Free transport for convex potentials

We construct non-commutative analogs of transport maps among free Gibbs state satisfying a certain convexity condition. Unlike previous constructions, our approach is non-perturbative in nature and thus can be used to construct transport maps between free Gibbs states associated to potentials which are far from quadratic, i.e., states which are far from the semicircle law. An essential technical ingredient in our approach is the extension of free stochastic analysis to non-commutative spaces of functions based on the Haagerup tensor product.


Introduction
A transport map between two probability measures is a function pushing the first measure onto the second. Finding transport maps which minimize a certain cost function is the central question in transportation theory. It was formalized by Monge in the eighteenth century, studied by Kantorovich during World War II and has known major advances in the last twenty years, starting with a work of Brenier [Bre91], see also the very inspiring book by Villani [Vil03]. In fact, the mere existence of a transport map is itself not completely trivial and was shown by von Neumann in 1930s, under very weak assumptions, as part of the program to classify measure spaces.
A central question is to find appropriate generalizations of this result to the noncommutative setting, where measures are replaced by non-commutative distributions, that is, tracial states. In this case, there is no notion of density but in certain instances arising in Voiculescu's free probability theory, integration by parts makes sense. It gives the adjoint in L 2 of Voiculescu's free difference quotient [Voi98], and is often a (cyclic) derivative of a non-commutative function that we call potential.
Non commutative laws which are characterized by such an integration by parts formula are called free Gibbs laws. In [GS12], two of the authors of this article constructed transport maps between a class of free Gibbs laws. They used ideas going back to Monge and Ampère, based on the remark that transport maps must satisfy an equation given by the change of variables formula. Solving this equation yields a transport map. Unfortunately, this equation was only solved in [GS12] in the case of potentials which are small perturbations of quadratic potentials, i.e., certain small perturbations of Voiculescu's free semicircular law. However, already this result yielded isomorphisms between the associated C * and von Neumann algebras in such perturbative situations, solving a number of open questions [Voi06]. In particular, this approach was used to show that the C * and von Neumann algebras of q-Gaussian laws [BS91] are isomorphic for sufficiently small values of q.
The goal of the present article is to consider non-perturbative situations. We will see that we can tackle situations where the potential is "strictly convex" (in a sense we will make precise later in the paper). The idea is once again to use a non-commutative version of the Monge-Ampère equation, but to solve it by interpolating the potential between the two given laws. This requires to solve a Poisson type equation. The latter, in strictly convex situations, can be solved by using the associated (free) semi-group. However, this program meets several difficulties in the non-commutative setting. First, smoothness properties of the semi-group were so far not studied. Furthermore, the appropriate notion of convexity has not yet been formulated. We detail our framework in Section 2, leaving to the appendix the elaboration of most of its properties. In Section 3, we study the semi-group defined in this framework and derive its properties. Based on this, we finally construct the transport map in Section 4.
In the rest of this section, we detail the classical construction of transport maps from which we took our inspiration, and explain how it generalizes to the case of a single noncommutative variable. We then consider the general non-commutative multi-variable case and state our main theorem.
1.1. Classical construction of transport maps. For any suitable real-valued function U from R d to R we define the probability measure We let V and V + W be two functions going fast enough to infinity so that Z V and Z V +W are finite. We would like to construct F : R d → R d so that µ V +W = F #µ V , i.e., so that for all test functions h h(F (y))dµ V (y) = h(x)dµ W (x) = h(F (y))Jac(F )(y)e −(V +W )(F (y)) dy/Z W where Jac(F ) denotes the Jacobian of F . We have simply performed the change of variables x = F (y) in the last line, assuming that F is C 1 . We therefore deduce that F should satisfy the transport equation: (1) V (y) = (V + W )(F (y)) − ln Jac(F )(y) + C for almost all y where we set C = ln Z V +W − ln Z V . If V − W is small we can seek a solution F which is close to identity, so that its Jacobian stays away from the zero and therefore does not get close to the singularity of the logarithm. The resulting equation can in turn be solved by the implicit function theorem. Such arguments were extended to the non-commutative setting in [GS12].
To solve the transport equation in a non-perturbative situation, we shall in this article proceed by interpolating the potential. Namely, let us consider potentials V α = αW + V and seek to construct a transport map F α of µ V onto µ Vα . The advantage of smooth interpolation is that transporting µ Vα onto µ V α+ε can a priori be solved for ε small enough by the previous pertubative arguments, and the full transport F 1 = F of µ V onto µ W can then be recovered by integration along the interpolation.
In fact, we shall solve the transport equation (1) under the additional restriction that F evolves according to a gradient flow: ∂ α F α = ∇g α (F α ). It turns out that g must then be a solution of the Poisson equation (2) L Vα g α = W + ∂ α ln Z Vα , with L Vα = ∆−∇V α .∇ the infinitesimal generator of the diffusion having µ Vα as its stationary measure. Solving the Poisson equation (2) amounts to inverting L Vα , that is, finding the Green function of the differential operator L Vα . This is a well known problem which can be solved under various boundary conditions or growth of V at infinity. To simplify we shall assume that V α (that is V and V + W ) are uniformly convex. This insures that the semigroup P α s = e sL Vα converges uniformly towards the Gibbs measure µ Vα as s goes to infinity.
More precisely, there exists some c > 0 such that for all Lipschitz functions f with bounded Lipschitz norm f L we have P α s f − µ Vα (f ) ∞ ≤ 2e −cs f L . As a consequence we can solve the Poisson equation (2) by setting where we noticed that µ Vα (W + ∂ α ln Z Vα ) = 0. Hence we see that in the classical setup (2) can be solved thanks to the associated semi-group. Moreover, by smoothness of x → P α s (W )(x), we see that g α is smooth if W is. To conclude, all that remains is to solve the transport equation ∂ α F α = ∇g α (F α ). In the rest of this article we generalize this strategy to the free probability framework.
Let us first investigate the free set-up in the one variable case. Typically, one should think about the non-commutative law of one variable as the asymptotic spectral measure of a random matrix, confined by a potential V : the joint law of these eigenvalues is given by It is then well known (see e.g. [AGZ10]) that the spectral measure L N = 1 N N i=1 δ λ i converges almost surely to the equilibrium measure µ V , which is characterized by the fact that the function (4) V (x) − 2 ln |x − y|dµ V (y) is equal to a constant c V on the support of µ V and is greater than this constant outside of the support. This equation implies the Schwinger-Dyson equation where P.V. denotes the principal value. We will call a free Gibbs law with potential V a solution to (5). It may not be unique; in fact, there is a continuum of solutions as soon as solutions have disconnected support: a solution corresponds to any choice of masses of the connected pieces of the support. This is not the case when V is uniformly convex. In this case, there is a unique solution and it has connected support. The interest in Schwinger-Dyson equation is that it can be interpreted as an integration by parts identity for the non-commutative derivative ∂f (x, y) := f (x)−f (y) x−y since it implies that As there is no notion of density in free probability, integration by parts can be seen as an important way to classify measures. Moreover, as we shall soon describe, there is a natural generalization of free Gibbs laws to the multi-variable setting. Let now V, W be two potentials. We would like to construct a transport map from the Gibbs law µ V with potential V to the Gibbs law µ V +W with potential V + W . We can follow the previous scheme and seek g α satisfying : ∂ α F α = g ′ α (F α ) and F α #µ V = µ Vα . By (4), we find that µ Vα almost surely we must have We recognize on the left hand side the infinitesimal generator ∆ Vα of the free diffusion driven by a free Brownian motion, [BS98]. More precisely, the infinitesimal generator of the free diffusion is given by if X has the same law as x.
The fact that this generator depends on the law of the variable complicates the resulting theory quite a lot. In particular, the operators e s∆ Vα acting on the obvious space of functions do not form a semigroup. To restore the semi-group property, we have to enlarge the set of test functions to be functions of not just the real variable x, but also of expectations of this random variable. Our idea here is similar to the one introduced in [Ceb13]. This in turn changes the generator of the diffusion to also involve differentiation under the expectation: We can now check that (e s(∆ Vα +δ Vα ) ) s≥0 is a semi-group so that we can apply the previous analysis.
As before, we shall solve (6) under a gradient form. Again, the natural gradient that we shall use also differentiates under expectation. Namely we let D to be given for any smooth functions f, f i , i ≥ 0 by Then, we shall find a function Dg α (of the variable x and the expectation, see Lemma 14), which satisfies a gradient form of (6) (after adding δ V to the generator and commuting D with ∆ Vα + δ Vα ) : Having obtained the solution g α , we finally solve To make things clearer, let us transport the measure P V N onto P W N and only afterwards take the large N-limit. Again, we consider the transport of P V N onto P Vα N . We may expect, by symmetry, the flow F α = (F α 1 , . . . , F α N ) for the transport map to be the gradient of a function of the empirical measure L N = 1 N δ λ N i : The infinitesimal generator L V = ∆ − ∇V.∇ acting on functions of the form F ( where the last term comes from differentiation of two different functions and is at most of order 1/N. Hence, when N goes to infinity we see that functions of the distribution of the λ i should not be taken as constant but also differentiated under the expectation. Taking the gradient in the Poisson equation (2) shows that we seek G α such that for each i Hence, taking the large N limit, we expect G α to be given at first order by the solution g α of (7). The final step to finish our construction of the transport map is to introduce a notion of uniform convexity of V such that the associated semi-group converges uniformly and sufficiently rapidly towards the invariant measure as time goes to infinity (to make sense of the integral over time from 0 to ∞), and such that if f is smooth then also x → e s(∆ V t +δ V t ) f (x) is smooth, uniformly in s (to be able to solve the transport equation). Our choice of the notion of uniform convexity of V is designed to guarantee such properties.
1.2. Construction of transport maps in free probability. We now want to explain our approach to the main goal of this article, which is to construct transport maps between non-commutative distributions of several non-commutative variables. In free probability theory, laws of non-commutative variables are defined as linear forms τ on the space C X 1 , . . . , X n of polynomials in the self-adjoint non-commutative letters X 1 , . . . , X n with coefficients in C which have mass one (so that τ (1) = 1), and which satisfy the traciality property (τ (P Q) = τ (QP )) and the state property (τ (P P * ) ≥ 0). Here * denotes the usual involution (zX i 1 · · · X i k ) * =zX i k · · · X i 1 .
An example one should keep in mind is the asymptotic law of several interacting random matrices with joint law given by dP V N (X N 1 , . . . , X N n ) = 1 Z V N exp{−NTr(V (X N 1 , . . . , X N n ))}dX N 1 · · · dX N n where dX N is the Lebesgue measure on the space of N × N Hermitian matrices and V is a self-adjoint polynomial in C X 1 , . . . , X n so that Z V N is finite. In this case τ X N (P ) = 1 N Tr(P (X N 1 , . . . , X N n )) is a non commutative law for any self-adjoint matrices X N 1 , . . . , X N n . So is its expectation under P V N and the limit of these expected value as N → ∞ (if the limit exists). Existence of such an (almost sure and L 1 (P V N )) limit was proven when V is a small perturbation of a quadratic potential [GMS06] and when V satisfies some property of convexity [GS09].
In this paper we will introduce a more suitable notion of convexity yielding as well existence and uniqueness of a limit τ V . We shall see that it includes the case of quartic potentials. By integration by parts, we see that the limit τ V must satisfy that for any polynomial P where ∂ i is the free difference quotient with respect to the ith derivative from C X 1 , . . . , X n to C X 1 , . . . , X n ⊗ C X 1 , . . . , X n given by and D i = m • ∂ i the cyclic derivative, m(a × b) = ba. When V = n i=1 X 2 i , σ n := τ n i=1 X 2 i is uniquely given recursively by (9) and is the law of n free semicircle variables. In general, we say that a non-commutative law τ V satisfying (9) is a free Gibbs law with potential V . Alternatively we say that the conjugate variables (∂ * i (1 ⊗ 1)) 1≤i≤n are equal to the cyclic gradient (D i V ) 1≤i≤n .
The goal of this paper is to construct non-commutative transport maps between τ V and σ n , following the ideas developed in the previous section. In fact, constructing the transport map as the solution of the transport equation (8) where g α is solution of a Poisson equation (7) is a natural analogue thanks to existence of free diffusion and free semi-groups. However, this program meets several issues that have to be addressed.
• One of the key point to construct the solution to Poisson equation was the fast convergence of the semi-group towards the free Gibbs law. In the free context, it is well known that semi-groups with deep double well potentials do not always converge. It is therefore natural to search for the appropriate notion of convexity in the noncommutative setting, which would imply convergence of the semi-group as time goes to infinity, uniformly on the initial condition. In [GS09], the notion of convexity that was used turns out to be too strong to include many examples. It assumed that for all n-tuples of self-adjoint variables (X, Y ) bounded by some R, is non-negative. This is not satisfied by V (X) = X 4 as can be checked by taking (X, Y ) to be two 2 × 2 matrices given by X 11 = 1, X 12 = X 21 = 0, X 22 = −6, Y 11 = 1, Y 12 = Y 21 = √ 11/4, Y 22 = −5. It would be more natural to assume that the Hessian of TrV (X N 1 , . . . , X N n ) is bounded below for any n-tuple of Hermitian matrices X N 1 , . . . , X N n . However, this Hessian lives in a tensor product space and saying that it is non-negative depends on the topology with which we equip the tensor product. We shall see that a good topology is given by the extended Haagerup tensor product and prove that our definition includes the case of quartic potentials.
• As in the one variable case, we have to consider functions not only of the variables but also of the expectation and the semi-group must also differentiate under expectation. Hence, we have to develop free stochastic calculus applied to such functions. • The solution of the Poisson equation is given in terms of the semi-group, and we need to show existence and smoothness of the transport maps which are the solution of the transport equation driven by this solution. This requires us to show that the semi-group acts smoothly on appropriate spaces of non-commutative functions, and also understand its image under the cyclic gradient.
We next state our result. In Section 2.4 we define several differential operators acting on functions of several non-commutative variables, some of them being well known in free probability, such as the difference quotient and the cyclic gradient. We extend their definition to functions which also depend on expectations, in order to define a proper semi-group on the appropriate function spaces. We then define the notion of (c, R) h-convexity of a function in Definition 3. It states that the Hessian of this function is bounded below by cI in the extended Haagerup tensor product, uniformly when evaluated on non-commutative variables bounded by R. An important point is that this notion is stable under addition. We then show in Proposition 5 that the free SDE with strictly h-convex potential converges as time goes to infinity towards a free Gibbs law. To construct the transport map between τ V and σ n , we shall need an additional technical assumption. First, as we proceed by interpolation of the potential, we need to assume that a nice bounded free Gibbs law exists for all potentials . This is the content of Assumption 16. We are now in position to state one of our main theorems, see Corollary 17 (with B = D = C and • For any α ∈ [0, 1], the von Neumann algebras associated to the free Gibbs law with potential V α are isomorphic; in particular, they are isomorphic to the von Neumann algebra generated by n free semicircular variables.
In the appendix, see Corollary 52, we show that the following perturbation of quartic potentials V satisfy all our hypotheses: cyclic variant which allows the action of cyclic permutations on these functions, space on which the cyclic gradient acts. Indeed, this gradient appears in the right hand side of the Dyson-Schwinger equation (9) and the non-commutative version of the transport equation (8), and is therefore key to our analysis. Our main result in this general situation is stated in Corollary 17. Our motivation for this generalization is two-fold. The first is to consider the crossed product F n ⋉ D of an action of the free group on D, as well as its q-deformation [JLU14]. At this point we did not verify that these deformations correspond to potentials that satisfy our assumptions (for q small enough). The motivation to also consider the algebra B comes from the analysis of the free product (Γ ⋉ D) * D (W * (S s , s ≤ t) ⊗ D): then B = Γ ⋉ D. Being able to construct transport maps in this setting would allow to construct solutions of free SDE's with initial conditions in B as the image by transport maps of some process S t 1 , . . . , S tn . For instance, one would want to obtain solutions of free SDE's similar to those considered in [Shl09] in the context of crossed product and for non-algebraic cocycles. Building such solutions in free products with amalgamation could enable the use of techniques similar to those in [DI16,Io15] and would lead to the study of algebras B by a free transport approach, for instance to answer questions such as uniqueness of Cartan decomposition up to unitary conjuguacy for non-trivial actions when Γ is a group with positive first ℓ 2 Betti number. Such interesting applications would thus require to consider non smooth potentials V , something which is still far from our reach. However, we feel that these potential applications outweigh the small additional difficulties involved in considering the more complex setting with nontrivial algebras B and D. Thus our article lays the groundwork for future developments in this direction and our main example of relative algebra B is exactly the kind of crossedproduct that could be interesting for the above-mentioned potential applications.
Acknowledgements. The authors would like to acknowledge the hospitality of the Focus Program on Noncommutative Distributions in Free Probability Theory held at the Fields institute in July 2013 where an early part of this work has been completed. We are also grateful to the Oberwolfach Workshop on Free Probability Theory held in June 2015 during which we were able to make further progress.

Definitions and framework
2.1. Spaces of analytic functions. We denote by M(X 1 , . . . , X n ) the set of monomials in X 1 , . . . , X n . Throughout this paper, B will denote a finite von Neumann algebra, and D a von Neumann subalgebra.
The extended Haagerup tensor product relative to D is denoted by For R > 0, we define formally Here R |m| E means the space E with standard norm multiplied by R |m| . This space can be regarded as the space of power series in X 1 , . . . , X n with coefficients in B and radius of convergence at least R by identifying a monomial b 0 X i 1 b 1 · · · X ip b p with the copy of the tensor b 0 ⊗· · ·⊗b p indexed by the monomial m = X i 1 · · · X ip . The definition of the Haagerup tensor product eh ⊗ D are discussed in section 1.2 and Lemma 5 of [Dab15] (see also [P, chaper 5], [M97, M05] for the general module case). The above definition requires a direct sum of D-modules in order that B X 1 , ..., X n : D, R eh ⊗ D B X 1 , ..., X n : D, R is well defined. Modulo this (important) property, we could have more simply considered the (ordinary operator space) ℓ 1 = ℓ 1 C direct sum (cf. [P, section 2.6]): we denote B X 1 , ..., X n : D, R, C the corresponding smaller space. We will only use this sum in the cyclic case.
Its cyclic variant B c X 1 , ..., X n : D, R, C is given by: where D ′ is the commutant of D and eh ⊗ D,c stands for the cyclic version of Haagerup tensor product defined in subsection 5.3. This space can be regarded as the space of power series in X 1 , . . . , X n with coefficients in B and radius of convergence at least R, and such that variables X j commute with D. As before, a monomial b 0 X i 1 b 1 · · · X ip b p is identified with the copy of the tensor b 0 ⊗ · · · ⊗ b p indexed by the monomial m = X i 1 · · · X ip . The use of the Haagerup tensor product eh ⊗ D,c ensures the possibility of cyclic permutation of various terms in the power series. C p+1 denotes the group of cyclic permutations acting on the cyclic tensor product, with generator ρ(b 0 ⊗ · · · ⊗ b p ) = b p ⊗ b 0 · · · ⊗ b p−1 . We will define in subsection 6.1 the cyclic gradient: it is roughly speaking a linear map on this space. We also define the analogue B c X 1 , ..., X n : D, R, C of B X 1 , ..., X n : D, R, C . B c X 1 , ..., X n : D, R, C and B X 1 , ..., X n : D, R are Banach algebras, see [Dab15, Theorem 39] and subsection 6.1.
Having those operadic compositions, which will be crucial for non-commutative calculus, is another reason for using variants of Haagerup tensor products. The reader should note ). We will also use the restriction to cyclic variants as defined in subsection 5.3: We will denote in short # for # 1 . We may also write for instance .#(., .) : We endow A eh ⊗ D,c 2 with the adjunction * so that (a⊗b) * = a * ⊗b * . Note that (a#b) * = b * #a * , so that (A eh ⊗ D,c 2 , * ) is a * -algebra.

2.2.
Spaces of analytic functions with expectations. We will need a generalization of analytic functions enabling functions of the conditional expectation E D on D. For example, we would like to consider functions of the type As the order in which conditional expectations are applied matters, we will label such a monomial by inserting an additional letter Y for each closing and opening parenthesis of the map. The matching between the closing and opening parenthesis then defines a non-crossing pair partitions of the set of positions of the letter Y . Conversely, given a non-commuting monomial in letters X 1 , . . . , X n and Y having even degree 2k in Y , and a non-crossing pair partition of the positions of the letter Y , we can define a unique expression of the type above. Thus, formally we set where M 2k (X 1 , ..., X n , Y ) is the set of non-commuting monomial in letters X 1 , . . . , X n and Y having even degree 2k in Y , |m| X denotes the degree in the letter X 1 , . . . , X n of m and |m| = |m| X + 2k. We call B k {X 1 , ..., X n : E D , R, C} the corresponding space with (nonmodule) operator space ℓ 1 sums (in the sense of [P, section 2.6]). Similarly, we define ; m ∈ M 2k (X 1 , ..., X n , Y ), π ∈ NC 2 (2k), |m| ≥ 1 .
We set B c,0 {X 1 , ..., X n : E D , R, C} = B c X 1 , ..., X n : D, R, C and B 0 {X 1 , ..., X n : E D , R} = B X 1 , ..., X n : D, R . Finally we define: Above, E D should be considered as a variable taken in the space of D-bilinear completely bounded maps. For P ∈ B c {X 1 , ..., X n : E D , R, C} and E : B X 1 , ..., X n : D, R → D unital Dbilinear completely-bounded map, we can define the map P → P (E) taking B c {X 1 , ..., X n : E D , R, C} to B X 1 , ..., X n : D, R by recursively replacing each sub-monomial E D (Q), Q ∈ B X 1 , ..., X n : D, R inside P by E(Q). A formal definition is explained in subsection 6.2 where all the technical Lemmas we will need about those analytic functions are proved.
2.3. Spaces of differentiable functions. Let A be a finite von Neumann algebra, B ⊂ A a von Neumann subalgebra. Set R be a closed subset of A n R . For convenience, we will first embed the algebra B c X 1 , ..., X n : D, R, C into a much larger algebra stands for the space of bounded continuous functions on U with values in a Banach space B. On this space we define the norm where by P (X 1 , . . . , X n ) we mean the value of P evaluated at (X 1 , . . . , X n ) ∈ U, itself evaluated as a power series in (X 1 , . . . , X n ) (see Proposition 29 for some details on those evaluations). We call the corresponding completion C * u (A, U : B, D) and C * u (A, R : B, D) .., X n : D, S, C ) the set of continuously differentiable functions on A n R with bounded first derivative, one can consider the differential ) should be thought of as a tangent space of A n R . As usual, one writes for X ∈ A n R and H ∈ D ′ ∩ A n sa , D H P (X) = dP (X).H and we see that (D H P : .., X n : D, S, C )). Likewise for .., X n : D, S, C ) an element of the set of k times coefficientwise continuously differentiable functions on A n R with bounded first k-th order differentials, one can consider the k-th order differential .., X n : D, S, C )). Hereø denotes the projective tensor product.
In this case D K D k−1 H P (X) = d k P (X).(K, H, ..., H) and .., X n : D, S, C ). We show in Proposition 30 that on B X 1 , ..., X n : D, R , the i-th free difference quotient is defined and is a canonical derivation satisfying ∂ i (X j ) = δ i=j 1ø1, ∂ i (b) = 0. This can be extended to B{X 1 , ..., X n : E D , R} by putting ∂ i • E D = 0.
We denote in short ∂ k (i 1 ,...,i k ) : B X 1 , ..., X n : D, R → B X 1 , ..., X n : Recall that D H stands for the directional derivative of a function in C 1 u (A, U : B, E D ), viewed as a function from U to the space of power series B c X 1 , ..., X n : D, R, C . However, this won't be the most convenient differential, since the non-commutative power series part will always be evaluated at the same X ∈ U and we will rather need the full differential which uses also the free difference quotient on the powers series part.
On the space of continuous differentiable functions C 1 (U, A) from U to A, denote by D X H the derivative in the direction H ∈ A n . Consider the map η : .., X n : D, S, C ) by η(P ) = (P (X))(X). Then one has We let d X be the differential associated with D X H . We will also write: We will consider the (separation) completion of with respect to the seminorms for (k, l) ∈ IN 2 given by This seminorm controls k free difference quotients and l full differentials. We will denote these (separation) completions by C k,l u (A, U : B, D), and C k,l u (A, R : B, D) when U = A n R . Note that the above map D p H , for p ≤ l extends continuously to a map . When in the definition of . k,X we replace .

13
Note that this require a supplementary assumption that U ⊂ A n R,U ltraApp where A n R,U ltraApp is defined before Proposition 32: this assumption is necessary to define evaluation into cyclic tensor products. This is crucial to see that the image of cyclic analytic functions by the free difference quotient belongs to the cyclic Haagerup tensor product, see also Proposition 30. More precisely we define A n R,U ltraApp the set of X 1 , ..., X n ∈ A, X i = X * i , [X i , D] = 0, ||X i || ≤ R and such that B, X 1 , ..., X n is the limit in E D -law (for the * -strong convergence of D) of variables in B c X 1 , ...X m : D, 2, C (S 1 , ..., S m ) with S i free semicircular variables over D. We will thus always assume U ⊂ A n R,U ltraApp when we deal with spaces with index c. Note that consistently, we will write C k c (A, R : B, D) when U = A n R,U ltraApp . For convenience later in writing estimates valid when there is at least one derivative, we also introduce a seminorm We next define differentiable functions depending on conditional expectations. Using the conditional expectation E D : A → D, we can define a completely bounded map E D,X : B X 1 , ..., X n : D, S → D by sending P to E D (P (X 1 , . . . , X n )), for any S > R.
We denote by Let H ∈ A n . Recall that D H stands for the directional derivative of a function in C 1 u (A, U : B, E D ), viewed as a function from U to the space of power series B c X 1 , ..., X n : D, R, C . Given P ∈ B c {X 1 , ..., X n : E D , R, C} a monomial involving E D , we note that D H (ω(P ) amounts to replacing each sub-monomial of the form E D (Q) with Q ∈ B c X 1 , ..., X n : ). In other words, D H corresponds to "differentiation under E D ". 14 2.4. Differential operators. For p, P ∈ B c {X 1 , ..., X n : E D , S, C}, we define recursively the cyclic gradient (D i,p (P ), 1 ≤ i ≤ n) by D i,p (X j ) = 1 j=i p, For instance, one computes D 1,p (X 2 E D (X 1 bX 2 )X 1 ) = pX 2 E D (X 1 bX 2 ) + bX 2 E D (X 1 pX 2 ). Moreover, observe that for polynomials P in {X 1 , . . . , X n }, We denote in short D i = D i,1 . Its restriction to polynomials in {X 1 , . . . , X n } corresponds to the usual cyclic derivative. We consider a flat Laplacian defined for P ∈ B{X 1 , ..., X n : We define δ ∆ a derivation on B{X 1 , ..., X n : E D , R} by requiring that it vanishes on B X 1 , ..., X n : D, R and satisfies Likewise, for any V ∈ B X 1 , ..., X n : D, R , the map produces a map δ V such that δ V (P ) = 0, for P ∈ B X 1 , ..., X n : D, R . Moreover, δ V is a derivation and for Q monomial in B{X 1 , ..., X n : E D , R}, δ V extends to B{X 1 , ..., X n : E D , R} (see Proposition 35). Moreover, we have for any g ∈ B c {X 1 , ..., X n : E D , R, C}, We extend ∆ V and δ V to V ∈ C 3 c (A, 2R : B, E D ) by adding the variables Z i to be evaluated at D i V (X), letting V 0 (Z) = 1 2 Z 2 i and setting for P ∈ B{X 1 , ..., X n : E D , R} (15) ∆ V (P )(E D,X )(X) := ∆ V 0 (Z) (P ) E D,X,DV (X) (X, DV (X)) .
∆ V (P ) belongs to C * tr (A, U). The extension of δ V is similar. We define, C k,l tr,V (A, U : .., X n ; E D , R} for the semi-norm (with ω(P ) = (X → P (E X,D ))): where (X) 1 denotes the unit ball around 0 of the normed space X. We also define a first order part seminorm ||P || C k,l tr,V (A,U :B,E D ),≥1 by replacing the first term in the sum with ω(P ) k,l,U ≥1 .
We also define the space C k,l tr,V,c (A, U : B, E D ) in the same way as before but considering everywhere cyclic extended-Haagerup tensor products.
To sum up we have introduced the following spaces where ⊂ means the existence of a canonical injective mapping, whereas → means the existence of a canonical map (with conditions written in index). We shall not discuss these mappings as we will not use them and leave the reader check them.
2.5. Free brownian motion. (S i t , t ≥ 0, 1 ≤ i ≤ n) will denote n free Brownian motions. Let U ⊂ A n R . We denote by * D the free product with amalgamation over D: see [VDN92] for a definition as well as for a definition of freeness with amalgamation over D.  .
We first recall a consequence of Hille-Yosida Theorem.
Proposition 2. The following are equivalent. ( ) has a semigroup of contraction e −tQ , ) has a resolvent family for all α > 0, α + Q is invertible in In this case we say Q ≥ 0.
Proof. We apply Hille-Yosida Theorem e.g. in the form of Theorem 1.12 in [MR], to each ) is a cone. Indeed, if α ≥ 0 and Q ≥ 0, clearly αQ ≥ 0. Moreover, Q ≥ 0 andQ ≥ 0 implies that Q +Q ≥ 0. Indeed, as Q andQ are bounded, they are defined everywhere as well as Q +Q, and one can use [T59] to see that is a contraction as the right hand side is. Moreover, this set is closed as follows easily from the characterization (2) (notice here that the set R,U ltraApp . We show below that (c, R) h-convex potentials have well behaved solutions of linear ODE.
) n of the following linear ODE for t ≥ s: Moreover, for any σ ∈ C n , the solution σ.(φ s,t (Y, X) j ) of the equation transformed by σ (that is the equation obtained by applying a cyclic permutation of the tensor indices) satisfies : ..øa * 1 ). The (unique) solution Φ s,t (Y, X) of the following linear ODE for t ≥ s: Proof. Proof of (a). Let X be a continuous self-adjoint process. The semigroup Θ X associated to Q = 1 2 (∂ k D j V (X)) kj gives a solution (Θ X s,t (Y )) t≥0 to Therefore we can define the solution to By assumption of (c, R) h-convexity, the semigroup e −t(Q− c 2 Id) = e c 2 t e −tQ is contractive, which gives the bound In particular, this sequence is bounded uniformly. By continuity of X, we can prove similarly that this sequence is Cauchy, and hence converges towards the solution of (16); the limit then clearly satisfies the bound (17). Uniqueness can be proved by Gronwall Lemma, as (∂ k D j V )(X . ) is uniformly bounded. Selfadjointness of φ s,t (Y, X) j follows from the uniqueness of the solution to the lin- Proof of (b). Using the notation of (a), define : Differentiating in t shows that Φ s,t is a solution of (18). The bounds follows readily from (a). Again, uniqueness follows from Gronwall's Lemma.
There exists T > 0 so that for any X 0 ∈ A n R,U ltraApp , there exists a unique solution to which is defined for all times t < T . Moreover, for all X 0 ,X 0 ∈ A n R,U ltraApp and t ≥ 0 (b) Assume that there exists X V = (X V 1 , . . . , X V n ) ∈ A n R/3,U ltraApp for which the conjugate variables are equal to D j V . Then part (a) holds with T = ∞ for any solution starting at X 0 ∈ A n R/3,U ltraApp . As a consequence, there is at most one free Gibbs law with potential V uniformly in A n R/3,U ltraApp .
Proof. Existence of X t (X 0 ) for all times t < T for which sup s<T X s (X 0 ) < R follows form the Picard iteration argument in [BS01]. The existence of T > 0 (depending only on the Lipschitz constant of DV ) is also shown there. Applying the same argument as in the proof of Lemma 4 by writing X 1 t = X t (X 0 ), X 0 t = X t (X 0 ), and arguing that ) is a closed cone, the estimate (20) follows from (17). Assuming the Assumption of part (b), we see that the solution X t (X V ) is stationary; in particular, its norm is constant. Part (a) and the estimate (20) then imply that any other solution starting at an element of A n R/3 stays in A n R , which means that T can be chosen to be infinite. Also, if there were two free Gibbs law with potential V , they would be stationary laws for the dynamics and (20) would imply that they are equal.
Throughout this paper we assume that Assumption 6. Let V, W ∈ C 3 c (A, 2R : B, E D ) be two non-commutative functions such that V and V + W are (c, 2R) h-convex for some c > 0. We assume that for any α ∈ [0, 1], there exists a solution (X V +αW 1 , . . . , X V +αW n ) ∈ A n R/3,U ltraApp with conjugate variables (D i (V + αW )) 1≤i≤n .
In subsection 6.9 we describe a class of quartic potentials satisfying this assumption. The existence of a solution to Schwinger-Dyson equations will be obtained from a random matrix model in the easiest case B = C and the convexity will be obtained by operator spaces techniques.
This Assumption insures that We consider the SDE where S is the free Brownian motion relative to D (with covariance map id D ). By Proposition 5, we deduce that there exists a unique solution X t satisfying X t < R for any X 0 ∈ A n R/3 . We denote it by X α t (X 0 , {S s , s ∈ [0, t]}), t ≥ 0, and X α t in short. We set for U ⊂ A n R , U α be the subset of its elements stable under the flow: comes from an element in C 1,0 tr,V,c (A, U α : B, E D : S ), and we have for any τ < t the relation (22) . Moreover, in each case t → X α t is continuous. Finally there exists a finite constant C k+l such that, for k + l ≥ 1 : above is a non-commutative function without expectation but can be thought of as an element of this larger space of functions, hence the reference to l. Note that most of the results only depends on k + l.
. We now prove that X α can be seen as a smooth function of X 0 , S, in the sense that it is an element of C k+l c (A, U α : B, D : S ). Fix T small enough, such that in particular 2 We construct by Picard iteration the process on [0, T ]. We let X [0,m] be defined recursively by X [0,0] . = X 0 and for m ≥ 1, Because X 0 ≤ R, one checks by induction on m that ||X c (A, 2R : B, D) n to prove that the Picard iteration procedure is first bounded (for T small) and then converges in the norm . k+l,0,Uα,c (for T even smaller so that the equation is locally lipschitz on the a priori bound obtained before in . k+l,0,Uα,c ). We let X s , s ≤ T be the limit : it belongs to C k+l c (A, U : B, D : S ) and is the unique solution of (21). By the definition of U α , for X 0 ∈ U α , X s ∈ U α , in particular ||X s || ≤ R . Hence, we can iterate the process by considering for s ∈ [0, T ] the sequence defined recursively by X [s,0] t = X s , t ≤ T and for m ≥ 1 Again this sequence converges in the norm . k+l,0,Uα,c to a limit X [s,∞] . As V is C k+l+2 c (A, 2R : B, D), such construction has a unique solution so that X for all s, s ′ ≤ t. We denote this solution X α . It satisfies (22). We continue by induction to construct X α ∈ C k+l c (A, U α : B, D : S ) for all times. The continuity of t → X t is clear, as a uniform limit of continuous functions.
We finally show (23). Using the first formula in the proof of Lemma 37 on the equation on Picard iterates and then taking the limit m → ∞, one gets for k ≥ 1: where the lower order terms (l.o.t) are with respect to the degree k of differentiation of X u . Evaluating the differentials and using Lemma 4 (b), one gets the exponentially decreasing 21 bound on their norms by induction over k (note that all the other lower order terms are nonlinear in derivatives of X t and thus bring more than one exponential decreasing enabling to compensate the increase via time integrals).
Lemma 8 (Itô's formula). Under Assumption 6, for P ∈ B{X 1 , ..., X n : E D , R} we have Proof. For P (later called polynomial) in the algebra generated by B, X 1 , ..., X n inside B X 1 , ..., X n : D, R , this is the standard Itô's formula, see [BS98,BS01]. By the norm continuity of all operations appearing, the extensions to ℓ 1 direct sums are obvious, so that it suffices to extend the formula to a monomial P ∈ B X 1 , ..., X n : D, R having only one term in the direct sum. Finally, using the standard decomposition of elements in extended Haagerup tensor products [M97] thanks to which P ∈ B eh ⊗ D n can be written with I j infinite indexing sets but I n = 1. We can truncate these infinite matrices by finite matrices, giving a net of approximation P n of P . All the terms in Itô's formula, once evaluated at a given time, will then converge in L 2 (M) (while staying bounded in M). Unfortunately, to get convergence of the time integrals we have to be a bit more careful.
Then considering constant functions with value in L 1 (A), it is easy to deduce the first line in the right hand side of Itô formula for P n weak-* converges to the one for P in A. To check the same result for the stochastic integral term, note that by Clarck-Ocone's formula and a priori boundedness of all the stochastic integrals, it suffices to check that for an adapted bounded U s , we have convergence to 0 of the pairing ) so that approximating it by a process with finitely many values and using weak-* convergence of the finitely many values of This completes the proof of the formula for P ∈ B X 1 , ..., X n : D, R .
For P in the algebra generated by B, X 1 , ..., X n , notice that the previous computations show that so that by induction over the number of conditional expectations, if P belongs to the algebra generated by B, X 1 , ..., X n , E D , Formula (25) follows for P polynomial in the algebra generated by B, X 1 , ..., X n , E D . The reduction from P ∈ B{X 1 , ..., X n : E D , R} to an element of the algebra generated by B, X 1 , ..., X n , E D is similar. Indeed, we can canonically embed ι : B{X 1 , ..., X n : E D , R} → B X 1 , ..., X n , S j , j ∈ N : D, R where the S i are free semi-circle, free with amalgamation over D. Each term in E D corresponds to a different set of S i and We can conclude by the previous considerations and the weak-* continuity of E W * (X 1 ,...,Xn,B) .

Semigroup.
Hereafter, we will often need a second technical assumption on D ⊂ B to apply Theorem 24.(3) and Proposition 28.(2) in the appendix. The appropriate definitions are given in the appendix in subsection 5.3.
As discussed in the appendix, the easiest non-trivial example of a pair (B, D) satisfying this assumption is B = Γ ⋉ D a crossed-product by a countable (or finite) discreate group Γ. In particular, when B = D this assumption is obviously satisfied.
We write A n R,App ⊂ A n R,U ltraApp the set A n R,U ltraApp if D = C and otherwise the set requiring additionally M = W * (B, X 1 , ..., X n ) ⊂ W * (B, S 1 , ...S m ) = B * D (D⊗W * (S 1 , ..., S m ) included into the algebra generated by m semicircular variables over D. Here, m can be infinite. This will be crucial when we will assume D ⊂ B satisfying the assumption of Theorem 24.(3) so that the conclusion of this Theorem and Proposition 28.
(2) will then be available for M in the sense that e D , .#e D will be a trace on We define: Theorem 27] (first for V polynomial and then for all V by density), one gets that for any X ∈ A n R,α , X α t ∈ A n R,α,conj for any t > 0. Hereafter we thus assume that X 0 ∈ A n R,α,conj . Denote A n R,α,conj1 = A n R,α,conj , A n R,α,conj0 = A n R,α . Hereafter, we will consider only functions of X and E D,X , we therefore drop the dependency in E D,X in the notations. Because we will need later to apply the cyclic gradient to the image of the semi-group, we will need the following ad'hoc space C k,l;−1 tr (A, A n R,α,conj ) which is the completion of B c {X 1 , . . . , X n : Generalizations of this norm are discussed in the appendix (44).
Proposition 10. Suppose Assumptions 6 and 9 hold. Let k ∈ {2, 3}, l ≥ 0 be given and assume V, W ∈ C k+l+2 c (A, 2R : B, D). The process X α t of Lemma 7 defines a strongly continuous semigroup ϕ α t on C k,l tr (A, A n R,α,conj : B, E D ) and, on C k,l;−1 . They are given by the formula . It satisfies the exponential bounds : , one gets strongly continuous one parameter families of maps t is well defined in all cases by composing the maps X α t from Lemma 7, the composition (P, X t ) → P (X t ), see Lemma 37, and expectations E B from Proposition 42. To get a semigroup we apply composition forŨ = U = A n R,α,conj , we have to check the consistency condition for composition, i.e. for any X 0 ∈ U, we have to check that X α t (X 0 , {S s , s ∈ [0, t]}) has one conjugate variable. This is proved in Proposition 46 in the appendix. Note this is where we need Assumption 9 and the condition A n R,α ⊂ A n R,App in order to apply Proposition The construction of ϕ α′ t and the consistency follow similarly. Let us check the semigroup property. It follows from the following formal computation : = ϕ α t (P ) where • u is the composition defined in Proposition 42. To justify this computation, the two first and last equations are the definitions of the "semigroup", third, fourth and next-to -last lines come from Proposition 42 and the fifth line from Lemma 7. We thus have the semigroup relation. The strong continuity on both spaces comes from continuity in u of X α u (see Lemma 7) and continuity of various compositions and E 0 (see Proposition 42).
Using the variant of (47) with U = V = A n R,α,conj , in the context of Proposition 42, that is adding Brownian motions filtration, we get Using contractivity of expectations in Proposition 42 and bounds in Lemmas 36 and 7, one gets the exponential bounds as claimed. The bounds for the seminorm ||P (X t )|| k,l,U ≥1 follow similarly.
Moreover, we get similar results for ϕ α′ by noticing that if The seminorm on this space is equivalent to ||P (X t )|| k,l,U ≥1 which we already estimated. Continuity of conditional expectation, see Proposition 42, allows to get the exponential bounds for ϕ α′ .
We next find the generator for the semi-group ϕ α′ t : it is given by L α = 1 2 (∆ Vα + δ Vα ) and we precise in the next Lemma some dense domains of this generator (without looking for the maximal one).
Proposition 11. Assume Assumption 6 and 9 and let k ∈ {2, 3}, l ≥ 2, be given with V, W ∈ C k+l+2 c (A, 2R : B, D) as before. We let ι ′ be the canonical map Proof. To compute the generator we start with Itô formula (25). Taking a conditional expectation, we deduce for P ∈ B c {X 1 , . . . , X n : E D , R, C}, We now want to check the same relation under a full cyclic gradient D. We need to check that all the terms above are in C k,1;−1 tr (A, R : B, E D ) for our chosen P . But we won't check that the relation (26) is valid in this space, we will only show this relation holds after application of the cyclic gradient in each representation. Indeed, we do not know if the full cyclic gradient D is closable, on the contrary to the free difference quotient.
From the definition of [(∆ Vα + δ Vα )P ] (see Def. (15)) as an evaluation of Note that all our terms are known to be in our expected space, we can apply (49) so that the equation (26) under D is true in any representation X 0 ∈ A n R,α if it is true under the differential d X 0 . Integrals are dealt with thanks to continuity of the semigroup with value in C k,l;−1 tr (A, R : B, E D ) from the previous Lemma. Seeing both sides of the equation (26) as a function of X 0 , one can differentiate both sides of (26) under d X 0 and obtain equality of both sides in each representation. We deduce the equality under the abstract d X 0 -differential in C * tr by injectivity of the map from C 0,l tr to C 0 (A n R,α , A) (contrary to the space C k,1;−1 tr before where this is unknown). We have thus deduced the equality in each representation : Applying Lemma 39 and seeing P as an element of C k,l tr,Vα (A, A n R,α,conj ), one knows that all the terms of the equality are in the domain of order k −2 free difference quotient and without having applied cyclic derivative, also in the domain of order k − 2 free difference quotient (since k, l ≥ 2). By closability, if X 0 ∈ A n R,α,conj we can apply the k − 2 order free difference quotient to the relation above and deduce corresponding relations. Therefore, the following bound extends for k ≥ 2 to P ∈ C k,l tr,Vα (A, A n R,α,conj ): || k−2,0;−1,A n R,α,conj → 0 goes to zero when t → 0 + , by the strong continuity of ϕ α s on C k−2,0;−1 tr (A, A n R,α,conj ). This gives the right derivative of ϕ α t at zero. Now for Q ∈ C k+l c (A, A n R,α,conj , B, E D ), by the semigroup property ϕ α s+t (Q) = ϕ α s (ι ′ ϕ α′ t (Q)) and applying the reasoning above to P = ϕ α′ t (Q), one gets the right derivative at any time. To compute the left derivative, we start similarly from the result of Itô Formula to P = ϕ α′ t−s Q starting at time t − s and using also the semigroup property Thus, using strong continuities of ϕ α and ϕ α′ , and reasoning as before in the more general spaces with some free difference quotient and cyclic derivative, we conclude that the left derivative is in C k−2,0;−1 tr (A, A n R,α,conj ).
We will need the following preparatory Lemma regarding conjugate variables. We will need a temporary technical assumption, satisfied under Assumption 9 if X 0 ∈ A n R,App as shown in the proof of Proposition 10. This will thus be the case for semicircular variables and then via our transport map for other models with h-convex potential.
Lemma 13. Assume Assumption 12. Fix such an Proof. The existence of the conjugate variable is a technical variant of [GS12] explained in the appendix, see Lemma 43. It is also shown there that Let us compute the time derivative of the above right hand side. From the elementary equation Since all the terms are in a matrix variant of D(∂ i ø D 1 eh ⊕ 1ø D ∂ i eh ) which is an algebra by Lemma 40.
(1) and J F α+h is differentiable in this space (using k ≥ 2), we can conclude from Lemma 40.(4) to the differentiability under J * so that we conclude after letting h going to zero that We have the chain rule for any g ∈ C 1,0 which completes the proof.
We will now proceed with the construction of the transport map F α .
Lemma 14. Assume that V, W, B, D, X = X 0 satisfy Assumptions 6, 9 and 12 and that V, W ∈ C 6 c (A, 2R : B, D). Fix such an X ∈ A n R/4,conj . Let Then Dg α satisfies the equation Moreover the differential equation has a unique solution in the space C 2,2 tr (A, A n R/4,conj : B, E D ) with the initial condition F 0 = X on a small time [0, α 0 ] for some α 0 ∈ (0, 1] which only depends on c, R, sup β∈[0,1] Dg β C 2,1 tr (A,A n R/3,conj :B,E D ) , non-increasing in the last variable. Proof. The integral defining Dg α exists in the space C 2,1 tr,c (A, A n R,α,conj : B, E D ) because of the exponential bound in Proposition 10 (with k = 2, l = 2). From the computation of the derivative in C 0,0;−1 tr,Vα (A, A n R,α,conj : B, E D ) in Proposition 11, one gets the derivative in where the last identity comes from Lemma 39 with g = ϕ α′ t (W ) and k = 0. Integrating in t and since D(ϕ α′ t (W )) tends to 0 when t → ∞, one gets the identity in C 0,0 tr (A, A n R,α,conj : B, E D ) : Fix α > 0. We next define an appropriate space on which the following map will be a contraction for α small enough. We take F β ∈ A n R/3,conj ⊂ A n R,β,conj to stay in a space independent of β. We set, for α to be chosen small enough and for any fixed By the previous Lemma (note that we don't need at this point α Thus for any X ∈ A n R/4,conj , F β (X) ∈ A n R/3,conj and we are in position to apply Lemma 37 to get Dg β (F β ) ∈ C 2,0 tr (A, A n R/3,conj : B, E D ) n . Moreover, applying Lemma 45 and the same exponential decay as before to deal with the tail of the .
Recalling J F 0 = 1 and using the continuity of J Dg β one can choose α = α(K) small enough such that χ is valued in E α,K . It remains to obtain a contraction, up to choose α even smaller.
Since Dg β lies in a bounded set in C 2,1 tr,c (A, A n R/3,conj : B, E D ) and E α,K is bounded, Dg β is uniformly Lipschitz by Lemma 37, with a Lipschitz norm which does not depend on β ∈ (0, 1). χ is thus a contraction on E α,K . It has therefore a unique fixed point which is our solution which is necessarily in Lemma 15. Assume the Assumption of Lemma 14. Let Then Υ α satisfies the differential equation in L ∞ ([0, α 0 ), W * (X)): As a consequence, In other words, for α ∈ [0, α 0 ], F α (X) has conjuguate variables DV α .
Proof. Using our previous computation of derivative of conjugate variables in Lemma 13, we compute We next rewrite the right hand side. To this end, notice that (50) yields Putting these equalities together give: . where we have finally used equation (29) and Lemma 39(1) applied to Dg α .
Hence, (30) yields We thus obtain the expected equation from which we deduce the bound : We have to reinforce slightly Assumption 6, a reinforcement which is still satisfied by our examples of quartic potentials.
Assumption 16. Let V, W ∈ C 3 c (A, 2R : B, E D ) be two non-commutative (c, 2R) h-convex functions satisfying Assumption 6 and moreover for any α ∈ [0, 1], there exists a solution ( Corollary 17. Let V, W, B, D satisfy Assumption 16 and 9 and V, W ∈ C 6 c (A, 2R : B, D). Assume also the pair (cV 0 , V −cV 0 ) satisfy Assumption 16. Fix an X ∈ A n R/4,conj and suppose it follows the free Gibbs law with potential V .
Let F α , 0 ≤ α ≤ α 0 be the solution constructed in Lemmas 14 and 15. Then: For any α ∈ [0, 1], the von Neumann algebras generated by B and generators of the free Gibbs law with potential V α = V + αW are isomorphic.
Proof. We first check that Assumption 12 is satisfied under our assumptions. First start with the case V = cV 0 , in which case Assumption 12 is satisfied thanks to Assumption 9 and Proposition 28 (2). Then building the transport map for the pair (cV 0 , V − cV 0 ) the same Assumption 12 is satisfied for X ∈ A n R/4,conj . By the previous Lemma 15, we find that Υ α = 0, which means that J Fα (1 ⊗ 1) = DV α for α ∈ [0, α 0 ]. Since V α is by assumption (c, 2R)-convex and F α < R/3 it follows that the law of F α is the free Gibbs law with potential V α . This proves (i). To , and consider the same ODE as in Lemma 14, and callF α the solution. V α replaced byV α . Note that F α 1 (X) ∈ A n R/4,U ltraApp by Assumption 16. It is not hard to see thatF α (F α 1 (X)), F α 1 −α (X) are solutions to the same ODE (only W is changed into −W as it should be since the time is reversed), and is thus the unique solution. Thus by what we proved, W * (F α (F α 1 (X)), B) ⊂ W * (F α 1 (X), B), which proves the reverse inclusion and thus W * (F α (X), Let us prove the last point of the Corollary. We have just checked the case α ∈ [0, α 0 ]. Moreover, (V α 0 , (1 − α 0 )W ) satisfies the same assumption as (V, W ) with the same constants (c, R). We can therefore perform the previous construction of a function F α with (V, W ) replaced by (V α 0 , (1 − α 0 )W ). This can be done until a parameter α ′ 0 which can be chosen to be equal to α 0 as the constants (c, R) are the same and the semi-groups under consideration are the same. Note also that Assumption 16 enables to check that ||F α 0 (X)|| ≤ R/4 and thus F α 0 (X) satisfies the same assumption as X.
Inductively, one concludes to the isomorphism for any α ∈ [0, 1[. To complete the proof, it suffices to note that for ǫ small enough, V, (1 + ǫ)W satisfy the same assumptions (a priori with a different convexity constant and replacing R/4 < R/3 by any larger value).

Appendix 1: Cyclic Haagerup Tensor Products
Let M be a finite von Neumann algebra and D ⊂ M be a von Neumann subalgebra.
Our goal is to define a notion of n-fold cyclic tensor product M eh ⊗ D,c n which will be a certain 31 subspace of the Haagerup tensor product M eh ⊗ D n . We start by considering the case n = 2, and then use amalgamated free products to build the more general cyclic tensor powers.
The inspiration for the construction comes from subfactor theory. Indeed, if M 0 ⊂ M 1 is a finite-index inclusion of II 1 factors and if M k denotes the k-th step in the iterated Jones basic construction, then (see e.g. [JS,Prop 4.4.1(ii)]) L 2 (M k ) are precisely the tensor powers of L 2 (M 1 ) regarded as an M 0 Hilbert bimodule: L 2 (M k ) = L 2 (M 1 ) ⊗ M 0 k . Moreover, the higher relative commutants M ′ 0 ∩ M k are precisely the cyclic tensor powers of M 1 . These ideas have been extended to the infinite-index case [B,Pe,Pe13]. In particular, the notion of Burns rotation will be useful for us to get a certain traciality property. 5.1. Preliminaries.
Finally, define the centralizer algebras  Haagerup tensor products and the basic construction. With these preliminaries recalled, we now turn to the definition of cyclic Haagerup tensor product. We start by a well-known technical result concerning the Jones basic construction.
If A is an operator space, we write A * for its dual as an operator space [P]. When A is a D − D bimodule, we write A ♮ for the dual operator D ′ − D ′ bimodule in the sense of Magajna [M05]. We will also denote by A ♮Dnorm the normal dual defined when A is itself a tensor product over D in [M05, Th 3.2]. While we will not recall the general definition of the normal dual here, we will mention that in the case that A is itself a tensor product over D (and therefore its dual can be viewed as the space of certain linear maps), the normal dual corresponds to maps that satisfy a normality condition on basic tensors. In the case that D = C, the bimodule dual is the same as the operator space dual A * .
Let D ⊂ M be finite von Neumann algebras, let e D be the Jones projection onto D, and denote by M, e D the basic construction for D ⊂ M. Let The map E D ′ is pointwise normal in D and thus its adjoint agreeing with the usual projection on L 1 ( M, e D ) ∩ L 2 (M)ø D L 2 (M), and giving an isomorphism Proof. The identification (Note that here the operator space structure D ( L 2 (D) L 2 (M) M ) C is the one of the indicated Hilbert module structure, not the one as a module over D op ). It remains to check but the last inclusion comes again from [M05, Th 3.2, Ex 3.15]. In this way we identify the compact ideal space with the norm closure of basic tensors in the extended Haagerup tensor product. This norm closure is exactly the Haagerup tensor product and thus we deduce the first isomorphism.
On the dense space . This gives the claimed isomorphism between the image of E D , D ′ ∩ I 0 ( M, e D ), and the quotient, as well as the identification with the Dixmier conditional expectation.
The key part of our Lemma is to check , is normal, one gets our result. The second quotient statement is analogous.
The reader should note that the identification L 1 ( M, e D ) ≃ L 2 (M) * ø hD op L 2 (M) is given on basic tensors by: This will be the key to various flips appearing naturally later.
5.2. The cyclic Haagerup tensor product, case n = 2. Recall that the spaces L p ( M, e D ) are made in compatible couples in the sense of interpolation theory [P]. We can see them as the inductive limit of L p (q M, e D q) for q finite projection. Thus these spaces are realized as an interpolation pair as a subspace of the topological direct sum .
We refer to [Dab15, Th 2] for a literature overview of the main algebraic operations available on module Haagerup tensor products. We will use them extensively. We single out several operations. The first is the map ⋆ (see Section 2) which is given on basic tensors M and a basic product U = xe D y ∈ M, e D we write: (inner action).
and if U ∈ D ′ ∩ M, e D : With these notations, we have the following statements, which we group into three Theorems for convenience of presentation.
(1c) The inner and outer multiplication actions give rise to inclusions σ 1 , σ 2 , which is our inner multiplication. Composing with E D ′ one induces a map The latter is isomorphic to D ′ ∩L 2 (M) * ⊗ ehD op L 2 (M), the last inclusion following for instance from the identification of this commutant with a quotient or because (32) and coincides with our inner action.
We first claim that for To show this, it suffices to take V ∈ A by density. We can also assume X is a finite sum. Indeed, if X = xø D y a standard decomposition for X [M05, (2.4),(2.5)] the ultrastrong Now for the remaining case V = ξø D op η, X = xø D y (without matrix tensor products), we note that the image of V in the identification with L 1 ( M, e D ) is ηe D ξ, as explained in (32) We now prove (1c); all we need to show is that σ 1 (X and on each variable when restricted to: Proof. Note that the intersection space M eh ⊗ D,c 2 is thus well-defined because of (1c).
We start by proving (1d). ). Since U and V are arbitrary in dense spaces this shows the desired relation and as a consequence that σ is involutive.
With the same notation and using the definitions, σ = σ ′ and the adjoint relation (33) several times, we have: by ⋆ . The commutation with σ also follows since we showed σ −1 To prove (1e), it remains to check the composition and the adjunction * give the expected Banach algebra structure. 37 We can reason similarly using our formula (33) and σ = σ ′ to check closure under the product: The middle relation then also shows σ(XY ) = σ(Y )σ(X). Similarly, (UV ) ⋆ = (U) ⋆ (V ) ⋆ which gives the only missing relation between * and product to get an involutive Banach algebra. We Indeed, by construction, the projection E D ′ (U) is built as a weak-* limit of convex combinations λ u uUu * converging thanks to the embedding Mø eh M ⊂ L 2 (M)øL 2 (M). Moreover, we have Because the injection is weak-* continuous, one also gets weak-* convergence of the convex combination in L 2 (M) * ø h L 2 (M) and thus, for any U ∈ Mø eh M, By density, E D ′ extends to a bounded map on L 2 (M) * ø h L 2 (M) which obviously induces a map L 2 (M) * ø hD op L 2 (M) → D ′ ∩ L 2 (M) * ø h L 2 (M), a cross-section to the quotient map (as seen first for U ∈ Mø eh M by the weak-* limit above). Then is indeed a row vector in L 2 (M) * , and similarly (v j y i ) is a column vector in L 2 (M). Thus we have indeed Moreover, by the definition of the norm, it is now easy to see This gives the outer action on L 1 ( M, e D ) as is easily seen using the identity σ −1 2 (XY ) = σ −1 2 (Y )#σ −1 2 (X). The stability and commutation are easy. We now turn to (2a)-(2d).
First note that σ T C : L 2 (M op ) * ø hD L 2 (M op ) → L 2 (M) * ø hD op L 2 (M), given by σ T C (aø D b) = bø D op a, is isometric. This uses that a row vector of L 2 (M op ) * is the same as a column vector of L 2 (M).
To prove (2a) column vector in L 2 (M) and (y k ) row vector in L 2 (M) * . Note that for ξ ∈ M, one can compute the evaluation with the formula above M) (one can first approximate y k , y ′ k by elements of M to establish the formula).
If we take (g j ) j∈J a basis of L 2 (M) as a right D-module (of elements of M if we want), then one can use the well-known formula We compute a term in the last formula. We continue our computation by applying Z ′ which also gives a map on L 1 (M) : Then since g j ∈ M, one can compute the trace : which could be expressed as a duality formula for Y #e D ∈ D ′ ∩ L 1 ( M ′ , e D ) since the sum in i is finite, thus one can use its commutativity with D : where we finally used one of our previous formulas with ξ = g j E D (g * j z ′ i ) instead of g j before. But looking again at (x ′ øx)#(Y #e D ) as the bounded operator on L 2 and using the relation for a right basis j g j E D (g * j z ′ i ) = z ′ i with convergence in L 2 , one may use operator weak-* convergence to replace (x ′ øx) by X = (x ′ l øx l ): , the predual of the weak-* Haagerup tensor product, one gets the claimed weak-* continuity and thus the proof of (2b) is complete.
To prove the density part in (2c), it is enough to show that for a finite sum, . More precisely, we will show that We thus want to prove, for any U, V ∈ D ′ ∩ M, e D ∩ L 1 ( M, e D ) : By density (simultaneous weak-* and L 1 using the agreeing conditional expectations) it But now we can use the weak-* continuity we just proved to replace the conditional expectations by the limit of a net of convex combinations of conjugates by unitaries of D, and thus by commutativity with D, the conditional expectations can be removed, and the relation then becomes obvious.
Finally, for (2d), taking bounded nets U n → U, V ν → V we note that U n #V, U#V ν are still bounded, thus weak-* precompact and it thus suffices to show that U#V is the unique cluster point, for instance by showing the nets converge weakly in L 2 (M)ø D L 2 (M) Theorem 24. We keep the assumptions and notation of Theorem 21 and Definition 22.
(3) Assume either that there exists a D-basis of L 2 (M) as a right D module (f i ) i∈I which is also a D-basis of L 2 (M) as a left D module or that D is a II 1 factor and that In particular, this realizes canonically isometrically L 2 (M M. Indeed, using the proof of the density and weak-* continuity in our part (2), we only need to consider X = E D ′ (x 1 øx 2 ), Y = E D ′ (y 1 øy 2 ) for x i , y i ∈ M. But from our previous computation, this reduces to: Now the key equality in the middle line comes from the extremality of L 2 (M) that gives from Theorem 19 a unitary Burns rotation. From unitarity it is easy to see that ρ(E D ′ (y 1 ø D y 2 )) = E D ′ (y 2 ø D y 1 ) so that the equality in the middle line comes from T r((V # L ∞ X)U) = T r(V (X# L 1 U)).
It thus remains to see these two actions agree on a common dense subspace.
We already noticed they form a dense subspace in both L 1 ( M, e D ) and (for the weak-* topology) in M, e D . Note that this indeed gives (even for For the last key equality, take a canonical representation of Y = y j ø D y ′ j , X = x i ø D x ′ i then we note that where we started by using the relations we just established, the adjoint relation (33) Thus (V # L ∞ X) = (Z#X)#e D = σ(X)# L 1 V and we can thus interpolate both maps to get the desired action. Finally, the agreement with the inner action on the commutant comes from the equality σ(X) 5.3. k-fold cyclic module extended Haagerup tensor products. We now turn to the construction of k-fold cyclic tensor powers M eh ⊗ D,c k extending the case k = 2 we have just dealt with. The desired properties of these tensor powers include the action of cyclic permutations, commutation with left-right actions of D as well as compatibility with various multiplication and evaluation operations. Elements in these modules will serve as coefficients for our generalized analytic functions, on which free difference quotient and cyclic gradients will be well-defined.
We will use free products with amalgamation as a convenient trick to reduce to the case of 2-fold cyclic modules we have already considered.
In particular, for any word n = n 1 ...n |n| in κ letters there is an embedding valued in L 2 (M) ø D |n|+1 ∩ N κ obtained by first sending the tensor x 0 ⊗ · · · ⊗ x |n| to x 0 S n 1 x 1 · · · S n |n| x |n| and then projecting onto the orthogonal complement of the space for the closure of the image of ι n .

Construction of intersection spaces.
To handle the action of a basic cyclic permutation, we need an intersection space similar to the intersection L 1 ( M, e D ) ∩ M, e D in the previous section (which corresponds to the case |n| + 1 = 2). For this, we will use We will be mostly interested in off-diagonal block matrices in these constructions, namely (for k = l), so that B(M : D, k, l) = T C (M : D, l, k) * and the duality can be seen as induced by T r above when they are seen as block matrices in the space above.
Let us start with a Lemma making explicit this relation. Consider, for n a word in κ letters, P n ∈ N κ , e D ∩ B(L 2 (N κ ), L 2 (M) ø D n ) the orthogonal projection on the n-th component in the decomposition L 2 (N κ ) ≃ ∞ k=0 |n|=k L 2 (M) ø D n . Note that we make the difference between the adjoint P * n ∈ B(L 2 (M) ø D n , L 2 (N κ )) and the map P n ∈ B(L 2 (N κ ) * , L 2 (M) ø D n * ): P n (ξ) = ξ • P * n = P n ξ, even though they may be conjugated by some isomorphisms above.
(1) Let X ∈ N κ , e D and Y ∈ L 1 ( N κ , e D ) ≃ L 2 (N) * ø hD op L 2 (N). X and Y agree in the classical intersection space, if and only if for all k, l words in κ letters, D, (k, l)), P * k pP k is finite in N κ , e D . Hence agreement of X and Y which boils down to the agreement for any finite projection of their compressions, gives P * l pP l XP * k pP k = (P * k pP k ø D op P * l pP l )(Y ) and thus the agreement after removing one application of P * i , i.e. as we said since this is for all finite projection p, P l XP k = (P k ø D op P l )(Y ). Conversely, since P ≤n = P 0 + ... + |m|=n P m increases to identity, it suffices to consider finite projection q ∈ N κ , e D with q ≤ P ≤n which readily reduces to compression by q ∧ P k = P * k (q ∧ P k )P k (on the right and q ∧ P l on the left) for a projection p on B(M : D, (k, l)). And we can then apply the converse reasoning.
(2) Note that For any φ ∈ (L 2 (M) ø D k ) we have a map φø hD op 1 : such that φ(x)P φ = 0, P φ y = y. Take P = φ∈L 2 (M ) ø D k P φ then P y = y and φ(xP ) = φ(x)P = φ(x)P φ P = 0 thus since φ is arbitrary in a space containing the dual of the space of x, xP = 0 and thus xø D op y = 0; thus we get the first claimed injectivity.
The agreement of intersections spaces comes from the fact that the intersection space of L 1 and L ∞ can be reduced to equality when compressed by rank 1 projections coming from elements in a fixed right-module basis. Then the agreement corresponds in the second picture to agreement when evaluating at this fixed basis (and evaluating by duality at this basis too). 45 5.3.2. Wick formula. We will also need a straightforward tensor variant of Wick formula. For k = k 1 ...k |k| , m = m 1 ...m |m| words in κ letters, we write k • m = k 1 ...k |k| m 1 ...m |m| for the concatenation, and also k • i m = k 1 ...k |k|−i m 1+i ...m |m| ,|k| ∧ |m| ≥ i ≥ 0 (defined only if the last i letters of k and the first i letters of m form identical words). Note that |k • i m| = |k| + |m| − 2i. We also write k = k |k| ...k 1 . Sometimes, we will need to emphasize the following isomorphism: Likewise, we have : given byι and P p [V #(P m XP * n )]P * q = 0 for either |q| > |n| + |l| or |q| < |n| + |l| − 2(|l| ∧ |n|) or |p| < |m| + |k| − 2(|k| ∧ |m|) or |p| > |m| + |k|.
These maps satisfy: Proof. The definition of the map m What really matters for us in the previous result is that the highest component of the product is a tensor product, while the remaining terms are then determined by applying multiplication and conditional expectations to its various components. For convenience for words m, n and k ≤ |m|, we write : m# K n = m 1 ...m K−1 n 1 ...n |n| m K+1 ...m |m| , m# K n = m 1 ...m K n 1 ...n |n| m K+1 ...m |m| .

5.3.3.
Flips and cyclic permutations. We start by interpreting a cyclic permutation σ = (l + 2, l + 3, . . . l + k + 2, 1, 2, . . . , l + 1) in C n , n = l + k + 2, as the flip (i.e., period two permutation) of the blocks [[l + 2, . . . , l + k + 2]] and [[1, . . . , l + 1]]. We mimic this point of view in terms of injections in our free product von Neumann algebra N κ . We thus make use of our results on the two-fold cyclic Haagerup tensor product in this context to construct a suitable intersection space, using which we then construct the n-fold cyclic Haagerup tensor product.
(iii) For the weak-* continuity of composition products, take bounded nets U n → U, V ν → V . By weak-* precompactness of U n # K V, U# K V ν , it suffices to show that they converge weakly in L 2 (M) ø D (|n|+|m|+1) to U# K V . By density it is enough to check convergence dually against any Z ∈ M eh ⊗ D (|n|+|m|+1) . Take any word o of length |o| = |m| − 1. We claim that N. This is obvious again by the isometric embedding 49 at L 2 level and since it suffices to check weak convergence in L 2 (N)ø D L 2 (N). Take similarly n = kl, |k| = K − 1, then (ι k ø D ι l )(U n − U) → 0. From the result in Theorem 23.(2), Since from the computation below coming from Lemma 37 we get the weak convergence by duality against (ι kǫ ø D ι ol )(Z)#e D . (iv) For the stability of composition products, consider first the situation of the third point, . But from the definitions, one easily gets for X ∈ N, e D ∩ L 1 ( N, e D ): ]#(P m XP * l )) P * l 2 •l 1 •l . and similarly : From the assumptions on U and V the two second lines are equal, and then, from (37) and (38), one deduces the conclusion we wanted, for all p, q : which, using Lemma 25.(1), implies our statement and : The other statements about composition product and product are similar, the first statement in each case always following from the second using the stability by adjoint proved before. We give a few details concerning the second point for the composition product.
As before it suffices to prove : But now, by assumption, we know : Moreover, by Lemma 26 they are valued respectively in ι −1 By Lemma 25.
(2), it suffices to see that the two elements we wish to prove equal ity we want can be obtained from the one we know by applying the multiplication .#V which is well defined on the appropriate extended Haagerup tensor powers of M in the range of our maps.
The reader should note that in this case, we thus actually proved

5.3.4.
Cyclic Haagerup tensor products: the general case. We are now ready to introduce our cyclic extended Haagerup tensor product as an intersection space with enough compatibility condition to have a cyclic group action on it. Once those cyclic group actions are obtained, our various products and actions leave stable our intersection space as expected. We also obtain a density result saying that our spaces are non-trivial as soon as D ′ ∩ L 2 (M) ø D n are.
We also obtain traciality and functoriality results crucial to build later evaluations maps.
By definition as an intersection, (J(σ)) defines an action on M eh ⊗ D,c n since on the intersection of kernels we exactly have J(σ 1 )J(σ 2 ) = J(σ 1 σ 2 ).
It mostly remains to show the density results. For, we prove that for any x 1 , ..., x N +2 ∈ M, . From the weak-* continuity on bounded sets of E D ′ , this implies the weak-* density. The L 2 density is even easier. More precisely, we show that and on this formula one reads it is also in the intersection of kernels defining M eh ⊗ D,c n .

This reduces to (35) if we show that
But we saw both sides can be further included in L 2 (M) ø D N +2 as a subspace with both E D ′ agreeing with the projection there. This concludes.
(2) From the action property in (1) on the dense set where J(σ) is defined, it suffices to consider σ a generator of the cyclic group. We thus extend J(σ) isometrically in the case σ is such that σ(1) = N + 2.
Moreover, by the density of (linear combinations of) vectors of the form E D ′ (x 1 ø D ...ø D x N +2 ) obtained in the proof of (1), it suffices to show that the restriction of J(σ) to those vectors is an isometry.
But note that with our fixed σ, we have obtained the relation : Moreover, assuming extremality, there is by Theorem 19 a unitary Burns rotation, and by its defining relation, it coincides with J(σ) −1 so that J(σ) is an isometry as stated. The case with a basis is left to the reader. , . One easily gets from the isometry relation : and one easily gets zero for various other injections. Finally, we know that linear combinations of E D ′ (nø D n ′ ), n, n ′ ∈ N κ are weak-* dense in , and then using the strong density of Span(ι k (M eh ⊗ D k ), k ∈ κ N , N ≥ 0) in N κ , we get the same result, with n, n ′ in this span. But now, we already noticed that E D ′ (ι k øι l (U)) = (ι k øι l (E D ′ (U)) thus proving the weak-* density of Since N κ is involved, we consider N ′ = W * (N κ , S ′ 1 , ..., S ′ κ ) and ι ′ k the corresponding evaluation for a word in κ letters (with primes), ι ′′ k the evaluation for M with a word k in 2κ letters. If |l| = p − 1 is a word in κ letters with primes , and k i 's are word in κ letters without prime as before, we write l • (k 1 , ..., k p ) = k 1 l 1 ...l p−1 k p and one then notices (using some orthogonality in free products) that ι ′ l • (ι k 1 ⊗ D ...ø D ι kp ) = ι ′′ l•(k 1 ,...,kp) . One easily deduces from this the stated stability, the boundedness following from the very definitions of norms involving more specific evaluation and from (N ′ , For the statement on conditional expectations, it suffices to prove the boundedness on E ø D p : N κ ø ehscD p → M ø ehscD p by the symmetry of this map which induces easily the stability of kernels involving the action of the cyclic group. It suffices to check that , which are easily checked on elementary tensors by freeness with amalgamation over M of N κ and W * (M, S ′ 1 , ..., S ′ κ ).

Appendix 2: Function spaces
In this section, we study several function spaces crucial to our constructions. We start by considering spaces of analytic functions as well as cyclic analytic functions (these can be regarded as enlargements of spaces of non-commutative polynomials and cyclically symmetrizable non-commutative polynomials). We then consider analytic functions that depend on expectations, i.e., enlargements of functions of the form where E is a (formal) conditional expectation. Finally, we consider analogues of spaces of C k -functions, defined as completions in certain C k norms. 6.1. Generalized Cyclic non-commutative analytic functions. In this section we study the properties of cyclic B c X 1 , ..., X n : D, R, C and ordinary B X 1 , ..., X n : D, R, C generalized analytic functions in n variables with radius of convergence at least R, defined in subsection 2. Here, as before, D ⊂ B are finite von Neumann algebras. We will also consider a variant with several radius of convergence R, S, B X 1 , ..., X n : D, R; Y 1 , ..., Y m : D, S , B c X 1 , ..., X n : D, R; Y 1 , ..., Y m : D, SC . We will use it freely later. If X = (X 1 , · · · , X n ), we also write B X : D, S for B X 1 , ..., X n : D, S , etc.
We have the following basic result: Proposition 29. Let X = (X 1 , . . . , X n ), Y = (Y 1 , . . . , Y m ). Then (a) The linear spaces B c X : D, R, C , B X : D, R, C (resp. B X : D, R ) are Banach * -algebras as well as operator spaces (resp. Banach algebra and strong operator D module). Moreover, B X : D, R, C , B X : D, R are dual operator spaces when seen as (module) duals of (module) c 0 direct sums of the fixed preduals of each term of the ℓ 1 direct sum. We always equip them with this weak-* topology. Finally the algebra generated by B, X is weak-* dense in those spaces.
(b) For P ∈ B X : D, R , Q 1 , ..., Q n ∈ D ′ ∩ B X : D, S, C , such that Q i ≤ R, there is a well defined composition obtained by evaluation at Q j : P (Q 1 , ..., Q n ) ∈ B X : D, S . The composition also makes sense on the cyclic variants and is compatible with canonical inclusion maps on these function spaces.
(c) If B ⊗k X : D, R, C (with C = C or C = D) is the subspace of B X, Y : D, R, C consisting of functions linear in each Y 1 , . . . , Y m and so that in each monomial each letter Y j only appears to the right of all letters Y i with i < j,then there are canonical maps .# (., ..., .) : .., X n ) ∈ W * (B, X 1 , ..., X n ). Proof. The fact that B X 1 , ..., X n : D, R is a Banach algebra is obtained in [Dab15,Th 39]. The dual operator space structure and weak-* density also come from this result. The stability by adjoint only works for direct sums over C (since adjoint is not a module map and would require the conjugate module structure). The stability by multiplication obtained in Proposition 28 gives the same result for B c X 1 , ..., X n : D, R, C . For the stability by composition, the well-known composition map in [Dab15] Theorem 2 is completely bounded in each of the middle variables and it is easy to see that the compositions built in Proposition 28 also are (since the intersection norm is obtained from Haagerup norms dealt with in the non-cyclic case). Thus, ℓ 1 direct sums are dealt with using universal property, the only key point is that we use operator space (and not module) ℓ 1 direct sum for composition in Q i variables since the multilinear map (P, Q 1 , ..., Q n ) → P (Q 1 , ..., Q n ) is a D − D module map only in the variable P . In this way, the previous complete contractivity can be used in each variable with the right universal property for each type of ℓ 1 direct sum. In order to use the universal property in P , one also needs to know the source and target modules are strong operator modules over D in the non-cyclic case, and they are since those extended Haagerup products are even normal dual operator modules. The statements for B ⊗k are obvious consequences. The evaluation map comes from the standard inclusion B (e) The following relations between derivatives and composition hold, denoting Q = (Q 1 , . . . , Q n ): ...,j k ) (P (Q)) = k l=2 n 1 ,...,n l 1≤i 1 <...<i l =k (∂ l (n 1 ,...,n l ) (P ))(Q)# and Let us write n X i (m) for the X i degree of a monomial m, i.e., the number of times the variable X i occurs in m. To define the free difference quotient and cyclic gradient, we start from the formal differentiation on monomial, add appropriate change of radius of convergences S < R to allow boundedness of the map and then gather the monomials at the ℓ 1 direct sum level by the universal property: C n X i (m) ; m ∈ M(X 1 , ..., X n ), |m| ≥ 1 , and similarly in the cyclic cases. In order to for the value to belong to the claimed space, we also need to specify a canonical map I with values in B ⊗2 X : D, S, C . Of course, we want it to send the j-th component in the ⊕ 1 C n X i (m) direct sum to the component of the monomial m X i ,j which is identical to m but with the j-th X i replaced by Y 1 . Since there is a bijection between the disjoint union over monomials of {m} × [[1, n X i (m)]] and the set of monomials in X and Y 1 linear in Y 1 , it is easy to see that I extend to a complete isomorphism of ℓ 1 C direct sums. We still write For the cyclic gradient, one can then apply a different cyclic permutation on each term of the direct sum and we gather them in a map σ : B ⊗2c X : D, S, C → B ⊗2c X : D, S, C and a multiplication map m d : B ⊗2c X : D, S, C → B c X : D, S, C (based on composition # at d on the appropriate term of the tensor product and extending m d (P ⊗ Q) = P dQ = (P ⊗ Q)#d) to get the expected cyclic gradient: For the free difference quotient, to see there is a canonical map to the range space B X 1 , ..., X n : D, R eh ⊗ D B X 1 , ..., X n : D, R , one applies the following Lemma to each term of the direct sum inductively, and then the universal property of ℓ 1 direct sums to combine them. (We of course apply after mapping ℓ 1 C to ℓ 1 D direct sums). The various relations then follow by construction from the various associativity properties of the compositions and multiplication defined in Proposition 28. We explain those associated to cyclic gradients. First, we obtain the derivation property of ∂ X i and ∂ X i (P Q) = ∂ X i (P )Q + P ∂ X i (Q) so that : and applying m d one gets (39). Similarly, one obtains first the relation and then and applying m d gives (41).
The following result is a module extended Haagerup variant of [OP97, Lemma 7], the proof is the same using universal property of ℓ 1 direct sums and [M97, Th 3.9]. We thus leave the details to the reader.
Let S be the closure of the subspace obtained by injectivity of Haagerup tensor product (E 1 . Then we have: completely isometrically.
We will also need a more subtle evaluation result for B ⊗kc X 1 , ..., X n : D, R, C . which require that our variables are nice functions of semi-circular variables.
We write A n R,U ltraApp for the set of X 1 , ..., X n ∈ A, X i = X * i , [X i , D] = 0, X i ≤ R and such that B, X 1 , ..., X n is the limit in E D -law (for the * -strong convergence of D) of variables in B c X 1 , ...X m : D, 2, C (S 1 , ..., S m ) with S i a family of semicircular variables over D, that is of elements in the set of analytic functions evaluated in S 1 , . . . , S m . Here m is some large enough fixed integer number. with N 1 = W * (B, S 1 , ..., S m ) but one easily deduces the more restricted space of value.
We now consider the more general case with X i ∈ C * ,+ (B, S 1 , ..., S m ) := C * (ev S 1 ,...,Sm (B X 1 , ..., X m : D, 2, C )), in the C * algebra generated in W * (B, S 1 , ..., S m ) by evaluations of our analytic functions at semicircular variables. There is a map φ 1 ø D ...ø D φ p on the extended Haagerup tensor product by functoriality and nothing is required to get a map on the intersection space To get the stated map and even first a map φ 1 ø D ...ø D φ p : B ø ehScD n → M ø ehScD p , we have to check various stability properties of kernels appearing in their definition as an intersection space. From the formula below describing the commutation of the cyclic action and various tensor products of the maps φ, this stability of kernels will become obvious. More precisely, let U ∈ B ø ehscD n for σ ∈ C p , we writeσ the induced permutations on blocks and V = J(σ)(U) and n = kl, σ(1) = |l| + 2, |l| = p − 2 − |k|. We want to show for any X ∈ N, e D ∩ L 1 ( N, e D ) with N = W * (M, S 1 , ..., S κ ) : For, it suffices to evaluate them to Y, Z ∈ [B X 1 , ..., X n : D, R, C (X 1 , ..., X n )] S 1 , ..., S κ =: C ⊂ L 2 (N) as in Lemma 25.
(2) and to take X ∈ Ce D C, and see equality in L 1 (D). The statement for X 1 , ..., X n analytic as above gives exactly this in this case. In the evaluated form, the convergence in E D -law is clearly enough to get the general case from this one. The evaluation map is then obtained by the universal property of ℓ 1 direct sums. It crucially uses the bound on the norm of the completely bounded map above R n−p that easily follows To prove various results on them, we need some formal notation to explain several computations combinatorially. First, since those spaces are defined as ℓ 1 direct sums over pairs of monomials m and non-crossing partitions σ ∈ NC 2 (2k) (indexing the parenthesizing where conditional expectations are inserted), we can write π m,σ for the projection on the corresponding component of the ℓ 1 direct sum, and ǫ m,σ for the corresponding injection.
Similarly, for P ∈ B{X 1 , ..., X n ; E D , R} and a unital D bimodular completely bounded linear map E ∈ UCB D−D (B X 1 , ..., X n : D, R , D), there is a canonical element P (E) ∈ B X 1 , ..., X n : D, R . Since P → P (E) will be completely bounded D − D bimodular on monomials, by the universal property of ℓ 1 direct sums, it suffices to define it for monomials P = π m,σ (P ), σ ∈ NC 2 (2k). It is defined by induction on k. Write σ − ∈ NC 2 (2(k − 1)) the , then Indeed the letters between the index j(i) and j(i+1) in m ′′ are only X's and we can thus apply E identifying B is well defined and we can apply E inductively.
In this way, we have a canonical map To state the algebraic and differential properties we will use, we also need the following variant (for C = C or C = D): B op(l) {X 1 , ..., X n : E D , R, C} where M ′ 2k (X 1 , ..., X n ; Z 1 , ..., Z l ; Y ) is the set of monomials linear in each Z i , without constraint on the order of appearance of Z 1 , ..., Z n and of order 2k in Y The blocks in Z i are made to evaluate a variable in D ′ ∩ N. We call B ⊗(l) {X 1 , ..., X n : E D , R, C} the subspace involving monomials with Z k ordered in increasing order of k and with all variables Z i having an even number of Y before them and with their pair partitions unions of those restricted to the intervals between them (thus Z i 's are interpreted as not being inside conditional expectations.) We write B ⊗(l)c {X 1 , ..., X n : E D , R, C} the cyclic variant generalizing B ⊗(l)c X 1 , ..., X n : E D , R, C .
The following result is obvious: Proposition 33. Let X = (X 1 , . . . , X n ).   1,1 ,...,j l 1,i 1 ) ) Q n 1 , d i 2 −i 1 X(j l 2,1 ,...,j l 2,i 2 −i 1 ) Q n 2 , ..., d k−i l−1 X(j l l,1 ,...,j l l,k−i l−1 and in particular: [Note the sum of l i,j in formula (42) is only a sum over partitions, the first term of the first set being written l 1,1 , the first term of the second set in the partition l 2,1 , the ordering between sets in the partition being by the ordering of the smallest element] Proof. For the most part, we only have to give a combinatorial formula for the derivations acting on monomials. Then by the bimodularity of the formula and explicit uniform bounds, the universal property of the ℓ 1 sum will extend them to module ℓ 1 direct sums. They will be moreover weak-* continuous as soon as they are weak-* continuous when restricted to monomial components since the c 0 sum of predual maps will then give a predual map. The derivation properties then determine d, ∂ on the E D -algebra generated by B, X 1 , ..., X n which is weak-* dense in the ℓ 1 direct sum (actually in each monomial space by properties of the extended Haagerup product and then, the finite sum of monomial spaces are normwise dense), thus weak-* continuity determine those maps everywhere.
For σ ∈ NC 2 (2k), m ∈ M 2k (X 1 , ..., X n , Y ), let us say a submonomial m ′ ⊂ m (with a fixed starting indexed, m ′ is thus formally a pair of the monomial and the starting index) is compatible with σ and write m ′ ∈ C(σ, m) if m ′ ∈ M 2l (X 1 , ..., X n , Y ), l ≤ k and l ′ the index in m of the first Y in m ′ , then σ| m ′ := σ| [[l ′ ,l ′ +2l−1]] ⊂ σ (which means there is no pairing in m broken in m ′ by our extraction of m ′ ). We then write Sub(σ, m ′ ) ∈ NC 2 (2l) the partition Then we define: Of course the sum is 0 if its indexing set is empty, this in particular explains ∂ i E D = 0 and the remaining properties are easy.
The definition of d is complementary. When m ′ or m ′′ are not both in C(σ, m) and m = m ′ X i m ′′ (some i), we write (m ′ , m ′′ ) ∈ IC(σ, m) (and this corresponds to a differentiation of X i below a conditional expectation).
Finally, we will need a second order operator and its commutation with cyclic gradients.
Proposition 35. There are continuous maps ∆, δ ∆ on B{X : E D , R}→ B{X : E D , S} for S < R uniquely defined as weak-* continuous map by the following properties (a) and (b): (a) For P ∈ B{X : E D , R} monomial and ∆E D = 0 (b) δ ∆ is a derivation, δ ∆ (P ) = 0 for any P ∈ B X : D, R , and for Q monomial in B{X : (c) Moreover, (d) Likewise, for any V ∈ B X : D, R , the map ∆ V = ∆ + i ∂ i (.)#D i V produces a derivation δ V such that δ V (P ) = 0 for P ∈ B X : D, R and for Q monomial in B{X : ). Moreover, for any g ∈ B X : D, R : Proof. Again it suffices to define those D − D bimodular maps on monomials spaces, i.e., at the level of extended Haagerup tensor products. Then the universal property of the direct sum will extend them as weak-* continuous maps as soon as each component map is weak-* continuous. The algebraic relation then determines the maps on the E D algebra generated by B, X 1 , ..., X n and weak-* density of this algebra implies the uniqueness of the weak-* continuous extension. For ∆ we use the formula above. Let σ ∈ NC 2 (2k), m ∈ M 2k (X 1 , ..., X n , Y ).
All properties but the last equation (43) are easy. By definition, we have: = n j=1 m=nX j n ′ X j n ′′ ,n ′ ∈C(σ,m) nY n ′ Y n ′′ =m ′ X i m ′′ ǫ m ′′ m ′ ,ρ |m ′′ | Y (Add(σ,n,n ′ ,n ′′ )) ((ρ |m ′′ | .(P ))) The sums can be divided into 3 cases depending on whether X i ∈ n, n ′ , n ′′ . Similarly, we have and there are also 3 cases depending X j 's are both in M ′′ , in M ′ or one in each. The proof of the equality is combinatorial, we check we have a bijection of the indexing sets of the sum, with equality of the terms summed in each case.
If X i ∈ n ′ , n ′ = oX i o ′ and m = nX j oX i o ′ X j n ′′ , m ′ = nY o, m ′′ = o ′ Y n ′′ . This suggests, M ′ = nX j o, M ′′ = o ′ X j n ′′ corresponding bijectively to a term where one X j is in M ′′ the other in M ′ with N = o ′ , N ′ = n ′′ n, N ′′ = o. Since N ′′ is related to a complement of n ′ , the relations imposed on n ′ , N ′ are equivalent after rotation. We also have m ′′ m ′ = NY N ′ Y N ′′ as expected, |M ′′ | = |m ′′ | implying the same rotation of P and Add(ρ |M ′′ | Y (σ), N, N ′ , N ′′ )) = ρ |m ′′ | Y (Add(σ, n, n ′ , n ′′ )), as is easily checked, implying the final equality.
6.3. Non-commutative C k,l -functions and their stability properties.
6.3.1. C k,l norms. As in the main text, we consider several variants C k,l;ǫ 1 ,ǫ 2 tr,V (A, U : B, E D ), ǫ 1 ∈ {0, 1}, ǫ 2 ∈ {−1, 0, 1, 2}: We of course also define a first order part seminorm P C k,l;ǫ 1 ,ǫ 2 tr,V (A,U :B,E D ),≥1 only replacing the first term in the sum by ι(P ) k,l,U ≥1 . Note that P In the last seminorm we considered P in variable X = (X 1 , ..., X n ) and Q in variable X ′ = (X ′ (1) , ..., X ′ (m−1) ) ∈ U m−1 and U m ⊂ A mn R = (A n R ) m . In order to get a consistent definition, we still have to check the last term is finite for P ∈ B c {X 1 , ..., X n ; E D , R, C}. We gather this and a complementary estimate in the following Lemma. A variant explains the inclusion C k,l c ⊂ C k,l tr,V,c at the end of subsection 2.4 with norm equivalent to the restricted norm (explaining why the completions are included in one another) Lemma 36. Assume U ⊂ A n R,appB−E D . For any P ∈ B c {X 1 , ..., X n ; E D , R, C}, we have sup and moreover if P ∈ B c X 1 , ..., X n ; D, R, C , for any p ≥ 0 we have: (1)1 , ..., X ′ (1)n , ..., X ′ (m)1 , ..., X ′ (m)n ; E D , R + , C}, m ≥ 1 X ′ = X ′ (1)1 , ..., X ′ (1)n , ..., X ′ (m)1 , ..., X ′ (m)n . We detail only the second estimate, since the first one mainly needs P monomial and is an easy extension.
To compute differentials we introduce partial differentials d s (X,X ′ )(r 1 ,...,rs) so that a full differential is Recall this d s (X,X ′ ) is the full differential so that d s X applied to P ∈ B c X 1 , ..., X n ; D, R, C is a certain expression involving free difference quotients but is not necessarily 0 (unlike d s by its definition).
We have to compute as easily checked on monomials, for s, l ≤ k − 1 and note there is no real sum to split the derivatives between P, Q (the sum can contain only one non-zero term) since the variables of Q and P are not the same.
Using this remark and the natural bound on products defined in Proposition 28, one gets the term in the seminorm to estimate for a fixed order s of differentials d s : The factor k appears for a the same reason as the sum over V , because in the sum over j (resp. over r) the position of differentials X, X ′ need to be determined by a starting point for the block of X ′ variables (resp. a set of X variables) and in the first case the number is less than l ≤ k.
Thus taking suprema in the definition of seminorms, one gets the concluding result for any p: and similarly The definition of the two bounded linear maps are then straightforward and injectivity comes from the fact that the bounds enable us to get equivalent norms on the image so that the separation completion defining the first space can be computed in the second. 6.3.2. Composition of functions. To understand the relationship between the Laplacian and composition of functions we need the following basic remark. Let P, Q 1 , ..., Q n ∈ ∪ R>0 B c {X 1 , ..., X n : E D , R, C}. Then: Thus we have a lack of stability of the form of the second order term so that it is natural to introduce for P ∈ B{X 1 , ..., X n : 2 )] n 2 the following expression: and similarly In this way one gets As before we can also define δ R as a derivation Finally to deal with our universal norms we need to consider in what space of variables our functions are valued to handle composition properly. For this consider U ⊂ A n R , V ⊂ A n S sets, S ≥ R and C a class of functions on U as before or one defined later, B C the space of analytic function (either B c {X 1 , ..., X n ; E D , R + , C} for classes with index tr or B c X : D, R, C or ∩ T >R C l+1 b (A n R , B c X 1 , ..., X n : D, T, C ) for classes with index u etc.) used to define it as a separation-completion with canonical map ι : B C → C. We define two candidates of sets admissible for composition which are subspaces of Comp(U, A n S , C) . We first define composition on the dense sub- ≤ T for some T ≥ S one can apply the definition of composition at analytic level from Propositions 29, 33. If P ∈ B c {X 1 , ..., X n ; E D , S + , C}, P defines X → P (E D,X ) on any V ⊂ A n S , so that we can define P (Q 1 , ..., Q n ) assuming only Q i (X) < S (case V = A n S above). We can now extend these maps. We first deal with the cases of stability by compositions and then deal with the variants we used in the main texts obtained via various compositions with canonical maps.
Lemma 37. Fix V, U as above with U ⊂ V (with V ⊂ A n R,U ltraApp as soon as a space with index c is involved). The above map (P, Q 1 , ..., Q n ) → P (Q 1 , ..., Q n ) extends continuously to Q 1 , ..., Q n ∈ Comp − (U, V,C k,l u (A, U : B, E D )) to give a map Although the case o ∈ [[1, max(0, l − 1)]] is not needed in this paper, it can be treated similarly but this is left to the reader.
Proof. Note first that for composition on Comp − we can extend the first definition of composition since then we have approximate Q ∈ ι(B) n with Q(X) ∈ V . For all extension to Comp we use the second definition since we can start from P ∈ B c {X 1 , ..., X n ; E D , S + , C} by density in the corresponding spaces. As we will see, we will always extend first in Q, and for P fixed as above this extension can be done with V = A n S , using Comp(U, A n S , C) = Comp − (U, A n S , C) (since A n S open and using compatibility with the topology of considered C) and then restrict this first extension to our space Comp(U, V, C) ⊂ Comp(U, A n S , C). We have to estimate various norms using (40)  one checks (using we started from one more derivative on U than necessary, namely l + 1 instead of l) that (Q 1 , ..., Q n ) → P (Q 1 , ..., Q n ) is uniformly continuous (on balls) thus extends by uniform continuity to Comp − (U, V, C k,l u (A, U : B, E D )). Obviously, if one does not care about constants, we have from the previous computation, a bound of the form P (Q 1 , ..., Q n ) k,l,U ≤ C(k, l, n) P k,l,V 1 + max i=1,...,n Q i k,l,U k+l thus P → P (Q 1 , ..., Q n ) is Lipschitz with value in the space continuous functions with supremum norm on Q i and thus extend to all P in the space C k,l u (A, V : B, E D ). This concludes to the extension part. Note that one deduces from the computations above the estimate of independent interest : (47) For the Lipschitz property, the only problematic term in the expression above is the composition d s X(r 1 ,...,rs) [(∂ l (o 1 ,...,o l ) (P ))(Q 1 , ..., Q n ). We note that under the supplementary assumption of differentiability for P , it is always differentiable with differential i d s+1 X(r 1 ,...,rs,i) [(∂ l (n 1 ,...,n l ) (P ))(Q 1 , ..., Q n )(·, ..., ·, H i ).
The conclusion follows by the fundamental Theorem of calculus. Now, the case of C k,l tr spaces is obvious because P (Q 1 , ..., Q n ) exactly comes from the composition in Proposition 33 and the discussion at the beginning of the proof to deal with Comp. C k,l tr,c is also a variant. We now turn to the spaces C k,l;0,ǫ 2 tr first with ǫ 2 = 1. For P fixed analytic, the extension in Q i is as easy as before (using the estimate below), it remains to prove uniform Lipschitz property in P . Recall the basic formula (41) and since in our case D Q i ,R (P )(Q 1 , ..., Q n ) ∈ C k,l tr (A, U m−1 : B, E D ) we have the following bound for p ≤ l : sup where we of course took the variables S = D Q j ,R(X ′ ) (P )(Q 1 , ...Q n ), X ′′ = (X ′ , X) ∈ U m , and used D i,D Q j ,R(X ′ ) (P )(Q 1 ,...Qn) (Q j )) k−1,p,U m ≤ D i,S(X ′′ ) (Q j )) k−1,p,U m+1 . And from a variant with parameter of our previous estimates for the change of variable (Q 1 (X), ..., Q n (X), X ′ ) (based on the fact that no additional sum related to composition is involved for the variables X ′ so that the constant C(k − 1, p, n) below only involves the number of variables of X), the last term is bounded by This gives the expected Lipschitz bound in P (using U ⊂ V in taking Q j (X) = X j ) for the part with cyclic gradients. The Lipschitz property in Q is dealt with as before.
We now consider the case ǫ 2 = 0. In this case the norm becomes and thus we can use the estimate (48) with R = 1 to conclude. We now turn to the case ǫ 2 = −1. In this case the norm becomes The term l p=0 n i=1 D i,1 (P ) k,p,U is controlled by the similar term (with summation up to l) in (44) which gives the norm of Q (noting that ǫ 2 ∨ 1 = 1). The other terms are treated as before.
Finally, we consider the case ǫ 2 = 2. This time 1 ∨ ǫ 2 = 2 and the summation over p goes up to l − 1; thus we can use essentially the same estimate as in (48)  so that one gets : This enables the extension in P after extension in Q if k, l ≥ 1 and gives the Lipschitz property in Q on bounded set as required (using o became o+1 for dealing with the annoying new term). The case ǫ 1 = −1 is possible because taking R kl = 1 k=l (1ø1)ø(1ø1) recovers the Laplacian and using a general R on the P variable enables to deal with the particular case (and remove the sup) for Q, P • Q variables.
Corollary 38. In the setting of the previous Lemma (in particular for U ⊂ V ⊂ A n R,U ltraApp ), for any l ≥ 1 (and k ≥ 2 in any case with W ) the map (P, Q 1 , ..., Q n ) → P (Q 1 , ..., Q n ) also of the definition of conjugate variables. The extension to C k,l tr,V (A, U), k ≥ 2, l ≥ 1 is then obvious by norm continuity of the various maps.
(2) We first need to extend (14) to V ∈ C k+1 c (A, A n R,conj : B, D), still for g = P ∈ B c {X 1 , ..., X n : E D , R, C}. If one uses the notation after this formula extending the definition of ∆ V + δ V to these values of V and notes from the formula (41) for cyclic gradient of compositions above (extended beyond analytic functions since [∆ V 0 (Z) + δ V 0 (Z) ](P ) is a noncommutative analytic function with expectation and we can use the composition Lemma as in the proof of Proposition 11), one gets the expected relation: where we used (14) for the extra variables Z to get It now remains to extend the relation in P to apply it to our g.
For the second statement we check that the map g → (∆ V + δ V )(g) is bounded for g analytic function with expectation between the spaces ∆ V + δ V : C k+2,2 tr,V (A, U : B, E D ) → C k,0;0,−1 tr,V (A, U∩A n R,conj1 : B, E D ), where the identity has just been checked. We need to bound the k-th order free difference quotient of h and Dh. We of course use Dg is controlled in C k+2,1 (A, U : B, E D ) thus by closability we can apply a k-th order free difference quotients to the relation for Dh (using Lemma 36 for the term with second order derivative on V ). We can also apply a k-th order free difference quotient to the formula for h, each time using the relation for δ V (g) in terms of differential. The bounds are now easy using for the term ∂δ V the identity checked before in (1) in any representation for δ V and commutation of ∂ and d.
6.4. Free Difference Quotient with value in extended Haagerup tensor products. We now consider closability properties of the free difference quotient with value in the extended Haagerup tensor product.
Finally, if F ∈ C k,0 tr (A, A n R,conj(1 k≥1 /2+1 k≥3 /2+1 k≥4 ) : B, E D ) and X i ≤ R then F (X) ∈ D(∂ k This shows the first result. The reasoning for ∂ i ø D 1, 1ø D ∂ i is similar. To check the derivation property it suffices to take bounded nets U n → U, V ν → V and to use the weak-* continuity of .# K . obtained in Proposition 27 from Theorem 23. (2) in order to take the limit successively in n, ν of (2) The second result is the relative variant of [Dab08, Remark 11, Lemma 12].
(3) The third result then follows similarly from the first using also the second result. It always suffices to show weak-* closability from M (or L 2 (M)) with value an L 2 tensor product, for which one needs densely defined adjoints with value L 1 (M) or L 2 (M) respectively.
We detail only the case k = 2, 3. From Voiculescu's formula, for a, b, c, d ∈ B X 1 , ..., X n , one deduces: where the second line is in M and the third in L 2 (M) by the second point as soon as the first order conjugate variables are in M (resp. both in L 1 (M) by the second point as soon as the first order conjugate variables are in L 2 (M)). This gives the various statements in case k = 2.
Likewise, we have : The higher order terms are then similar to this last case when we have both first and second conjugate variables in M. All the higher adjoints are then valued in L 2 (M) on basic tensors from B X 1 , ..., X n .
For the compatibility with C k spaces, the non-commutative analytic functionals are clearly in the domain and the extension by density is straightforward (even with norm instead of weak-* convergence which is used at the analytic function level though).
(4) For the fourth statement the M valued extension only involves application of canonical maps associated to Haagerup tensor product to mimic the formula above. For the second part of the fourth statement, we extend each term of the formula above. First we Lemma 43.(2)]). We next write down explicit bounds for the last L 2 (M) valued extension. From the Cauchy-Schwarz inequality for Hilbert modules one gets ( j a j ξ and moreover We also have similarly for u > 0 (2) Moreover, the extension result of Corollary 38 is also valid for any U ⊂ A n R,conj0 , V ⊂ A n S,conj0 giving composition maps •: Proof. By density, it suffices to prove contractivity restricting to the polynomial variant of the space C 0 b,tr (U, B X 1 , ..., X n : D, R ). But if P is in the partial evaluation η S (B c {X 1 , ..., X n , S t 1 , ..., S tm − S t m−1 : B, E D , max[R, max i=2,n 2(t i − t i−1 )]C})}, it is easy to see by definition of free semicircular variables with amalgamation that E A (P (E D,X )) = Q(E D,X ) for some Q ∈ B{X 1 , ..., X n : B, E D , R}. Q is the same as P where brownian variables are replaced by sums over formal conditional expectations.
Then, the conditional expectation is obtained by replacing with pairings and conditional expectations the brownian variables in an appropriate way so that we define with for convenience t 0 = 0 : so that the relation above E A (P (E D,X )(X)) = (E B (P ))(E D,X )(X), X ∈ A n R is easy to check by definition of free Brownian motions. Note that (51) (∆ + δ ∆ )(E 0 (P )) = E 0 ((∆ + δ ∆ )(P )) (where of course ∆ only applies on X i variables) since, using the definition in the proof of Proposition 35, both expressions correspond to having a supplementary sum over pairs of X i variables giving a partition not crossing the previous ones and replaced by a formal E.
Using relation (49) with e, H i in the smaller algebra A, one sees that for e ∈ A, Indeed, for e, P monomials, since e has no dependence in S t 's, there is a bijection between pairs of S t 's appearing in each monomial after and before applying D i,e . Since cyclic permutations keep non-crossing partitions the result is thus an easy combinatorial rewriting. It thus remains to check contractivity estimates to extend E 0 to spaces of C k functions. For X ∈ U, P as before ∂ l i (Q)(E D,X )(X) = ∂ l ∂ l i (E 0 (P ))(X) A ø ehD(l+1) ≤ (∂ l i (P ))(X) A ø ehD(l+1) . Here it is crucial to note that for all cyclic variants that by Proposition 28.(3) if ∂ l i eh [(P (E D,X )(X))] is in a cyclic extended Haagerup tensor product, it remains there after application of (E ø eh l+1 A ).
Likewise, the full differential commute with conditional expectation (which is a linear bounded map, we thus get the bound for all parts of the seminorm involving free difference quotients and full differentials. We thus proved contractivity on C k,l tr -spaces. Since (∆ V + δ V )(P ) = (∆ 0 + δ ∆ )(P ) + d X P. (D 1 V, ..., D n V ) the previous results give (∆ V + δ V )(E B (P )) = E B ((∆ V + δ V )(P )) so that since in this case k ≥ 2, the choice of the seminorm chosen with this term is compatible with contractivity. The contractivity of the term with cyclic gradients is also easy with the previous established commutation relation, so that one gets the stated contractivity on C k,l tr,V -spaces. Obtaining multiplication maps is as easy as before in this context and by arguments of stability of subspaces for C k c -spaces. The variant E u and its relations are obvious. 6.6. Regular Change of variables for Conjugate variables. The computation of conjugate variables along change of variables we used to identify conjugate variables of our transport maps are explained in the next Lemma 43 with the differentiation along a path of such change of variables.
Let M = W * (X 1 , ..., X n , B) for (X 1 , ..., X n ) ∈ (A, τ ). We will soon assume those variables have enough conjugate variables relative to D in presence of B.
Take P ∈ B X 1 , ..., X n , D, R, C , R ≥ max(S, sup X∈U ′ F i (X) ), then P (Y ) satisfies the natural extension of formula (45) from the proof of Lemma 37 and so we get the equation (∂ j (P ))(Y )#∂ i,X Y j .
Note that from the assumption on (J F ) ij = ∂ j,X Y i , one deduces that J F is invertible in M n (M eh ⊗ D,c M) so that one gets: Thus applying the weak-* continuity of Theorem 23.
Then, if both sides extend to compact operators, one obtains the claimed equality. We already said the left hand side does (for instance by our previous bound on e −tH(X 1 ) obtained from the series expansion) and the right hand side will by our next bound giving the contractivity property. Thus, for instance from [M05], when µ ≥ 3λ 2 8ν : c n (X 1 )c n (X 1 ) * || But note that for ξ ∈ L 2 (M), with (e j ) j∈IN an orthonormal basis of this space, we first get using Parseval equality and Tonelli Fubini Theorem to switch the sum over j: IR dγ(σ) e −tY 2 1 +i √ tνσ(X 1 + λ 4ν ) ξ, e −tY 2 1 +i √ tνσ(X 1 + λ 4ν ) ξ = j IR dγ(σ) e −tY 2 1 +i √ tνσ(X 1 + λ 4ν ) ξ, e j e j , e −tY 2 1 +i √ tνσ(X 1 + λ 4ν ) ξ = j n | e j , c n (X 1 )ξ | 2 = n j | e j , c n (X 1 )ξ | 2 = n c n (X 1 )ξ 2 = ξ, ∞ n=0 c n (X 1 ) * c n (X 1 )ξ 88 where the third line is obtained by using Parseval equality again this time on L 2 (dγ), and again Fubini-Tonelli and Parseval. Thus, we got, since ν > 0: ∞ n=0 c n (X 1 ) * c n (X 1 ) = 1 √ 2π IR dσe −σ 2 /2 (e −tY 2 1 +i √ tνσ(X 1 + λ 4ν ) ) * e −tY 2 1 +i √ tνσ(X 1 + λ 4ν ) = 1 √ 2π IR dσe −σ 2 /2 e −2tY 2 1 is a contraction and so is ∞ n=0 c n (X 1 )c n (X 1 ) * . Finally, from (35), it is easy to see in truncating the series that σ(e −tH(X 1 ) ) = e −tH(X 1 ) and this concludes to : In order to deduce a more general example, we need to describe more explicitly the norm is of course the one we built in Proposition 28 (1). By density of the algebraic tensor product, it suffices to get a contractivity on basic tensors. Since the target norm is induced form M ø eshcD m , it suffices to get the contractivity with this target space. This decomposes in various contractivity for each flip (using the fonctoriality of nuclear tensor product). We thus have to see that # : This is obvious from complete contractivity of composition of CB maps.
Moreover, let P = P * ∈ C u 1 , ..., u n a * -polynomial in unitary variables, and define for ε > 0 Then, for any R > 0 and any c ′ ∈ [0, c), there exists ε R > 0 so that for ε ∈ [−ε R , ε R ], W ∈ C 6 c (A, R : B, D) is (c ′ , R) h-convex. Proof. From the additivity of positivity, the positivity elements form a cone, so that it suffices to consider k = 1 and even to show that W (X) = υ 1 ( n i=1 λ i,1 X i ) is (0,R) convex. But with the notation of the previous proof ∂ i D j W = λ j,1 λ i,1 H( n i=1 λ i,1 X i ). We finally consider the case where the polynomial is pertubed. In order to check that V ∈ C 6 c (A, R : B, D), since this space is obviously an algebra, it suffices to check P t (X) = 1 t √ −1−X 1 ∈ C 6 c (A, R : B, D) for t > 0. For t large enough, a geometric series converging in C 6 c (A, R : B, D) shows this. The set of such t is thus non-empty, it is easy to check that C 6 c (A, R : B, D) has an equivalent Banach algebra norm, then, a Neumann series gives the set of t is open. It remains to see it is closed in ]0, ∞[ to get the result by connectivity. An easy computation shows that ||P t || 6,0,A n R ≤ 6 k=0 1/t k+1 as soon as we showed P t is in the space above, since ∂ k (1,...,1) P t (X 1 ) = (k!)P t (X 1 ) ⊗k+1 . When t n → t > 0, and using one easily gets the convergence ||P tn − P t || 6,0,A n R → 0 (in getting a Cauchy sequence and identifying the limit with P t ). It only remains to check the stated h-convexity. It suffices to take the coefficients of P small enough so that b = (∂ i D j (V − V )) j,i has a norm ||b|| := ||b|| Consider the probability on (M N (C) sa ) n given (for some normalization constant Z V,N ) by : As in [GMS06], we use an integration by parts formula on µ V,N which gives ∀P ∈ C X 1 , ..., X m : and the second concentration result in Proposition 50 implies that the right hand side converges when N → ω to (τ X ω ⊗ τ X ω )(∂ i (P )). One thus obtains the relation in taking of limit to ω of the integration by parts relation. Moreover, note that this implies τ X ω has finite Fisher information.
Step 3 : Properties and use of the SDE.
Let X 0 = X ω or a R ω -embeddable solution of (SD V ), which ensures X 0 ∈ A n R/3,App in the scalar case B = C. The application of our Proposition 5 thus gives a unique solution X t (X 0 ) on [0, ∞[ solving Considering another solution starting at Y 0 , one obtains: ||X t (X 0 ) − X t (Y 0 )|| 2 2 ≤ e −ct ||X 0 − Y 0 || 2 2 . Then exponential decay implies that the laws τ Xt(X ω ) and τ Xt(X ω ′ ) are arbitrarily close for t → ∞ and since they are equal to τ X ω and τ X ω ′ by stationarity, one deduces that X ω have the same law for any ultrafilter. Similarly, (SD V ) has a unique R ω -embeddable solution and the exponential decay implies a stationary state for the SDE is unique too.
Step 4 : Conclusion on the limit of E µ V,N • τ . .
The law E µ V,N • 1 N T r is close to E µ V,N • τ B N for N large enough and this second law lies in the compact set S n C (tracial state space of the universal free product C([−C, C]) * n with the weak-* topology) and from the result on ultrafilter limits the sequence has a unique limit point there (any such limit point being a τ X ω ). We thus deduce by compactness the claimed convergence.
Corollary 52. Let V, V + W be of the form of V in Lemma 49, and thus (c, R) h-convex for all R and some c > 0. Then they satisfy Assumption 16.
Proof. The application of the previous Theorem gives existence of solution of (SD Vα ), α ∈ [0, 1] which is R ω -embeddable or equivalently L(F ∞ ) ω -embeddable which is a reformulation of A n R,U ltraApp in the case B = C. Everything else comes from Lemma 49 and stability of (c, R) h-convexity under taking convex combinations.