theorem

4.3.

Remark 4.6. (1) According to [Jun4] this inequality also holds for p = 1. We conjec-

ture that there is is continuous passage for p→ 1 improving the constant above.

(2) The same proof works for q-Araki Wood factors defined by Shlyahktenko for q = 0

and for q ∈ (−1, 1) by Hiai [Hia]. For q = 1 we refer to [Jun4] for the appropriate

gaussian substitute.

(3) Using this Khintchine type inequality 4.5, we can deduce, as in [Jun4], that every

quotient of Rp ⊕ Cp completely embeds into Lp(RIII1) the Lp-spaces associated

to the hyperfinite III1 factor. In particular, the operator space OH embeds into

Lp(N ) where N is a hyperfinite factor of type IIIλ, 0 < λ ≤ 1. We refer to [Xu]

for more details.

Remark 4.7. There is also a version of the Rosenthal inequality for the non-faithful

case in the range 1 < p ≤ 2. We assume that the algebras (Ak) are independent over

EM : N → M , the support of E is e and φ is faithful on eNe. The main technical

difference is that we have to introduce two extra spaces

Xp(se) = ∑k

skxke : xk ∈ Lp(Ak) and Xp(es) = ∑k

exks : xk ∈ Lp(Ak)

It is easily shown that that Xp(se) and Xp(es) are complemented in

Xp = ∑k

xk : xk ∈ Lp(Ak) .

Thanks to Lemma 2.1, we are able to describe the (anti-linear) dual Xp′(se) of Xp(se) as

an intersection of two terms, an `p′-term and a column square function. Using the duality

argument from Theorem 4.3 this yields

‖∑k

skxke‖p ≤ infskxke=skxk(c)e+skxk(d)e

‖(∑k

EM(xk(c)∗skxk(c)))

12‖p +

(∑k

‖skxk(d)e‖pp

) 1p

≤ C√p′‖∑k

skxke‖p .

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 33

A similar result holds for Xp(es) and Theorem 4.3 applies to the faithful part: Let xk ∈Lp(Ak) and yk = xk − EM(xk). First we have

‖∑k

xk‖p ∼5

max

‖∑k

EM(xk)‖p,

(∑k

‖skxksk‖pp

) 1p

, ‖∑k

eyke‖p, ‖∑k

skxke‖p, ‖∑k

exksk‖p.

According to Theorem (4.3) we have that

‖∑k

eyke‖p ∼Cp′ infyk = yk(c)+yk(r)+yk(d)

‖(eyk(c)e)‖Lp(N,EM ,`c2) + ‖(eyk(r)e)‖Lp(N,EM ,`r2) + ‖(eyk(d)e)‖`p(Lp)) .

Here the infimum is taken over yk(c), yk(r), yk(d) ∈ Lp(Ak). This complicated expression

involving maximum and infimum is particulary interesting in the connection with inde-

pendent copies (as in [Jun4]). In this case, the expressions are symmetric. This formula

can be used to prove that subsymmetric sequences in Lp(N), 1 < p ≤ 2 are symmetric

(see [JR] for more details).

5. Symmetric subspaces of Lp

In this section, we will prove a quantitative version of the Kadec-Pe lczynski alternative,

the noncommutative analogue of the corresponding result in [JMST]. It will be convenient

to state these result in the Banach space and operator space setting. We refer to [Pis2] for

a general background on operator spaces and completely bounded maps and to [JNRX]

for the operator space structure of noncommutative Lp spaces. In this paper we focus on

subspaces X, Y of Lp(N). In this situation the cb-norm of a linear map T : X → Y is

given by

‖T‖cb = ‖idLp(B(`2)) ⊗ T : Lp(B(`2);X) → Lp(B(`2);Y )‖ .

Here Lp(B(`2);X) ⊂ Lp(B(`2)⊗N) consist of the matrices with coefficients in X. In other

words, the cb-norm is calculated with matrix valued coefficients instead of scalar valued

coefficients. Note that martingale inequalities often automatically extend to the matrix

valued case.

Lemma 5.1. Let N be a hyperfinite type IIIλ factor where 0 ≤ λ ≤ 1. Let 1 < p < ∞,

then Lp(N) has a completely unconditional finite dimensional decomposition (see below for

a definition).

34 M. JUNGE AND Q. XU

Proof. In the range 0 < λ ≤ 1, we may assume that N is an ITPFI factor. In general

(including λ = 0), we can always find a normal faithful state φ, and an increasing sequence

of finite subalgebras Nn with φ-invariant conditional expectation En : N → Nn see [JRX1].

We define the difference operators dn = En − En−1 where E0 = 0. Note that the spaces

Fn = dn(N) are finite dimensional and every element can be written uniquely as x =∑n dn(x). Thus Lp(N) has a finite dimensional decomposition. Such a decomposition is

called completely unconditional if all the maps Tε(∑

n dn) =∑

n εndn with εn = ±1 are

uniformly completely bounded. This means that the maps idLp(B(`2)) ⊗ Tε are uniformly

bounded, i.e. there exists a constant c(p) such that

‖∑n

εn(id⊗ dn)(x)‖p ≤ c(p)‖∑n

(id⊗ dn)(x)‖p .(5.1)

holds for all choices of signs (εn) and x ∈ Lp(B(`2) ⊗ N). Equation (5.1) is a direct

consequence of the Burkholder-Gundy inequalities [PX1, JX1].

Theorem 5.2. Let N be a hyperfinite von Neumann algebra. Let 2 < p <∞ and (xk) ⊂Lp(N) be a sequence of norm one vectors, which converges weakly to 0. Then there exist

constants 0 ≤ c1, c2 ≤ 1, a subsequence (xn) of (xn) such that

‖∑n

an ⊗ xn‖p ∼c(p)

(∑n

‖an‖pp

) 1p

+ c1‖(∑n

a∗nan)12‖p + c2‖(

∑n

ana∗n)

12‖p

holds for all finitely supported sequence (an) ⊂ Sp.

Proof. Using Remark 1.4 we may assume that N∗ is separable. We will first use a standard

trick in order to ensure that we may work with a factor. Indeed, let φ be a normal faithful

state. We consider the crossed product M = ⊗n∈N(N, φ) o G between the infinite tensor

product ⊗n∈N(N, φ) and the discrete group G of all finite permutations on N. Any finite

permutation acts on the infinite tensor product by shuffling the corresponding coordinates.

Clearly, we also have a conditional expectation E0 : M → N obtained by first projecting

onto the identity element e in G and then to the first component in the infinite tensor

product. According to [HW, Proof of Theorem 2.6] we know that M is a hyperfinite factor.

According to Connes’ characterization M is type IIIλ for some 0 ≤ λ ≤ 1 [Con]. According

to Lemma 5.1, we have a normal faithful state, conditional expectations Ek : N → Nk onto

finite dimensional subalgebras Nk. Now, the proof follows very closely its commutative

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 35

model see [JMST, Theorem 1.14,p=50]. Using the gliding hump procedure, we may find

a perturbation of a subsequence (xn) and a subsequence Ek such that

i) En(xn) = xn,

ii) En(xk) = 0 for all k > n,

iii) limk En(x∗kxk) = yn and ‖En(x∗kxk)− yn‖ p2≤ ε2−k for k > n,

iv) limk En(xkx∗k) = zn and ‖En(xkx

∗k)− zn‖ p

2≤ ε2−k for k > n.

Here ε > 0 is arbitrary and will be chosen after knowing the yn’s. It follows immediately

from iii) that (yn) is a martingale, namely En(yn+1) = yn. Since 1 < p2, we deduce from

the Burkholder-Gundy inequalities [JX1] that (yn) is convergent to some y ∈ Lp(M).

Similarly, we obtain that (zn) is convergent to some z ∈ Lp(N). We define c1 = ‖y‖1/2p/2

and c2 = ‖z‖1/2p/2. Passing to another subsequence denoted by (xn) (En), we may assume

‖En(x∗n+1xn+1)− y‖ p2≤ 2−(n+2)‖y‖ p

2and ‖En(xn+1x

∗n+1)− z‖ p

2≤ 2−(n+2)‖z‖ p

2.

We apply Burkholder’s inequality [JX1] and find

‖∑n

an ⊗ xn‖p ∼c(p)

(∑n

‖an ⊗ xn‖pp

) 1p

+ ‖∑n

a∗nan ⊗ En(x∗n+1xn+1)‖12p2

+ ‖∑n

a∗nan ⊗ En(xn+1x∗n+1)‖

12p2.

From perturbation we have 12≤ ‖xn‖ ≤ 2. The triangle inequality implies

‖∑n

a∗nan ⊗ En(x∗n+1xn+1)−∑n

a∗nan ⊗ y‖ p2≤∑n

‖a∗nan‖‖En(x∗n+1xn+1)− y‖ p2

≤ 1

2‖y‖ p

2supn‖a∗nan‖ p

2≤ 1

2c21‖∑n

a∗nan‖ p2.

Therefore, we get

c21‖∑n

a∗nan‖ p2

= ‖∑n

a∗nan ⊗ y‖ p2

≤ ‖∑n

a∗nan ⊗ En(x∗n+1xn+1)‖+ ‖∑n

a∗nan ⊗ En(x∗n+1xn+1)−∑n

a∗nan ⊗ y‖ p2

≤ 3

2c21‖∑n

a∗nan‖ .

The same argument applies to the last term and the assertion follows.

36 M. JUNGE AND Q. XU

Let us recall some notation from the theory of operator spaces. The spaces Cp and Rp

are defined as subspaces of Sp given by

Cp = spanek1 : k ∈ N and Rp = spane1k : k ∈ N .

As an application, we obtain an operator space version of the Kadec-Pe lzsinski alternative.

Corollary 5.3. Let N be hyperfinite. Let 2 ≤ p < ∞ and (xn) be a sequence which con-

verges to 0 weakly. Then (xn) contains a subsequence (x′n) which is completely equivalent

to `p, Rp, Cp or Rp ∩ Cp.

Proof. Let (x′n) the subsequence from Theorem 5.2. If c1 = c2 = 0, then (x′n) is completely

equivalent to the unit vector basis of `p. If c1 = 0 and c2 > 0, then we find a copy of Rp.

Similarly, if c2 = 0 and c1 > 0 it turns out to be Cp. The case c1 > 0 and c2 > 0 yields

Rp ∩ Cp.

Remark 5.4. We deduce in particular that every infinite dimensional subspace X ⊂Lp(N) contains a completely symmetric subspace, i.e. a basic sequence (xk) such that

(5.2) ‖∑k

εkaπ(k) ⊗ xk‖ ≤ C‖∑k

ak ⊗ ek‖

holds for all ak ∈ Lp(M), εk = ±1 and permutations π of the integers. This problem (even

for scalar coefficients) is open for 1 ≤ p < 2. The problem is also open for 2 < p < ∞without assuming that N is hyperfinite. On the Banach space level we refer to [RX] and

[Ran] for different versions of the Kadec-Pe lczinski alternative.

We will now show that conversely the only symmetric subspaces of Lp(N) are the one

found in (5.3). The next result is a our starting point.

Theorem 5.5. Let 2 ≤ p <∞, N a von Neumann algebra and xij ∈ Lp(N). Then(E‖

n∑i=1

εixiπ(i)‖pp

) 1p

∼cp

(1

n

n∑i,j=1

‖xij‖pp

) 1p

+

∥∥∥∥∥∥(

1

n

n∑i,j=1

(x∗ijxij + xijx∗ij)

) 12

∥∥∥∥∥∥p

.

Here the expectation is taken over all choices of sign εi = ±1 and all permutation π on

1, .., n.

Proof. By approximation, we may assume that N is σ-finite and ψ is normal faithful state.

We consider Ω = −1, 1n×Πn, where Πn is the set of all permutations on 1, ..., n. The

Haar measure on this group is the product measure µ = ε⊗ ν of the normalized counting

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 37

measures ε and ν on −1, 1n, Πn, respectively. The underlying von Neumann algebra is

then given by N = L∞(Ω, 2Ω, µ)⊗N with respect to the state

φ(x) =

∫Ω

ψ(x(ω)) dµ(ω) .

In order to apply Burkholder’s inequality we have to use the right filtrations which we have

taken from [JMST]. For k = 1, ..., n we consider the functions fk : Πn → R, fk(π) = π(k).

The σ-algebra Σ2k is defined as the smallest σ-algebra on Πn making f1, ..., fk measurable.

By Σ1k we denote the smallest σ-algebra on −1, 1n making ε1, ..., εk measurable. Let Σk

be the σ-algebra generated by Σ1k × Σ2

k. We find the martingale filtration (Nk)k where

Nk = L∞(Ω,Σk, µ)⊗N .

We denote by Ek the conditional expectation on L∞(Ω,Σk, µ). Then Ek = Ek ⊗ id is a

φ-invariant conditional expectation onto Nk. After these preliminaries we consider

X =n∑i=1

εixiπ(i) ∈ Lp(N ) .

We assume in addition xkj = 0 for k > n2. The upper estimate for general matrices (xij)

then follows from the triangle inequality. We note that dk = εkxkπ(k) lies in Nk and

Ek−1(εkxkπ(k)) = E(εk)Ek−1(xkπ(k)) = 0 .

Therefore, Burkholder’s inequality [JX1] implies

‖X‖p ≤ cp

(n∑k=1

‖xkπ(k)‖pp

) 1p

+ ‖n∑k=1

Ek−1(x∗kπ(k)xkπ(k) + xkπ(k)x

∗kπ(k))‖

12p2.

Clearly, for all k = 1, .., n, we have

‖εkxkπ(k)‖pp =n∑j=1

Prob(π(k) = j)‖xkj‖pp =n∑j=1

(n− 1)!

n!‖xkj‖pp =

1

n

n∑j=1

‖xkj‖pp .

Hence, we get (n∑k=1

‖xkπ(k)‖pp

) 1p

=

(1

n

n∑kj=1

‖xkj‖pp

) 1p

.

Let E2k−1 be the conditional expectation onto L∞(Πn,Σ

2k, µ). We observe that

Ek−1(x∗kπ(k)xkπ(k)) = (E2

k−1 ⊗ id)(x∗kπ(k)xkπ(k)) .

The atoms in Σ2k−1 are indexed by a tuples (i1, ..., ik−1) such that the ij’s are mutually

different numbers in 1, .., n. More precisely,

A(i1,...,ik−1) = π |π(1) = i1, · · ·π(k − 1) = ik−1 .

38 M. JUNGE AND Q. XU

Clearly, the cardinality of A(i1,...,ik−1) is the cardinality of Πn−(k−1), i.e. (n − k + 1)!.

Therefore letting αk = (n−k+1)!n!

, we get

(E2k−1 ⊗ id)(x∗kπ(k)xkπ(k)) =

∑(i1,...,ik)

1A(i1,...,ik−1)(π)α−1

k

∫A(i1,...,ik−1)

x∗kπ(k)xkπ(k) dν(π) .

For fixed (i1, .., ik−1), we consider B = i1, ..., ik−1 and deduce for k ≤ n2

α−1k

∫A(i1,...,ik−1)

x∗kπ(k)xkπ(k) dν(π) =1

n− k + 1

∑j /∈B

x∗kjxkj ≤2

n

n∑j=1

x∗kjxkj .

Hence for all k ≤ n2

we have

(E2k−1 ⊗ id)(x∗kπ(k)xkπ(k)) ≤

2

n

n∑j=1

x∗kjxkj .

Therefore xkj = 0 for k ≥ n2

implies

n∑k=1

Ek−1(x∗kπ(k)xkπ(k)) ≤

2

n

∑kj

x∗kjxkj

for all π. The same argument applies for xkπ(k)x∗kπ(k). Therefore the upper estimate follows

from the triangle inequality. For the lower estimate we use p2≥ 1 and the orthogonality

of the Rademacher variables:

‖X‖2p = (E‖X∗X‖ p

2)

2p ≥ ‖E(X∗X)‖ p

2= ‖

n∑k=1

∫Πn

x∗kπ(k)xkπ(k)‖ p2

= ‖n∑

kj=1

1

nx∗kjxkj‖ p

2.

The same calculation involving XX∗ yields the other square function estimate. Since

Lp(N ) has cotype p we trivially find the missing `p-estimate.

Remark 5.6. The same argument applies for subgroups G = G1 × · · · ×Gm ⊂ Πn where

Gi is the permutation group of an interval Ii ⊂ 1, ..., n of cardinality ni . If the intervals

Ii are disjoint, we obtain(E‖

n∑i=1

εixiπ(i)‖pp

) 1p

∼cp

(m∑i=1

1

ni

∑k,l∈Ii

‖xkl‖pp

) 1p

+ ‖

(m∑i=1

∑k,l∈Ii

1

ni

∑kl∈Ii

(x∗klxkl + xklx∗kl)

) 12

‖p .

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 39

A symmetric space is a Banach space X with a (finite or infinite) basis (ek) such that

for all (αk) ⊂ C, all signs εk and all permutations π

‖∑k

εkαπ(k)ek‖X ≤ C ‖∑k

αkek‖X .

Let us denote Sym(X) the infimum over all constant C such that there exists a basis with

this property. The well-known space Jp(a, b) is a weighted intersection of `p and `2.

‖x‖Jp(a,b) = maxa‖x‖p, b‖x‖2 .

We recall the Banach-Mazur distance

d(E,F ) = infT :E→F isomorphism

‖T‖‖T−1‖ .

Theorem 5.5 implies the noncommutative analogue of [JMST, Theorem 1.1]:

Corollary 5.7. Let 2 ≤ p < ∞ and X be an n dimensional subspace of Lp(N). Then

there exists a, b such that

d(Jp(a, b), X) ≤ cp Sym(X)2 .

Proof. Let e1, ..., en ∈ Lp(N) be a symmetric basis in X, then for all coefficients α1, ..., αn,

we have

1

C‖

n∑k=1

αkek‖X ≤

(E‖

n∑k=1

εkαkeπ(k)‖pp

) 1p

≤ C ‖n∑k=1

αkek‖X .

According to Theorem 5.5, we deduce(E‖

n∑k=1

εkαkeπ(k)‖pp

) 1p

∼cp max

a(

n∑k=1

|αk|p) 1

p

, b

(n∑k=1

|αk|2) 1

2

,

where

a = a(e1, .., en) =

(1

n

n∑k=1

‖ek‖pp

) 1p

and b = b(e1, .., en) = ‖

(1

n

n∑k=1

(e∗kek + eke∗k)

) 12

‖p .

We can always assume ‖ek‖p = 1 for all k and deduce 0 ≤ a, b ≤ 1. Taking the infimum

over all normalized symmetric bases, we may find a sequence (e1(j), ..., en(j))j∈N such

that the corresponding parameters a(j) and b(j) converge to a, b, respectively. Using

d(Jp(a, b), Jp(a′, b′)) ≤ max a

a′, bb′maxa′

a, b

′

b we deduce the assertion.

40 M. JUNGE AND Q. XU

Proof of Theorem 0.2. We define

a =1

n

(n∑k=1

‖ek‖pp

) 12

, c = ‖( 1

n

n∑k=1

e∗kek)12‖p and r = ‖( 1

n

n∑k=1

eke∗k)

12‖p .

Then Theorem 5.5 implies (0.4). The second assertion is a standard reformulation from

the theory of operator spaces (see the following Remark (5.8) below).

Remark 5.8. In the theory of operator space the analogue of the Banach Mazur distance

is defined as dcb(X,Y ) = inf ‖T‖cb‖T−1‖cb (see [Pis2] for details). Therefore equation (0.4)

implies that

dcb(spanek : 1 ≤ k ≤ n, a`np ∩ cCnp ∩ rRn

p ) ≤ c0p2C2 .

Corollary 5.9. Let 2 ≤ p < ∞ and X ⊂ Lp(N) an infinite dimensional subspace with a

symmetric basis, then X is isomorphic to `p or `2. If X has a completely symmetric basis,

i.e. (5.2) is satisfied for all m ∈ N and ak ∈ Lp(Mm), then X is completely isomorphic to

`p, Rp, Cp or Rp ∩ Cp.

Proof. Let (ek) ⊂ X be a (completely) symmetric basis. For fixed n ∈ N, we consider

an =1

n

(n∑k=1

‖ek‖pp

) 12

, cn = ‖( 1

n

n∑k=1

e∗kek)12‖p and rn = ‖( 1

n

n∑k=1

eke∗k)

12‖p .

Note that supn an, supn cn and supn rn can be estimated by supk ‖ek‖. Passing to a subse-

quence, we assume that the three sequences are converging to a, c and r. We deduce that

for any finitely supported sequence (ak) we have

‖∑k

ak ⊗ ek‖p ∼C2cp a

(n∑k=1

‖ak‖pp

) 1p

+ b‖(n∑k=1

a∗kak)12‖p + c‖(

n∑k=1

aka∗k)

12‖p .

In the Banach space case this only holds for scalar sequences and then X is isomorphic

to `p if c = 0 = r and to `2 if maxc, r > 0. In the operator space case, we obtain a

complete isomorphism with `p if c = 0 = r. If r = 0, we obtain Cp. If c = 0, we get Rp.

Finally Rp ∩ Cp occurs if c and r are both positive.

Remark 5.10. In [JR] we will show that every subsymmetric basic sequence is symmetric.

A sequence (ek) is called subsymmetric if

‖∑k

εkak ⊗ ejk‖p ∼C ‖∑k

ak ⊗ ek‖

holds for all increasing sequences (jk) of integers. Therefore Corollary 0.3 yields a charac-

terization of spaces with a subsymmetric basis.

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 41

Corollary 5.11. Let 2 < p < ∞ and N be a finite von Neumann algebra. Then there is

no sequence (ek) ⊂ Lp(N) such that

(5.3) ‖∑k

ak ⊗ ek‖ ∼C ‖(∑k

a∗kak)12‖p

holds for some constant C and all m ∈ N and (ak) ⊂ Lp(Mm). In particular, Cp and Rp

do not completely embed into Lp(N).

Proof. Let us assume τ is a normal faithful trace on N with τ(1) = 1. Let (ek) ⊂ Lp(N, τ)

and C > 0 such that (5.3) holds. Then we see that

‖[aik]‖Sp ∼C ‖∑ik

aike1,i ⊗ ek‖Lp(B(`2)⊗N)

holds for any matrix [aik]. Note that for x =∑

ik aik⊗e1,i⊗ek and xi =∑

k aikek Holder’s

inequality implies

‖x‖22 = ‖xx∗‖1 = ‖

∑i

xix∗i ‖L1(N,τ) ≤ ‖

∑i

xix∗i ‖ p

2= ‖x‖2

p .

This tells us that on the subspace Y = spane1,i⊗ek the norm in Lp∩L2 and Lp coincide.

Thus we have found an embedding of Sp in Lp(B(`2)⊗N) ∩ L2(B(`2)⊗N). According to

[Jun3] the latter space embeds into Lp(M) for a finite von Neumann algebra M . Thus we

obtain an embedding of Sp into Lp(M). This is absurd in view of the results in [Suk]. The

proof for Rp is identically the same.

6. Unitary ideals embedding in Lp(N) for p ≥ 2

In this section we will characterize those unitary ideals which can be embedded into

some noncommutative Lp space for 2 ≤ p <∞. In a preliminary step we consider spaces

with a bisymmetric basis:

Proposition 6.1. Let 2 ≤ p < ∞ and N and M be von Neumann algebras and n ∈ N.

Let xij ∈ Lp(N ) and yij ∈ Lp(M) be two n × n matrices. Then there exist constants

c1, ..., c9 depending on the matrix xij such that(E‖

n∑i,j=1

εiε′jxπ(i),π′(j) ⊗ yij‖pp

) 1p

∼cp

c1

(n∑

i,j=1

‖yij‖pp

) 1p

+ c2

n∑j=1

‖

(n∑i=1

y∗ijyij

) 12

‖pp

1p

+ c3

n∑j=1

‖

(n∑i=1

yijy∗ij

) 12

‖pp

1p

42 M. JUNGE AND Q. XU

+ c4

n∑i=1

‖

(n∑j=1

y∗ijyij

) 12

‖pp

1p

+ c5

n∑i=1

‖

(n∑j=1

yijy∗ij

) 12

‖pp

1p

+ c6‖

(n∑

i,j=1

y∗ijyij

) 12

‖p + c7‖

(n∑

i,j=1

yijy∗ij

) 12

‖p

+ c8‖n∑

i,j=1

eji ⊗ yij‖Lp(Mn⊗M) + c9‖n∑

i,j=1

eij ⊗ yij‖Lp(Mn⊗M) .

Here the expectation is taken over independent copies εi, ε′i of Rademacher variables and

independent independent of copies π and π′ of permutations.

Proof. We apply Theorem 5.5 and get(Eε′π′Eε,π‖

∑ij

εiε′jxπ(i)π′(j) ⊗ yij‖p

)∼cp n

− 1p

(Eε′π′

∑il

‖∑j

xlπ′(j) ⊗ yij‖pp

) 1p

+ n−12

(Eε′π′‖

∑i,l,j

e1,il ⊗ xlπ′(j) ⊗ yij‖pp

) 1p

+ n−12

(Eε′π′‖

∑i,l,j

eil,1 ⊗ xlπ′(j) ⊗ yij‖pp

) 1p

.

We apply Theorem 5.5 for a second time to the first term and find

n−1p

(Eε′π′

∑il

‖∑j

xlπ′(j) ⊗ yij‖pp

) 1p

∼cp n− 2

p

(∑l,k

‖xlk‖pp

) 1p(∑

i,j

‖yij‖pp

) 1p

+ n−( 1p+ 1

2)

(∑il

‖∑jk

e1,jk ⊗ xlk ⊗ yij‖pp

) 1p

+ n−( 1p+ 1

2)

(∑il

‖∑jk

ejk,1 ⊗ xlk ⊗ yij‖pp

) 1p

.

(6.1)

Therefore, we have identified c1 = n−2p

(∑l,k ‖xlk‖pp

) 1p. We observe that e1,jk and e1,k⊗e1,j

are conjugated by unitaries. This implies

‖∑jk

e1,jk ⊗ xlk ⊗ yij‖p = ‖(∑k

e1,k ⊗ xlk)⊗ (∑j

e1,jyij)‖p

= ‖(∑k

xlkx∗lk)

12‖p ‖(

∑j

yijy∗ij)

12‖p .

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 43

Therefore, we find

c3 = n−1p− 1

2

(∑l

‖(∑k

xlkx∗lk)

12‖pp

) 1p

.

The same calculation yields c2 = n−1p− 1

2

(∑l ‖(∑

k x∗lkxlk)

12‖pp) 1

p. Applying Theorem 5.5

to the second and third term in (6.1) we may similarly identify the other constants c4 up

to c9. They are always given by the same expression with the yij’s replaced by the xij’s

and an additional normalizing factor depending on n.

Remark 6.2. Let us say that (xij) ⊂ Lp(N) is a bi-symmetric basis if there exist a

constant C > 0 such that

‖∑ij

εiε′jaπ(i)π(j)xij‖p ≤ C‖

∑ij

aijxij‖p

holds for all scalar matrices (aij) with finite entries, all εi = ±1, ε′j = ±1 and all permuta-

tion π and π′. For scalar coefficients the terms corresponding to c2 and c3 coincide. Also

they coincide for c4, c5, for c6, c7 and c8, c9. Therefore we find the building blocks

`p(N2), `p(`2), `p(`2)⊥, Sp, `2(N2)

and all possible intersections. Here ⊥ means that we interchange (i, j) to (ji). This yields

a lengthy but complete characterization of all Banach spaces with bi-symmetric basis

embedding into Lp(N) for 2 < p < ∞. On the operator space level, we have 9 building

blocks

Sp Rp(N2) Cp(N2) S⊥p

↓ ↓

`p(Rp) `p(Cp)⊥ `p(Rp)

⊥ `p(Cp)

`p(N2) .

Again one has to consider all possible intersection. Note that some of them simplify.

The flashes indicate complete contractions, e.g. Sp ⊂ `p(Rp) ∩ `p(Cp)⊥ and Rp(N2) ⊂`p(Rp) ∩ `p(Rp)

⊥. We leave the task to calculate the number of spaces obtained in that

way and further details to the interested reader.

44 M. JUNGE AND Q. XU

Let E be a Banach space with a symmetric, normalized basis (ek). The unitary ideal

SE is defined to be the closure of finite rank matrices with respect to the norm

‖a‖SE= ‖

∑k

sk(a)ek‖E ,

where (sk(a)) = (λk(|a|)) is the sequence of singular numbers.

Corollary 6.3. Let 2 ≤ p < ∞ and E a symmetric space. Then SE is isomorphic to

Lp(N ) for some von Neumann algebra N if and only if E = `2, E = `p.

Proof. If SE is isomorphic to a subspace Lp(N), then this is also true for the subspace

spanned by the diagonal. According to Corollary 0.3 we have E = `2 or E = `p. In both

cases S`p = Sp and S`2 = S2 are complemented subspaces of Sp.

Remark 6.4. Let X be an rearrangement invariant space on (0,∞) and N be a semifinite

von Neumann algebra with a normal faithful trace. Then one can define LX(N, τ) (see e.g.

[DDdP]). The same argument as above shows that if LX(N, τ) is isomorphic to a subspace

Y ⊂ Lp(M), then LX(N, τ) is isomorphic to Lp(N, τ), L2(N, τ) or Lp(N, τ)∩L2(N, τ). In

[Jun3] we show that indeed all these three spaces embed into Lp(M) for a suitable choice

of M . Thus the above characterizations ‘extends’ to the continuous case.

We conclude this section with a characterization of unitary ideals which embed Lp(N)

in the category of operator spaces. We need the following observation.

Lemma 6.5. Let 2 ≤ p <∞. Let y1, ..., yn ∈ Lp(M). Then(E‖

n∑i=1

uijyj‖pp

) 1p

∼c√p maxn−

12 (‖(

∑j

y∗j yj)12‖p, ‖(

∑j

yjy∗j )

12‖p) .

Here the expectation is taken over the normalized Haar measure on the complex unitary

group.

Proof. According to [MP], we may replace (uij)bygij√n

where gij are complex gaussian.

In combination with the noncommutative Khintchine inequality (see [LPP], [Pis2]), this

yields

E

(‖

n∑i=1

uijyj‖pp

) 1p

∼c n− 1

2 E

(‖

n∑i=1

gijyj ⊗ xi‖pp

) 1p

∼c√p n

− 12 (‖(

∑j

y∗j yj)12‖p + ‖(

∑j

yjy∗j )

12‖p)

NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 45

Theorem 6.6. Let 2 ≤ p <∞ and (xij) ⊂ Lp(N) and C > 0 such that

‖∑ijkl

uikxklvlj ⊗ yij‖p ≤ C‖∑ij

xij ⊗ yij‖p

holds for all yij ∈ Lp(Mm), m ∈ N and all unitary matrices u and v. Then there are

constant c6, ..., c9 such that

‖∑ij

xij ⊗ yij‖Lp(N⊗M) ∼fp(C) c6‖n∑

i,j=1

eij,1 ⊗ yij‖Lp(Mn2⊗M)+ c7‖n∑

i,j=1

e1,ij ⊗ yij‖Lp(Mn2⊗M)

+ c8‖n∑

i,j=1

eji ⊗ yij‖Lp(Mn⊗M) + c9‖n∑

i,j=1

eij ⊗ yij‖Lp(Mn⊗M) .

Proof. Note that permutation matrices and the diagonal metrices with entries ε1, ..., εn are

unitaries. This implies(E‖∑ij

εkεlxπ(i),π′(j) ⊗ (∑kl

uikyijvlj)‖p) 1

p

∼C2 ‖∑ij

xij ⊗ yij‖p .

We first fix an integer n ∈ N and integrate with respect to the Haar in ε, ε′, π and π′.

According to Proposition 6.1 we obtain 9 terms. The terms corresponding to c6 to c9 are

invariant under unitary transformation from the right an the left. Let us consider the term

corresponding to c2. Using the unitary invariance of the column space and Lemma (6.5),

we get(E

n∑l=1

‖∑i,j,k

ek,1 ⊗ (uikyijvlj)‖pp

) 1p

=

(E

n∑l=1

‖∑i

(∑k

uikek,1)⊗ (∑j

yijvlj)‖pp

) 1p

=

(E

n∑l=1

‖∑i

ei,1 ⊗ (∑j

yijvlj)‖pp

) 1p

=

(n∑l=1

E‖∑j

(∑i

ei,1 ⊗ yij)vlj‖pp

) 1p

∼c√p n

− 12

(n∑l=1

‖∑i,j

ej1 ⊗ ei1 ⊗ yij‖pp

) 1p

+ n−12

(n∑l=1

‖∑i,j

e1j ⊗ ei1 ⊗ yij‖pp

) 1p

= n1p− 1

2‖∑ij

eij,1 ⊗ yij‖p + n1p− 1

2‖∑ij

eij ⊗ yij‖p .

In the finite dimensional case, we find additional contributions c4 to c9. In the infinite

dimensional case these terms converge to 0 for n→∞. Indeed, from the proof of Propo-

sition 6.1 we see that the constant c1, ..., c9 are uniformly bounded provided the (xij) are

uniformly bounded. The uniform bound follows by applying the assumption to scalar

46 M. JUNGE AND Q. XU

coefficients. Therefore, we may pass to a subsequence such that c1(n), ..., c9(n) converge

to constant c1, ..., c9 such that c1 = c2 = · · · = c5 = 0.

Remark 6.7. The result shows that the building block for operator space unitary ideals

embedding in Lp(N) for 2 ≤ p <∞ are

Sp, S⊥p , Cp(N2), Rp(N2) .

Since there are no trivial inclusions this amounts to 16 spaces obtained by intersections.

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NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 49

Department of Mathematics, University of Illinois, Urbana, IL 61801, USA

E-mail address, Marius Junge: [email protected]

Departement de Mathematiques, Universite de France-Comte, 16 Route de Gray, 25030

Besancon Cedex, France

E-mail address, Quanhua Xu: [email protected]