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A CHARACTERIZATION OF FREENESS BY A FACTORIZATION PROPERTY OF R-TRANSFORM. 1 0 0 ALEXANDRU NICA*, DIMITRI SHLYAKHTENKO†,AND ROLAND SPEICHER 2 n ABSTRACT. Let M be a B-probability space. Assume that B itself is a D-probability a space;thenM canbeviewedasaD-probabilityspaceaswell. LetX ∈ M. Wecharac- J terizefreenessofX fromB withamalgamationoverDintermsofacertainfactorization 7 conditionlinkingtheB-valuedandD-valuedR-transformsofX. Wegiveanapplication 1 torandommatrices. ] A O h. 1. OPERATOR-VALUED R-TRANSFORM. t a 1.1. Multiplicative functions. Let B be a unital algebra. Recall that a B-probability m space (M,E : M → B) (see e.g. [8], [6]) is a pair consistingof an algebra M containing [ B as a unital subalgebra, and a conditional expectation E : M → B. In other words, 1 E : M → B isunitaland B-bilinear: v 6 E(1) = 1, E(bmb′) = bE(m)b′, ∀b,b′ ∈ B,∀m ∈ M. 4 1 ElementsofM arecalled B-valuedrandom variables. 1 0 Recall that a multiplicative map h···i : Mn → B is a C-multilinear map, (i.e., a 1 sequence of maps Mn ∋ m ,...,m 7→ hmS ,...,m i ∈ B) satisfying the B-linearity 0 1 n 1 n / conditions h t a hbm1,...,mni = bhm1,...,mni m hm ,...,m bi = hm ,...,m ib 1 n 1 n : v hm ,...,m b,m ,...,m i = hm ,...,m ,bm ,...,m i. i 1 k k+1 n 1 k k+1 n X (Here m ∈ M, b ∈ B). r i a Given a non-crossing partition π ∈ NC(n) and an arbitrary multiplicative map h···i, we can constructabracketingπh···i : Mn → B, defined recursivelyby 1 hm ,...,m i = hm ,...,m i k 1 k 1 k (π ⊔ρ)hm ,...,m i = πhm ,...,m i·ρhm ,...,m i 1 k 1 p p+1 k ins(p,ρ → π)(m ,...,m ) = πhm ,...,m ρhm ,...,m i, 1 k 1 p p+1 p+q m ,...,m i. p+q+1 k Here 1 denotes the partition with the sole class {1,...,k}, π ⊔ρ denotes disjoint union k (with the equivalence classes of ρ placed after those of π), and ins(p,ρ → π) denotes the *ResearchsupportedbyagrantofNSERC,Canada. †ResearchsupportedbyanNSFpostdoctoralfellowship. 1 2 A.NICA,D.SHLYAKHTENKO,ANDR.SPEICHER partition obtained from π by inserting the partition ρ after the p-th element of the set on which π determinesapartition. Inotherwords,each partitionπ isinterpretedasarecipeforplacingbracketsh···i,and πh···i isthevalueoftheresultingexpression. 1.2. Moments and R-transform. The B-probability space structure of the algebra M givesriseto oneexampleofsuch amultiplicativemap,namely,themomentsmap [···] : Mn → B, E [ givenby [m ,...,m ] = E(m ···m ). 1 n E 1 n The reason for the name is that, having fixed B-random variables X ,...,X ∈ M, the 1 p followingvaluesof[···] , E [b X b ,··· ,X b ,X b ] 0 i1 1 in−1 n−1 in n E are called B-valued momentsofthefamilyX ,...,X . 1 n In [6] and [7] the notion of B-valued R-transform was introduced (we follow the com- binatorial approach of [6], see also [5]). Like the map [···], the R-transform map is a multiplicativemap {···} : Mn → B. E [ Thefollowingcombinatorialformulaactuallydetermines{···} uniquely: E [m ,...,m ] = all possiblebracketingsinvolving{···} 1 n E E X = π{m ,...,m } . 1 n E X π∈NC(n) The uniqueness of the definition can be easily seen by observing that the right-hand side of the equation above involves {m ,...,m } and that the rest of the terms are products 1 n E offactors ofsmallerorder (i.e.,restrictionsof{···} toMk,k < n). E It is important to note that [···] determines {···} and vice-versa. Moreover, the value of[···]| depends onlyon{···}| , andvice-versa. Mn M∪···∪Mn 1.3. Moment and cumulant series. Givena multiplicativefunction h···i : Mk → B, and X ,...,X ∈ B, considerthefamilyofmultilinearmaps S 1 n Mh···i : Bk−1 → B, i ,...,i ∈ {1,...,n} i1,...,ik 1 k givenby h···i M (b ,...,b ) = hX b ,··· ,b X i. i1,...,ik 1 k−1 i1 1 k−1 ik In theparticularexamplesabove,weget themoment seriesofX ,...,X , 1 n µX1,...,Xn = M[···]E i1,...,ik i1,...,ik and thecumulantseries, kX1,...,Xn = M{···}E . i1,...,ik i1,...,ik CHARACTERIZATIONOFFREENESS 3 We willsometimeswrite kX1,...,Xn B;i1,...,ik to emphasizethat theseries isvaluedin B. 1.4. Freeness with amalgamation. Let M ,M ⊂ M be two subalgebras, each contain- 1 2 ing B. The importance of R-transform is apparent from the following theorem ([6], see also [1]): Theorem. Let S ,S be two subsets of M. Let M be thealgebra generatedby S and B, 1 2 i i i = 1,2.Then M and M arefree with amalgamationover B iff whenever X ,...,X ∈ 1 2 1 n S ∪S , 1 2 {X ,...,X } = 0 1 n E unlesseither allX ,...,X ∈ S , orallX ,...,X ∈ S . 1 n 1 1 n 2 1.5. Canonicalrandom variables. Let X ,...,X ∈ M befixed. 1 n Thenby[7]thereexistsaB-probabilityspace(F,E : F → B),elementsλ∗,...,λ∗ ∈ B 1 n F, and elementsλk satisfyingthefollowingproperties: p (i) λ∗ b λ∗ b ...λ∗ b λk = kX1,...,Xn(b ,...,b ),b ,...,b ∈ B; j1 1 j2 2 jk k j j1,...,jk,j 1 k 1 k (ii) Let w = b a b a b ...a b , where b ∈ B, a = λk or a = λ∗. Then E (w) = 0 0 1 1 2 2 n n i i p i p B unlessw can bereduced toan element ofB usingrelation 1.5(i). (By convention, we take λ0 = E (X ) ∈ B). We shouldclarify that for each k, λk is just p B p p aformalvariable,andwedonotassumeanyrelationsbetween{λp} : forexample,λk is k k,p p not thek-th powerofλ1. p It is not hard to show that the properties listed abovedetermine the restriction of E to B thealgebragenerated by λ∗ and {λk} . p p,k Let Y = λ∗ + λk. j j j X k≥0 (This series is formal; however, Y ,...,Y have moments, since each such moment in- 1 n volvesonlyafinitenumberoftermsfrom theseriesdefining Y ). j It turnsoutthat kX1,...,Xn = kY1,...,Yn, and µX1,...,Xn = µY1,...,Yn. i1,...,ik i1,...,ik i1,...,ik i1,...,ik In other words, given a cumulant series, Y ,...,Y is an explicit family of B-valued ran- 1 n domvariables,whosecumulantseriesis equalto theonegiven. 2. FREENESS FROM A SUBALGEBRA. 2.1. D-cumulants vs. B-cumulants. Let now D ⊂ B be a unital subalgebra, and let F : B → D be a conditional expectation. If (M,E : M → B) is a B-probability space, then (M,F ◦E : M → D)is aD-probabilityspace. 4 A.NICA,D.SHLYAKHTENKO,ANDR.SPEICHER Theorem 2.1. Let X ,...,X ∈ M. Assumethat the B-valued cumulants of X ,...,X 1 n 1 n satisfy kX1,...,Xn(d ,...,d ) ∈ D, ∀d ,...,d ∈ D. B;i1,...,ik 1 k−1 1 k−1 Then the D-valued cumulantsof X ,...,X are given by the restrictionsof the B-valued 1 n cumulants: kX1,...,Xn(d ,...,d ) = kX1,...,Xn(d ,...,d ), ∀d ,...,d ∈ D. D;i1,...,ik 1 k−1 B;i1,...,ik 1 k−1 1 k−1 Proof. LetN bethealgebrageneratedbyD andX ,...,X . Theconditiononcumulants 1 n impliesthat {···} | E Np S isvaluedinD. Itfollowsfromthemoment-cumulantformulathat[···] | isvaluedin E Np D, and hencethat S [···] | = [···] | = π{···} | . F◦E Np E Np E Np S S X S π∈NC(p) Since themoment-cumulantformuladetermines{···} | ,it followsthat F◦E Np S {···} | = {···} | . F◦E Np E Np S S We record an equivalent formulation of the theorem above (which was implicit in the proof): Theorem 2.2. Let N ⊂ M be a subalgebra, containing D. Assume that {···} | is E Np valued inD. Then S {···} | = {···} | . F◦E Np E Np S S Ingeneral,intheabsenceoftheconditionthatkX1,...,Xn restrictedto∪Dp isvaluedinD, B the expression of D-valued cumulants of X ,...,X in terms of the B-valued cumulants 1 n is quite complicated. Note, for example, that if X is a B-valued random variable, and b ∈ B, then the B-valued cumulant series of bX are very easy to describe. On the other hand,theD-valuedcumulantseriesofbX canhaveaverycomplicatedexpressioninterms oftheD-valuedcumulantseriesofX and b. The sufficient condition in the theorem above is actually quite close to being necessary in the case that the conditional expectations are positive maps of ∗-algebras. As an illus- tration, consider thecase that D ⊂ B consists of scalar multiplesof 1, and F : B → D is such thatτ = F ◦E isatraceon M, satisfyingτ(xy) = τ(yx) forallx,y ∈ M. Recall thatX iscalled aB-semicircularvariableifitscumulantseries isgivenby kX (b ,...,b ) = δ η(b ) 1,...1 1 p−1 p,2 1 p for some map η : B → B.|I{tzis}easily seen that if X is B-semicircular, then η(b) = E(XbX). CHARACTERIZATIONOFFREENESS 5 Theorem 2.3. Let (M,E : M → B) be a B-probability space, such that M and B are C∗-algebras. Let F : B → C = D ⊂ B be a faithful state. Assume that τ = F ◦ E is a faithfultraceonM. Let X beaB-semicircularvariableinM. ThenthedistributionofX with respect toτ isthesemicirclelawiffE(X2) ∈ C. Proof. If E(X2) = kX(1) ∈ C, it followsthat the B-valued cumulantsof X, restricted to 11 D = C are valued in D. Hence by Theorem 2.2, the D-valued cumulant series of X are the same as the restriction of the B-valued cumulant series; hence the only scalar-valued cumulant of X which is nonzero is the second cumulant kX, so that the distribution of X 11 is thesemicirclelaw. Conversely, assume that the distribution of X is the semicircle law. Let η(b) = kX(b), 11 b ∈ B. Thenwehave 2τ(η(1))2 = 2τ(X2)2 = τ(X4) = F ◦E(X4) = F(E(η(η(1))))+F(E(η(1)η(1))) = τ(Xη(1)X)+τ(η(1)2) = τ(η(1)XX)+τ(η(1)2) = 2τ(η(1)2), so that τ(η(1)) = τ(η(1)2)1/2. By theCauchy-Schwartz inequality,wehavethat ifη(1) ∈/ C, τ(η(1)) = hη(1),1i < kη(1)k ·k1k = τ(η(1)2)1/2, 2 2 which isacontradiction. Henceη(1) ∈ C. Wementionacorollary,whichisofinteresttorandommatrixtheory. Letσ(x,y) = σ(y,x) be a non-negative function on [0,1]2, having at most a finite number of discontinuities in eachverticalline. LetG(n)beann×nrandommatrixwithentriesg ,sothat{g : i ≤ j} ij ij areindependentcomplexGaussianrandomvariables,g = g ,theexpectationE(g ) = 0 ij ji ij and the variance E(|g |2) = 1σ(i, j). The matrices G(n) are called Gaussian Random ij n n n Band Matrices. Let µ betheexpectedeigenvaluedistributionofG(n), i.e., n 1 µ ([a,b]) = ×expectednumberofeigenvaluesofG(n)in[a,b]. n n Corollary 2.4. The eigenvalue distribution measures µ of the Gaussian Random Band n 1 Matrices G(n) converge weakly to the semicircle law iff σ(x,y)dy is a.e. a constant, 0 independentof x. R The proof of this relies on a result from [2], showing that G(n) has limit eigenvalue distributionµ,givenasfollows. LetX betheL∞[0,1]-semicircularvariableinanL∞[0,1]- probability space (M,E : M → L∞[0,1]), so that E(XfX)(x) = 1f(y)σ(x,y)dy. Let 0 F : L∞[0,1] → C denote the linear functional F(f) = 1f(x)dxR, and denote by τ the 0 R 6 A.NICA,D.SHLYAKHTENKO,ANDR.SPEICHER traceF◦E onW∗(X,L∞[0,1]). Thenµisthescalar-valueddistributionofX withrespect to τ, i.e., tkdµ(t) = τ(Xk). Z It remains to apply Theorem 2.3, to conclude that µ is a semicircle law iff E(X2) ∈ C, 1 i.e., σ(x,y)dy is aconstantfunctionofx. 0 R 2.2. A characterization of freeness. We are now ready to state the main theorem of this note. ThefollowingtheoremwasearlierprovedforB-valuedsemicircularvariablesin[3], and found manyuses inoperatoralgebratheory. Theorem 2.5. Let X ,...,X ∈ M. Assume that F : B → D satisfies the faithfullness 1 n condition that if b ∈ B and if F(b b ) = 0 for all b ∈ B, then b = 0. Then X ,...,X 1 1 2 2 1 1 n arefree fromB with amalgamationover D ifftheirB-valued cumulantseriessatisfies (2.1) kX1,...,Xn(b ,...,b ) = F(kX1,...,Xn(F(b ),...,F(b )). B;i1,...,ik 1 k−1 B;i1,...,ik 1 k−1 forallb ,... ,b ∈ B. Inshort,k = F ◦k◦F. (Herethecumulantseriesarecomputed 1 k−1 in theB-probabilityspace(M,E : M → B)). Equivalently, kX1,...,Xn(b ,...,b ) = kX1,...,Xn(F(b ),...,F(b )). B;i1,...,ik 1 k−1 D;i1,...,ik 1 k−1 We note that in the case that M is a C∗-probability space, the faithfulness assumption aboveis exactly the conditionthat the GNS representation of B with respect to thecondi- tionalexpectationF isfaithful. SinceX ,...,X arefreefromBwithamalgamationoverDiffthealgebraN generated 1 n byX ,...,X andD isfreefromB overD,thetheoremabovecanbeequivalentlystated 1 n as Theorem 2.6. LetN ⊂ M beasubalgebraofM,containingD. AssumethatF : B → D satisfies the faithfullness condition that if b ∈ B and if F(b b ) = 0 for all b ∈ B, then 1 1 2 2 b = 0. Then N isfreefromB over D iff 1 {n b ,n b ,...,n } = F{n F(b ),n F(b ),...,n } 1 1 2 2 k E 1 1 2 2 k E for allk andn ,...,n ∈ N, b ,...,b ∈ B. 1 k 1 k−1 Proof. Weprovethetheorem inthefirst formulation. Assume that the condition (2.1) is satisfied by the cumulant series of X ,...,X . Let 1 n Y ,...,Y beas in Section1.5. Sincethefreeness ofX ,...,X from B withamalgama- 1 n 1 n tionoverD isaconditionontheB-momentseriesofX ,...,X ,andY ,...,Y havethe 1 n 1 n sameB-momentseriesasX ,...,X ,itissufficienttoprovethatY ,...,Y ∈ F arefree 1 n 1 n withamalgamationoverD from B. Since (2.1) is satisfied, λ0 = E (Y ) ∈ D and hence Y ,...,Y belong to the algebra j B j 1 n L generated in F by λ∗ and λq, 1 ≤ j,p ≤ n, q ≥ 1 and D. Therefore, it is sufficient to j p provethat Lis freefrom B withamalgamationoverD. Let w ,...,w ∈ L, so that F ◦E (w ) = 0, and let b ,...,b ∈ B, so that F(b ) = 0 1 s B j 0 s j (allowingalsob and/orb tobeequal to1). Wemustprovethat 0 s F ◦E (b w b ···w b ) = 0. B 0 1 1 s s CHARACTERIZATIONOFFREENESS 7 Note that the factorization condition (2.1) as well as the definition of the generators of L (see 1.5(i) and 1.5(ii)) imply that E | has values in D. It follows that we may assume B L that E (w ) = 0 (since F ◦E (w ) = E (w ) ∈ D). By the definition of E , its kernel B j B j B j B is spannedby irreduciblenon-trivialwords inthegenerators λ∗ and λq. Then j p W = b w ···w b 0 1 s s is again a linear combination of words in the generators λ∗ and λq. By linearity, we may j p reducetothecasethatW isasingleword. IfW isirreducible,itmustbenon-trivial(since each w is non-trivial), hence E (W) = 0, so that F ◦E (W) = 0. So assume that W is i B B notirreducible. Sinceeach w isirreducible,thismeansthatW containsasub-wordofthe i form W = W ·d λ∗ d λ∗ d ··· 1 0 i1 1 i2 2 d b d λ∗ ···b ···λ∗ d b d λ∗ s 1 s+1 is+2 j ik k r k+1 ik+1 ···d λ∗ d λp ·W . p ip p+1 j 2 Usingtherelation(1.5(i))and thefactorizationcondition(2.1), weget that W = W d k (d ,...,F(d b d ),...,d )W = 0, 1 0 i1,...,ip,j 1 k r k+1 p+1 2 sinceF(d b d ) = d F(b )d = 0. Thusinany case, F ◦E (W) = 0. k r k+1 k r k+1 B Wehavethereforeseenthatthefactorizationconditionimpliesfreenesswithamalgama- tion. Toprovetheotherimplication,assumethatX ,...,X arefreewithamalgamationover 1 n D from B. LetZ ,...,Z beB-valuedrandomvariables,so that 1 n kZ1,...,Zn (b ,...,b ) = F(kX1,...,Xn(F(b ),...,F(b ))). B;i1,...,ik 1 k−1 D,i1,...,ik 1 k−1 where k denote D-valued cumulants. (Note that the first occurrence of F is actually D redundant, as kX1,...,Xn(F(b ),...,F(b ) ∈ D). Then by the first part of the proof, D,i1,...,ik 1 k−1 Z ,...,Z are free from B with amalgamation over D. Moreover, by Theorem 2.2, 1 n the D-valued distributions of Z ,...,Z and X ,...,X are the same. By assumption, 1 n 1 n X ,...,X are free with amalgamation over D from B. This freeness, together with the 1 n D-valued distribution of X ,...,X , determines their B-valued distribution. Indeed, the 1 n freeness assumptionsdetermine F ◦E(b′b X b X ···b X b ), b′,b ∈ B 0 i1 1 i2 n−1 in n i which inviewoftheassumptionson F determines E(b X b X ···b X b ), b ∈ B. 0 i1 1 i2 n−1 in n i It follows that the B-valued distributions of X ,...,X and Z ,...,Z coincide. Hence 1 n 1 n theB-valuedcumulantsofX ,...,X satisfy(2.1). 1 n As an application,wehavethefollowingproposition(see e.g. [4, Lemma2.7]): Proposition2.7. Let N ⊂ M be a subalgebra. Let B ⊂ C ⊂ D ⊂ M be subalgebras, and E : M → C, E : M → B, E : M → D be conditional expectations, so that C B D E = E ◦E , E = E ◦E . AssumethatE andE andE satisfythefaithfullness B B C C C D C B D assumptionsofTheorem2.5. AssumethatN isfreefromC withamalgamationoverB,and 8 A.NICA,D.SHLYAKHTENKO,ANDR.SPEICHER also free from D with amalgamation over C. Then N is free from D with amalgamation over B. Proof. SinceN isfreefromD withamalgamationoverC,wehavethatforalln ∈ N and j d ∈ D, j {n d ,n d ,...,n } = E {n E (d ),...,n } . 1 1 2 2 k ED C 1 C 1 k EC Since N isfree from C withamalgamationoverD,weget similarlythatforall c ∈ C, j {n c ,n c ,...,n } = E {n E (c ),...,n } . 1 1 2 2 k EC B 1 B 1 k EB Applyingthiswithc = E (d ) andcombiningwiththepreviousequationgives j C j {n d ,n d ,...,n } = E {n E (d ),...,n } , 1 1 2 2 k ED B 1 B 1 k EB sinceE = E ◦E . Hence N isfree from D withamalgamationoverB. B B C We mention as a corollary the following identity. Let B ⊂ C ⊂ D be C∗-algebras, EN : N → B, EC : C → B and ED : D → C be conditional expectations, having B B C faithful GNSrepresentations. Considerthereduced free product (((N,EN)∗ (C,EC)),EN ∗id)∗ (D,ED), B B B B C C where EN ∗ id denotes the canonical conditional expectation from the free product B (N,EN)∗ (C,EC)ontoC. Then B B B (((N,EN)∗ (C,EC)),EN ∗id)∗ (D,ED) B B B B C C ∼= (N,EN)∗ (D,EC ◦ED). B B B C Toseethis,itissufficienttoprovethatN ⊂ (((N,EN)∗ (C,EC)),EN ∗id)∗ (D,ED) B B B B C C isfreefromD withamalgamationoverB,sinceboth(((N,EN)∗ (C,EC)),EN ∗id)∗ B B B B C (D,ED) and (N,EN) ∗ (D,EC ◦ ED) are generated by N and D as C∗-algebras. But C B B B C N is free from D over C, and from C over D, by construction. Hence by the proposition above,N isfree fromD overB. REFERENCES [1] A.Nica,R-transformsoffreejointdistributionsandnon-crossingpartitions,JournalofFunctionalAnal- ysis135(1996),271–296. [2] D. 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CHARACTERIZATIONOFFREENESS 9 DEPARTMENT OF PURE MATHEMATICS, UNIVERSITY OF WATERLOO, WATERLOO, ONTARIO, N2L 3G1, CANADA E-mailaddress:[email protected] DEPARTMENTOF MATHEMATICS,UCLA, LOS ANGELES,CA 90095,USA E-mailaddress:[email protected] DEPARTMENT OF MATHEMATICS AND STATISTICS, QUEEN’S UNIVERSITY, KINGSTON, ONTARIO K7L 3N6,CANADA E-mailaddress:[email protected]