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Decomposition of an Order Isomorphism Between Matrix-Ordered Hilbert Spaces

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Decomposition of an Order Isomorphism Between Matrix-Ordered Hilbert Spaces 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 2 N H H+ 2 N H~ H~ 2 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 2 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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1974 YASUHIDE MIURA 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 = UjAj of A we obtain the following properties: (1) There exists a unique invertible operator B in M+ such that jAj = 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 2M,ξ 2 H~+. Proof. (1) Let M be non-σ-finite. Choose an increasing net fpigi2I 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 2 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 2 qiM qi is an order automorphism on qiH . By [3, Theorem 3.3] there exists a + unique invertible operator Bi 2 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 fAig is bounded, flog Big 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;X2Ms.a.: Thus flog Big is bounded. It follows that fpiBipig is bounded because piMpi and qiMqi are ∗-isomorphic. Therefore, one can find a subnet of fpi log Bipig that converges to some element C 2M+ in the σ-weak topology. We may index the subnet as the same i 2 I.Wethenhaveforξ,η 2H, ((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 2M, and un- f g ∗ f g der the strong topologyS Ai converges to A A; hence qi(log Ai)qi converges to ∗ H H ∗ log A A.Since i2I qi is dense in , we obtain the equality C + JCJ =logA A. Therefore, eCJeC J = A∗A. Thus there exists an element B 2M+ such that jAj = BJBJ. Since, in addition, qiBqiJqiBqiJqi = qijAjqi, 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 2 M~ H+ 0H~+ 0 0 0 projection pi such that U(qi )=qi with qi = piJpiJ. Using also [1, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DECOMPOSITION OF AN ORDER ISOMORPHISM 1975 ∗ 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) 2M 2 0H~+ 2M 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 2 p Ms.a.p such that Yij 0 H~ = i i qi αi(qiXqi). Using an isometry between the Jordan algebras, one sees that fαi(qiXqi)g is bounded, because k αi(qiXqi) k = k qiXqi k≤kX k;i 2 I.Thus fYig is bounded. We may then say that fYig converges to some operator Y 2 M~ s.a. in the σ-weak topology. We then have for ξ 2 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 2 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 f g 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 2M. It is now immediate 2 2 that α(X )=α(X) for all X 2Ms.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 Mj 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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