PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 7, Pages 1973–1977 S 0002-9939(04)07454-4 Article electronically published on February 6, 2004

DECOMPOSITION OF AN ORDER ISOMORPHISM BETWEEN MATRIX-ORDERED HILBERT SPACES

YASUHIDE MIURA

(Communicated by David R. Larson)

Abstract. The purpose of this note is to show that any order isomorphism between noncommutative L2-spaces associated with von Neumann algebras is decomposed into a sum of a completely positive map and a completely co- positive map. The result is an L2 version of a theorem of Kadison for a Jordan isomorphism on operator algebras.

1. Introduction In the theory of operator algebras, a notion of self-dual cones was studied by A. Connes [1], and he characterized a standard Hilbert space. In [9] L. M. Schmitt and G. Wittstock introduced a matrix-ordered Hilbert space to handle a non- commutative order and characterized it using the face property of the family of self-dual cones. From the point of view of the complete positivity of the maps, we shall consider a decomposition theorem of an order isomorphism on matrix-ordered Hilbert spaces. Let H be a Hilbert space over a complex number field C,andletH+ be a self- dual cone in H.Asetofalln × n matrices is denoted by Mn.PutHn = H⊗Mn(= H ∈ N H H+ ∈ N H˜ H˜ ∈ N Mn( )) for n . Suppose that ( , n ,n )and( , n, n )arematrix- ordered Hilbert spaces. A linear map A of H into H˜ is said to be n-positive (resp. ⊗ H+ ⊂ H˜+ n-co-positive) when the multiplicity map An(= A idn)satisfiesAn n n t H+ ⊂ H˜+ t · (resp. (An n ) n ). Here ( ) denotes a set of all transposed matrices. When A is n-positive (resp. n-co-positive) for all n ∈ N, A is said to be completely positive (resp. completely co-positive). We refer mainly to [11] for standard results in the theory of operator algebras. We use the notation as introduced in [9] with respect to matrix-ordered standard forms.

2. Results We first generalize a theorem of A. Connes [1] for the polar decomposition of an order isomorphism to the case where a von Neumann algebra is non-σ-finite.

Received by the editors March 6, 2003. 2000 Mathematics Subject Classification. Primary 46L10, 46L40. Key words and phrases. Order isomorphism, completely positive map, matrix-ordered Hilbert space. This research was partially supported by the Grants-in-Aid for Scientific Research, The Min- istry of Education, Culture, Sports, Science and Technology, Japan.

c 2004 American Mathematical Society 1973

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Proposition 1. Let (M, H,J,H+) and (M˜ , H˜, J˜, H˜+) be standard forms, and let A be a linear bijection of H onto H˜ satisfying AH+ = H˜+. Then for a polar decomposition A = U|A| of A we obtain the following properties: (1) There exists a unique invertible operator B in M+ such that |A| = BJBJ (cf. [4, Corollary II.3.2]). (2) There exists a unique Jordan ∗-isomorphism α of M onto M˜ such that (α(X)ξ,ξ)=(XU−1ξ,U−1ξ) for all X ∈M,ξ ∈ H˜+.

Proof. (1) Let M be non-σ-finite. Choose an increasing net {pi}i∈I of σ-finite projections in M converging strongly to 1. Put qi = piJpiJ.By[1,Theorem4.2] + + + qiH is a closed face of H˜ .SinceA is an order isomorphism, A(qiH ) is a closed H˜+ 0 ∈ M˜ H+ face of . There then exists a σ-finite projection pi such that A(qi )= 0H˜+ 0 0 0 0 H+ qi where qi denotes piJpiJ. Hence qiAqi is an order isomorphism of qi onto 0H˜+ M qi . These cones appear respectively in the reduced standard forms (qi qi, H H+ 0M˜ 0 0H˜ 0 0 0H˜+ 0 ∗ 0 qi , qiJqi, qi )and(qi qi, qi , qiJqi, qi ). Put Ai =(qiAqi) qiAqi.Then + + Ai ∈ qiM qi is an order automorphism on qiH . By [3, Theorem 3.3] there exists a + unique invertible operator Bi ∈ qiM qi such that Ai = BiJiBiJi,whereJi denotes qiJqi. Taking a logarithm of both sides, we have log Ai =logBi + Ji(logi Bi)Ji. Since {Ai} is bounded, {log Bi} is bounded. Indeed, we have in a standard form that a map 7→ 1 X δX = 2 (X + JXJ) is a Jordan isomorphism of a selfadjoint part of M into a selfadjoint part of a set of all order derivations D(H+) by [4, Corollary VI.2.3]. It is known that any isomor- phism of a JB-algebra into another JB-algebra is an isometry (see [3, Proposition 3,4.3]). Hence

k δX k = k X k,X∈Ms.a..

Thus {log Bi} is bounded. It follows that {piBipi} is bounded because piMpi and qiMqi are ∗-isomorphic. Therefore, one can find a subnet of {pi log Bipi} that converges to some element C ∈M+ in the σ-weak topology. We may index the subnet as the same i ∈ I.Wethenhaveforξ,η ∈H,

((C + JCJ)qj ξ,qjη) = lim((pi(log Bi)pi + Jpi(log Bi)piJ)qj ξ,qjη) i = ((log Bj + Jj(log Bj)Jj )qjξ,qjη)

= lim(log Aiqj ξ,qjη) i ∗ =(logA Aqjξ,qjη),

using the facts that qiXqiJqiXqiJqi = piXpiJpiXpiJqi for all X ∈M, and un- { } ∗ { } der the strong topologyS Ai converges to A A; hence qi(log Ai)qi converges to ∗ H H ∗ log A A.Since i∈I qi is dense in , we obtain the equality C + JCJ =logA A. Therefore, eCJeC J = A∗A. Thus there exists an element B ∈M+ such that |A| = BJBJ. Since, in addition, qiBqiJqiBqiJqi = qi|A|qi, one easily sees the invertibility and the unicity of B using the same properties as in the σ-finite case. (2) From (1) we have U = AB−1JB−1J. It follows that U is an isometry + + satisfying UH = H˜ .Letpi and qi be as in (1). There then exists a σ-finite 0 ∈ M˜ H+ 0H˜+ 0 0 0 projection pi such that U(qi )=qi with qi = piJpiJ. Using also [1,

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∗ M 0M˜ 0 Theorem 3.3], one can find a unique Jordan -isomorphism αi of qi qi onto qi qi such that −1 −1 (αi(qiXqi)ξ,ξ)=(qiXqiU ξ,U ξ) ∈M ∈ 0H˜+ ∈M 0 M˜ 0 0M˜ 0 for all X ,ξ qi .NowfixX s.a..Sincepi pi and qi qi are 0 ˜ 0 ∗-isomorphic, there exists a unique operator Yi ∈ p Ms.a.p such that Yi| 0 H˜ = i i qi αi(qiXqi). Using an isometry between the Jordan algebras, one sees that {αi(qiXqi)} is bounded, because k αi(qiXqi) k = k qiXqi k≤kX k,i ∈ I.Thus {Yi} is bounded. We may then say that {Yi} converges to some operator Y ∈ M˜ s.a. in the σ-weak topology. We then have for ξ ∈ H˜+, 0 0 0 0 0 0 (Yqjξ,qjξ) = lim(Yiqjξ,qjξ) = lim(αi(qiXqi)qj ξ,qjξ) i i −1 0 −1 0 = lim(qiXqiU qjξ,U qjξ) i −1 0 −1 0 =(XU qjξ,U qjξ). Taking a limit with respect to j,weobtain (Yξ,ξ)=(XU−1ξ,U−1ξ) for all ξ ∈ H˜+. It is known that any normal state on the von Neumann algebra M˜ is represented by a vector state with respect to an element of H˜+ (see [2, Lemma 2.10 (1)]). Therefore, the above element Y is uniquely determined. Moreover, we 0 0 { } have qiYqi = αi(qiXqi). It follows that α(qiXqi) converges to Y in the strong topology. Hence one can define α(X)=Y for all X ∈M. It is now immediate 2 2 that α(X )=α(X) for all X ∈Ms.a.. Considering the inverse order isomorphism U −1,wehaveα(M)=M˜ . This completes the proof.  In the following lemma we deal with a reduced matrix-ordered standard form by a completely positive projection. M H H+ Lemma 2. With ( , , n ) a matrix-ordered standard form, let E be a completely H M H H+ positive projection on .Then(E E, E , En n ) is a matrix-ordered standard form. Proof. The statement was shown in [8, Lemma 3] where M is σ-finite. In the case where M is not σ-finite, since E is a completely positive projection, there N N H H+ exists a von Neumann algebra such that ( , E , En n ) is a matrix-ordered M| N M H H+ standard form by [6, Lemma 3]. Hence E EH = and (E E, E , En n ) is a matrix-ordered standard form by using the same discussion as in the proof in [7].  Now, we shall state the decomposition theorem for an order isomorphism between noncommutative L2-spaces. M H H+ M˜ H˜ H˜+ Theorem 3. Let ( , , n ) and ( , , n ) be matrix-ordered standard forms. Suppose that A is a 1-positive map of H into H˜ such that AH+ is a self-dual cone in the closed range of A. If both the support projection E and the range projection F of A are completely positive, then there exists a central projection P of EME such that AP is completely positive and A(E − P ) is completely co-positive. In particular, if A is an order isomorphism of H onto H˜, then there exists a central projection P of M such that AP is completely positive and A(1 − P ) is completely co-positive.

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Proof. We first consider the case where A is an order isomorphism. Let U, B and α be as in Proposition 1. It follows from a theorem of Kadison [5] that there exists a central projection P of M satisfying ˜ α : MP → Mα(P ), onto ∗-isomorphism and ˜ α : M1−P → Mα(1−P ), onto ∗-anti-isomorphism. Indeed, α(P ) is a central projection of M˜ .Sinceα preserves a ∗-operation and power, α(P ) is a projection. Suppose that Q is an arbitrary projection in M. Since α is order preserving, we have α(QP ) ≤ α(P )andα(Q(1−P )) ≤ α(1−P ). It follows that two projections α(P )andα(QP ) are commutative, and so are α(1−P ) and α(Q(1 − P )). Hence, α(Q)=α(QP + Q(1 − P )) and α(P )commute.Since α is bijective, a set α(Q) generates a von Neumann algebra M˜ . Therefore, α(P ) belongs to a center of M˜ . Now, there then exists a unique completely positive isometry u : P H→α(P )H˜ such that

+ + −1 u(P H )=α(P )H˜ )andα(x)=uxu ,x∈MP by [7, Proposition 2.4], which is also valid for the non-σ-finite case. Hence (UxU−1ξ, −1 + ξ)=(uxu ξ,ξ),x∈MP ,ξ ∈ α(P )H˜ . We have from the unicity of a completely positive isometry that UP = u.Notethatα(P )UP = UP. Indeed,wehavefor ξ ∈ α(1 − P )H˜+ the equality k PU−1ξ k2=(UPU−1ξ,ξ)=(α(P )ξ,ξ)=0. This yields PU−1α(1 − P )=O,andsoPU−1 = PU−1α(P ). Therefore, we obtain that AP = UBJBJP = uBJBJP and AP is completely positive. ∗ 0 M → M˜ 0 0 We next consider a -isomorphism α : 1−P 1−α(P ) defined by α (X)= ∗ Jα˜ (X) J,X˜ ∈M1−P . There then exists a unique completely positive isometry v :(1− P )H→α(1 − P )H˜ such that

+ + 0 −1 v(1 − P )H =(1− α(P ))H˜ and α (x)=vxv ,x∈M1−P . ∗ −1 Then we have α(x)=Jvx˜ v J,x˜ ∈M1−P . Note that the complete positivity − H+ − H˜+0 H˜+0 above means vn(1 P )n n =(1 α(P ))n n ,where n denotes the self-dual cones associated with M˜ 0 (cf. [10]). Hence v is a completely co-positive map under M H H+ M˜ H˜ H˜+ the setting ( , , n )and( , , n ). Hence (UxU−1ξ,ξ)=(Jvx˜ ∗v−1Jξ,ξ˜ ) =(Jξ,vx˜ ∗v−1Jξ˜ ) =(vxv−1ξ,ξ)

+ for all x ∈M1−P ,ξ ∈ (1 − P )H . It follows that U(1 − P )=v. We conclude by the equality A(1 − P )=vBJBJ(1 − P )thatA(1 − P ) is completely co-positive. We now consider a general A.SinceAH+ ⊂ H˜+,wehaveAH+ ⊂ F H˜+. Since F is a projection, F H˜+ is a self-dual cone in F H˜. It follows from the self- duality of AH+ that AH+ = F H˜+. This yields from Lemma 2 that FAE is an order isomorphism of EH onto F H˜ in the sense of matrix-ordered standard forms M H H+ M˜ H˜ H˜+ (E E, E , En n )and(F F , F , Fn n ). Using the first part of the proof, we obtain the desired result. Indeed, there exists a central projection P ∈ EME

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such that FAP is completely positive and FA(E − P ) is completely co-positive under the reduced matrix-ordered standard forms. We obtain the inclusion t − H+ t − H+ ⊂ H˜+ ⊂ H˜+ (An(En Pn) n )= (FnAn(En Pn) n ) Fn n n . This completes the proof.  Finally, the author wishes to express his sincere gratitude to Professor Y. Kata- yama for having pointed out this problem to him. He also thanks the members of Sendai Seminar, especially Professors T. Okayasu and K. Saitˆofortheiruseful suggestions. References

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Department of Mathematics, Faculty of Humanities and Social Sciences, Iwate Uni- versity, Morioka, 020-8550, Japan E-mail address: [email protected]

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