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Noncommutative Lp-spaces
Qunahua Xu
Universite de Franche-Comte
Besancon – France
Lectures in the Summer School on
Banach spaces and Operator spaces
Nankai University – China
July 16 - July 20, 2007
Plan:
• Noncommutative Lp-spaces
• Noncommutative martingale inequalities
• Complete embedding of OH into a noncommutative L1
von Neumann algebras (VNA)
• B(H): the C*-algebra of bounded operators on a Hilbert space H. As aBanach space, B(H) is a dual space, whose predual is the class of traceoperators, which will be described later.
• VNA: a VNA on H is a C*-subalgebra M of B(H) which contains 1 andis w*-closed. So M is also a dual space whose predual is denoted by M∗.The functionals in M∗ are the normal functionals on M.
• Let M be a VNA. M+ denotes the positive part of M. Each element of Mis a linear combination of four positive elements.
• The modulus (or absolute value) of an operator x ∈ M is defined as|x| = (x∗x)1/2. Note that we have a second modulus: |x∗| = (xx∗)1/2.Because of the noncommutativity, |x| 6= |x∗| in general.
Traces
Now we come to our central notion: traces.
Definition. Let τ : M → C be a linear functional. τ is called
• positive if τ(M+) ⊂ R+;
• normal if τ ∈ M∗;
• faithful if x ∈ M+ , τ(x) = 0 ⇒ x = 0;
• tracial if τ(xy) = τ(yx) for all x, y ∈ M.
We consider sometimes infinite traces too. Below is the definition
Definition.
(i) A trace on M is a map τ : M+ → [0,∞] satisfying
• ∀ x, y ∈ M+, ∀ λ ∈ R+, τ(x + λy) = τ(x) + λτ(y);
• ∀ x ∈ M, τ(x∗x) = τ(xx∗).
(ii) A trace τ is said to be
• normal if supi τ(xi) = τ(supi xi) for any bounded increasing net (xi)in M+;
• faithful if τ(x) = 0 implies x = 0;
• finite if τ(1) < ∞;
• semifinite if for any non-zero x ∈ M+ there exists a non-zero y ∈ M+
such that y ≤ x and τ(y) < ∞.
(iii) M is called semifinite if it admits a normal semifinite faithful (n.s.f.)trace τ . In this case, we call M, (M, τ) = a noncommutative measurespace.
Properties of a trace.
Let τ be a n.s.f. trace on M.
• τ is nondecreasing: 0 ≤ x ≤ y ⇒ τ(x) ≤ τ(y).
• If τ is finite, τ(x) < ∞ for any x ∈ M+. In this case, τ extends to a linearfunctional on M. Moreover, τ(xy) = τ(yx) for all x, y ∈ M because of thepolarization identity:
xy =1
4
3∑
k=0
(−i)k(x∗ + iky)∗(x∗ + iky) .
Thus a finite trace is nothing but a tracial positive functional.
• For any family (ei)i∈I of projections in M
τ( ∨
i∈I
ei
) ≤∑i∈I
τ(ei).
• If (en) ⊂ is a decreasing sequence of projections, then
τ(∧n
en) = limn→∞
τ(en).
Convention. In the sequel, τ will be always a normal faithful tracial stateon M. We then call (M, τ) a noncommutative probability space.
Noncommutative Lp and examples
Definition. Let 0 < p < ∞ and x ∈ M. Define
‖x‖p =(τ(|x|p))1/p
.
Fact. ‖x‖p = ‖|x|‖p = ‖x∗‖p for any x ∈ M.
Proof. Let x = u|x| be the polar decomposition of x. Then u|x|u∗ = |x∗|.Then by induction, u|x|nu∗ = |x∗|n for every positive integer n, and souP (|x|)u∗ = P (|x∗|) for any polynomial P . It thus follows that u|x|pu∗ =|x∗|p. Therefore,
‖x∗‖pp = τ(|x∗|p) = τ(u|x|pu∗) = τ(u∗u|x|p) = τ(|x|p) = ‖x‖p
p .
2
We will see that ‖ ‖p is a norm or p-norm according to p ≥ 1 or p < 1. Thisallows us to introduce the noncommutative Lp:
Definition. Let p < ∞. The noncommutative Lp-space Lp(M, τ) asso-ciated with (M, τ) is defined as the completion of (M, ‖ ‖p).Set L∞(M) = M equipped with the operator norm, i.e., ‖x‖∞ = ‖x‖.Remark. The elements of Lp(M) can be described as measurable operators
like in the commutative case.
Example 1: Commutative case.
• (Ω, P ): a probability space.
• M = L∞(Ω): a VNA on H = L2(Ω) by multiplication.
• τ : M → C:
τ(f) =
∫
Ω
f dP.
• For p < ∞ and f ∈ L∞(Ω)
‖f‖pp =
∫
Ω
|f |p .
We thus recover the usual Lp-spaces.
Example 2: Schatten classes.
• M = B(H).
• τ : B(H)+ → [0,∞] the usual trace: given (ξi)i∈I an orthonormal basis ofH, we have
Tr(x) =∑
i
〈xξi, ξi〉.
• If x ∈ B(H)+ is of finite trace, i.e. Tr(x) < ∞, then x is compact. Thusthere exists an orthonormal sequence (ξn) ⊂ H such that
xξ =∑n≥0
λn(x) 〈ξ, ξn〉, ξ ∈ H.
It thus follows thatTr(x) =
∑n≥0
λn(x),
where the λn(x) denote the eigenvalues of x, repeated according to theirmultiplicities.
• We then deduce that Lp(B(H)) consists of all compact operators on Hsuch that
Tr(|x|p) =∑n≥0
λn(|x|p) =∑n≥0
(λn(|x|))p
< ∞.
Set Sp(H) = Lp(B(H)), the Schatten classes associated with H.Sp = Sp(`2) and Sn
p = Sp(`n2 ). Sp is the noncommutative analogue of `p.
• S1(H) = trace class and S2(H) = Hilbert-Schmidt class.
• The duality theorem implies
S1(H)∗ = B(H).
• Let (αi) ⊂ C. Then∥∥ ∑
i
αieij
∥∥p
=( ∑
i
|αi|2)1/2
=∥∥ ∑
i
αieji
∥∥p
for every fixed j and∥∥ ∑
i
αieii
∥∥p
=( ∑
i
|αi|p)1/p
.
Example 3: Tensor product.
• (M1, τ1) and (M2, τ2): two noncommutative probability spaces.
• M = M1⊗M2: the VNA tensor product.
• τ = τ1 ⊗ τ2: the tensor trace, determined by
τ(x1 ⊗ x2) = τ1(x1) τ2(x2), x1 ∈ M1, x2 ∈ M2.
• (M, τ) is again a noncommutative probability space.
Let us consider two special cases.
Case 1.
• (M, τ) a noncommutative probability space; (Ω, P ) a probability space.
• (N, ν) = L∞(Ω, P )⊗(M, τ) with ν given by
ν(x) =
∫
Ω
τ(x(ω)) dP (ω), x ∈ N.
• For 0 < p < ∞
ν(|x|p) =
∫
Ω
τ(|x(ω)|p) dP (ω) =
∫
Ω
‖x(ω)‖pp dP (ω).
Thus Lp(N) coincides isometrically with Lp(Ω; Lp(M)), the usual Lp-spaceof p-integrable functions from Ω to Lp(M). This example is of particularimportance M = Mn equipped with the normalized trace. Then we getrandom matrices, which play a paramount role in quantum probability,quantum mechanics and mathematical physics.
Case 2.
• (N, ν) = (B(`2), Tr)⊗(M, τ). Let x = (xij) ∈ N+. Then
ν(x) =∑
i
τ(xii) .
• Let (ai) be a finite sequence in Lp(M). Then
∥∥ ∑i
ei1 ⊗ ai
∥∥Lp(N)
=∥∥( ∑
i
a∗i ai
)1/2∥∥Lp(M)
,
∥∥ ∑i
e1i ⊗ ai
∥∥Lp(N)
=∥∥( ∑
i
aia∗i
)1/2∥∥Lp(M)
,
∥∥ ∑i
eii ⊗ ai
∥∥Lp(N)
=( ∑
i
‖ai‖pLp(M)
)1/p.
Extension to infinite tensor product.
The previous tensor product construction is readily extended to a finite num-ber of von Neumann algebras. The resulting tensor product is clearly asso-ciative. We can also consider infinite tensor product. So let (Mn, τn)n≥1 bea sequence of noncommutative probability spaces. Using GNS construction,we can define the infinite tensor product
(N, ν) =⊗
n≥1(Mn, τn).
Usually, Mn is regarded as a von Neumann subalgebra of N via the inclusionxn 7→ 1M1 ⊗ · · · 1Mn−1 ⊗ xn ⊗ 1Mn+1 · · · . Then the trace ν coincides with τn
on Mn.
A particular case.
• Mn = M2, τn = tr2, the normalized trace on M2.
• Then(M2, tr2)
⊗n = (M2n tr2n);
The embedding of M2n into M2n+1 is realized by
M2n 3 x 7→ x⊕ x =
(x 00 x
)∈M2n+1 .
• (N, τ) = (M2, tr2)⊗N is the hyperfinite II1 factor.
Example 3: Clifford von Neumann algebras.
• Let
P =
(1 00 −1
)and Q =
(0 11 0
).
These are two Pauli matrices. Note that PQ = −QP .
• Let (N, τ) = (M2, tr2)N. Consider the following operators in N:
ε1 = P and εn = Q⊗ · · · ⊗Q︸ ︷︷ ︸(n− 1) times
⊗P, n ≥ 2.
Then ε∗n = εn and
(CAR) εm εn + εn εm = 2δm,n 1 , m, n ∈ N.
• For A ⊂ N set
ε∅ = 1 if A = ∅ and εA = εk1 · · · εkn ,
where k1, ..., kn are the elements of A ranged in increasing order.
• We have: τ(εA) = 0 if A 6= ∅ and τ(ε∅) = 1.
• Let R0 be the ∗-subalgebra of N generated by the εn. Then
R0 = spanεA : A ⊂ N, |A| < ∞.
• Let R be the w*-closure of R0. R is a Clifford von Neumann algebra. So(R, τ
∣∣R) becomes a noncommutative probability space.
• (CAR) implies that the εA form an orthonormal basis of L2(R).
• Given any finite sequence (αn) ⊂ R, using (CAR), we find( ∑
n
αn εn
)∗( ∑n
αn εn
)=
∑n
α2n , so
∥∥ ∑n
αn εn
∥∥∞ =
( ∑n
α2n
)1/2.
It then follows that∥∥ ∑
n
αn εn
∥∥p∼cp
( ∑n
|αn|2)1/2
.
for any finite sequence (αn) ⊂ C. Consequently, the subspace of Lp(R)generated by (εn) is isomorphic to `2.
Holder inequality and Minkowski inequalities
Theorem (Holder inequality).Let 1 ≤ p < ∞ and p′ be the conjugate index of p. Let ‖ ‖∞ be the operatornorm. Then
|τ(xy)| ≤ ‖x‖p‖y‖p′ , x, y ∈ M.
Proof. Consider the special case: x and y are positive linear combinationsof mutually orthogonal projections. Then we can write:
x =n∑
k=1
αkek and y =n∑
k=1
βkfk,
where αk, βk ∈ R+ and (ek) and (fk) are increasing sequences of projections.Then
τ(xy) =n∑
j,k=1
αjβkτ(ejfk) ≤n∑
j,k=1
αjβk min(τ(ej), τ(fk)
).
Now let x, y : R+ → R+ be defined by
x =n∑
k=1
αk1l[0,τ(ek)] and y =n∑
k=1
βk1l[0,τ(fk)] .
Then‖x‖p = ‖x‖Lp(R+) and ‖y‖p′ = ‖y‖Lp′ (R+) .
On the other hand,
∫ ∞
0
x(t)y(t)dt =n∑
j,k=1
αjβk min(τ(ej), τ(fk)
).
Thus using the classical Holder inequality for functions, we deduce
τ(xy) ≤ ‖x‖Lp(R+)‖y‖Lp′ (R+) = ‖x‖p‖y‖p′ .
2
Corollary. We have
‖x‖p = sup|τ(xy)| : y ∈ M, ‖y‖p′ ≤ 1, x ∈ M.
Consequently, the Minkowski inequality holds:
‖x + y‖p ≤ ‖x‖p + ‖y‖p , x, y ∈ M,
so ‖ ‖p is a norm on M.
Proof. The Holder inequality implies
sup|τ(xy)| : y ∈ M, ‖y‖p′ ≤ 1 ≤ ‖x‖p .
We claim that the supremum above is attained for y = |x|p−1u∗/‖x‖p−1p ,
where u is the partial isometry in the polar decomposition of x. Indeed, wehave
yy∗ =|x|p−1u∗u|x|p−1
‖x‖2p−2p
=|x|2p−2
‖x‖2p−2p
.
It then follows that ‖y‖p′p′ = τ(|y∗|p′) = 1. On the other hand,
τ(xy) =τ(x|x|p−1u∗)
‖x‖p−1p
=τ(|x|p)‖x‖p−1
p
= ‖x‖p .
Thus we get our claim. 2
Remark.
(i) The Holder inequality implies that the trace τ can be extended to acontractive linear functional on L1(M). The extension possesses allproperties of the initial τ on M:
• τ(x) ≥ 0, x ∈ L+1 (M);
• τ(x∗) = τ(x), x ∈ L1(M);
• τ(xy) = τ(yx), x ∈ Lp(M), y ∈ Lp′(M).
(ii) L2(M) is a Hilbert space with respect to the scalar product: 〈x, y〉 =τ(y∗x).
Duality
Theorem. Let 1 < p < ∞. Then (Lp(M))∗ = Lp′(M) isometrically withrespect to the following duality
(x, y) = τ(xy), x ∈ Lp(M), y ∈ Lp′(M).
Consequently, Lp(M) is reflexive for 1 < p < ∞.
Proof. Case p = 1. Given x ∈ L1(M) define ϕx : M → C by ϕx(y) = τ(xy).Then ϕx is a continuous linear functional on M and ‖ϕx‖ = ‖x‖1. Thus themap x 7→ ϕx is isometric from L1(M) into the dual space M∗ of M. By thenormality of τ , it is easy to show that ϕx ∈ M∗. Thus
ϕx : x ∈ L1(M) ⊂ M∗.
We claim that this inclusion is an equality. Indeed, let y ∈ (M∗)∗ = Msuch that ϕx(y) = 0 for all x ∈ L1(M). In particular, ϕy∗(y) = 0. Namely,τ(y∗y) = 0. Then the faithfulness of τ implies y∗y = 0, i.e., y = 0. Thisyields our claim. Therefore, the map x 7→ ϕx establishes an isometry fromL1(M) onto M∗. Since (M∗)∗ = M, we deduce the desired duality equalitythat (L1(M))∗ = M.
In the case 1 < p < ∞ the same argument shows that Lp′(M) is an isometricsubspace of the dual (Lp(M))∗ of Lp(M). To prove this isometry is surjective,we need to show that Lp(M) is reflexive. This latter result will be an imme-diate consequence of the following noncommutative Clarkson inequality.
Theorem (Clarkson inequality). Let 2 ≤ p ≤ ∞. Then
(∥∥x + y
2
∥∥p+
∥∥x− y
2
∥∥p)1/p ≤ (1
2(‖x‖p + ‖y‖p)
)1/p, x, y ∈ Lp(M).
Proof. Let N = M ⊕ M and define ν : N → C by ν(x, y) = τ(x) + τ(y).Then Lp(N) = Lp(M)⊕p Lp(M). Now consider T : Lp(N) → Lp(N)
T (x, y) =(x + y
2,
x− y
2
).
Then
∥∥T : L∞(N) → L∞(N)∥∥ ≤ 1 (triangle inequality)
∥∥T : L2(N) → L2(N)∥∥ ≤ 1√
2(parallelogram identity)
Therefore, by the Riesz interpolation
∥∥T : Lp(N) → Lp(N)∥∥ ≤ 1
21/p, 2 < p < ∞.
2
Corollary. Lp(M) is reflexive for any 2 ≤ p < ∞.
Theorem (Riesz interpolation theorem). Let 1 ≤ p0 < p1 ≤ ∞. Let T : M →L1(M) be a linear map such that
Ck = sup‖Tx‖pk: x ∈ M, ‖x‖pk
≤ 1 < ∞, k = 0, 1.
Then T extends to a bounded map on Lp(M) for every p ∈ (p0, p1). Moreover,the norm of T on Lp(M) is controlled by C1−θ
0 Cθ1 , where θ is determined by
1/p = (1− θ)/p0 + θ/p1 .
End of the proof of the duality theorem: 1 < p < ∞. Let x ∈ Lp′(M) anddefine ϕx : Lp(M) → C by ϕx(y) = τ(xy). Then ‖ϕx‖ = ‖x‖p′ , and thus themap x 7→ ϕx identifies isometrically Lp′(M) as a closed subspace of (Lp(M))∗.Suppose for the moment that p ≥ 2. Then by the previous corollary, Lp(M)is reflexive. Let y ∈ ((Lp(M))∗)∗ = Lp(M) be an element annihilating thesubspace Lp′(M), i.e.,
τ(xy) = ϕx(y) = 0, ∀ x ∈ Lp′(M).
Then ‖y‖p = 0. Thus y = 0. It follows that Lp′(M) = (Lp(M))∗. Conse-quently, Lp′(M) is also reflexive, so the last duality equality holds for p < 2too. 2
More properties
Noncommutative Lp-spaces share many properties with the usual (commu-tative) Lp. Below is a list of such properties:
• Interpolation. Let 1 ≤ p0 < p1 ≤ ∞, 0 < θ < 1, and let p be determinedby 1/p = (1− θ)/p0 + θ/p1. Then
(Lp0(M), Lp1(M)
)θ
= Lp(M) with equal norms.
We have a similar result for real interpolation.
• Uniform convexity and uniform smoothness. Lp(M) is uniform con-vex and uniform smooth for 1 < p < ∞.
• Type and cotype. Let 1 ≤ p < ∞. Then Lp(M) is of type min(p, 2)and cotype max(p, 2).
• UMD property. Lp(M) is a UMD space for 1 < p < ∞.
Differences from commutative Lp-spaces
Despite the strong similarity between noncommutative Lp-spaces and com-mutative counterparts. They are, however, very different. Below are tworemarkable differences:
• Unconditional basis. The Schatten Sp fails to have any unconditionalbasis when p 6= 2, in sharp contrast with `p. Moreover, Sp doe not haveany local unconditional structure.
• Stability. Let 1 ≤ p < ∞, p 6= 2. Then Lp(M) is stable iff M is of type I.
A VNA is of type I if it is a direct sum of algebras of the form L∞(Ω)⊗B(H).
Conditional expectations
(M, τ) = a noncommutative probability space.
• N ⊂ M : a von Neumann subalgebra, i.e., N is a C*-subalgebra of M whichis w*-closed and contains the identity of M.
• τ∣∣N: a normal faithful tracial state on N; so (N, τ
∣∣N) is again a noncommu-
tative probability space.
• It is clear that the noncommutative Lp-space Lp(N) = Lp(N, τ∣∣N) ⊂ Lp(M)
isometrically.
Let ι : L1(N) → L1(M) be the natural inclusion and E = ι∗ : M → N be theadjoint of ι.
Proposition and Definition.E : M → N satisfies the following properties:
• E is a normal contractive positive projection;
• E(axb) = aE(x)b for any x ∈ M and a, b ∈ N;
• τ E = τ .
The map E is called the conditional expectation of M with respect toN and denoted by E( ·
∣∣N).
Proposition. E extends to a contractive positive projection from Lp(M)onto Lp(N) for any 1 ≤ p < ∞, still denoted by E .
Proof. Let x ∈ M and y ∈ N. Then
|τ(E(x)y)| = |τ(xy)| ≤ ‖x‖p ‖y‖p′ .
It follows that‖E(x)‖p ≤ ‖x‖p .
Therefore, E extends to a contraction on Lp(M). 2
Remark.
• E is the orthogonal projection from L2(M) onto L2(N).
• E : M → N satisfies the Cauchy-Schwarz inequality:
E(x)∗E(x) ≤ E(x∗x), ∀ x ∈ M.
Noncommutative martingales
We consider
• (Mn)n≥1: an increasing sequence of von Neumann subalgebras of M suchthat ∪nMn is w*-dense in M. This is called filtration of M.
• En = E(·∣∣Mn): conditional expectation relative to Mn. Then
Em En = En Em = Emin(m,n), ∀ m,n ≥ 1.
Definition.
• A sequence x = (xn) ⊂ L1(M) is called a martingale with respect to(Mn) if En(xn+1) = xn for every n ≥ 1.
• If in addition xn ∈ Lp(M) with p ≥ 1, x is called an Lp-martingale withrespect to (Mn). In this case we set
‖x‖p = supn≥1
‖xn‖p .
If ‖x‖p < ∞, x is called a bounded Lp-martingale.
• Let x = (xn) be a martingale with respect to (Mn). Define dxn = xn−xn−1
for n ≥ 1 with the convention that x−1 = 0, and let dx = (dxn)n≥1. Thedxn are called the martingale differences of x, and dx the martingaledifference sequence of x.
Convention. If non confusion can occur, we will simply say martingales ornoncommutative martingales instead of martingales with respect to (Mn).
Remark.
• Let x = (xn) be an Lp-martingale. Then (xn) is adapted in the sense thatxn ∈ Lp(Mn) for all n. Moreover, by induction,
Em(xn) = xm , ∀ m < n.
• Since En is contractive on Lp(M),
‖xn‖p = ‖En(xn+1)‖p ≤ ‖xn+1‖p .
Thus the sequence (‖xn‖p) is increasing, so
‖x‖p = limn→∞
‖xn‖p .
• dx satisfies the following:
(i) (dxn) is adapted;
(ii) En−1(dxn) = 0 for every n ≥ 2.
Conversely, any sequence (dn) in L1(M) verifying these two properties isthe martingale difference sequence of a martingale: Indeed, let
xn =n∑
k=1
dk.
• dx is L2(M) is orthogonal. Indeed, using trace preserving of conditionalexpectations, one has, for m < n
τ(dx∗mdxn) = τ(Em(dx∗mdxn)) = τ(dx∗mEm(dxn)) = 0.
Mean convergence of martingales.
• Let x∞ ∈ Lp(M) with 1 ≤ p ≤ ∞, and let xn = En(x∞). Then x = (xn) is abounded Lp-martingale and xn converges to x∞ in Lp(M) (in w*-topologyfor p = ∞). Moreover, ‖x‖p = ‖x∞‖p.
• Conversely, let x = (xn) be a bounded Lp-martingale with 1 < p ≤ ∞.Then there exists x∞ ∈ Lp(M) such that xn = En(x∞) for every n.
Proof. Consider only the case of p < ∞. Since⋃
n Mn is w*-closed in M,⋃n Mn is dense in Lp(M), and so is ∪nLp(Mn). Therefore, using the fact that
the En are contractions on Lp(M), we need to prove the assertion only in thecase where x∞ ∈ ⋃
n Lp(Mn). Thus assume x∞ ∈ Lp(MN) for some N . Thenxn = xN for all n ≥ N , so xn → x∞.
For the second part, recall that Lp(M) is a dual space for p > 1. Thus theunit ball of Lp(M) is w*-compact. Consequently, if x = (xn) is a boundedLp-martingale with 1 < p ≤ ∞, (xn) has a w*-limit x∞ in Lp(M). Then it iseasy to show that xn = En(x∞) 2
Remark. The space of all bounded Lp-martingales, equipped with ‖ ‖p, isisometric to Lp(M) for p > 1. This permits us to not distinguish a martingalex and its final value x∞ (if the latter exists).
Examples
Example 1: Semi-noncommutative case.
• (Ω,F , P ) = a probability space; (N, ν) = a noncommutative probabilityspace. Set M = L∞(Ω,F , P )⊗N and τ = P ⊗ν.
• (Fn): a filtration of σ-subalgebras of F such that the union⋃
nFn gener-ates F .
• Mn = L∞(Ω,Fn, P )⊗N. Then (Mn) is a filtration of M.
• ThenEn = En ⊗ idLp(N) with En = E(·
∣∣Fn) .
• Thus in this case the noncommutative martingale theory reduces to theusual martingale theory but with values in noncommutative Lp-spaces. IfN = C, we recover the theory of usual martingales.
Example 2: Filtration of matrix algebras.
• M = B(`2) equipped with the usual trace Tr; so we are no longer ina noncommutative probability space. However,by approximation we canconsider only finite matrices of an arbitrary order N and the normalizedtrace on B(`N
2 ).
• Set Mn = B(`n2 ) and M∞ = B(`2). Then Mn ⊂ M∞ by viewing an n × n
matrix as an infinite one whose left upper corner of size n × n is thegiven n × n matrix and all other entries are zero. Note that Mn is not aunital subalgebra of M∞. The unit of Mn is the projection en ∈M∞ whichprojects a sequence in `2 into its n first coordinates.
• Let en be the natural projection from `2 onto `n2 . Then
En(x) = enxen, x ∈M∞ .
Again, En extends to a contractive projection from Sp onto Snp .
Example 3: Clifford martingales.
Let R be the Clifford von Neumann algebra constructed generated by a(CAR) sequence (εn). Let Rn be the subalgebra generated by ε1, · · · , εn.Then (Rn) is a filtration of R. The corresponding martingales are calledClifford martingales. Their structure is quite simple.
Noncommutative martingale inequalities
• The theory of noncommutative martingale inequalities is not new and goesback to 70’s.
• Very few major results were obtained before 90’s.
• Difficulties: the usual techniques from the classical martingale theory, suchas maximal functions, stopping times, etc., are no longer available.
• New period of the development started since the work of Pisier/Xu in 96.
• Since then many inequalities in the classical martingale theory have beentransferred into the noncommutative setting. Below are some names in-volved in the theory: Junge, Lust-Piquard, Mei, Musat, Parcet, Pisier,Randrianantoanina, Xu.
• Reasons of this remarkable new development: motivations from and inter-actions with operator space theory and quantum probability.
Throughout this lecture, (M, τ) always denotes a noncommutative probabilityequipped with a filtration (Mn). All martingales are with respect to (Mn).
Noncommutative Burkholder-Gundy inequalities
Recall the classical Burkholder-Gundy inequalities. Let 1 < p < ∞ andf = (fn) be a bounded Lp-martingale on a probability space (Ω,F , P ). Then
‖f‖p ∼cp
∥∥( ∑n
|dfn|2)1/2∥∥
p.
The function( ∑
n |dfn|2)1/2
is traditionally called the square function off and denoted by S(f).
Theorem. (Burkholder-Gundy inequality: p ≥ 2) Let 2 ≤ p < ∞ and x be afinite Lp-martingale. Then
‖x‖p ∼cp
∥∥( ∑n
dx∗ndxn
)1/2∥∥p,
∥∥( ∑n
dxndx∗n)1/2∥∥
p
.
Square functions. Now we have two square functions:
Sc(x) =( ∑
n
dx∗ndxn
)1/2and Sr(x) =
( ∑n
dxndx∗k)1/2
.
Then the previous Burkholder-Gundy inequality can be restated as
‖x‖p ∼cp
∥∥Sc(x)∥∥
p,
∥∥Sr(x)∥∥
p
.
Warn. Although ‖a‖p = ‖a∗‖p for a single operator a ∈ Lp(M), for a finitesequence (an) ⊂ Lp(M) the two norms
∥∥( ∑n
|an|2)1/2∥∥
pand
∥∥( ∑n
|a∗n|2)1/2∥∥
p
are not comparable at all if p 6= 2. For example, let M = B(`2) with theusual trace, and let an = en 1. Then clearly
∥∥( N∑n=1
a∗nan
)1/2∥∥p
= N1/2 and∥∥( N∑
n=1
ana∗n)1/2∥∥
p= N1/p .
Theorem. (Burkholder-Gundy inequality: p < 2) Let 1 < p < 2 and x be afinite Lp-martingale. Then
‖x‖p ∼cp inf∥∥( ∑
n
dy∗ndyn
)1/2∥∥p+
∥∥( ∑n
dzndz∗n)1/2∥∥
p
,
where the infimum runs over all finite Lp-martingales y and z such that x = y+z.
Remark.
• This form for p < 2 is more much complicated and subtler than that forp ≥ 2. It is the source of many difficulties.
• The Burkholder-Gundy inequalities were proved by Pisier/Xu in 96 by acombinatory method. An alternate approach was later given Randrianan-toanina. The optimal orders of the best constants are also determined byJunge/Xu and Randrianantoanina.
Martingale transforms and UMD property
Since Sc(x) and Sr(x) are unconditional in dx, the Burkholder-Gundy in-equalities immediately imply the following
Theorem. (Martingale transforms)Let 1 < p < ∞ and x = (xn) be a bounded Lp-martingale. Then
(UMDp)∥∥ ∑
n
εndxn
∥∥p≤ cp
∥∥ ∑n
dxn
∥∥p, ∀ εn = ±1.
Theorem. (Weak type (1, 1) inequality of Martingale transforms;Randrianantoanina)For any bounded L1-martingale x = (xn)
τ[1l(λ,∞)(
∣∣n∑
k=1
εkdxk
∣∣)] ≤ c ‖x‖1
λ, ∀ εn = ±1 , ∀ n ∈ N, ∀ λ > 0.
Remark. This weak type (1, 1) inequality implies (UMDp) by virtue of:
• the type (2, 2) inequality: (UMDp) is trivial for p = 2.
• the Marcinkiewicz interpolation theorem.
UMD property
Now we specify the martingale transforms to the case of commutative mar-tingales with values in noncommutative Lp-spaces:
• (Ω,F , P ) = a probability space; (N, ν) = a noncommutative probabilityspace. Set M = L∞(Ω,F , P )⊗N and τ = P ⊗ν.
• (Fn): a filtration of σ-subalgebras of F such that the union⋃
nFn gener-ates F .
• Mn = L∞(Ω,Fn, P )⊗N. Then (Mn) is a filtration of M.
• ThenEn = En ⊗ idLp(N) with En = E(·
∣∣Fn) .
• Thus in this case the noncommutative martingale theory reduces to theusual martingale theory but with values in noncommutative Lp-spaces.
• Then (UMDp) yields: for any bounded Lp-martingale f = (fn) with respectto (Fn) with values in Lp(N), 1 < p < ∞
∥∥ ∑n
εndfn
∥∥p≤ cp
∥∥f∥∥
p, ∀ εn = ±1.
• This means that Lp(N) is a UMD space for every 1 < p < ∞ with constantdepending only on p.
Noncommutative Khintchine inequalities
(εn)n≥1: a Rademacher sequence on a probability space (Ω, P ).
• The classical Khintchine inequalities: for any finite sequence (αn) ⊂ C∥∥ ∑
n
αnεn
∥∥p∼cp
( ∑n
|αn|2)1/2
=∥∥ ∑
n
αnεn
∥∥2.
• By Fubini, we deduce the following (with Lp = Lp(0, 1)): for all finitesequences (an) in Lp
∥∥ ∑n≥1
anεn
∥∥Lp(Ω;Lp)
∼cp
∥∥( ∑n≥1
|an|2)1/2∥∥
Lp.
Aim: Extend this last inequality to the noncommutative setting.
Theorem. (Noncommutative Khintchine inequalities)Let 1 ≤ p < ∞ and (an) be a finite sequence in Lp(M).
• if 2 ≤ p < ∞, then∥∥ ∑
n
εnan
∥∥Lp(Ω;Lp(M))
∼cp max∥∥(an)
∥∥Lp(M;`c
2),
∥∥(an)∥∥
Lp(M;`r2)
;
• if 1 ≤ p ≤ 2, then∥∥ ∑
n
εnan
∥∥Lp(Ω;Lp(M))
∼c inf∥∥(bn)
∥∥Lp(M;`c
2)+
∥∥(cn)∥∥
Lp(M;`r2)
,
where the infimun runs over all decompositions an = bn + cn with bn and cn
in Lp(M).
This theorem is due to Lust-Piquard in the case 1 < p < ∞ and to Lust-Piquard and Pisier for p = 1.
Proof of the case 1 < p < ∞: Note that (εnan) is a martingale differ-ence sequence with respect to the filtration generated by the εn. Thus thenoncommutative Burkholder-Gudndy inequalities imply the noncommutativeKhintchine inequalities. The revelant constants depend on p. 2
Proposition. Let 1 ≤ p ≤ ∞ and (an) ⊂ Lp(M) be a finite sequence.
• If 2 ≤ p ≤ ∞, then
max∥∥(an)
∥∥Lp(M;`c
2),
∥∥(an)∥∥
Lp(M;`r2)
≤ ( ∑n
‖an‖2p
)1/2.
• If 1 ≤ p ≤ 2, then
min∥∥(an)
∥∥Lp(M;`c
2),
∥∥(an)∥∥
Lp(M;`r2)
≥ ( ∑n
‖an‖2p
)1/2.
Proof. Let p ≥ 2. By the triangle inequality in Lp/2(M) we have
∥∥(an)∥∥2
Lp(M;`c2)
=∥∥ ∑
n
|an|2∥∥
p/2≤
∑n
‖an‖2p .
Passing to adjoints yields the row norm inequality. The case p < 2 thenfollows by duality. 2
Corollary. Let (ϕn) be an orthonormal sequence in L2(Ω) and (an) a finitesequence in Lp(M). Then for 2 ≤ p ≤ ∞
max∥∥(an)
∥∥Lp(M;`c
2),
∥∥(an)∥∥
Lp(M;`r2)
≤∥∥ ∑
n
ϕnan
∥∥L2(Ω;Lp(M))
and for 1 ≤ p ≤ 2
∥∥ ∑n
ϕnan
∥∥L2(Ω;Lp(M))
≤ inf∥∥(bn)
∥∥Lp(M;`c
2)+
∥∥(cn)∥∥
Lp(M;`r2)
,
where the infimun runs over all decompositions an = bn + cn with bn and cn inLp(M).
Remark.
• The Rademacher sequence (εk) in the noncommutative Khintchine inequal-ities can be replaced by a standard Gaussian sequence.
• Khintchine type inequalities for noncommutative random variables: a se-quence of free generators, a semicircular system and a CAR sequence.
Noncommutative Doob maximal inequality
• The classical Doob maximal inequality: for any bounded Lp-martingalef = (fn) on a probability space (Ω,F , P ) with 1 < p ≤ ∞
∥∥ supn≥1
|fn|∥∥
p≤ c
p− 1‖f‖p .
• A first difficulty in the noncommutative setting: the noncommutative ana-logue of the pointwise maximal function is no longer available.
• Instead of sup |fn|, we can consider upper bounds of the |fn|.
Theorem. (Noncommutative Doob maximal inequality; Junge)Let x = (xn) be a positive bounded Lp-martingale, 1 < p ≤ ∞. Then thereexists a ∈ L+
p (M) such that
xn ≤ a, ∀ n ≥ 1 and ‖a‖p ≤ cp ‖x‖p .
Remark.
• By decomposing a bounded Lp-martingale into four positive martingales,we can, of course, extend this result to the general case. But this approachhas a drawback because of the two different moduli of operators.
• It is more convenient to use the space Lp(M; `∞) to treat general martin-gales.
The spaces Lp(M; `∞)
Let 1 ≤ p ≤ ∞. Define Lp(M; `∞) as the space of all sequences x = (xn)n≥1
in Lp(M) which admit a factorization of the following form: there are a, b ∈L2p(M) and a bounded sequence y = (yn) ⊂ L∞(M) such that
xn = aynb , ∀ n ≥ 1
with the norm
‖x‖Lp(M;`∞) = infxn=aynb
‖a‖2p supn≥1
‖yn‖∞ ‖b‖2p
.
Lp(M; `∞) is a vector-valued noncommutative Lp-space. It is introduced firstby Pisier for hyperfinite M and then by Junge in the general case.
Remark. Consider a positive sequence x = (xn) in Lp(M). Assume thatthere exists a ∈ L+
p (M) such that
xn ≤ a, ∀ n ≥ 1.
Then x1/2n = una1/2 for some contraction un ∈ M, so xn = a1/2u∗nuna1/2. Thus
x ∈ Lp(M; `∞) and‖x‖Lp(M;`∞) ≤ ‖a‖p.
The converse is also true. Therefore, a positive sequence x belongs toLp(M; `∞) iff there exists a ∈ L+
p (M) such that
xn ≤ a, ∀ n ≥ 1.
If this is the case, then
‖x‖Lp(M;`∞) = inf‖a‖p : a ∈ L+
p (M) s.t. xn ≤ a, ∀ n ≥ 1.
Remark. Lp(M; `∞) is not closed under absolute value, so
‖(xn)‖Lp(M;`∞) 6= ‖(|xn|)‖Lp(M;`∞) in general.
Theorem. (Noncommutative Doob maximal inequality continued)Let x = (xn) be a bounded Lp-martingale, 1 < p ≤ ∞. Then
‖x‖Lp(M;`∞) ≤ cp supn‖xn‖p .
Proof of the maximal Doob inequality
We sketch the alternate proof by Junge-Xu. It is based on Cuculescus weaktype (1, 1) inequality via a Marcinkiewicz type interpolation.
Theorem. (Cuculescu) Let x = (xn) be a bounded positive L1-martingale andλ > 0. Then there exists a projection e ∈ M such that
exne ≤ λe , ∀ n ≥ 1 and τ(e⊥) ≤ ‖x‖1
λ.
Proof. We define a sequence of projections (en)n≥0 by induction. First lete0 = 1. Then for every n ≥ 1 define
en = 1l(0,λ](en−1xnen−1).
Then (en) is decreasing. Let e =∧
n≥1 en. Then
exne = e(enxnen)e ≤ λ e ≤ λ.
Thus it remains to show the trace estimate of e⊥. To this end, fix n ≥ 1.Since xn ≥ 0, τ(enxn) ≥ 0. On the other hand, by the definition of ek,
(ek−1 − ek)(ek−1xkek−1)(ek−1 − ek) ≥ λ (ek−1 − ek).
Therefore,
τ(xn) = τ(enxn) +n∑
k=1
τ((ek−1 − ek)xn
)
≥n∑
k=1
τ(Ek[(ek−1 − ek)xn]
)
=n∑
k=1
τ((ek−1 − ek)xk
)
=n∑
k=1
τ((ek−1 − ek)(ek−1xkek−1)(ek−1 − ek)
)
≥ λ
n∑
k=1
τ(ek−1 − ek) = s τ(1− en).
Theorem. (Marcinkiewicz type interpolation; Junge/Xu) Let Sn : L+1 (M) →
L+1 (M) be a map for every n ≥ 1, and let S = (Sn)n≥1. Assume
• S is subadditive: Sn(x + y) ≤ Sn(x) + Sn(y) for all n ≥ 1.
• S is of weak type (1, 1): ∃ c1 s.t. ∀ x ∈ L+1 (M), ∀ λ > 0 ∃ e ∈ M s.t.
e(Sn(x)
)e ≤ λ, ∀ n ≥ 1 and τ(e⊥) ≤ c1 ‖x‖1
λ.
• S is of type (∞,∞): ∃ c∞ s.t.
‖Sn(x)‖∞ ≤ c∞ ‖x‖∞ , ∀ x ∈ L+∞(M), ∀ n ∈ N.
Then S is of type (p, p) for any 1 < p < ∞.
Proof of the maximal Doob inequality. First consider a positive bounded Lp-martingale x. Since p > 1, x has a final value x∞ in Lp(M): xn = En(x∞).Thus inequality (D+
p ) follows immediately from Cuculescu’s inequality andthe Marcinkiewicz type interpolation.
Decomposing any bounded Lp-martingale into a linear combination of fourpositive martingales, we deduce the general case. 2
Remark. Junge-Xu establish noncommutative maximal ergodic inequalities.
Operator space structure on noncommutative Lp
We describe the natural operator space structure on Lp introduced by Pisier.
• p = ∞: L∞(M) = M has its natural o.s.s. as a VNA. This
• p = 1: Consider the opposite VNA Mop. If M acts on H, then Mop acts onH∗ and
Mop = xt : x ∈ M.The map x 7→ x∗ is an isomorphism between M and Mop. Thus L1(M) isisometric to L1(M
op) at the Banach space level. This allows us to equipL1(M) with the o.s.s. inherited from (Mop)∗.
• 1 < p < ∞: The o.s.s. of Lp(M) is obtained by complex interpolation:
Lp(M) =(L∞(M), L1(M)
)1p
.
• L2(M) is an OH.
Two particular cases:
• Commutative Lp. Lp(Ω) is equipped with its natural o.s.s., so does `p.We have `2 = OH.
• Schatten classes. The same happens to the Schatten classes Sp. If1 ≤ p < ∞, the dual space of Sp is Sp′ with respect to the so-calledparallel duality bracket
〈x, y〉 = Tr(xyt) =∑i,j
xijyij, x = (xij) ∈ Sp, y = (yij) ∈ Sp′ .
Column and row subspaces of Sp:
• Cp: the first column subspace of Sp; so C∞ = C.
• Rp: the first row subspace of Sp; so R∞ = R.
• Cp ' Rp ' `2 isometrically as Banach spaces.
• Cp and Rp are not completely isomorphic for p 6= 2.
• We have the following completely isometric identities:
(Cp)∗ ∼= Cp′
∼= Rp and (Rp)∗ ∼= Rp′
∼= Cp
by identifying the canonical bases in question.
Vector-valued Schatten classes
Aim: Define Sp[E] for an o.s. E.
• S∞[E]: S∞[E] = S∞ ⊗min E. Note that Sn∞[E] = Mn(E).
• S1[E]: S1[E] = S1⊗E (o.s. projective tensor product). Alternately, wecan use duality. Let u ∈ S1 ⊗ E. Write
u =∑
k
ak ⊗ xk with ak ∈ S1, xk ∈ E.
Consider u as a linear functional on S∞[E∗]:
〈u, v〉 =∑
k,j
〈ak, bj〉〈xk, ξj〉 =∑
k,j
Tr(akbtj) ξj(xk)
for v =∑
j bk ⊗ ξj ∈ S∞ ⊗ E∗. Define
‖u‖ = ‖u‖(S∞[E∗])∗ = ‖u‖CB(S∞[E∗], C) .
Define S1[E] to be the closure of S1 ⊗E with respect to this norm. Next,we have to introduce a norm ‖ · ‖n on Mn(S1[E]) for any n. This is noweasy. As before, every u ∈Mn(S1 ⊗E) induces a linear map from S∞[E∗]to Mn. Then define
‖u‖n = ‖u‖CB(S∞[E∗], Mn).
It is then routine to check that these norms satisfy Ruan’s axioms. There-fore, S1[E] becomes an operator space. By definition,
S1[E]∗ = S∞[E∗].
• Sp[E]: For any 1 < p < ∞:
Sp[E] =(S∞[E], S1[E]
)1/p
.
Theorem. (Pisier) Let E and F be two operator spaces. Let 1 ≤ p < ∞.Then a linear map u : E → F is c.b. iff ISp ⊗u extends to a bounded map fromSp[E] to Sp[F ]. In this case, we have
‖u‖cb =∥∥ISp ⊗ u : Sp[E] → Sp[F ]
∥∥ .
This theorem implies that the o.s.s. of E is completely determined by theBanach space structure of Sp[E].
Schatten classes with values in noncommutative Lp.
• We haveSp[Lp(M)] = Lp(B(`2)⊗M), 1 ≤ p < ∞.
If E ⊂ Lp(M), then Sp[E] is the closure of Sp ⊗ E in Lp(B(`2)⊗M).
• Cp[E] and Rp[E]: Cp[E] is the closure of Cp ⊗ E in Sp[E]. If E ⊂ Lp(M),then for (ak) ⊂ E
∥∥ ∑
k
ak ⊗ ek
∥∥Cp[E]
=∥∥( ∑
k
a∗kak
)1/2∥∥Lp(M)
,
where (ek) denotes the canonical basis of Cp. We also have a similardescription for Rp[E].
An important reformulation of noncommutative Khintchine.
• Cp ∩Rp: the diagonal subspace of Cp ⊕p Rp. Then for (ak) ⊂ Sp
∥∥ ∑
k
ak ⊗ ek
∥∥Sp[Cp∩Rp]
=∥∥ ∑
k
ak ⊗ ek
∥∥Cp[Sp]∩Rp[Sp]
= max∥∥( ∑
k
a∗kak
)1/2∥∥p,
∥∥( ∑
k
aka∗k
)1/2∥∥p
.
• Cp + Rp: the quotient of Cp ⊕p Rp by the subspace (x, y) ; x + y = 0.Then for (ak) ⊂ Sp
∥∥(ak)∥∥
Sp[Cp+Rp]= inf
∥∥(bk)∥∥
Sp[Cp]+
∥∥(ck)∥∥
Sp[Rp]
= inf∥∥(
∑
k
(b∗kbk
)1/2)∥∥
p+
∥∥(∑
k
(ckc
∗k
)1/2)∥∥
p
,
where the infimum runs over all decompositions of ak = bk + ck in Sp. Itfollows that Sp[Cp + Rp] = Cp[Sp] + Rp[Sp].
• Define
CRp = Cp + Rp if p < 2 and CRp = Cp ∩Rp if p ≥ 2.
Then the noncommutative Khintchine inequalities assert that the closedsubspace of Lp(Ω) spanned by (εn) is completely isomorphic toCRp.
Isomorphic classification of noncommutative Lp
Aim: Classify noncommutative Lp
• up to isomorphism
• up to complete isomorphism
• according to the different types of the underlying von Neumann algebras
Very hard task.
Considerable progress has been achieved during the last few years, mainlyby Haagerup, Rosenthal and Sokochev.
Below is a list of major results. All VNA are assumed to be separable (i.e.with separable predual) and 1 ≤ p < ∞, p 6= 2.
• 9 non-isomorphic Lp associated with type I VNA. The building blocks arethe Lp associated with the following algebras
`∞, L∞(0, 1),⊕
n
Mn , B(`2).
The others are obtained by direct sum or tensor product. (Sukochev)
• 13 non-isomorphic Lp associated with hyperfinite semifinite VNA. Thebuilding blocks are the Lp associated with the following algebras
`∞, L∞(0, 1),⊕
n
Mn B(`2), R,
where R is the hyperfinite II1 factor. The others are obtained by directsum or tensor product. (Haagerup/Rosenthal/Sokochev)
• M finite and N infinite⇒ Lp(M) 6' Lp(N). (Haagerup/Rosenthal/Sokochev)
• M semifinite factor, N type III factors and both M and N hyperfinite ⇒Lp(M) 6' cb Lp(N). (Haagerup/Musat)
Open problems
Many open problems about the linear classification of noncommutative Lp,or more generally, their linear structures. Below is a partial list.
• Let M be type II∞ and N type III. Are Lp(M) and Lp(N) isomorphic?Same question in the category of operator spaces.
• Characterize the image of a completely contractively projection on Lp(M).(See Ng/Ozawa for the case p = 1 and Le Merdy/Ricard/Roydor for thecase M = B(`2).)The same problem for completely positive and completely contractive pro-jections.
• Let (xn) ⊂ Lp(M) be a weakly null sequence (1 < p < ∞). Does (xn)admit a unconditional basic subsequence? (Yes if M is hyperfinite becauseof the noncommutative Burkholder-Gundy.)
Some open problems related to this course:
• Let x = (xn) be a finite Clifford Lp-martingale with 2 ≤ p < ∞. ByBurkholder-Gundy, we have
α−1p max‖Sc(x)‖p , ‖Sr(x)‖p ≤ ‖x‖p ≤ βp max‖Sc(x)‖p , ‖Sr(x)‖p.
It is known that max(αp, βp) ≤ c p for some universal constant c. Doesone have max(αp, βp) ≤ c
√p ?
• Does Sp[C] have the UMD property for 1 < p < ∞? (See Musat’s thesisfor related results and Pisier’s Asterisque book for more open problems ofsimilar nature.)
• Does OH completely isometrically embed into an L1?
• Let T be the unit circle with normalized Haar measure. For p > 1, doesthere exist a constant cp such that
∥∥(Sn(f)
)n≥1
∥∥Lp(L∞(T)⊗B(`2);`∞)
≤ cp ‖f‖Lp(L∞(T)⊗B(`2);`∞)
for all f ∈ Lp(L∞(T)⊗B(`2); `∞) ?
