NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS M. JUNGE * AND Q. XU Abstract. We show norm estimates for the sum of independent random variables in noncommutative L p spaces for 1 <p< ∞ following previous work by the authors. These estimates 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 independent entries, and characterize those symmetric subspaces and unitary ideal spaces which can be realized as subspaces of noncommutative L p 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 L p spaces. Our estimates on random matrices are motivated by the noncommutative Khintchine inequality. In [LP] Lust-Piquard showed that for 2 ≤ p< ∞ and scalar coefficients (a ij ) one has Eij ε ij a ij e ij p ∼ c(p) i j |a ij | 2 p 2 1 p + j i |a ij | 2 p 2 1 p . (0.1) * The first author is partially supported by the National Science Foundation Foundation DMS-0301116. 1
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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
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
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
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
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
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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