Serban T Belinschi

Victor Vinnikov

Victor Vinnikov Noncommutative

Noncommutative hyperbolic metrics

published or not. The documents may come

Introduction

A noncommutative function is a function, defined on a domain in the disjoint union of square matrices of all sizes over a vector space, that satisfies natural compatibility conditions: it respects direct sums and similarities. Noncommutative function theory is the free analogue of classical function theory, much like operator space theory [START_REF] Effros | Operator Spaces[END_REF][START_REF] Paulsen | Completely Bounded Maps and Operator Algebras[END_REF] is the free analogue of classical Banach space theory. Noncommutative functions were first introduced by Taylor [START_REF] Taylor | A General Framework for a Multi-Operator Functional Calculus[END_REF][START_REF] Taylor | Functions of several noncommuting variables[END_REF] in his monumental work on noncommutative spectral theory. Their theory was further developed by Voiculescu [START_REF] Voiculescu | Free Analysis Questions I: Duality Transform for the Coalgebra of ∂ X:B Internat[END_REF][START_REF] Voiculescu | Free analysis questions II: The Grassmannian completion and the series expansions at the origin[END_REF], and, in a systematic fashion, by Kalyuzhnyi-Verbovetskyi and the second author [START_REF] Kaliuzhnyi-Verbovetskyi | Foundations of free noncommutative function theory[END_REF]. We mention also the work of Helton, Klep, and McCullough (see, e.g., [START_REF] Helton | Analytic mappings between noncommutative pencil balls[END_REF][START_REF] Helton | Proper Analytic Free Maps[END_REF]), of Popescu (see, e.g., [START_REF] Popescu | Free holomorphic functions on the unit ball of B(H) n[END_REF][START_REF] Popescu | Free holomorphic automorphisms of the unit ball of B(H) n[END_REF]), of Muhly and Solel (see, e.g., [START_REF] Muhly | Progress in noncommutative function theory[END_REF]), and of Agler and McCarthy (see, e.g., [START_REF] Agler | Aspects of Non-commutative Function Theory[END_REF]). Noncommutative functions appear naturally in a large variety of settings: noncommutative algebra, systems and control, spectral theory, and free probability. They possess very strong regularity properties (reminiscent of the regularity properties of usual analytic functions) and admit a good difference-differential calculus.

At least some analytic aspects of noncommutative function theory are now quite well understood (see, e.g., [START_REF] Pascoe | The noncommutative Löwner theorem for matrix monotone functions over operator systems[END_REF][START_REF] Shamovich | On fixed points of self maps of the free ball[END_REF][START_REF] Ball | Interpolation and transfer-function realization for the noncommutative Schur-Agler class[END_REF] and the references therein for some recent developments). The purpose of this paper is to move in the direction of noncommutative geometric function theory by introducing a free analogue of Kobayashi (pseudo)metric for general noncommutative domains.

We review the basics of noncommutative functions, noncommutative kernels, and topologies for noncommutative sets in Section 2. Following that, we introduce in Section 3 a "noncommutative length function" δ D for a noncommutative set D (satisfying some mild assumptions); δ D reflects in a natural way the "noncommutative geometry" of D. It follows immediately that any noncommutative function between two noncommutative sets is contractive with respect to the corresponding noncommutative length functions. We show that the noncommutative length function δ D is nondegenerate if and only if there are no nonconstant noncommutative functions from the whole noncommutative space over C to the multiples of a given matrix level k in D, for all k ≥ 1. We use the noncommutative length function to define a pseudodistance on D in several ways; we mimick the definition of the Kobayashi pseudodistance using either the Lempert This work was initiated, and largely completed, during the 2016 Thematic Semester in Analysis, funded by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02.

function or the infinitesimal Kobayashi pseudometric, and possibly allowing to go to higher matrix levels.

Our definition of the noncommutative length function is motivated by [START_REF] Belinschi | A noncommutative Version of the Julia-Wolff-Carathéodory Theorem[END_REF]; it would be interesting to clarify possible connections with [START_REF] Agler | Operator theory and the Carathéodory metric[END_REF].

In Section 4, we consider the large class of noncommutative sets that are defined by noncommutative kernels. These can be thought of as analogues of the sets {ρ(z, z) > 0} for some (global) defining function ρ in several complex variables. In this case it is possible to obtain explicit formulae for the noncommutative length function by applying the noncommutative difference-differential operators to the kernel. This applies in particular to generalized noncommutative balls (or halfplanes) defined by a noncommutative function that were investigated in [START_REF] Agler | Global holomorphic functions in several non-commuting variables[END_REF][START_REF] Agler | Pick interpolation for free holomorphic functions[END_REF][START_REF] Agler | Global holomorphic functions in several non-commuting variables II[END_REF][START_REF] Ball | Interpolation and transfer-function realization for the noncommutative Schur-Agler class[END_REF] as suitable domains for interpolation problems. For these, the noncommutative distance turns out to be dominated by the usual Kobayashi distance on every matrix level, with equality for noncommutative balls (or halfplanes) over a C * algebra.

In Section 5, we seek to apply the noncommutative metric to noncommutative function theory. We show that under some assumptions, isometries with respect the noncommutative length function are exactly bijective noncommutative mappings, and a noncommutative mapping from a noncommutative set to a subset that is at a positive distance from the boundary of the set has a unique attracting fixed point.

Finally, in Section 6, we use the noncommutative metric on the noncommutative upper half plane over a finite von Neumann algebra to solve the functional equation ω(b) = b + h(ω(b)) for h a noncommutative self map of the noncommutative upper halfplane to itself satisfying a rather mild vanishing condition at infinity. This solves the problem of defining free convolution powers of distributions of unbounded selfadjoint random variables in free probability.

2. Noncommutative domains, functions and kernels 2.1. Noncommutative functions. Noncommutative functions originate in Joseph L. Taylor's work [START_REF] Taylor | A General Framework for a Multi-Operator Functional Calculus[END_REF][START_REF] Taylor | Functions of several noncommuting variables[END_REF] on spectral theory and functional calculus for k-tuples of noncommuting operators. We largely follow [START_REF] Kaliuzhnyi-Verbovetskyi | Foundations of free noncommutative function theory[END_REF] in our presentation of noncommutative sets and functions. We refer to [START_REF] Kaliuzhnyi-Verbovetskyi | Foundations of free noncommutative function theory[END_REF] for details on, and proofs of, the statements below.

Let us introduce the following notation: if S is a nonempty set, we denote by S m×n the set of all matrices with m rows and n columns having entries from S. If S = F is a field, then we use the standard notation GL n (F) for the group of matrices X in F n×n which are invertible (that is, there exists X -1 ∈ F n×n such that XX -1 = X -1 X = I n , where I n is the diagonal matrix having the multiplicative unit of F on the diagonal and zero elsewhere). We will work almost exclusively with subsets of operator spaces and operator systems (linear subspaces of the algebra B(H) of bounded operators over a Hilbert space H -which we assume to be separable -which contain the unit 1 of B(H), are norm-closed and selfadjoint -see [START_REF] Effros | Operator Spaces[END_REF]). However some of our definitions hold in much broader generality. Given a complex vector space V, a noncommutative set is a family Ω nc := (Ω n ) n∈N such that (a) for each n ∈ N, Ω n ⊆ V n×n ; (b) for each m, n ∈ N, we have Ω m ⊕ Ω n ⊆ Ω m+n .

The noncommutative set Ω nc is called right admissible if in addition the condition (c) below is satisfied:

(c) for each m, n ∈ N and a ∈ Ω m , b ∈ Ω n , w ∈ V m×n , there is an ǫ > 0 such that a zw 0 b ∈ Ω m+n for all z ∈ C, |z| < ǫ.

Left admissible sets are defined similarly, except that zw appears in the lower left corner of the matrix. Given complex vector spaces V, W and a noncommutative set Ω nc ⊆ ∞ n=1 V n×n , a noncommutative function is a family f := (f n ) n∈N such that f n : Ω n → W n×n and

(1)

f m (a) ⊕ f n (b) = f m+n (a ⊕ b) for all m, n ∈ N, a ∈ Ω m , b ∈ Ω n ;
(2) for all n ∈ N, f n (T -1 aT ) = T -1 f n (a)T whenever a ∈ Ω n and T ∈ GL n (C) are such that T -1 aT belongs to the domain of definition of f n . These two conditions are equivalent to the requirement that f respects intertwinings by scalar matrices:

(I) For all m, n ∈ N, a ∈ Ω m , b ∈ Ω n , S ∈ C m×n , we have If V, W are operator spaces, it is shown in [START_REF] Kaliuzhnyi-Verbovetskyi | Foundations of free noncommutative function theory[END_REF]Theorem 7.2]) that, under very mild openness conditions on Ω nc , local boundedness for f implies each f n is analytic as a map between Banach spaces. More specifically, if Ω nc is finitely open (that is, for all n ∈ N, the intersection of Ω n with any finite dimensional complex subspace is open) and f is locally bounded on slices (that is, for every n ∈ N, for every a ∈ Ω n and b ∈ V n×n , there exists an ε > 0 such that the set {f n (a+zb) : z ∈ C, |z| < ε} is bounded in W n×n ), then each f n is Gâteaux complex differentiable on Ω n (see Section 2.3 below). Indeed, this is a consequence of the following essential property of noncommutative functions: This implies in particular that f n+m extends to the set of all elements a b 0 c such that [START_REF] Kaliuzhnyi-Verbovetskyi | Foundations of free noncommutative function theory[END_REF]Section 2.2]). Two properties of this operator that are important for us are

if Ω nc is admissible, a ∈ Ω n , c ∈ Ω m , b ∈ V n×m
a ∈ Ω n , c ∈ Ω m , b ∈ V n×m (see
(3) ∆f n,n (a, c)(ac) = f (a)f (c) = ∆f n,n (c, a)(ac), ∆f n,n (a, a)(b) = f ′ n (a)(b), the derivative of f n in a aplied to the element b ∈ V n×m . Moreover, ∆f (a, c) as functions of a and c, respectively, satisfy properties similar to the ones described in items (1), (2) above -see [START_REF] Kaliuzhnyi-Verbovetskyi | Foundations of free noncommutative function theory[END_REF] for details (for convenience, from now on we shall suppress the indices denoting the level for noncommutative functions, as it will almost always be obvious from the context).

Example 2.1. There are many examples of noncommutative functions. We provide here three.

(1) The best known is provided by the classical theory of analytic functions of one complex variable: if D is a simply connected domain in C and f :

D → C is analytic, then f is the first level of an nc map f : ∞ n=1 {A ∈ C n×n : σ(A) ⊂ D} → ∞
n=1 C n×n given by the classical analytic functional calculus:

f n (A) = (2πi) -1 γ (A -ζI n ) -1 f (ζ)
dζ, for some simple closed curve γ which surrounds once counterclockwise the spectrum σ(A) of A.

(2) If P (X 1 , . . . , X k ) is a polynomial in k non-commuting indeterminates X 1 , . . . , X k and A is a C * -algebra, then the evaluation P (a 1 , . . . , a k ), a j ∈ A n×n , n ∈ N, is an nc function. More generally, this can be extended to power series P with (finite or infinite) radius of convergence (see, for instance, [START_REF] Popescu | Noncommutative disc algebras and their representations[END_REF]). (3) If A is a unital C * -algebra and B ⊆ A is an inclusion of C * -algebras which share the same unit, assume that E :

A → B is a unit-preserving conditional expectation. If X = X * ∈ A, then the map G X defined by G X,n (b) = (Id C n×n ⊗ E) (b -I n ⊗ X) -1 , b ∈ B n×n
, is an nc function (see [START_REF] Voiculescu | The coalgebra of the free difference quotient and free probability[END_REF][START_REF] Voiculescu | Free Analysis Questions I: Duality Transform for the Coalgebra of ∂ X:B Internat[END_REF]). Its domain is the set of all b such that b -I n ⊗ X is invertible. The noncommutative upper half-plane =⇒ SK(a, c)(P )T = K(ã, c)(SP T ). ( 6) Conditions ( 4) and ( 6) are equivalent to conditions (4), [START_REF] Agler | Pick interpolation for free holomorphic functions[END_REF] and [START_REF] Kaliuzhnyi-Verbovetskyi | Foundations of free noncommutative function theory[END_REF]Proposition 2.15]).

a, ã ∈ Ω m , S ∈ GL m (C) are such that SaS -1 = ã, c, c ∈ Ω n , T ∈ GL n (C) are such that T -1 cT = c, P ∈ V m×n 0 =⇒ K(ã, c)(P ) = SK(a, c)(S -1 P T -1 )T. (7) If f : Ω nc → W nc is a noncommutative map, then Ω nc × Ω nc ∋ (a, c) → ∆f (a, c) ∈ L(V, W) nc satisfies the above conditions (see
We call K a noncommutative (nc) kernel if K satisfies (4) and respects intertwinings in the following sense:

a ∈ Ω m , ã ∈ Ω m, S ∈ C m×m are such that Sa = ãS, c ∈ Ω n , c ∈ Ω ñ, T ∈ C ñ×n are such that T c = cT, P ∈ V m×n 0 =⇒ SK(a, c)(P )T * = K(ã, c)(SP T * ). (8)
Conditions ( 4) and ( 8) are equivalent to conditions (4), [START_REF] Agler | Pick interpolation for free holomorphic functions[END_REF] and

a, ã ∈ Ω m , S ∈ GL m (C) are such that SaS -1 = ã, c, c ∈ Ω n , T ∈ GL n (C) are such that T cT -1 = c, P ∈ V m×n 0 =⇒ K(ã, c)(P ) = SK(a, c)(S -1 P (T -1 ) * )T * . (9)
Observe that if K is an affine nc kernel, then (a, c) → K(a, c * ) is an nc kernel.

We say that a noncommutative kernel K is a completely positive noncommutative (cp nc) kernel if in addition [START_REF] Ball | Interpolation and transfer-function realization for the noncommutative Schur-Agler class[END_REF] a

∈ Ω m , P ≥ 0 in V m×m 0 =⇒ K(a, a)(P ) ≥ 0 in V m×m 1 for all m ∈ N. If V 0 , V 1 are C * -algebras, then (10) is equivalent to requiring that for all N ∈ N, m 1 , m 2 , . . . , m N ∈ N, (11)
a (j) ∈ Ω mj , P j ∈ V N ×mj 0 , b j ∈ V mj 1 , 1 ≤ j ≤ N =⇒ N i,j=1 b * i K(a (i) , a (j) )(P * i P j )b j ≥ 0 (see [9, Proposition 2.2]). If K(a, a) is completely positive, then it is also completely bounded and K(a, a) = K(a, a) cb = K(a, a)(1) . Example 2.2. Let A be a C * -algebra. The simplest non-constant nc kernel is A nc × A nc ∋ (a, c) → a • c * ∈ L(A, A) nc . That is, for m, n ∈ N, a ∈ A m×m , c ∈ A n×n and P ∈ A m×n , we have (a, c) → (P → aP c * ). More generally, if G, H are nc functions from Ω nc ⊆ V nc to A nc , then (a, c) → G(a) • H(c)
* is an nc kernel. One can further pre-compose this kernel with a completely bounded map Ψ : A → A:

Ω m × Ω n ∋ (a, c) → A m×n ∋ P → G(a)(Id C m×n ⊗ Ψ)(P )H(c) * ∈ A m×n
is an nc kernel. If G = H and Ψ is completely positive, then this is a cp nc kernel. In a certain sense, all nc kernels are of this form (we refer to [9, Theorem 3.1] for the precise statement). Note also that (a, c)

→ [P → G(a)(Id C m×n ⊗ Ψ)(P )H(c * ) * ] is an affine nc kernel.
Example 2.3. One of the main objectives of this paper is to analyze certain metric properties of noncommutative sets. An important class of such sets is given precisely by noncommutative kernels. Let A be a C * -algebra, V be an operator space and Ω nc ⊂ V nc be an nc set. Assume that K : Ω nc × Ω nc → L(A) nc is a noncommutative kernel. We may define the set

D K := ∞ n=1 {a ∈ Ω n : K(a, a)(I n ) > 0} Dn .
Observe that if K were assumed instead to be an affine nc kernel, then the above definition would change to

D n = {a ∈ Ω n : K(a, a * )(I n ) > 0}. Clearly D K may be empty or equal to Ω nc . If a ∈ Ω m , ã ∈ Ω m,
then, by ( 4) and ( 5), K(a ⊕ ã, a ⊕ ã) ∈ L(A (m+ m)×(m+ m) ) and

K(a ⊕ ã, a ⊕ ã)(I m+ m) = K(a, a)(I m ) K(a, ã)(0) K(ã, a)(0) K(ã, ã)(I m) = K(a, a)(I m ) 0 0 K(ã, ã)(I m) > 0.
Thus, under the weaker assumptions that K is a global kernel, we are guaranteed that D K is a noncommutative set. Under our assumption that K is a noncommutative kernel, we have in addition that for any S ∈ GL m (C),

K(SaS -1 , (S -1 ) * aS * )(I m ) = SK(a, a)(S -1 I m S)S -1 = SK(a, a)(I m )S -1 .
Thus, if S is unitary (that is, S * = S -1 ), then K(SaS * , SaS * )(I m ) > 0 whenever K(a, a)(I m ) > 0. We conclude that if K is an nc kernel on Ω nc , then D K is a noncommutative set which is invariant with respect to conjugation by scalar unitary matrices.

Some of the more famous examples of noncommutative sets are given by nc kernels: (i) The noncommutative upper half-plane

H + (A) = ∞ n=1 H + (A n×n ))
, where

H + (A n×n ) = {a ∈ A n×n : ℑa > 0} (we remind the reader that ℑb = (b - b * )/2i, ℜb = (b + b * )/2, so that b = ℜb + iℑb). The kernel in this case is K(a, c)(P ) = (aP -(cP * ) * )/2i, a ∈ A m×m , c ∈ A n×n , P ∈ A m×n .
It is easy to verify that this is a globally defined nc kernel. This set is important in free probability (see [START_REF] Voiculescu | The coalgebra of the free difference quotient and free probability[END_REF][START_REF] Voiculescu | Free Analysis Questions I: Duality Transform for the Coalgebra of ∂ X:B Internat[END_REF]). (ii) The unit ball

B 1 (A) = ∞ n=1 B 1 (A n×n ), where B 1 (A n×n ) = {a ∈ A n×n : a < 1} (the norm considered being the C * -norm on A n×n ).
Here the kernel is even simpler: K(a, c)(P ) = 1 -aP c * . (iii) More generally, if G is a noncommutative function with values in A, we could define H + (A) G by using the kernel K(a, c)(P ) = (G(a)P -(G(c)P * ) * )/2i and B 1 (A) G by using the kernel K(a, c)(P ) = 1 -G(a)P G(c) * . However, some are not: (iv) Consider N (A) = ∞ n=1 {a ∈ A n×n : a n = 0}. Clearly N (A) is closed under direct sums, and, moreover, if S ∈ GL n (C) and a ∈ {a ∈ A n×n : a n = 0}, then (SaS -1 ) n = Sa n S -1 = 0. So this set is in fact invariant under conjugation by all of GL n (C), not just by the unitary group. This is because N (A) is "thin," in the sense that it has empty interior in all the natural topologies on nc sets (see below). Thus, one cannot expect that N (A) is of the form D K for an nc kernel K.

2.3.

Three topologies on noncommutative sets. As already stated, operator spaces constitute the natural framework for noncommutative function theory. We recall that (see, for instance, [START_REF] Effros | Operator Spaces[END_REF]) if V is an operator space, then

a ⊕ ã m+ m = max{ a m , ã m}, m, m ∈ N, a ∈ V m×m , ã ∈ V m× m,
and

SaT n ≤ S a m T , m, n ∈ N, a ∈ V m×m , S ∈ C n×m , T ∈ C m×n .
A topology naturally compatible with these norm conditions is the uniformly-open topology. It has as basis balls defined the following way: if c ∈ V s×s and r ∈ (0, +∞), then [START_REF] Kaliuzhnyi-Verbovetskyi | Foundations of free noncommutative function theory[END_REF]Theorem 7.2] that if f is a noncommutative function defined on Ω nc which is locally bounded on slices, then f is analytic on slices, in the sense that for any n ∈ N and any finite dimensional subspace X of V n×n , f | X is analytic as a function of several complex variables.

B nc (c, r) = ∞ n=1 a ∈ V sn×sn : a -⊕ n j=1 c sn < r .
Finally, one can also consider the topology in which a set Ω nc is open in V nc if and only if Ω n is open in the topological vector space topology of V n×n for all n ∈ N. Observe that such a set is also finitely open. We refer to it as the level topology.

A (pseudo)distance on noncommutative sets

Let V be a complex topological vector space. As we progress through the paper, we put more and more structure on V, but for our first definition, we need nothing more than the axioms of a complex topological vector space. For now we endow V n×m , n, m ∈ N, with the usual (product) topology. Let D be a noncommutative subset of V nc and consider the following properties:

(1) For any n ∈ N, D n is open in V n×n ; (2) If U is a unitary n × n complex matrix and a ∈ D n , then U aU * ∈ D n ; (3) If a ∈ V n×n , c ∈ V m×m are such that a 0 0 c ∈ D n+m ,
Indeed, if δ D (a, c)(b) = 0 or +∞, then the statement is obvious. Else, if a sb 0 c ∈ D n+m for all s ∈ [0, δ D (a, c)(b) -1 ), then a sb 0 c = a s s0 s 0 b 0 c , so that a r(s 0 b) 0 c ∈ D n+m for all r ∈ [0, (s 0 δ D (a, c)(b)) -1 ), which shows that s 0 δ D (a, c)(b) = δ D (a, c)(s 0 b).
Remark 3.1. Given a complex vector space V endowed with a topology for which the multiplication with positive scalars is continuous (a requirement automatically satisfied by a topological vector space), the quantity δ is upper semicontinuous in its three variables whenever it is defined on an nc set which satisfies property (1) above. Indeed, consider such an nc set Ω ⊆ V nc . It is enough to prove the statement at level one. Thus, consider three nets {a ι } ι∈I , {c ι } ι∈I , and {b ι } ι∈I converging to a, c ∈ Ω 1 and b ∈ V, respectively. Let t ∈ (0, +∞) be chosen so that a sb 0 c ∈ Ω 2 for all s ≤ t. Since

Ω 2 is open in the topology of V 2×2 , there exists an ι 0 ∈ I such that

a ι [0, t]b ι 0 c ι ⊂ Ω 2
for all ι ≥ ι 0 (we have used here the compactness of [0, t]). Thus, t -1 > δ(a, c)(b) implies that t -1 > δ(a ι , c ι )(b ι ) for all ι large enough. This implies that

(13) lim sup ι∈I δ(a ι , c ι )(b ι ) ≤ δ(a, c)(b), a, c ∈ Ω 1 , b ∈ V.
This shows that δ is upper semicontinuous on nc sets that satisfy property (1) under very mild conditions on the topology of the underlying vector space. Remarkably, under the supplementary hypothesis that the intersection

∂Ω 2k ∩ a R + b 0 c is discrete for all b ∈ V k×k , a, c
∈ Ω k , the exact same argument applied to the complement of Ω shows that δ is lower semicontinuous, and thus continuous.

The following proposition is straightforward, but, unless some of the hypotheses (1) -( 3) from above are assumed, it may well be vacuous. Proposition 3.2. Let V, W be two complex topological vector spaces and let D and E be two noncommutative subsets of V nc and W nc , respectively. Assume that f : D → E is a function such that Then

δ D (a, c)(b) ≥ δ E (f (a), f (c))(∆f (a, c)(b)), a ∈ D n , c ∈ D m , b ∈ V n×m .
Note that the hypothesis on the homogeneity of ∆f (a, c)(b) in b is meaningful only if there exists some interval (t, r) such that a sb 0 c ∈ D n+m for all s ∈ (t, r). Otherwise, one can simply define ∆f (a, c)(sb) as s∆f (a, c)(b).

Proof. The statement is tautological:

consider a ∈ D n , c ∈ D m and b ∈ V n×m such that a sb 0 c ∈ D n+m for all s ∈ [0, t 0 ). If t 0 = +∞, then δ D (a, c)(b) = 0 and f a sb 0 c = f (a) ∆f (a, c)(sb) 0 f (c) = f (a) s∆f (a, c)(b) 0 f (c) for all s ∈ [0, +∞), so that δ E (f (a), f (c))(∆f (a, c)(b)) = 0. If t 0 = 0 (i.e. δ D (a, c)(b) = +∞), then the inequality δ D (a, c)(b) ≥ δ E (f (a), f (c))(∆f (a, c)(b)) is obvious thanks to hypothesis (b). Finally, if t 0 = δ D (a, c)(b) -1 ∈ (0, +∞), then f a sb 0 c = f (a) ∆f (a, c)(sb) 0 f (c) = f (a) s∆f (a, c)(b) 0 f (c) ∈ E n+m for all s ∈ [0, t 0 ), which implies t 0 = δ D (a, c)(b) -1 ≤ δ E (f (a), f (c))(∆f (a, c)(b)) -1
. This concludes the proof.

Remark 3.3.

(1) If we assume hypotheses (1) for D, then for any a ∈ D n , c ∈ D m , b ∈ V n×m we are guaranteed that there exists a t 0 ∈ (0, +∞] such that a sb 0 c ∈ D n+m for all s ∈ [0, t 0 ). Thus, under a very mild assumption of openness in a complex topological vector space, we are guaranteed that δ D (a, c)(b) is finite (possibly zero). ( 2) Assumption (2) on D is sufficient (although not necessary) in order to guarantee that a zb

0 c ∈ D n+m for all z ∈ C, |z| < δ D (a, c)(b) -1 . Indeed,
(f (a), f (c))(∆f (a, c)(b)) ≤ δ D (a, c)(b), a ∈ D n , c ∈ D m , b ∈ V n×m , m, n ∈ N.
Next, we study some of the properties of δ D in more detail.

Lemma 3.5. Assume that the noncommutative subset D of V nc satisfies properties (2) and (3). For any unitary matrices

U ∈ C n×n , V ∈ C m×m , a 1 , a 2 ∈ D n , c 1 , c 2 ∈ D m , b 11 ∈ V n×n , b 12 ∈ V n×m , b 21 ∈ V m×n , b 22 ∈ V m×m , we have (14) δ D (U a 1 U * , V c 2 V * )(U b 12 V * ) = δ D (a 1 , c 2 )(b 12 ) (15) δ D a 1 0 0 c 1 , a 2 0 0 c 2 b 11 0 0 b 22 = max{δ D (a 1 , c 1 )(b 11 ), δ D (a 2 , c 2 )(b 22 )}, ( 16
) δ D a 1 0 0 c 1 , a 2 0 0 c 2 0 b 12 b 21 0 = max{δ D (a 1 , c 2 )(b 12 ), δ D (c 1 , a 2 )(b 21 )}.
Proof. Relation ( 14) follows trivially from hypothesis (2): for s ≥ 0, we have the chain of equivalences

a 1 sb 12 0 c 2 ∈ D n+m ⇐⇒ U 0 0 V a 1 sb 12 0 c 2 U * 0 0 V * ∈ D n+m ⇐⇒ U a 1 U * sU b 12 V * 0 V c 2 V * ∈ D n+m .
A slight variation of this trick proves [START_REF] Bercovici | Free convolutions of measures with unbounded support[END_REF] and [START_REF] Ph | On the Free Convolution with a Semi-circular Distribution[END_REF]. Let

U 0 =     0 m×n I m 0 m×n 0 m×m 0 n×n 0 n×m I n 0 n×m I n 0 n×m 0 n×n 0 n×m 0 m×n 0 m×m 0 m×n I m     ,
a complex (2n + 2m) × (2n + 2m) unitary matrix. For any s ≥ 0, we have

    a 1 0 0 sb 12 0 c 1 sb 21 0 0 0 a 2 0 0 0 0 c 2     ∈ D 2(n+m) ⇐⇒ U 0     a 1 0 0 sb 12 0 c 1 sb 21 0 0 0 a 2 0 0 0 0 c 2     U * 0 ∈ D 2(n+m) ⇐⇒     c 1 sb 21 0 0 0 a 2 0 0 0 0 a 1 sb 12 0 0 0 c 2     ∈ D m+n+n+m ⇐⇒ c 1 sb 21 0 a 2 , a 1 sb 12 0 c 2 ∈ D n+m ,
where we have used property (3) in the last equivalence and property (2) in the first. This proves [START_REF] Ph | On the Free Convolution with a Semi-circular Distribution[END_REF]. Relation ( 15) is proved the same way.

The next lemma shows that, in a certain way, δ D is itself a sort of noncommutative function.

Lemma 3.6. Assume that D ⊆ V nc satisfies properties (2) and (3). For any n, m ∈ N, a ∈ D n , c ∈ D m , b ∈ V n×m , and any k ∈ N, we have

δ D (I k ⊗ a, I k ⊗ c)(Z ⊗ b) = δ D (a, c)(b) ZZ * 1 2 , Z ∈ C k×k .
Proof. We shall prove this lemma in two steps. In the first step, we assume that a = c (and implicitly m = n). Consider unitary matrices U, V * ∈ C k×k which diagonalize Z:

U ZV * = diag(λ 1 , . . . , λ k ), where 0 ≤ λ 1 ≤ • • • ≤ λ k = Z * Z 1 2 are the singular values of Z. Then U ⊗ 1 0 0 V ⊗ 1 I k ⊗ a Z ⊗ b 0 I k ⊗ a U * ⊗ 1 0 0 V * ⊗ 1 = I k ⊗ a U ZV * ⊗ b 0 I k ⊗ a .
Thus, by property (2),

I k ⊗ a sZ ⊗ b 0 I k ⊗ a ∈ D 2kn if and only if the matrix               a 0 • • • 0 sλ 1 b 0 • • • 0 0 a • • • 0 0 sλ 2 b • • • 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 • • • a 0 0 • • • sλ k b 0 0 • • • 0 a 0 • • • 0 0 0 • • • 0 0 a • • • 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 • • • 0 0 • • • a               ∈ D 2kn .
Successive permutations transform this into the condition

              a sλ 1 b • • • 0 0 • • • 0 0 0 a • • • 0 0 • • • 0 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 • • • a sλ j b • • • 0 0 0 0 • • • 0 a • • • 0 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 • • • 0 0 • • • a sλ k b 0 0 0 • • • 0 • • • 0 a               ∈ D 2kn , i.e. diag a sλ 1 b 0 a , . . . , a sλ k b 0 a ∈ D 2kn .
By property (3) we have that this happens if and only if each block a sλ j b 0 a belongs to D 2n . Since the largest singular value

λ k of Z equals Z * Z 1 2
, the first step is proved. In order to prove the second step, we use equation ( 16) of Lemma 3.5, which guarantees that

δ D (I k ⊗ a, I k ⊗ c)(Z ⊗ b) = δ D I k ⊗ a 0 0 I k ⊗ c , I k ⊗ a 0 0 I k ⊗ c 0 Z ⊗ b 0 0 .
By conjugating with a permutation matrix, it follows, again via Lemma 3.5 and the first step, that

δ D I k ⊗ a 0 0 I k ⊗ c , I k ⊗ a 0 0 I k ⊗ c 0 Z ⊗ b 0 0 = δ D I k ⊗ a 0 0 c , I k ⊗ a 0 0 c Z ⊗ 0 b 0 0 = δ D a 0 0 c , a 0 0 c 0 b 0 0 Z * Z 1 2 = δ D (a, c) (b) Z * Z 1 2 .
With these two lemmas, we can prove now the main result of this section. For simplicity, denote δD (a, c)

:= δ D (a, c)(a -c).
Theorem 3.7. Assume that D ⊆ V nc satisfies properties (2) and (3). The following statements are equivalent for any m, n ∈ N:

(i) For any a ∈ D n , c ∈ D m , δ D (a, c)(b) = 0 =⇒ b = 0; (ii) For any a ∈ D n , δ D (a, a)(b) = 0 =⇒ b = 0; (iii) For any a, c ∈ D n , δD (a, c) = 0 =⇒ a = c; (iv) For any k ∈ N, there exists no non-constant noncommutative function f : C nc → (D k ) nc .
By (D k ) nc we denote all the levels of D which are multiples of k.

Proof. Implications (i) =⇒ (ii) and (i) =⇒ (iii) are obvious. If there exists a noncommutative f : C nc → (D k ) nc for some k ∈ N, then, by Proposition 3.2, it follows that for any n ∈ N, Z, W ∈ C n×n , δD (f (Z), f (W )) ≤ δCnc (Z, W ) = 0. Thus, if f is not constant, (iii) is violated. Thus, (iii) =⇒ (iv). The implication (ii) =⇒ (i) follows from equation ( 16) of Lemma 3.5 by writing

δ D a 0 0 c , a 0 0 c 0 b 0 0 = δ D (a, c) (b).
Finally, to prove (iv) =⇒ (ii), assume that we found

a 0 ∈ D n0 and b 0 ∈ V n0×n0 \ {0} such that δ D (a 0 , a 0 )(b 0 ) = 0. We build the linear noncommutative function f (Z) = I p ⊗ a 0 0 0 I p ⊗ a 0 + Z ⊗ 0 b 0 0 0 , Z ∈ C p×p .
By a conjugation with a permutation matrix and an application of Lemma 3.6, we conclude that f takes values in D 2pn0 , so that (iv) does not hold. This completes the proof.

The function δD allows us to define two distances (possibly degenerate) on D, by mimicking the definition of the Kobayashi distance, with δD playing the role of Lempert function.

Definition 3.8. If D is a noncommutative set in an operator space satisfying assumptions ( 2) and ( 3), then for any

n ∈ N, a, c ∈ D n , dD (a, c) = inf    N j=1 δD (a j-1 , a j ) : a j ∈ D n , 0 ≤ j ≤ N, a 0 = a, a N = c, N ∈ N    .
Wel call such a finite sequence a = a 0 , a 1 , . . . , a N = c a division of dD (a, c).

The function dD : D × D → [0, +∞] fails to separate the points of D if one (and hence all) the conditions of Theorem 3.7 are satisfied.

It is quite easy to show that d is a distance. Indeed, since δD (a, c) = δD (c, a), it follows that dD (a, c) = dD (c, a). So only the triangle inequality remains to be proved. Let a, c, v ∈ D n . If a 0 = a, a 1 , . . . , a N = c and a N = c, . . . , a N +p-1 , a N +p = v are divisions for dD (a, c) and dD (c, v), respectively, then a 0 = a, a 1 , . . . , a N , a N +1 , . . . , a N +p = v is a division for dD (a, v). In particular,

N j=1 δD (a j-1 , a j ) + N +p j=N +1 δD (a j-1 , a j ) ≥ inf    M j=1 δD (d j-1 , d j ) : d 0 , . . . , d j division of dD (a, v)    for all divisions a 0 = a, a 1 , . . . , a N = c for dD (a, c) and a N = c, . . . , a N +p-1 , a N +p = v for dD (c, v). Taking infimum separately after each division provides dD (a, c) + dD (c, v) ≥ dD (a, v).
A slight drawback of dD is that it may depend to a certain extent on the level n in which the points a and c live. The following modification of Definition 3.8 seems to be more natural in the noncommutative framework. Definition 3.9. If D is a noncommutative set in an operator space satisfying assumptions ( 2) and (3), then for any

n ∈ N, a, c ∈ D n , dD,∞ (a, c) = inf    N j=1 δD (a j-1 , a j ) : a j ∈ D np , 0 ≤ j ≤ N, a 0 = I p ⊗ a, a N = I p ⊗ c, N, p ∈ N    .
Clearly, according to Lemma 3.6, dD,∞ (a, c) ≤ dD (a, c). At this moment we do not know for what sets D the two distances are equal. It will be seen below that they coincide on the unit ball of a C * -algebra.

The most general version of the Schwarz-Pick Lemma tells us that an analytic map between two hyperbolic domains is a contraction with respect to the corresponding Kobayashi metrics. The following corollary is a direct consequence of Proposition 3.2 and the above definition.

Corollary 3.10. Let D, E be two noncommutative sets satisfying assumptions (2) and (3). Let f : D → E be a noncommutative function. Then f is a contraction with respect to the above-defined distances:

dE (f (a), f (c)) ≤ dD (a, c) and dE,∞ (f (a), f (c)) ≤ dD,∞ (a, c), a, c ∈ D n , n ∈ N.
Note that assuming also hypothesis (1) in the above corollary guarantees that the two sides of the inequality above are both finite (possibly zero).

Until now we have made no assumptions on the openness of D. As seen in Remark 3.1, hypotheses (1) -(3) guarantee that δ D is upper semicontinuous in its three variables, and in particular so is δD . Thus, we may define infinitesimal versions of dD and dD,∞ .

d D (a, c) = inf [0,1] δ(a(t), a(t))(a ′ (t)) dt : a : [0, 1] → D n continuously differentiable, a(0) = a, a(1) = c . Similarly, d D,∞ (a, c) = inf [0,1] δ(a(t), a(t))(a ′ (t)) dt : p ∈ N, a : [0, 1] → D np continuously differentiable, a(0) = I p ⊗ a, a(1) = I p ⊗ c .
The openness of D n implies that δ(a(t), a(t))(a ′ (t)) is finite for all t ∈ [0, 1]. Since an upper semicontinuous function attains its supremum on a compact set, it follows that the set {δ(a(t), a(t))(a ′ (t)) : t ∈ [0, 1]} is bounded. Thus, the integrals defining d D are finite, so that d D is well-defined and finite (possibly zero). Clearly d D (a, c) ≥ d D,∞ (a, c). Both d D and d D,∞ are (possibly degenerate) distances: as before, it is only the triangle inequality that needs to be verified. If a, v, c ∈ D n , then the above infimum over all paths from a to c is necessarily no greater than the infimum over all paths from a to c which go through v. Since δ D is continuous and paths which are continuous and differentiable everywhere except at one point can be approximated arbitrarily well by paths which are differentiable everywhere, it follows that d D (a, c) ≤ d D (a, v) + d D (v, c), and the same for d D,∞ .

Another application of Proposition 3.2 shows that noncommutative functions are contractions also with respect to d D and d D,∞ . We record this fact below. Corollary 3.12. Let D, E be two noncommutative sets satisfying assumptions (1)-( 3). Let f : D → E be a noncommutative function. Then f is a contraction with respect to the above-defined distances:

d E (f (a), f (c)) ≤ d D (a, c) and d E,∞ (f (a), f (c)) ≤ d D,∞ (a, c), a, c ∈ D n , n ∈ N.
We establish next the relation between dD and d D under the assumptions (1) -( 3). As a direct consequence of the upper semicontinuity of δ (Remark 3.1), we obtain for any differentiable path a defined on [0, 1] and any t ∈ [0, 1] the relation

lim sup h→0 δ D (a(t), a(t + h)) a(t + h) -a(t) h ≤ δ D (a(t), a(t))(a ′ (t)).
(When t = 0 or t = 1, the limit should of course be taken one-sided.) In particular given an arbitrary path a, a division of [0, 1] translates into a division of d(a, c).

Given ε > 0, for any t ∈ [0, 1] there exists η t,ε > 0 such that δ D (a(t), a(t + h)) a(t+h)-a(t) h < δ D (a(t), a(t))(a ′ (t))+ε for any |h| < η t,ε . The family {(t-η t,ε , t+η t,ε )} 0≤t≤1 is an open cover of [0, 1]
, so that we may extract a finite subcover (t

1 -η t1,ε , t 1 + η t1,ε ), . . . , (t N - η tN ,ε , t N + η tN ,ε ), t 1 < • • • < t N . Let t 0 = 0, t N +1 = 1.
By choosing the smallest among η tj ,ε , 1 ≤ j ≤ N , and increasing the number of points t j if necessary, we may assume that

η t1,ε = • • • = η tN ,ε = η ε > 0 and t j ∈ (t j-1 -η ε , t j-1 + η ε ) ∩ (t j+1 -η ε , t j+1 + η ε ). Then dD (a, c) ≤ N j=0 δD (a(t j ), a(t j+1 )) (17) = N j=0 (t j+1 -t j )δ D (a(t j ), a(t j+1 )) a(t j+1 ) -a(t j ) t j+1 -t j < N j=0 (t j+1 -t j )δ D (a(s j ), a(s j ))(a ′ (s j )) + ε (s j ∈ [t j , t j+1 ]) ≤ N j=0 (t j+1 -t j ) [tj,tj+1] δ D (a(t), a(t))(a ′ (t)) dt + ε (18) = [0,1] δ(a(t), a(t))(a ′ (t)) dt + ε.
We have used in [START_REF] Dineen | The Schwarz Lemma[END_REF] the definition of dD , and in relation [START_REF] Earle | A fixed point theorem forholomorphic mappings[END_REF] the fact that we may choose s j arbitrarily in [t j , t j+1 ], and we decide to choose an s j such that

δ D (a(s j ), a(s j ))(a ′ (s j )) ≤ 1 t j+1 -t j [tj ,tj+1] δ D (a(t), a(t))(a ′ (t)) dt.
Since a has been arbitrarily chosen, it follows that dD (a, c) ≤ d D (a, c) for all a, c belonging to the same level of D. Thus, Since we have shown that δ and δ generate distances, it is natural to ask what topology one may expect those distances to determine on the original space. In the most general case, we are able to make only the following statement: Proposition 3.13. Assume that D is a noncommutative subset of a topological vector space V which satisfies assumptions Thus, lim ι∈I dD (a ι , a) = 0 whenever {a ι } ι∈I ⊆ D n converges to a in the product topology of V n×n . This completes our proof.

A smooth (pseudo)metric

We have seen above that simple properties of noncommutative sets allow us to define a distance which is often nondegenerate, and with respect to which analytic noncommutative functions are natural contractions. These results have a "metric space" flavour. In this section we consider the case when the distance defined has a "differential geometry" flavour. 4.1. Hypotheses. Let V be an operator space and J , K be C * -algebras. Let O nc ⊆ V nc be a noncommutative set, and assume that G : O nc × O nc → L(J , K) is an affine noncommutative kernel. Recall that if (a, c) → G(a, c) is an affine nc kernel, then (a, c) → G(a, c * ) is an nc kernel. We prefer to work with the affine kernel G because we will often need to take its derivative (or, rather, difference-differential) on both the first and second coordinate (which we denote by 0 ∆G(a; a ′ , c) and 1 ∆G(a, c; c ′ ), respectively). Consider the following properties:

(1) O nc is uniformly open and G is locally uniformly bounded. Thus, G is uniformly analytic in each of its two variables. (8) The function G is analytic on a neighbourhood of Ω n × Ω n for each n ∈ N. In our results below, we will assume various subsets of the above hypotheses. We would like to emphasize at this moment already that they are not very restrictive, and important families of kernels satisfy all of them.

O n such that a * , c * ∈ O n , we have G(a, c)(v) * = G(c * , a * )(v) for v = v * ∈ J n×n ; (5) {a ∈ O nc : G(a, a * )(1) > 0} = ∅.
Pick a point a 0 ∈ {a ∈ O nc : G(a, a * )(1) > 0} at the first nonempty level. Let D G,nc be the connected component of a 0 (i.e. at each multiple k of the level in which a 0 occurs, we consider the connected component of a 0 ⊗ 1 k ). In all applications we are currently aware of, the set O nc is considerably bigger than D G,nc . It seems in fact that at the present level of knowledge in this field, analyticity of G on the boundary of D G,nc is necessary in order to obtain powerful results about arbitrary functions defined on it. Given the case of single-variable analytic functions, that is probably not so surprising. However, for the purposes of the next section, this hypothesis is not needed.

We would like to emphasize that if G(a, a * ) is completely positive, then the condition G(a, a * )(1) > 0 can be replaced by the condition G(a, a * )(x) > 0 for any x > 0. Indeed, one implication is obvious. Conversely, if x > 0, then it is invertible and x ≥ x -1 -1 1 > 0, so that 0 < x -1 -1 G(a, a * )(1) ≤ G(a, a * )(x). For our purposes, completely positive kernels are "bad": they generate a degenerate pseudometric. However, in the following, to the extent possible, we shall perform our computations in such a way as to be able to draw conclusions for both the case G(a, a * ) completely positive and G(a, a * )(1) > 0 (without the assumption that G(a, a * ) is positive).

4.2.

The smooth pseudometric. The following proposition gives a noncommutative version of a hyperbolic pseudometric. This version is given in terms of the defining functions of the domains in question and its definition is purely algebraic. It is clear that noncommutative domains admit hyperbolic pseudometrics level-by-level. However, there would be apropri no reason to think that they are related to the pseudometric we define here. We will see later that in some cases our pseudometric indeed generates the Kobayashi metric, while in others it does not. As in Theorem 3.7, and as in the classical theory of several complex variables, for the pseudometric to be nondegenerate, it is necessary that the domains do not contain holomorphic images of complex lines (i.e. copies of C) at any level.

Consider G satisfying properties [(1), (2) or (3)], (4), and (5), and define D G,nc as above. Without loss of generality, we assume that D G,1 = ∅. Recall that the spectrum of an operator V on a Hilbert space is denoted by σ

(V ). For a ∈ D G,n , c ∈ D G,m , b ∈ V n×m , we have δ DG,nc (a, c)(b) = max 0, sup σ G(a, a * )(1) -1/2 0 ∆G(a; c, c * )(b, 1) [G(c, c * )(1)] -1 × 1 ∆G(c, c * ; a * )(1, b * ) -1 ∆ 0 ∆G(a; c, c * ; a * )(b, 1, b * ) G(a, a * )(1) -1/2 1 2 , (21)
and, when m = n, [START_REF] Haagerup | A new application of random matrices: Ext(C * red (F 2)) is not a group[END_REF] δDG,nc (a, c) = δ DG,nc (a, c)(ac).

It will be seen below that δDG,nc (a, c) = max 0, sup σ G(a, a * )(1)

-1 2 G(a, c * )(1)G(c, c * )(1) -1 × G(c, a * )(1)G(a, a * )(1) -1 2 -1 1 2 .
It will be apparent that these two objects coincide with the ones defined in Section 3 for the particular case of domains defined via inequalities of the type described in hypothesis (5) above.

Consider another function H defined on some noncommutative subset of W nc , which satisfies the same properties as G. We define D H,nc the same way as D G,nc Proposition 4.1. Let D G,nc , D H,nc be two domains defined as above. Let f :

D G,nc → D H,nc be a noncommutative map. For any n, m ∈ N, a ∈ D G,n , c ∈ D G,m , b ∈ V n×m , we have (23) δ DH,nc (f (a), f (c))(∆f (a, c)(b)) ≤ δ DG,nc (a, c)(b). If m = n, then δDH,nc (f (a), f (c)) ≤ δDG,nc (a, c).
and

H(f (a), f (a) * )(1) -1 2 H(f (a), f (c) * )(1)H(f (c), f (c) * )(1) -1 H(f (c), f (a) * )(1) × H(f (a), f (a) * )(1) -1 2 ≤ G(a, a * )(1) -1 2 G(a, c * )(1)G(c, c * )(1) -1 G(c, a * )(1)G(a, a * )(1) -1 2 . ( 24
)
In addition,

H(f (a), f (c) * )(1)H(f (c), f (c) * )(1) -1 H(f (c), f (a) * )(1) -H(f (a), f (a) * )(1) ≤ H(f (a), f (a) * )(1) × G(a, a * )(1) -1 2 G(a, c * )(1)G(c, c * )(1) -1 G(c, a * )(1)G(a, a * )(1) -1 2 -1 . (25) Remark 4.2. If in addition H(u, v * )(1)H(v, v * )(1) -1 H(v, u * )(1) -H(u, u * )(1) ≥ 0 for all u, v ∈ D H,nc , then relation (25) is equivalent to H(f (a), f (a) * )(1) -1 2 H(f (a), f (c) * )(1) × H(f (c), f (c) * )(1) -1 H(f (c), f (a) * )(1)H(f (a), f (a) * )(1) -1 2 -1 (26) ≤ G(a, a * )(1) -1 2 G(a, c * )(1)G(c, c * )(1) -1 G(c, a * )(1)G(a, a * )(1) -1 2 -1 . Remark 4.3. The condition H(u, v * )(1)H(v, v * )(1) -1 H(v, u * )(1) -H(u, u * )(1) ≥ 0 for all u, v ∈ D H,
(A n×n ) = {b ∈ A n×n : ℑb > 0}.
It is given by the affine kernel H(a, c)(P ) = (2i) -1 (aP -P c). Then the inequality reduces to

a -c * 2i c -c * 2i -1 c -a * 2i - a -a * 2i ≥ 0. But a -c * 2i c -c * 2i -1 c -a * 2i - a -a * 2i = 1 4 (a -c) c -c * 2i -1
(ac) * ≥ 0 whenever a = c in the upper half-plane. One can generalize this to kernels of the form H(a, c)(P ) = (2i) -1 (h(a)P -P h(c * ) * ), for some noncommutative function h.

A better-known class of kernels is given by the formula H(a, c)(P ) = 1h(a)P h(c * ) * with h a noncommutative function. Then H(a, c * )(1) = 1-h(a)h(c) * , so from the point of view of the above inequality it is enough to consider the case when h is the identity function. If H(a, c)(P ) = 1 -aP c, this comes down to:

(1 -ac * )(1 -cc * ) -1 (1 -ca * ) -(1 -aa * ) = 1 -ca * + (c -a)c * (1 -cc * ) -1 c(c -a) * + (c -a)c * -1 + aa * = (c -a)c * (1 -cc * ) -1 c(c -a) * + cc * + aa * -ac * -ca * = (c -a)c * (1 -cc * ) -1 c(c -a) * + (c -a)(c -a) * ≥ 0 whenever a = c satisfy a , c < 1.
Note that this also proves that

H(a, a * )(1) -1 2 H(a, c * )(1)H(c, c * )(1) -1 H(c, a * )(1)H(a, a * )(1) -1 2 -1 = 0
for H(a, c)(P ) = 1h(a)P h(c * ) * if and only if h(a) = h(c). So the pseudodistance defined by this formula separates points if and only if h is injective. The same fact holds for the generalized half-plane.

Proof of Proposition 4.1. The proof is based on showing that formula [START_REF] Garnett | Bounded Analytic Functions[END_REF] for δ DG,nc coincides with the definition [START_REF] Belinschi | A noncommutative Version of the Julia-Wolff-Carathéodory Theorem[END_REF] in the particular case of a domain defined by a noncommutative kernel as in assumption ( 5). An application of Proposition 3.2 will allow us to conclude. On the way to proving (23), we will obtain formulas allowing us to argue that ( 24) and ( 25) hold by applying the same principle as in the proof of Proposition 3.2. In some cases, for future reference, we will perform computations which are slightly more involved than absolutely necessary.

Thus, let us start by evaluating G on elements

a b 0 c , a ∈ D G,n , c ∈ D G,m . We have G a b 0 c , a * 0 b * c * (I n+m ) > 0 whenever a b 0 c ∈ D G,n+m . As a ∈ D G,n , c ∈ D G,m
, we can use the properties of nc functions/kernels to write explicitly the entries of this matrix. For future reference, we consider the general case, with P 11 ∈ J n×n , P 12 ∈ J n×m , P 21 ∈ J m×n , P

22 ∈ J m×m , a 1 , a 2 ∈ D G1,n , c 1 , c 2 ∈ D G1,m , b 1 ∈ V n×m , b 2 ∈ V m×n .
According to condition [START_REF] Akhieser | The classical moment problem and some related questions in analysis[END_REF] in the definition of affine nc kernels,

G a 1 b 1 0 m×n c 1 , a 2 0 n×m b 2 c 2 P 11 P 12 P 21 P 22 = G a 1 b 1 0 c 1 , 0 1 n 1 m 0 c 2 b 2 0 n×m a 2 0 1 m 1 n 0 P 12 P 11 P 22 P 21 0 1 m 1 n 0 = G a 1 b 1 0 c 1 , c 2
=     G a 1 , c 2 b 2 0 a 2 P 12 P 11 + 0 ∆G a 1 ; c 1 , c 2 b 2 0 a 2 b 1 , P 22 P 21 G c 1 , c 2 b 2 0 a 2 ) P 22 P 21     .
We identify each of the two components of this column vector.

G a 1 , c 2 b 2 0 a 2 P 12 P 11 = G(a 1 , c 2 )([P 12 ]) 1 ∆G(a 1 , c 2 ; a 2 )([P 12 ], b 2 ) + G(a 1 , a 2 )([P 11 ]) , and
G c 1 , c 2 b 2 0 a 2 P 22 P 21 = G(c 1 , c 2 )([P 22 ]) 1 ∆G(c 1 , c 2 ; a 2 )([P 22 ], b 2 ) + G(c 1 , a 2 )([P 21 ]) .
Finally,

0 ∆G a 1 , c 1 , c 2 b 2 0 a 2 b 1 , P 22 P 21 = 0 ∆G (a 1 ; c 1 , c 2 ) (b 1 , [P 22 ]) 1 ∆ 0 ∆G(a 1 ; c 1 , c 2 ; a 2 )(b 1 , [P 22 ], b 2 ) + 0 ∆G(a 1 ; c 1 , a 2 )(b 1 , [P 21 ]
) , a row vector with two components. To centralize all results, if

G a 1 b 1 0 c 1 , c 2 b 2 0 a 2 P 12 P 11 P 22 P 21 = G 11 G 12 G 21 G 22 , then (27)
G 11 = G(a 1 , c 2 )([P 12 ]) + 0 ∆G (a 1 ; c 1 , c 2 ) (b 1 , [P 22 ]), G 12 = 1 ∆G(a 1 , c 2 ; a 2 )([P 12 ], b 2 ) (28) + G(a 1 , a 2 )([P 11 ]) + 1 ∆ 0 ∆G(a 1 ; c 1 , c 2 ; a 2 )(b 1 , [P 22 ], b 2 ) + 0 ∆G(a 1 ; c 1 , a 2 )(b 1 , [P 21 ]), (29)
G 21 = G(c 1 , c 2 )([P 22 ]), (30)
G 22 = 1 ∆G(c 1 , c 2 ; a 2 )([P 22 ], b 2 ) + G(c 1 , a 2 )([P 21 ]).
For any C * -algebra A, u v v * w ∈ A (n+m)×(n+m) is strictly positive if and only if u > 0, w > 0 and v * u -1 v < w (or, equivalently, u > vw -1 v * -see [START_REF] Paulsen | Completely Bounded Maps and Operator Algebras[END_REF]Chapter 3]).

Thus, ) for all n, m, then, according to the above formula applied to -1

a b 0 c ∈ D G,n+m if and only if 1 a ∈ D G,n , c ∈ D G,m and G a b 0 c , c * b * 0 a * 0 1 n 1 m 0 0 1 m 1 n 0 > 0.
a 1 = a, a 2 = a * , c 1 = c, c 2 = c * , b 1 = b, b 2 = b * , the requirement G 12 G 11 G 22 G 21 > 0,
1 ∆G(c, c * ; a * )([P 22 ], b * ) < G(a, a * )([P 11 ]) + 1 ∆ 0 ∆G(a; c, c * ; a * )(b, [P 22 ], b * ). ( 32
)
(Note that if G 1 (x, x * ) were cp, by letting P 11 go to zero in the above, we'd conclude that the map

P 22 → 1 ∆ 0 ∆G 1 (a, c, c * , a * )(b, [P 22 ], b * ) is necessarily a completely positive map whenever b is so that a b 0 c ∈ D G,n+m .)
Given a, c as above, by the openness of O nc , which is a consequence of condition ( 5) and of the analyticity of G, we know that there is an ǫ > 0 depending on a, c so that

a b 0 c ∈ D G,n+m for all b ∈ V n×m with b < ǫ. Fix a direction b 0 ∈ V n×m . Then,
recalling the definition (12) for δ,

ε 0 := δ DG,nc (a, c)(b) -1 = sup t ∈ (0, +∞) : a sb 0 0 c ∈ D G,n+m ∀s < t ∈ (0, +∞].
Observe that if

1 ∆ 0 ∆G(a; c, c * ; a * )(b 0 , 1, b * 0 ) ≥ 0 ∆G(a; c, c * )(b 0 , 1)G(c, c * )(1) -1 1 ∆G(c, c * ; a * )(1, b * 0 ) then D G,
n+m contains a complex line. Indeed, one simply divides by |z| 2 in [START_REF] Popescu | Noncommutative disc algebras and their representations[END_REF].

We argue that δ DG,nc (a, c)(b) is indeed given in this case by formula [START_REF] Garnett | Bounded Analytic Functions[END_REF]. If ε 0 < +∞, then it can be written as [START_REF] Garnett | Bounded Analytic Functions[END_REF] holds. By Proposition 3.2, we conclude relation [START_REF] Harris | Fixed point theorems for infinite dimensional holomorphic functions[END_REF]. If ε 0 = +∞, there is nothing to prove.

δ DG,nc (a, c)(b) 2 = ε -2 0 = sup ϕ G(a, a * )(1) -1/2 0 ∆G(a; c, c * )(b 0 , 1) [G(c, c * )(1)] -1 1 ∆G(c, c * ; a * )(1, b * 0 ) -1 ∆ 0 ∆G(a; c, c * ; a * )(b 0 , 1, b * 0 ) G(a, a * )(1) -1/2 : ϕ : K n×n → C state . Thus, formula
Consider now the case b 0 = ǫ(ac) for some arbitrary ǫ ∈ C. We apply Equation (3) to write

• 0 ∆G(a; c, a * )(ǫ(a -c), [P 21 ]) = G(a, a * )([ǫP 21 ]) -G(c, a * )([ǫP 21 ]); • 1 ∆G(a, c * ; a * )([P 12 ], ǫ(a -c) * ) = G(a, a * )([ǫP 12 ]) -G(a, c * )([ǫP 12 ]); • 0 ∆G(a; c, c * )(ǫ(a -c), [P 22 ]) = G(a, c * )([ǫP 22 ]) -G(c, c * )([ǫP 22 ]); • 1 ∆G(c, c * ; a * )([P 22 ], ǫ(a -c) * ) = G(c, a * )([ǫP 22 ]) -G(c, c * )([ǫP 22 ]); and [G(a, c * )([P 12 ]) + 0 ∆G(a; c, c * )(b, [P 22 ])] [G(c, c * )([P 22 ])] -1 × [ 1 ∆G(c, c * ; a * )([P 22 ], b * ) + G(c, a * )([P 21 ])] < G(a, a * )([P 11 ]) + 0 ∆G(a; c, a * )(b, [P 21 ]) + 1 ∆G(a, c * ; a * )([P 12 ], b * ) + 1 ∆ 0 ∆G(a; c, c * ; a * )(b, [P 22 ], b * ). (31) • 1 ∆ 0 ∆G(a; c, c * ; a * )(ǫ(a -c), [P 22 ], ǫ(a -c) * ) = G(a, a * )([|ǫ| 2 P 22 ]) -G(c, a * )([|ǫ| 2 P 22 ]) -G(a, c * )([|ǫ| 2 P 22 ]) + G(c, c * )([|ǫ| 2 P 22 ]).
We record for future reference the expressions for G ij corresponding to b 0 = ǫ(ac).

G 11 = G(a, c * )([P 12 ]) + ǫ(G(a, c * )([P 22 ]) -G(c, c * )([P 22 ])) G 12 = G(a, a * )([ǫP 21 ] + [ǫP 12 ]) -G(a, c * )([ǫP 12 ]) -G(c, a * )([ǫP 21 ]) + G(a, a * )([P 11 ]) (33) + |ǫ| 2 (G(a, a * )([P 22 ]) + G(c, c * )([P 22 ]) -G(a, c * )([P 22 ]) -G(c, a * )([P 22 ])) G 21 = G(c, c * )([P 22 ]) G 22 = G(c, a * )([P 21 ]) + ǫ(G(c, a * )([P 22 ]) -G(c, c * )([P 22 ]))
For ǫ = 1, we obtain that for any state ψ on K n×n and ε > 0, there is a state ϕ on K n×n depending on ε such that

ψ H(f (a), f (a) * )(1) -1 2 H(f (a), f (c) * )(1) [H(f (c), f (c) * )(1)] -1 × H(f (c), f (a) * )(1) H(f (a), f (a) * )(1) -1 2 -1 -ε ≤ ϕ G(a, a * )(1) -1 2 G(a, c * )(1) [G(c, c * )(1)] -1 G(c, a * )(1)G(a, a * )(1) -1 2 -1 .
Recall that ψ, ϕ are states, so that ϕ(1) = ψ(1) = 1, which implies that

ψ H(f (a), f (a) * )(1) -1 2 H(f (a), f (c) * )(1) [H(f (c), f (c) * )(1)] -1 × H(f (c), f (a) * )(1) H(f (a), f (a) * )(1) -1 2 -ε ≤ ϕ G(a, a * )(1) -1 2 G(a, c * )(1) [G(c, c * )(1)] -1 G(c, a * )(1)G(a, a * )(1) -1 2 .
Clearly the elements under the states above are nonnegative, so this reduces to

H(f (a), f (a) * )(1) -1 2 H(f (a), f (c) * )(1) [H(f (c), f (c) * )(1)] -1 × H(f (c), f (a) * )(1) H(f (a), f (a) * )(1) -1 2 ≤ G(a, a * )(1) -1 2 G(a, c * )(1) [G(c, c * )(1)] -1 G(c, a * )(1)G(a, a * )(1) -1 2 .
The last inequality of our proposition, [START_REF] Helton | Proper Analytic Free Maps[END_REF], is a trivial consequence of the selfadointness of the elements involved, together with the previous results.

We are not automatically able to conclude the norm-inequality (26) only because the norm of the left-hand side might be achieved at the lower bound of the spectrum. However, assuming the hypothesis of Remark 4.2 guarantees this is not the case. It wouldn't be unreasonable to suppose that this hypothesis is satisfied in most cases of interest. So we discuss next three things related to it. Remark 4.4. First, not surprisingly, the inequality G(a, a * )(1)-G(a, c * )(1)G(c, c * )(1) -1 G(c, a * )(1) ≥ 0, opposite to the one introduced in Remark 4.2, cannot hold under the assumption of no complex lines in D G,nc . Indeed, if we put b = ǫ(ac) in formulas ( 27), ( 28), ( 29) and (30) for a 1 = a * 2 = a, c 1 = c * 2 = c, we obtain, according to (33) with ǫ > 0, P 11 = P 22 = 1, P 12 = P 21 = 0, the matrix inequality

G(a, a * )(1 + ǫ 2 ) -ǫ 2 G(a, c * )(1) -ǫ 2 G(c, a * )(1) + ǫ 2 G(c, c * )(1) ǫ[G(a, c * )(1) -G(c, c * )(1)] ǫ[G(c, a * )(1) -G(c, c * )(1)] G(c, c * )(1) > 0.
Multiplying with 1 ǫ 0 1 left and its adjoint right does not change the positivity of the matrix, so that

(34) (1 + ǫ 2 )G(a, a * )(1) ǫG(a, c * )(1) ǫG(c, a * )(1) G(c, c * )(1) > 0,
for any ǫ > 0 such that a ǫ(ac) 0 c ∈ D G,2n . Since D G,2n contains no complex line, it follows that there is an ǫ 0 (a, c) > 0 maximal beyond which the matrix inequality above fails. Thus, necessarily G(a, a

* )(1) -G(a, c * )(1)G(c, c * )(1) -1 G(c, a * )(
f (t) = [G(a, c * )(1) -t1] [G(c, c * )(1) -t1] -1 [G(c, a * )(1) -t1] -[G(a, a * )(1) -t1], then f ′ (t) = (G(c, c * )(1) -t1) -1 (G(c, a * )(1) -t1) -1 × (G(a, c * )(1) -t1)(G(c, c * )(1) -t1) -1 -1 ≥ 0, (recall hypothesis (3) which states that G(a, c)(1) * = G(c * , a * )(1)
), and

f ′′ (t) = 2 1 -(G(a, c * )(1) -t1)(G(c, c * )(1) -t1) -1 × (G(c, c * )(1) -t1) -1 1 -(G(c, c * )(1) -t1) -1 (G(c, a * )(1) -t1) ≥ 0,
I k ⊗ a b 0 I k ⊗ c ∈ D km+kn .
It is quite easy to see that this set is noncommutative:

if b = b 1 ⊕ b 2 with b j ∈ (a, c) kj , j = 1, 2, then I k1+k2 ⊗ a b 0 I k1+k2 ⊗ c =     I k1 ⊗ a 0 b 1 0 0 I k2 ⊗ a 0 b 2 0 0 I k1 ⊗ c 0 0 0 0 I k2 ⊗ c     ,
By permuting rows 2 and 3 and columns 2 and 3 (which comes to the conjugation with a scalar matrix), we obtain

    I k1 ⊗ a b 1 0 0 0 I k1 ⊗ c 0 0 0 0 I k2 ⊗ a b 2 0 0 0 I k2 ⊗ c     , which belongs to D (n+m)(k1+k2) because D is a noncommutative set.
Similarly, (a, c) nc is invariant by conjugation with scalar unitary matrices: if b ∈ (V n×m ) k×k , then for any unitary matrix U ∈ C k×k ,

I k ⊗ a U bU * 0 I k ⊗ c = U ⊗ I n 0 0 U ⊗ I m I k ⊗ a b 0 I k ⊗ c U * ⊗ I n 0 0 U * ⊗ I m ,
which belongs to D k(m+n) by assumption (2) on the set D.

Given the unitary invariance of the set (a, c) nc and Lemma 3.6, one is justified in asking whether (a, c) nc is in fact matrix convex. That turns out to be false in general.

Let us recall Wittstock's definition of a matrix convex set (see [START_REF] Effros | Matrix Convexity: Operator Analogues of the Bipolar and Hahn-Banach Theorems[END_REF]Section 3]): a matrix convex set is a noncommutative set K = n K n such that for any S ∈ C r×n satisfying S * S = I n , we have S * K r S ⊆ K n . Since (a, c) nc is invariant by conjugation with scalar unitary matrices, matrix convexity of (a, c) nc is equivalent to the following statement: for any k < k ′ ∈ N and b ∈ (a, c) k ′ , we have

I k 0 b I k 0 ∈ (a, c) k ,
i.e. the upper right k × k corner of b is an element of (a, c) nc whenever b is. There is a simple counterexample to this statement: consider the unit disk D in the complex plane, and the noncommutative set

D = k∈N {A ∈ C k×k : σ(A) ⊂ D, A < k}.

This is clearly a noncommutative set (if

A j ∈ D kj , then A 1 ⊕A 2 = max{ A 1 , A 2 } < max{k 1 , k 2 } < k 1 + k 2 ) which is unitarily invariant ( U * AU = A ). However,     0 0 3 0 0 0 0 1 2 0 0 0 0 0 0 0 0     ∈ D 4 , while 0 3 0 0 ∈ D 2 ,
which means that 3 0 0 1 2 ∈ (0, 0) 2 , while 3 ∈ (0, 0) 1 .

However, there are important classes of nc sets for which the set is matrix convex. One such example is the class of generalized half-planes (see Remark 4.3). Consider an injective nc map h : V nc → A nc for some unital C * -algebra A. Recall that a generalized half-plane is

H + h (V) = ∞ n=1 {a ∈ V n×n : h(a) + h(a) * > 0}.
Then elements b ∈ (a, c) nc must satisfy

(ℜh(a)) -1/2 ∆h(a, c)(b)(ℜh(c)) -1 ∆h(a, c)(b) * (ℜh(a)) -1/2 < 4 • 1.
That is, for any k ′ ∈ N,

(I k ′ ⊗ ℜh(a)) -1/2 ∆h(I k ′ ⊗ a, I k ′ ⊗ c)(b)(I k ′ ⊗ ℜh(c)) -1 × ∆h(I k ′ ⊗ a, I k ′ ⊗ c)(b) * (I k ′ ⊗ ℜh(a)) -1/2 =   k ′ l=1 (ℜh(a)) -1/2 ∆h(a, c)(b il )(ℜh(c)) -1 ∆h(a, c)(b jl ) * (ℜh(a)) -1/2   1≤i,j≤k ′ < 4I k ′ ⊗ 1.
If one fixes such a k ′ > 1 in N and a b ∈ (a, c) k ′ , proving matrix convexity comes to proving that the upper right

k × k corner of b is in (a, c) k for all 0 < k < k ′ . That is, k l=1 (ℜh(a)) -1/2 ∆h(a, c)(b il )(ℜh(c)) -1 ∆h(a, c)(b jl ) * (ℜh(a)) -1/2 1≤i,j≤k < 4I k ⊗ 1.
Denoting P k the projection onto the first k coordinates of C k ′ ×k ′ , the above relation is equivalent to

(P k I k ′ ⊗ ℜh(a)) -1/2 ∆h(I k ′ ⊗ a, I k ′ ⊗ c)(b)(P k I k ′ ⊗ ℜh(c)) -1 × ∆h(I k ′ ⊗ a, I k ′ ⊗ c)(b) * (P k I k ′ ⊗ ℜh(a)) -1/2 < 4P k I k ′ ⊗ 1 = 4I k ⊗ 1.
This is implied by the general fact that AA * < 4I k ′ =⇒ P AP A * P ≤ 4P = 4I k . Indeed, clearly AA * < 4I k ′ =⇒ P AA * P ≤ 4P = 4I k and P ≤ I k ′ =⇒ (P A)P (P * A) * ≤ P AA * P ≤ 4I k . Thus, P k bP k ∈ (a, c) k for all 0 < k < k ′ . Clearly this proof applies as well to generalized balls.

As mentioned in Section 3, the definition of dD is similar to the definition of the Kobayashi distance. We show next that generally d D is dominated by the Kobayashi distance, with equality for the unit ball or half-plane of a C * -algebra. We denote by k D,n the Kobayashi distance on the domain D n , and by κ D,n the infinitesimal Kobayashi metric on D n . Proposition 4.7. Let V be an operator system and A a C * -algebra. Consider an injective noncommutative function h defined on an open noncommutative subset of V nc whose range contains the unit ball of A nc . Define

D h,nc = ∞ n=1 {a ∈ V n×n : h(a)h(a) * < 1} ⊆ V nc . Then d D h,n (a, c) ≤ k D h,n (a, c) for all a, c ∈ D h,n , n ∈ N.
Proof. According to relation [START_REF] Taylor | Functions of several noncommuting variables[END_REF], the infinitesimal Poincaré (or hyperbolic) metric on the unit disk D, κ D (z, v), coincides with δ D (z, z)(v): they both equal |v| 1-zz . Thus, the metric generated by δ D (z, z)(v) coincides with the one generated by κ D (z, v), the Poincaré metric. According to the definition of the Kobayashi metric,

κ D h,n (a, b) = inf{|w| : ∃f : D → D h,n analytic, f (0) = a, f ′ (0)(w) = b} = inf{κ D (0, w) : ∃f : D → D h,n analytic, f (0) = a, f ′ (0)(w) = b}. ( 35
)
By von Neumann's inequality, any holomorphic function f : D → D h,n has a noncommutative extension f :

D nc → (D h,n ) nc : h(f (T ))h(f (T )) * < 1 whenever T ∈ C n×n , T < 1. This implies δ D h (f (z), f (z))(f ′ (z)(w)) = δ D h (f (z), f (z))(∆f (z, z)(w)) ≤ δ Dnc (z, z)(w) = κ D (z, w)
for all z, w ∈ D. For any ε > 0, we choose

f ε : D → D h,n such that f ε (0) = a, f ′ ε (0) ∈ Span C {b}, and b f ′ ε (0) = κ D (0, b f ′ ε (0) ) ≤ κ D h,n (f ε (0), f ′ ε (0)( b f ′ ε (0) )) + ε = κ D h,n (a, b) + ε. Thus, δ D h (a, a)(b) = δ D h (f ε (0), f ε (0)) f ′ ε (0) b f ′ ε (0) ≤ δ Dnc (0, 0) b f ′ ε (0) = κ D 0, b f ′ ε (0) ≤ κ D h,n (a, b) + ε,
for any ε > 0. Thus, by integrating and taking the infimum,

d D h,n ≤ k D h,n .
Remark 4.8. Expression [START_REF] Taylor | Functions of several noncommuting variables[END_REF] and the proof of the above proposition guarantee that d and k coincide level-by-level on the noncommutative unit ball. Of course, this statement remains true for any noncommutative domain which is biholomorphically equivalent to the noncommutative unit ball, including the noncommutative upper half-plane (see also Theorem 5.1 below). In general, however, one should not expect equality in the inequality from Proposition 4.7. The essential ingredient in the proof of Proposition 4.7 is the fact that any analytic map f : D → D h,n lifts to a nc map on D nc (the conclusion of Proposition 4.7 holds for any domain D with this property). This ingredient is missing when we try to obtain the reverse inequality in Proposition 4.7: maps f : D h,n → D need not satisfy a von Neumann inequality. Indeed, in most cases, one cannot even extend such an f to a nc map.

The definition of the classical infinitesimal Kobayashi metric [START_REF] Popescu | Free holomorphic automorphisms of the unit ball of B(H) n[END_REF] on a domain is given in terms of the infimum of the norms of derivatives of analytic functions from the Poincaré disk into the domain. From the infinitesimal metric, one builds the Kobayashi distance by integrating along curves. In the noncommutative context, distance d D,∞ from Definition 3.11 allows for a similar definition in terms of derivatives of noncommutative functions on the noncommutative unit disk. Consider again a nc domain D ⊂ V nc and let x ∈ D n , b ∈ V n×n . We claim that

δ D (x, x)(b) = sup t ≥ 0 : ∃f : D nc → (D 2n ) nc , f (0) = I 2 ⊗ x, ∆f (0, 0)(1) = t 0 b 0 0 -1 .
As above, by (D n ) nc we denote the subset of D formed of all levels which are multiples of n. Indeed, inequality ≤ follows easily: as shown in Proposition 3.2, and Lemma 3.5, if f is as in the right-hand side of the above relation, then

tδ D (x, x)(b) = δ D (x, x)(tb) = δ D x 0 0 x , x 0 0 x 0 tb 0 0 = δ D (f (0), f (0))(∆f (0, 0)(1)) ≤ δ D (0, 0)(1) = 1.
We show the reverse inequality by finding an "extremal" function. Let ι =

1 δD(x,x)(b) . Consider the nc function f = {f p } p∈N , f p (Z) = I p ⊗ x 0 0 I p ⊗ x + Z ⊗ 0 ιb 0 0 , Z ∈ C p×p , Z < 1, p ∈ N.
According to Lemmas 3.6 and 3.5, we have

δ D (x, x)(b) > δ D (x, x)(b) Z = δ D I p ⊗ x 0 0 I p ⊗ x , I p ⊗ x 0 0 I p ⊗ x Z ⊗ 0 b 0 0 ,
whenever Z is a contraction. Thus, f takes values in (D H,2n ) nc and for Z = 1 ∈ C, we actually reach the supremum in the above equality.

A classification of noncommutative domains of holomorphy

We turn now towards a classification of noncommutative domains (with respect to the level topology) which contain no complex lines at any level, up to noncommutative holomorphic equivalence, in terms of δ. In this section we assume that V is a Banach space, and when dealing with domains defined by kernels, we further assume that V is an operator space. Observe that if f : D → E is a noncommutative automorphism (i.e. a map which is bijective at each level, with analytic inverse), then inequality stated in Corollary 3.4 must hold in both directions (for f and f -1 ), so they must become equalities. That is,

δD (a, c) = δE (f (a), f (c)), a, c ∈ D n , n ∈ N. (36)
Conversely, assume that there is a function f as above such that equality [START_REF] Rudin | Function theory in the unit ball of C n[END_REF] holds for all a, c ∈ D n , n ∈ N. Then it follows trivially that f is injective. Indeed, if not, there would be an n ∈ N and points a = c ∈ D n such that f (a) = f (c). Then we would have 0 = δE (f (a), f (c)) = δD (a, c), a contradiction, according to Theorem 3.7, to the hypothesis that D contains no complex lines.

Proving the surjectivity of f as a consequence of equality ( 36) is not possible in full generality. We make the following assumption about our domains:

Given a noncommutative set D in the noncommutative extension V nc of a Banach space V, which is invariant under conjugation with scalar matrices,

For any n ∈ N and a ∈ D n , if {c k } k∈N ⊂ D n satisfies lim k→∞ inf x∈D c n x -c k = 0, then (37) lim k→∞ δD (a, c k ) = +∞.
This hypothesis does not exclude the possibility that δD ≡ +∞.

Theorem 5.1. Consider two noncommutative domains D and E in a given space V nc which are invariant under conjugation by unitary scalar matrices and contain no complex lines, and a noncommutative function f : D → E. Assume that both D and E satisfy hypothesis [START_REF] Shamovich | On fixed points of self maps of the free ball[END_REF]. Then the following are equivalent:

(1) f satisfies δD (a, c) = δE (f (a), f (c)), a, c ∈ D.
(2) f is a bijective noncommutative map, with noncommutative inverse.

The reader might worry about a trivial counterexample: the map from the nc disk to the nc bidisk sending z to (z, 0). However, we excluded this possibility by the way we formulated our statement: in this case, the nc disk is equal to its boundary in the "environment" in which the bidisk lives, so according to [START_REF] Shamovich | On fixed points of self maps of the free ball[END_REF], its δ would have to be constantly equal to infinity.

Proof. (2) =⇒ ( 1): This implication is trivial. We know that δD (a, c)

≥ δE (f (a), f (c)). If a ′ , c ′ ∈ E, then by (2) there exist a, c ∈ D such that f (a) = a ′ , f (c) = c ′ , which means f -1 (a ′ ) = a, f -1 (c ′ ) = c. By Proposition 3.2, δE (f (a), f (c)) = δE (a ′ , c ′ ) ≥ δD f -1 (a ′ ), f -1 (c ′ ) = δD (a, c).
(1) =⇒ ( 2): We have already seen that under condition (1), f is injective. Thus, we need to show that f is also surjective. Once we showed that, the noncommutativity of the correspondence a ′ → f -1 (a ′ ) allows us to conclude. The essential part of the proof is in the following quite obvious lemma, which we nevertheless state separately, since it might be of independent interest. Proof. The proof of this lemma is utterly trivial: assume towards contradiction that there exist points in D \ D ′ . Pick a point x ∈ D ∩ ∂D ′ (by ∂D ′ we understand the boundary of the set D ′ at the corresponding level n in the norm topology of the Banach space V n×n ) and a point a ∈ D ′ . By the definition of the boundary, there exists a sequence {c k } k∈N ⊂ D ′ converging to x in norm. In particular, {c k } k∈N satisfies the condition of hypothesis [START_REF] Shamovich | On fixed points of self maps of the free ball[END_REF], so that δD

′ (a, c k ) → +∞ as k → ∞. By Remark 3.1, we have ∞ > δD (a, x) ≥ lim sup n→∞ δD (a, c k ) = lim sup n→∞ δD ′ (a, c k ) = ∞,
((1 -c * c) -1 -1)b * < 1 -aa * -bb * , which is in its own turn equivalent to (38) (1 -aa * ) -1 2 b(1 -c * c) -1 b * (1 -aa * ) -1 2 < 1.
Thus,

(39) δ D (a, c)(b) = (1 -aa * ) -1 2 b(1 -c * c) -1 2
However, for any choice of selfadjoints a and c, we have that δD (a, c) ≤ 4 3 . Thus, δ stays bounded (by 4/3) on the intersection of the selfadjoints with D.

On the other hand, we have D D nc , the nc unit ball of C, and

δ Dnc | D = δ D .
In the context of Lemma 5.2, we record here the "opposite" case: we show that, under certain conditions, strict inclusion of domains leads to strict inequalities between the associated distances.

Proposition 5.4. Consider an operator space V. Let D ′ ⊂ D be an inclusion of noncommutative domains in V nc . Assume that

(1)

M := sup n∈N sup x∈D ′ n x < +∞; (2) m := inf n∈N inf{ x -w : x ∈ D ′ n , w ∈ V n×n \ D n } > 0. Then there exists a constant k ∈ [0, 1) such that kδ D ′ ≥ δ D . In particular, k δD ′ ≥ δD .
Proof. Let n be a fixed level, and pick a, c

∈ D ′ n , b ∈ V n×n , b = 0. By definition, δ D ′ (a, c)(b) -1 = sup t > 0 : a sb 0 c ∈ D ′ 2n for all s < t , δ D (a, c)(b) -1 = sup t > 0 : a rb 0 c ∈ D 2n for all r < t .
We know that the distance from D ′ 2n to V 2n×2n \ D 2n is at least m, so that

(δ D (a, c)(b) -1 -s) b = a sb 0 c - a δ D (a, c)(b) -1 b 0 c ≥ m whenever a sb 0 c ∈ D ′ 2n , and thus, δ D (a, c)(b) -1 -δ D ′ (a, c)(b) -1 ≥ m b for any a, c ∈ D ′ n , b = 0. It follows that δ D ′ (a, c)(b) δ D (a, c)(b) ≥ 1 + mδ D ′ (a, c)(b) b = 1 + mδ D ′ (a, c) b b . We bound from below δ D ′ (a, c)(b) when b = 1 and a, c ∈ D ′ 2n . We have δ D ′ (a, c)(b) > ξ ⇐⇒ δ D ′ (a, c)(b) -1 < ξ -1 ; but any element in D ′ 2n has norm bounded from above by M , so a sb 0 c ∈ D ′ 2n implies |s| = sb ≤ M . Thus, δ D ′ (a, c)(b) ≥ M -1 . We obtain δD ′ (a,c) δD (a,c) ≥ 1 + m M , for the constant k = M m+M < 1.
Taking b = ac, a = c, yields the result for δ.

For our four distances, we have Corollary 5.5. Under the assumptions, and with the notations, of Proposition 5.4, we have kd

D ′ ≥ d D | D ′ , kd D ′ ,∞ ≥ d D,∞ | D ′ , k dD ′ ≥ dD | D ′ , and k dD ′ ,∞ ≥ dD,∞ | D ′ .
Proof. For any n ∈ N, a, c ∈ D ′ n , p ∈ N, and division

I p ⊗ a = a 0 , a 1 , . . . , a N = I p ⊗ c ∈ D ′ pn , we have k N j=1 δD ′ (a j-1 , a j ) > N j=1
δD (a j-1 , a j ).

Taking infimum after all divisions at all levels p ∈ N in the left hand side provides k dD ′ ,∞ (a, c). Increasing the number of divisions in the right hand side can only decrease the infimum, so that k dD ′ ,∞ (a, c) ≥ dD,∞ (a, c). Same argument, with "divisions" replaced by "continuously differentiable paths" yields the result for

d D ′ ,∞ (a, c), d D,∞ (a, c).
The result for d, d is proved the same way, except we do not take infimum after p.

As a side benefit, we obtain from the proof of Proposition 5.4 that on bounded domains in operator spaces, δ and the norm are locally equivalent. We have already seen in Proposition 3.13 that if a ka → 0, then δD (a k , a) → 0 and thus dD (a k , a) → 0. Now assume that in a bounded domain D we have a sequence {a k } k∈N ⊂ D and a point a ∈ D n so that dD (a k , a) → 0 as k → ∞. We have seen in the proof of Proposition 5.

4 that δ D (a, c)(b) ≥ M -1 if D is included in a norm-ball of radius M , uniformly in a, c ∈ D n , b ∈ V n×n , b = 1, n ∈ N. Thus, δD (a, c) ≥ M -1 a -c , so that for any division a = a 0 , a 1 , . . . , a N = c of δD (a, c), we have N j=1 δD (a j-1 , a j ) ≥ M -1 N j=1 a j -a j-1 ≥ M -1 a -c . Thus, dD (a, c) ≥ M -1 a -c
. Applying this to c = a k yields lim k→∞ aa k = 0. Obviously, nothing changes if we amplify a, c by I p . We have proved Proposition 5.6. If D is a bounded nc domain in an operator space V and n ∈ N, then on any subset A ⊂ D n which is at a positive distance from D c n , the topologies induced by d D , d D,∞ , dD , dD,∞ , and the norm of V n×n coincide.

Remark 5.7. A very similar proof shows that the result stated in Proposition 5.6 holds also for bounded strict subsets of half-planes.

These results yield easily a noncommutative version of the Earle-Hamilton Theorem [START_REF] Earle | A fixed point theorem forholomorphic mappings[END_REF]. The proof of [START_REF] Harris | Fixed point theorems for infinite dimensional holomorphic functions[END_REF]Theorem 2] translates verbatim (with, in the notations of [START_REF] Harris | Fixed point theorems for infinite dimensional holomorphic functions[END_REF], α(x, v) replaced by δ D (x, x)(v) and the norm unit ball in a Banach space replaced by the nc unit ball in an operator space). Proposition 5.8. Let V, D, and D ′ be as in Proposition 5.4. If f : D → D ′ is a noncommutative function, then there exists a unique attracting fixed point w ∈ D ′ for f in the sense of [START_REF] Abduvalieva | Fixed point theorems for noncommutative functions[END_REF].

An application to a problem in free probability

In this section, we use some of the tools introduced before in order to study a problem in free probability. We consider a W * -noncommutative probability space (M, E, B), where B ⊆ M is a unital inclusion of W * -algebras and E : M → B is a weakly-continuous unit-preserving conditional expectation. Elements in M are called operator-valued (or, sometimes, B-valued) random variables. If X = X * ∈ M , we define the distribution of X with respect to E to be the collection of multilinear maps

µ X = {m n,X , n ∈ N}, called moments, where m 0,X = 1 ∈ B ⊆ M , m 1,X = E[X] ∈ B, and m n,X : B × • • • × B n-1 times → B, m n,X (b 1 , . . . , b n-1 ) = E[Xb 1 Xb 2 • • • Xb n-1 X], n > 1.
Such distributions are encoded analytically by the noncommutative Cauchy-Stieltjes transform (see Example 2.1(3)):

G X,n (b) = (Id C n×n ⊗ E) (b -I n ⊗ X) -1 , n ∈ N, b ∈ B n×n , ℑb > 0.
This is a noncommutative function mapping the noncommutative upper half-plane of B into the noncommutative lower half-plane (see, for instance, [START_REF] Voiculescu | The coalgebra of the free difference quotient and free probability[END_REF]). It has several good properties, including the fact that ℑG X,n (b) < 0, so that F X,n (b) := G X,n (b) -1 exists and maps elements of positive imaginary part into elements of positive imaginary part. Moreover, it has been shown in [START_REF] Belinschi | Infinite divisibility and a noncommutative Boolean-tofree Bercovici-Pata bijection[END_REF] that ℑF X,n (b) ≥ ℑb, so that h X,n (b) := F X,n (b)b, ℑb > 0, takes values elements of nonnegative imaginary part.

It has been shown in [START_REF] Anshelevich | Convolution powers in the operator-valued framework[END_REF] that for any given selfadjoint X ∈ M and completely positive map ρ : B → B such that ρ -Id B is still completely positive on B, there exists a selfadjoint X ρ in a possibly larger W * -algebra containing M such that E extends to this possibly larger algebra and the following relations hold:

(40) G Xρ,n (b) = G X,n (ω ρ (b)), ω ρ (b) = b + (ρ -Id B )h X,n (ω ρ (b)), ℑb > 0, n ∈ N.
In terms of the free probability significance of X ρ , we only mention that µ Xρ = µ ⊞ρ X , and refer the interested reader to [START_REF] Anshelevich | Convolution powers in the operator-valued framework[END_REF] for details. We wish to mention, however, that, thanks to a trick due to Hari Bercovici, understanding free convolution powers indexed by completely positive maps suffices in order to understand free additive convolutions of operator-valued distributions, so, in a certain sense, {µ ⊞ρ X : ρ and ρ -Id B completely positive} is the most general object to be understood in the context of free convolutions of operator-valued distributions.

All of the above has been done for selfadjoint operators that belong to M , that is, bounded selfadjoint operators. We will apply our results in order to show that, under certain hypotheses, this can be also done for unbounded operators X = X * affiliated to M , generalizing the results of [START_REF] Bercovici | Free convolutions of measures with unbounded support[END_REF] to the operator-valued context. We make the following hypotheses regarding X, M , and B:

(H1) B and X generate a * -algebra of possibly unbounded densely defined operators B X , such that the spectral projections of any selfadjoint element of B X belong to M . In particular, the (classical) distribution of any selfadjoint element from B X with respect to any weakly continuous state on M must be a probability measure (we assume weakly continuity for states from now on);

(H2) E ℑ(b -X) -1 < 0 whenever ℑb > 0 in B.
Hypothesis (H1) is very natural, and allows us to extend the notion of B-valued distribution of X with respect to E to the case when X = X * is unbounded, but affiliated to M . Thus, Definition 6.1. Let (M, E, B) be a W * -noncommutative probability space and X = X * be affiliated to M , possibly unbounded. Assume that X, M, and B satisfy Hypothesis (H1). We define the B-valued distribution µ X of X with respect to E to be the collection of Borel probability measures on R, {µ P,ϕ : P = P * ∈ B X , ϕ : B → C state}, where µ P,ϕ is determined via the following equality:

R 1 z -t dµ P,ϕ (t) = ϕ E (z -P ) -1 , z ∈ C \ R.
The above objects are indeed well-defined: by (H1), (z -P ) -1 ∈ M is a bounded, normal operator. Thus, one may apply E to it in order to obtain an element of B. Since B is a C * -algebra, the set of states on B separates its elements: if b, b ′ ∈ B are such that ϕ(b) = ϕ(b ′ ) for all states ϕ on B, then b = b ′ . Since for any fixed state θ on M we have lim y→0 θ(P (iy -P ) -1 ) = 0, it follows that µ P,ϕ is a probability measure. Moreover, if X = X * is bounded, then the notion of distribution from Definition 6.1 coincides with the classical notion of B-valued distribution. Indeed, as any element in a C * -algebra is the linear combination of two selfadjoint elements of the same C * -algebra, it follows that the collection of moments of µ X is determined by the values E takes on selfadjoint polynomials P ∈ B X , and vice-versa. Since elements b ∈ B are uniquely determined by {ϕ(b) : ϕ : B → C state}, we conclude that the two definitions are equivalent for bounded variables X = X * ∈ M .

Hypothesis (H1) is clearly satisfied under the assumption that M is a finite factor and E is the trace-preserving conditional expectation.

Hypothesis (H2) deserves a few more comments. It is natural in terms of allowing for the analytic functions tools (including the R-transform of Voiculescu -see [START_REF] Voiculescu | Operations on certain non-commutative operator-valued random variables[END_REF][START_REF] Voiculescu | Free analysis questions II: The Grassmannian completion and the series expansions at the origin[END_REF]) to be deployed. But it can be also viewed as a measure of nondegeneracy of E: indeed,

let b = u + iv, u = u * , v > 0. Then E ℑ(b -X) -1 = E ℑ((u -X) + iv) -1 = -v -1 2 E v -1 2 (u -X)v -1 2 2 + 1 -1 v -1 2 , so that E ℑ(b -X) -1 < 0 if and only if E v -1 2 (u -X)v -1 2 2 + 1 -1 > 0. It is clear that, since X is unbounded, min σ v -1/2 (u -X)v -1/2 2 + 1 -1 = 0. Also, k := v -1/2 (u -X)v -1/2 2 + 1 -1 ≤ 1. Thus, non-invertibility of E ℑ(b -X) -1
becomes equivalent to the equality

E k - v -1 2 (u -X)v -1 2 2 + 1 -1 = k - v -1 2 (u -X)v -1 2 2 + 1 -1 .
That is, E is isometric on a positive element which is not in B. Thinking in terms of the duals of M and B, respectively, this tells us that there exists an element ϕ of norm one in the dual

B * of B such that v -1 2 (u -X)v -1 2 2 + 1 -1
k reaches its norm on ϕ • E. Thus, hypothesis (H2) is implied by the requirement that positive elements in M but not in B do not reach their norms on B * • E. It may be worth mentioning that in the case of a tracial W * -probability space with normal faithful trace state τ which is left invariant by E, hypothesis (H2) comes to stating that

v -1 2 (u -X)v -1 2 2 + 1 -1 -k
does not reach its norm on L 2 (B, τ ), and in the case when B is finite dimensional, (H2) is equivalent to not allowing algebraic relations between X and elements in B.

In this section, we shall show that the fixed point equation ( 40) has a nontrivial solution also when X is unbounded, but still satisfies hypotheses (H1) and (H2) above. While, according to [START_REF] Williams | B-Valued free convolution for unbounded operators[END_REF][START_REF] Pascoe | Representation of free Herglotz functions[END_REF], the noncommutative extension G X,n (b) = (Id C n×n ⊗ E) (b -I n ⊗ X) -1 , n ∈ N, does not characterize the distribution of X for all possible unbounded random variables X as above, it follows quite easily (see, for instance, [START_REF] Haagerup | A new application of random matrices: Ext(C * red (F 2)) is not a group[END_REF]) that the extension {α ∈

C n×n : α = α * } × H + n (B) ∋ (α, b) → G α⊗X (b) = (Id C n×n ⊗ E) (b -α ⊗ X) -1
, n ∈ N, does encode the distribution of X with respect to E, as defined in Definition 6.1. For the reader's comfort, we will sketch the argument in Remark 6.4 below.

ε 0 1) -1 ∆h 0 (a, c)(b) * (ℑh 0 (a) -ε 0 1) -1/2 ≤ (ℑa) -1/2 b(ℑc) -1/2 2 • 1 for all a, c ∈ H + (B), b ∈ B.
We re-write this as

∆h 0 (a, c)(b)(ℑh 0 (c) -ε 0 1) -1 ∆h 0 (a, c)(b) * ≤ (ℑa) -1 2 b(ℑc) -1 2 2 (ℑh 0 (a) -ε 0 1).
≤ (ℑa) -1 2 b(ℑc) -1 2 2 (1 -ε 0 (ℑh 0 (a)) -1 ) ≤ (ℑa) -1 2 b(ℑc) -1 2 2 1 -ε 0 (ℑh 0 (a)) -1 = (ℑa) -1 2 b(ℑc) -1 2 2 (1 -ε 0 (ℑh 0 (a)) -1 ). Since xx * ≤ M • 1 ⇐⇒ x * x ≤ M • 1, we immediately obtain (ℑh 0 (a)) -1/2 ∆h 0 (a, c)(b)(ℑh 0 (c)) -1 ∆h 0 (a, c)(b) * (ℑh 0 (a)) -1/2 ≤ (ℑa) -1 2 b(ℑc) -1 2 2 (1 -ε 0 (ℑh 0 (a)) -1 )(1 -ε 0 (ℑh 0 (c)) -1 ). (41) Applying this to b = a -c yields (ℑh 0 (a)) -1/2 (h 0 (a) -h 0 (c))(ℑh 0 (c)) -1 (h 0 (a) -h 0 (c)) * (ℑh 0 (a)) -1/2 ≤ (ℑa) -1 2 (a -c)(ℑc) -1 2 2 (1 -ε 0 (ℑh 0 (a)) -1 )(1 -ε 0 (ℑh 0 (c)) -1 ). ( 42
)
It thus follows that if ω(b 0 ) ∈ H + (B) + iε 0 1 is a fixed point for h 0 , then it must be the unique and attracting fixed point of h 0 . Indeed, for an arbitrary a ∈ H + (B), if we let r = (ℑa) -1/2 (aω(b 0 ))(ℑω(b 0 )) -1/2 , it follows that Thus, for any N ∈ N, we have, by an iteration of ( 42),

(ℑh •N 0 (a)) -1 2 (h •N 0 (a) -ω(b 0 ))(ℑω(b 0 )) -1 2 2 = (ℑh •N 0 (a)) -1 2 (h •N 0 (a) -h •N 0 (ω(b 0 )))(ℑh •N 0 (ω(b 0 ))) -1 2 2 ≤ (ℑa) -1 2 (a -ω(b 0 ))(ℑω(b 0 )) -1 2 2 × N j=1 (1 -ε 0 (ℑh •j 0 (a)) -1 )(1 -ε 0 (ℑh •j 0 (ω(b 0 ))) -1 ) = (ℑa) -1 2 (a -ω(b 0 ))(ℑω(b 0 )) -1 2 2 × (1 -ε 0 (ℑω(b 0 )) -1 ) N N j=1 (1 -ε 0 (ℑh •j 0 (a)) -1 ). ( 44
)
Letting N go to infinity sends (1ε 0 (ℑω(b 0 )) -1 ) N to zero, so that lim

N →∞ (ℑh •N 0 (a)) -1 2 (h •N 0 (a) -ω(b 0 ))(ℑω(b 0 )) -1 2 2 = 0.
Recall that x -1 ≥ 1

x for any positive operator x. Since

(2 + 4r 2 ) -1 ℑω(b 0 ) ≤ ℑh •N 0 (a) ≤ ℜω(b 0 ) + ℑω(b 0 ) (4r + 1) 2 ,
we have

(ℑh •N 0 (a)) -1 2 (h •N 0 (a) -ω(b 0 ))(ℑω(b 0 )) -1 2 2 ≥ h •N 0 (a) -ω(b 0 ) 2 ℑω(b 0 ) ℑh •N 0 (a) ≥ h •N 0 (a) -ω(b 0 ) 2 ℑω(b 0 ) ℜω(b 0 ) + ℑω(b 0 ) 2 (4r + 1) 2 ,
which allows us to conclude that lim

N →∞ h •N 0 (a) -ω(b 0 )) = 0,
uniformly on bounded sets which are at strictly positive norm-distance from B \H + (B). Iterating in relation [START_REF] Voiculescu | The coalgebra of the free difference quotient and free probability[END_REF] for

a = c = ω(b 0 ) yields [h ′ 0 (ω(b 0 ))] •N (b) ℑω(b 0 ) ≤ (ℑω(b 0 )) -1 2 [h ′ 0 (ω(b 0 ))] •N (b)(ℑω(b 0 )) -1 2 ≤ (ℑω(b 0 )) -1 2 b(ℑω(b 0 )) -1 2 (1 -ε 0 (ℑω(b 0 )) -1 ) N ≤ (ℑω(b 0 )) -1 (1 -ε 0 (ℑω(b 0 )) -1 ) N b , which implies that [h ′ 0 (ω(b 0 ))] •N (b) ≤ ℑω(b 0 ) (ℑω(b 0 )) -1 (1 -ε 0 (ℑω(b 0 )) -1 ) N b for all b ∈ B, so that [h ′ 0 (ω(b 0 ))] •N ≤ ℑω(b 0 ) (ℑω(b 0 )) -1 (1 -ε 0 (ℑω(b 0 )) -1 ) N , the norm of [h ′ 0 (ω(b 0 ))]
•N being the norm of a bounded linear self-map of the C *algebra B. Thus, for N > log( ℑω(b0) (ℑω(b0)) -1 )

-log(1-ε0 (ℑω(b0)) -1 ) , we have [h ′ 0 (ω(b 0 ))] •N < 1. In general, if a linear operator T on a Banach space B satisfies T N < 1, we may write

kN -1 j=0 T j = (1 + T + • • • + T N -1 ) + T N (1 + T + • • • + T N -1 ) + T 2N (1 + T + • • • + T N -1 ) + • • • + T (k-1)N (1 + T + • • • + T N -1 ) = (1 + T + • • • + T N -1 ) k-1 j=0 (T N ) j , which tends to (1 + T + • • • + T N -1 )(1 -T N ) -1 as k → ∞.
Since N is fixed, it follows easily that in fact so does k j=0 T j . A simple algebraic manipulation shows that (1

+ T + • • • + T N -1 )(1 -T N ) -1 = (1 -T ) -1 . Thus, Id B -h ′ 0 (ω(b 0 )
) is invertible as a linear self-map of the Banach space B. By the implicit function theorem for analytic maps on Banach spaces, it follows that ω depents analytically on b 0 . This result, together with the properties of fixed points for noncommutative maps proved in [START_REF] Abduvalieva | Fixed point theorems for noncommutative functions[END_REF] allow us to conclude that ω is a noncommutative map on a noncommutative neighbourhood of b 0 .

All of the above has been established under the assumption that a fixed point ω(b 0 ) exists. We have not proved its existence, though. Relation [START_REF] Voiculescu | Free Analysis Questions I: Duality Transform for the Coalgebra of ∂ X:B Internat[END_REF] would allow us easily to prove such an existence along the lines of the above proof if we could somehow guarantee the boundedness of the iterates {h •N 0 (a)} N ∈N for some given a ∈ H + (B). Unfortunately, this does not seem possible to do in a direct way. Thus, we show the existence of the fixed point ω(b 0 ) by a perturbative argument, most of which is contained in the following proposition, which, we believe, might be of independent interest. Define (45) k 0 (a) = -h 0 (-a -1 ) -1 , a ∈ H + (B).

As ℑh(a) > ε 0 1, it follows that k 0 (H + (B)) ⊆ {w : wi(2ε 0 ) -1 1 < (2ε 0 ) -1 }, the noncommutative ball centered at an imaginary multiple of the identity. Proof. Note that the set a + k 0 (H + (B)) is bounded and bounded away from the complement of H + (B). Thus, the argument used above allows us to conclude the existence, uniqueness and analyticity of x on H + (B). The existence of x(0) in H + (B) is the only difficult part of the proof. For this, we shall use some results from [START_REF] Belinschi | Some Geometric Properties of the Subordination Function Associated to an Operator-Valued Free Convolution Semigroup[END_REF], specifically Proposition 3.

]). These results have been formulated for functions of a slightly different nature, but it is very easy to see that all elements of the proofs involved adapt to bounded functions like k 0 which satisfy k 0 (a * ) * = k 0 (a). We claim that x(H + (B)) = {m + in : m = m * , n > ℑk 0 (m + in)}. Since x(a) = a + k 0 (x(a)), the inclusion ⊆ is quite obvious. To prove ⊇, recall that the map B sa ∋ p → ℜx(p + iq) ∈ B sa is a bijection for any given q > 0 (see [START_REF] Belinschi | Some Geometric Properties of the Subordination Function Associated to an Operator-Valued Free Convolution Semigroup[END_REF]Corollary 3.3]). We also know that there exists a smooth function g q : B sa → {b ∈ B : ℑb > 0} such that g q (ℜx(p + iq)) = ℑx(p + iq). In particular, for any m ∈ B sa , there exists a unique n > 0 such that g q (m) = n: we have

m + in = x(p + iq) = p + iq + k 0 (x(p + iq)) = p + iq + k 0 (m + in),
so that p = m -ℜk 0 (m + in), q = n -ℑk 0 (m + in). This proves ⊇.

Since k 0 (H + (B)) is bounded, it folows that for any pair m = m * , n > 0, we have yn > k 0 (m + iyn) for all sufficiently large y ∈ (0, +∞). Thus, we may define 0 ≤ t m,n = inf{y > 0 : sn > ℑk 0 (m + isn) for all s > y}.

We argue that for all s > t m,n , we have sn > ℑk 0 (m + isn), and for all 0 ≤ s ≤ t m,n , we have sn > ℑk 0 (m + isn). The argument is virtually identical to the one in [13, Lemma 5.8] and is based on related works in the case of scalar, classical distributions by Biane [START_REF] Ph | On the Free Convolution with a Semi-circular Distribution[END_REF] and by Huang [START_REF] Huang | Supports of Measures in a Free Additive Convolution Semigroup[END_REF], so we will only sketch it. We consider the map ≥ y for all y ∈ (0, y 0 ]. Since this holds for any state ϕ, our claim follows.

C + ∋ z → ϕ(m+ zn -k 0 (m + zn)) ∈ C for an arbitrary state ϕ. If H(z) = ϕ(m) ϕ(n) + z -ϕ(k0(m+zn)) ϕ(n) , then
Obviously, there are two possibilities: either t m,n > 0 or t m,n = 0. Consider first the case when t m,n = 0. Pick a state ϕ on B and n ′ > 0. We have

(yn) -1 2 (yn -yn ′ )(yn ′ ) -1 2 2 ≥ |ϕ(k 0 (m + iyn)) -ϕ(k 0 (m + iyn ′ ))| 2 ℑϕ(k 0 (m + iyn))ℑϕ(k 0 (m + iyn ′ )) ,
which in its own turn implies

n -1 2 (n -n ′ )(n ′ ) -1 2 2 ≥ ℑϕ(k 0 (m + iyn)) ℑϕ(k 0 (m + iyn ′ )) - ℑϕ(k 0 (m + iyn ′ )) ℑϕ(k 0 (m + iyn)) 2 .
As t m,n = 0, we have ϕ(ℑk0(m+iyn)) y < ϕ(n) for all y > 0, so that necessarily

0 ≤ lim inf y→0 ℑϕ(k 0 (m + iyn ′ )) y ≤ ϕ(n) 2   2 + n -1 2 (n -n ′ )(n ′ ) -1 2 2 + 2 + n -1 2 (n -n ′ )(n ′ ) -1 2 2 2 -4   .
As this holds for any n ′ > 0 and any state ϕ on B, we conclude that k 0 satisfies the hypotheses of [START_REF] Belinschi | A noncommutative Version of the Julia-Wolff-Carathéodory Theorem[END_REF]Theorem 2.3]. Thus, if there exists a pair m = m * , n > 0 such that t m,n = 0, then for any n ′ > 0,

lim y→0 k 0 (m + iyn ′ ) = α = α *
exists in the norm topology. However, observe that since k 0 (H + (B)) ⊆ {w : wi(2ε 0 ) -1 1 < (2ε 0 ) -1 }, and the limit is in norm, we must have α = 0. Now consider the case when t m,n > 0. As seen above, for any y > t m,n , there exist p y = m -ℜk 0 (m + iyn), q y = yn -ℑk 0 (m + iyn) such that x(p y + iq y ) = p y + iq y + k 0 (x(p y + iq y )) = p y + iq y + k 0 (m + iyn) (in particular, q y > 0). This provides the expression of lim y→tm,n x(p y + iq y ) = m + it m,n n ∈ H + (B). Simple continuity guarantees that x(p tm,n +iq tm,n ) = p tm,n +iq tm,n +k 0 (x(p tm,n +iq tm,n )). Thus, H + (B) ∋ w → p tm,n + iq tm,n + k 0 (w) ∈ H + (B) has a fixed point in H + (B). Since the range of this map is bounded in the unbounded set H + (B), the fixed point is necessarily unique and attracting (indeed, one can apply the argument from the first part of the proof of Theorem 6.2, for ex., to the map {w + p tm,n + iq tm,n : wiε -1 0 1 < ε -1 0 } ∋ w → p tm,n + iq tm,n + k 0 (w) ∈ {w : wi(2ε 0 ) -1 1 < (2ε 0 ) -1 } to conclude uniqueness and norm-convergence of iterates to the fixed point, or one can appeal to Proposition 5.4). Thus, x extends to a norm-neighbourhood of p tm,n + iq tm,n .

To summarize: either t m,n = 0, and then, by an application of [12, Theorem 2.3], k 0 has a Julia-Carathéodory derivative at m, and lim y→0 k 0 (m + iyn ′ ) = 0 in norm for all n ′ > 0, or t m,n > 0, and then x extends analytically around p tm,n + iq tm,n = m -ℜk 0 (m + it m,n n) + i(t m,n n -ℑk 0 (m + it m,n n)). We apply this to m = 0. Assume towards contradiction that t 0,n = 0 for some n > 0. Recall from [START_REF] Belinschi | Some Geometric Properties of the Subordination Function Associated to an Operator-Valued Free Convolution Semigroup[END_REF]Relation (22)] the definition of the pseudo-horodisks at zero in "direction" n > 0:

H(0, n) = {w ∈ H + (B) : (w -0) * (ℑw) -1 (w -0) ≤ n} = {w ∈ H + (B) : n -1/2 ℑwn -1/2 + n -1/2 ℜw(ℑw) -1 ℜwn -1/2 ≤ 1}, and
H(0, n) = {w ∈ H + (B) : (w -0) * (ℑw) -1 (w -0) < n} = {w ∈ H + (B) : n -1/2 ℑwn -1/2 + n -1/2 ℜw(ℑw) -1 ℜwn -1/2 < 1}.
Note that the only selfadjoint element in H(0, n) is zero. Indeed, by definition, if w ∈ H(0, n), then n ≥ ℜw(ℑw) -1 ℜw + ℑw, so that if ℑw → 0, then necessarily ℜw → 0 (in fact one can easily obtain the estimate ℑw ≥ ℑw (n -ℑw) > ℑw ℜw(ℑw) -1 ℜw ≥ (ℜw) 2 ). Consider B(iyn, y -1/2 ), y > 0, with B defined in relation [START_REF] Voiculescu | Free analysis questions II: The Grassmannian completion and the series expansions at the origin[END_REF]. We have:

H(0, n) ⊆ 0<t<1 0<y<t B(iyn, y -1/2 ) ⊆ H(0, n).
This has been shown in [START_REF] Belinschi | Some Geometric Properties of the Subordination Function Associated to an Operator-Valued Free Convolution Semigroup[END_REF], but we will provide a sketch of the proof below. Thus, assume towards contradiction that a ∈ H(0, n), but a ∈ 0<t<1 0<y<t B(iyn, y -1/2 ). Then there exist a t 0 ∈ (0, 1) such that a ∈ B(iyn, y -1/2 ) for any y ∈ (0, t 0 ). That is, (aiyn) * (ℑa) -1 (aiyn) ≤ n for all y ∈ (0, t 0 ). At the same time, there exists an ǫ a,n ∈ (0, +∞) such that a * (ℑa) -1 a ≤ n-ǫ a,n •1. However, (a-iyn) * (ℑa) -1 (a-iyn) = a * (ℑa) -1 a+y(in(ℑa) -1 a-ia * (ℑa) -1 n+yn(ℑa) -1 n) ≤ a * (ℑa) -1 a+y(2 n (ℑa) -1 a + y n 2 (ℑa) -1 ) < a * (ℑa) -1 a + ǫ a,n • 1 ≤ n for all y ∈ (0, n (ℑa) -1 ( a 2 + ǫ a,na )). This is a contradiction. Thus the first inclusion holds. The second inclusion is equally simple: a ∈ B(iy j n, y -1/2 j ) for some sequence y j decreasing to zero is equivalent to a * (ℑa) -1 a + y j (in(ℑa) -1 aia * (ℑa) -1 n + y j n(ℑa) -1 n) ≤ n for all j ∈ N, which implies a * (ℑa) -1 a ≤ n, that is, a ∈ H(0, n). We have

(46) k 0 H(0, n) ⊆ k 0 0<t<1 0<y<t B(iyn, y -1/2 ) ⊆ 0<t<1 k 0 0<y<t B(iyn, y -1/2 ) = 0<t<1 0<y<t k 0 (B(iyn, y -1/2 )).
Recall that iyn = x(p y + iq y ) = p y + iq y + k 0 (x(p y + iq y )) = p y + iq y + k 0 (iyn), that is, iyn = x(p y + iq y ) is a fixed point for w → p y + iq y + k 0 (w). Thus, (47) k 0 (B(yn, y -1/2 )) ⊆ B(iyn, y -1/2 ) -(p y + iq y ).

We have seen that p y = -ℜk 0 (iyn) tends to zero in norm (in fact p y /y is bounded as y → 0), and q y = yn -ℑk 0 (iyn) → 0 in norm as y → 0 (in fact, q y /y is uniformly bounded for y ∈ (0, 1)).

We claim that 0<t<1 0<y<t

(B(iyn, y -1/2 ) -(p y + iq y )) ⊆ H(0, n).

Assume that is not the case. Then there exists

a 0 ∈ 0<t<1 0<y<t (B(iyn, y -1/2 ) -(p y + iq y )) \ H(0, n),
that is, for all t ∈ (0, 1), there exists 0 < y < t such that a 0 ∈ B(iyn, y -1/2 ) -(p y + iq y ), and yet a * 0 (ℑa 0 ) -1 a 0 ≤ n. So (representing B on a Hilbert space via the GNS construction), there exists a unit vector ξ and a number η > 0 such that (48) (ℑa 0 ) -1 a 0 ξ, a 0 ξ > nξ, ξ + η and a 0 = α 0p yiq y , where (α 0iyn) * (ℑα 0 ) -1 (α 0iyn) ≤ n. Thus, we found a sequence {y j } j∈N decreasing to zero such that (a 0 + p yj + iq yjiy j n) * (ℑa 0 + q yj ) -1 (a 0 + p yj + iq yjiy j n) ≤ n;

in particular, (ℑa 0 + q yj ) -1 (a 0 + p yj + iq yjiy j n)ξ, (a 0 + p yj + iq yjiy j n)ξ ≤ nξ, ξ .

Expanding, we obtain (ℑa 0 + q yj ) -1 a 0 ξ, a 0 ξ + 2ℜ (ℑa 0 + q yj ) -1 a 0 ξ, (p yj + iq yjiy j n)ξ + (ℑa 0 + q yj ) -1 (p yj + iq yjiy j n)ξ, (p yj + iq yjiy j n)ξ ≤ nξ, ξ . (49) From ( 48) and (49) together we obtain (by cancelling nξ, ξ ) (ℑa 0 ) -1 a 0 ξ, a 0 ξη > (ℑa 0 + q yj ) -1 a 0 ξ, a 0 ξ + 2ℜ (ℑa 0 + q yj ) -1 a 0 ξ, (p yj + iq yjiy j n)ξ + (ℑa 0 + q yj ) -1 (p yj + iq yjiy j n)ξ, (p yj + iq yjiy j n)ξ .

We re-arrange this relation to get (ℑa 0 ) -1 a 0 ξ, a 0 ξ -(ℑa 0 + q yj ) -1 a 0 ξ, a 0 ξη = (ℑa 0 + q yj ) -1 q yj (ℑa 0 ) -1 a 0 ξ, a 0 ξη > 2ℜ (ℑa 0 + q yj ) -1 a 0 ξ, (p yj + iq yjiy j n)ξ + (ℑa 0 + q yj ) -1 (p yj + iq yjiy j n)ξ, (p yj + iq yjiy j n)ξ .

Since lim j→∞ q yj = lim j→∞ p yj = lim j→∞ y j = 0, when we take limit as j → ∞ in the above inequality, we obtain -η > 0, a contradiction. Thus, 0<t<1 0<y<t (B(iyn, y -1/2 ) -(p y + iq y )) ⊆ H(0, n).

Combining this with relations (46) and (47), we obtain k 0 (H(0, n)) ⊆ H(0, n).

Quite trivially,

a ∈ H(0, n) ⇐⇒ n -1/2 a * (ℑa) -1 an -1/2 ≤ 1 ⇐⇒ n -1/2 ℑan -1/2 + n -1/2 ℜa(ℑa) -1 ℜan -1/2 ≤ 1 ⇐⇒ (ℑa + ℜa(ℑa) -1 ℜa) -1 ≥ n -1 ⇐⇒ ℑ(-a -1 ) ≥ n -1 .

Thus, H(0, n) is mapped bijectively (and as a noncommutative set) onto the set {a ∈ H + (B) : ℑa ≥ n -1 } by the correspondence a → -a -1 . By the definition of k 0 (see (45)), it follows that h({a ∈ H + (B) : ℑa ≥ n -1 }) ⊆ {a ∈ H + (B) : ℑa ≥ n -1 }.

However, our hypothesis on h states that lim y→∞ ℑh(ℜa+iyℑa)ξ,ξ y = 0 for any a ∈ H + (B) and unit vector ξ. That is, given u = u * , v > 0, there exists an y u,v,ξ > 0 depending on u, v and ξ such that ℑh(u + iyv)ξ, ξ < y vξ, ξ /2 whenever y ≥ y u,v,ξ . But, choosing n -1 = yv, we obtain ℑh(u + iyv) ≥ ℑ(u + iyv) = yv, a contradiction. This concludes the proof of our proposition. By Proposition 6.3, a → -(b 0 + h(-a -1 )) -1 has an attracting fixed point in H + (B), call it a(b 0 ). Then ω(b 0 ) := -a(b 0 ) -1 is the fixed point of w → b 0 + h(w). The results of [START_REF] Abduvalieva | Fixed point theorems for noncommutative functions[END_REF] allow us to conclude the proof of Theorem 6.2.

In order to argue that Theorem 6.2 solves the problem of defining free convolution powers of distributions of unbounded selfadjoint random variables in the context of Definition 6.1 and Hypotheses (H1), (H2), let us show that if X = X * ∈ M , then h X (b) = E[(b -X) -1 ] -1b satisfies the hypothesis of Theorem 6.2. Fix b = u + iv, u = u * , v > 0. Then h X (u + zv) = E (u -X + zv) -1 -1uzv

= v 1/2 E z + v -1/2 (u -X)v -1/2 -1 -1 v 1/2 -u -zv = v 1 2 E z + v -1 2 (u -X)v -1 2 -1 -1 -z -v -1/2 uv -1/2 v 1 2 .
We argue that h X satisfies the hypothesis of Theorem 6.2. This means (via a polarization argument) to show that lim y→+∞ ℑhX (u+iyv)ξ,ξ y = 0. Denote Y = v -1 2 (u-X)v -1 2 . Since ℑ(m + in) -1 = -(mn -1 m + n) -1 , ℜ(m + in) -1 = n -1 m(mn -1 m + n) -1 , we have

ℑE 1 z -Y = -E y y 2 + (x -Y ) 2 < 0, ℜE 1 z -Y = E x -Y y 2 + (x -Y ) 2 ,
where z = x + iy. Thus, Dividing by y provides us with the majorizing term E -1. This, as a function of y, is decreasing, as it can be seen by taking the (classical) derivative with respect to y:

ℑE (z -Y ) -1 -1 = E x -Y y 2 + (x -Y ) 2 E y y 2 + (x -Y ) 2 -1 E x -Y y 2 + (x -Y ) 2 + E y y 2 + (x -Y ) 2
∂ y E y 2 y 2 + (x -Y ) 2 -1 = -E y 2 y 2 + (x -Y ) 2 -1 E 2y(x -Y ) 2 (y 2 + (x -Y ) 2 ) 2 E y 2 y 2 + (x -Y ) 2 -1 ≤ 0.
Thus, E Composing this with any wo-continuous state ϕ on the algebra of B provides us with a state ϕ • E on M with respect to which the distribution of Y is not a probability, contradicting (H1). If z u j v j Q j are realizations as above of the monomials m j , j = 1, 2, then

  z u 1 u 2 v 1 Q 1 0 v 2 0 Q 2  
is a realization of m 1 + m 2 : its inverse has (zm 1m 2 ) -1 as its (1, 1) entry, and its lower right corner is invertible regardless of the choice of m 1 and m 2 . Finally, if z u v Q is a realization of the polynomial P ∈ B X , then

  z u v * u * 0 Q * v Q 0   is a selfad-
joint realization of the selfadjoint polynomial P + P * . Of course, it is quite possible that applying the expectation E entrywise to the inverse of such a matrix is impossible (for instance, E[X] might not be defined for a given unbounded X affiliated with M ).

However, if Q = Q * , then for any number ǫ > 0, the matrix

z v * v Q + iǫ is invertible
in the space of matrices over M of the corresponding size, and its inverse is bounded in M . We take lim ǫ→0 E z v * v Q + iǫ 1,1

to obtain E (z -P ) -1 . Composing with any state ϕ on B yields the claimed result.

(2) In view of the above, applying Theorem 6.2 to solve our problem in free probability requires only one more ingredient: we must show that if X = X * satisfies Hypotheses (H1) and (H2), then so does α ⊗ X for any α = α * ∈ C n×n , n ∈ N. It is quite easy to show that α ⊗ X satisfies (H1): elementary matrix arithmetics rules guarantee that B n×n and α ⊗ X form a * -algebra of possibly unbounded operators affiliated to M n×n . We show next that (Id C n×n ⊗ E) ℑ(bα ⊗ X) -1 < 0 for any b ∈ H + n (B). This is again quite straightforward. First, since α is a scalar selfadjoint matrix, it can be diagonalized by conjugating with a unitary matrix U ∈ C n×n : U * (Id C n×n ⊗E) ℑ(b-α ⊗ X) -1 U = (Id C n×n ⊗E) ℑ(U * bU -diag(α 1 X, . . . , α n X)) -1 , where α 1 , . . . , α n are the real eigenvalues of α. Thus, it is enough to prove the statement for diagonal matrices α. Second, by a simple maximum principle argument, if this inequality happens at iv for some v > 0, then it must happen everywhere on H + n (B). Thus, it is enough to prove the statement for b = iI n ⊗ 1. But in this case, the statement is immediate: (Id C n×n ⊗E) ℑ(iI n ⊗ 1diag(α 1 X, . . . , α n X)) -1 = diag(E (i1α 1 X) -1 , . . . , E (i1α n X) -1 ) < 0, according to (H2).

Denote (Id

C n×n ⊗ E) (b -α ⊗ X) -1 , α = α * ∈ C n×n , n ∈ N,
by G X , and the same for F and h. Corollary 6.5. Under hypotheses (H1) and (H2), µ ⊞ρ X is well-defined for all cp maps ρ : B → B such that ρ -Id B is still cp.

Proof. Apply Theorem 6.2 to h(w) = (ρ -Id B )h X (w).

Let us conclude with a brief comment on the Nevanlinna representation of h X . If X ∈ M , results of [START_REF] Popa | Non-commutative functions and non-commutative free Lévy-Hinçin formula[END_REF] guarantee the existence of an extension B X of B in which there exists a bounded selfadjoint element X , and of a completely positive map ρ : B X → B such that h X (b) = -E[X] + ρ (Xb) -1 , b ∈ H + (B). As in the case of the classical Nevanlinna representation, for unbounded operators X, the cp map ρ is not the appropriate completely positive map anymore. We define η : B X → B by η[a] = ρ (Xi) -1 a(X + i) -1 . The correspondence becomes now h X (b) = ℜh X (i) + η (Xb) -1 + b + b(Xb) -1 b , ℑb > 0.

Observe that indeed ℑh(i) = η[i]. Rewriting this map as (50) h X (b) = ℜh X (i) + η (Xb) -1 -X + X (Xb) -1 X makes it clear that it maps H + (B) in its closure. It would be interesting to determine whether this expression is equivalent in some sense to the one obtained in [START_REF] Pascoe | Representation of free Herglotz functions[END_REF] for maps from the nc unit ball of a C * -algebra to its right half-plane.

( 1 )

 1 aS = Sb =⇒ f m (a)S = Sf n (b).

  such that a b 0 c∈ Ω n+m , then there exists a linear map ∆f n,m (a, c) :V n×m → W n×m such that (2) f n+m a b 0 c = f n (a) ∆f n,m (a, c)(b) 0 f m (c) .

1 ,

 1 then a ∈ D n , c ∈ D m . (Note that this is a sort of "converse" of part (b) of the definition of noncommutative sets.) Let S n,m = {g : V n×m → [0, +∞] : g(tb) = tg(b)∀t ≥ 0} (with the convention 0 × (+∞) = +∞), and define S = n,m∈N S n,m . Define a function δ D : D × D → S such that δ D (a, c) ∈ S n,m whenever a ∈ D n , c ∈ D m , by (12) δ D (a, c)(b) = sup t ∈ [0, +∞] : a sb 0 c ∈ D n+m for all s ∈ [0, t] with the convention 1/0 = +∞. Observe first that δ D (a, c) is indeed well-defined because noncommutative sets respect direct sums: a sb 0 c ∈ D n+m at least for s = 0. Second, δ D (a, c)(b) = 0 ⇐⇒ a sb 0 c ∈ D n+m for all s ∈ [0, +∞). Third, if s 0 ∈ (0, +∞) is given, then, as indicated in the definition, δ D (a, c)(s 0 b) = s 0 δ D (a, c)(b).

  (a) for any a ∈ D n , we have f (a) ∈ E n ; (b) f respects direct sums; (c) if a ∈ D n , c ∈ D m and b ∈ V n×m are such that a b 0 c ∈ D n+m , then there exists a function of three variables denoted ∆f (a, c)(b) such that ∆f (a, c)(tb) = t∆f (a, c)(b) for all t ∈ [0, +∞) with the property that tb is in the domain of ∆f (a, c)(•), and f satisfies

Definition 3 . 11 .

 311 If D is a noncommutative set in an operator space satisfying assumptions (1)-(3), then for any n ∈ N, a, c ∈ D n ,

  [START_REF] Effros | Matrix Convexity: Operator Analogues of the Bipolar and Hahn-Banach Theorems[END_REF] dD ≤ d D for all D ⊂ V nc satisfying hypotheses (1) -(3). All of the above arguments hold when dD is replaced by dD,∞ and d D by d D,∞ . We record the result below: (20) dD,∞ ≤ d D,∞ for all D ⊂ V nc satisfying hypotheses (1) -(3).

  (1)-(3). If n ∈ N is given and a subset A of D n is open in the topology generated by dD , then it is open in the product topology induced by V on V n×n . Proof. Assume that a ∈ D n is given. For any net {a ι } ι∈I ⊆ D n which converges to a in the product topology of V n×n , we have by Remark 3.1 that 0 = δD (a, a) ≥ lim sup ι∈I δD (a ι , a) ≥ lim sup ι∈I dD (a ι , a) ≥ 0.

( 2 )

 2 O nc is finitely open and G is locally bounded on slices. Thus, G is analytic on slices in each of its two variables.(3) O nc is open in the level topology and G is locally bounded on slices. Thus, G is analytic on slices in each of its two variables. (4) For any n ∈ N and a, c ∈

( 6 )

 6 At each level at which the set {a ∈ O nc : G(a, a * )(1) > 0} is nonempty, we have G(a, a * )(1) -1 → +∞ as a tends to the norm-topology boundary of {a ∈ O nc : G(a, a * )(1) > 0}.[START_REF] Akhieser | The classical moment problem and some related questions in analysis[END_REF] Let Ω be a connected component of {a ∈ O nc : G(a, a * )(1) > 0}. For any given a ∈ Ω n , c ∈ Ω n , we have G(a, c * )(1) invertible as an element in the C * -algebra K n×n .

  would become 0 < G(c, c * )([P 22 ]), 0 < G(a, a * )([P 11 ]) + 1 ∆ 0 ∆G(a; c, c * ; a * )(b, [P 22 ], b * ) + 1 ∆G(a, c * ; a * )([P 12 ], b * ) + 0 ∆G(a; c, a * )(b, [P 21 ]), The requirement of positivity for G applied to a block-diagonal P = P 11 P 12 P 21 P 22 = (c, c * )([P 22 ]), 0 < G(a, a * )([P 11 ]) + 1 ∆ 0 ∆G(a; c, c * ; a * )(b, [P 22 ], b * ), and 0 ∆G(a; c, c * )(b, [P 22 ]) [G(c, c * )([P 22 ])]

Remark 4 . 6 .

 46 for all t ∈ R \ σ(G(c, c * )[START_REF] Abduvalieva | Fixed point theorems for noncommutative functions[END_REF]. This means that for any state ϕ, the map t → ϕ • f is convex and increasing on each connected component of R \ σ(G(c, c * )(1). Clearly,lim t→±∞ f (t)-(G(a, c * )(1)+G(c, c * )(1)+G(c, a * )(1)-G(a, a * )(1)) = 0, so if a is such that G(a, c * )(1) + G(c, c * )(1) + G(c, a * )(1) -G(a, a * )(1) ≥ 0 (for example, c ∈ D G,n and a close to c), then f (t) ≥ 0 for all real t in the connected component of -∞. Conversely, if G(a, c * )(1) + G(c, c * )(1) + G(c, a * )(1)-G(a, a * )(1) ≥ 0, then f (t) ≥ 0 for all real t in the connected component of +∞. Finally, the function δ has been defined in terms of the length of a "ray" in a given direction. In this remark we look at the whole set of points b for which the upper triangular matrix a b 0 c belongs to the chosen noncommutative set. Consider a nc set D which satisfies property (2). Fix m, n ∈ N and a ∈ D n , c ∈ D m . Let (a, c) nc = k∈N b ∈ V nk×mk :

Lemma 5 . 2 .

 52 Consider a noncommutative domain D and a noncommutative subset D ′ ⊆ D. Assume that both D and D ′ are invariant under conjugation by scalar unitary matrices and satisfy hypothesis (37). If δD (a, c) = δD ′ (a, c) for all a, c ∈ D ′ , then D = D ′ .

Remark 5 . 3 .

 53 an obvious contradiction. Thus, D ′ = D, as claimed. Consider the set f (D) ⊂ E. For any x, y ∈ f (D), there exist unique a, c ∈ D such that f (a) = x, f (c) = y. It follows from Proposition 3.2 that δ f (D) (x, y)(xy) ≤ δ D (a, c)(ac). Since f (D) ⊂ E, we necessarily have δf(D) (x, y) ≥ δE (x, y). Together with the hypothesis of (1), we obtain δD (a, c) = δE (f (a), f (c)) = δE (x, y) ≤ δf(D) (x, y) ≤ δD (a, c), so that δf(D) (x, y) = δE (x, y) for all x, y ∈ f (D). By Lemma 5.2, we conclude that f (D) = E. It turns out that in Theorem 5.1 we cannot dispense with the requirement that δ blows up at the boundary. The following counterexample, similar to the one in Remark 4.6, shows what goes wrong if this requirement is dropped. Consider a domain D ⊆ 1 2 D ⊂ C and define the nc set D = ∞ n=1 {A ∈ C n×n : σ(A) ⊂ D, A < 1}. The proof from Remark 4.6 applies to show that D is a unitarily invariant noncommutative set which is open at each level. However, a direct computation shows that a b 0 c < 1 if and only if aa * + bb * < 1, cc * < 1, and bc * (1cc * ) -1 cb * < 1aa *bb * (this holds in an arbitrary C * -algebra). Since the other restriction in the definition of D is on the spectrum of the matrix A, it only affects a and c; there is no other restriction on b. The last inequality is equivalent to b

Theorem 6 . 2 .

 62 Consider a noncommutative function h : H + (B) → B such that ℑh(b) ≥ 0 and lim y→+∞ ℑh(ℜb+iyℑb) y = 0 in the wo-topology for all b ∈ H + (B). For any given b > 0, the map w → b + h(w) has a unique attracting fixed point in H + (B), to be denoted by ω(b), and the correspondence b → ω(b) is a noncommutative self-map of H + (B), hence, in particular, analytic. Proof. It is clearly enough to prove the theorem at level one: fix b 0 ∈ B such that ℑb 0 > ε 0 1 > 0. For any n ≥ 1, the map h 0 : w → b 0 ⊗ I n + h(w) sends H + (B n×n ) into H + (B n×n )+ iε 0 1, so that, as a noncommutative map, it sends H + (B) to H + (B)+ iε 0 1. We re-write the proof of Corollary 3.4 for this context: if ℑ a b 0 c ∈ H + (B 2×2 ), then ℑh 0 a b 0 c ∈ H + (B 2×2 )+iε 0 1. That means (ℑh 0 (a)-ε 0 1) -1/2 ∆h 0 (a, c)(b)(ℑh 0 (c)-

  Multiplying left and right by (ℑh 0 (a)) -1/2 , we obtain (ℑh 0 (a)) -1/2 ∆h 0 (a, c)(b)(ℑh 0 (c)ε 0 1) -1 ∆h 0 (a, c)(b) * (ℑh 0 (a)) -1/2

  h 0 (B(ω(b 0 ), 2r)) ⊂ B(ω(b 0 ), 2r), where, as in [12, Proposition 3.2], we denote (43) B(c, t) = {a ∈ B : (ℑa) -1/2 (ac)(ℑc) -1/2 ≤ t}. As shown in [12, Proposition 3.2], B(ω(b 0 ), 2r) is bounded in norm in the sense that d ≤ ℜω(b 0 ) + ℑω(b 0 ) 2r 2 + 1 + 2r r 2 + 1 + 2r 2r 2 + 1 + 2r r 2 + 1 , and that it is bounded away from the boundary of H + (B) in the sense that ℑd ≥ (2 + 4r 2 ) -1 ℑω(b 0 ), d ∈ B(ω(b 0 ), 2r).

Proposition 6 . 3 .

 63 For any a ∈ H + (B) ∪ {0}, the fixed-point equation x = a + k 0 (x) has a unique solution x(a) in H + (B). x(a) is a noncommutative function of a whenever a ∈ H + (B), and x(0 m ⊕ 0 n ) = x(0 m ) ⊕ x(0 n ) for all m, n ∈ N.

y 2 y 2 + 2 - 1 - 2 - 1 = 1 +

 221211 (x-Y ) 1 is a decreasing function of y. If it does not decrease to zero, then there exists a positive operator 0 = c ≥ 0 which belongs to the von Neumann algebra B such that lim y→∞ E y 2 y 2 +(x-Y ) c in the strong operator topology. Multiplying left and right by (1 + c) -1/2 allows us to conclude that (1 + c) 1/2 E (z -Y ) -1 (1 + c) 1/2 belongs to the norm-ball of center -i/(2y) and radius 1/(2y). Taking the imaginary part and multiplying by y yields lim y→∞ (1 + c) 1/2 E y 2 y 2 + (x -Y ) 2 (1 + c) 1/2 = 1 in the so-topology. Thus 2 lim y→∞ E y 2 y 2 + (x -Y ) 2 = (1 + c) -1 .

Remark 6 . 4 .2 2 j

 642 [START_REF] Abduvalieva | Fixed point theorems for noncommutative functions[END_REF] We provide here a very brief sketch of a selfadjoint version of the classical realization-linearization trick that allows us to conclude that, under Hypotheses (H1), (H2), the noncommutative map{α ∈ C n×n : α = α * }×H + n (B) ∋ (α, b) → G α⊗X (b) = (Id C n×n ⊗E) (bα ⊗ X) -1, n ∈ N, determines the distribution of X as defined in Definition 6.1. Direct computation using the Schur complement shows that the matrix  We use here that if 0 < b -1 j decreases to 1, then 0 < b j increases to 1; this can be seen by evaluating(1b j ) 1/2 ξ, ξ 2 = b 1/(b -1 j -1) 1/2 ξ, ξ 2 ≤ b j ξ, ξ (b -1 j -1)ξ, ξ .is invertible, and its inverse has (zb 0 X • • • b n-1 Xb n ) -1as its (1, 1) entry. We call this matrix the realization of the monomial b 0 X • • • b n-1 Xb n . We note that its lower right (2n)×(2n) corner of this matrix is invertible regardless of the monomial b 0 X • • • b n-1 Xb n .

  This topology is not Hausdorff. A noncommutative function f defined on a noncommutative set Ω nc ⊆ V nc with values in an operator space is said to be uniformly analytic if Ω nc is uniformly open, and f is uniformly locally bounded and complex differentiable at each level. It is shown in[START_REF] Kaliuzhnyi-Verbovetskyi | Foundations of free noncommutative function theory[END_REF] Corollary 7.28] that f is analytic if and only if it is uniformly locally bounded (that is, the requirement of complex differentiability at each level is automatically satisfied by an nc function which is uniformly locally bounded on a uniformly open nc set).The second important topology (already mentioned above) is the finitely open topology: a set Ω nc ⊆ V nc is called finitely open if for any n ∈ N, the intersection of Ω n with any finite dimensional subspace X of V n×n is open in the Euclidean topology of X . It is shown in

  nc is satisfied by a large and important class of noncommutative domains. In particular, it is satisfied by generalized noncommutative half-planes and generalized noncommutative balls. Indeed, consider the upper half-plane from Example 2.3(i), H +

  1) ≥ 0.Remark 4.5. Second, we observe that certain obvious transformations of G have similarly obvious effects on D G,nc . For example, composing G with a completely positive unital map increases D G,nc . Indeed, if Φ is such a map, then G(a, a

* )(1) > ε1 =⇒ Φ(G(a, a * )(1)) > εΦ(1) = ε1, so that D G,nc ⊆ D Φ•G,nc .

Subtracting a positive multiple of 1 from G decreases D G,nc , adding increases it. However, if for any t ∈ R \ σ(G(c, c * )(1)) we let

  1, Remark 3.2(2), and Corollary 3.3, together with the definition of a noncommutative version of horodisks in the noncommutative upper half-plane (see [11, Relation