NONCOMMUTATIVE Lp -SPACE AND OPERATOR SYSTEM
arXiv:0904.0518v2 [math.OA] 28 Jun 2009
KYUNG HOON HAN Abstract. We show that noncommutative Lp -spaces satisfy the axioms of the (nonuni1 tal) operator system with a dominating constant 2 p . Therefore, noncommutative Lp 1 spaces can be embedded into B pH q 2 p -completely isomorphically and complete order isomorphically.
1. Introducton A unital involutive subspace of B pH q had been abstractly characterized by Choi and Effros [CE]. Their axioms are based on observations of the relationship between the unit, the matrix order, and the matrix norm of the unital involutive subspace of B pH q. Indeed, the unit and the matrix order can be used to determine the matrix norm by applying
0 x }x} inf tλ ¡ 0 : λI ¤ x 0 ¤ λI u for a unital involutive subspace X of B pH q and x P Mn pX q. The abstract characterization of a nonunital involutive subspace of B pH q was completed by Werner [W]. The axioms are based on observations of the relationship between the matrix norm and the matrix order of the involutive subspace of B pH q. Since the unit may be absent, the above equality is replaced by
0 x }x} supt|ϕp x 0 q| : ϕ P M2n pX q1, u
for an involutive subspace X of B pH q and x P Mn pX q. We denote by M2n pX q1, the set of positive contractive functionals on M2n pX q. A complex involutive vector space X is called a matrix ordered vector space if for each n P N, there is a set Mn pX q Mn pX qsa such that (1) (2) (3)
Mn pX q X rMn pX q s t0u for all n P N, Mn pX q ` Mm pX q Mn m pX q for all m, n P N, γ Mm pX q γ Mn pX q for each m, n P N and all γ
P Mm.,npCq.
One might infer from these conditions that Mn pX q is actually a cone. An operator space X is called a matrix ordered operator space iff X is a matrix ordered vector space and for every n P N, (1) the -operation is an isometry on Mn pX q and (2) the cones Mn pX q are closed.
2000 Mathematics Subject Classification. 46L07, 46L52, 47L07. Key words and phrases. noncommutative Lp -space, operator system. This work was supported by the BK21 project of the Ministry of Education, Korea. 1
2
KYUNG HOON HAN
Suppose X is a matrix ordered operator space. For x P Mn pX q, the modified numerical radius is defined by
0 x q| : ϕ P M2npX q1, u. νX pxq supt|ϕp x 0
We call a matrix ordered operator space an operator system iff there is a k that for all n P N and x P Mn pxq,
¡ 0 such
}x} ¤ kνX pxq.
Since νX pxq ¤ }x} always holds, we can say that an operator system is a matrix ordered operator space such that the operator space norm and the modified numerical radius are equivalent uniformly for all n P N. Werner showed that X is an operator system if and only if there is a complete order isomorphism Φ from X onto an involutive subspace of B pH q, which is a complete topological onto-isomorphism [W]. Hence, the operator system is an abstract characterization of the involutive subspace of B pH q in the completely isomorphic and complete order isomorphic sense. In this paper, noncommutative Lp -space is meant in the sense of Haagerup, which is based on Tomita-Takesaki theory [Te1]. While the noncommutative Lp -spaces arising from semifinite von Neumann algebras are a natural generalization of classical Lp -spaces [S, Ta], noncommutative Lp -spaces arising from type III von Neumann algebras are quite complicated. Recently, the reduction method approximating a general noncommutative Lp -space by tracial noncommutative Lp -spaces has been developed [HJX]. Noncommutative Lp -spaces can also be obtained by the complex interpolation method. There exist a canonical matrix order and a canonical operator space structure on each noncommutative Lp -space. The matrix order is given by the positive cones Lp pMn pMqq , and the operator space structure is given by the complex interpolation Mn pLp pMqq pMn pMq, Mn pMop qq 1 . p
We refer to [K, Pi1, Te2] for the details, and the reference [JRX, Section 4] is recommended for the summary. In particular, its discrete noncommutative vector valued Lp -space Spn pLp pMqq is given by Spn pLp pMqq Lp pMn pMqq.
We refer to [Pi2] for general information on noncommutative vector valued Lp -spaces. The purpose of this paper is to show that noncommutative Lp -spaces satisfy the 1 axioms of the operator system with a dominating constant 2 p . As a corollary, we obtain the following embedding theorem: noncommutative Lp -spaces can be embedded 1 into B pH q 2 p -completely isomorphically and complete order isomorphically. 2. Noncommutative Lp -space and operator system If X is a matrix ordered operator space, we regard Spn pX q as an operator space having the same matrix order structure with Mn pX q. That is, we do not distinguish between Spn pX q and Mn pX q as matrix ordered vector spaces. For a noncommutative Lp -space Lp pMq, it is more efficient to describe Spn pLp pMqq rather than Mn pLp pMqq. So, we change the modified numerical radius into a form more
NONCOMMUTATIVE Lp -SPACE AND OPERATOR SYSTEM
3
adequate to noncommutative Lp -spaces. For a matrix ordered operator space X and p x P Mn pX q, we define νX pxq as p νX
pxq 2 p1 supt|ϕp x0 x0
q| : ϕ P Sp2npX q1, u,
where Sp2n pX q1, denotes the set of positive contractive functionals on Sp2n pX q. Note that a functional ϕ is positive on Sp2n pX q if and only if it is positive on M2n pX q because Sp2n pX q and M2n pX q have the same order structure. According to the following lemma, the uniform equivalence of a matrix norm and νX can be confirmed by showing the p uniform equivalence of the Spn -norm and νX . Lemma 2.1. Suppose X is a matrix ordered operator space, and there is a constant p k ¡ 0 satisfying }x}SpnpX q ¤ kνX pxq for any x P Mn pX q. Then we have
}x}M pX q ¤ kνX pxq n
for any x P Mn pX q. Hence, X is an operator system. n Proof. For a, b P pS2p q1 , we have
}
a 0 0 b
}S p}a}2pS 2n 2p
From [Pi2, Theorem 1.5], the map
x11 x12 x21 x22
n 2p
Þ 2 p1 ϕp a0 b0 Ñ
}b}2pS q ¤ 2 n 2p
1 2p
x11 x12 x21 x22
1 2p
.
a 0 0 b
q
for ϕ P Sp2n pX q1, defines a positive contractive functional on M2n pX q. It follows that
| : ϕ P M2npX q1, u
a 0 0 x ¥ 2 supt|ϕp 0 b x 0 a0 0b q| : ϕ P Sp2npX q1, , a, b P pS2pn q1u
0 axb 2 supt|ϕp b xa 0 | : ϕ P Sp2npX q1, , a, b P pS2pn q1 u suptνXp paxbq : a, b P pS2pn q1 u ¥ k1 supt}axb}S pX q : a, b P pS2pn q1u k1}x}M pX q for x P Mn pX q. For the last equality, see [Pi2, Lemma 1.7]. Lemma 2.2. Suppose a is a bounded linear operator on a Hilbert space and a v |a| is its polar Let f be a continuous function defined on r0, 8q with f p0q 0. decomposition.
| a | a Then is a positive operator and the equality a |a|
1 |a | a 1 f p|a |q f p|a |qv fp q 2 vf p|a|q f p|a|q 2 a |a| νX pxq supt|ϕp
0 x x 0
1 p
1 p
n p
n
holds.
4
KYUNG HOON HAN
Proof. The positivity follows from [Pa, Exercise 8.8 (vi)]. By using the Weierstrass theorem and the right polar decomposition a |a |v, it is sufficient to show that the equality n
|a | a 2n1 |a|n |a|n1a a |a| |a|n a |a |n1 holds for n ¥ 1. We proceed by mathematical induction. Since a paa qk a pa aqk 1 for k ¥ 0, we have n1 a |a |n1a pa aq 2 1 |a|n 1. Similarly, apa aqk paa qk a implies a|a|n
It follows that
|a|na.
|a|n |a|n1a a |a|n a |a |n1 n 1 n 1
a | | a | | a |n a a|a|n | n1 2 a|a|n |a|a|a|n1 a|a|n1a |a|n 1 n 1 n
n |a | 2 a|a|n ||aa|n| a1 .
|a|
a |a|
2
n 1
In order to prove that the Schatten p-class Sp is an operator system, Lemma 2.2 is sufficient. However, in order to prove that a general noncommutative Lp -space is an operator system, an unbounded version of Lemma 2.2 is needed. We denote the Dirac measure massed at the zero point by δ0 . Lemma 2.3. Let a be a closed densely defined ³8 operator on a Hilbert space. Suppose a v |a| is a polar decomposition and |a| 0 tdE is a spectral decomposition. Then
1 1 vv v 2 2 δ0 1 v I 12 vv 2 2 |a | a . is a spectral measure corresponding to 12 a |a| Proof. Let
1 vEv vE 1 E . 2 Ev v vE Since v v E pp0, 8qq, v v commutes with E p∆q for any Borel set ∆ in r0, 8q. For all Borel sets ∆1 and ∆2 in r0, 8q, we have E 1 p∆1 qE 1 p∆2 q
1 vE p∆1 qv vE p∆1 q vE p∆2 qv vE p∆2 q 4 E p∆1qv vvE p∆1q E p∆2qv vvE p∆2q
1 vE p∆1 qv vE p∆2 qv vE p∆1 qE p∆2 qv vE p∆1 qv vE p∆2 q vE p∆1 qv vE p∆2 q 4 E p∆1 qvvE p∆2qv vvE p∆1 qE p∆2 qv E p∆1 qvvE p∆2 q vvE p∆1qvvE p∆2q
1 vE p∆1 X ∆2 qv vE p∆1 X ∆2 q 2 E p∆1 X ∆2qv vvE p∆1 X ∆2 q E 1p∆1 X ∆2 q. r1 E
vEv vE Ev v vE
I
NONCOMMUTATIVE Lp -SPACE AND OPERATOR SYSTEM
5
Taking ∆1 ∆2 , we see that E 1 is projection valued. The countable additivity with respect to strong operator topology is obvious. Since E pt0uq I v v, we have E 1 pt0uq 0.
r satisfies all the conditions of a spectral measure. From these, it is easy to check that E We still need to show that
» 8 1 |a | a h h x | y zdErξ,ξ a k k | a | 2 0
for ξ
hk and h P Dpaq, k P Dpaq. For the left-hand side, we compute
x |aa | |aa| hk | hk y x|a|h|hy xak|hy xah|ky x|a|k|ky xv|a|vh|hy xv|a|k|hy x|a|vh|ky xvv|a|k|ky »8 »8 tdEk,v h tdEv h,vh 0 0 »8 »8 tdEv h,k
0
For the right-hand side, we compute
1 vE p∆qv 1 Eξ,ξ p∆q x
vE p∆q v vE p∆q
tdEk,v vk .
0
h h | y k k
2 E p∆qv 1 pxvE p∆qvh|hy xvE p∆qk|hy xE p∆qvh|ky xvvE p∆qk|kyq 2 1 pEv h,vhp∆q Ek,vhp∆q Ev h,k p∆q Ek,vvk p∆qq. 2
Lemma 2.4. Suppose M is a semifinite von Neumann algebra with a faithful semifinite normal trace τ and a is a τ -measurable operator with its polar decomposition a v|a|. |a | a is a Let f be a continuous function defined on r0, 8q with f p0q 0. Then a |a| positive self-adjoint τ2 -measurable operator and the equality 1 fp 2
|a| a
a |a|
q
1 f p|a |q f p|a |qv 2 v f p|a |q f p|a|q
holds. Proof. Because we have
|a|
1 τ2 pE˜ ppn, 8qq τ p vE ppn, 8qqv q 2 τ pE pn, 8qq Ñ 0,
1 τ p v vE pn, 8qq 2
a a |a| is τ2-measurable. Suppose E is a spectral measure corresponding to |a|. We let an
aE pr0, nsq.
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KYUNG HOON HAN
Then we have
|an| pE pr0, nsaaE pr0, nsq |a|E pr0, nsq. Since an v v |a|E pr0, nsqv ¥ 0 and pan v q2 aE pr0, nsqv vE pr0, nsqa aE pr0, nsqa , 1 2
we have
an v
We compute 1 |a | 2 a 1 |a | 4 a
paE pr0, nsqaq |an|. 1 2
a r |a| E pr0, nsq
a |a|
vE pr0, nsqv vE pr0, nsq E pr0, nsqv v vE pr0, nsq
|vE pr0, nsqv aE pr0, nsqv |a|vE pr0, nsq av vE pr0, nsq 14 a|a vE pr0, nsqv |a|E pr0, nsqv avE pr0, nsq |a|vvE pr0, nsq
1 |an | an 2 a |an| . n
Since |an | is bounded, we can apply Lemma 2.2. The partial isometry in the polar decomposition of an is vn : vE pr0, nsq. It follows that 1 fp 2
Since v |an |k v
|a|
a |a|
a
qErpr0, nsq f p 12 |aan| |aann| q n
1 f p|an |q f p|an |qvn 2 vf p|a |q f p|an|q . n n
vpanvqk v |an|k for any k ¥ 1, we have v f p|an |qv f p|an |q.
Similarly, we also have
vf p|an |qv f p|an |q. The spectral measure for |a | is vEv δ0 pI vv q. It follows that
1 f p|a |q f p|a |qv r E pr0, nsq 2 v f p|a |q f p|a|q
1 2f p|a |qvE pr0, nsqv 2f p|a |qvE pr0, nsq 4 vf p|a|qvE pr0, nsqv f p|a|qE pr0, nsqv vf p|a|qvE pr0, nsq f p|a|qvvE pr0, nsq
1 2f p|a |vE pr0, nsqv q 2f p|a |vE pr0, nsqv qvn 4 vf p|a|vE pr0, nsqvq vvf p|a|E pr0, nsqqv vf p|a|vE pr0, nsqvqv f p|a|E pr0, nsqq n n
12 vffp|p|aan|q |q ffp|p|aan|qn|qvn . n n
rpr0, nsq is τ2 -dense, we obtain the desired result from [Te1, Because the set n81 ranE Proposition I.12]. Lemma 2.5. Suppose M is a von Neumann algebra with a faithful semifinite normal weight ϕ. If a is an element of Haagerup’s noncommutative Lp -space Lp pMq, then we have
a | a |a | a P LppM2 pMqq | and } a |a| }LppM2 pMqq 2}a}LppMq . a |a|
NONCOMMUTATIVE Lp -SPACE AND OPERATOR SYSTEM
7
Proof. Let θ be the dual action of R on M σϕ R. Then idM2 b θ is the dual action of R on M2 pMq σtrbϕ R. Because
pidM
2
θθssp|paa q|q e |a |
b θ q |a | a
s
s p
we have
a From Lemma 2.4, it follows that
} |a | a
a |a|
}
|a|
a |a|
a |a| ,
a
P LppM2 pMqq
.
p |a | a qq } a |a| }L pM pMqq p p
p1 2 } v|a|a||p |a|a||pv }L pM pMqq 2p1p}|a|p}L pMq }|a|p}L pMq q 2p}a}pL pMq .
p Lp M2 M
p p
θs paq θs p|a|q
a |a|
1
2
1
1
2
1
p
Theorem 2.6. Suppose M is a von Neumann algebra with a faithful semifinite normal weight ϕ. Its noncommutative Lp -space Lp pMq is an operator system with
}a}M pL pMqq ¤ 2 νL pMq paq, a P MnpLppMqq. Proof. Let a P Lp pMn pMqq with its right polar decomposition a |a |v. Because }a}L pM pMqq }|a|v}L pM pMqq ¤ }a}L pM pMqq , the involution is an isometry on Lp pMn pMqq, which can be identified with Spn pLp pMqq. From [Pi2, Lemma 1.7], the involution is also an isometry on Mn pLp pMqq. The H:older inequality and [Te1, Proposition II.33] show that the positive cone Lp pMn pMqq is 1 p
n
p
p
p
n
p
p
n
n
closed. Hence, the noncommutative Lp -space is a matrix ordered operator space. Let a P Spn pLp pMqq and 1p 1q 1. From Lemma 2.5, it follows that
}a}L pM pMqq supt|trMpbaq| : b P Lq pMnpMqq1 u suptRe trMpb aq : b P Lq pMnpMqq1u supt 12 ptrMpb aq trMpba qq : b P Lq pMnpMqq1u
1 |b |a : b P Lq pMnpMqq1u supt 2 trM pMq |ba b|a b a
1 | b | b 0 a supt 2 trM pMq p b |b| a 0 q : b P Lq pMnpMqq1u
0 a ¤ suptϕp a 0 q : ϕ P LppM2npMqq1, u p
n
2
2
2
1 p
νLp p pMq paq.
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KYUNG HOON HAN
Thus, from Lemma 2.1, we obtain the desired result.
By combining this with [W, Theorem 4.15], we obtain the following embedding theorem. Corollary 2.7. Noncommutative Lp -spaces can be embedded into B pH q 2 p -completely isomorphically and complete order isomorphically. 1
For a C -algebra A, we define an involution on its dual space A as ϕ paq ϕpa q ,
ϕ P A , a P A.
Along the line of operator space duality, we identify Mn pA q and Mn pAq algebraically by using the parallel duality pairing
xrϕij s, raij sy
¸
ϕij paij q,
ϕij
P A , aij P A.
i,j
We can now define the positive cone Mn pA q . However, the duality between the von Neumann algebra Mn pMq and its noncommutative L1 -space L1 pMn pMqq is given by the trace duality pairing
xraij s, rbij sy trM b trMpraij srbij sq
¸
n
trM paij bji q.
i,j
Hence, we must be more careful. For an operator space V , its opposite operator space V op is defined by
}rvijops}M pV q }rvjis}M pV q. op
n
n
op
Note that V and V are the same normed spaces. For a matrix ordered operator space X, we define its opposite matrix ordered operator space X op as the opposite operator space with the matrix order
rxopijs ¥ 0 rxjis ¥ 0.
A functional ϕ : Mn pX q Ñ C can be written as ϕ rϕij s,
ϕprxij sq
¸
ϕij pxij q.
i,j
Let ϕ˜ rϕjis. Then we have
ϕprxij sq ϕ˜prxji sq. A functional ϕ : Mn pX q Ñ C is positive and contractive if and only if ϕ˜ : Mn pX op q Ñ C is positive and contractive. From this, we see that X is an operator system if and only if X op is an operator system. Corollary 2.8. The dual space of a C -algebra is an operator system. Proof. The bidual A of a C -algebra A can be identified with the enveloping von Neumann algebra of A. The opposite matrix ordered operator space pA qop is completely isometric and completely order isomorphic to L1 pA q by [Te1, Theorem II.7]. Hence, the dual space A is an operator system. 1
Finally, we show that the constant 2 p in the inequality
}a}L pM pMqq ¤ 2 p
n
1 p
νLp p pMq paq
NONCOMMUTATIVE Lp -SPACE AND OPERATOR SYSTEM 1
9
is the best one. However, this does not imply that 2 p is the best constant in Theorem 2.6. Unexpectedly, it is achieved by the 2-dimensional commutative case. We consider an element p1, 1q in ℓ2p . It is sufficient to show that 1 p
1 q
νℓp2p pp1, 1qq ¤ 1.
1, we let
a1 b1 a2 b2 Æ 4 ¯b1 d1 Æ P pSq q1, . ¯b2 d2 We have a1 b1 1 1 a2 b2 ÆÆ 1 1ÆÆq 0 ¤ trp ¯b1 d1 1 1 ¯b2 d2 1 1 a1 a2 d1 d2 b1 b2 ¯b1 ¯b2 . It follows that a1 b1 1 a2 b2 ÆÆ 1ÆÆq b1 b2 ¯b1 ¯b2 trp ¯b1 d1 1 ¯b2 d2 1 ¤ 12 pa1 a2 d1 d2 b1 b2 ¯b1 ¯b2 q a1 b1 1 1 a2 b2 ÆÆ 1 1Æ 12 trp ¯b1 d1 1 1 Æ q ¯b2 d2 1 1 1 1 1 1ÆÆ ¤ 12 } 1 1 }S 1 1 1 1 1 1 ÆÆ 12 } 1 1 }S 1 1 2 . For
4 p
4 p
1 p
In the same manner, we can also show that
a1 b1 1 Æ a2 b2 trp 1 ¯b1 d1 Æ ¯ b2 d2 1
We obtain the inequality
νℓp2p pp1, 1qq ¤ 1.
1Æ Æ q ¥ 2
1 p
.
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KYUNG HOON HAN
References [CE] M.-D. Choi and E. Effros, Injectivity and operator spaces, J. Funct. Anal. 24 (1997), 156–209. [HJX] U. Haagerup M. Junge, and Q. Xu, A reduction method for noncommutative Lp -spaces and applications , Trans. Amer. Math. Soc., to appear. [JRX] M. Junge, Z-J. Ruan and Q. Xu, Rigid OLp structures of non-commutative Lp -spaces associated with hyperfinite von Neumann algebras, Math. Scand. 96 (2005), 63–95. [K] H. Kosaki, Applications of the complex interpolation method to a von Neumann algebra: Noncommutative Lp -spaces, J. Funct. Anal. 56 (1984), 29–78 [Pa] V.I. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Studies in Advanced Mathematics, 78. Cambridge University Press, Cambridge, UK, 2002. [Pi1] G. Pisier, The operator Hilbert space OH, complex interpolation and tensor norms, Mem. Amer. Math. Soc. 122, No. 585, 1996. [Pi2] G. Pisier, Non-commutative vector valued Lp -spaces and completely p-summing maps, Ast´erisque 247, 1998. [S] I. E. Segal, A non-commutative extension of abstract integration, Ann. Math. (2) 57 (1953), 401–457. [Ta] M. Takesaki, Theory of Operator Algebras II, Encyclopaedia of Mathematical Sciences, Vol. 125, Springer-Verlag, 2002 [Te1] M. Terp, Lp spaces associated with von Neumann algebras, Math. Institute, Copenhagen University, 1981. [Te2] M. Terp, Interpolation spaces between a von Neumann algebra and its predual, J. Operator Theory 8 (1982), 327-360. [W] W. Werner, Subspaces of LpH q that are -invariant, J. Funct. Anal. 193 (2002), 207–223. Department of Mathematical Sciences, Seoul National University, San 56-1 ShinRimDong, KwanAk-Gu, Seoul 151-747, Korea E-mail address:
[email protected]
