L -space is a complete Jordan invariant The positive contractive part of a noncommutative Linear Algebra and its Applications

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Linear Algebra and its Applications

www.elsevier.com/locate/laa

The positive contractive part of a noncommutative L

-space is a complete Jordan invariant

Chi-Wai Leung^a, Chi-Keung Ng^b, Ngai-Ching Wong^c,∗

a DepartmentofMathematics,TheChineseUniversityofHongKong,HongKong

b ChernInstituteofMathematicsandLPMC,NankaiUniversity,Tianjin300071, China

cDepartmentofAppliedMathematics,NationalSunYat-senUniversity, Kaohsiung,80424,Taiwan

a r t i c l e i n f o a bs t r a c t

Article history:

Received22October2016 Accepted22December2016 Availableonline28December2016 Submittedby P.Semrl

MSC:

primary46L10,46L52 secondary54E35

Keywords:

Non-commutativeL^p-spaces Positivecontractiveelements Metricspaces

Bijectiveisometries Jordan^∗-isomorphisms

Let 1 ≤ p ≤ +∞. We show that the positive part of the closed unitballofanon-commutativeL^p-space,as ametric space,isacompleteJordan^∗-invariantfortheunderlyingvon Neumannalgebra.

©2016ElsevierInc.Allrightsreserved.

* Correspondingauthor.

E-mailaddresses:[email protected](C.-W. Leung),[email protected](C.-K. Ng), [email protected](N.-C. Wong).

http://dx.doi.org/10.1016/j.laa.2016.12.033 0024-3795/©2016ElsevierInc.Allrightsreserved.

1. Introduction

Given avon Neumann algebra M, celebrated results of R.V. Kadison showed that several partial structures of M can recover the von Neumann algebra up to Jordan

∗-isomorphisms. In particular, each of the following is a complete Jordan ^∗-invariant of M: the Banachspace structure of the self-adjoint partM_sa of M ([4, Theorem 2]), theordered vectorspace structure ofM_sa([4,Corollary 5])and thetopological convex setstructure ofthenormalstatespaceofM ([5,Theorem 4.5]).

Letp∈ [1,+∞], and letL^p(M) be the non-commutativeL^p-space associated to M withthecanonicalconeL^p(M)₊.IfM issemi-finite,P.-K.Tamshowedin[14]thatthe orderedBanachspace(L^p(M)_sa,L^p(M)₊) characterisesMuptoJordan^∗-isomorphisms.

In the case when M is σ-finite (but not necessarily semi-finite) and p = 2, the corre- sponding result follows from a result of A. Connes (namely, [2, Théorème 3.3]). For a general W^∗-algebra M, results of L.M. Schmitt in [12] show that the ordered Banach space(L^p(M)_sa,L^p(M)₊) determinestherealLiealgebraM/Z(M),whereZ(M) isthe centerof M. On theother hand,extending resultsof B.Russo ([11])and F.J. Yeadon ([16]),D.Shermanshowedin[13]thattheBanachspaceL^p(M) isalsoacompleteJordan

∗-invariantforageneralvonNeumannalgebraM whenp= 2.

Alongthese lines,wewillshowinthisarticlethattheunderlyingmetricspacestruc- tureofthepositivecontractivepart

L^p(M)¹₊:=L^p(M)₊∩L^p(M)¹ (1≤p≤+∞)

ofL^p(M) isalso acompleteJordan ^∗-invariantof M, where L^p(M)¹ istheclosed unit ball.Moreprecisely,weobtaininTheorem 3.1thattwoarbitraryvonNeumannalgebras M and N are Jordan ^∗-isomorphic whenever there exists a bijection Φ from L^p(M)¹₊ ontoL^p(N)¹₊ whichisisometricinthesensethat

Φ(x)−Φ(y)=x−y (x, y∈L^p(M)¹₊).

Notice that, when p = 2, the closed unit ball L²(M)¹ itself is not a complete Jordan

∗-invariant(sinceforanyinfinitedimensionalvonNeumannalgebraMwithaseparable predual,onehasL²(M)∼=²),butitspositivepartisaJordan ^∗-invariant.

The ideas of our proof go as follows. In the case of p = +∞, we employ a strong form of the Mazur–Ulam theorem (whichwas first proved by P. Mankiewicz) to show thata“shifting” Ψ ofΦ extends to alinearbijective isometry from M_sa onto N_sa (see Proposition 3.2),andthemapx →Ψ(1)Ψ(x) inducesaJordan^∗-isomorphismfromMto N (thankstoaresultofR.V. Kadison).Inthecaseofp= 1,weuseLemma 3.6toshow thatΦ(0)= 0,exceptforafewfinitedimensionalcases.Wethenuseourpreviousresult in[6] concerning normal statespaces to obtain the conclusion. For the remaining few finitedimensionalcases,weuseaHausdorffdimensionargumenttoshowthatMandN are^∗-isomorphic.Inthecaseofp∈(1,+∞),wefirstusethestrictconvexityofL^p(N)sa

toshowthatΦ isaffine(Lemma 3.10)andhenceΦ(0)= 0 (Proposition 3.9).Thenweuse severalpropertiesofnon-commutativeL^p-spaces(seeStatement(S1)–(S4))torelatethe restriction ofΦ onthepositivepartoftheunitsphereofL^p(M)sa to abiorthogonality preserving bijection between thenormal statespaces of M and N. Finally,we use our previousresultsin[6]to finishtheproof.

2. Preliminaries

Throughout thisarticle,ifE isasubsetofanormedspaceX andλ>0,we set E^λ:={x∈E:x ≤λ}.

In thefollowing,we will brieflyrecall (mainly from [15] and[10])notationsconcerning non-commutativeL^p-spaces.LetMbea(complex)vonNeumannalgebraona(complex) Hilbert space H. Let ϕ be a fixed normal semi-finite faithful weight on M and α : R →Aut(M) bethe modular automorphism groupcorresponding to ϕ. Then the von NeumannalgebracrossedproductMˇ :=M¯αRissemi-finiteandwefixanormalfaithful semi-finitetraceτ onM.ˇ ThemeasuretopologyonMˇ (asintroducedbyE. Nelsonin[8]) is givenbyaneighborhoodbasisat0 oftheform

U(, δ) :={x∈Mˇ :xp ≤and τ(1−p)≤δ, for a projectionp∈Mˇ}.

The completion,L0( ˇM ,τ),ofMˇ with respectto this topologyisa^∗-algebraextending the^∗-algebrastructureonM.ˇ

OnemayidentifyL₀( ˇM ,τ) withacollectionofclosedand denselydefinedoperators onL²(R;H) affiliatedwithM.ˇ Moreprecisely,supposethatTissuchaclosedoperatoron L²(R;H) andthat|T|istheabsolutevalueofT withthespectralmeasureE_|T|.ThenT corresponds(uniquely)toanelementinL₀( ˇM ,τ) ifandonlyifτ

1−E_|T|([0,λ])

<+∞ when λ is large enough. In this case, the ^∗-operation on L₀( ˇM ,τ) coincides with the adjoint.Moreover,theadditionandthemultiplicationonL₀( ˇM ,τ) aretheclosuresofthe corresponding operationsfordenselydefinedclosedoperators.WedenotebyL₀( ˇM ,τ)₊ theset ofallpositiveself-adjoint(but notnecessarilybounded)operators inL0( ˇM ,τ).

The dual action αˆ : R → Aut( ˇM) of α extends to an action on L0( ˇM ,τ) by

∗-automorphisms. For any p ∈ [1,+∞], we set (with the convention that e^−s/p = 1 when p= +∞)

L^p(M) :={T ∈L₀( ˇM , τ) : ˆα_s(T) =e^−s/pT, for alls∈R}. Denote byL^p(M)sathesetof allself-adjoint operatorsinL^p(M) andput

L^p(M)+:=L^p(M)∩L0( ˇM , τ)+.

If T ∈L0( ˇM ,τ) andT =u|T| isthepolardecomposition,then T ∈L^p(M) ifandonly if|T|∈L^p(M).

Inthecasewhenp∈(1,+∞),themapthatsendsx∈Mˇ+ tox^p extendstoamap Λ_p:L₀( ˇM , τ)₊→L₀( ˇM , τ)₊.

Forany T ∈L₀( ˇM ,τ)₊, onehasT ∈L^p(M) ifand onlyifΛ_p(T)∈L¹(M).There isa canonicalidentificationofM_∗withL¹(M) thatsendsthepositivepartM_∗,+ ofM_∗onto L¹(M)+,andthisinducesaBanachspacenorm · 1 onL¹(M) (seee.g.Theorem 7in Chapter IIof [15]).Thefunctiondefinedby

Tp:=Λp(|T|)^1/p1 (2.1) isanorm onL^p(M) thatturnsitinto aBanachspace.Letusalsodenote

S^p(M) :={T ∈L^p(M)+:Tp= 1}. (2.2) Itisknownthat(L^p(M),L^p(M)+) isindependentofthechoiceofϕuptoanisometric orderisomorphism(seee.g.Theorem 37andCorollary38inChapterIIof[15]).Onthe other hand,onemayidentifyM with L^∞(M) (asorderedBanachspaces) throughthe canonicalinclusionM ⊆Mˇ ⊆L0( ˇM ,τ) (seeProposition 10inChapter IIof[15]).

3. Themainresult

Theorem3.1. LetM andN be twovonNeumann algebrasandletp∈[1,+∞].If there isabijectiveisometryΦ:L^p(M)¹₊→L^p(N)¹₊,thenM andN areJordan^∗-isomorphic.

Inordertoprovethisresult,weshallgivesomepreparationsinPropositions 3.2,3.5 and3.9forthecasesp= +∞,p= 1 andp∈(1,+∞),respectively.

3.1. The caseof p= +∞

Proposition 3.2. If Φ:M₊¹ →N₊¹ is abijectiveisometry, then Ψ:Msa^1/2 →Nsa^1/2 given byΨ(x):= Φ(x+¹₂)−¹₂ extendstoalinear isometryfromMsa ontoNsa.

Itthen followsfrom [4,Theorem2]thatΨ(1) isaself-adjointunitaryandΨ further inducesaJordanisomorphismx →Ψ(1)Ψ(x) fromM_sa ontoN_sa.

Example3.3.LetM=C⊕∞C.ThesetM₊¹ equalsthesquareinR⊕∞Rwithvertices (0,0),(0,1),(1,1) and(1,0).IfΦ0:R⊕∞R→R⊕∞Ristheclockwiserotationby90 degreeaboutthecenter(¹₂,¹₂),thentherestrictionΦ ofΦ0onM₊¹ isabijectiveisometry ontoM₊¹ thatsends(0,0) to(0,1).Hence,Φ itself cannotbeextendedtoalinearmap.

However,ifΨ isdefinedasinProposition 3.2,thenΨ(1,1)= Φ₃

2,³₂

−₁

2,¹₂

= (1,−1) andthemap

(x, y) →Ψ(1,1)Ψ(x, y) = (1,−1)

Φ0(x+ 1/2, y+ 1/2)−(1/2,1/2)

= (y, x) extendstoa^∗-automorphismofM.

In order to establish Proposition 3.2, we need the following strongerversion of the Mazur–Ulam theorem, which was first proved in [7, Theorem 2] (see also [1, Theo- rem 14.1]).

Lemma 3.4. Let U be a non-empty openconnected subset of anormed spaceX andW be an open subset of a normed space Y. Then every isometry from U onto W can be extended uniquelytoanaffine isometry fromX ontoY.

ProofofProposition 3.2. Letusfirstnotethatforanyx∈M_sa,onehasx∈M₊¹ ifand only if x−¹₂ ≤ ¹₂ (by considering the C^∗-subalgebra generated by x and 1).Thus, x →x−¹₂ isabijectiveisometryfromM₊¹ ontoM

1

sa2 andthemapΨ inthestatementis abijectiveisometryfromM

1

sa2 ontoN

1

sa2. Ifx∈M

1

sa2, thenx=¹₂ ifandonlyifthereexistsx∈M

1

sa2 withx−x= 1.This implies

Ψ

{x∈Msa:x= 1/2}

={y∈Nsa:y= 1/2}. Consequently,Ψ(0)= 0 andΨ willsendtheinterior,B_M

0,¹₂

,ofMsa¹² ontotheinterior ofN

1

sa2.ByLemma 3.4,themapΨ|BM

0,¹₂extendstoalinearisometryΨ from˜ Msaonto NsaandthecontinuityofΨ tellsus thatΨ˜|

M

1 sa2

= Ψ. 2

3.2. Thecase ofp= 1

Proposition3.5.If thereexistsabijectiveisometryΦfromM_∗¹_,+ ontoN_∗¹_,+,thenM and N areJordan^∗-isomorphic.

Note, firstofall,thatonecannotuseLemma 3.4forthiscase,becausetheinteriorof M_∗,+¹ couldbeanemptyset;e.g.,whenM =L^∞([0,1]).

Foranyμ∈M_∗,+,wedenotebysuppμthesupportprojectionofμinM.Recallthat forany μ,ν∈M_∗,+, wehave

μ−ν=μ+ν if and only if suppμ·suppν = 0. (3.1) Inorder toobtainProposition 3.5, weneedthefollowinglemma.

Lemma 3.6. If N containsthree non-zero projections, q₁, q₂ andq₃,orthogonal toeach other, thenthebijectiveisometry ΦinProposition 3.5will send 0to0.

Proof. Suppose on the contrary that Φ(0) = 0. Let us first show thatsupp Φ(0) = 1.

Indeed,ifitisnotthecase,onecanfindμ∈M_∗,+¹ suchthatΦ(μ)= 1 andsupp Φ(μ)≤ 1−supp Φ(0),which,together with(3.1),givesthecontradiction that

1≥ μ−0=Φ(μ)−Φ(0)=Φ(μ)+Φ(0)>1.

Asaresult,Φ(0)(qk)>0 fork= 1,2,3.Wemayalsoassume,withoutlossofgenerality, thatΦ(0)(q₁)≤ Φ(0)/3 because

3

k=1Φ(0)(q_k)≤ Φ(0).

Now,pickanyν ∈M_∗¹_,+ withΦ(ν)= 1 andsupp Φ(ν)≤q1.Since2q1−1 isaunitary andΦ(ν)−Φ(0)=ν≤1,onearrivesat thefollowing contradiction:

1≥ |(Φ(ν)−Φ(0))(q₁−(1−q₁))|=|1−Φ(0)(q₁) + Φ(0)(1−q₁)|

= 1 +Φ(0) −2Φ(0)(q1)>1. 2

ByLemma 3.6, if N contains three non-zeroprojections orthogonalto one another, then Φ induces an isometric bijection from the normal state space of M to that of N, and hence, we may conclude that M and N are Jordan ^∗-isomorphic by using [6, Theorem 3.4]. For the benefit of the readers, we will instead go through briefly the argumentof[6,Theorem 3.4]byrecallingthefollowingtwolemmas. Thesetwolemmas willalsobe neededinthecaseofp∈(1,+∞) below.

LetusrecallthatabijectionΓ fromthelatticeofprojectionsinMtothatofN isan orthoisomorphism ifforanyprojectionspandq inM, onehas

p q= 0 if and only if Γ(p)Γ(q) = 0.

Lemma 3.7. ([6, Lemma 3.1(a)]) Suppose that Ψ is a bijection from the normal state spaceof M tothat of N, whichis biorthogonality preserving in thesense that forany normalstates μandν ofM,onehas

suppμ·suppν = 0 if and only if supp Ψ(μ)·supp Ψ(ν) = 0.

Thenthere isanorthoisomorphism Ψˇ from thelattice ofprojections in M tothat of N satisfying Ψ(suppˇ μ)= supp Ψ(μ)forany normal stateμon M.

Asecondlemmathatweneedisthefollowingpossiblywell-knownvariantofatheorem ofH.A. Dye in[3] (seee.g.[6,Lemma2.2(a)]).Note thatanassumption ofnothaving type I₂ summand is needed for the original version of Dye’s theorem. However, the variantherehasaweakerconclusionand doesnotneedtheassumption concerningthe absenceoftypeI2 summand.

Lemma 3.8. If thereexists anorthoisomorphism from thelattice ofprojections in M to that of N,then M andN areJordan^∗-isomorphic.

ProofofProposition 3.5. LetusfirstconsiderthecasewhenN containsthree non-zero projections orthogonal to each other. Then by Lemma 3.6, the map Φ restricts to an isometric bijection Ψ fromthenormal statespaceof M tothatof N.Hence,(3.1) im- pliesthatΨ isbiorthogonalitypreserving. TheconclusionnowfollowsfromLemmas 3.7 and 3.8. In the case when M contains three non-zero projections orthogonal to one another,onemayobtainthesameconclusionbyconsideringthebijectiveisometryΦ⁻¹. SupposethatneitherM norN containsthreenon-zeroprojectionsorthogonaltoone another. ThenM andN canonlybe C,C⊕∞Cor M₂(C).ObservethattheHausdorff dimensionsofthequasi-statespaceofC,C⊕∞CandM₂(C) are1,2 and4 respectively.

Since abijective isometry preserves Hausdorffdimensions,we concludethatM and N are ^∗-isomorphic. 2

3.3. Apreparationforthecaseof p∈(1,+∞)

Proposition3.9.Letp∈(1,+∞).SupposethatM andN aretwovonNeumannalgebras such that M =Cor N =C.If Φ:L^p(M)¹₊ →L^p(N)¹₊ is abijectiveisometry, then Φ is anaffine mapsending0to0.

Notice thatL^p(M)sa andL^p(N)sa arestrictlyconvexBanachspacesforp∈(1,+∞) (see e.g.,Section5of[9]).The followinglemma ispossibly well-known,butwegive its simpleproofhereforcompleteness.

Lemma 3.10. Let X1 and X2 be Banach spaces such that X2 is strictly convex. Then every isometry fromaconvex subsetK of X1 toX2 isautomaticallyan affinemap.

Proof. Weneedtoverifythatf

(x+y)/2

=

f(x)+f(y)

/2 foreveryx,y∈K.Suppose thatxandy aretwoarbitrarilychosenfixedelementsinK withx=y.Byreplacing K with K−yand f with themap fromK−y toX2thatsendszto f(z+y)−f(y),one mayassumethaty= 0 andf(0)= 0. Thus,wehavef(z)=z(z∈K)and

f(x)−f(x/2)=x/2=f(x)/2 =f(x) − x/2 =f(x) − f(x/2). Now, thestrictconvexityofX₂ givesf(x)−f(x/2)=t·f(x/2) forsomet∈R+.This, together with f(x)/2 = f(x)− f(x/2), will produce t = 1. Hencewe have the requiredrelationf(x)/2=f(x/2). 2

Proof of Proposition 3.9. We note, first of all, that if M = C, then the Hausdorff dimensionofL^p(M)¹₊ isoneandhencesoistheHausdorffdimensionofL^p(N)¹₊,which gives N =C.Therefore, we mayassumethatthedimensions ofboth Msa andNsa are at leasttwo.

SinceL^p(M)saisstrictlyconvex,thesetofextremepointsofL^p(M)¹₊ isS^p(M)∪ {0} (see (2.2)). The same is true for L^p(N)¹₊. By Lemma 3.10, the map Φ is affine and hence Φ(0) ∈ S^p(N)∪ {0}. Suppose on the contrary that Φ(0) ∈ S^p(N). Then, as dimL^p(N)_sa>1,thereisasequence{v_k}k∈N inS^p(N)\ {Φ(0)}withv_k−Φ(0)→0, and hence {Φ⁻¹(v_k)}k∈N is a sequence in S^p(M) norm-converging to 0, which is ab- surd. 2

3.4. The finishingof theproof

Forany T ∈L^p(M)sa, wedenote bysuppT thesupportprojection ofT, i.e.suppT is thesmallestprojection pinM satisfying T·p=T (or equivalently, p·T =T). Let usrecallthefollowingstatementsconcerningS,T ∈L^p(M)+ fromFact 1.2andFact 1.3 of[10]:

S1). supp Λ_p(T)= suppT;

S2). S·T = 0 ifandonlyifsuppS·suppT = 0;

S3). ifsuppS·suppT = 0,thenS+T^pp=S−T^pp=S^pp+T^pp;

S4). ifp= 2 andS+T^pp=S−T^pp=S^pp+T^pp,thensuppS·suppT = 0.

Proofof Theorem 3.1. Thecases of p= +∞ and p= 1 are proved inProposition 3.2 (togetherwith[4,Theorem2])andProposition 3.5,throughthecanonicalidentifications of L¹(M) and L^∞(M) with M_∗ and M, respectively. Moreover, the case of p = 2 is alreadyestablishedin[6,Corollary3.11](duetoProposition 3.9and[6,Proposition 3.7]).

Now,weconsider p∈(1,+∞)\ {2}.Without lossofgenerality, wemayassumethat M = C or N = C. By Proposition 3.9, the map Φ is affine and Φ(0) = 0. On the otherhand,itfollows fromRelation (2.1)thatΛp restrictstoabijectionfrom L^p(M)¹₊ ontoL¹(M)¹₊ with Λp(S^p(M))=S¹(M) (see(2.2)). Therefore, Φ induces abijection Φ :ˆ S¹(M)→ S¹(N) withΦ(A)ˆ = Λ_p(Φ(Λ⁻¹_p (A)). For any A,B ∈S¹(M),it follows from(S1),(S3)and (S4)that

suppA·suppB= 0 if and only if

Λ⁻¹_p (A)

2 +Λ⁻¹_p (B) 2

p

p

=

Λ⁻¹_p (A)

2 −Λ⁻¹_p (B) 2

p

p

= 2^1−p.

As Φ is an isometric affine map satisfying Φ(0) = 0, the map Φ canˆ be regarded as a biorthogonality preserving bijection between the normal state spaces of M and N (through the identification L¹(M) = M_∗ and (S2)). The conclusion now follows from Lemmas 3.7 and3.8. 2

Remark3.11.SupposethatM C.Whenp= 2,onecanuse[6,Theorem3.8] andthe argument of [6, Proposition 3.7] to obtain a Jordan ^∗-isomorphism Θ : N → M with

Λ2◦Φ = Θ^∗◦Λ2|L²(M)¹₊. In the case of p ∈ [1,+∞)\ {2}, let us state the following question:

Does there exist a Jordan ^∗-isomorphism Θ : N → M satisfying Λ_p ◦Φ = Θ^∗ ◦ Λ_p|L^p(M)¹+?

In thecaseofp= +∞,we havealreadyseeninExample 3.3thatthis strongerversion doesnothold.

Acknowledgements

The authors are supported by National Natural Science Foundation of China (11471168)andTaiwanMOSTgrant(104-2115-M-110-009-MY2).Theyalsoappreciate forthehelpfulcommentsfrom thereferee.

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