Analytic subordination for bi-free convolution
Serban Belinschi, Hari Bercovici, Yinzheng Gu, Paul Skoufranis
aa r X i v : . [ m a t h . OA ] J a n ANALYTIC SUBORDINATION FOR BI-FREE CONVOLUTION
S.T. BELINSCHI, H. BERCOVICI, Y. GU, AND P. SKOUFRANIS
Abstract.
In this paper we study some analytic properties of bi-free additiveconvolution, both scalar- and operator-valued. We show that using propertiesof Voiculescu’s subordination functions associated to free additive convolutionof operator-valued distributions, simpler formulas for bi-free convolutions canbe derived. We use these formulas in order to prove several results about atomsof bi-free additive convolutions. Introduction
In his second paper on bi-free independence [34], Voiculescu provided alinearizing transform for the bi-free additive convolution of compactly supportedprobability measures in the plane R . This formula may be re-written to naturallyinvolve the subordination functions of the free additive convolutions of the two faces(marginals) of the two probability measures on R (see [34, Remark 2.3] or Equa-tion (4) below). Motivated by the recent work of two of us [20] (see Remark 3.4below), we note that this connection can be more directly justified by appealing tothe operator-valued subordination as introduced in [29], applied to the restrictionto upper triangular 2 × × × This work was started during Y. Gu’s visit to the IMT, partially supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02. H. Bercovici was partially supported bya grant from the National Science Foundation. P. Skoufranis was supported in part by DiscoveryGrant RGPIN-2017-05711 from NSERC (Canada). Notations and background
Noncommutative random variables and (bi)freeness.
For the purposesof this paper, we only need to consider the definition of the bi-freeness of twopairs of random variables. In that, we follow [34, Section 1.1], and refer to [15,33] for full details of the analytic and combinatorial aspects of bi-free probability.Consider a noncommutative probability space ( A , ϕ ), where A is a unital algebraover the field of complex numbers C and ϕ : A → C is a unit-preserving linearfunctional. We assume that A is endowed with an involution ∗ and that ϕ ( x ∗ ) = ϕ ( x ) for all x ∈ A . We will always assume that ϕ is positive and faithful, meaningthat ϕ ( x ∗ x ) ≥
0, with equality if and only if x = 0. In this case, we refer to( A , ϕ ) as a ∗ -noncommutative probability space. If, in addition, A is a C ∗ -algebra(respectively, a W ∗ -algebra) and ϕ is a (normal) state, then ( A , ϕ ) is said to be a C ∗ -noncommutative probability space (respectively, a W ∗ -noncommutative probabilityspace). Elements of A are called random variables. The distribution of a k -tupleof random variables ( a , . . . , a k ) ∈ A k is by definition the collection of all mixedmoments of the k -tuple: µ ( a ,...,a k ) = { ϕ ( a i · · · a i m ) : m ∈ N , i , . . . , i m ∈ { , . . . , k }} . A two-faced pair of noncommutative random variables in ( A , ϕ ) is a pair( a, b ) ∈ A . We consider a as the left random variable and b as the right randomvariable of the pair. A pair { ( a , b ) , ( a , b ) } of two-faced noncommutative randomvariables in ( A , ϕ ) is said to be bi-free if their distribution with respect to ϕ satisfiesthe following property: there are two vector spaces X , X with distinguished statevectors ξ , ξ (i.e. X j = C ξ j ⊕ ker ψ j , with ψ j : X j → C linear, ψ j ( ξ j ) = 1),so that if ( X , ker ψ, ξ ) = ( X , ker ψ , ξ ) ∗ ( X , ker ψ , ξ ), and λ j , ρ j are the leftand right representations of L ( X j ) on L ( X ), respectively, j = 1 , , then the jointdistribution of a , a , b , b with respect to ϕ in A equals the joint distribution ofvariables λ ( a ) , λ ( a ) , ρ ( b ) , ρ ( b ) with respect to ϕ ξ in L ( X ). Here ϕ ξ ( T ) = ψ ( T ( ξ )) , T ∈ L ( X ). It follows from the definition of bi-freeness that left and rightrandom variables of different pairs are classically independent with respect to ϕ (see [33, Proposition 2.16]) so that, in particular, they commute.In [34], Voiculescu shows that if the joint distribution of ( a j , b j ) is de-termined by two-bands moments (i.e. moments of the form ϕ ( LR ), where L runsthrough all monomials in the left face and R runs through all monomials in theright face), then the same remains true for ( a + a , b + b ). If in addition ( A , ϕ )is a C ∗ -noncommutative probability space in which a j = a ∗ j , b j = b ∗ j , a j b j = b j a j ,then the joint distribution of ( a j , b j ) coincides with the moments of a compactlysupported probability measure η j in the plane. We will follow [34] and refer to suchvariables as bi-partite . The correspondence is given via the relation ϕ ( a mj b nj ) = Z R t m s n d η j ( t, s ) , m, n ∈ N , j = 1 , . Obviously, under this hypothesis, the distribution of ( a + a , b + b ) is itself thejoint distribution of two commuting self-adjoint random variables, so that thereexists a compactly supported probability measure η on R whose moments coincidewith it. The measure η depends only on η and η via formulae provided, forexample, in [15]. The notation η = η ⊞⊞ η was introduced in [33] and is called thebi-free additive convolution of η and η . The measure η has the property that its NALYTIC SUBORDINATION FOR BI-FREE CONVOLUTION 3 marginals are the free additive convolutions of the marginals of η and η . Morespecifically, if µ j is the distribution of a j and ν j is the distribution of b j , then thefirst marginal of η is µ ⊞ µ and the second marginal of η is ν ⊞ ν .2.2. Analytic transforms.
In order to linearize the bi-free additive convolution,Voiculescu introduced the partial bi-free R -transform, a function of two complexvariables defined on a neighbourhood of zero in C . We introduce this function,together with its single-variable analogue, and indicate how it allows one to interpretthe operation ⊞⊞ in terms of the single-variable analytic subordination functions[27, 13, 7].First, define G η j ( z, w ) = ϕ (cid:0) ( z − a j ) − ( w − b j ) − (cid:1) = Z R d η j ( t, s )( z − t )( w − s ) ,G µ j ( z ) = ϕ (cid:0) ( z − a j ) − (cid:1) = Z R d µ j ( t ) z − t , G ν j ( w ) = ϕ (cid:0) ( w − b j ) − (cid:1) = Z R d ν j ( t ) w − s , for z ∈ C \ σ ( a j ) , w ∈ C \ σ ( b j ), j = 1 ,
2. Here σ ( T ) denotes the spectrum of theoperator T . We shall refer to all three of these functions as the Cauchy transforms ofthe corresponding measures. Observe that they determine uniquely the probabilitymeasures in question, and depend only on the distribution of ( a j , b j ) with respectto ϕ . Nevertheless, in the following, we will sometimes write G ( a j ,b j ) for G η j , or G a j for G µ j (respectively, G b j for G ν j ).We remind the reader of some properties of Cauchy transforms of positivemeasures. Let us start with the simpler one-variable Cauchy transform: it is ananalytic function G µ sending the upper half-plane C + of the complex plane intothe lower half-plane C − , G µ ( z ) = G µ ( z ), and we have µ ( R ) = lim y → + ∞ iyG µ ( iy ) . The topological support of the measure µ , denoted by supp( µ ), is characterizedby the fact that G µ extends analytically with real values to its complement. Oneeasily sees that G µ is decreasing on each connected component of R \ supp( µ ). Itis negative on ( −∞ , inf supp( µ )) and positive on (sup supp( µ ) , + ∞ ). However, G µ may pass through zero on a bounded component of R \ supp( µ ). If µ is a probabilitymeasure (that is, µ ( R ) = 1), then a simple geometric argument shows that G µ ( z ) ∈ ( u + iv ∈ C : u + (cid:18) v + 12 ℑ z (cid:19) ≤ ℑ z ) ,v ≤ − min (cid:26) ℑ z ( ℜ z − m ) + ( ℑ z ) , ℑ z ( ℜ z − M ) + ( ℑ z ) (cid:27)(cid:27) whenever supp( µ ) ⊆ [ m, M ]. Indeed, more precisely, G µ ( z ) is a limit of convexcombinations of points ( z − t ) − , m ≤ t ≤ M , and these are points which lay onthe arc of the circle centered at − / ℑ z and of radius 1 / ℑ z which is bordered bythe points ( z − m ) − and ( z − M ) − and does not contain zero.Unfortunately, geometric properties of the two-variable Cauchy trans-form are nowhere near as nice as those of the one-variable Cauchy transform. How-ever, given a compactly supported Borel probability measure η on R , one can stilldeduce some useful properties of G η ( z, w ). It is quite obvious that G η ( z, w ) = G η ( z, w ), and that G η is analytic as a function of two complex variables on theset { ( z, w ) ∈ C : ( { z } × R ) ∩ supp( η ) = ( R × { w } ) ∩ supp( η ) = ∅ } . That is tosay, if µ, ν are the marginals of η , then the only part of C on which G η may not S.T. BELINSCHI, H. BERCOVICI, Y. GU, AND P. SKOUFRANIS be analytic is the union of two strips, namely supp( µ ) × C and C × supp( ν ). Inparticular, while the domain of analyticity of G η may not be simply connected, itis a connected open subset of C whose complement is a closed set of (Hausdorff)dimension at most 3. Regrettably, G η does not preserve half-planes, and may mapelements from C + × C + in R , including possibly zero. The zero set of a nonconstanttwo-variable analytic function is an analytic set which is either empty or of com-plex dimension one (we refer to [16] for definition and properties of analytic sets).Specifically, if there exists a point ( z , w ) in the domain of analyticity of G η suchthat G η ( z , w ) = 0, and the map z G η ( z, w ) has a finite number of zeros in agiven bounded neighborhood of ( z , w ) , then, by Weiestrass’ preparation theorem,there exist a nonempty open bidisk U centered at ( z , w ), an integer k ∈ N andone-variable analytic functions c , . . . , c k defined on the first coordinate of U suchthat G η ( z, w ) = (( z − z ) k + c ( w )( z − z ) k − + · · · + c k ( w )) φ ( z, w ), where φ ( z, w )is a zero-free analytic function on U . For k = 1 we recover the classical analyticimplicit function theorem: z = z − c ( w ) is the implicit function. Thus, the zeroset of G η cannot be compactly contained in its domain of analyticity. We would liketo make this statement more precise for the case in which both z and w belongto a half-plane (upper or lower - not necessarily the same). Say z ∈ H , w ∈ H , H j ∈ { C ± } . In this case it is clear that neither z G η ( z, w ) nor w G η ( z , w )is constantly equal to zero, so Weierstrass’ preparation theorem applies to it forboth coordinates. Consider the restriction of G η to H × H . By [16, Definition2.1.2], { ( z, w ) ∈ H × H : G η ( z, w ) = 0 } is an analytic subset of H × H . Let Z be a connected component of this set. This is a principal analytic set (the zero setof a two-variable analytic function) of dimension and codimension one, so, as [16,Theorem 2.3 and Corollary 2.8.2] inform us, its singular points form a discrete set(a point of Z is regular if it has a neighbourhood U such that U ∩ Z is a manifold,and it is singular if it is not regular). Of course, these singular points may verywell accumulate near the boundary of the natural domain of G η . Tautologically, Z is also irreducible, so that the intersection of Z with any other one-dimensionalanalytic subset of H × H is either a discrete (possibly empty) set, or contains Z (see [16, Sections 5.3-5.6]). This observation will be useful later on when we needto compare zero sets of different two-variable Cauchy transforms.It is known from [26] that if one defines K µ j ( z ) as the inverse of G µ j ( z )on a neighbourhood of infinity (so that K µ j (0) = ∞ ), then the function R µ j ( z ) = K µ j ( z ) − z , called the (free) R -transform of µ j , is analytic (instead of just mero-morphic) on the same neighbourhood of zero and satisfies the relation R µ ⊞ µ ( z ) = R µ ( z ) + R µ ( z ) for z in a small enough neighbourhood of zero. Observe that if wedefine ω a ( z ) = K µ ( G µ ⊞ µ ( z )) and ω a ( z ) = K µ ( G µ ⊞ µ ( z )), then the relationsatisfied by the R -transforms can be re-written as(1) ω a ( z ) + ω a ( z ) − z = 1 G µ ⊞ µ ( z ) = 1 G µ ( ω a ( z )) = 1 G µ ( ω a ( z )) . It has been shown that the functions ω a j , j = 1 ,
2, called the subordination func-tions , extend analytically as self-maps of the complex upper half-plane C + andEquation (1) holds for all z ∈ C + (see [27, 13, 7]). Of course, a similar relationholds for R ν j ( w ) , G ν j ( w ) and ω b j ( w ), j = 1 , NALYTIC SUBORDINATION FOR BI-FREE CONVOLUTION 5
For the measure η j (which is the distribution of the pair ( a j , b j ) withrespect to ϕ ), Voiculescu introduces in [34, Theorem 2.1] the function(2) R ( a j ,b j ) ( z, w ) = R η j ( z, w ) = 1 + zR µ j ( z ) + wR ν j ( w ) − zwG η j ( K µ j ( z ) , K ν j ( w )) , for z, w in a small enough bi-disk centred at zero (also see [23, Section 7.2]). Thisfunction is called the partial bi-free R -transform of ( a j , b j ) (or of η j ). Observe firstthat this function is indeed well-defined, including at zero, sincelim w → lim z → G η j ( K µ j ( z ) , K ν j ( w )) zw = lim w → G ν j ( K ν j ( w )) w = 1 . The limits can clearly be permuted. In particular, R η j (0 ,
0) = 0. Theorem 2.1combined with Section 1.2 from [34] provide the following:(3) R η ( z, w ) + R η ( z, w ) = R η ⊞⊞ η ( z, w ) , | z | + | w | sufficiently small . Given the linearizing property of the one-variable R -transform, this is equivalentto zwG η ( K µ ( z ) , K ν ( w )) + zwG η ( K µ ( z ) , K ν ( w )) − zwG η ⊞⊞ η ( K µ ⊞ µ ( z ) , K ν ⊞ ν ( w )) . This relation and Equation (1) allow us to write a formula for G η ⊞⊞ η defined onall of ( C \ σ ( a + a )) × ( C \ σ ( b + b )) involving the subordination functions of freeadditive convolution. If we divide by zw and replace in the above z by G µ ⊞ µ ( z )and w by G ν ⊞ µ ( w ), then1 G η ( ω a ( z ) , ω b ( w )) + 1 G η ( ω a ( z ) , ω b ( w ))(4) = 1 G µ ⊞ µ ( z ) G ν ⊞ ν ( w ) + 1 G η ⊞⊞ η ( z, w ) . This relation clearly holds for | z | and | w | sufficiently large as an equality of analyticfunctions. If rewritten as G η ⊞⊞ η ( z, w ) ( G η ( ω a ( z ) , ω b ( w )) + G η ( ω a ( z ) , ω b ( w )))(5) = G η ( ω a ( z ) , ω b ( w )) (cid:18) G η ⊞⊞ η ( z, w ) G µ ⊞ µ ( z ) G ν ⊞ ν ( w ) + 1 (cid:19) G η ( ω a ( z ) , ω b ( w )) , then it holds for any z ∈ C \ σ ( a + a ) , w ∈ C \ σ ( b + b ), as an equality ofmeromorphic functions (see [34, Remark 2.3 and Lemma 2.6] ). In fact, the onlypoles may come from zeros of G µ ⊞ µ and of G ν ⊞ ν . These functions can havezeros only in co( σ ( a + a )) \ σ ( a + a ) (respectively co( σ ( b + b )) \ σ ( b + b )- we have denoted by co( A ) the convex hull of the set A ). However, Equation (1)guarantees that exactly one of ω a , ω a has a simple pole at the zero of G µ ⊞ µ (with a similar statement for b and ν ). The behavior of the two-variable Cauchytransform at infinity guarantees that the right-hand side of the above equality isactually analytic in such a point, and thus the equality is an equality of analyticfunctions on ( C \ σ ( a + a )) × ( C \ σ ( b + b )).We will see below that in fact Equation (4) extends to ( C + × C + ) ∪ ( C − × C − ) as an equality of meromorphic functions, and { ( z, w ) ∈ C + × C + : G η ⊞⊞ η ( z, w ) =0 } ⊇ { ( z, w ) ∈ C + × C + : G η j ( ω a j ( z ) , ω b j ( w )) = 0 } , j = 1 , S.T. BELINSCHI, H. BERCOVICI, Y. GU, AND P. SKOUFRANIS
Operator-valued random variables, their analytic transforms, and(bi)freeness with amalgamation.
Most importantly for us, Voiculescu extendedin [29] the analytic subordination results from above to self-adjoint random variableswhich are free with amalgamation over some subalgebra. We outline his resultbelow. Let (
M, E, B ) be an operator-valued W ∗ -noncommutative probabilityspace; that is, B ⊆ M is a unital inclusion of (unital) W ∗ -algebras, and E : M → B is a unit-preserving conditional expectation. Let X = X ∗ , X = X ∗ ∈ M be freeover B with respect to E . Then there exists a countable family ω = { ω n } n ∈ N , witheach ω n defined on a subset of M n ( B ), such that( E ⊗ Id M n ( B ) ) (cid:2) ( v − ( X + X ) ⊗ I n ) − (cid:3) = ( E ⊗ Id M n ( B ) ) (cid:2) ( ω n ( v ) − X ⊗ I n ) − (cid:3) , for all v ∈ M n ( B ) with strictly positive imaginary part or of inverse of sufficientlysmall norm. The functions ω n increase the imaginary part of v if ℑ v >
0, and thedependence on n satisfies certain compatibility conditions (see [29, 31]). As it willusually be clear from the context, from now on we supress the level n from ournotation. The functions of the type G X = { G X,n } n ∈ N , G X,n ( v ) = ( E ⊗ Id M n ( B ) ) (cid:2) ( v − X ⊗ I n ) − (cid:3) are natural extensions of the classical Cauchy transforms and share many of theproperties of their classical, complex-valued counterparts. Voiculescu showed in[28] that they allow the definition of B -valued R -transforms via the exact sameprocedure as for the complex-valued R -transforms (for a discussion of the nat-ural domain of K X , see [9]), and that these R -transforms satisfy R X + X ,n ( v ) = R X ,n ( v )+ R X ,n ( v ), n ∈ N , on a small neighbourhood of zero in M n ( B ), for X , X free with respect to E (see [32]). As for the functions ω above, in the following wewill suppress the index n from the notations of G and R whenever the space onwhich they are defined is clear from the context.An argument similar to the one used to prove (1) shows that the B -valuedsubordination functions satisfy precisely the same equation (1), but with variables v ∈ M n ( B ) , ℑ v > z ∈ C + (see [9] for details).An operator-valued version of the analytic transforms of bi-freeness hasbeen elaborated by one of us in [24]. We present a version of Equation (4) foroperator-valued transforms. We consider a C ∗ - B - B -noncommutative probabilityspace; the case when B is finite dimensional is of a special interest to us (see[24, Definitions 2.5 and 5.1] for details). An important difference from the caseof C -valued case comes from the fact that a noncommutative algebra may re-ceive a natural “opposite” structure. Thus, for instance, if analytic transformsof left random variables in a B - B -noncommutative probability space coincide withVoiculescu’s analytic transforms introduced above, analytic transforms of the rightrandom variables, while defined the same way (and thus having the same analyticproperties), are viewed as being defined on (open subsets of) B op , the algebra hav-ing the same underlying set and vector space structure as B , but with multiplicationdefined by b · op b ′ = b ′ b . Consequently, a C ∗ -, or a W ∗ - B - B noncommutative prob-ability space requires, beyond the data ( M, E, B ) described above, a way to view B and B op simultaneously as subalgebras of M , that is, a linear homomorphism ε : B ⊗ B op → M . This homomorphism satisfies certain conditions for which werefer to [24, Definition 2.5] (see also [15]). For our study, it is important to note that NALYTIC SUBORDINATION FOR BI-FREE CONVOLUTION 7 we must add an ℓ or an r to each analytic transform defined on B , correspondingto whether it is viewed as being defined on B or B op . In addition, the B - B -valuedequivalent of G η ( z, w ) (or, more precisely, of G η ( z − , w − ) /zw ), becomes a func-tion of three variables M ( X,Y ) ( b, c, d ), linear in c ∈ B , and for which b ∈ B is a“left” indeterminate and d ∈ B op is a “right” indeterminate (whether c is viewedas a left or right indeterminate is irrelevant). More specifically, for a bi-randomvariable ( X, Y ) we define the B -valued partial moment generating function M ( X,Y ) ( b, c, d ) := X n,m ≥ E ((L b X ) n (R d Y ) m R c ) , b, c, d ∈ B, k b k , k d k small . The moment generating functions of the left and right variables are M ℓX ( b ) = P n ≥ E ((L b X ) n ) and M rX ( d ) = P n ≥ E ((R d X ) n ), respectively. For the purposesof this paper, the reader is invited to see L b and R d just as special ways of viewingthe scalar algebras B and B op embedded in the noncommutative algebra M . Thus,Voiculescu’s subordination relations from above are re-written in terms of the twomoment generating functions as G X + X ( b − ) = M ℓX + X ( b ) b = M ℓX ( ω ( b − ) − ) ω ( b − ) − ,G Y + Y ( d − ) = dM rY + Y ( d ) = ω ( d − ) − M rY ( ω ( d − ) − ) , respectively (the convention from [24] is slightly different from ours: the G ( b ) from[24] is G ( b − ) here).For our purposes, we prefer to view M ( X,Y ) as an analytic function from B × B with values in L ( B ), the space of continuous linear operators from B toitself. Viewed as such, we have M ( X,Y ) (0 , c,
0) = c , i.e. M ( X,Y ) (0 , · ,
0) = Id B .Since the correspondence B × B ∋ ( b, d ) M ( X,Y ) ( b, · , d ) ∈ L ( B ) is analytic,we conclude that on a small enough norm-neighbourhood of (0 , M ( X,Y ) ( b, · , d ) ∈ L ( B ) is invertible as a linear map from B to B . We defineΨ ( X,Y ) ( b, · , d ) = b − M ( X,Y ) ( b, · , d ) h− i d − ∈ L ( B ), where M ( X,Y ) ( b, · , d ) h− i is theinverse of M ( X,Y ) ( b, · , d ) in L ( B ).With these notations, the partial R -transform of ( X, Y ) defined in [24,Section 5] is the analytic map B × B ∋ ( b, d ) R ( X,Y ) ( b, · , d ) ∈ L ( B ) uniquelydetermined on a neighbourhood of (0 ,
0) by the initial condition R ( X,Y ) (0 , · ,
0) = 0(the zero element in L ( B )), and the functional equation R ( X,Y ) ( M ℓX ( b ) b, c, dM rY ( d )) = M ℓX ( b ) c + cM rY ( d ) − M ℓX ( b ) b Ψ ( X,Y ) ( b, c, d ) dM rY ( d ) − c. If ( X , Y ) and ( X , Y ) are bi-free over B with respect to E , then the partial R -transform satisfies R ( X + X ,Y + Y ) ( b, c, d ) = R ( X ,Y ) ( b, c, d ) + R ( X ,Y ) ( b, c, d ) . Using the functional equation defining R ( X,Y ) , we obtain M ℓX + X ( b ) c + cM rY + Y ( d ) − M ℓX + X ( b ) b Ψ ( X + X ,Y + Y ) ( b, c, d ) dM rY + Y ( d ) − c = R ( X + X ,Y + Y ) ( M ℓX + X ( b ) b, c, dM rY + Y ( d ))= R ( X ,Y ) ( M ℓX + X ( b ) b, c, dM rY + Y ( d )) + R ( X ,Y ) ( M ℓX + X ( b ) b, c, dM rY + Y ( d )) . S.T. BELINSCHI, H. BERCOVICI, Y. GU, AND P. SKOUFRANIS
The subordination relation provides R ( X j ,Y j ) ( M ℓX + X ( b ) b, c, dM rY + Y ( d ))= R ( X j ,Y j ) ( M ℓX j ( ω X j ( b − ) − ) ω X j ( b − ) − , c, ω Y j ( d − ) − M rY j ( ω Y j ( d − ) − ))= M ℓX j ( ω X j ( b − ) − ) c + cM rY j ( ω Y j ( d − ) − ) − c − M ℓX j ( ω X j ( b − ) − ) ω X j ( b − ) − × Ψ ( X j ,Y j ) (cid:0) ω X j ( b − ) − , c, ω Y j ( d − ) − (cid:1) ω Y j ( d − ) − M rY j ( ω Y j ( d − ) − ) . This provides the following operator-valued analogue of relation (4): M ℓX + X ( b ) c + cM rY + Y ( d ) − M ℓX + X ( b ) b Ψ ( X + X ,Y + Y ) ( b, c, d ) dM rY + Y ( d ) − c = M ℓX ( ω X ( b − ) − ) c + cM rY ( ω Y ( d − ) − ) − c − M ℓX ( ω X ( b − ) − ) ω X ( b − ) − × Ψ ( X ,Y ) (cid:0) ω X ( b − ) − , c, ω Y ( d − ) − (cid:1) ω Y ( d − ) − M rY ( ω Y ( d − ) − )+ M ℓX ( ω X ( b − ) − ) c + cM rY ( ω Y ( d − ) − ) − c − M ℓX ( ω X ( b − ) − ) ω X ( b − ) − × Ψ ( X ,Y ) (cid:0) ω X ( b − ) − , c, ω Y ( d − ) − (cid:1) ω Y ( d − ) − M rY ( ω Y ( d − ) − ) . (6)Furthermore, recalling that G X ( b ) = M ℓX ( b − ) b − , the B -valued version of (1)written under the form G X + X ( b − ) b − = G X ( ω X ( b − )) ω X ( b − ) + G X ( ω X ( b − )) ω X ( b − ) − M ℓX + X ( b ) b Ψ ( X + X ,Y + Y ) ( b, c, d ) dM rY + Y ( d ) + c = M ℓX ( ω X ( b − ) − ) ω X ( b − ) − × Ψ ( X ,Y ) (cid:0) ω X ( b − ) − , c, ω Y ( d − ) − (cid:1) ω Y ( d − ) − M rY ( ω Y ( d − ) − )+ M ℓX ( ω X ( b − ) − ) ω X ( b − ) − × Ψ ( X ,Y ) (cid:0) ω X ( b − ) − , c, ω Y ( d − ) − (cid:1) ω Y ( d − ) − M rY ( ω Y ( d − ) − ) . (7)Since we usually prefer to deal with G rather than M , we perform in the above thechanges of variable b b − and d d − in order to obtain G X + X ( b )Ψ ( X + X ,Y + Y ) ( b − , c, d − ) G Y + Y ( d ) + c = G X ( ω X ( b ))Ψ ( X ,Y ) (cid:0) ω X ( b ) − , c, ω Y ( d ) − (cid:1) G Y ( ω Y ( d ))+ G X ( ω X ( b ))Ψ ( X ,Y ) (cid:0) ω X ( b ) − , c, ω Y ( d ) − (cid:1) G Y ( ω Y ( d )) , (8)a precise analogue of Equation (4) (the parallel is made more obvious by the factthat Ψ( b − , · , d − ) = G ( b, · , d ) h− i , as G ( b − , b − cd − , d − ) := M ( b, c, d )). Thisrelation holds for all b, d for which the functions Ψ are defined as analytic extensionsfrom the set of elements b, d satisfying the requirement that k b − k and k d − k aresufficiently small. This is an open set, so that the above relation does determineΨ ( X + X ,Y + Y ) uniquely from the knowledge of Ψ ( X j ,Y j ) , j = 1 , , and of the freeadditive convolution of operator-valued distributions. We record next a slightlydifferent version of (8), which resembles the scalar reduced partial R -transform. Its NALYTIC SUBORDINATION FOR BI-FREE CONVOLUTION 9 validity follows trivially from (8):Ψ ( X + X ,Y + Y ) ( K X + X ( b ) − , c, K Y + Y ( d ) − ) − b − cd − =Ψ ( X ,Y ) (cid:0) K X ( b ) − , c, K Y ( d ) − (cid:1) − b − cd − + Ψ ( X ,Y ) (cid:0) K X ( b ) − , c, K Y ( d ) − (cid:1) − b − cd − , (9) The transforms G, M,
Ψ and R do not fully characterize the joint dis-tribution of ( X, Y ), but only the “band moments.” However, this will suffice forour main purpose of studying general distributions of bi-partite bi-free randomvariables (we remind the reader that by “bi-partite” we mean that all left randomvariables commute with all right random variables). In this context, our interestis mainly in bi-freeness with amalgamation over finite dimensional algebras. Theconstruction providing an M n ( C )- M n ( C )-noncommutative probability space from aclassical noncommutative probability space ( A , ϕ ) is the following (see [23, Section6] or [24, Section 4]). First, one defines the left and right actionsL b ( T ) = " n X k =1 b ik T kj ni,j =1 , R d ( T ) = " n X k =1 d kj T ik ni,j =1 , for all b, d ∈ M n ( C ) , T ∈ M n ( A ) . Both L b and R d are indeed bounded linear mapson M n ( A ). The correspondences b L b and d R d are algebra homomorphismsfrom M n ( C ) and M n ( C ) op , respectively, into L ( M n ( A )) . We define the right algebra L ( M n ( A )) r as the set { Z ∈ L ( M n ( A )) : Z L b = L b Z for all b ∈ M n ( C ) } , and L ( M n ( A )) ℓ the same way, but with L replaced by R. One embeds M n ( A ) in L ( M n ( A )) r via Z R( Z ) , R( Z )( T ) = " n X k =1 Z kj T ik ni,j =1 for all T ∈ M n ( A ) , and in L ( M n ( A )) ℓ via Z L( Z ) , L( Z )( T ) = " n X k =1 Z ik T kj ni,j =1 for all T ∈ M n ( A ) . The map Z L( Z ) is an injective algebra ∗ -homomorphism from M n ( A ) into L ( M n ( A )), and Z R( Z ) from M n ( A op ) op into L ( M n ( A )). The conditional ex-pectation E n : L ( M n ( A )) → M n ( C ) defined by E n [ W ] = [ ϕ ( W ( I n ) ij )] ni,j =1 satisfies E n [L( Z )] = [ ϕ ( Z ij )] ni,j =1 and E n [R( Z )] = [ ϕ ( Z ij )] ni,j =1 . That is, the distributionof Z ∈ M n ( A ) with respect to ϕ ⊗ Id M n ( C ) is the same as the distribution ofL( Z ) (respectively, R( Z )) with respect to E n . We note that R d R( Y ) = R( Y d ),L b L( X ) = L( bX ), and R I n = L I n = Id M n ( A ) . Then the map M ( X,Y ) defined above is written as M ( X,Y ) ( b, c, d ) = E n (cid:2) (1 − L b L( X )) − (1 − R d R( Y )) − R c (cid:3) = ( ϕ ⊗ Id M n ( C ) ) (cid:2) (1 − L( bX )) − (1 − R( Y d )) − R( c )( I n ) (cid:3) = ( ϕ ⊗ Id M n ( C ) ) (cid:2) L((1 − bX ) − )R((1 − Y d ) − )R( c )( I n ) (cid:3) = ( ϕ ⊗ Id M n ( C ) ) (cid:2) L((1 − bX ) − )R( c (1 − Y d ) − )( I n ) (cid:3) = ( ϕ ⊗ Id M n ( C ) ) (cid:2) L((1 − bX ) − )( c (1 − Y d ) − ) (cid:3) = ( ϕ ⊗ Id M n ( C ) ) (cid:2) (1 − bX ) − c (1 − Y d ) − (cid:3) . Thus, vitally for us, the band-moment generating function for a pair of faces(L( X ) , R( Y )) coincides with the band moment generating function of ( X, Y ). Thisallows us to extend the map M ( X,Y ) ( b, · , d ), as its scalar-valued analogue, to { b ∈ M n ( C ) : ± ℑ b > } × { d ∈ M n ( C ) op : ± ℑ d > } .To conclude this section, we state a particular case of [23, Theorem 6.3.1](alternatively see [24, Theorem 4.1]): Lemma 2.1.
Let ( A , ϕ ) be a noncommutative probability space and n ∈ N . As-sume that ( a , b ) and ( a , b ) are bi-free with respect to ϕ . Then ( L ( X ) , R ( Y )) and ( L ( X ) , R ( Y )) are bi-free with amalgamation over M n ( C ) whenever X j ∈ M n ( C [ a j ]) , Y j ∈ M n ( C [ b j ]) , j = 1 , . Bi-free analytic subordination
In this section we establish our main subordination result for scalar-valued bi-free random variables. Fix a C ∗ -noncommutative probability space ( A , ϕ )and two pairs ( a , b ) , ( a , b ) ∈ A that are bi-free with respect to ϕ and bi-partite.Thus, a j (respectively, b j ) is the left (respectively, right) variable in the pair ( a j , b j ),and a j b j = b j a j , j = 1 ,
2. (Many of the computations below are valid under weakerhypotheses. In many circumstances, our computations with analytic transformsalso hold for operator-valued bi-free pairs of random variables. We indicate belowthose cases in which this extension is valid.)Define X j ∈ M ( A ) by X j = (cid:20) a j b j (cid:21) , j = 1 ,
2. In the context wewill consider below, it is important that the left face is in the upper left, andthe right face in the lower right, corner. We define the conditional expectation M ( ϕ ) = ϕ ⊗ Id M ( C ) from M ( A ) onto M ( C ). With respect to this expectation,we consider G X j ( v ) = E (cid:2) ( v − X j ) − (cid:3) for v ∈ M ( C ) such that ℑ v > k v − k < k X j k − . As usual, the R -transform is defined via the functional equation G X j ( v − + R X j ( v )) = v . As before, denote K X j ( v ) = v − + R X j ( v ). We restrict all of thesefunctions to upper triangular matrices in M ( C ). Direct computation using theSchur complement yields G X j (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = M ( ϕ ) (cid:20) ( z − a j ) − − ( z − a j ) − ζ ( w − b j ) − w − b j ) − (cid:21) = (cid:20) ϕ (cid:0) ( z − a j ) − (cid:1) − ϕ (cid:0) ( z − a j ) − ζ ( w − b j ) − (cid:1) ϕ (cid:0) ( w − b j ) − (cid:1) (cid:21) = (cid:20) G µ j ( z ) − ζG η j ( z, w )0 G ν j ( w ) (cid:21) . NALYTIC SUBORDINATION FOR BI-FREE CONVOLUTION 11 (First two equalities hold for operator-valued random variables ( a j , b j ), with themap ϕ (cid:0) ( z − a j ) − ζ ( w − b j ) − (cid:1) replaced by ζ M ( a j ,b j ) ( z − , z − ζw − , w − ) = G ( a j ,b j ) ( z, ζ, w ) - see Section 2.) The compositional inverse of an analytic mapthat maps upper triangular matrices onto upper triangular matrices preserves up-per triangular matrices. That is, K X j (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = (cid:20) K a j ( z ) − k ( z, ζ, w )0 K b j ( w ) (cid:21) = (cid:20) K µ j ( z ) − k ( z, ζ, w )0 K ν j ( w ) (cid:21) for some function k ( z, ζ, w ). Then, by the above formula, (cid:20) z ζ w (cid:21) = G X j (cid:18)(cid:20) K a j ( z ) − k ( z, ζ, w )0 K b j ( w ) (cid:21)(cid:19) = (cid:20) G a j ( K a j ( z )) ϕ (( K a j ( z ) − a j ) − k ( z, ζ, w )( K b j ( w ) − b j ) − ))0 G b j ( K b j ( w )) (cid:21) = (cid:20) G a j ( K a j ( z )) G ( a j ,b j ) ( K a j ( z ) , k ( z, ζ, w ) , K b j ( w )))0 G b j ( K b j ( w )) (cid:21) = (cid:20) G µ j ( K µ j ( z )) k ( z, ζ, w ) G η j ( K µ j ( z ) , K ν j ( w ))0 G ν j ( K ν j ( w )) (cid:21) . (Again, the first three equalities hold for operator-valued random variables.) The(1 ,
2) entry shows that k ( z, ζ, w ) is linear in ζ , and has as inverse the linear map ξ G ( a j ,b j ) ( K a j ( z ) , ξ, K b j ( w )). Thus, k ( z, · , w ) = G ( a j ,b j ) ( K a j ( z ) , · , K b j ( w )) h− i = K µ j ( z ) M ( a j ,b j ) ( K µ j ( z ) − , · , K ν j ( w ) − ) h− i K ν j ( w )= Ψ ( a j ,b j ) ( K µ j ( z ) − , · , K ν j ( w ) − ) , and hence k ( z, ζ, w ) = Ψ ( a j ,b j ) ( K µ j ( z ) − , ζ, K ν j ( w ) − ) = ζG η j ( K µ j ( z ) , K ν j ( w )) . (The last quantity above only makes sense for scalar-valued variables.) In particu-lar, R X j (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = (cid:20) K µ j ( z ) − z − − k ( z, ζ, w ) + z − ζw − K ν j ( w ) − w − (cid:21) = (cid:20) R µ j ( z ) z − ζw − − Ψ ( a j ,b j ) ( K µ j ( z ) − , ζ, K ν j ( w ) − )0 R ν j ( w ) (cid:21) = " R µ j ( z ) ζ (cid:16) zw − G ηj ( K µj ( z ) ,K νj ( w )) (cid:17) R ν j ( w ) (10)(with the first two equalities making sense for operator-valued variables). Observethat in all the computations this far we have not used the fact that a j b j = b j a j . If weagree to consider only the band moments of type ϕ ( LR ) ( L and R being monomialsin the left, respectively right, variable), all the above computations remain valid,with the possible difference that ϕ (cid:0) ( z − a j ) − ( w − b j ) − (cid:1) might not be the Cauchytransform of a probability measure η j on R . We record our conclusion in thefollowing: Lemma 3.1.
Let ( A , ϕ ) be a C ∗ -noncommutative probability space and ( a, b ) ∈ A be a two-faced pair of noncommutative random variables. Define the M ( C ) -valuedrandom variable X = (cid:20) a b (cid:21) ∈ M ( A ) . Then R X (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = (cid:20) R a ( z ) ζzw ( R ( a,b ) ( z, w ) − zR a ( z ) − wR b ( w ))0 R b ( w ) (cid:21) . In particular, if ( a , b ) , ( a , b ) ∈ A are bi-free with respect to ϕ , then, with thenotations from the beginning of this section, (11) R X + X (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = R X (cid:18)(cid:20) z ζ w (cid:21)(cid:19) + R X (cid:18)(cid:20) z ζ w (cid:21)(cid:19) for all z, w ∈ C of sufficiently small absolute value and all ζ ∈ C . (Lemma 3.1 holds for operator-valued random variables in a B - B - C ∗ -noncommutative probability space ( M, E, B ). More precisely, if ( a, b ) ∈ M is atwo-faced pair of B -valued random variables, then, defining X = (cid:20) a b (cid:21) ∈ M ( M ),we have R X (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = (cid:20) R a ( z ) z − ζw − − Ψ ( a,b ) ( K a ( z ) − , ζ, K b ( w ) − )0 R b ( w ) (cid:21) , where Ψ ( a,b ) has been defined in Section 2, and Equation (11) also holds in thiscontext.) We would like to emphasize again that R X above denotes the R -transformof the M ( C ) (or M ( B ))-valued random variable X , as introduced in [28]. Thus,Equation (11) implies that X and X “mimic” freeness in terms of the relationsbetween their analytic transforms when restricted to upper triangular matrices.More specifically: Remark 3.2.
Let ( A , ϕ ) be a C ∗ -noncommutative probability space. Assumethat ( a , b ) and ( a , b ) are self-adjoint and bi-free with respect to ϕ . Define X j = (cid:20) a j b j (cid:21) , j = 1 , M ( A ) , ϕ ⊗ Id M ( C ) , M ( C )). Considertwo random variables Y , Y which are free with respect to M ( ϕ ) := ϕ ⊗ Id M ( C ) ,and such that the ∗ -distribution of X j and Y j with respect to M ( ϕ ) coincide for j = 1 ,
2. Then the restrictions of the Cauchy and R -transforms of X , X , X + X and Y , Y , Y + Y , respectively, to the upper triangular 2 × × ω Y j is the M ( C )-valued subordination function satisfying G Y j ◦ ω Y j = G Y + Y , then( G X j ◦ ω Y j ) (cid:18)(cid:20) z ζ w (cid:21)(cid:19) − = G X + X (cid:18)(cid:20) z ζ w (cid:21)(cid:19) − = ( ω Y + ω Y ) (cid:18)(cid:20) z ζ w (cid:21)(cid:19) − (cid:20) z ζ w (cid:21) , for all z, w ∈ C + , ζ ∈ C , j = 1 ,
2. Thus, ω Y j = G h− i Y j ◦ G Y + Y = G h− i X j ◦ G X + X when restricted to matrices (cid:20) z ζ w (cid:21) with z, w having inverses of small norm. (Allrelations above hold as well for operator-valued variables.) NALYTIC SUBORDINATION FOR BI-FREE CONVOLUTION 13
In light of Lemma 3.1 and Equation (1), the proof of the above remark isobvious as soon as one accounts for the fact that the M ( C )-valued distribution ofthe variable X determines the M ( C )-valued Cauchy transform of X . This remarkallows us to recover Equation (4). Indeed, as it follows from [9, Theorem 2.7] thatthe functions ω Y j map upper triangular matrices to upper triangular matrices, wehave that ω Y j (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = (cid:20) f f f (cid:21) . Since X j and Y j have the same distributionand X j is diagonal and self-adjoint, so must be Y j , and its diagonal entries musthave the same (joint) distribution as ( a j , b j ). Thus, we obtain that G Y + Y (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = G X j (cid:18)(cid:20) f f f (cid:21)(cid:19) = (cid:20) ϕ (cid:0) ( f − a j ) − (cid:1) − ϕ (cid:0) ( f − a j ) − f ( f − b j ) − (cid:1) ϕ (cid:0) ( f − b j ) − (cid:1) (cid:21) , which guarantees that f = ω a j ( z ) , f = ω b j ( w ). Using again the previous remark,we obtain for f = f ( z, ζ, w ) and z, w having small inverse f = Ψ ( a j ,b j ) (cid:0) ω a ( z ) − , ϕ (cid:0) ( z − a − a ) − ζ ( w − b − b ) − (cid:1) , ω b ( w ) − (cid:1) = ζϕ (cid:0) ( z − a − a ) − ( w − b − b ) − (cid:1) ϕ (cid:0) ( ω a j ( z ) − a j ) − ( ω b j ( w ) − b j ) − (cid:1) . (Again, the first equality is true for operator-valued random variables.) Recallingthat ω Y j is defined and analytic on all matrices (cid:20) z ζ w (cid:21) with positive imaginarypart, we can write f ( z, ζ, w ) G η j ( ω a j ( z ) , ω b j ( w )) = ζG η ⊞⊞ η ( z, w ). By analyticcontinuation, this relation holds for all z, w ∈ C + and ζ ∈ C . In particular, wehave G η ⊞⊞ η ( z, w ) = 0 whenever G η j ( ω a j ( z ) , ω b j ( w )) = 0, so that each connectedcomponent in C + × C + of the zero set of G η j ( ω a j ( z ) , ω b j ( w )) equals a connectedcomponent of the zero set of G η ⊞⊞ η . Replacing this in the upper triangular matrix-valued analogue (provided above) of (1) provides a slightly modified version ofEquation (4):(12) G η ⊞⊞ η ( z, w ) G η ( ω a ( z ) , ω b ( w )) + G η ⊞⊞ η ( z, w ) G η ( ω a ( z ) , ω b ( w )) = G η ⊞⊞ η ( z, w ) G µ ⊞ µ ( z ) G ν ⊞ ν ( w ) + 1 , for all z, w ∈ C + , guaranteeing analytic extension through the zero sets (in C + × C + )of G η j ( ω a j ( z ) , ω b j ( w )) , j = 1 ,
2. (Observe that, while in order to obtain a relationbetween Cauchy transforms of measures in the plane, we need to assume that a j and b j commute, the formal calculations above hold even in the absence ofthis hypothesis.) Thus, relation (12) holds on the connected set ( C + × C + ) ∪ ( C − × C − ) ∪ { ( z, w ) ∈ C : | z | > k a + a k and | w | > k b + b k} . There is oneobstacle to this relation extending to all of C + × C − , namely the possibility that G ηj ( ω aj ( z ) ,ω bj ( w )) G η ⊞⊞ η ( z ,w ) = 0 for some z ∈ C + , w ∈ C − , j ∈ { , } . In this case, we haveversion (5) of equations (4) and (12) for computing G η ⊞⊞ η in terms of G η j , ω a j ,and ω b j , j = 1 , Example 3.3.
Relation (11) does not extend to arbitrary 2 × (cid:20) z ζζ ′ w (cid:21) .In other words, ( a , b ) and ( a , b ) being bi-free with respect to ϕ does not neces-sarily imply that X and X are free with amalgamation over M ( C ), as it can be seen by computing some moments. Consider M ( ϕ ) (cid:18)(cid:20) a b (cid:21) (cid:20) a b (cid:21) (cid:20) (cid:21) (cid:20) a b (cid:21) (cid:20) a b (cid:21)(cid:19) = (cid:20) ϕ ( a a b b )0 0 (cid:21) . If ( a , b ) and ( a , b ) are bi-free with respect to ϕ , then(13) ϕ ( a a b b ) = ϕ ( a b ) ϕ ( a ) ϕ ( b ) + ϕ ( a b ) ϕ ( a ) ϕ ( b ) − ϕ ( a ) ϕ ( a ) ϕ ( b ) ϕ ( b ) . On the other hand, if (cid:20) a b (cid:21) and (cid:20) a b (cid:21) are free with amalgamation over M ( C ), then we would have M ( ϕ ) (cid:18)(cid:20) a b (cid:21) (cid:20) a b (cid:21) (cid:20) (cid:21) (cid:20) a b (cid:21) (cid:20) a b (cid:21)(cid:19) , = ϕ ( a b ) ϕ ( a b ) , which is generally different from the right hand-side of (13). Remark 3.4.
As mentioned in the introduction, the idea of constructing an M ( C )-valued random variable X j from the two-faced pair ( a j , b j ) and considering theoperator-valued transforms of X j was motivated by the study of bi-Boolean inde-pendence [20]. We would like to elaborate a bit on this point. Given a probabilitymeasure µ on R , define the analytic function(14) h µ : C + → C + ∪ R , h µ ( z ) = 1 G µ ( z ) − z. It was shown by Speicher and Woroudi in [25] that h µ linearizes the Boolean additiveconvolution ⊎ in the sense that h µ ⊎ µ ( z ) = h µ ( z ) + h µ ( z ) , z ∈ C + . Moreover, itwas shown in [25, Theorem 3.6] that every probability measure on R is ⊎ -infinitelydivisible. This result was later extended by Popa and Vinnikov to operator-valueddistributions in [22]. Given X = X ∗ in a W ∗ -noncommutative probability space( M, E, B ), define h X by(15) h X ( b ) = G X ( b ) − − b b ∈ B, ℑ b > k b − k < k X k − . If X and X are Boolean independent with respect to E , then h X + X ( b ) = h X ( b ) + h X ( b ). More importantly, it follows from [22, Theorem 3.5] that theresult on ⊎ -infinite divisibility also holds in the B -valued setting (i.e. for every n ∈ N , there exists X n = X ∗ n such that nh X n ( b ) = h X ( b )).On the other hand, given a probability measure η on R with marginals µ and ν , the function e E η was introduced in [20, Section 4] by(16) e E η ( z, w ) = G η ( z, w ) G µ ( z ) G ν ( w ) − , ( z, w ) ∈ ( C \ R ) , to study bi-Boolean independence. The function e E η is the (reduced part of) thepartial bi-Boolean self-energy which linearizes the bi-Boolean additive convolution ⊎⊎ in the sense that e E η ( z, w ) = e E η ( z, w ) + e E η ( z, w ) if η = η ⊎ ⊎ η . Due to thesimple form of e E η , it was natural to hypothesize that every probability measureon R is ⊎⊎ -infinitely divisible. When trying to prove the conjecture, we observedthat if ( a, b ) has joint distribution η and if we consider the M ( C )-valued random NALYTIC SUBORDINATION FOR BI-FREE CONVOLUTION 15 variable X = (cid:20) a b (cid:21) , then h X (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = " h µ ( z ) ζ (cid:16) G η ( z,w ) G µ ( z ) G ν ( w ) − (cid:17) h ν ( w ) , so that the three components of the partial bi-Boolean self-energy E η ( z, w ) = e E η ( z, w ) − z h µ ( z ) − w h ν ( w ) appear as the three non-zero entries of h X evaluated atthe upper triangular matrix (cid:20) z ζ w (cid:21) . This, in connection to the above-mentionedresult of Popa and Vinnikov, suggested that the conjecture might be true. However,that is not the case, as shown by the counterexample in [20, Example 5.13].The following consequence of Remark 3.2 establishes some analytic prop-erties of the Cauchy transform of the bi-free additive convolution of two probabilitymeasures in R that will help us prove a converse of the characterization of bi-freeextreme values of Voiculescu. Proposition 3.5.
Let ( a , b ) and ( a , b ) be bi-free pairs of random variables.With the notations from Section , we have ℑ G a j ( z ) ℑ G b j ( w ) ≥ ℑ z ℑ w | G ( a j ,b j ) ( z, w ) | , and ℑ ω a j ( z ) ℑ ω b j ( w ) | G η j ( ω a j ( z ) , ω b j ( w )) | ≥ ℑ z ℑ w | G η ⊞⊞ η ( z, w ) | , z, w ∈ C + . Proof.
The proof is based on a simple trick, which is a particular case of [6,Proposition 3.1]. Observe that ℑ (cid:20) z ζ w (cid:21) > ℑ z > , ℑ w > ℑ z ℑ w > | ζ | . As G X j maps elements from M ( C ) with positive imagi-nary part into elements from M ( C ) with negative imaginary part, it follows that4 ℑ G a j ( z ) ℑ G b j ( w ) > | ζ | | G ( a j ,b j ) ( z, w ) | whenever ℑ (cid:20) z ζ w (cid:21) >
0. Letting | ζ | tendto 2 √ℑ z ℑ w from below yields ℑ G a j ( z ) ℑ G b j ( w ) ≥ ℑ z ℑ w | G ( a j ,b j ) ( z, w ) | , z, w ∈ C + . The second relation follows from the fact that ℑ ω X j (cid:18)(cid:20) z ζ w (cid:21)(cid:19) > ℑ (cid:20) z ζ w (cid:21) >
0, and the fact that ω X j (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = " ω a j ( z ) ζG η ⊞⊞ η ( z,w ) G ηj ( ω aj ( z ) ,ω bj ( w )) ω b j ( w ) , as shown in the computations following Remark 3.2. Indeed, it is known from [29]that the maps ω introduced in (1) send the set of elements with positive imaginarypart into itself. We obtain again that ℑ z > , ℑ w > ℑ z ℑ w > | ζ | implytogether that 4 ℑ ω a j ( z ) ℑ ω b j ( w ) > (cid:12)(cid:12)(cid:12) ζG η ⊞⊞ η ( z,w ) G ηj ( ω aj ( z ) ,ω bj ( w )) (cid:12)(cid:12)(cid:12) . Letting | ζ | tend to 2 √ℑ z ℑ w from below allows us to conclude. (cid:3) We emphasize that this proposition allows us to make sense of Equa-tion (4) as an equality of meromorphic functions on C + × C + , and, given theproperties of the one- and two-variables Cauchy transform, also on C − × C − .For the purposes of this article, we call a function H ( z, w ) meromorphic if forany ( z , w ) in its domain we either have that H is holomorphic around ( z , w ),or both z H ( z, w ) and w H ( z , w ) are one-variable meromorphic func-tions on some neighborhood of z , and w , respectively. We have shown that theembodiment of Equation (4) in the shape of Equation (12) makes sense as anequality of analytic functions; in particular, the zero set of G η ⊞⊞ η includes thezero sets of both G η j ( ω a j ( z ) , ω b j ( w )) , j = 1 ,
2. However, thanks to the previousproposition and Equation (12), we can conclude more: if G η ⊞⊞ η ( z , w ) = 0,then G η ( ω a ( z ) , ω b ( w )) G η ( ω a ( z ) , ω b ( w )) = 0. Indeed, let us assume that G η ( ω a ( z ) , ω b ( w )) = 0. By approaching ( z , w ) from the complement of thezero sets of the three functions involved, we obtainlim ( z,w ) → ( z ,w ) G η ⊞⊞ η ( z, w ) G η ( ω a ( z ) , ω b ( w )) = 1 , so that G η ( ω a ( z ) , ω b ( w )) = 0. As shown in Section 2.2, z G η ⊞⊞ η ( z, w ), z G η j ( ω a j ( z ) , ω b j ( w )) are not identically zero, so that z is a zero of finiteorder for all these functions. The same statement holds for w . If both z G η j ( ω a j ( z ) , ω b j ( w )), j = 1 ,
2, have a zero of precisely the same order at z ,then so does z G η ⊞⊞ η ( z, w ), and relation (4) makes sense for those mero-morphic functions when written as Laurent series around z = z . If one (say z G η ( ω a ( z ) , ω b ( w )) has a zero of higher order than the other, then z G η ⊞⊞ η ( z, w ) has a zero of the same order as z G η ( ω a ( z ) , ω b ( w )), so again(4) makes sense.The main result of this section follows easily from the above considera-tions, Remark 3.2 and Equation (8). Theorem 3.6.
Let ( M, E, B ) be a C ∗ - B - B -noncommutative probability space, fora C ∗ -algebra B . Assume that ( a , b ) and ( a , b ) are self-adjoint random variablesin M which are bi-free over B with respect to E . Denote X j = (cid:20) a j b j (cid:21) , j = 1 , .Then M ( E ) "(cid:18)(cid:20) z ζ w (cid:21) − X − X (cid:19) − = M ( E ) "(cid:18)(cid:20) ω a ( z ) Π ( z, ζ, w )0 ω b ( w ) (cid:21) − X (cid:19) − = (cid:20) G a ( ω a ( z )) G ( a ,b ) ( ω a ( z ) , Π ( z, ζ, w ) , ω b ( w ))0 G b ( ω b ( w )) (cid:21) , where Π ( z, · , w ) is the composition of the following linear maps depending analyt-ically on z, w : Π ( z, ζ, w ) = (cid:2) G ( a ,b ) ( ω a ( z ) , · , ω b ( w )) + G ( a ,b ) ( ω a ( z ) , · , ω b ( w )) − G ( a ,b ) ( ω a ( z ) , G a ( ω a ( z )) − G ( a ,b ) ( ω a ( z ) , · , ω b ( w )) G b ( ω b ( w )) − , ω b ( w )) (cid:3) h− i ◦ G ( a ,b ) ( ω a ( z ) , ζ, ω b ( w )) . If B = C , then E is a state and the following simpler expression for Π holds Π ( z, ζ, w ) = ζG ( a ,b ( ω a ( z ) ,ω b ( w ))1 G ( a ,b ( ω a ( z ) ,ω b ( w )) + G ( a ,b ( ω a ( z ) ,ω b ( w )) − G a ( ω a ( z )) G b ( ω b ( w ))NALYTIC SUBORDINATION FOR BI-FREE CONVOLUTION 17 for all z, w ∈ C + , ζ ∈ C . Here ω a and ω b are the subordination functions intro-duced in Section 2, Equation (1) , and M ( E ) is the conditional expectation onto M ( B ) given by E ⊗ Id M ( B ) .Proof. Given Equation (4), we only need to argue that the expression for Π makessense. This follows directly from the considerations before the statement of ourTheorem. (cid:3) Note that while the existence and analyticity of the operator-valued ver-sion of Π is shown in Remark 3.2 and the considerations following it, the ingredientsof its expression as provided by the above theorem are guaranteed to be analyticonly for z − , w − small in norm.The following corollary is the converse of [35, Theorem 2.1]. Corollary 3.7.
Assume that ( a , b ) and ( a , b ) are bi-free bi-partite self-adjointtwo-faced pairs. We denote by η j the distribution of ( a j , b j ) and by µ j and ν j itsfirst and second marginal, respectively. Assume that there exists a point ( ξ, ζ ) ∈ R such that ( η ⊞⊞ η )( { ( ξ, ζ ) } ) > . Then there exist ( ξ j , ζ j ) ∈ R , j = 1 , , such that ( ξ + ξ , ζ + ζ ) = ( ξ, ζ ) and η j ( { ( ξ j , ζ j ) } ) > . Moreover, µ ⊞ µ )( { ξ } )( ν ⊞ ν )( { ζ } )( η ⊞⊞ η )( { ( ξ, ζ ) } ) = µ ( { ξ } ) ν ( { ζ } ) η ( { ( ξ , ζ ) } ) + µ ( { ξ } ) ν ( { ζ } ) η ( { ( ξ , ζ ) } ) . Proof.
Observe first that( µ ⊞ µ )( { ξ } ) = Z R { ξ }× R ( s, t ) d( η ⊞⊞ η )( s, t ) ≥ ( η ⊞⊞ η )( { ( ξ, ζ ) } ) , with a similar result for ν ⊞ ν . Thus, [11, Theorem 7.4] indicates that µ j and ν j all have atoms. More precise, there are ξ j , ζ j ∈ R , j = 1 , , such that(1) 1 < µ ( { ξ } ) + µ ( { ξ } ) = ( µ ⊞ µ )( { ξ } ) + 1;(2) 1 < ν ( { ζ } ) + ν ( { ζ } ) = ( ν ⊞ ν )( { ζ } ) + 1;(3) ξ + ξ = ξ ;(4) ζ + ζ = ζ .According to the same article [11], from this it follows that ω a j ( iy + ξ ), ω b j ( iy + ζ )tend nontangentially to ξ j and ζ j , respectively, as y ↓
0, andlim y ↓ ℑ ω a j ( ξ + iy ) y = µ j ( { ξ j } )( µ ⊞ µ )( { ξ } ) , lim y ↓ ℑ ω b j ( ζ + iy ) y = ν j ( { ζ j } )( ν ⊞ ν )( { ζ } ) . From the dominated convergence theorem, we know that for any finite measure η in the plane and any sequences { z n } n , { w n } n ⊂ C + which converge nontangentiallyto ξ and ζ , respectively, we havelim n →∞ ( z n − ξ )( w n − ζ ) G η ( z n , w n ) = η ( { ( ξ, ζ ) } ) . In particular, under the hypothesis of our corollary, lim y → ( iy ) G η ⊞⊞ η ( ξ + iy, ζ + iy ) = ( η ⊞⊞ η )( { ( ξ, ζ ) } ) >
0, and so, for y > G η ⊞⊞ η ( ξ + iy, ζ + iy ) = 0. Applying Proposition 3.5 with z = ξ + iy, w = ζ + iy and letting y ↓ (cid:18) ( µ ⊞ µ )( { ξ } )( ν ⊞ ν )( { ζ } ) µ j ( { ξ j } ) ν j ( { ζ j } ) (cid:19) η j ( { ( ξ j , ζ j ) } ) ≥ ( η ⊞⊞ η )( { ( ξ, ζ ) } ) > . We divide by y in Equation (4) to conclude that1( η ⊞⊞ η )( { ( ξ, ζ ) } ) + 1( µ ⊞ µ )( { ξ } )( ν ⊞ ν )( { ζ } )= lim y ↓ y G η ⊞⊞ η ( ξ + iy, ζ + iy ) + 1 yG µ ⊞ µ ( ξ + iy ) yG ν ⊞ ν ( ζ + iy )= X j =1 (cid:20) lim y ↓ ℑ ω a j ( ξ + iy ) ℑ ω b j ( ζ + iy ) y × lim y ↓ ℑ ω a j ( ξ + iy ) ℑ ω b j ( ζ + iy ) G η j ( ω a j ( ξ + iy ) , ω b j ( ζ + iy )) (cid:21) = 1( µ ⊞ µ )( { ξ } )( ν ⊞ ν )( { ζ ) } ) (cid:18) µ ( { ξ } ) ν ( { ζ } ) η ( { ( ξ , ζ ) } ) + µ ( { ξ } ) ν ( { ζ } ) η ( { ( ξ , ζ ) } ) (cid:19) . (cid:3) Remark 3.8.
As an immediate consequence, [11, Corollary 7.5] holds for proba-bility measures on R and bi-free convolution: that is, for any compactly supportedBorel probability measure η on R , η ⊞⊞ η has at most one atom, and if η is ⊞⊞ -infinitely divisible (see [18]), then it has at most one atom.Significantly stronger results are known for free additive convolution:no singular continuous part is present in the Lebesgue decomposition of the freeconvolution of two probability measures whose supports contain more than onepoint, and its absolutely continuous part has a density which is continuous whereverfinite (see [3]). For compactly supported measures, it has been shown in [5] thatthe Cauchy transform of the free convolution of two probability measures µ and ν isunbounded if and only if µ ( { α } ) + ν ( { β } ) ≥ α, β ∈ R . Thus, only atomsof µ and ν can make the density of µ ⊞ ν unbounded. Of course, a similar resultcannot be expected to hold for bi-free convolutions (indeed, there are examples ofcentral limits which are not absolutely continuous with respect to the Lebesguemeasure on R - see [33]). However, Corollary 3.7 can be significantly (and easily)improved in this direction, using the tools provided by Proposition 3.5. Corollary 3.9.
Assume that ( a , b ) and ( a , b ) are bi-free bi-partite self-adjointtwo-faced pairs, neither of a , b , a , b being a multiple of the identity. We denoteby η j the distribution of ( a j , b j ) and by µ j and ν j its first and second marginal,respectively, j = 1 , . Assume that there exist sequences { z n } n ∈ N , { w n } n ∈ N ⊂ C + such that (17) lim n →∞ p ℑ z n ℑ w n | G η ⊞⊞ η ( z n , w n ) | = + ∞ . Then at least one of the two marginals of η ⊞ ⊞ η has an unbounded Cauchytransform. In particular, there exists a pair α, β ∈ R such that either µ ( { α } ) + µ ( { β } ) ≥ , or ν ( { α } ) + ν ( { β } ) ≥ . Note that this result provides a large class of probability measures in R which cannot arise as non-trivial bi-free convolutions. Proof.
The proof is straightforward: recall from Proposition 3.5 that q ℑ G µ ⊞ µ ( z n ) ℑ G ν ⊞ ν ( w n ) ≥ p ℑ z n ℑ w n | G η ⊞⊞ η ( z n , w n ) | . NALYTIC SUBORDINATION FOR BI-FREE CONVOLUTION 19
Thus, at least one of { G µ ⊞ µ ( z n ) } n ∈ N , { G ν ⊞ ν ( w n ) } n ∈ N is unbounded. An appli-cation of [5, Theorem 7] concludes the proof. (cid:3) Corollary 3.10.
With the notations, and under the hypotheses, of the previouscorollary, assume that there exist u, x ∈ R and α > / such that lim ( y ,y ) → (0 , ( y y ) α | G η ⊞⊞ η ( x + iy , u + iy ) | = + ∞ . Then there exist pairs α , α , β , β ∈ R such that α + β = x, α + β = u , and µ ( { α } ) + µ ( { β } ) ≥ , ν ( { α } ) + ν ( { β } ) ≥ .Proof. As seen above, we have( y y ) α − (cid:0) ℑ G µ ⊞ µ ( x + iy ) ℑ G ν ⊞ ν ( u + iy ) (cid:1) ≥ ( y y ) α | G η ⊞⊞ η ( x + iy , u + iy ) | . Taking limit as ( y , y ) → (0 ,
0) in this inequality yieldslim ( y ,y ) → (0 , ( y y ) α − ℑ G µ ⊞ µ ( x + iy ) ℑ G ν ⊞ ν ( u + iy ) = + ∞ Since the two coordinates y , y in the above limit are independent of each other, weconclude that there exists a t ∈ (0 , α −
1] such that the sets { y t ℑ G µ ⊞ µ ( x + iy ) : y > } and { y t ℑ G ν ⊞ ν ( x + iy ) : y > } are both unbounded. An application of thesame [5, Theorem 7] allows us to conclude. (cid:3) We conclude this section with a simple remark generalizing the lineariza-tion result from [9] to bi-free bi-partite self-adjoint random variables.
Proposition 3.11.
Assume that ( a , b ) , ( a , b ) ∈ A self-adjoint random vari-ables in the C ∗ -noncommutative probability space ( A , ϕ ) which are bi-free with re-spect to ϕ and satisfy a j b j = b j a j , j = 1 , . Consider self-adjoint polynomials p, q intwo noncommuting indeterminates. Then there exist m ∈ N , α j , β j , γ j ∈ M m +1 ( C ) which are self-adjoint such that ϕ (cid:0) ( z − p ( a , a )) − ( w − q ( b , b )) − (cid:1) =[ G ( a ⊗ α + a ⊗ α ,b ⊗ β + b ⊗ β ) ( ze , + γ , − e , , we , + γ )] ,m +2 . Moreover, G ( a ⊗ α + a ⊗ α ,b ⊗ β + b ⊗ β ) can be computed via the analytic subordina-tion functions provided by Theorem Proof.
As shown in [1] (see also [9, Section 3]), for any self-adjoint polynomial intwo noncommuting self-adjoint indeterminates p ∈ C h X , X i , one can find m p ∈ N and a self-adjoint matrix L p = (cid:20) − u ∗ p − u p − Q p (cid:21) ∈ M m p +1 ( C h X , X i ) such that(1) each entry of L p is of degree less than or equal to one,(2) u p ∈ M m p × ( C h X , X i ),(3) Q p ∈ M m p ( C h X , X i ) is invertible with Q − p ∈ M m p ( C h X , X i ), and(4) (cid:2) ( ze , − L p ) − (cid:3) , = (cid:20) z u ∗ p u p Q p (cid:21) − ! , = ( z − p ) − , that is, p = u ∗ p Q − p u p .Clearly, such an L p needs not be unique.Choose now such a matrix L p ∈ M m p +1 ( C h X , X i ), and another matrix L q ∈ M m q +1 ( C h X , X i ) satisfying the same properties. We would like to evaluate L p in X = a , X = a and L q in X = b , X = b and apply Lemma 2.1 and Remark 3.2. In order to be able to do that, we need that m p = m q . Unfortunatelythere is no apriori reason for that to happen. Assume without loss of generalitythat m p < m q . We show next that we can modify L p such that it still satisfiesitems (1)–(4) above, but with m p replaced by m p + r for any r ∈ N . Indeed, if L p satisfies (1)–(4) above, then (cid:18) ze , − (cid:20) L p m p × r r × m p r × r (cid:21)(cid:19) − = (cid:20) ( ze , − L p ) − m p × r r × m p − r × r (cid:21) . Thus, if our first choice of L p and L q have different sizes, we complete the smallerone with an identity matrix of the desired size in the lower right corner (and zeroelsewhere) in order to make them of equal size m + 1. Then z u ∗ p − × m u p Q p m × m × m × m w u ∗ q m × m × m u q Q q − = ( z − p ) − ⋆ ( z − p ) − ( w − q ) − ⋆⋆ ⋆ ⋆ ⋆ × m ( w − q ) − ⋆ m × m × m ⋆ ⋆ We evaluate u p , Q p in a and a , u q , Q q in b and b , and apply ϕ . The propositionfollows now easily from Lemma 2.1 and Remark 3.2. (cid:3) Bi-free convolution semigroups
One remarkable feature of free additive convolution is the existence ofpartially defined free convolution semigroups: for any Borel probability measure µ on R , there exists a family ( µ t ) t ≥ of Borel probability measures on R such that µ = µ and µ s + t = µ s ⊞ µ t for all s, t ≥
1. This phenomenon was first notedin [12] for t large enough, and proved in [21] for all t ≥
1. In [8], an analyticsubordination formula for µ t to µ is provided: for any t ≥ z ∈ C + , thereexists ω µ ( t, z ) ∈ C + such that ω µ ( t, z ) = zt + (cid:18) − t (cid:19) G µ ( ω µ ( t, z )) . Moreover, G µ ( ω µ ( t, z )) = G µ t ( z ), z ∈ C , and the correspondence z ω µ ( t, z )is analytic on C + . It is easy to see that the above equation uniquely determines ω µ ( t, z ), and hence µ t . The paper [21] provides also an operatorial construction of µ t : if a = a ∗ in some ∗ -noncommutative probability space ( A , ϕ ) has distribution µ with respect to ϕ and p = p ∗ = p is a projection which is free from a and satisfies ϕ ( p ) = 1 /t , then the distribution of pap in the reduced algebra ( p A p, ϕ ( p ) ϕ ( p · p ))is µ t . Using this construction, one of us has generalized, together with Huang andMingo, the result of Nica and Speicher to bi-free additive convolution.More precisely, it has been shown in [18, Theorem 5.3] that for anycompactly supported Borel probability measure η on R , there exists a partiallydefined bi-free convolution semigroup ( η t ) t ≥ satisfying the conditions η = η and η s + t = η s ⊞ ⊞ η t for all s, t ≥
1. As expected, we have R η t ( z, w ) = tR η ( z, w ), t ≥
1. The partial semigroup ( η t ) t ≥ extends to a full weakly continuous semigroup NALYTIC SUBORDINATION FOR BI-FREE CONVOLUTION 21 [0 , + ∞ ) ∋ t η t with η = δ (0 , if and only if η is bi-freely infinitely divisible (see[18, Theorem 4.2]).Consider a C ∗ -noncommutative probability space ( A , ϕ ) and let ( a , b ) ∈A be a bi-partite two-faced pair of self-adjoint random variables whose joint dis-tribution with respect to ϕ is η . For any t ≥
1, let ( a t , b t ) ∈ A be a two-facedpair of noncommutative random variables such that a t b t = b t a t , a t = a ∗ t , and b t = b ∗ t , and the distribution of ( a t , b t ) with respect to ϕ equals η t . Denote by µ t the distribution of a t and by ν t the distribution of b t with respect to ϕ . Define X t = (cid:20) a t b t (cid:21) ∈ M ( A ). As seen in Lemma 3.1, we have R X t (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = (cid:20) R a t ( z ) z − ζw − ( R ( a t ,b t ) ( z, w ) − zR a t ( z ) − wR b t ( w ))0 R b t ( w ) (cid:21) = t (cid:20) R a ( z ) z − ζw − ( R ( a ,b ) ( z, w ) − zR a ( z ) − wR b ( w ))0 R b ( w ) (cid:21) = tR X (cid:18)(cid:20) z ζ w (cid:21)(cid:19) . As shown in [2, Theorem 7.9], for a given X = X ∗ ∈ M ( A ) and t ≥
1, thereexists an ˜ X t = ˜ X ∗ t such that R ˜ X t = tR X . By restricting R ˜ X t to the set ofupper triangular matrices and applying Lemma 3.1, we see that R ˜ X t (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = R X t (cid:18)(cid:20) z ζ w (cid:21)(cid:19) for all z, w, ζ ∈ C of sufficiently small absolute value. In particular,it follows that the subordination formula of [2, Theorem 8.4] holds for X t : thereexists a function ω X t defined on the set of elements (cid:20) z ζ w (cid:21) with strictly positiveimaginary part which satisfies the functional equation ω X t (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = 1 t (cid:20) z ζ w (cid:21) + (cid:18) − t (cid:19) G X (cid:18) ω X t (cid:18)(cid:20) z ζ w (cid:21)(cid:19)(cid:19) − , and G X ◦ ω X t = G X t . The point ω X t (cid:18)(cid:20) z ζ w (cid:21)(cid:19) is the unique attracting fixedpoint of the map v t (cid:20) z ζ w (cid:21) + (cid:0) − t (cid:1) G X ( v ) − . Since this map sends uppertriangular matrices to upper triangular matrices, it follows that ω X t (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = (cid:20) f ( z, ζ, w ) f ( z, ζ, w )0 f ( z, ζ, w ) (cid:21) itself is upper triangular. Its entries are easily determinedby using the above-displayed equation: (cid:20) f ( z, ζ, w ) f ( z, ζ, w )0 f ( z, ζ, w ) (cid:21) = 1 t (cid:20) z ζ w (cid:21) + (cid:18) − t (cid:19) " G a ( f ( z, ζ, w )) − ϕ ( ( f ( z,ζ,w ) − a ) − f ( z,ζ,w )( f ( z,ζ,w ) − b ) − ) G a ( f ( z,ζ,w )) G b ( f ( z,ζ,w )) G b ( f ( z, ζ, w )) − = " zt + (cid:0) − t (cid:1) G a ( f ( z,ζ,w )) 1 t ζ + (cid:0) − t (cid:1) f ( z,ζ,w ) G ( a ,b ( f ( z,ζ,w ) ,f ( z,ζ,w )) G a ( f ( z,ζ,w )) G b ( f ( z,ζ,w )) wt + (cid:0) − t (cid:1) G b ( f ( z,ζ,w )) . The equalities corresponding to entries (1 ,
1) and (2 ,
2) provide as indicated atthe beginning of this section, via [8, Theorem 2.5], that f ( z, ζ, w ) = ω µ ( t, z ), f ( z, ζ, w ) = ω ν ( t, w ). The (1 ,
2) corner provides the relation f ( z, ζ, w ) = ζ G a ( ω µ ( t, z )) G b ( ω ν ( t, w )) tG a ( ω µ ( t, z )) G b ( ω ν ( t, w )) + (1 − t ) G ( a ,b ) ( ω µ ( t, z ) , ω ν ( t, w ))= ζ G a t ( z ) G b t ( w ) tG a t ( z ) G b t ( w ) + (1 − t ) G ( a ,b ) ( ω µ ( t, z ) , ω ν ( t, w )) . Thus, using G X ◦ ω X t = G X t we obtain a formula for the Cauchy transform of ameasure in a partial bi-free additive convolution semigroup:(18) G ( a t ,b t ) ( z, w ) = 1 tG ( a ,b ( ω µ ( t,z ) ,ω ν ( t,w )) + − tG a ( ω µ ( t,z )) G b ( ω ν ( t,w )) , z, w ∈ C + . An analogue of Proposition 3.5 now easily follows:
Proposition 4.1.
Let ( A , ϕ ) be a C ∗ -noncommutative probability space. Assumethat for any t ≥ , there is a commuting self-adjoint two-faced pair ( a t , b t ) ∈ A ofnoncommutative random variables such that the distribution of ( a t , b t ) with respectto ϕ equals η t , and η s + t = η s ⊞⊞ η t , s, t ≥ . Denote µ t the distribution of a t and ν t the distribution of b t . Then ℑ ω µ ( t, z ) ℑ ω ν ( t, w ) | G ( a ,b ) ( ω µ ( t, z ) , ω ν ( t, w )) | ≥ ℑ z ℑ w | G ( a t ,b t ) ( z, w ) | , for all z, w ∈ C + . Proof.
The inequality follows from the fact that ℑ ω X t (cid:18)(cid:20) z ζ w (cid:21)(cid:19) > ℑ (cid:20) z ζ w (cid:21) > M ( C ) and from the relation G ( a t ,b t ) ( z, w ) = f ( z, ζ, w ) G ( a ,b ) ( ω µ ( t, z ) , ω ν ( t, w )) . The proof is identical to the proof of Proposition 3.5 and is left as an exercise tothe reader. (cid:3)
Corollary 4.2.
Consider a compactly supported Borel probability measure η on R and let t > be given. Let ( η t ) t ≥ be its partial bi-free convolution semi-group. Assume that there is a point ( ξ, ζ ) ∈ R so that η t ( { ( ξ, ζ ) } ) > . Then η ( { ( ξ/t, ζ/t ) } ) > and η t ( { ( ξ, ζ ) } ) = ( tµ ( { ξ/t } ) + 1 − t )( tν ( { ζ/t } ) + 1 − t ) η ( { ( ξ/t, ζ/t ) } ) tµ ( { ξ/t } ) ν ( { ζ/t } ) + (1 − t ) η ( { ( ξ/t, ζ/t ) } ) Proof.
The presence of an atom of η t at ( ξ, ζ ) implies the presence of atoms for themarginals µ t (at ξ ) and ν t (at ζ ), respectively. Thus, as shown in [8, Theorem 3.1],we have(1) lim y ↓ ω µ ( t, ξ + iy ) = ξ/t , lim y ↓ ω ν ( t, ζ + iy ) = ζ/t ;(2) µ t ( { ξ } ) = tµ ( { ξ/t } ) + 1 − t, ν t ( { ζ } ) = tν ( { ζ/t } ) + 1 − t ;(3) lim y ↓ ℑ ω µ ( t,ξ + iy ) y = t + (cid:0) − t (cid:1) µ t ( { ξ } ) = µ ( { ξ/t } ) tµ ( { ξ/t } )+1 − t andlim y ↓ ℑ ω ν ( t,ζ + iy ) y = t + (cid:0) − t (cid:1) ν t ( { ζ } ) = ν ( { ζ/t } ) tν ( { ζ/t } )+1 − t . NALYTIC SUBORDINATION FOR BI-FREE CONVOLUTION 23
In particular, µ ( { ξ/t } ) > − /t and ν ( { ζ/t } ) > − /t. Applying Proposition 4.1to z = ξ + iy, w = ζ + iy and taking limit as y → (cid:18) ( tµ ( { ξ/t } ) + 1 − t )( tν ( { ζ/t } ) + 1 − t ) µ ( { ξ/t } ) ν ( { ζ/t } ) (cid:19) η ( { ( ξ/t, ζ/t ) } ) ≥ η t ( { ( ξ, ζ ) } ) > . Thus, η ( { ( ξ/t, ζ/t ) } ) >
0. Multiplying by y in (18) evaluated in z = ξ + iy, w = ζ + iy and taking limits as y decreases to zero yields η t ( { ( ξ, ζ ) } )= lim y ↓ y G ( a t ,b t ) ( ξ + iy, ζ + iy )= lim y ↓ t ℑ ω µ ( t,ξ + iy ) ℑ ω ν ( t,ζ + iy ) y ℑ ω µ ( t,ξ + iy ) ℑ ω ν ( t,ζ + iy ) G ( a ,b ( ω µ ( t,ξ + iy ) ,ω ν ( t,ζ + iy )) + − tyG at ( ξ + iy ) yG bt ( ζ + iy ) = 1 tµ ( { ξ/t } ) ν ( { ζ/t } )( tµ ( { ξ/t } )+1 − t )( tν ( { ζ/t } )+1 − t ) η ( { ( ξ/t,ζ/t ) } ) + − t ( tµ ( { ξ/t } )+1 − t )( tν ( { ζ/t } )+1 − t ) = ( tµ ( { ξ/t } ) + 1 − t )( tν ( { ζ/t } ) + 1 − t ) η ( { ( ξ/t, ζ/t ) } ) tµ ( { ξ/t } ) ν ( { ζ/t } ) + (1 − t ) η ( { ( ξ/t, ζ/t ) } ) , which concludes our proof. (cid:3) We record a more elegant version of the relation from the above corollary:(19) µ t ( { ξ } ) ν t ( { ζ } ) η t ( { ( ξ, ζ ) } ) = t µ ( { ξ/t } ) ν ( { ζ/t } ) η ( { ( ξ/t, ζ/t ) } ) + 1 − t. Example 4.3.
We compute a simple example: let η = δ (1 , + δ (0 , + δ (1 , .Then µ = δ + δ , ν = δ + δ . The longest an atom can hope to survive isfor as long as t <
4. Indeed, µ t ( { } ) = max { , tµ ( { } ) + 1 − t } = max { , − t } , µ t ( { t } ) = max { , − t } , ν t ( { } ) = max { , − t } , ν t ( { t } ) = max { , − t } . Soif t < /
7, then η t ( { (0 , } ) = (cid:0) − t (cid:1) (cid:0) − t (cid:1) t + − t = (cid:18) − t (cid:19) , if t <
4, then η t ( { ( t, t ) } ) = (cid:0) − t (cid:1) (cid:0) − t (cid:1) t + (1 − t ) = (cid:18) − t (cid:19) , and if t < /
3, then η t ( { ( t, } ) = (cid:0) − t (cid:1) (cid:0) − t (cid:1) t + − t = 18 (8 − t )(4 − t )4 + 3 t . A direct computation shows that the sum of the mass of the three atoms is strictlyless than one for any t >
1, so that a nonatomic part occurs immediately after t = 1, as in the case of free convolution of measures on R .Unlike for free convolution semigroups, the expression for the non-atomicpart of η t is much more unwieldy. Indeed, while in principle formula (18) allowsfor a direct computation of G η t , the actual computation, even for such a simplemeasure as the one from Example 4.3, becomes uncomfortably long. We providehere just the necessary ingredients: The reciprocals of the Cauchy transforms ofthe marginals at t = 1 are G µ ( z ) − = z − z z − and G ν ( w ) − = w − w w − , and of the reciprocal of the Cauchy transform of η is G η ( z, w ) − = zw ( z − w − zw − z − w +1 . Forgiven t >
1, the subordination functions associated to the two marginals are ω µ ( t, z ) = 8 z + 8 − t + q [8 z − t + 6 − p t − z − t + 6 + 2 p t − , and ω ν ( t, w ) = 4 w + 4 − t + q [4 w + 2 − t − p t − w + 2 − t + 2 p t − . Replacing in (18) provides the explicit (algebraic) expression for G η t .The case t = 4, the time when the last atom disappears, provides ω µ (4 , z ) = 2 z − √ z − z + 254 , ω ν (4 , w ) = w − √ w − w + 42 , so that 1 G µ ( ω µ (4 , z )) = z − √ z − z + 253 , G ν ( ω ν (4 , w )) = w − √ w − w + 43 . Then 1 − G µ ( ω µ (4 , z )) G ν ( ω ν (4 , w )) = − (cid:0) z − √ z − z + 25 (cid:1) (cid:0) w − √ w − w + 4 (cid:1) G η ( ω µ (4 , z ) , ω ν (4 , w ))= (cid:16) z − p z − z + 25 (cid:17) (cid:16) z − p z − z + 25 (cid:17) × (cid:0) w − √ w − w + 4 (cid:1) (cid:0) w − √ w − w + 4 (cid:1)(cid:2) (cid:0) z − √ z − z + 25 (cid:1) − (cid:3) (cid:2) w − √ w − w + 4 − (cid:3) + . Substituting − G µ ( ω µ (4 ,z )) G ν ( ω ν (4 ,w )) and G η ( ω µ (4 ,z ) ,ω ν (4 ,w )) into Equation (18) pro-duces G η ( z, w ).5. Conditionally bi-free analytic subordination
In this section, we discuss how the method of Section 3 can be used tostudy the conditionally bi-free additive convolution in the scalar-valued setting.Motivated by the universal constructions for conditionally free independence [14]and bi-free independence [33], two of us introduced in [19] the notion of condition-ally bi-free independence for pairs of algebras in the setting of a two-state noncom-mutative probability space ( A , ϕ, ψ ) such that conditionally bi-freeness reduces tobi-freeness when ϕ = ψ and reduces to conditionally freeness when only left or onlyright algebras are considered. For the theoretical definition in terms of actions ona reduced free product space, we refer to [19, Definition 3.4]. NALYTIC SUBORDINATION FOR BI-FREE CONVOLUTION 25
Let ( a , b ) and ( a , b ) be bi-partite self-adjoint two-faced pairs in atwo-state C ∗ -noncommutative probability space ( A , ϕ, ψ ) such that the joint dis-tributions of ( a j , b j ) with respect to ( ϕ, ψ ) coincide with the moments of a pair( θ j , η j ) of compactly supported probability meausres on R via ϕ ( a mj b nj ) = Z R t m s n d θ j ( t, s ) and ψ ( a mj b nj ) = Z R t m s n d η j ( t, s ) , j = 1 , . If ( a , b ) and ( a , b ) are conditionally bi-free with respect to ( ϕ, ψ ), then thejoint distribution of ( a + a , b + b ) is again a pair ( θ, η ) of compactly supportedprobability measures on R . The pair ( θ, η ) depends only on the pairs ( θ , η ) and( θ , η ), and is called the conditionally bi-free additive convolution of ( θ , η ) and( θ , η ), denoted ( θ, η ) = ( θ , η ) ⊞⊞ c ( θ , η ). More precisely, η = η ⊞⊞ η is the bi-free additive convolution of η and η , and the moments of θ can be computed usingthe moments of θ , θ , η , and η via the formula provided by [19, Theorem 4.8]. Asin Section 2, we denote by σ j , τ j the marginals of θ j and µ j , ν j the marginals of η j so that σ j and µ j are the distributions of a j with respect to ϕ and ψ , respectively,and τ j and ν j are the distributions of b j with respect to ϕ and ψ, respectively. If σ, τ denote the marginals of θ , then ( σ, µ ) is the conditionally free convolution of( σ , µ ) and ( σ , µ ), denoted ( σ, µ ) = ( σ , µ ) ⊞ c ( σ , µ ), where µ = µ ⊞ µ , andsimilarly ( τ, ν ) = ( τ , ν ) ⊞ c ( τ , ν ), where τ = τ ⊞ τ .To linearize the conditionally bi-free additive convolution, the partialconditionally bi-free R -transform was introduced in [19, Section 5] as the analogueof the partial bi-free R -transform, which is also a function of two complex variablesdefined on a neighbourhood of zero in C . To introduce this function, we shall firstreview how the conditionally free additive convolution is calculated (see [14, 4]). Let G σ j and G µ j be the (one-dimensional) Cauchy transforms of σ j and µ j , respectively,and let K µ j be the inverse of G µ j on a neighbourhood of infinity as in Section 2.The conditionally free R -transform of ( σ j , µ j ) is defined by R ( σ j ,µ j ) ( z ) = K µ j ( z ) − G σ j ( K µ j ( z ))on a small neighbourhood of zero where K µ j is defined, and satisfies the relation R ( σ,µ ) ( z ) = R ( σ ,µ ) ( z ) + R ( σ ,µ ) ( z )for z in a small enough neighbourhood of zero. (While K µ j has a simple pole atzero, the function R ( σ j ,µ j ) ( z ) extends holomorphically, not meromorphically, in 0.)For the pair ( θ j , η j ) of measures on R , let G θ j and G η j be the Cauchytransforms of θ j and η j , respectively. The partial conditionally bi-free R -transform of ( θ j , η j ) is defined by R ( θ j ,η j ) ( z, w ) = zR ( σ j ,µ j ) ( z ) + wR ( τ j ,ν j ) ( w ) + e R ( θ j ,η j ) ( z, w ) , where e R ( θ j ,η j ) ( z, w ) = zwG θ j ( K µ j ( z ) , K ν j ( w )) G σ j ( K µ j ( z )) G τ j ( K ν j ( w )) G η j ( K µ j ( z ) , K ν j ( w )) − zwG η j ( K µ j ( z ) , K ν j ( w )) , for z, w in a small enough bi-disk centred at zero (see [19, Corollary 5.7 and Def-inition 5.8]). The crucial property of the partial conditionally bi-free R -transform is R ( θ,η ) ( z, w ) = R ( θ ,η ) ( z, w ) + R ( θ ,η ) ( z, w ) , | z | + | w | sufficiently small , if ( θ, η ) = ( θ , η ) ⊞⊞ c ( θ , η ). In terms of random variables, the conditionally freeand bi-free R -transforms are defined by exactly the same formulas as above exceptthe notations are slightly different, which we summarize as follows. For a randomvariable a in a two-state noncommutative probability space ( A , ϕ, ψ ), let G ϕa and G ψa be the Cauchy transforms of a with respect to ϕ and ψ , respectively. Thenthe conditionally free R -transform of a is denoted by R c a . Similarly, for a two-facedpair ( a, b ) in ( A , ϕ, ψ ), let G ϕ ( a,b ) and G ψ ( a,b ) be the two-variable Cauchy transformsof ( a, b ) with respect to ϕ and ψ , respectively. The partial conditionally bi-free R -transform of ( a, b ) is denoted by R c( a,b ) .The notion of conditionally free R -transform can be generalized to theoperator-valued setting as follows (see, e.g., [10, 22]). Let ( M, E, F, B, D ) be a C ∗ -( B, D )-noncommutative probability space. That is, B ⊆ M and B ⊆ D areunital inclusions of unital C ∗ -algebras, E : M → B is a unit-preserving conditionalexpectation, and F : M → D is a unital B -bimodule map. For X = X ∗ ∈ M ,the B -valued and D -valued Cauchy transforms of X with respect to E and F aredefined by G X ( b ) = E (cid:2) ( b − X ) − (cid:3) and G X ( b ) = F (cid:2) ( b − X ) − (cid:3) , respectively, for b ∈ B, ℑ b > k b − k < k X k − . As seen in Section 2 when discussing the B -valued free R -transform, the function G X has a compositional inverse, denoted K X ,defined on an open set of B which contains zero in its norm-closure. The D -valuedconditionally free R -transform of X is defined by R c X ( b ) = K X ( b ) − G X ( K X ( b )) − , k b k small , where the exponent − G X ( K X ( b )) in the algebra D . The D -valued conditionally free R -transform plays the same role as its scalar-valuedcounterpart: if X = X ∗ , X = X ∗ ∈ M are conditionally free over ( B, D ) withrespect to ( E, F ), then R c X + X ( b ) = R c X ( b ) + R c X ( b ) for all b ∈ B of sufficientlysmall norm. We consider now the conditionally free analogue of Lemma 3.1.Let ( a, b ) be a self-adjoint two-faced pair in a two-state C ∗ -noncommutativeprobability space ( A , ϕ, ψ ). Define X ∈ M ( A ) by X = (cid:20) a b (cid:21) , let B = D = M ( C ), and define M ( ϕ ) , M ( ψ ) : M ( A ) → M ( C ) by M ( ϕ ) = ϕ ⊗ Id M ( C ) and M ( ψ ) = ψ ⊗ Id M ( C ) , respectively. Furthermore, let G X and G X be the Cauchytransforms of X with respect to M ( ψ ) and M ( ϕ ), respectively, and let K X be the NALYTIC SUBORDINATION FOR BI-FREE CONVOLUTION 27 compositional inverse of G X . Using the results from Section 3 above, we have R c X (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = K X (cid:18)(cid:20) z ζ w (cid:21)(cid:19) − G X (cid:18) K X (cid:18)(cid:20) z ζ w (cid:21)(cid:19)(cid:19) − = " K a ( z ) − ζG ψ ( a,b ) ( K a ( z ) ,K b ( w )) K b ( w ) − G X " K a ( z ) − ζG ψ ( a,b ) ( K a ( z ) ,K b ( w )) K b ( w ) − = " K a ( z ) − ζG ψ ( a,b ) ( K a ( z ) ,K b ( w )) K b ( w ) − G ϕa ( K a ( z )) ζG ϕ ( a,b ) ( K a ( z ) ,K b ( w )) G ψ ( a,b ) ( K a ( z ) ,K b ( w )) G ϕb ( K b ( w )) − = K a ( z ) − G ϕa ( K a ( z )) ζ G ϕ ( a,b ) ( K a ( z ) ,K b ( w )) − G ϕa ( K a ( z )) G ϕb ( K b ( w )) G ϕa ( K a ( z )) G ϕb ( K b ( w )) G ψ ( a,b ) ( K a ( z ) ,K b ( w )) K b ( w ) − G ϕb ( K b ( w )) = " R c a ( z ) ζzw e R c( a,b ) ( z, w )0 R c b ( w ) , a perfect analogue of Lemma 3.1. We record our conclusion in the following: Lemma 5.1.
Let ( a, b ) be a self-adjoint two-faced pair in a two-state C ∗ -noncommutativeprobability space ( A , ϕ, ψ ) . Define the M ( C ) -valued random variable X = (cid:20) a b (cid:21) ∈ M ( A ) and let M ( ϕ ) , M ( ψ ) be as above. Then R c X (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = (cid:20) R c a ( z ) ζzw ( R c( a,b ) ( z, w ) − zR c a ( z ) − wR c b ( w ))0 R c b ( w ) (cid:21) . In particular, if ( a , b ) and ( a , b ) in ( A , ϕ, ψ ) are conditionally bi-free with respectto ( ϕ, ψ ) , then, under the above notation, (20) R c X + X (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = R c X (cid:18)(cid:20) z ζ w (cid:21)(cid:19) + R c X (cid:18)(cid:20) z ζ w (cid:21)(cid:19) for all z, w ∈ C of sufficiently small absolute value and all ζ ∈ C . Remark 5.2.
As with the bi-free case, ( a , b ) and ( a , b ) being conditionally bi-free with respect to ( ϕ, ψ ) does not necessarily imply X and X are conditionallyfree with amalgamation over ( M ( C ) , M ( C )) with respect to ( M ( ϕ ) , M ( ψ )), sothat Equation (20) does not extend to arbitrary 2 × (cid:20) z ζζ ′ w (cid:21) .Next, we discuss how the single-variable analytic subordination functionsfor ⊞ can be used to compute ⊞⊞ c . Let ( θ , η ) and ( θ , η ) be as above withmarginals σ j , τ j for θ j and µ j , ν j for η j , and let ( θ, η ) = ( θ , η ) ⊞ ⊞ c ( θ , η ) withmarginals σ, τ for θ and µ, ν for η . Given a Borel probability measure λ on R ,recall the function h λ defined in (14) by h λ ( z ) = G λ ( z ) − z . It was shown in [4,Proposition 3] that(21) h σ ( z ) = h σ ( ω a ( z )) + h σ ( ω a ( z )) , z ∈ C + , where ( σ, µ ) = ( σ , µ ) ⊞ c ( σ , µ ) and ω a , ω a are the single-variable subordinationfunctions related to µ , µ (see (1)). Of course, we also have h τ ( w ) = h τ ( ω b ( w )) + h τ ( ω b ( w )), where ( τ, µ ) = ( τ , ν ) ⊞ c ( τ , ν ) with subordination functions ω b , ω b .In the following, we present a two-dimensional analogue of Equation (21).Since the partial conditionally bi-free R -transform linearizes ⊞⊞ c , we have X j =1 (cid:20) G θ j ( K µ j ( z ) , K ν j ( w )) G σ j ( K µ j ( z )) G τ j ( K ν j ( w )) G η j ( K µ j ( z ) , K ν j ( w )) − G η j ( K µ j ( z ) , K ν j ( w )) (cid:21) = G θ ( K µ ( z ) , K ν ( w )) G σ ( K µ ( z )) G τ ( K ν ( w )) G η ( K µ ( z ) , K ν ( w )) − G η ( K µ ( z ) , K ν ( w )) . Replacing in the above z by G µ ( z ) and w by G ν ( w ) yields X j =1 (cid:20) G θ j ( ω a j ( z ) , ω b j ( w )) G σ j ( ω a j ( z )) G τ j ( ω b j ( w )) G η j ( ω a j ( z ) , ω b j ( w )) − G η j ( ω a j ( z ) , ω b j ( w )) (cid:21) = G θ ( z, w ) G σ ( z ) G τ ( w ) G η ( z, w ) − G η ( z, w ) . (22)For a probability measure ρ on R with marginals ρ (1) and ρ (2) , recall the definition(16) of the analytic function e E ρ ( z, w ) = G ρ ( z,w ) G ρ (1) ( z ) G ρ (2) ( w ) − , ( z, w ) ∈ ( C \ R ) . Usingthis function, Equation (22) can be written as(23) e E θ ( z, w ) G η ( z, w ) = e E θ ( ω a ( z ) , ω b ( w )) G η ( ω a ( z ) , ω b ( w )) + e E θ ( ω a ( z ) , ω b ( w )) G η ( ω a ( z ) , ω b ( w ))as the conditionally bi-free analogue of Equation (21). As shown in Section 3, theabove equation can be viewed as an equality of analytic functions on C + × C + as soonas we multiply both sides by G η ( z, w ). Otherwise, it is an equality of meromorphicfunctions, as explained in the comments following the proof of Proposition 3.5.Let ( M, E, F, B, D ) be a C ∗ -( B, D )-noncommutative probability space.For Y = Y ∗ ∈ M , define(24) h Y ( b ) = G Y ( b ) − − b for b ∈ B with ℑ ( b ) > B ⊆ D ). The following operator-valuedanalogue of Equation (21) was proved in [10, Lemma 2.14]: h Y + Y ( b ) = h Y ( ω Y ( b )) + h Y ( ω Y ( b )) , whenever Y , Y ∈ M are self-adjoint, free over ( B, D ), where ω Y j is the B -valuedsubordination function satisfying G Y + Y = G Y j ◦ ω Y j .In view of Lemma 5.1 and Equation (24), we obtain the following ana-logue of Remark 3.2. Remark 5.3.
Under the assumptions and notation of Remark 3.2 and the abovediscussions, if ( a , b ) and ( a , b ) in ( A , ϕ, ψ ) are conditionally bi-free with respectto ( ϕ, ψ ), and if Y and Y in ( M ( A ) , M ( ϕ ) , M ( ψ ) , M ( C ) , M ( C )) are condition-ally free with respect to ( M ( ϕ ) , M ( ψ )), then(25) G X + X (cid:18)(cid:20) z ζ w (cid:21)(cid:19) − + (cid:20) z ζ w (cid:21) = ( G X ◦ ω Y ) (cid:18)(cid:20) z ζ w (cid:21)(cid:19) − +( G X ◦ ω Y ) (cid:18)(cid:20) z ζ w (cid:21)(cid:19) − for all z, w ∈ C + , ζ ∈ C . Moreover, as seen in the discussions following Remark3.2, ω Y j is given by ω Y j (cid:18)(cid:20) z ζ w (cid:21)(cid:19) = (cid:20) ω a j ( z ) Π j ( z, ζ, w )0 ω b j ( w ) (cid:21) , where ω a j and ω b j are the single-variable subordination functions with respect to ψ (i.e. G ψa + a = G ψa j ◦ ω a j and G ψb + b = G ψb j ◦ ω b j , j = 1 ,
2, and Π j ( z, ζ, w ) is thefunction introduced in Theorem 3.6). Consequently, we recover Equation (23) fromEquation (25).We can now state the following analogue of Theorem 3.6 in the scalar-valued setting, which easily follows from the above considerations. Theorem 5.4.
Let ( A , ϕ, ψ ) be a two-state C ∗ -noncommutative probability space,and let ( a , b ) and ( a , b ) be self-adjoint two-faced pairs in ( A , ϕ, ψ ) which areconditionally bi-free with respect to ( ϕ, ψ ) . Denote X j = (cid:20) a j b j (cid:21) , j = 1 , . Then M ( ϕ ) "(cid:18)(cid:20) z ζ w (cid:21) − X − X (cid:19) − − + (cid:20) z ζ w (cid:21) = M ( ϕ ) "(cid:18)(cid:20) ω a ( z ) Π ( z, ζ, w )0 ω b ( w ) (cid:21) − X (cid:19) − − + M ( ϕ ) "(cid:18)(cid:20) ω a ( z ) Π ( z, ζ, w )0 ω b ( w ) (cid:21) − X (cid:19) − − for all z, w ∈ C + , ζ ∈ C , where Π ( z, ζ, w ) and Π ( z, ζ, w ) are the functions intro-duced in Theorem . Remark 5.5.
We conclude this section with a remark. In noncommutative prob-ability theory, there is another notion of independence, called monotonic inde-pendence, introduced by Muraki which, together with tensor, free, Boolean, andanti-monotonic independences, form the only five notions of natural independence.In the operator-valued setting, if X = X ∗ and X = X ∗ in ( M, E, B ) are monoton-ically independent with amalgamation over B , then G X + X ( b ) = G X ( G X ( b ) − ).Now, if ( a , b ) and ( a , b ) are self-adjoint two-faced pairs in ( A , ϕ ) and if we define X j = (cid:20) a j b j (cid:21) for j = 1 , G X + X (cid:18)(cid:20) z ζ w (cid:21)(cid:19) and G X G X (cid:18)(cid:20) z ζ w (cid:21)(cid:19) − ! , and comparing the (1 ,
2) entries, we obtain an expression for G ( a + a ,b + b ) ( z, w )in terms of the Cauchy transforms of the two pairs and the marginals. This hasbeen shown in [17] to lead to one of the two natural notions of bi-monotonic inde-pendence, generalizing monotonic independence to the two-faced setting.6. No conditional expectations of the resolvent
The analytic subordination result of Biane is stronger than the resultstated in (1): it is shown in [13] that if a , a are free self-adjoint random variables in the tracial C ∗ -noncommutative probability space ( A , ϕ ), then E C [ a j ] (cid:2) ( z − a − a ) − (cid:3) = ( ω a j ( z ) − a j ) − , z ∈ C + , where E C [ a j ] denotes the unique trace-preserving conditional expectation from thevon Neumann algebra generated by a and a onto the von Neumann algebra gen-erated by a j . Voiculescu generalized this result to self-adjoint random variableswhich are free with amalgamation with respect to a trace-preserving conditionalexpectation. However, in order to prove formula (1) alone, both in its scalar-and operator-valued version, only analytic function theory methods are needed, asshown in [7, 9]. It is remarkable that one can use this same analytic functionsmachinery to prove Biane’s result, at the cost of an amplification to 3 × A , ϕ ) is a tracial W ∗ -noncommutative probability spaceand B ⊆ A is a von Neumann subalgebra, then there exists a unique trace-preserving conditional expectation E : A → B . This expectation is defined viathe following relation: for any x ∈ A , E [ x ] is the unique element in B so that ϕ ( xξ ∗ ) = ϕ ( E [ x ] ξ ∗ ) for all ξ ∈ B . Clearly, given the hypothesis of weak ∗ -continuityand faithfulness on ϕ , it is enough to verify the equality ϕ ( xξ ∗ ) = ϕ ( E [ x ] ξ ∗ ) for allelements ξ in a subset of B whose linear span is dense in B . Thus, in order to provethe relation E C [ a j ] (cid:2) ( z − a − a ) − (cid:3) = ( ω a j ( z ) − a j ) − , it suffices to show that forany v ∈ C [ a j ] , v >
0, we have ϕ (cid:0) ( z − a − a ) − v (cid:1) = ϕ (cid:0) ( ω a j ( z ) − a j ) − v (cid:1) . In order to do this, we use a linearization trick similar to the one used in [9] (whichoriginates in Anderson’s paper [1]) and Lemma 2.1, with the right variable equal tozero. Consider A = − v − v a ∈ M ( C [ a ]) , A = a ∈ M ( C [ a ])(recall that v is an arbitrary positive element in the von Neumann algebra generatedby a ). Lemma 2.1 implies that A and A are free with amalgamation over M ( C )with respect to M ( ϕ ) := ϕ ⊗ Id M ( C ) . According to [9, Theorem 2.7], relation (1)holds for the M ( C )-valued Cauchy transforms of A , A and A + A . We have G A + A z = M ( ϕ ) z − A − A − = ϕ ( v ( z − a − a ) − v ) 1 − ϕ ( v ( z − a − a ) − )1 0 0 − ϕ (( z − a − a ) − v ) 0 ϕ (( z − a − a ) − ) . (26)For simplicity, we denote R = ( z − a − a ) − , so that G a + a ( z ) = ϕ ( R ). Then G A + A z − = ϕ ( vR ) ϕ ( Rv ) ϕ ( R ) − ϕ ( vRv ) ϕ ( vR ) ϕ ( R ) ϕ ( Rv ) ϕ ( R ) 1 ϕ ( R ) . NALYTIC SUBORDINATION FOR BI-FREE CONVOLUTION 31
Theorem 2.7 of [9] guarantees (through purely function-theoretic arguments) theexistence of subordination functions ω A and ω A satisfying (1). This relationimplies via a few arithmetic manipulations and a few applications of the identityprinciple for analytic functions, that there are functions θ , θ , τ , τ dependinganalytically on z such that ω A z = θ ( z ) θ ( z )0 θ ( z ) ω a ( z ) ,ω A z = τ ( z ) τ ( z )0 τ ( z ) ω a ( z ) , (the functions ω a j are the subordination functions from formula (1) associated to a j , j = 1 , G A + A = G A ◦ ω A and G a + a = G a ◦ ω a translate into ϕ ( vRv ) 1 − ϕ ( vR )1 0 0 − ϕ ( Rv ) 0 ϕ ( R ) = τ ( z ) ϕ ( R ) − τ ( z ) 1 − τ ( z ) ϕ ( R )1 0 0 − τ ( z ) ϕ ( R ) 0 ϕ ( R ) Thus, τ ( z ) = ϕ ( vR ) ϕ ( R ) . Together with relation (1) applied to the matrix-valuedfunctions, this provides us with the equality θ ( z ) = 0 (and, as an added bonus, τ ( z ) = ϕ ( vR ) ϕ ( R ) − ϕ ( vRv )). The subordination relations corresponding to A and a yield ϕ ( vRv ) 1 − ϕ ( vR )1 0 0 − ϕ ( Rv ) 0 ϕ ( R ) = ϕ (cid:16) v ω a ( z ) − a (cid:17) − θ ( z ) 1 − ϕ (cid:16) vω a ( z ) − a (cid:17) − ϕ (cid:16) vω a ( z ) − a (cid:17) ϕ (cid:16) ω a ( z ) − a (cid:17) , The equality of (1 ,
3) entries completes the proof of Biane’s result. As an addedbonus, traciality of ϕ allows us to conclude also that θ ( z ) = 0, which determines ω A , ω A on our variable.Based on Lemma 3.1 and Remark 3.2, it is tempting to use the same trickin order to find E C [ a j ,b j ] (cid:2) ( z − a − a ) − ( w − b − b ) − (cid:3) for ( a , b ) and ( a , b ) in A bi-free with respect to ϕ and bi-partite (that is, a j b j = b j a j , j = 1 , × V = v v z − a − a u u w − b − b , where v ∈ C [ a ] , u ∈ C [ b ] are both strictly positive. Lemma 2.1 guarantees that( A , B ) = − v − v a , − u − u b and ( A , B ) = a , b are bi-free with amalgamation over M ( C ). For simplicity, let Z = z , W = w , and e , = Proposition 3.6 and Remark 3.2 apply to X j = (cid:20) A j B j (cid:21) , j = 1 ,
2, and the scalar matrix (cid:20)
Z e , W (cid:21) . On the other hand,inverting the matrix V , we obtain on the two 3 × b , b , w . In the upper right 3 × v ( z − a − a ) − ( w − b − b ) − u − v ( z − a − a ) − ( w − b − b ) − z − a − a ) − ( w − b − b ) − u z − a − a ) − ( w − b − b ) − . If ϕ were tracial, applying ϕ on the above and comparing with the correspondingmatrix entry from G X (cid:18) ω X (cid:18)(cid:20) Z e , W (cid:21)(cid:19)(cid:19) would provide the bi-free analogue ofBiane’s result. However, it turns out that ϕ is tracial only in the relatively trivialcase in which the two faces are independent. We emphasize that a formula giving E C [ a j ,b j ] (cid:2) ( z − a − a ) − ( w − b − b ) − (cid:3) as a product of resolvents of a j and b j would imply that ϕ ( v ( z − a − a ) − ( w − b − b ) − u ) = ϕ ( uv ( z − a − a ) − ( w − b − b ) − ) for v ∈ C [ a j ] , u ∈ C [ b j ]. Theorem 6.1.
Let ( a , b ) and ( a , b ) be pairs of self-adjoint operators that are bi-free in a ∗ -noncommutative probability space ( A , ϕ ) . Suppose that τ := ϕ | alg( a ,a ,b ,b ) is tracial and that for each k ∈ { , } there does not exists α k , β k ∈ R such that ( ϕ ( a nk ) , ϕ ( b nk )) = ( α nk , β nk ) for all n ∈ N (i.e. neither pair is scalars in distribution).Then alg( a , a ) and alg( b , b ) are independent. In particular, ϕ decomposes asthe tensor product of tracial states on alg( a , a ) and alg( b , b ) .Proof. By the combinatorial theory of bi-free independence (see [15]) it suffices toprove the following: for all n ∈ N , for all non-constant χ : { , . . . , n } → { ℓ, r } , andfor all k ∈ { , } we have κ χ ( c , . . . , c n ) = 0where c m = a k if χ ( m ) = ℓ and c m = b k if χ ( m ) = r . We will only verify the abovewhen k = 1 as the case k = 2 follows by symmetry. We proceed by induction on n .As there does not exists α, β ∈ R such that ( ϕ ( a n ) , ϕ ( b n )) = ( α n , β n )for all n ∈ N , there exists n , n ∈ N such that ϕ ( a n + n ) = ϕ ( a n ) ϕ ( a n ) or ϕ ( b n + n ) = ϕ ( b n ) ϕ ( b n ). We will assume that ϕ ( a n + n ) = ϕ ( a n ) ϕ ( a n ) as theother case will follow by similar arguments. NALYTIC SUBORDINATION FOR BI-FREE CONVOLUTION 33
The case n = 1 is trivial so we begin with the case n = 2. Here( χ (1) , χ (2)) ∈ { ( ℓ, r ) , ( r, ℓ ) } . By bi-freeness and traciality, we know that ϕ ( a n + n ) ϕ ( a b ) = ϕ ( a n + n a b )= ϕ ( a n a b a n )= ϕ ( a n + n ) ϕ ( a ) ϕ ( b ) + ϕ ( a n ) ϕ ( a n ) κ ( ℓ,r ) ( a , b ) . Thus, as κ ( ℓ,r ) ( a , b ) = ϕ ( a b ) − ϕ ( a ) ϕ ( b )we obtain that ϕ ( a n + n ) κ ( ℓ,r ) ( a , b ) = ϕ ( a n ) ϕ ( a n ) κ ( ℓ,r ) ( a , b ) . As ϕ ( a n + n ) = ϕ ( a n ) ϕ ( a n ), this implies κ ( ℓ,r ) ( a , b ) = 0. Similarly, ϕ ( a n + n ) ϕ ( b a ) = ϕ ( a n + n b a )= ϕ ( a n b a a n )= ϕ ( a n + n ) ϕ ( a ) ϕ ( b ) + ϕ ( a n ) ϕ ( a n ) κ ( r,ℓ ) ( b , a ) . Thus the same argument implies κ ( r,ℓ ) ( b , a ) = 0.For the inductive step, suppose we have verified the result for n − n ∈ N . Let χ : { , . . . , n } → { ℓ, r } be non-constant and let where c m = a if χ ( m ) = ℓ and c m = b if χ ( m ) = r . Let m = | χ − ( { ℓ } ) | and let m = | χ − ( { r } ) | .By bi-freeness and traciality, we know that ϕ ( a n + n ) ϕ ( c · · · c n ) = ϕ ( a n + n c · · · c n )= ϕ ( a n c · · · c n a n )= ϕ ( a n + n ) ϕ ( a m ) ϕ ( b m ) + ϕ ( a n ) ϕ ( a n ) κ χ ( c , . . . , c n )(where we have used the induction hypothesis to deduce any cumulant involving a and b of length at most n − ϕ ( c · · · c n ) = ϕ ( a m ) ϕ ( b m ) + κ χ ( c , . . . , c n )(where we have used the induction hypothesis to deduce any cumulant involving a and b of length at most n − ϕ ( a n + n ) κ χ ( c , . . . , c n ) = ϕ ( a n ) ϕ ( a n ) κ χ ( c , . . . , c n ) . As ϕ ( a n + n ) = ϕ ( a n ) ϕ ( a n ), this implies κ χ ( c , . . . , c n ) = 0. Hence the resultfollows. (cid:3) References [1] G. W. Anderson,
Convergence of the largest singular value of a polynomial in independentWigner matrices . Ann. Probab. (2013), no. 3B, 2103-2181.[2] M. Anshelevich, S.T. Belinschi, M. F´evrier, and A. Nica, Convolution powers in the operator-valued framework . Trans. Amer. Math. Soc. (2013), no. 4, 2063–2097.[3] S.T. Belinschi,
The Lebesgue decomposition of the free additive convolution of two probabilitydistributions . Probab. Theory Related Fields (1–2), 125–150 (2008).[4] S.T. Belinschi,
C-free convolution for measures with unbounded support , in: Von Neumannalgebras in Sibiu, Theta Ser. Adv. Math., (2008), Theta, Bucharest, 1–7.[5] S.T. Belinschi, L ∞ -boundedness of density for free additive convolutions , Rev. RoumaineMath. Pures Appl. (2), 173–184 (2014).[6] S.T. Belinschi, A noncommutative version of the Julia-Wolff-Carath´eodory Theorem . J. Lon-don Math. Soc. (2) (2017), no. 2, 541–566. [7] S.T. Belinschi and H. Bercovici, A new approach to subordination results in free probability .J. Anal. Math. (2007), 357–365.[8] S.T. Belinschi and H. Bercovici,
Atoms and regularity for measures in a partially defined freeconvolution semigroup . Math. Z. (2004), no. 4, 665–674.[9] S.T. Belinschi, T. Mai, and R. Speicher,
Analytic subordination theory of operator-valued freeadditive convolution and the solution of a general random matrix problem.
Journal f¨ur diereine und angewandte Mathematik, (2017), 21–53.[10] S.T. Belinschi, M. Popa, and V. Vinnikov,
Infinite divisibility and a non-commutativeBoolean-to-free Bercovici-Pata bijection . J. Funct. Anal. (2012), no. 1, 94–123.[11] H. Bercovici and D.V. Voiculescu,
Regularity questions for free convolution.
Nonselfadjointoperator algebras, operator theory, and related topics, 37–47, Oper. Theory Adv. Appl. 104,Birkh¨auser, Basel, 1998.[12] H. Bercovici and D.V. Voiculescu,
Superconvergence to the central limit and failure of theCram´er theorem for free random variables.
Probab. Theory Related Fields (1995), no.2, 215–222.[13] P. Biane,
Processes with free increments . Math. Z. (1998), 143–174.[14] M. Bo˙zejko, M. Leinert, and R. Speicher,
Convolution and limit theorems for conditionallyfree random variables . Pacific J. Math. (1996), no. 2, 357–388.[15] I. Charlesworth, B. Nelson, and P. Skoufranis,
On Two-Faced Families of Non-CommutativeRandom Variables . Canad. J. Math. (2015), no. 6, 1290–1325.[16] E.M. Chirka, Complex Analytic Sets. (Translated from Russian by R.A.M. Hoksbergen.)Kluwer Academic Publishers 1989, Dordrecht.[17] Y. Gu, T. Hasebe, and P. Skoufranis, Bi-monotonic independence for pairs of algebras .Preprint (2017), arXiv:1708.05334v1 [math.OA].[18] Y. Gu, H-W. Huang and J. Mingo,
An analogue of the L´evy-Hinˇcin formula for bi-freeinfinitely divisible distributions . Indiana Univ. Math. J. (2016), 1795–1831.[19] Y. Gu and P. Skoufranis, Conditionally bi-free independence for pairs of faces . J. Funct.Anal. (2017), no. 5, 1663–1733.[20] Y. Gu and P. Skoufranis,
Bi-Boolean independence for pairs of algebras . Preprint (2017),arXiv:1703.03072v1 [math.OA].[21] A. Nica and R. Speicher,
On the multiplication of free N -tuples of noncommutative randomvariables. Amer. J. Math. (1996), no. 4, 799–837.[22] M. Popa and V. Vinnikov,
Non-commutative functions and the non-commutative free L´evy-Hinˇcin formula . Adv. Math. (2013), 131–157.[23] P. Skoufranis,
Independences and Partial R -Transforms in Bi-Free Probability . Ann. Inst.Henri Poincar´e Probab. Stat. (2016), no. 3, 1437–1473.[24] P. Skoufranis, On operator-valued bi-free distributions . Adv. Math. (2016), 638–715.[25] R. Speicher and R. Woroudi,
Boolean convolution , in: Free Probability Theory, Fields Inst.Commun., (1997), 267–279.[26] D.V. Voiculescu, Addition of certain non-commutative random variables , J. Funct. Anal. (1986), 323–346.[27] D.V. Voiculescu, The analogues of entropy and of Fisher’s information measure in free prob-ability theory. I . Comm. Math. Phys. (1993), 411–440.[28] D.V. Voiculescu,
Operations on certain non-commutative operator-valued random variables .Ast´erisque (1995), 243–275.[29] D.V. Voiculescu,
The coalgebra of the free difference quotient and free probability . Internat.Math. Res. Notices (2000), , 79–106.[30] D.V. Voiculescu, Analytic subordination consequences of free Markovianity . Indiana Univ.Math. J. (2002), 1161–1166.[31] D.V. Voiculescu, Free Analysis Questions I: Duality Transform for the Coalgebra of ∂ X : B Internat. Math. Res. Not. (2004), no. 16, 793–822.[32] D.V. Voiculescu,
Free analysis questions II : The Grassmannian completion and the seriesexpansions at the origin , J. reine angew. Math. (2010), 155–236.[33] D.V. Voiculescu, Free probability for pairs of faces I.
Comm. Math. Phys. (2014), no. 3,955–980.[34] D.V. Voiculescu,
Free probability for pairs of faces II: 2-variables bi-free partial R-transformand systems with rank ≤ commutation . Ann. Inst. Henri Poincar´e Probab. Stat. (2016),No. 1, 1–15. NALYTIC SUBORDINATION FOR BI-FREE CONVOLUTION 35 [35] D.V. Voiculescu,
Free probability for pairs of faces IV: Bi-free Extremes in the Plane . J.Theor. Probab. (2017) no. 1, 222–240. CNRS - Institut de Math´ematiques de Toulouse, 118 Route de Narbonne,31062 Toulouse, France
E-mail address : [email protected] Department of Mathematics, Indiana University, 831 E. Third St., Blooming-ton, IN 47405, USA
E-mail address : [email protected] Department of Mathematics and Statistics, Queen’s University, Jeffery Hall,48 University Ave. Kingston, ON K7L 3N6, Canada
E-mail address : [email protected] Department of Mathematics and Statistics York University, N520 Ross 4700Keele Street, Toronto, ON M3J 1P3, Canada
E-mail address ::