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arXiv:math/0206062v1
[math.O
A]6Jun2002
DOOBS INEQUALITY FOR NON-COMMUTATIVEMARTINGALES
MARIUS JUNGE
Abstract. Let 1 p < and (xn)nN be a sequence of positive elements in a non-commutative Lp space and (En)nN be an increasing sequence of conditional expectations,
then n
En(xn)
p
cpn
xn
p
.
This inequality is due to Burkholder, Davis and Gundy in the commutative case. By
duality, we obtain a version of Doobs maximal inequality for 1 < p .
Introduction:
Inspired by quantum mechanics and probability, non-commutative probability has be-
come an independent field of mathematical research. We refer to P.A. Meyers exposition
[Me], the successive conferences on quantum probability [AvW], the lecture notes by Jajte
[Ja1, Ja2] on almost sure and uniform convergence and finally the work of Voiculescu,
Dykema, Nica [VDN] and of Biane, Speicher [BS] concerning the recent progress in free
probability and free Brownian motion. Doobs inequality is a classical tool in probability
and analysis. Transferring classical inequalities into the non-commutative setting theory
often requires an additional insight. Pisier, Xu [PX, Ps3] use functional analytic and com-
binatorial methods to establish the non-commutative versions of the Burkholder-Gundy
square function inequality. The absence of stopping time arguments, at least until the
time of this writing, imposes one of the main difficulties in this recent branch of martin-
gale theory.
The formulation of Doobs inequality for non-commutative martingales faces the following
problem. For an increasing sequence of conditional expectations (En)nN and a positiveoperator x in Lp, there is no reason why supn En(x) or supn |En(x)| should be an elementin Lp or represent a (possibly unbounded) operator at all. Using Pisiers non-commutative
vector-valued Lp-space Lp(N; ) we can overcome this problem and at least guess the right
formulation of Doobs inequality. However, Pisiers definition is restricted to von Neumann
algebras with a -weakly dense net of finite dimensional subalgebras, so-called hyperfinite
Junge is partially supported by the National Science Foundation.1
http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v1http://arxiv.org/abs/math/0206062v17/27/2019 Martingales Junge
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von Neumann algebras. But maximal inequalities are also interesting for free stochastic
processes where the underlying von Neumann algebra is genuinely not hyperfinite. All
these obstacles disappear for the so-called dual version of Doobs inequality: For everysequence (xn)nN of positive operators
n
En(xn)
p
cpn
xn
p
. (DDp)
In the commutative case this inequality is due to Burkholder, Davis and Gundy [ BDG]
(even in the more general setting of Orlicz norms). Since it is crucial to understand
our approach to Doobs inequality, let us indicate the duality argument relating (DDp)
and Doobs inequality in the commutative case. Indeed, (DDp) implies that T(xn) =n En(xn) defines a continuous linear map between Lp(1) and Lp. The norm ofT
yields the best constant in Doobs inequality for the conjugate index p = p
p1.sup
n
|En(x)|p
= T(x)Lp () T xp = cp xp .
Personally, I learned this argument after reading Dilworths paper [Di]. But I am sure it
is known to experts in the field, see Garcias [Ga] for the general theory (and [AMS] for
the explicit equivalence). (DDp) admits an entirely elementary proof in the commutative
case (see again [AMS]). This elementary proof still works in the non-commutative case
for p = 2, see Lemma 3.1. It is the starting point of our investigation. We recommend
the reader (not familiar with modular theory) to start in section 3 where interpolation
is used to extend (DDp) to 1 p 2 and suitable norms are introduced to make theabove duality argument work in the non-commutative case. In section 4, we establish
the dual version (DDp) in the range 2 p < using duality arguments which rely onPisier/Xus version of Steins inequality in combination with techniques from Hilbert C-
modules. By duality, we obtain the non-commutative Doob inequality in the more delicate
range 1 < p 2. The heart of our arguments rely on the (apparently new) connectionbetween Hilbert C-modules and non-commutative Lp spaces presented in section 2. These
duality techniques are necessary because p > 1 and 0
a
b implies 0
ap
bp only for
commuting operators. This is very often used in the elementary approach to commutativemartingales inequalities as in Garsias book [Ga].
Let us formulate our main results for finite von Neumann algebras. If : N C isa normal, tracial state, i.e. (xy) = (yx), then the space Lp(N, ) is defined by the
completion of N with respect to the norm
xp = ((xx)p2 )
1p .
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NON-COMMUTATIVE DOOB INEQUALITY 3
We refer to the first section for more precise definitions and references. Given a subalgebra
M N, the embedding : L1(M, ) L1(N, ) is isometric because |x| =
xx M forall x M. The dual map E = : N M yields a conditional expectation satisfying
E(axb) = aE(x)b
for all a, b M and x N, see [Tk, Theorem 3.4.]. Since E is trace preserving, E extendsto a contraction E : Lp(N, ) Lp(N, ) with range Lp(M, ). In the following, we con-sider an increasing sequence (Nn)nN N of von Neumann subalgebras with conditionalexpectations (En)nN. We recall that an element x is positive if it is of the form x = y
y.
Theorem 0.1. Let1 p < , then there exists a constant cp depending only on p such
that for every sequence of positive elements (xn)nN Lp(N, )
n
En(xn)
p
cpn
xn
p
. (DDp)
Note the close relation to the non-commutative Stein inequality, see [ PX, Theorem 2.3.],n
En(xn)En(xn)
p
22pn
xnxn
p
.
Using Kadisons inequality En(xn)En(xn) En(xnxn), it turns out that (DDp) is stronger
than Steins inequality. However, Steins inequality combined with the theory of HilbertC-modules yields one of the fundamental inequalities in the proof of Theorem 0.1. Using a
Hahn-Banach separation argument a la Grothendieck-Pietsch, we deduce Doobs maximal
inequality.
Theorem 0.2. [Doobs maximal inequality] Let 1 < p and x Lp(N, ), thenthere exist a, b L2p(N, ) and a sequence of contractions (yn) N such that
En(x) = aynb and a2p b2p cp xp
Here cp
is the constant in Theorem 0.1 for the conjugate index p
=
p
p1. In particular,for every positive x Lp(N, ), there exists a positive b Lp(N, ) such thatEn(x) b
for all n N.
In the case of hyperfinite von Neumann algebras this is equivalent to the corresponding
vector-valued inequality, see [Ps2], and therefore justifies the name maximal inequality.
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In the semi-commutative case where Nn = L(, n, ) M and n are increasing -algebras this inequality is stronger than the vector-valued Doob inequality, see Remark
5.5. In particular, this applies for random matrices. The inequality can be extendedto a continuous index set under suitable density assumptions, for examples for Clifford
martingales or free stochastical processes. Since these modifications are rather obvious,
we omit the details.
Clearly, maximal inequalities immediately imply almost sure convergence. Therefore it is
not surprising that Theorem 0.2 implies almost uniform convergence convergence of the
martingale truncations (En(x)) for p 2 and bilateral convergence for 1 < p 2 incase of a tracial state. We refer to [Ja1, Ja2] for the definition of these notions and more
details. In the tracial case the bilateral convergence of the martingale truncations is knownby a result of Cuculescu [Cu] even for martingales in L1(N, ). Therefore Theorem 0.2
provides a alternative approach to these results but only for p > 1. However, the maximal
inequality discussed in [Ja1] cannot easily be interpolated to obtain Theorem 0.2 as in the
real case. In a subsequent paper [DJ2], we will apply the maximal inequality of Theorem
0.2 in Haagerup Lp spaces in order to obtain (bilateral) almost sure convergence for all
states thus underlining the strength of these maximal inequalities.
Preliminary results and notation are contained in section 1. Section 5 contains immedi-
ate applications to submartingales and conditional expectations associated to actions of
groups.
I am indebted to Q. Xu for many discussion and support. I want to thank A. Defant for
the discussion about almost everywhere convergence. The knowledge of similar results for
almost sure convergence of unconditional sequences, see [DJ1], have been very encouraging.
I would like to thank Stanislaw Goldstein for initiating a correction in the proofs of Lemma
2.3 and Lemma 3.2.
1. Notation and preliminary results
As a shortcut, we use (xn), (xnk) instead of (xn)nN, (xnk)n,kN for sequences indexed by
the natural numbers N or its cartesian product N2. We use standard notation in operator
algebras, as in [Tk, KR]. In particular, B(H) denotes the algebra of bounded operators
on a Hilbert space H and K(H), K denote the subalgebra of compact operators on H, 2,respectively. The letters N, M will be used for von Neumann algebras, i.e. subalgebras
of some B(H) which are closed with respect to the -weak operator topology. We refer
to [Dx, Tk] for the different locally convex topologies relevant to operator algebras. For
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NON-COMMUTATIVE DOOB INEQUALITY 5
n N we denote by Mn(N) the von Neumann algebra of n n matrices with values inN. We will briefly use Mn for Mn(C). Given C
-algebras A, B, we denote by A Btheminimal tensor product. For von Neumann algebras N B(H1), M B(H2), we useNM for the closure of N M B(H1 H2) in the -weak operator topology. Let usrecall that a von Neumann algebra is semifinite if there exists a normal, semifinite faithful
trace. A trace is a positive homogeneous, additive function on N+ = {xx |x N}, thecone of positive elements ofN, such that for all increasing nets (xi)i with supremum in N
and for all x Nn) (supi xi) = supi (xi).
s) For every 0 < x there exists 0 < y < x such that 0 < (y) < .f) (x) = 0 implies x = 0.
t) For all unitaries u N: (uxu) = (x).A positive homogeneous, additive function w : N+ [0, ] satisfying n), s), f), but notthe last property t), is called a n.s.f. (normal, semifinite, faithful) weight.
It will be worthwhile to clarify the different notions of non-commutative Lp-spaces. If
is a trace then
m() =
n
i=1
yixi
n N,n
i=1
[(yi yi) + (xi xi)] <
is the definition ideal on which there exists a unique linear extension : m() Csatisfying (xy) = (yx). The Lp-(quasi)-norm is defined for x m() by
xp = ((xx)p2 )
1p .
Then Lp(N, ) is the completion of m() with respect to the Lp-norm. (For p < 1 smaller
ideal is needed in order to guarantee that (|x|p) is finite.) We refer to [Ne, Te, KF, Ye] formore on this and the fact that Lp(N, ) can be realized as unbounded operators affiliated
to N.
The starting point of Kosakis [Ko] definition of an Lp-space is a normal faithful state on a von Neumann algebra N. Then N acts on the Hilbert space L2(N, ) obtained by
completing N with respect to the norm
xL2(N,) = (xx)12 .
The modular operator is an (unbounded) operator obtained from the polar decomposi-
tion S = J12 of the antilinear operator S(x) = x on L2(N, ) , see [KR, Section 9.2.]. We
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denote by t : N N the modular automorphism group defined by t (x) = itxit.Let us recall the standard notation
x.(y) = (xy) .
For each t there is a natural map It : N N (N the unique predual of N) defined by
It(x) = t(x). .
According to [Ko, Theorem 2.5.], there is a unique extension Iz : N N such that forfixed x the function fx : {z| 1 Im(z) 0} N, fx(z) = Iz(x) is analytic andsatisfies
fx(t)(y) = (y
t (x)) and fx(i + t)(y) = (
t (x)y) .
The density of the algebra of analytic elements shows that for 0 1 the map Ii isinjective. By complex interpolation, the Banach space
Lp(N , , ) = [Ii(N), N] 1p
is defined by specifying x0 =I1i(x)N and x1 = xN . We refer to [Te, Fi] for
further information.
Haagerups abstract Lp space [Ha1, Te] is defined for every von Neumann algebra N using
the crossed product Nw R with respect to the modular automorphism group of a n.s.f.
weight w. (In our applications, we can assume w = for a n.f. state.) IfN acts faithfully
on a Hilbert space H, then the crossed product Nw R is defined as the von Neumann
algebra defined on L2(R, H) and generated by
(x)((t)) = wt((t)) and (s)(t) = (t s) .
Then Nw R, see [PTA], is semifinite and admits a unique trace such that the dual
action
s(x) = W(s)xW(s)
satisfies (s(x)) = es(x). Here W(s) is defined by the phase shift
W(s)(t) = eist(t) .
The dual action satisfies s((x)) = (x) and moreover
(N) = {x Nw R | s(x) = x , for all s R}(1.1)
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NON-COMMUTATIVE DOOB INEQUALITY 7
Let us agree to identify N with (N) in the following. Lp(N) is defined to be the space
of unbounded, -measurable operators affiliated to Nw R such that for all s Rs(x) = e
sp x .
Note that the intersection Lp(N) Lq(N) is {0} for different values p = q. There is anatural isomorphism between N and L1(N) such that for every normal functional Nthere is a unique a L1(N) associated satisfying
(ax) = (
R
s(x))
for all positive x Nwt R. The key point in this construction is the definition of thetrace function tr : L1(N)
C (corresponding to the integral in the commutative case)
given by
tr(a) = (1) .
Let 1p
+ 1p
= 1 and x Lp(N), y Lp(N). Then we have the trace propertytr(xy) = tr(yx) .
The polar decomposition x = u|x| of x Lp(N) satisfies u N andxp = tr(|x|p)
1p .
N acts as a left and right module on Lp(N) and more generally Holders inequality
xyr xp yq(1.2)holds whenever 1
p+ 1
q= 1
r. As for semifinite von Neumann algebras, there is a positive
cone Lp(N)+ in Lp(N) consisting of elements in Lp(N) which are positive as unbounded
operators affiliated to Nw R. Following [Te, Proposition 33, Theorem 32], we deduce
for 0 x y Lp(N) and 1p + 1p = 1xp = sup
zLp(N)+,zp1
tr(zx) supzLp(N)+,zp1
tr(zy) = yp .(1.3)
In the sequel, we will often use the following simple observation.
Lemma 1.1. Let 0 < p and x, y Lp(N) such that xx yy. If p is the leftsupport projection of y, then xy1p is a well-defined element in N of norm less than one.
Proof: We note that a = xy1p is affiliated with Nw R and
p(y1)xxy1p p(y1)yyy1p p
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shows that a is a contraction and in particular -measurable. Moreover, we have
p = s(p) = s(yy1p) = e
sp ys(y
1p)
and hence
s(y1p) = e
sp y1p .(1.4)
Therefore, the equality
s(a) = s(x)s(y1p) = e
sp xe
sp y1p = a
shows with (1.1) that a is a contraction in N.
In the -finite case, Kosakis Lp-space is isomorphic to Haagerups Lp space, see [Ko,
section 8]. Indeed, given a n. f. state with corresponding density D in L1(N) = N,then
t (x) = DitxDit
and for all x NIi(x)[Ii(N),N] 1
p
=D p xD 1p
Lp(N).
We recall that an element N is analytic, ift t (x) extends to an analytic function withvalues in N. The -closed subalgebra of analytic elements will be denoted by
A. The
following Lemma is probably well-known, see [Ko], [JX, Lemma 1.1.]. We add a shortproof for the convenience of the reader.
Lemma 1.2. Let0 < p < , then D 12pA+D 12p is dense in Lp(N)+ and N D1p is dense in
Lp(N) and for 1 p the map Jp : Lp(N) L1(N), Jp(x) = xD11p is injective.
Proof: D is a -measurable operator with support projection 1. Therefore, xD11p = 0
implies that x = xD11p D
1p1 is a well-defined -measurable operator and equals 0. Hence,
Jp is injective for 1
p
. Let 1
p+ 1
p= 1. We show the density ofD
12p
A+D
12p in Lp(N)+
for 1 p < . If this is not the case, the Hahn-Banach theorem implies the existenceof x Lp(N)+ and y Lp(N)sa such that tr(yD
12p aD
12p ) = tr(D
12p yD
12p a) 0 for all
a A+ and tr(xy) > 0. Using the -strong density ofA+ in N+, see [PTA], we deduce thaty = D
12p D
12p yD
12p D
12p is negative and hence tr(xy) 0, a contradiction. Let 1
2 p 1
and y Lp(N)+, then we can approximate y 12 by an element x = D14p aD
14p , a A+ and
hence Holders inequality implies that x2 y = x(x y 12 ) + (x y 12 )y 12 approximates y.Since a is analytic, we observe that x2 = D
14p aD
14p D
14p aD
14p = D
12p i
4p(a) i
4p(a)D
12p
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NON-COMMUTATIVE DOOB INEQUALITY 9
is in D12pA+D 12p . By induction, we deduce the density of D 12pA+D 12p in Lp(N)+ for all
0 < p k
tr(En(xn)Ek(xk))
=nk
tr(Ek(En(xn)xk)) +n>k
tr(En(xnEk(xk)))
=nk
tr(En(xn)xk) +n>k
tr(xnEk(xk))
=k
tr(
nk
En(xn)
xk) +
n
tr(xn
k
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Lemma 3.2. If (DD2) holds with constant c2 and 1 p 2, then for all sequences (xn)and (yn) in L2p(N)
n
En(xnyn)
p
c2(p1)
p
2
n
xnxn
12
p
n
ynyn
12
p
.
Proof: Let us first prove the assertion for finite sequences and p = 2 or p = 1. We start
with (DD1). Using Lemma 2.1, we get
n
En(xnxn)
1
= tr(n
En(xnxn)) =
n
tr(En(xnxn))
=n
tr(xnxn) =
n
xnxn1
.
By the density of elements of the form xn = anD1p , the Cauchy-Schwarz inequality, see
Theorem 2.17, implies with Lemma 3.1 for p {1, 2}n
En(xnyn)
p
n
En(xnxn)
12
p
n
En(ynyn)
12
p
cpn
xnxn
12
p
n
ynyn
12
p
,
(3.1)
where c2 is the constant given in the assumption and c1 = 1. To use interpolation, we
consider finite sequences (xn) and (yn) such thatn
xnxn
p
= 1 =
n
ynyn
p
.
We define X =
n xnxn, Y =
n y
nyn. Their support projections are denoted by qX
and qY and are in N, see [Te, Proposition 4. 2) c), Proposition 12]. Since X1
2 qX, qYY1
2
are well-defined unbounded operators, we can define
vn = xnX 1
2 qX , wn = ynY 1
2 qY
Note that xnxn X and ynyn Y implies xn = xnqX, yn = ynqY, respectively andaccording to Lemma 1.1 we have vn N, wn N. Then, we observe
n
vnvn = qXX1
2 XX12 qX qX 1 ,
n
wnwn = qYY1
2 Y Y12 qY qY 1 .
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NON-COMMUTATIVE DOOB INEQUALITY 25
Let be determined by 1p
= 11
+ 2
. According to Kosakis interpolation [Ko]
[L2,L(N, ), L4,L(N, )] = L2p,L(N, ) and [L2,R(N, ), L4,R(N, )] = L2p,R(N, )
with respect to the state (x) = tr(Dx). By approximation, we may assume that there are
continuous functions X(z), Y(z) on the strip {0 Re(z) 1} with values in N, analyticin the interior, such that X
12 = D
12p X(), Y
12 = Y()D
12p and
supt
max{D 12 X(it)2, D 14 X(1 + it)4} 1,
supt
max{Y(it)D 122, Y(1 + it)D 144} 1.
Now, we note Kosakis symmetric interpolation result
[L1,sym(N, ), L2,sym(N, )] = Lp,sym(N, ) .
Hence, by (3.1) and Holders inequality we getn
En(X12 vnwnY
12 )
p
=
n
D12p En(X()v
nwnY())D
12p
p
supt
n
D12 En(X(it)v
nwnY(it))D
12
1
1
supt
n
D14 E
n(X(1 + it)v
nw
nY(1 + it))D
14
2
supt
n
D12 X(it)vnvnX(it)
D12
12
1
n
D12 Y(it)wnwnY(it)
D12
12
1
c2 supt
n
D14 X(1 + it)vnvnX(1 + it)
D14
2
1
supt
n
D14 Y(1 + it)wnwnY(1 + it)D
14
2
2
supt
D 12 X(it)X(it)D 12 121
supt
D 12 Y(it)Y(it)D 12 121
supt
D 14 X(1 + it)X(1 + it)D 14 122
c2 supt
D 14 Y(1 + it)Y(1 + it)D 14 122
c2 = c2(p1)
p
2 .
The assertion is proved.
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26 MARIUS JUNGE
Proof of Theorem 0.1 in the case 1 p 2: For 1 p 2 and a sequence ofpositive elements (zn) Lp(N), we can apply Lemma 3.2 to xn = yn = z
12n and deduce
the assertion for the sequence (zn).
The duality argument relies on the following norm for sequences (xn) Lp(N)
(xn)Lp(N;1) = infnj
vnjvnj
12
p
nj
wjnwjn
12
p
.
Here the infimum is taken over all (double indexed) sequences ( vnj) and (wnj) such that
for all n
xn =j
vnjwjn .
We require norm convergence for p < and convergence in the -weak operator topologyfor p = . In fact, we think ofxn being obtained by matrix multiplication of a row with acolumn vector. We denote by Lp(N; 1) the set of all sequences admitting a decompositionas above.
Remark 3.3. This norm is motivated by the following characterization of a normal, de-
composable map T :
N, see [Pa]. Indeed, a normal map is decomposable if and only
if there are sequences (xn) N, (yn) N such that T(en) = ynxn andn
ynyn
N
n
xnxn
N
< .
Lemma 3.4. If (DDp) holds, then
n
En(xnyn)
p cp
n
xnxn12
p
n
ynyn12
p
,
The linear map T : Lp(N; 1) Lp(N; 1), T((xn)) = (En(xn)) satisfies
T cp .
Proof: We have seen in (3.1) that the first inequality is an immediate consequence of
the Cauchy-Schwarz inequality, see Theorem 2.17. As for the second assertion, we can
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NON-COMMUTATIVE DOOB INEQUALITY 27
assume that N is separable and use the Kasparov maps unp : Lp(N, En) Lp(Nn; C2 )
from Proposition 2.8. Let xn = j vnjwnj , then(En(xn))Lp(N;1) =
(nj
unp(vnj)
unp (wnj))
Lp(N;1)
nj
unp(vnj)
unp(vnj)
12
p
nj
unp (wnj)unp(wnj)
12
p
=
nj
En(vnjvnj)
12
p
nj
En(wnjwnj)
12
p
cpnj
vnjvnj
12
p
nj
wnjwnj
12
p
.
Taking the infimum over all these decompositions, we obtain the assertion.
Let us state some elementary properties of the space Lp(N; 1).
Lemma 3.5. For 1 p the set Lp(N; 1) is a Banach space. For 1 p < , the
set L0
p of elements
xn =j
vnjwjn
such thatcard{(j, n)|vnj = 0 or wjn = 0} < is dense inLp(N; 1). If (xn) is a sequencethen
n
xn
p
(xn)Lp(N;1)
and equality holds if all the xns are positive.
Proof: The proof of the triangle inequality is completely elementary, see also [Ps2], but
essential. Indeed, let > 0 and
xn =j1
vnj1wj1n and yn =j2
vnj2wj2n ,
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28 MARIUS JUNGE
such that
nj1
vnj1vnj1p
=
nj1
wj1nwj1np
(1 + ) (xn)Lp(N;1) ,nj2
vnj2vnj2
p
=
nj2
wj2nwj2n
p
(1 + ) (yn)Lp(N;1) .
We have
xn + yn =j1
vnj1wj1n +j2
vnj2wj2n
and the triangle inequality in Lp(N) impliesnj1
vnj1vnj1
+nj2
vnj2vnj2
p
nj1
vnj1vnj1
p
+
nj2
vnj2vnj2
p
(1 + )((xn)Lp(N;1) + (yn)Lp(N;1)) .
Similarly,
nj1
w
j1nvj1n +nj2
w
j2nwj2np
(1 + )((xn)Lp(N;1) + (yn)Lp(N;1))
and the assertion follows with 0. We consider the spaces of column matrices, rowmatrices, Lp(N;
C2 (N
2)), Lp(N; R2 (N
2)) Lp(B(2(N2))N), respectively. Using,
((xnk) (ynk) ) = (k
xnkynk)nN ,
we deduce that Lp(N; 1) is isomorphic to a quotient space of the projective tensor productLp(N;
R2 (N
2))
Lp(N;
C2 (N
2)), see [DF] for a definition and basic properties of the
projective tensor product. Hence Lp(N; 1) is complete. The image under of pairs offinite sequences generates L0p. According to Corollary 2.4 finite sequences are dense in thecolumn and row spaces and hence L0p is dense in Lp(N; 1). Finally, let (xn) be a sequenceof positive elements. Clearly, xn = x
12nx
12n and hence
(xn)Lp(N;1) n
xn
p
.
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NON-COMMUTATIVE DOOB INEQUALITY 29
On the other hand if xn =
j vnjwnj, we deduce from Holders inequality
n
xn
p
=
nj
(e1,nj vnj)(enj,1 wnj)p
nj
e1,nj vnj2p
nj
enj,1 wnj2p
=
nj
vnjvnj
12
p
nj
wnjwnj
12
p
.
Taking the infimum yields the assertion.
Inspired by Pisiers vector-valued Lp(N, ; ) space, we define for 0 < p
supn
|xn|p
= (xn)Lp(N;) = inf xn=aynb a2p b2p supn ynN ,
where the infimum is taken over all a, b L2p(N) and all bounded sequences (yk). If Nis a hyperfinite, finite von Neumann algebra, this space coincides with Lp(N, ; ) in thesense of Pisier [Ps2]. The first (formal) notation is suggestive and facilitates understand-
ing our inequalities in view of the commutative theory. For positive elements, we will
drop the absolute value. Let us note that Haagerups work [Ha2] shows that the equal-
ity L1(N; ) = L1(N) (operator space projective tensor product) only holds forinjective von Neumann algebras. However, this does not affect the following factorization
result which is, nowadays, a standard application of the Grothendieck-Pietsch version of
the Hahn-Banach theorem, see [Ps1, Ps2].
Proposition 3.6. Let1 p < . If p = 1, thenL1(N; 1) = 1(L1(N)) holds with equalnorms. If 1 < p , p < satisfy 1
p+ 1
p= 1, then
Lp(N; 1) = Lp(N; )
holds isometrically.
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30 MARIUS JUNGE
Proof: If zn = aynb and (yn) is a bounded sequence, we deduce from Holders inequality
for all (vnj) L2p(N), (wjn) L2p(N)n
tr(znj
vnjwjn)
=nj
tr(aynbvnjwjn)
=nj
tr(ynbvnjwjna)
sup
n
ynnj
bvnjwjna1
supn
yn
nj
bvnj22 1
2
nj
wjna22 1
2
= supn yn bnj
vnjvjnb
12
1
anj
wnjwjna
12
1
supn
yn b2pnj
vnjvnj
12
p
a2pnj
wjnwjn
12
p
.
Hence, Lp(N; ) Lp(N; 1). Using 1(L1(N)) L1(N; 1), we deduce the equality1(L1(N)) = L1(N; 1). Now we show that for 1 < p < all the functionals are inLp(N; ). Let : Lp(N; 1) C be a norm one functional. Using 1(Lp(N))
Lp(N; 1), we can assume that there exists a sequence (zn)
Lp(N) such that
[(xn)] = (zn)[(xn)] =n
tr(znxn) .
Let us denote by B = B+Lp(N)
the positive part of the unit ball in Lp(N). B is compact
when equipped with the (Lp(N), Lp(N))-topology. The definition of Lp(N; 1) implieswith the geometric/arithmetric mean inequality
nj
tr(znvnjwjn)
=
[(
j
vnjwjn)n]
nj
vnjvnj
12
p
nj
wjnwjn
12
p
12
supc,dB
[nj
tr(vnjvnjc) +
nj
tr(wjnwjnd)] .
Since the right hand side remains unchanged under multiplication with signs nj, we deduce
nj
|tr(znvnjwjn)| 12
supc,dB
[nj
tr(vnjvnjc) +
nj
tr(wjnwjnd)] .
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NON-COMMUTATIVE DOOB INEQUALITY 31
Following the Grothendieck-Pietsch separation argument as in [Ps1], we observe that the
C given by the functions
fv,w(c, d) =n
[tr(vnvnc) + tr(wnwnd) 2|tr(znvnwn)|]
is disjoint from the cone C = {g| sup g < 0}. Here v = (vn) and w = (wn) are finitesequences and hence fv,w is continuous with respect to the product topology on B B.Since fv,w + fv,w can be obtained by taking the (vn, wn)s to the right of the finite sequence
(vn, wn), we deduce that C is a cone. Hence, there exist a measure on B B and ascalar t such that for all g C and f C
BB
g d < t
BB
f d .
Since we are dealing with cones, it turns out that t = 0 and is positive. Therefore, we
can and will assume that is a probability measure. We define the positive elements c
and d by their projections
a =
BB
c d(c, d) , b =
BB
d d(c, d) .
By convexity of B, we deduce a, b B. Hence, we obtain
n
2 |tr(znvnwn)| BB
n
[tr(vnvnc) + tr(wnv
nd)] d(c, d)
=n
BB
tr(vnvnc) d(c, d) +
BB
tr(wnwnd) d(c, d)
=n
[tr(vnvna) + tr(w
nwnb)] .
Using once more 2st = infr>0(rs)2 + (r1t)2, we get
n
|tr(znvnwn)|
n
tr(vnvna)
12
n
tr(wnwnb)
12
=
n
a 12 vn22
12
n
b12wn22
12
.
(3.2)
Let qa, qb N be the support projections of a, b, respectively. Consider, da = ap + (1 qa)D(1 qa), D the density of . Then da(x) = tr(dax) is a normal, faithful state on Nand according to Lemma 1.2 d
12a N is dense in L2(N). Hence,
qad12a N = a
p
2 N = a12 a
p
2p N a 12 L2p(N)
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32 MARIUS JUNGE
shows that a12 L2p(N) is dense in qaL2(N). Similarly, b
12 L2p(N) is dense in qbL2(N) and
therefore (3.2) implies that for every n
N there is a contraction Tn : qaL2(N)
qbL2(N)
such that for all v, w L2p(N)tr(wznv) = (b
12 w, Tn(a
12 v)) = tr(wb
12 Tn(a
12 v)) .
This means Tn is a bounded extension of the densely defined hermitian form
(b12 qb(h
), zna1
2 qa(h) ) = (b1
2 qbh, zna
12 qah)
Using the density of a12 L2p(N) and b
12L2p(N) it is easily checked that qbTnqa is affiliated
with N. Since Tn is bounded, we deduce qbTnqa N. On the other hand, we have forv L2p(N) and w L2p(N)
|tr(zn(1 qa)vw)| tr(a(1 qa)vv) 12 tr(wwb) 12 = 0and
|tr(znvw(1 qb)| tr(avv) 12 tr(ww(1 qb)b) 12 = 0 .This shows zn = qbznqa and therefore
zn = qbznqa = qbb12 qbb
12 zna
12 qaa
12 qa = b
12 yna
12 .
The assertion is proved because L0p is dense in Lp(N; 1) and hence the functional isuniquely determined by the sequence (zn).
Remark 3.7. Let1 p < and (zn) Lp(N) a sequence of positive elements, thensupn
zn
p
= sup
n
tr(znxn)
xn 0 ,
n
xn
p
1 .
Moreover, there exists a positive element a L2p(N) and a sequence of positive elementsyn such that
zn
= ayn
a and
a2
2p sup
n yn
= supn
znp
.
For positive elements (xn) Lp(N)+, we also haven
xn
p
= sup
n
tr(xnzn)
zn 0 , (zn)Lp(N;) 1
and therefore the cones of positive sequences in Lp(N; 1) respectively Lp(N; ) are induality.
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NON-COMMUTATIVE DOOB INEQUALITY 33
Proof: For positive elements (zn) satisfying
n
tr(znxn)
nxnp
,
for all sequences of positive elements (xn) Lp(N)+, we deducej,n
|tr(znvnjvnj)| n
tr(zn(j
vnjvnj))
nj
vnjvnj
p
= supcB
j,n
tr(vnjvnjc) .
Using the Hahn-Banach separation argument in the space of continuous functions on B,
we obtain a positive element a in the unit ball of Lp(N) such thatn
|tr(znvnvn)| n
tr(vnvna) .
Since a12 zna
12 is positive, this inequality still ensures that all the yn = a
12 zna
12 are
positive contractions in N. The last equality follows immediately from Lemma 3.5 and
the duality between the positive parts of Lp(N) and Lp(N).
Remark 3.8. Proposition3.6and Remark3.7 can easily be modified for uncountable (or-
dered) index sets by requiring the inequality for all countable (ordered) subsets or for an
essential supremum. This is helpful in the context of continuous filtrations.
The required duality argument is now very simple.
Lemma 3.9. Let1 < p and 1p
+ 1p
= 1 assume that (DDp) holds with constant cp,
then for every y Lp(N)
supn
|En(y)|p
cp yp .
Moreover, for every sequence of positive elements (yn)supn |jn
En(yj)|p
cpn
yn
p
.
Proof: Indeed, as observed in Lemma 3.4 and using Lemma 3.5, we deduce that (DDp)
implies that the linear map T : Lp(N; 1) Lp(N), T((xn)) =
n En(xn) satisfies
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34 MARIUS JUNGE
T cp. By duality and Proposition 3.6, we deduce for all y Lp(N)
supn |En(y)|p = T(y)Lp(N;1) T yp cp yp .Given a sequence of positive elements (yn), we consider y =
j yj and obtainsup
n
|En(y)|p
cp yp = cpj
yj
p
.
However, for positive elements (xn) Lp(N), we deduce by positivity
n tr(En(jn yj)xn) n j tr(En(yj)xn) (En(y))nNLp(N;)n xn
p
.
Hence, Remark 3.7 impliessupnjn
En(yj)
p
sup
n
En(y))nN
p
cp yp = cpj
yj
p
.
The assertion is proved.
Theorem 3.10. For 1 < p there exists a constant cp such that for every sequence(Nn) of von Neumann subalgebras with sequence of -invariant conditional expectations
(En) satisfying EnEm = Emin(n,m) and for every x Lp(N) there exist a, b L2p(N) anda bounded sequence (yn) N such that
En(x) = aynb and a2p b2p supn
yn cp xp .
If x is positive, one can in addition assume that b = a and all the yns are positive.
Proof for 2 p : This follows immediately from Lemma 3.9, Lemma 3.2 and Lemma3.1. Using that En(x) is positive for positive x, the addition follows from Remark 3.7.
4. The dual version of Doobs inequality for 2 p <
In our approach to (DDp) in the range 2 p < , our aim is to obtain the same kindof inequalities for the maximal function as in Garsias book [Ga]. As mentioned in the
introduction, we are forced to use more duality arguments because 0 a b impliesa b only for 0 1 and therefore most of the elementary proofs in Garsias bookare no longer valid in the non-commutative case. We will make the same assumptions
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36 MARIUS JUNGE
The following proposition is a modification of the Theorem in the appendix of Pisier, Xus
paper [PX] and enables us to apply duality.
Proposition 4.2. Let 1 r < 2 < r , and 1r
+ 1r
= 1, then for all (xj) Lr(N)and (yn) Lr(N, (En); C2 )
n
tr(ynxn)
2 (yn)Lr(N,(En);C2 )j
xjxj
12
r2
.
Moreover,
n
tr(ynxn)
2 (yn)Lr(N,(En);C2 ) supn jn
En(xjxj)
12
r2
.
Proof: Let us assume that both sequences are finite, i.e. xj = 0 = yj for j m. Bydensity, we can moreover assume that yj = ajD
1r with aj N and
mj=1
D1r Ej(a
jaj)D
1r
r
2
< 1 .
By continuity, we can assume that there is an > 0 such that
D 2r +m
j=1
D1r Ej(a
jaj)D
1r
r
2
1 .
Let 1 q such that 1q
+ 2r
= 1. For n N, we define
Sn =
D
2r +
nj=1
D1r Ej(a
jaj)D
1r
r2q
Lq(Nn) Lq(N) .
The support projection of Sn is 1 and D1q Sn. According to Lemma 1.1, we deduce
that wn := D12q S
12
n Nn. HenceynS
12
n = anD1r S
12
n = anD12 D
12q S
12
n = anD12 wn
L2(N) .
In particular, S 1
2n yn L2(N) and
S 1
2n y
nynS
12
n = wnD
12 ananD
12 wn L1(N) .
Moreover,
En(S 1
2n y
nynS
12
n ) = En(wnD
12 ananD
12 wn) = w
nD
12 En(a
nan)D
12 wn
= S1
2n D
1r En(a
nan)D
1r S
12
n .
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NON-COMMUTATIVE DOOB INEQUALITY 37
This implies with the Cauchy-Schwarz inequality
n
tr(ynxn) =
ntr(xnyn)
= n
tr((xnS
1
2n )(S1
2n yn))
=
n
tr(S1
2n y
nxnS
12n )
n
tr(S1
2n y
nynS
12
n )
12
n
tr(xnxnSn)
12
=
n
tr(En(S 1
2n y
nynS
12
n ))
12n
tr(xnxnSn)
12
=
n
tr(S1
2n D
1r En(a
nan)D
1r S
12
n )
12
n
tr(En(xnxn)Sn)
12
.
To estimate the first term, we define = 2r
[1, 2] and notice that
1 = 1 2r
= 1 2 + 2r
= 1q
.
For fixed n, we define x = Sqn1 and z = Sqn. Since
r
2 1, we have
x =
D2r +
n1j=1
D1r Ej(a
jaj)D
1r
r2
D2r +
nj=1
D1r Ej(a
jaj)D
1r
r2= z .
Then, we note that z12 = z
12q = S
12
n . Hence Lemma 4.1 implies
tr(S 1
2n D
1r En(a
nan)D
1r S
12
n ) = tr(z12 (z x)z12 ) 2tr(z x)
= 2tr(Sqn Sqn1) .Therefore, we obtain
n
tr(S
12
n D
1r
En(a
nan)D
1r
S
12
n ) =n
2tr(Sq
n Sq
n1) = 2tr(Sq
m)
= 2
D 2r +m
j=1
D2r Ej(a
jaj)D
2r
r
2
r
2
2 .
Now, we want to estimate the second term. Let us define
j = Sj Sj1 .
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38 MARIUS JUNGE
and note that j Lq(Nj). As usual, we set S1 = 0. Moreover, r 2q implies that j ispositive. Then, we deduce with EjEn = Emin(j,n)
n
tr(En(xnxn)Sn) =
jn
tr(En(xnxn)j)
=j
tr(nj
En(xnxn)j)
=j
tr(Ej
nj
En(xnxn)
j)
=j
tr(nj
Ej(xnxn)j)
Now, we can continue in two different ways
j
tr(nj
Ej(xnxn)j) supj Ej(nj
xnxn)r2
j
jq
=
supj Ej(nj
xnxn)
r2
Smq
supj Ej(
nj
xnxn)
r2
.
By homogeneity, we obtain the second assertion
n
tr(ynxn)
2
supj Ej(nj
xnxn)
12
r2
(yn)Lr(N,(En);C2 ) .
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NON-COMMUTATIVE DOOB INEQUALITY 39
Using for all j N that Ej(j) = j , we also get by positivity
j
tr(nj
Ej(xnxn)j) =
j
tr(nj
xnxnEj(j))
=j
tr(nj
xnxnj)
tr(n
xnxnj
j)
n
xnxn
r2
j
j
q
n x
n
xnr2
Sm
q n x
n
xnr2
.
Again by homogeneity, we deducen
tr(ynxn)
2
n
xnxn
12
r2
(yn)Lr(N,(En);C2 ) .
Remark 4.3. For r = r = 2 the assertion is trivially true because L2(N, (En); C2 ) =
2(L2(N)) and
n
xn2
2
= n
x
n
xn1
= tr(n
x
n
xn) = tr(E1(n
x
n
xn))
=
E1(n
xnxn)
1
supj Ej(
nj
xnxn)
1
.
Proposition 4.2 also shows that the BM OC and HC1 duality from [PX] is still valid in the
non tracial case.
Corollary 4.4. Let 1 q < and 2q the constant in Steins inequality, then for all(xn)
L2q(N)
n
En(xnxn)
q
222qsupn En(
jn
xjxj)
q
.
Proof: We define r = 2q and noten
En(xnxn)
12
r
= (xn)Lp(N,(En);C2 ) .
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40 MARIUS JUNGE
Therefore the assertion follows from Proposition 4.2 and Theorem 2.17.
Proof of (DDp) for 1 < p < : We define r = 2p > 2 Let (zn) Lp(N) be a sequenceof positive elements and define xn = z
12n . Hence by Theorem 2.17 and Proposition 4.2, we
deduce
n
En(zn)
12
p
=
n
En(xnxn)
12
p
r sup(yn)L
r(N,(En);C2 )
1
n
tr(ynxn)
r
2
n
xnxn
12
p
.
The assertion follows with cp 222p.
Proof of Theorem 0.2 and 3.10 for 1 < p 2: This follows from (DDp) via Lemma3.9 and Remark 3.7.
Remark 4.5. Let N be separable and be a functional on Lp(N, (En); C2 ), then there
exists a sequence (xn) such that
((yn)) =n
tr(xnyn) and
supn En(jn
xjxj)
12
p2
d p2Lp (N,(En);C2 ) .
Here d p2
is the constant in Doobs inequality from Theorem 0.2. The assertion yields anextension of the BM OC-H
C1 duality for 2 < p < and fails for p = 2. The proof uses
the Kasparov isomorphism from Proposition 2.15. We leave it to the interested reader.
Answering a question by G. Pisier, we can even produce an asymmetric version of Doobs
inequality. Indeed, let 1 < p and consider 1 r,s < such that 2p
= 1s
+ 1t.
Given x Lp(N) and sequences (vnj) Lr(N), (wjn) Ls(N), we deduce from the
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NON-COMMUTATIVE DOOB INEQUALITY 41
Cauchy-Schwartz inequality 2.13, (DDs) and (DDt) that
n,j
tr(En(x)vnjwjn) =
n,j
tr(xEn(vnjwjn))
xpnj
En(vnjwjn)
p
xpnj
En(vnjvnj)
12
s
nj
En(wjnwjn)
12
t
c12s c
12t
x
p
nj
vnjvnj
12
s
nj
wjnwjn12
t
.
Similar as in Proposition 3.6, we deduce the existence of bounded a sequence (zn) and
elements a L2s(N), b L2t(N) such that En(x) = bzna and
a2s b2t supn
zn c12s c
12t xp .
Note that 12t
+ 12s
= 1p
and therefore we have proved the following asymmetric version of
Theorem 0.2. (The assumption 2 < q, r is indeed necessary [DJ1].)
Corollary 4.6. Let 1 < p and 2 < q , r such that1
q +1
r =1
p . Then forevery x Lp(N) there exists a sequence (zn) N and a Lq(N), b Lr(N) such thatEn(x) = aznb and
aq br supn
zn c(p, q, r) xp .
5. Applications
In this section, we present first applications of Doobs inequality in terms of submartin-
gales, Doob decomposition. We make the same assumptions about N, (Nn), (En), and
D as in the previous section and start with almost immediate consequences of the dualversion of Doobs inequality.
Corollary 5.1. Let1 < p and (zn) be an adapted sequence of positive elements, i.e.zn Lp(Nn)+. If for all n N
zn En(zn+1) and supm
zmp < ,
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42 MARIUS JUNGE
then there exist a positive element a L2p(N)+ and a sequence of positive contractions(yn) N such that
zn = ayna and a22p cp supm
zmp .
Proof: It suffices to consider p < . We note that for n mzn En(zn+1) En(En+1(zn+2)) = En(zn+2) En(zm) .
Let 1p
+ 1p
= 1. In order to estimate the norm in Lp(N; ), we refer to Remark 3.7. Givena finite sequence (xn) of positive elements such that xn = 0 for n m, we deduce from(DDp)
n
tr(znxn) =
n
tr(En(znxn)) =n
tr(En(zn)En(xn))
n
tr(En(zm)En(xn)) =n
tr(En(zmEn(xn)))
=n
tr(zmEn(xn)) tr(zmn
En(xn))
zmp
n
En(xn)
p
cp zmp
n
xn
p
Hence, the assertion follows from Remark 3.7.
Corollary 5.2. Let2 < p and (zn) be an adapted seqeunce, i.e. zn Lp(Nn). If forall n N
znzn En(zn+1zn+1) ,then there exist a positive element a Lp(N) and a sequence (yn) of contractions suchthat
zn = yna and ap c12p
p2supm
zmp .
Proof: We apply Corollary 5.1 to zn = znzn L p2 (Nn) and obtain positive contractions
(vn) N and a Lp(N) such thatznzn = avna and a2p c pp2 sup
m
zmzm p2
.
If qa is the support projection of a, we see that yn = v12n a
1qa satisfies the assertion.
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NON-COMMUTATIVE DOOB INEQUALITY 43
Let us mention an immediate application of (DDp) in terms of the Doob decomposition of
the square function. Given a martingale sequence x = k dk(x), where dk(x) = Ek(x) Ek1(x) (and E0(x) = 0), we recall that one of the square functions is given by
sc(x) =k
dk(x)dk(x)
The square function is the discrete analogue of the quadratic variation term see [BS, Ko]
for more details. The Doob decomposition of sc(x) is given by the martingale part
Vn(x) =n
k=1
dk(x)dk(x) Ek1(dk(x)dk(x))
and the predictable part
Wn(x) =n
k=2
Ek1(dk(x)dk(x)) Lp(Mk1) .
Note that
Vn(x) + Wn(x) =n
k=1
dk(x)dk(x) .
Corollary 5.3. Let2 < p < and x Lp(N), then
supn
max{
Wn
(x)
12p
2
,
Vn
(x)
12p
2} p
(1 + c p2
)12
xp
.
Here p is an absolute constant. In particular, there exists an element V(x) in L p2
(N)
such that Vn(x) = En(V(x)).
Proof: Using (DD p2
) for the sequence (Ek1) and the non-commutative Burkholder-
Gundy inequality [PX, JX], we deduce
Wn(x) p2
c p2
nk=2
dk(x)dk(x)
p2
c p2
2p x2p .
Hence the triangle implies
Vn(x) p2
Wn(x) p2
+
n
k=1
dk(x)dk(x)
p2
(c p2
2p + 1) x2p .
By uniform convexity of Lp2
(N), we obtain the limit value V(x) = limn Vn(x) with the
desired properties.
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44 MARIUS JUNGE
The next application yields norm estimates for
npnEn(x)qn with respect to a sequence
(pn), (qn) of disjoint projections. This corresponds to a double sided non-adapted stopping
time.
Corollary 5.4. Let1 < p , (vn), (wn) be sequences of bounded elements, then for allx Lp(N)
n
vnEn(x)wn
p
cp xp maxn
vnvn
12
,
n
vnvn
12
maxn
wnwn
12
,n
wnwn12
.
Proof: The case p = is obvious. Hence, we assume 1 < p < . Let y Lp(N) andchoose y1 L2p(N), y2 L2p(N) such that y = y1y2 and
yp = y122p = y222p .
Then, according to Lemma 3.4, we deduce from (DDp):
tr(n
vnEn(x)wny) = n
tr(xEn(wnyvn)) xp n En(wny1y2vn)p xp cp
n
wny1y1w
n
12
p
n
vny2y2wn
12
p
.
To conclude, we use [PX, Lemma 1.1], see also [JX],
n
wny1y1w
n
p
y1y1p max
n
wnwn
,
n
wnwn
= yp maxn
wnwn
,
nwnwn
.
A similar argument applies for the other term and therefore taking the supremum over all
y of norm 1 implies the assertion.
Remark 5.5. Let1 p < . The non-commutative Doob inequality from Theorem 0.2implies the vector-valued Doob inequality in Lp(, , ; Lp(N)).
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NON-COMMUTATIVE DOOB INEQUALITY 45
Proof: It suffices to treat the discrete case. Let N = L(, , )N and (n)nN bean increasing sequence of -subalgebras with conditional expectations (En). Let Nn =L(, , )N. Then the conditional expectation En onto Nn is given by En = En id. Let f Lp(, , ; Lp(N)) = Lp(N). According to Theorem 0.2 there exist a L2p(N), b L2p(N) and contractions (zn) Nsuch that
En idLp(N)(f) = aznb and a2p b2p cp fp .Hence, for every and n N
En(f)()Lp(N) = a()zn()b()Lp(N) a()L2p(N) z()Nb()L2p(N) a()L2p(N) b()L2p(N) .
Holders inequality implies the assertion
supn
En(f)()pLp(N) d()
1p
a()pL2p(N) b()p
L2p(N)d()
1p
aL2p(,,;L2p(N) bL2p(,,;L2p(N)= a2p b2p cp fp .
In the next application we want to relate group actions with (DDp). To illustrate this, we
consider a finite von Neumann algebra N and an increasing sequence (An) N of finitedimensional subalgebras with 1N An. Let Nn = An be the relative commutant of An inN. If Gn denotes the unitary group ofAn, we have a natural action : Gn B(Lp(N))
n(u)(x) = uxu
such that the conditional expectation on the commutant Nn is given by
En(x) = ENn(x) =
Gn
uxudn(u) .
Let G =
n Gn and the product measure, then
n
En(xn)
p
G
n
n(un)(xn)
p
p
d(u1, u2, ..)
1
p
.
We will show that for a sequence (xn) of positive elements even the right hand side can be
estimated by n xnp. For simplicity let us use the random variables n(x) : G Lp(N),given for = (u1, u2...) by
n(x)() = unxun .
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46 MARIUS JUNGE
For the special case of tensor products of finite dimensional von Neumann algebras the
following theorem implies (DDp).
Theorem 5.6. Let1 < p < and N, (An), (Nn) be as above and (xn) be a sequence ofpositive elements, then
G
n
n(xn)
p
p
d
1p
pn
xn
p
.
Here p is a constant which only depends on p.
Proof: The assertion is obvious for p = 1 and by interpolation as in Lemma 3.2 it suffices
to prove the assertion for p 4. Let (xn) be a finite sequence of positive elements. Then,we observe
G
n
n(xn)
p
p
d
1p
n
En(xn)
p
+
G
n
n(xn) En(xn)p
p
d
1p
cpn
xn
p
+
G
n
n(xn) En(xn)p
p
d
1p
.
Let n be the -algebra generated by the first n coordinates in G =
k Gk and Nn =L(G, n, )N L(G, , )N with the corresponding conditional expectation En.Then, we note
En1(n(xn)) = knGk
unxnun dn(un)dn+1(un+1) = En(xn) .
Hence the n-th martingale difference of
n n(xn) satisfies
dn = En(k
k(xk)) En1(k
k(xk)) = n(xn) En(xn) .
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NON-COMMUTATIVE DOOB INEQUALITY 47
We apply the non-commutative Rosenthal inequality, see [JX], in this case and obtain
1rp
IE n
n(xn) En(xn)p
p
1p
n
n(xn) En(xn)pp 1
p
+
n
En1(dndn + dndn)
12
p2
2
n
xnpp 1
p
+ 212
n
En(xnxn) + En(x
nxn)
12
p2
2n
xnpp1p
+ 21
2 c
1
2p2
n
xnxn + xnxn
12
p2
Let (n) be a sequence of independent Rademacher variables. Using the triangle inequality
and the orthogonality of the (n)s, we deduce as in [LP]
max
n
xnxn
12
p2
,
n
xnxn
12
p2
IE
n
nxn
2
p
12
.
By interpolation, we have n
xnpp
1p
n
xnxn
12
p2
.
However, since the xn are positive, we deduce for any choice of signs n with positivityn
nxn
p
n=1
xn
p
+
n=1
xn
p
2n
xn
p
.
Hence, we obtain
IE n
n(xn) En(xn)p
p
1p
rp(2 + 4c12p2
)
n
xn
p
.
The assertion is proved.
Remark 5.7. These methods can also be used to show that for every f Lp(G; Lp(N))there exist a, b L2p(; L2p(N)) and a sequence of contractions (yn) L()N such
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48 MARIUS JUNGE
that
Fn(f) = aynb and
a
2p
b
p
c2p
f
p
where
Fn(f)(g1, g2,...) = n(gn)f(g1,...,gn, ..) .
Note that the crossed product N(n)
Gn acts on Lp(; Lp(N)) and Fn somehow removes
the action of Gn on f. Similar results hold for a von Neumann algebra with a faithful
normal state and -invariant, strongly continuous group actions n : Gn Aut(N) ofcompact groups such that the centralizer algebras are increasing or decreasing.
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2000 Mathematics Subject Classification: 46L53, 46L52, 47L25.
Key-words: Non-commutative Lp-spaces, Doobs inequality
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL