Completely 1-complemented subspaces of Schatten spaces

0.43 MB·English

by  Christian Le Merdy

COMPLETELY 1-COMPLEMENTED SUBSPACES OF SCHATTEN
SPACES
CHRISTIAN LE MERDY, E´RIC RICARD AND JEAN ROYDOR
9
0 Abstract. We considerthe Schatten spacesSp in the frameworkofoperatorspace theory
0 and for any 1 p = 2 < , we characterize the completely 1-complemented subspaces of
2 ≤ 6 ∞
Sp. They turn out to be the direct sums of spaces of the form Sp(H,K), where H,K are

n Hilbert spaces. This result is related to some previous work of Arazy-Friedman giving a
a description of all 1-complemented subspaces of Sp in terms of the Cartan factors of types
J
1-4. We use operator space structures on these Cartan factors regarded as subspaces of
8
appropriate noncommutative Lp-spaces. Also we show that for any n 2, there is a triple

] isomorphism on some Cartan factor of type 4 and of dimension 2n whi≥ch is not completely
A isometric, and we investigate Lp-versions of such isomorphisms.
O
.
h
t
a
2000 Mathematics Subject Classification : 46L07, 46L89, 17C65
m
[

2 1. Introduction
v
8 Let , be Hilbert spaces. For any p 1, let Sp( , ) be the Schatten space of all
0 H K ≥ H K
1
4 operators x: such that x = tr( x p) p is finite. Let X Sp( , ) be a
p
4 H → K k k | | ⊂ H K
(closed) subspace. We say that X is 1-complemented in Sp( , ) if it is the range of a
.
3 (cid:0) (cid:1) H K
contractive projection P: Sp( , ) Sp( , ). In their remarkable memoirs [2, 3], Arazy
0
H K → H K
8 and Friedman gave a complete classification of all such subspaces (for p = 2), in terms of
0 6
Cartan factors of types 1-4.
:
v In this paper we consider Schatten spaces and their complemented subspaces in the frame-
i
X work of operator spaces and completely bounded maps. Following Pisier’s work [17], we
r regard Sp( , ) as an operator space and we give a complete description of the completely
a H K
1-complemented subspaces of Sp( , ), that is, spaces X Sp( , ) which are the range
H K ⊂ H K
of a completely contractive projection of Sp( , ).
H K
The statement of our main result, Theorem 1.1 below, requires some tensor product defi-
nitions and some notation. For any Hilbert spaces H,H′,K,K′, we will consider the natural
embedding
2 2
Sp(H′,K′) Sp(H,K) Sp(H′ H,K′ K),
⊗ ⊂ ⊗ ⊗
2
where denotes the Hilbertian tensor product. Thus for any subspace Z Sp(H′,K′) and
⊗ ⊂
any a Sp(H,K), we will regard
∈
Z a := z a : z Z
⊗ { ⊗ ∈ }
Date: January 8, 2009.
1
2 CHRISTIANLE MERDY,E´RIC RICARDANDJEAN ROYDOR
2 2
as a subspace of Sp(H′ H,K′ K).
If I,J are two index⊗sets, we⊗set Sp = Sp(ℓ2,ℓ2) and we write Sp = Sp . With this
I,J J I I I,I
2
notation, Sp Sp(H,K) Sp(ℓ2(H),ℓ2(K)), where ℓ2(H) = ℓ2 H is the 2-direct sum of
I,J ⊗ ⊂ J I J J ⊗
J copies of H.
Next we recall that if (H ) and (K ) are two families of Hilbert spaces, then we have a
α α α α
natural isometric embedding
p 2 2
Sp(H ,K ) Sp H , K ,
α α α α
⊕α ⊂ ⊕α ⊕α
p (cid:0) (cid:1) 2
where Sp(H ,K ) denotes the p-direct sum of the Sp(H ,K )’s and H denotes the
α α α α α
⊕α ⊕α
p
2-direct sum of the H ’s. This is obtained by identifying any (x ) in Sp(H ,K ) with
α α α α α
⊕α
2 2
the ‘diagonal’ operator H K taking any (ξ ) to (x (ξ )) .
α α α α α α α
⊕α → ⊕α
Theorem 1.1. Let , be Hilbert spaces, let 1 p = 2 < and let X Sp( , ) be a
H K ≤ 6 ∞ ⊂ H K
subspace. The following are equivalent.
(i) X is completely 1-complemented in Sp( , ).
H K
(ii) X is [2]-1-complemented in Sp( , ).
H K
(iii) There exist, for some set A, two families of indices (I ) and (J ) , a family
α α∈A α α∈A
(H ) of Hilbert spaces, as wellas operatorsa Sp(H ), andtwo linearisometries
α α∈A α α
∈
2 2
U: ℓ2 (H ) and V : ℓ2 (H )
α⊕∈A Jα α −→ H α⊕∈A Iα α −→ K
such that
p
X = V Sp a U∗.
Iα,Jα ⊗ α
(cid:16)Mα (cid:17)
(iv) There exist, for some set A, two families of indices (I ) and (J ) such that X
α α∈A α α∈A
p
is completely isometric to the p-direct sum Sp .
⊕α Iα,Jα
See Definition 2.1 below for the meaning of (ii). In the above statement, the main im-
plication is (i) (iii). The starting point of its proof is the Arazy-Friedman work [2, 3]
⇒
giving a list of all 1-complemented subspaces of Sp( , ). In Section 2, we give some back-
H K
ground on this classification and some preliminary results, as well as a brief account on the
matricial structure of Schatten spaces and completely bounded maps on their subspaces.
The strategy to prove (ii) (iii) consists in taking any 1-complemented X Sp( , )
⇒ ⊂ H K
from the Arazy-Friedman list, to exhibit a canonical contractive projection onto X, and to
determine whether that projection is completely contractive (or [2]-1-contractive). This is
mostly achieved in Sections 3-5. Theorem 1.1 is eventually proved in Section 6.
Let n 1 be an integer, let be the Clifford algebra generated by a collection
2n
≥ C
(ω ,...,ω ) of Fermions, and let F be the linear span of 1,ω ,...,ω ,ω ω .
1 2n n 2n 1 2n 1 2n
⊂ C { ··· }
Then let τ: F F bethe linear mapping such thatτ(ω ω ) = ω ω andτ is the
n n 1 2n 1 2n
→ ··· − ···
identity onthelinearspanof 1,ω ,...,ω . ThespaceF isaCartanfactoroftype4andτ
1 2n n
{ }
3
isatripleisomorphism. This‘transposemap’playsakeyroleinthestudyof1-complemented
subspaces of Sp( , ) (see Section 5). In Section 7, we investigate further properties of τ
H K
in the framework of operator space theory. First we show that τ = (n + 1)/n. Then
cb
k k
let Fp Lp( ) be the space F regarded an a subspace of the noncommutative Lp-space
n ⊂ C2n n
associated to . We determine when τ: Fp Fp is completely contractive (it depends on
C2n n → n
n and p), and we give applications and complements.
We refer the reader to [11, 12, 13] for some work on contractive and completely contractive
projections on some Cartan factors, which is somehow related to the present paper. We
also mention the Ng-Ozawa paper [14] for a description of the completely 1-complemented
subspaces of noncommutative L1-spaces.
2. Background on complete boundedness and 1-complemented subspaces
We start with some preliminary facts concerning completely bounded maps on Schatten
spaces and their subspaces. Let 1 p < , let , ′, , ′ be Hilbert spaces and consider
≤ ∞ H H K K
subspaces X Sp( , ) and Y Sp( ′, ′). For any index set I, we let
⊂ H K ⊂ H K
p
Sp X Sp(ℓ2( ),ℓ2( ))
I ⊗ ⊂ I H I K
denote the completion of Sp X induced by the embedding of Sp Sp( , ) into the space
I ⊗ I ⊗ H K
Sp(ℓ2( ),ℓ2( )).
I H I K
Note that for any integer n 1, Sp X coincides with the space of all n n matrices
≥ n ⊗ ×
with entries in X. Let u: X Y be a bounded linear map. We set
→
p p
(2.1) kukn = ISnp ⊗u: Snp⊗X −→ Snp⊗Y
for any n 1, and we say that u is(cid:13)[n]-contractive if u n 1. T(cid:13)his is equivalent to
≥ (cid:13) k k ≤ (cid:13)
(2.2) [u(x )] [x ] , x X, 1 i,j n.
ij Sp(ℓ2n(H′),ℓ2n(K′)) ≤ ij Sp(ℓ2n(H),ℓ2n(K)) ij ∈ ≤ ≤
Next we set(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
(2.3) u = sup u .
cb n
k k k k
n≥1
By definition, u is completely bounded if u < , and we say that u is a complete
cb
k k ∞
contraction (or is completely contractive) if u 1. Also we say that u is a complete
cb
k k ≤
isometry if ISnp ⊗u is an isometry for any n ≥ 1.
The above definitions come from Pisier’s fundamental work [17] and we wish to point out
that they are consistent with the usual terminology of operator space theory. Indeed, assume
that Schatten spaces areequipped with their ‘natural’operator space structure introduced in
Pisier’s memoir. Then equip any subspace of a Schatten space with the inherited structure.
p p
With these conventions it is easy to check that the spaces Sp X and Sp Y coincide with
n⊗ n⊗
the operator space valued Schatten spaces Sp[X] and Sp[Y] from [17, Chapter 1]. Hence it
n n
follows from [17, Lem. 1.7] that the definitions of and given by (2.1) and (2.3)
n cb
k k k k
coincide with the ones obtained by regarding X Sp( , ) and Y Sp( ′, ′) as operator
⊂ H K ⊂ H K
spaces. We shall not use much of operator space theory and we refer the interested reader
to [18], [5] or [16] for basic definitions and background.
4 CHRISTIANLE MERDY,E´RIC RICARDANDJEAN ROYDOR
Definition 2.1. Let n 1 be an integer. We say that X Sp( , ) is [n]-1-complemented
≥ ⊂ H K
if X is the range of an [n]-contractive projection P: Sp( , ) Sp( , ). Then we say
H K → H K
that X is completely 1-complemented if it is the range of a completely contractive projection
P: Sp( , ) Sp( , ).
H K → H K
Note that S2( , ) is ‘homogeneous’, that is, any bounded linear map u: S2( , )
H K H K →
S2( , ) is automatically completely bounded, with u = u (see [18, Chap. 7]). Con-
cb
H K k k k k
sequently, any X S2( , ) is completely 1-complemented. Thus we will only focus on
⊂ H K
1 p = 2 < in the sequel.
≤ 6 ∞
We say that X Sp( , ) and Y Sp( ′, ′) are equivalent, and we write
⊂ H K ⊂ H K
X Y,
∼
if there exist two partial isometries U: ′ and V : ′ such that
H → H K → K
(2.4) X = VYU∗ and Y = V∗XU.
Note that if X = VYU∗, then Y = V∗XU if and only if y = V∗VyU∗U for any y Y, if
∈
and only if the mapping y VyU∗ is one-to-one on Y.
7→
Lemma 2.2. Let , ′, , ′ be Hilbert spaces, and let W : ′ and W : ′ be
1 2
H H K K H → H K → K
two contractions. Then the linear mapping Sp( ′, ′) Sp( , ) taking any z Sp( ′, ′)
H K → H K ∈ H K
to W∗zW is a complete contraction.
2 1
(cid:3)
Proof. This is clear using (2.2).
Lemma 2.3. Assume that X Sp( , ) and Y Sp( ′, ′) are equivalent. Then X and
⊂ H K ⊂ H K
Y are completely isometric and for any n 1, X is [n]-1-complemented in Sp( , ) if and
≥ H K
only if Y is [n]-1-complemented in Sp( ′, ′). Also, X is completely 1-complemented in
H K
Sp( , ) if and only if Y is completely 1-complemented in Sp( ′, ′).
H K H K
Proof. Lemma 2.2 ensures that y VyU∗ is a complete isometry from Y onto X. Now
7→
suppose that P: Sp( , ) Sp( , ) is a contractive projection whose range is equal to
H K → H K
X, and that X and Y satisfy (2.4). Then the mapping Q: Sp( ′, ′) Sp( ′, ′) defined
H K → H K
by
Q(z) = V∗P(VzU∗)U, z Sp( ′, ′),
∈ H K
is a contractive projection whose range is equal to Y. Moreover it follows from Lemma 2.2
that Q P for any integer n 1. This implies the second part of the statement. (cid:3)
n n
k k ≤ k k ≥
Remark 2.4. Although it is not appearent in the notation, the property X Y depends
∼
on the embeddings X Sp( , ) and Y Sp( ′, ′), and not only on the operator space
⊂ H K ⊂ H K
structures of X and Y. Namely, X and Y may be completely isometric without being
equivalent. This subtlety should not lead to any confusion, since the embeddings considered
for various spaces studied below will be clear from the context. Note also that if we have
Hilbert spaces H and K , then Sp(H,K) regarded as a subspace of Sp( , ) is
⊂ H ⊂ K H K
equivalent to Sp(H,K) regarded as a subspace of itself.
5
Inthesecond partofthissection, wereview theclassificationof1-complemented subspaces
of Sp( , ) obtained by Arazy-Friedman [2, 3]. We fix some 1 p = 2 < throughout.
H K ≤ 6 ∞
Let X ,X Sp( , ) be two subspaces. We say that X and X are orthogonal if
1 2 1 2
⊂ H K
x∗x = 0 and x x∗ = 0, x X , x X .
1 2 1 2 1 ∈ 1 2 ∈ 2
As observed in [3, p. 18], this is equivalent to the identity
(2.5) x +x p = x p + x p, x X , x X .
1 2 1 2 1 1 2 2
k k k k k k ∈ ∈
Alsoitiseasytocheck(lefttothereader)thatthisisequivalenttotheexistenceoforthogonal
2 2
decompositions = H H and = K K such that X Sp(H ,K ) for i = 1,2.
1 2 1 2 i i i
H ⊕ K ⊕ ⊂
Consequently, if (X ) is a family of pairwise orthogonal subspaces of Sp( , ), the closed
α α
p H K
subspace ofSp( , )generatedbytheX ’sisequaltotheirp-direct sum X . Furthermore
α α
H K ⊕α
we have
p p p p
(2.6) Sp X = Sp X
n⊗ ⊕α α ⊕α n⊗ α
for any n 1. (cid:0) (cid:1) (cid:0) (cid:1)
≥
We say that X Sp( , ) is indecomposable if it cannot be written as the direct sum
⊂ H K
of two non trivial orthogonal subspaces. According to [3, Prop. 2.2], any subspace X of
p
Sp( , ) is equal to a direct sum X of pairwise orthogonal indecomposable subspaces.
α
H K ⊕α
For that reason we will concentrate on indecomposable subspaces in the rest of this section
andin thenext three sections. We notethat ifX andY aretwo subspaces ofsome Sp-spaces,
and if X and Y are isometric, then X is indecomposable if and only if Y is indecomposable.
Indeed, this follows from (2.5).
For any two index sets I and J, we regard elements of Sp as scalar matrices [t ]
I,J ij i∈I,j∈J
in the usual way. Then we let σ: Sp Sp be the transpose map, defined by
I,J → J,I
σ [t ] = [t ].
ij ji
This is an isometry. In the case J = I, w(cid:0)e le(cid:1)t
p = w Sp : σ(w) = w and p = w Sp : σ(w) = w
SI { ∈ I } AI { ∈ I − }
be the spaces of symmetric and anti-symmetric matrices, respectively.
It is clear that p and p are 1-complemented subspaces of Sp. Indeed,
SI AI I
1 1
(2.7) P = (Id+σ) and P = (Id σ)
s a
2 2 −
arecontractiveprojectionswhoserangeareequalto p and p respectively. Likewise, forany
operator a Sp(H) in some Sp-space, the two spaceSsI p aAaInd p a are 1-complemented
∈ SI⊗ AI⊗
subspaces of Sp(ℓ2(H)).
I
Definition 2.5. We say that X Sp( , ) is a space of symmetric matrices (resp. of
anti-symmetric matrices) if it is e⊂quivaleHntKto a space of the form p a (resp. p a),
SI ⊗ AI ⊗
where I is an index set and a Sp(H).
∈
6 CHRISTIANLE MERDY,E´RIC RICARDANDJEAN ROYDOR
Let a Sp(H ) and a Sp(H ) be two operators, and consider the spaces
1 1 2 2
∈ ∈
Y = Sp a Sp(ℓ2(H ),ℓ2(H )) and Y = Sp a Sp(ℓ2(H ),ℓ2(H )),
1 I,J ⊗ 1 ⊂ J 1 I 1 2 J,I ⊗ 2 ⊂ I 2 J 2
as well as
p
(2.8) Y = (w a ,σ(w) a ) : w Sp Y Y .
{ ⊗ 1 ⊗ 2 ∈ I,J} ⊂ 1⊕ 2
In this definition we assume that (a ,a ) = (0,0), excluding the trivial case Y = 0 .
1 2
6 { }
However a or a can be equal to 0.
1 2
p
The space Y is 1-complemented in Y Y . To check this fact, and also for further pur-
1 2
poses, it is convenient to use matrix nota⊕tion. In the sequel, for any z Sp , z Sp we
1 ∈ I,J 2 ∈ J,I
p
identify (z a ,z a ) Y Y with the 2 2 diagonal matrix
1 1 2 2 1 2
⊗ ⊗ ∈ ⊕ ×
z 0
1 .
0 z
2
(cid:20) (cid:21)
We may assume that a p+ a p = 1, and we let t = a p. Then z a = z a =
k 1kp k 2kp k 1kp k 1⊗ 1k k 1kk 1k
1 1 p
tp z , whereas z a = (1 t)p z . Hence the norm on Y Y in the above identifi-
1 2 2 2 1 2
k k k ⊗ k − k k ⊕
cation is given by
z 0 1
1 = t z p +(1 t) z p p.
0 z k 1k − k 2k
2
(cid:13)(cid:20) (cid:21)(cid:13)
Furthermore (cid:13) (cid:13) (cid:0) (cid:1)
(cid:13) (cid:13)
(cid:13) (cid:13)w 0
p
Y = : w S .
0 σ(w) ∈ I,J
(cid:26)(cid:20) (cid:21) (cid:27)
p p
Let P: Y Y Y Y be the linear mapping defined by
1 2 1 2
⊕ → ⊕
z 0 tz +(1 t)σ−1(z ) 0
(2.9) P 1 = 1 − 2 .
0 z 0 tσ(z )+(1 t)z
2 1 2
(cid:18)(cid:20) (cid:21)(cid:19) (cid:20) − (cid:21)
It is plain that P is a projection onto Y. Moreover by convexity we have
tz +(1 t)σ−1(z ) p t z +(1 t) σ−1(z ) p
1 2 1 2
k − k ≤ k k − k k
p
z 0
(cid:0)t z p +(1 t) σ−1(z ) p(cid:1) 1
≤ k 1k − k 2 k ≤ 0 z
2
(cid:13)(cid:20) (cid:21)(cid:13)
p p (cid:13) (cid:13)
for any z S , z S . Likewise, (cid:13) (cid:13)
1 ∈ I,J 2 ∈ J,I (cid:13) (cid:13)
z 0
tσ(z )+(1 t)z 1 ,
k 1 − 2k ≤ 0 z
2
(cid:13)(cid:20) (cid:21)(cid:13)
(cid:13) (cid:13)
which shows that the projection P is contractive. T(cid:13)his implies(cid:13)that Z is 1-complemented in
(cid:13) (cid:13)
the p-direct sum of Sp(ℓ2(H ),ℓ2(H )) and Sp(ℓ2(H ),ℓ2(H )), and hence in the Sp-space
J 1 I 1 I 2 I 2
2 2
Sp ℓ2(H ) ℓ2(H ),ℓ2(H ) ℓ2(H ) .
J 1 ⊕ I 2 I 1 ⊕ I 2
Definition 2.6. We say tha(cid:0)t X Sp( , ) is a space of (cid:1)rectangular matrices if it is
⊂ H K
equivalent to a space Y of the form (2.8).
7
We now turn to the construction of operator spaces acting on anti-symmetric Fock spaces.
We refer the reader to [4, 20] for general information on these spaces. Let n 1 be an
≥
integer. For any k = 0,...,n, we let Λ denote the k-fold anti-symmetric tensor product
n,k
of the Hilbert space ℓ2, equipped with the canonical inner product given by
n
ξ ξ , ξ′ ξ′ = det ξ ,ξ′ , ξ , ξ′ ℓ2.
1 ∧···∧ k 1 ∧···∧ k h i ji i j ∈ n
By convention, Λ = C. We let Ω be a particular unit element of Λ , which is called the
(cid:10)n,0 (cid:11) (cid:2) (cid:3) n,0
vacuum vector. Then the anti-symmetric Fock space over ℓ2 is the Hilbertian direct sum
n
2
Λ = Λ .
n n,k
0≤⊕k≤n
Throughout we let (e ,...,e ) denote the canonical basis of ℓ2 and we let be the set of
1 n n Pn
all subsets of 1,...,n . Let A with cardinal A = k, and let 1 j < < j n
n 1 k
{ } ∈ P | | ≤ ··· ≤
be the increasing enumeration of the elements of A. Then we set
(2.10) e = e e .
A j1 ∧···∧ jk
By convention, e = Ω. Clearly the system e : A = k is an orthonormal basis of Λ .
∅ A n,k
{ | | }
We will call it ‘canonical’ in the sequel. Note that dim(Λ ) = n and that dim(Λ ) = 2n.
n,k k n
For any 1 j n, we let
≤ ≤ (cid:0) (cid:1)
c : Λ Λ
n,j n n
−→
be the so-called creation operator defined by letting c (Ω) = e , and
n,j j
c (ξ ξ ) = e ξ ξ , ξ ,...,ξ ℓ2.
n,j 1 ∧···∧ k j ∧ 1 ∧···∧ k 1 k ∈ n
Next we denote by P : Λ Λ the orthogonal projection onto the space
n n n
→
Λeven = Λ
n 0≤⊕k≤n n,k
k even
generated by tensor products of even rank. Following [2, p. 24], we let
x = c P and x = c∗ P
n,j n,j n n,j n,j n
be the restrictions of c and c∗ to Λeven for any 1 j n. Then we set
n,j n,j n ≤ ≤
e
AH = Span x ,x : j = 1,...,n .
n n,j n,j
{ }
This is a 2n-dimensional operator space. Next we let
e
BH = x∗ : x AH = Span x∗ ,x∗ : j = 1,...,n
n { ∈ n} { n,j n,j }
be the adjoint space of AH . Note that c = P c +c P . Consequently,
n n,j n n,j n,j n
e
x∗ = c (Id P ) and x∗ = c∗ (Id P )
n,j n,j − n n,j n,j − n
are the restrictions of c and c∗ to the space
n,j n,j
e
Λodd = Λ Λeven = Λ
n n ⊖ n 0≤⊕k≤n n,k
k odd
generated by the tensor products of odd rank.
In the sequel we let AHp and BHp denote the spaces AH and BH respectively, regarded
n n n n
as subspaces of Sp(Λ ).
n
8 CHRISTIANLE MERDY,E´RIC RICARDANDJEAN ROYDOR
Let κ: AH BH be the exchange map defined by letting
n n
→
(2.11) κ(x ) = x∗ and κ(x ) = x∗ , j = 1,...,n.
n,j n,j n,j n,j
It follows from the calculations in [2, Chap. 2] that κ is an isometry from AHp onto BHp.
n n
e e
(An explicit proof of this fact will be given in Section 5, see Remark 5.12). For any operators
a Sp(H ) and a Sp(H ), with (a ,a ) = (0,0), we will consider
1 1 2 2 1 2
∈ ∈ 6
2 p 2
(2.12) Z = (x a ,κ(x) a ) : x AHp Sp(Λ H ) Sp(Λ H ).
⊗ 1 ⊗ 2 ∈ n ⊂ n⊗ 1 ⊕ n⊗ 2
(cid:8) (cid:9) 2
According to [2, Prop. 2.9], this space is 1-complemented in the p-direct sum of Sp(Λ H )
n 1
⊗
2
and Sp(Λ H ).
n 2
⊗
Now following [2, p. 33] we consider the (2n 1)-dimensional operator space
−
DAH = Span x +x ; x ,x : j = 1,...,n 1 ,
n n,n n,n n,j n,j
{ − }
and we let DAHp be that space regarded as a subspace of Sp(Λ ). Then for any a Sp(H),
n e 2e n ∈
the space DAHp a is 1-complemented in Sp(Λ H), by [2, Prop. 2.13].
n ⊗ n⊗
Simple proofs of the above mentioned 1-complementation results will be given later on in
Section 5, see Remark 5.12.
Definition 2.7. Let X Sp( , ) be a finite dimensional space of dimension N 1. If
⊂ H K ≥
N = 2n 1 is odd, we say that X is a spinorial space if it is equivalent to a space of the
−
form DAHp a, for some a Sp(H). If N = 2n is even, we say that X is a spinorial space
n⊗ ∈
if it is equivalent to a space of the form (2.12).
See the end of this section for more on this terminology.
We shall now define a class of finite dimensional Hilbert spaces which are 1-complemented
subspaces of Sp. Let 1 k n. It is clear that the creation operators c map Λ into
n,j n,k−1
≤ ≤
Λ . For any j = 1,...,n, we let
n,k
c : Λ Λ
n,j,k n,k−1 n,k
−→
be the restriction of c to Λ . A quick examination of the definition of the c ’s shows
n,j n,k−1 n,j
that the matrix of c in the canonical bases of Λ and Λ has its entries in 1,0,1 ,
n,j,k n,k−1 n,k
{− }
with at most one non zero element oneach row and oneach column. Moreover the 1 entries
appear exactly n−1 times. Hence c p = n−1 for any j. We let ±
k−1 k n,j,kkp k−1
(cid:0) (cid:1) H = Span c (cid:0) : j(cid:1)= 1,...,n ,
n,k n,j,k
{ }
and we let Hp be that space regarded as a subspace of Sp(Λ ). By [3, Chap. 7], Hp is an
n,k n n,k
n-dimensional Hilbert space. More precisely, the linear mapping
(2.13) ϕ : ℓ2 Hp , ϕ (e ) = c , 1 j n,
k n −→ n,k k j n,j,k ≤ ≤
isamultipleofanisometry, i.e. n−1 −p1ϕ isanisometry. Wereferthereaderto[11,12,13]
k−1 k
for more on these Hilbert spaces and their operator space properties.
(cid:2)(cid:0) (cid:1)(cid:3)
9
For any operators a Sp(H ),...,a Sp(H ), with (a ,...,a ) = (0,...,0), we will
1 1 n n 1 n
∈ ∈ 6
consider the space
p 2
(2.14) E = (ϕ (s) a ,...,ϕ (s) a ) : s ℓ2 Sp(Λ H ).
1 ⊗ 1 n ⊗ n ∈ n ⊂ 1≤⊕k≤n n⊗ k
(cid:8) (cid:9)
Clearly E is a Hilbert space. Indeed if we assume (after normalisation) that
n−1 a p = 1,
k−1 k kkp
k
X(cid:0) (cid:1)
then the linear mapping ℓ2 E taking any s ℓ2 to (ϕ (s) a ,...,ϕ (s) a ) is an
n → ∈ n 1 ⊗ 1 n ⊗ n
isometry. According to [2, Prop. 2.5], the space E is 1-complemented.
Theorem 2.8. (Arazy-Friedman) Let , be Hilbert spaces, and let X Sp( , ) be an
H K ⊂ H K
indecomposable subspace, with 1 p = 2 < . The following are equivalent.
≤ 6 ∞
(i) X is 1-complemented in Sp( , ).
H K
(ii) X is either a space of symmetric matrices, or a space of anti-symmetric matrices
(in the sense of Definition 2.5), or a space of rectangular matrices (in the sense of
Definition 2.6), or a spinorial space (in the sense of Definition 2.7) of dimension
5, or a finite dimensional Hilbertian space equivalent to a space of the form (2.14).
≥
By Lemma 2.3 and the results we have recorded along this section, all the spaces in the
list (ii) are 1-complemented. The hard implication ‘(i) (ii)’ is proved in [3, Chap. 7] in
⇒
the case p > 1 and in [2, Chap. 5] in the case p = 1.
After reducing to the case of indecomposable spaces, the proof of Theorem 1.1 will mainly
consist in showing that the spaces in the list (ii) above are not completely 1-complemented,
except the ones which are equivalent to some Sp a. This will be achieved in the next
I,J ⊗
three sections.
It should be noticed that the classes of 1-complemented subspaces considered above do
p p
not exclude each other. For instance, the Hilbert space S is equivalent to H , whereas
n,1 n,1
Sp is equivalent to Hp . On the other hand, AHp = ℓp and it follows from [2, Chap. 2]
1,n n,n 1 2
that AHp is equivalent to Sp, AHp is equivalent to p, DAHp is equivalent to p and p is
equivalen2t to Hp . 2 3 A4 2 S2 A3
3,2
Remark 2.9. Suppose thatp > 1. IfX Sp( , )is1-complemented, thenthecontractive
⊂ H K
projection P : Sp( , ) Sp( , ) whose range is equal to X is unique (see [3, Prop. 1.2]).
H K → H K
This uniqueness property is false in the case p = 1 (see e.g. [2, p. 36]). However a
similar result holds true, as follows. Let X S1( , ) be a subspace and let r B( )
⊂ H K ∈ H
and ℓ B( ) be the smallest orthogonal projections such that ℓxr = x for any x X. Let
∈ K ∈
H andK bethe ranges ofr andℓ, respectively. Thus X S1(H,K), andH,K are
⊂ H ⊂ K ⊂
the smallest subspaces of , withthat property. Wesay thatX is nondegenerate if H =
H K H
and K = . It is proved in [2, Th. 2.15] that if X is 1-complemented and nondegenerate,
K
then the contractive projection P on S1( , ) with range equal to X is unique.
H K
Note that X regarded as a subspace of S1( , ) is equivalent to X regarded as a subspace
H K
of S1(H,K). Thus if we wish to determine whether X is [n]-1-complemented (for some
n 1), there is no loss of generality in assuming that X is nondegenerate.
≥
10 CHRISTIANLE MERDY,E´RIC RICARDANDJEAN ROYDOR
We end this section with some terminology and notions which play a central role in the
work of Arazy-Friedman [2, 3], and some basic facts.
Let , be Hilbert spaces and let X B( , ) be a closed subspace. By definition X is
H K ⊂ H K
a JC∗-triple if xx∗x belongs to X for any x X. Next a linear map u: X Y between two
∈ →
JC∗-triples X and Y is called a triple homomorphism if u(xx∗x) = u(x)u(x)∗u(x) for any
x X. If u is one-to-one, we say that u is a triple monomorphism. If further u is a bijection,
∈
then u−1 also is a triple homomorphism and we say that u is a triple isomorphism in this
case. It is well-known that a bijection u: X Y between two JC∗-triples is an isometry if
→
and only if it is a triple isomorphism (see [7]). We say that X and Y are triple equivalent if
there is a triple isomorphism from X onto Y.
We now turn to Cartan factors of types 1-4. We mainly follow [7] (see also [11]). By
definition,aCartanfactoroftype1isaJC∗-triplewhichistripleequivalenttosomeB( , ),
H K
where , are Hilbert spaces. Next let be a Hilbert space with a distinguished Hilbertian
H K H
basis, and let w tw denote the associated transpose map on B( ). Then the space of
7→ H
anti-symmetric operators
( ) = w B( ) : tw = w
A H { ∈ H − }
is a JC∗-triple, and we call Cartan factor of type 2 any JC∗-triple which is triple equivalent
to some ( ). Likewise, the space of symmetric operators
A H
( ) = w B( ) : tw = w
S H { ∈ H }
is a JC∗-triple, and we call Cartan factor of type 3 any JC∗-triple which is triple equivalent
to some ( ). Lastly, let X B( ) be a closed subspace such that x∗ X for any x X
S H ⊂ H ∈ ∈
and x2 is a scalar multiple of the identity operator for any x X. Then X is a JC∗-triple,
∈
and we call Cartan factor of type 4 any JC∗-triple which is triple equivalent to such a space.
Let n 2 be an integer. An n-tuple (s ,...,s ) of operators in some B( ) is called a
1 n
≥ H
spin system if each s is a selfadjoint unitary and
j
sjsj′ +sj′sj = 0, 1 j = j′ n.
≤ 6 ≤
In this case, the n-dimensional space
X = Span s ,...,s B( )
1 n
{ } ⊂ H
is a Cartan factor of type 4.
Let w ,...,w be the operators on Λ defined by
1 n n
ω = c +c∗ , j = 1,...,n.
j n,j n,j
These operators are called Fermions and they form a spin system (see e.g. [4]). Hence
their linear span is an n-dimensional Cartan factor of type 4. It is well-known that all
n-dimensional Cartan factors of type 4 are mutually triple equivalent. Thus the space
Span ω ,...,ω is actually a model for such spaces.
1 n
{ }
ItturnsoutthatthespacesAH andDAH consideredinthissectionareCartanfactorsof
n n
type 4. This will be implicitly shown along the proof of Theorem 5.11. We refer the reader to
[3] for details on this, and for a deeper analysis of the relationship between 1-complemented
subspaces of Sp-spaces and Cartan factors.