Operator-Valued Monotone Convolution Semigroups

and an Extension of the Bercovici-Pata Bijection.

Michael Anshelevich^{1}, John D. Williams

Received: May 19, 2015 Revised: May 17, 2016

Communicated by Friedrich Götze

Abstract. In a 1999 paper, Bercovici and Pata showed that a natural bijection between the classically, free and Boolean infinitely divisible measures held at the level of limit theorems of triangular arrays. This result was extended to include monotone convolution by the authors in [AW14]. In recent years, operator-valued versions of free, Boolean and monotone probability have also been developed. Belinschi, Popa and Vinnikov showed that the Bercovici-Pata bijection holds for the operator-valued versions of free and Boolean probability. In this article, we extend the bijection to include monotone probability theory even in the operator-valued case. To prove this result, we develop the general theory of composition semigroups of noncommutative functions and largely recapture Berkson and Porta’s classical results on composition semigroups of complex functions in operator-valued setting. As a byproduct, we deduce that operator-valued monotonically infinitely divisible distributions belong to monotone convolution semigroups. Finally, in the appendix, we extend the result of the second author on the classification of Cauchy transforms for non-commutative distributions to the Cauchy transforms associated to more general completely positive maps.

2010 Mathematics Subject Classification: Primary 46L53; Secondary 30D05, 47D06, 60E07

1The first author was supported in part by NSF grant DMS-1160849.

1. Introduction

It is a remarkable fact that there are natural bijections between the classes of infinitely divisible measures in each of the four universal non-commutative probability theories, which not only arise from the Lévy-Hinc̆in representations of the measures, but are maintained at the level of limit theorems of triangular arrays. This is made precise in the following theorem:

Theorem 1.1. Fix a finite positive Borel measure σ on ℝ, a real number γ, a sequence of probability measures {μ_{n}}_{n∈ℕ}, and a sequence of positive integers k_{1} < k_{2} < · · ·. The following assertions are equivalent:

(a) (Classical / tensor) The sequence \underbrace{μ_{n} ∗ μ_{n} ∗ ⋯ ∗ μ_{n}}_{k_{n}} converges weakly to

ν_{∗}^{γ,σ};

(b) (Free) The sequence \underbrace{μ_{n} ⊞ μ_{n} ⊞ ⋯ ⊞ μ_{n}}_{k_{n}} converges weakly to ν_{⊞}^{γ,σ};

(c) (Boolean) The sequence \underbrace{μ_{n} ⊎ μ_{n} ⊎ ⋯ ⊎ μ_{n}}_{k_{n}} converges weakly to ν_{⊎}^{γ,σ};

(d) (Monotone) The sequence \underbrace{μ_{n} ▷ μ_{n} ▷ ⋯ ▷ μ_{n}}_{k_{n}} converges weakly to ν_{▷}^{γ,σ};

(e) The measures

k_{n} \frac{x^{2}}{x^{2} + 1}dμ_{n}(x) → σ

weakly, and

lim_{n↑∞} k_{n} ∫_{ℝ} \frac{x}{x^{2} + 1}dμ_{n}(x) = γ.

Here ν_{∗}^{γ,σ}, ν_{⊞}^{γ,σ}, ν_{⊎}^{γ,σ}, ν_{▷}^{γ,σ} are probability measures defined explicitly through their complex-analytic transforms. The equivalence of classical, free, and Boolean limit theorems in parts (a), (b), (c) and (e) was proven in a by now classic paper due to Bercovici and Pata [BP99]. The monotone non-commutative probability theory is of more recent provenance [Mur00, Mur01]. The inclusion of part (d) was proven in our recent paper [AW14]. Voiculescu developed operator-valued notions of non-commutative probability [Voi87] where probability measures are replaced by certain completely positive maps from the ring of non-commutative polynomials over a C^{∗}-algebra. An analogous theorem in this more general setting, namely the equivalence of parts (b) and (c), was proven in [BPV12]. The first main result in this paper is the inclusion of (d) at this level of generality. In order to study monotone infinitely divisible B-valued distributions, we must first develop the theory of composition semigroups of non-commutative functions in a manner analogous to Berkson and Porta’s study of these semigroups at the level of complex functions [BPo78]. This stems from the fact that the convolution operation for monotone probability theory satisfies the following relation for the associated F - transforms,

F_{μ▷ν} = F_{μ} ∘ F_{ν},

Operator-Valued Monotone Convolution Semigroups 843

so that infinitely-divisible distributions form such a composition semigroup. In the second main result of the paper, we prove that any monotone infinitely-divisible distribution can be included in such a semigroup. Note that even in the scalar-valued case, this is a recent result, proved by Serban Belinschi in his thesis. Finally, we characterize generators of such composition semigroups, and a smaller set of generators of composition semigroups of F -transforms. In Section 2, we provide background and preliminary results. In section 3, we study composition semigroups of vector-valued and non-commutative analytic functions. The main results of this section are Proposition 3.3, which shows that there is a natural notion of a time derivative for semigroups of vector-valued analytic functions {f_{t}}_{t≥0}, and Theorem 3.5, which proves that, in the case of F -transforms and more general self-maps of the complex upper half plane, these semi-groups are in bijection with certain classes of functions defined through their analytic and asymptotic properties. This bijection provides a Lévy-Hinc̆in representation for these infinitely divisible distributions. In section 4 we prove the main result of the paper, namely the extension of Theorem 1.1 to the operator-valued case. In contrast to the previous section, this is achieved through a combinatorial methodology. We close the paper with the Appendix, which is primarily concerned with the extension of the main result in [Wil13], namely the classification of the Cauchy transforms associated to B-valued distributions, to a more general class of functions including the Cauchy transforms associated to more general CP maps. Acknowledgements. We are grateful to the referee for helpful comments.

2. Preliminaries

Let B denote a unital C^{∗}-algebra and X a self-adjoint symbol. We will define the ring of noncommutative polynomials B⟨X⟩ as the algebraic free product of B and X. B_{0}⟨X⟩ are polynomials in B⟨X⟩ with zero constant term.

Definition 2.1. Let μ : B⟨X⟩ → B denote a linear map. We say that μ is exponentially bounded with constant M if

(1)  ∥μ(b_{1}Xb_{2} ⋯ Xb_{n+1})∥ ≤ M^{n}∥b_{1}∥∥b_{2}∥ ⋯ ∥b_{n+1}∥

We abuse terminology and say that the map μ is completely positive (CP) if

(2)  (μ ⊗ 1_{n}) ([P_{i}(X)P_{j}^{∗}(X)]_{i,j=1}^{n}) ≥ 0

for every family P_{i}(X) ∈ B⟨X⟩. We define a set Σ_{0} to be those B-bimodular linear maps μ satisfying (1) and (2).

For a general introduction to non-commutative functions, we refer to [KVV14]. Throughout, B, A shall denote unital C^{∗}-algebras. Let M_{n}(B) denote the n × n matrices with entries in B. We define the noncommutative space over B to be the set B_{nc} = {M_{n}(B)}_{n=1}^{∞}. A non-commutative set is a subset Ω ⊂ B_{nc} that respects direct sums. That is, for X ∈ Ω ∩ M_{n}(B) and Y ∈ Ω ∩ M_{p}(B) we have that X ⊕ Y ∈ Ω ∩ M_{n+p}(B). We note that these definitions apply to the more general case of B being any unital, commutative ring, but we focus on the C^{∗}-algebraic setting. Given b ∈ M_{n}(B), the non-commutative ball of radius δ about b is the set B_{δ}^{nc}(b) := ⊔_{k=1}^{∞}B_{δ}(⊕^{k}b) where B_{δ}(⊕^{k}b) ⊂ M_{nk}(B) is the standard ball of radius δ. A non-commutative function is a map f : Ω → A_{nc} with the following properties:

(a) f(Ω_{n}) ⊂ M_{n}(A)

(b) f respects direct sums : f(X ⊕ Y) = f(X) ⊕ f(Y)

(c) f respects similarities: For X ∈ Ω_{n} and S ∈ M_{n}(ℂ) invertible we have that

f(SXS^{−1}) = Sf(X)S^{−1}

provided that SXS^{−1} ∈ Ω_{n}. A non-commutative function is said to be locally bounded in slices if, for every n and element x ∈ Ω_{n}, f|_{Ω_{n}} is bounded on some neighborhood of x in the norm topology. It is a remarkable fact originally due to Taylor ([Tay72], [Tay73]) that a noncommutative function that is Gâteaux differentiable and locally bounded in slices is in fact analytic. A non-commutative function is uniformly analytic at b ∈ M_{n}(B) if it is analytic and bounded on B_{r}^{nc}(b) for some r > 0. Let M_{n}^{+,ϵ}(B) ⊂ M_{n}(B) denote those element b ∈ M_{n}(B) with ℑ(b) > ϵ1_{n} and M_{n}^{+}(B) = ∪_{ϵ>0}M_{n}^{+,ϵ}. We form a non-commutative set

H^{+}(B) = ⊔_{n=1}^{∞}M_{n}^{+}(B)

and refer to this set as the non-commutative upper half plane. We define a family of sets in H^{+}(B). For α, ϵ > 0 define a non-commutative Stolz angle to be

Γ_{α,ϵ}^{(n)} := {b ∈ M_{n}^{+,ϵ}(B) : ℑ(b) > αℜ(b)}.

Let μ ∈ Σ_{0}. We define the Cauchy transform of μ to be the analytic, non-commutative function G_{μ} = {G_{μ}^{(n)}}_{n=1}^{∞} such that

G_{μ}^{(n)}(b) := (μ ⊗ 1_{n})((b − X ⊗ 1_{n})^{−1}) : H^{+}(B) ↦ H^{−}(B).

From this map, we may construct the moment generating function, the F-transform, the Voiculescu transform and the R-transform respectively through the following equalities:

H^{(n)}(b) := G^{(n)}(b^{−1}) : H^{−}(B) ↦ H^{−}(B)
F^{(n)}(b) := G^{(n)}(b)^{−1} : H^{+}(B) ↦ H^{+}(B)
φ_{μ}^{(n)}(b) := (F_{μ}^{(n)})^{⟨−1⟩}(b) − b
R_{μ}^{(n)}(b) := φ_{μ}^{(n)}(b^{−1})

where the superscript ⟨−1⟩ refers to the composition inverse. We also note that the moment generating function extend to a neighborhood of 0 for μ ∈ Σ_{0} and that the Voiculescu-transform is only defined on a subset of H^{+}(B). The following result, proven in [Wil13] and [PV13], classifies the F -transforms in terms of their analytic and asymptotic properties.

Theorem 2.1. Let f = (f^{(n)}) : H^{+}(B) → H^{+}(B) denote an analytic, noncommutative function. The following conditions are equivalent.

(a) f = F_{μ} for some μ ∈ Σ_{0}.

(b) The noncommutative function k = (k^{(n)})_{n=1}^{∞} defined by k^{(n)}(b) := (f^{(n)}(b^{−1}))^{−1} has uniformly analytic extension to a neighborhood of 0. Moreover, for any sequence {b_{k}}_{k∈ℕ} with ∥b_{k}^{−1}∥ ↓ 0, b_{k}^{−1}f^{(n)}(b_{k}) → 1_{n} in norm.

(c) There exists an α ∈ B and a σ : B⟨X⟩ → B which satisfies (1) and (2) such that, for all n ∈ ℕ,

f^{(n)}(b) = α1_{n} + b − (σ ⊗ 1_{n})(b(1 − Xb)^{−1}).

Moreover, the map σ in (c) is of the form σ(P (X)) = ρ(XP (X)X) for ρ such that its restriction to B_{0}⟨X⟩ is positive.

We will require several classical results in complex function theory to prove our results. Theorem 3.16.3 in [HP74] is a useful analogue of the classical Cauchy estimates in complex analysis. We also refer to this reference for an overview of the differential structure of vector valued functions, including the higher order derivative δ^{n} utilized below.

Theorem 2.2. Let f be Gâteaux differentiable in U and assume that ∥f(x)∥ ≤ M for x ∈ U. Then

∥δ^{n}f(a; h)∥ ≤ Mn!

for a + h ∈ U.

Further, theorem 3.17.17 in [HP74] provides Lipschitz estimates for analytic functions. Indeed, for an analytic function f that is locally bounded by M(a) in a neighborhood of radius r_{a}, we have that

(3)  ∥f(y) − f(x)∥ ≤ \frac{2M(a)∥x − y∥}{r_{a} − 2∥x − y∥}

Notation 2.2. We define a family Λ of functions Φ : H^{+}(B) → \overline{H^{−}(B)} through the following properties:

(i) The map R(b) := Φ(b^{−1}) has uniformly analytic continuation to a noncommutative ball about 0 with R(b)^{∗} = R(b)

(ii) For any sequence {b_{k}}_{k∈ℕ} ∈ B with ∥b_{k}^{−1}∥ ↓ 0, we have that b_{k}^{−1}Φ(b_{k}) → 0.

We also define a larger family of functions Λ̃ by replacing (i) and (ii) with the following weaker conditions

(I) For any ϵ > 0, Φ is uniformly bounded on ⊔_{n=1}^{∞}M_{n}^{+,ϵ}(B). (II) For any α, ϵ > 0 and a sequence {b_{k}}_{k∈ℕ} ∈ Γ_{α,ϵ}^{(n)} with ∥b_{k}^{−1}∥ ↓ 0, we have that

b_{k}^{−1}Φ(b_{k}) → 0.

Definition 2.3. Let μ, ν ∈ Σ_{0}. We define the monotone convolution to be the noncommutative operation (μ, ν) ↦ μ ▷ ν ∈ Σ_{0} defined implicitly though the equality

F_{μ▷ν} := F_{μ} ∘ F_{ν}.

Note that this definition uses Theorem 2.1 in an essential way, to show that a composition of F -transforms is an F -transform. See Section 4 and references [Pop08, HS11, Pop12, HS14] for the relation between this definition and monotone independence of Muraki.

Definition 2.4. We say that μ is a ▷-infinitely divisible distribution if, for every n, there exists a distribution μ_{n} ∈ Σ_{0} such that

(4)  μ = \underbrace{μ_{n} ▷ μ_{n} ▷ ⋯ ▷ μ_{n}}_{n times}

We define a composition semigroup of F -transforms {F_{t}}_{t∈ℚ^{+}} by letting F_{p/q} := F_{μ_{q}}^{∘p} where μ = μ_{q}^{▷q} for all p, q ∈ ℕ. We will show in Theorem 3.5that this semigroup extends to an ℝ^{+} semigroup, which moreover is generated by a function Φ ∈ Λ in a sense that will be made specific. Moreover, one of the main results in [Wil13] is that the set Λ is exactly the set of Voiculescu transforms associated to ⊞-infinitely divisible distributions. This is not a coincidence and will drive the main result of this paper.

3. Lévy-Hinc̆in Representations for Semigroups of Non-Commutative Functions.

We begin this section with a result showing that the divisors of ▷-infinitely divisible distributions maintain the same exponential bound. A similar result can be proven in the combinatorial setting of Section 4 in an easier manner, but the bound is less sharp.

Proposition 3.1. Let μ denote a ▷-infinitely divisible distribution with exponential bound M. Then, for each k, the distribution μ_{k} satisfying μ = μ_{k}^{▷k} has exponential bound M.

Proof. Let Xb_{1}Xb_{2} ⋯ b_{n−1}X = Q(X) ∈ B⟨X⟩ such that ∥b_{1}∥ = ∥b_{2}∥ = ⋯ ∥b_{n−1}∥ = 1 and assume, for the sake of contradiction, that ∥μ_{k}(Q(X))∥ > M^{n}. Then, using the Schwarz inequality for 2-positive maps, we have that

∥μ_{k}(Q^{∗}(X)Q(X))∥∥μ_{k}(1)∥ ≥ ∥μ_{k}(Q(X))μ_{k}(Q^{∗}(X))∥
= ∥μ_{k}(Q(X))∥^{2} > M^{2n}

Since μ_{k}(1) = 1, we may assume that our monomial P (X) = Xb_{1}Xb_{2} ⋯ b_{n−1}X^{2}b_{n−1}^{∗}X ⋯ b_{1}^{∗}X has the property that μ_{k}(P (X)) > M^{2n}. Define an element B ∈ M_{2n}(B) by

B = ( \begin{matrix}0 & 1 & 0 & 0 & 0 & 0 & 0 & ⋯ & 0 \\ 1 & 0 & b_{1} & 0 & 0 & 0 & 0 & ⋯ & 0 \\ 0 & b_{1}^{∗} & 0 & 1 & 0 & 0 & 0 & ⋯ & 0 \\ 0 & 0 & 1 & 0 & b_{2} & 0 & 0 & ⋯ & 0 \\ 0 & 0 & 0 & b_{2}^{∗} & 0 & 1 & 0 & ⋯ & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & b_{3} & ⋯ & 0 \\ & ⋮ & & & ⋮ & & & & ⋮ \\ 0 & 0 & 0 & 0 & 0 & ⋯ & b_{n−1}^{∗} & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & ⋯ & 0 & 1 & 0 \end{matrix} ) .

That is, the superdiagonal alternates between 1 and b_{i}, the subdiagonal alternates between 1 and b_{i}^{∗}. Now, let 0 < ϵ, δ and

B_{δ,ϵ} = δB + ϵ ( ∑_{i=1}^{2n−1} e_{i,i}) + \frac{e_{2n,2n}}{δ^{n−1}}

where ϵ is arbitrarily small and δ is chosen so that B_{δ,ϵ} is a strictly positive element. Moreover, we have that

e_{1,1}(B_{δ,ϵ}(X ⊗ 1_{2n})B_{δ,ϵ})^{2n}e_{1,1} = e_{1,1}B_{δ,ϵ}[(X ⊗ 1_{2n})B_{δ,ϵ}^{2}]^{2n−1}(X ⊗ 1_{2n})B_{δ,ϵ}e_{1,1} (5)
= P (X) + O(max (δ, ϵ)).

To see this, note that a non-trivial contribution to (5) must be of the form

b_{1,2}Xb_{2,j_{3}}b_{j_{3},j_{4}}Xb_{j_{4},j_{5}}X ⋯ b_{j_{4n−2},j_{4n−1}}b_{j_{4n−1},2}Xb_{2,1}

where b_{i,j} denotes the i, j entry of B_{δ,ϵ}. Now, such a non-zero term is not O(max (δ, ϵ)) means that b_{j_{ℓ},j_{ℓ+1}} must equal b_{2n,2n} for two distinct ℓ. However, the only possible way for this to occur is if j_{k} = k for k = 2, …, 2n, j_{2n} = j_{2n+1} = j_{2n+2} = 2n and j_{p} = 4n + 2 − p for p = 2n + 2, …, 4n − 1. By assumption, there exists a state ϕ ∈ B^{∗} such that ϕ(μ_{k}(P (X))) > M^{2k}. Thus, for ϵ small enough, we have that

(6)  ϕ_{1,1} ∘ (μ_{k} ⊗ 1_{2n})((B_{δ,ϵ}(X ⊗ 1_{2n})B_{δ,ϵ})^{2n}) > M^{2n}

(here ϕ⊗e_{1,1} = ϕ_{1,1}). This implies that the scalar valued Cauchy transform associated to this random variable,

G_{μ_{k}}^{δ,ϵ}(z) = ϕ_{1,1} ∘ (μ_{k} ⊗ 1_{2n})((z1_{2n} − B_{δ,ϵ}(X ⊗ 1_{2n})B_{δ,ϵ})^{−1})

arises from a measure whose support has non-trivial intersection with ℝ \ [−M, M], whereas the (similarly defined) G_{μ}^{δ,ϵ} has support contained in [−M, M] (since its moments have growth rate smaller than powers of M). Using Stieltjes inversion, this implies that

(7)  lim_{t↓0} −ℑG_{μ_{k}}^{δ,ϵ}(x + it) > 0

for some x > M (or the limit simply does not exist in the atomic case). Calculating the imaginary part of this Cauchy transform, we have

ℑ([μ_{k}((z1_{2n} − B_{δ,ϵ}XB_{δ,ϵ})^{−1})]^{−1}) = B_{δ,ϵ}^{−1}ℑ([μ_{k}(B_{δ,ϵ}^{−2}z − X)^{−1}]^{−1})B_{δ,ϵ}^{−1}
= B_{δ,ϵ}^{−1}ℑF_{μ_{k}}^{(n)}(zB_{δ,ϵ}^{−2})B_{δ,ϵ}^{−1}
≤ B_{δ,ϵ}^{−1}ℑF_{μ}^{(n)}(zB_{δ,ϵ}^{−2})B_{δ,ϵ}^{−1}

= ℑ([μ((z1_{2n} − B_{δ,ϵ}XB_{δ,ϵ})^{−1})]^{−1}) (8)

where the inequality follows from the fact that F_{μ} = F_{μ_{k}}^{∘k−1} ∘ F_{μ_{k}} and F -transforms increase the imaginary part. Rewriting the right hand side of (8), we have that

ℑ([μ((z1_{2n} − B_{δ,ϵ}XB_{δ,ϵ})^{−1})]^{−1})
= [μ((z1_{2n} − B_{δ,ϵ}XB_{δ,ϵ})^{−1})^{∗}]^{−1}·

ℑ(μ((z1_{2n} − B_{δ,ϵ}XB_{δ,ϵ})^{−1}))[μ((z1_{2n} − B_{δ,ϵ}XB_{δ,ϵ})^{−1})]^{−1} (9)

= F_{μ}^{δ,ϵ}(z)^{∗}ℑ(F_{μ}^{δ,ϵ}(z))F_{μ}^{δ,ϵ}(z)

We conclude that

(10)  ℑ([μ((z1_{2n} − B_{δ,ϵ}XB_{δ,ϵ})^{−1})]^{−1}) ≤ F_{μ}^{δ,ϵ}(z)^{∗}ℑ(F_{μ}^{δ,ϵ}(z))F_{μ}^{δ,ϵ}(z).

Since F_{μ}^{δ,ϵ} extends to ℝ \ [−M, M]

lim_{t↓0} G_{μ}^{δ,ϵ}(x + it)

converges to a positive element in B and

lim_{t↓0} ℑ(F_{μ}^{δ,ϵ}(x + it)) → 0

it follows that the right hand side or (10) converges to 0 in norm, contradicting (7). This completes our proof. □

Proposition 3.2. Let μ, μ_{k} be as in the preceding proposition. We have that F_{μ_{k}} → Id in norm as k ↑ ∞ uniformly on M_{n}^{+,ϵ}(B), and this convergence is also uniform over n . Moreover, the functions F_{μ_{k}}^{(n)}(b^{−1}) − b^{−1} and F_{μ_{k}}^{(n)}(b^{−1})^{−1} extend analytically to B_{r}^{nc}(0), where the radius r is dependent only on M from Proposition 3.1, and satisfy

(11)  F_{μ_{k}}^{(n)}(b^{−1}) − b^{−1} → 0_{n}
(12)  F_{μ_{k}}^{(n)}(b^{−1})^{−1} = H_{μ_{k}}^{(n)}(b) → b

where this convergence is uniform on B_{r}^{nc}(0).

Proof. Consider the Nevanlinna representations of each of these functions

(13)  F_{μ_{k}}^{(n)}(b) = α_{k} ⊗ 1_{n} + b − G_{ρ_{k}}^{(n)}(b)

defined in Theorem 2.1 , where we have adopted the notation that μ = μ_{1}. We claim that the distributions ρ_{k} share a common exponential bound N for all k ∈ ℕ. To prove this claim, first observe that, by Theorem 4.1 in [Wil13], there exist distributions ν_{k} such that

b − F_{μ_{k}}^{(n)}(b) = φ_{ν_{k}}^{(n)}(b) = −α_{k} ⊗ 1_{n} + G_{ρ_{k}}^{(n)}(b).

Moreover, it was shown in [PV13] that if the ν and the ν_{k} have a common exponential bound N then the distributions ρ and ρ_{k} have a common exponential bound N^{2} + 1. Focusing on the ν_{k}, we may manipulate equations 13 to conclude that

(14)  R_{ν_{k}}(b^{−1}) = φ_{ν_{k}}(b) = b^{−1} − F_{μ_{k}}(b^{−1}).

Now, expand the moment series

(15)  F_{μ_{k}}^{(n)}(b^{−1})^{−1} = H_{μ_{k}}^{(n)}(b) = ∑_{p=0}^{∞} μ_{k}((bX)^{p}b).

Note that Proposition 3.1 implies that this function is convergent and uniformly bounded for b ∈ B_{r}^{nc}(0), independent of k. Observe that the moment generating function satisfies (16) [H_{μ_{k}}^{(n)}(b)]^{−1} = b^{−1} −μ_{k}(X)+μ_{k}(X)bμ_{k}(X)−μ_{k}(XbX)+⋯ = b^{−1} +f^{(n)}(b, X) where f^{(n)}(b, X) is analytic in b and converges for ∥b∥ small, where the radius of convergence is only dependent on M. Thus, [H_{μ_{k}}^{(n)}(b)]^{−1} − b^{−1} extends to a neighborhood of 0 whose radius is independent of n and k and agrees with F_{μ_{k}}^{(n)}(b^{−1})−b^{−1} when b is invertible. Moreover, these observations, combined with (14) imply that the functions R_{ν_{k}} have a common R, C > 0 such that the functions extend to a common domain B_{R}^{nc}(0) with a common bound C. Now a careful look at the Kantorovich argument in part II of the proof of Theorem 4.1 in [Wil13] allows us to conclude that the exponential bound on the distributions ν_{k} depend only on R, proving our claim. Recall that F_{μ_{k}} ∘ ⋯ ∘ F_{μ_{k}} = F_{μ} we have that

(17) G_{ρ}^{(n)}(b) = G_{ρ_{k}}^{(n)}(b) + G_{ρ_{k}}^{(n)} ∘ F_{μ_{k}}^{(n)}(b) + ⋯ + G_{ρ_{k}}^{(n)} ∘ \underbrace{F_{μ_{k}}^{(n)} ∘ ⋯ ∘ F_{μ_{k}}^{(n)}}_{k−1 times}(b)

Letting b = z1_{n} for z ∈ ℂ, we have that

lim_{|z|↑∞} zH_{ρ}^{(n)} (\frac{1}{z}1_{n}) = lim_{|z|↑∞} zG_{ρ}^{(n)}(z1_{n})

= lim_{|z|↑∞} ∑_{ℓ=1}^{k−1} zG_{ρ_{k}}^{(n)} ∘ (F_{μ_{k}}^{(n)})^{∘ℓ}(z1_{n})

= lim_{|z|↑∞} ∑_{ℓ=1}^{k−1} zH_{ρ_{k}}^{(n)} ([(F_{μ_{k}}^{(n)})^{∘ℓ}(z1_{n})]^{−1})

= lim_{|z|↑∞} ∑_{ℓ=1}^{k−1} zH_{ρ_{k}}^{(n)} ∘ G_{ν_{ℓ}}^{(n)} (z1_{n})

= lim_{|w|↓0} ∑_{ℓ=1}^{k−1} \frac{1}{w} H_{ρ_{k}}^{(n)} ∘ G_{ν_{ℓ}}^{(n)} (\frac{1}{w} 1_{n})

= lim_{|w|↓0} ∑_{ℓ=1}^{k−1} \frac{1}{w} H_{ρ_{k}}^{(n)} ∘ H_{ν_{ℓ}}^{(n)} (w1_{n})

where [(F_{μ_{k}}^{(n)})^{∘ℓ}]^{−1} = G_{ν_{ℓ}} is the Cauchy transform of a distribution ν_{ℓ} ∈ Σ_{0} (this follows from Theorem 2.1). Moreover, we have that

lim_{|w|↓0} \frac{1}{w} H_{ν_{ℓ}}^{(n)} (w1_{n}) = 1_{n}

so that, passing to limits and utilizing the chain rule and the fact that H_{ν_{ℓ}}^{(n)}(0_{n}) = 0_{n} , we have that

δH_{ρ}^{(n)}(0_{n}; 1_{n}) = kδH_{ρ_{k}}^{(n)}(0_{n}; 1_{n})

Utilizing the main result in our appendix, Theorem A.1, we conclude that

(18)  ρ(1) = μ(X^{2}) = kμ_{k}(X^{2}) = kρ_{k}(1).

so that ρ_{k}(1) = O(1/k).

Now, assume that b ∈ M_{n}^{+,ϵ}(B). We claim that ∥b^{−1}∥ ≤ 1/ϵ. Indeed, observe that, for b = x + iy with y > ϵ1_{n},

(19)  b = \sqrt{y}(i + (\sqrt{y})^{−1}x(\sqrt{y})^{−1})\sqrt{y}

(it follows easily from this equation that b is invertible, but this is known). Thus,

(20)  b^{−1} = (\sqrt{y})^{−1}(i + (\sqrt{y})^{−1}x(\sqrt{y})^{−1})^{−1}(\sqrt{y})^{−1}.

Now, utilizing the spectral mapping theorem and the fact that the spectral radius agrees with the norm for normal operators, we have that ∥(\sqrt{y})^{−1}∥ ≤ (\sqrt{ϵ})^{−1}. Moreover, since i + (\sqrt{y})^{−1}x(\sqrt{y})^{−1} is normal and has spectrum with imaginary part larger than 1, we have that (i + (\sqrt{y})^{−1}x(\sqrt{y})^{−1})^{−1} is normal and, by the same spectral considerations, has norm bounded by 1. These observations, combined with (19) imply our claim. Thus, for b ∈ M_{n}^{+}(B), we have

∥F_{μ_{k}}^{(n)}(b) − b∥ ≤ ∥α_{k}∥ + ∥(ρ_{k} ⊗ 1_{n})((b − X)^{−1}∥

≤ ∥α∥/k + ∥(b − X)^{−1}∥∥(ρ_{k} ⊗ 1_{n})(1_{n})∥

≤ \frac{∥α∥}{k} + \frac{∥ρ_{k}(1)∥}{ϵ} = \frac{∥α∥ + ρ(1)/ϵ}{k}

and the right hand side converges to zero uniformly over M_{n}^{+,ϵ}(B), independent of n. Regarding the second part of our Proposition, we first observe that each of the moments of μ_{k} converges to 0. Indeed, utilizing the Schwarz inequality for 2-positive maps as well as Proposition 3.1, we have that

∥μ_{k}(Xb_{1}Xb_{2}X ⋯ b_{ℓ}X)∥^{2} ≤ ∥μ_{k}(X^{2})∥∥μ_{k}(Xb_{ℓ}^{∗}X ⋯ b_{2}^{∗}Xb_{1}^{∗}b_{1}Xb_{2}X ⋯ b_{ℓ}X)∥

≤ \frac{∥μ(X^{2})∥M^{2ℓ}∥b_{1}∥^{2}∥b_{2}∥^{2} ⋯ ∥b_{ℓ}∥^{2}}{k}

Moreover, the tail of the series expansion of f^{(n)}(b, X) is bounded in norm independent of n and k . the individual entries all go to 0 so the we conclude that f^{(n)}(b, X) → 0 uniformly on b ∈ B_{r}^{nc}(0) as k ↑ ∞ so that we can immediately conclude that (12) holds. This completes our proof. □

We next prove a differentiation result for vector valued functions. We adapt a proof found in [BPo78] of a similar result for complex functions.

Proposition 3.3. Let A and B denote unital Banach algebras. Consider an open subset Ω ⊂ A. Let f_{t} : Ω ↦ B for all t ≥ 0 be a composition semigroup of analytic functions. Assume that for every b′ ∈ Ω, there exists a δ > 0 such that

(a) lim_{t↓0} f_{t}(b) − b → 0 uniformly over b ∈ B_{δ}(b′)

(b) For any T > 0, we have that f_{t}(b) − b is uniformly bounded over b ∈ B_{δ}(b′) and t ∈ [0, T].

Then, there exists an analytic Φ : Ω ↦ B such that

(21)  \frac{df_{t}(b)}{dt} = −Φ(f_{t}(b)).

Proof. Fix b′ ∈ Ω. We first claim that there exists an α > 0 such that

(22)  ∥f_{2t}(b) − 2f_{t}(b) + b∥ ≤ \frac{1}{10}∥f_{t}(b) − b∥.

for all t ∈ [0, α] and b ∈ B_{δ/2}(b′) where the value of δ comes from the statement . Indeed, fix b ∈ B_{δ/2}(b′). We first consider the simple case when there exists a sequence t_{n} ↓ 0 such that f_{t_{n}}(b) = b. Since {f_{t}} form a composition semigroup, this property then holds for a dense set of t’s, and by continuity assumption in part (a), for all t > 0. So (22) holds trivially. Thus, suppose that f_{t}(b) ≠ b for t ∈ [0, α]. Define a family of complex functions g_{t} through the following equalities:

h_{t} := \frac{f_{t}(b) − b}{∥f_{t}(b) − b∥}; g_{t}(ζ) := f_{t}(b + ζh_{t}) − b : B_{δ/2}(0) ↦ B.

where B_{δ/2}(0) refers to the neighborhood of zero in the complex plane. Note that, since we are taking a ball of radius δ/2, we may define h_{t} for all such b provided that our choice of α is small enough. Consider the vector valued complex integral

(23)  ∫_{0}^{∥f_{t}(b)−b∥} \frac{d}{dζ} [g_{t}(ζ) − ζh_{t}]dζ.

By (a) and the Cauchy estimates in Theorem 2.2, the integrand can be made arbitrarily small for t small. By the fundamental theorem, this integral is equal to

g_{t}(∥f_{t}(b) − b∥) − g_{t}(0) − (f_{t}(b) − b) = = f_{t}(b + (f_{t}(b) − b)) − b − 2(f_{t}(b) − b) = f_{2t}(b) − 2f_{t}(b) + b.

Using our bound on the integrand, equation (22) follows immediately. We now use (22) to prove that for α > 0 there exists an M > 0 such that

(24)  ∥f_{t}(b) − b∥ ≤ Mt^{2/3}

for all t ∈ [0, α] and b ∈ B_{δ/2}(b′). Indeed, pick t ∈ [0, α] and m ∈ ℕ such that 2^{m}t ≤ α < 2^{m+1}t. Note that inequality (22) and the triangle inequality imply that

2∥f_{t}(b) − b∥ − ∥f_{2t}(b) − b∥ ≤ ∥f_{2t}(b) − 2f_{t}(b) + b∥ ≤ \frac{1}{10}∥f_{t}(b) − b∥

so that

(25)  ∥f_{t}(b) − b∥ ≤ \frac{10}{19}∥f_{2t}(b) − b∥ ≤ 2^{−2/3}∥f_{2t}(b) − b∥

Using this estimate inductively, we have

∥f_{t}(b)−b∥ ≤ 2^{−2/3}∥f_{2t}(b)−b∥ ≤ ⋯ ≤ 2^{−2m/3}∥f_{2^{m}t}(b)−b∥ = t^{2/3} (\frac{1}{2^{m}t})^{2/3} M′

where M′ is a bound on ∥f_{s}(b) − b∥ for s ≤ 2 which exists by (b). Equation (24) follows with M = 2^{2/3}M′/α. Now, revisiting the argument for (22), inequality (24) implies that the integrand in (23) has bound equal to

2Mt^{2/3}

as a result of the Cauchy estimates. Thus, we have the following:

(26)  ∥f_{2t}(b) − 2f_{t}(b) + b∥ ≤ 2t^{2/3}∥f_{t}(b) − b∥ ≤ 2Mt^{4/3}.

We may further conclude that

(27)  ∥\frac{f_{2t}(b) − b}{2t} − \frac{f_{t}(b) − b}{t}∥ ≤ Mt^{1/3}

Thus, we have that

(28)  lim_{k↑∞} 2^{k}(f_{2^{−k}}(b) − b)

converges uniformly on B_{δ/2}(b′) and we refer to this limit as −Φ(b). Using (27), we note that Φ is locally bounded. Indeed, we have that

∥2^{p}(f_{1/2^{p}}(b) − b) + Φ(b)∥ ≤ ∑_{k=p}^{∞} ∥2^{k}(f_{1/2^{k}}(b) − b) − 2^{k+1}(f_{1/2^{k+1}}(b) − b)∥

≤ \frac{M}{2} ∑_{k=p}^{∞} (\frac{1}{2^{1/3}} )^{k} = MC(p). (29)

for all b ∈ B_{δ/2}(b′). Local boundedness of Φ follows since (f_{1/2^{p}}(b) − b) is locally bounded. Also note that C(p) → 0 as p ↑ ∞. Regarding analyticity of Φ, consider a state φ ∈ B^{∗} , b ∈ B_{δ/2}(b′), and an element h ∈ B with ∥h∥ ≤ 1. We define complex maps

H_{m}(z) : B_{δ/2}(0) ⊂ ℂ → ℂ

for m ≥ 0 through the equalities:

H_{0}(z) := φ ∘ Φ(b + zh); H_{m}(z) := 2^{m}φ ∘ (f_{2^{−m}}(b + zh) − (b + zh)).

By (28), H_{m} → H_{0} for z ∈ B_{δ/2}(0), and by (29), the limit is bounded on this set. Thus, H_{0} is analytic in z. By Dunford’s theorem ([Dun38]), it follows that Φ(b + zh) is analytic in z and, therefore, Gâteaux differentiable. As this function is locally bounded, it is analytic. Regarding (21), observe that {f_{t}(b)}_{t≥0} is compact since it is the continuous image of [0, t]. As (a) and (b) hold on neighborhoods of every point in this set, taking a finite cover, we have that (a) and (b) holds uniformly on a neighborhood of this set and, after a close look at the relevant constants, (29) is also maintained on this set. Now, fix t ≥ 0 and let ℓ_{p}/2^{p} → t as p ↑ ∞.

f_{t}(b) − b = (f_{t}(b) − f_{t−ℓ_{p}/2^{p}}(b)) + ∑_{j=1}^{ℓ_{p}}(f_{j/2^{p}}(b) − f_{(j−1)/2^{p}}(b))

= (f_{t}(b) − f_{t−ℓ_{p}/2^{p}}(b)) + ∑_{j=1}^{ℓ_{p}} \frac{1}{2^{p}} (2^{p}[f_{j/2^{p}}(b) − f_{(j−1)/2^{p}}(b)])

As p ↑ ∞,

f_{t}(b) − f_{t−ℓ_{p}/2^{p}}(b) = f_{ℓ_{p}/2^{p}} ∘ f_{t−ℓ_{p}/2^{p}}(b) − f_{t−ℓ_{p}/2^{p}}(b) → 0

since (a) holds on the entire path. Moreover, the remaining summand is simply a Riemann sum approximation of a sequence of functions converging uniformly to −Φ∘ f_{s}(b) for s ∈ [0, t]. The following equation follows immediately:

f_{t}(b) = b − ∫_{0}^{t} Φ ∘ f_{s}(b)ds.

We conclude that (21) holds, completing our proof. □

Corollary 3.4. Let A and B denote Banach algebras and Ω ⊂ ⊔_{n=1}^{∞}M_{n}(A) a noncommutative set. Let F_{t} : Ω ↦ ⊔_{n=1}^{∞}M_{n}(B) for all t ≥ 0 and assume that they form a composition semigroup of analytic non-commutative functions. Assume that, for each n, the composition semigroup of vector valued analytic functions {F_{t}^{(n)}}_{t≥0} satisfies the hypotheses of Proposition 3.3. Then there exists an analytic, noncommutative map Φ : Ω ↦ ⊔_{n=1}^{∞}M_{n}(B) such that

(30)  \frac{dF_{t}^{(n)}(b)}{dt} = −Φ^{(n)}(F_{t}^{(n)}(b))

for all n ∈ ℕ, b ∈ Ω_{n}. Moreover, if we strengthen these assumptions so that, for any n and b ∈ M_{n}(B), there exists a δ > 0 with

(a) lim_{t↓0} F_{t} − Id → 0 uniformly over B_{δ}^{nc}(b).

(b) For any T > 0, we have that f_{t}(b) − b is uniformly bounded on B_{δ}^{nc}(b) and

t ∈ [0, T]. then Φ is uniformly analytic.

Proof. We showed in Proposition 3.3 this map Φ exists. We must show that it is a non-commutative function. However, this is immediate since, for b_{1} ∈ M_{n}(B) and b_{2} ∈ M_{p}(B), we have

Φ^{(n+p)}(b_{1} ⊕ b_{2}) = lim_{k↑∞} 2^{k}(F_{2^{−k}}^{(n+p)}(b_{1} ⊕ b_{2}) − b_{1} ⊕ b_{2})

= lim_{k↑∞} 2^{k}([F_{2^{−k}}^{(n)}(b_{1}) − b_{1}] ⊕ [F_{2^{−k}}^{(n)}(b_{2}) − b_{2}])

= Φ^{(n)}(b_{1}) ⊕ Φ^{(p)}(b_{2}).

A similar proof shows that it also satisfies the defining invariance property so that our first claim holds. With respect to the uniform analyticity, we refer to the proof of Proposition 3.3. Observe that inequality (22) holds for α small enough. This α is only dependent on the convergence of the integrand in (23). This converges to 0 uniformly on B_{δ}^{nc}(b) by assumption (a) and the same Cauchy estimate so that the choice of α is also uniform on this set. Moreover, the constant M in (24) is equal to 2^{2/3}M′/α where M′ is the upper bound on F_{s} − Id for s ≤ α. Assumption (b) implies that this bound is uniform on B_{δ}^{nc}(b). Thus, inequality (29) holds on all of this set, implying uniform analyticity. □

Theorem 3.5. Let {F_{t}}_{t∈ℚ^{+}} denote a composition semigroup of non-commutative functions F_{t} : H^{+}(B) ↦ H^{+}(B) such that (i) ∥F_{t}^{(n)}(b) − b∥ → 0 uniformly on M_{n}^{+,ϵ}(B) for all ϵ > 0, independent of n as

t ↓ 0. (ii) For any α, ϵ > 0 and sequence b_{k} ∈ Γ_{α,ϵ}^{(n)} with ∥b_{k}^{−1}∥ ↓ 0, we have that

b_{k}^{−1}F_{t}^{(n)}(b_{k}) → 1_{n} as k ↑ ∞ (iii) ℑF_{t}^{(n)}(b) ≥ ℑb for all b ∈ M_{n}^{+}(B) and t ≥ 0. Then {F_{t}}_{t∈ℚ^{+}} extends to a semigroup {F_{t}}_{t≥0} and the map Φ from Proposition 3.4 is an element of Λ̃. Since, by Proposition 3.2, the conditions above are satisfied by F -transforms, this implies that a ▷-infinitely divisible distribution μ as in Definition 2.4 can be realized as μ = μ_{1} for a monotone convolution semigroup {μ_{t}}_{t≥0}. For such a semigroup, Φ ∈ Λ. Conversely, given a map Φ ∈ Λ̃ we may construct a semigroup of non-commutative functions satisfying the hypotheses above as well as the differential equation

(31)  \frac{dF_{t}(b)}{dt} = −Φ(F_{t}(b))

If Φ ∈ Λ then the semigroup arises from a ▷-infinitely divisible distribution.

We shall refer to this element Φ as the generator or the semigroup {F_{t}}_{t≥0}.

Proof. First, let Φ ∈ Λ̃. We will produce the semigroup it generates by the method of successive approximations. Consider a sequence of non-commutative functions {f_{k}(t, ·)}_{t≥0, k∈ℕ} defined as follows:

(32)  f_{1}^{(n)}(t, b) = b; f_{k+1}^{(n)}(t, b) = b − ∫_{0}^{t} Φ(f_{k}^{(n)}(s, b))ds.

We claim that f_{k}(t, ·) is convergent and satisfies the semigroup property with generator Φ. Observe that since Φ is uniformly bounded by a constant M on set M_{n}^{+,ϵ/2}(B) and f_{k}(t, ·) maps the set M_{n}^{+,ϵ}(B) to itself since

Φ : H^{+}(B) ↦ H^{−}(B)

we have that

(33)  ℑf_{k}^{(n)}(t, b) ≥ ℑ(b).

By (3), this implies that f_{k}^{(n)}(t, ·) is Lipschitz on the set B_{ϵ/2}(b) ⊂ M_{n}^{+,ϵ/2}(B) for all b ∈ M_{n}^{+,ϵ}(B), and the Lipschitz constant L is uniform over both k, b and bounded t. Moreover, we may extend the Lipschitz inequality

∥f_{k}(t, b) − f_{k}(t, b′)∥ ≤ L∥b − b′∥

to all b, b′ ∈ M_{n}^{+,ϵ}(B) by taking a path b + s(b′ − b) for s ∈ [0, 1] and using the Lipschitz estimate on intervals of distance ϵ/2 since the distances are additive on this path. Using this Lipschitz estimate in the integrand of (32), we conclude that

(34)  ∥f_{2}^{(n)}(t, b) − f_{1}^{(n)}(t, b)∥ = t∥Φ(b)∥ ≤ tML

and we may conclude that

∥f_{3}^{(n)}(t, b) − f_{2}^{(n)}(t, b)∥ = ∥∫_{0}^{t}[Φ(f_{2}^{(n)}(s, b)) − Φ(f_{1}^{(n)}(s, b))]ds∥

≤ L ∥∫_{0}^{t}[f_{2}^{(n)}(s, b) − f_{1}^{(n)}(s, b)]ds∥

≤ L ∫_{0}^{t}[LMs]ds ≤ \frac{t^{2}L^{2}M}{2}

Continuing inductively, we have that

(35)  ∥f_{k+1}^{(n)}(t, b) − f_{k}^{(n)}(t, b)∥ ≤ \frac{M(Lt)^{k+1}}{L(k + 1)!} .

For any choice of t ∈ [0, α], we have that

(36)  f_{N+1}^{(n)}(t, b) − b = ∑_{k=0}^{N} (f_{k+1}^{(n)}(t, b) − f_{k}^{(n)}(t, b))

is a convergent series as N ↑ ∞ and we may conclude that f_{N}(t, ·) converges to a function f(t, ·) uniformly on M_{n}^{+,ϵ}(B), independent of n. It is clear that f(t, ·) satisfies (31). Regarding the asymptotics, let α, ϵ > 0 and fix a sequence b_{ℓ} ∈ Γ_{α,ϵ}^{(n)} with ∥b_{ℓ}^{−1}∥ ↓ 0. Note that b_{ℓ}^{−1}f_{1}^{(n)}(t, b_{ℓ}) ≡ 1_{n} and satisfies ∥f_{1}^{(n)}(t, b_{ℓ})∥^{−1} ↓ 0 as ∥b_{ℓ}^{−1}∥ ↓ 0. We claim b_{ℓ}^{−1}f_{k}^{(n)}(t, b_{ℓ}) → 1_{n} and satisfies ∥f_{k}^{(n)}(t, b_{ℓ})∥^{−1} ↓ 0 as ∥b_{ℓ}^{−1}∥ ↓ 0 for all k, uniformly over t ∈ [0, α]. Proceeding by induction, we have that for fixed k

(37) b_{ℓ}^{−1}f_{k+1}^{(n)}(t, b_{ℓ}) = 1_{n} − ∫_{0}^{t}[b_{ℓ}^{−1}f_{k}^{(n)}(s, b_{ℓ})](f_{k}^{(n)}(s, b_{ℓ}))^{−1}Φ(f_{k}^{(n)}(s, b_{ℓ}))ds.

We bound the integrand by

∥[b_{ℓ}^{−1}f_{k}^{(n)}(s, b_{ℓ})]∥∥(f_{k}^{(n)}(s, b_{ℓ}))^{−1}Φ(f_{k}^{(n)}(s, b_{ℓ}))∥

which converges to 0 uniformly over s ∈ [0, α] by induction, so that (37) converges to 1_{n}. Moreover,

∥[f_{k+1}^{(n)}(t, b_{ℓ})]^{−1}∥ ≤ ∥b_{ℓ}^{−1}∥∥b_{ℓ}[f_{k+1}^{(n)}(t, b_{ℓ})]^{−1}∥ → 0.

Thus, each f_{k}(t, ·) has the appropriate asymptotics and, since f(t, ·) is a uniform limit of these functions on M_{n}^{+,ϵ}, our claim holds Condition (iii) follows from (33). In order to complete our proof, we further assume that Φ ∈ Λ and prove that the functions f(t, ·) are in fact the F -transforms of noncommutative distributions μ_{t} ∈ Σ_{0}. To do so we must show that the function f(t, b^{−1})^{−1} has a uniformly analytic extension to a neighborhood of 0 for all t ≥ 0. Note that, since Φ ∈ Λ, there exists a δ > 0 and constants M, L > 0 such that Φ^{(n)}(b^{−1}) extends to B_{δ}^{nc}(0) with upper bound M and Lipschitz constant L. Now fix α > 0. We claim that, for γ > 0 small enough we have that f_{k}^{(n)}(t, b^{−1})^{−1} extends to B_{γ}(0_{n}) ⊂ M_{n}(B) for all n and satisfies f_{k}^{(n)}(t, b^{−1})^{−1} ∈ B_{δ}(0_{n}) for all b ∈ B_{γ}(0_{n}). Choose any t ∈ [0, α] and b ∈ B_{γ}(0_{n}) where γ < δ is yet unspecified. We have

∥f_{2}^{(n)}(t, b^{−1})^{−1} − f_{1}^{(n)}(t, b^{−1})^{−1}∥ = ∥[(1_{n} − ∫_{0}^{t} bΦ(b^{−1})ds)^{−1} − 1_{n}] b∥

≤ ∑_{n=1}^{∞} ∥∫_{0}^{t} bΦ(b^{−1})ds∥^{n} ∥b∥

≤ γ ∑_{n=1}^{∞}(γMα)^{n}

= \frac{γ^{2}Mα}{1 − γMα}

Deriving a similar inequality for general k, we have that

\begin{matrix}∥ & f_{k+1}^{(n)}(t, b^{−1})^{−1} − f_{k}^{(n)}(t, b^{−1})^{−1}∥ \\ = ∥(b^{−1} − ∫_{0}^{t} Φ ∘ f_{k}^{(n)}(s, b^{−1})ds)^{−1} − (b^{−1} − ∫_{0}^{t} Φ ∘ f_{k−1}^{(n)}(s, b^{−1})ds)^{−1}∥ \\ = ∥(1_{n} − ∫_{0}^{t} bΦ(f_{k}^{(n)}(t, b^{−1})))^{−1} (b ∫_{0}^{t} Φ(f_{k−1}^{(n)}(t, b^{−1})) − Φ(f_{k}^{(n)}(t, b^{−1}))) \\ (1_{n} − ∫_{0}^{t} bΦ(f_{k−1}^{(n)}(t, b^{−1})))^{−1} b∥ \\ ≤ (\frac{1}{1 − γMα})^{2} (γ^{2}Lα)∥f_{k}^{(n)}(t, b^{−1})^{−1} − f_{k−1}^{(n)}(t, b^{−1})^{−1}∥ \end{matrix} (38)

By induction, we have that

∥f_{k+1}^{(n)}(t, b^{−1})^{−1} − b∥ = ∑_{ℓ=1}^{k} \frac{Mγ^{2ℓ}L^{ℓ−1}α^{ℓ}}{(1 − γMα)^{2ℓ−1}}

This is convergent as k ↑ ∞ for γ small and converges to 0 as γ ↓ 0. Thus, for γ small enough, we have that f_{k+1}^{(n)}(t, b^{−1}) ∈ B_{δ}(0_{n}) for all k and n and, therefore, converges to a limit function on B_{γ}(0_{n}) (since the differences in (38) are Cauchy). This limit function must agree with f(t, ·) by analytic continuation. This completes our proof that f(t, ·) is an F -transform for all t. To address the converse, consider a semigroup {F_{t}}_{t∈ℚ^{+}} satisfying the (i) and (ii) in the statement of the theorem. First note that this easily extends to an ℝ^{+} composition semigroup. Indeed, define F_{t}(b) = lim_{p/q→t} F_{p/q}(b). To see that this is well defined, note that, as p/q, p′/q′ → t, we have

∥F_{p/q}^{(n)}(b) − F_{p′/q′}^{(n)}(b)∥ = ∥F_{p/q−p′/q′}^{(n)} ∘ F_{p′/q′}^{(n)}(b) − F_{p′/q′}^{(n)}(b)∥ → 0

uniformly on M_{n}^{+,ϵ}(B) by property (i) and (iii) . It is immediate that this is a composition semigroup over ℝ^{+} satisfying (i), (ii) and (iii).

By Corollary 3.4, this semigroup may be differentiated to produce a non-commutative function Φ . Regarding the asymptotics of Φ, consider the inequality

(39) ∥b^{−1}Φ^{(n)}(b)∥ ≤ ∥\frac{b^{−1}(F_{t}^{(n)}(b) − b)}{t}∥ + ∥b^{−1}∥ ∥\frac{(F_{t}^{(n)}(b) − b)}{t} − Φ^{(n)}(b)∥ .

Utilizing inequality (29) in the proof of Proposition 3.3 produces

(40)  ∥\frac{(F_{2^{N}}^{(n)}(b) − b)}{2^{N}} − Φ^{(n)}(b)∥ ≤ M ∑_{k=N+1}^{∞} (\frac{1}{2^{1/3}} )^{k}

where this M = 2M′/α . As was noted in the proof of Corollary 3.4, uniform convergence in the sense of (i) and (ii) implies a uniform bound on M. Thus, (40) converges to 0 uniformly on M_{n}^{+,ϵ}(B) so that, for fixed t small enough, second term on the right hand side of (39) is smaller than any δ > 0 for b ∈ M_{n}^{+,ϵ}(B). Letting b_{k} ∈ Γ_{α,ϵ}^{(n)} satisfy ∥b_{k}^{−1}∥ ↓ 0, the first term on the right hand side of (39) converges to 0 by assumption (ii), and it follows that Φ ∈ Λ̃. If {F_{t}}_{t≥0} arises from a ▷-infinitely divisible measure, then it follows from Proposition 3.1 and Theorem 2.1 that b_{k}^{−1}F_{μ_{t}}^{(n)}(b_{k}) → 1_{n} for any sequence b_{k} ∈ M_{n}(B) with ∥b_{k}^{−1}∥ ↓ 0 and a similar proof allows one to conclude that Φ satisfies condition (ii) in the definition of Λ. It remains to show that Φ satisfies (i). However, Proposition 3.2 implies that there exists a fixed r > 0 such that each function F_{μ_{t}}^{(n)}(b^{−1}) − b^{−1} extends to B_{r}({0}) and converges to 0 uniformly on this set. Thus, the strengthened hypotheses in Corollary 3.3 hold so that the non-commutative function defined by the equalities

R^{(n)}(b) = lim_{t↓0} \frac{F_{μ_{t}}^{(n)}(b^{−1}) − b^{−1}}{t}

is uniformly analytic at 0 and, by continuation, is an extension of Φ^{(n)}(b^{−1}) for each n. Thus, Φ ∈ Λ, completing our proof. □

The following proposition establishes continuity in generating the semigroups, and may be useful in future applications.

Proposition 3.6. Assume that Φ_{1}, Φ_{2} ∈ Λ̃ generate the semigroups of noncommutative functions {F_{1}(t, ·)}_{t≥0} and {F_{2}(t, ·)}_{t≥0}. If we assume that ∥Φ_{1}^{(n)}(b) − Φ_{2}^{(n)}(b)∥ < ϵ for all b ∈ B_{δ}(b′) ⊂ M_{n}(B), a ball of radius δ where ℑ(b′) > δ1_{n}, then we may conclude that ∥F_{1}^{(n)}(1, b) − F_{2}^{(n)}(1, b)∥ < Cϵ for all b ∈ B_{δ}(b′) where C depends only on Φ_{1}.

Proof. To prove our claim, we first note that, by the vector-valued chain rule,

\frac{δ^{2}F_{i}^{(n)}(t, b)}{δt^{2}} = δΦ^{(n)} (F_{i}^{(n)}(t, b), \frac{δ}{δt}F^{(n)}(b, t))

so that F_{i}(t, b) is twice differentiable in t and has uniformly bounded derivative for b ∈ H^{+,ϵ}(B) and t ∈ [0, 1]. We refer to the maximum of this bound over i = 1, 2 as M_{2}.

Using the remainder estimates for the Taylor series associated to F_{i}, we have the following:

(41)  ∥F_{i}(b, t + γ) − F_{i}(b, t) − γΦ(F_{i}(b, t))∥ ≤ \frac{M_{2}γ^{2}}{2}

Let M_{1} = sup_{b∈M_{n}^{+,ϵ}(B), n∈ℕ} ∥δΦ^{(n)}(b, ·)∥. Utilizing the estimate (41) with γ = 1/N, we produce the following inequalities:

∥F_{1}^{(n)}(b, t_{0} + 1/N) − F_{2}^{(n)}(b, t_{0} + 1/N)∥

≤ \frac{M_{2}}{N^{2}} + \frac{1}{N} ∥Φ_{1}^{(n)}(F_{1}^{(n)}(b, t_{0})) − Φ_{2}^{(n)}(F_{2}^{(n)}(b, t_{0}))∥

+ ∥F_{1}^{(n)}(b, t_{0}) − F_{2}^{(n)}(b, t_{0})∥

≤ \frac{M_{2}}{N^{2}} + \frac{1}{N} ∥Φ_{1}^{(n)}(F_{1}^{(n)}(b, t_{0})) − Φ_{1}^{(n)}(F_{2}^{(n)}(b, t_{0}))∥

+ \frac{1}{N} ∥Φ_{1}^{(n)}(F_{2}^{(n)}(b, t_{0})) − Φ_{2}^{(n)}(F_{2}^{(n)}(b, t_{0}))∥ + ∥F_{1}^{(n)}(b, t_{0}) − F_{2}^{(n)}(b, t_{0})∥

≤ \frac{M_{2}}{N^{2}} + \frac{ϵ}{N} + (1 + \frac{M_{1}}{N} ) ∥F_{1}^{(n)}(b, t_{0}) − F_{2}^{(n)}(b, t_{0})∥

Using this estimate inductively, we have that

∥F_{1}^{(n)}(b, 1) − F_{2}^{(n)}(b, 1)∥ ≤ (\frac{ϵ}{N} + \frac{M_{2}}{N^{2}} ) ∑_{k=0}^{N−1} (1 + \frac{M_{1}}{N} )^{k} → \frac{e^{M_{1}} − 1}{M_{1}}ϵ

where the convergence occurs as N ↑ ∞. This implies our result.

□

4. The Bercovici-Pata Bijection.

Definition 4.1. Let (S, ≺) be a poset (partially ordered set). An order on S is an order-preserving bijection

f : (S, ≺) → ({1, 2, …, |S|} , <) .

Denote by o(S) the number of different orders on S.

Lemma 4.2. Let (S, ≺) be a poset, and S = U ⊔ V a partition of S. U and V are posets with the induced order.

(a) Suppose that for all u ∈ U and v ∈ V, u ≺ v. Then

o(S) = o(U)o(V).

(b) Suppose that for all u ∈ U and v ∈ V, u and v are unrelated to each other. Then

\frac{o(S)}{|S|!} = \frac{o(U)}{|U|!} \frac{o(V)}{|V |!}.

Proof. Part (a) is obvious. It is also clear that under the assumptions of part (b), there is a bijection between the orders on S and triples

{order on U, order on V, a subset of {1, 2, …, |S|} of cardinality |U|} .

Therefore

o(S) = \binom{|S|}{|U|}o(U)o(V).

This implies part (b). □

Definition 4.3. For a non-crossing partition π = {V_{1}, V_{2}, …, V_{k}}, define a partial order on it as follows: for U, V ∈ π, U ≺ V if for some i, j ∈ U and any v ∈ V, we have i < v < j. In this case we say that U covers V. Minimal elements with respect to this order are called the outer blocks of π; the rest are the inner blocks.

See [HS11, HS14] for more on orders on non-crossing partitions.

Definition 4.4. Let μ : B⟨X⟩ → B be a B-bimodule map; at this point no positivity assumptions are made. Its monotone cumulant functional is the B-bimodule map K^{μ} : B_{0}⟨X⟩ → B defined implicitly by

(42)  μ[b_{0}Xb_{1}X … b_{n−1}Xb_{n}] = ∑_{π∈NC(n)} \frac{o(π)}{|π|!}K_{π}^{μ}[b_{0}Xb_{1}X … b_{n−1}Xb_{n}].

Here for a non-crossing partition π, K_{π}^{μ} is defined in terms of K^{μ} in the usual way as in [Spe98] (see Section 3 of [ABFN13] for a detailed discussion), and o(π) is the number of orders on π considered as a poset (as in the preceding definition). The implicit definition determines the monotone cumulants uniquely since (43)

K^{μ}[b_{0}X … b_{n−1}Xb_{n}] = μ[b_{0}X … b_{n−1}Xb_{n}]− ∑_{\begin{matrix}π∈NC(n) \\ π≠ˆ1_{n} \end{matrix}} \frac{o(π)}{|π|!}K_{π}^{μ}[b_{0}X … b_{n−1}Xb_{n}],

and the second term on the right-hand side can be expressed in terms of lower-order moments.

Remark 4.5. For N ∈ ℕ, we note that

K^{μ⊗1_{N}} = K^{μ} ⊗ 1_{N}.

The proof of this fact is identical to that of Proposition 6.3 of [PV13]. It follows that the generating function arguments in the rest of this section work equally well for each μ ⊗ 1_{N}, and so the corresponding generating functions completely determine the states.

Lemma 4.6. For B-bimodule maps, μ_{i} → μ if and only if K^{μ_{i}} → K^{μ}.

Proof. By assumption, μ_{i}[b] = b = μ[b]. For n ≥ 1, clearly if

K^{μ_{i}}[b_{0}Xb_{1}X … b_{n−1}Xb_{n}] → K^{μ}[b_{0}Xb_{1}X … b_{n−1}Xb_{n}]

then

μ_{i}[b_{0}Xb_{1}X … b_{n−1}Xb_{n}] → μ[b_{0}Xb_{1}X … b_{n−1}Xb_{n}]

from equation (42). The other implication follows by induction on n, using equation (43). □

Definition 4.7. For μ as above and η : B → B a linear map, define μ^{▷η} via

K^{μ^{▷η}}[b_{0}Xb_{1}X … b_{n−1}Xb_{n}] = b_{0}η (K^{μ}[Xb_{1}X … b_{n−1}X]) b_{n}.

Define the formal generating functions

H^{μ}(b) = ∑_{n=0}^{∞} μ[b(Xb)^{n}]

and

K^{μ}(b) = ∑_{n=1}^{∞} K^{μ}[b(Xb)^{n}].

Note that as formal series,

H^{μ}(b) = G^{μ}(b^{−1}),

so our notation is consistent with the analytic function notation in the rest of the article, except that we use superscripts for formal series. Note also that these generating functions differ by a factor of b from the more standard ones, and are more appropriate for the computations with monotone convolution.

Remark 4.8. Fix n ∈ ℕ and π ∈ NC(n). Denote by V_{1}, …, V_{k} the outer blocks of π, by c(V_{i}) the partition consisting of V_{i} and the inner blocks it covers, and by c_{j}(V_{i}), j = 1, 2, …, |V_{i}| − 1 the partition consisting of the inner blocks lying between the jth and the (j + 1)st elements of V_{i}. By Lemma 4.2 part (b),

(44)  \frac{o(π)}{|π|!} = ∏_{i=1}^{k} \frac{o(c(V_{i}))}{|c(V_{i})|!}.

By part (a) of that lemma,

o(c(V_{i})) = o({V_{i}})o ( ⋃_{j=1}^{|V_{i}|−1} c_{j}(V_{i})) = o ( ⋃_{j=1}^{|V_{i}|−1} c_{j}(V_{i}))

and so by part (b),

(45)  \frac{o(c(V_{i}))}{(|c(V_{i})| − 1)!} = ∏_{j=1}^{|V_{i}|−1} \frac{o(c_{j}(V_{i}))}{|c_{j}(V_{i})|!}.

The following results may be contained in [Pop08], and are closely related to Proposition 3.5 in [HS14]. We provide a purely combinatorial direct proof.

Proposition 4.9. Let μ : B⟨X⟩ → B be an exponentially bounded B-bimodule map. Then for each n

\frac{dH^{(μ⊗1_{N})^{▷t}}(b)}{dt} = K^{μ⊗1_{N}} (H^{(μ⊗1_{N})^{▷t}}(b)).

Documenta Mathematica 21 (2016) 841–871

Proof. It suffices to prove the result for N = 1. We begin by proving this equality for each of the coefficients of the series expansions of H^{μ^{▷t}} and K^{μ} ∘ H^{μ^{▷t}}. Since

\begin{aligned}\frac{d}{dt}μ^{▷t}[b(Xb)^{n}] & = \frac{d}{dt} ∑_{π∈NC(n)} t^{|π|} \frac{o(π)}{|π|!}K_{π}^{μ}[bXbX … bXb] \\ & = ∑_{π∈NC(n)} t^{|π|−1} \frac{o(π)}{(|π| − 1)!}K_{π}^{μ}[bXbX … bXb], \end{aligned} ; (46)

the coefficient of K_{π}^{μ}[b(Xb)^{n}] in its expansion is t^{|π|−1} \frac{o(π)}{(|π|−1)!}. On the other hand,

K^{μ} [H^{μ^{▷t}}(b) (XH^{μ^{▷t}}(b))^{l}]

= K^{μ} [H^{μ^{▷t}}(b)XH^{μ^{▷t}}(b)X … H^{μ^{▷t}}(b)XH^{μ^{▷t}}(b)]

= ∑_{k_{0},...,k_{l}≥0} K^{μ} [ ∑_{π_{0}∈NC(k_{0})} t^{|π_{0}|} \frac{o(π_{0})}{|π_{0}|!}K_{π_{0}}^{μ}X

∑_{π_{1}∈NC(k_{1})} t^{|π_{1}|} \frac{o(π_{1})}{|π_{1}|!}K_{π_{1}}^{μ}X … X ∑_{π_{l}∈NC(k_{l})} t^{|π_{l}|} \frac{o(π_{l})}{|π_{l}|!}K_{π_{l}}^{μ} ]

= ∑_{k_{0},...,k_{l}≥0} ∑_{\begin{matrix}π_{i}∈NC(k_{i}), \\ 0≤i≤l \end{matrix}} \frac{o(π_{0})}{|π_{0}|!} \frac{o(π_{1})}{|π_{1}|!} … \frac{o(π_{l})}{|π_{l}|!}K^{μ}

[K_{π_{0}}^{μ}XK_{π_{1}}^{μ}X … XK_{π_{l}}^{μ}] t^{|π_{0}|+|π_{1}|+...+|π_{l}|},

where K_{∅}(b) = b. Fixing n = k_{0} + … + k_{l} + l, each term in this expansion is a multiple of K_{π}^{μ}[b(Xb)^{n}], where π is constructed from partitions π_{0}, π_{1}, …, π_{k} and an additional outer block of l elements:

V = {k_{0} + 1, k_{0} + k_{1} + 2, …, k_{0} + … + k_{l−1} + l} ∈ π

and

π_{i} = restriction of π to [k_{0} + … + k_{i−1} + i + 1, k_{0} + … + k_{i} + i], i = 0, 1, …, l.

Note that |π_{0}| + |π_{1}| + … + |π_{l}| = |π| − 1. This identification has an inverse, which requires first choosing one of the k outer blocks of π. Order the outer blocks leftto-right and call the specially chosen block V_{i}. Using the notation from Remark 4.8, we see that the coefficient of K_{π}^{μ}[b(Xb)^{n}] in the expansion of K^{μ}(H^{μ^{▷t}}(b)) is t^{|π|−1} times

∑_{i=1}^{k} \frac{o (⋃_{j<i} c(V_{j}))}{|⋃_{j<i} c(V_{j})|!} ( ∏_{j=1}^{|V_{i}|−1} \frac{o(c_{j}(V_{i}))}{|c_{j}(V_{i})|!}) \frac{o (⋃_{j>i} c(V_{j}))}{|⋃_{j>i} c(V_{j})|!}

= ∑_{i=1}^{k} \frac{o(c(V_{i}))}{(|c(V_{i})| − 1)!} ∏_{j≠i} \frac{o(c(V_{j}))}{|c(V_{j})|!}

= ∏_{j=1}^{k} \frac{o(c(V_{j}))}{|c(V_{j})|!} ∑_{i=1}^{k} |c(V_{i})|

= |π| ∏_{j=1}^{k} \frac{o(c(V_{j}))}{|c(V_{j})|!}

= \frac{o(π)}{(|π| − 1)!}.

Here we used equation (45), and equation (44) applied to partitions ⋃_{j<i} c(V_{j}) and ⋃_{j>i} c(V_{j}), in the first line, and again (44) in the last line. Since we obtained the same coefficient as in expansion (46), the result is proved for each of the individual components of the respective series expansions for each n ∈ ℕ. Extending this to the series expansions and, therefore, the functions, observe that all of the sets over which the sums occur have cardinality whose growth rate is exponential over n. Thus, for ∥b∥ small enough, the exponential boundedness of μ implies that the respective series are absolutely convergent. We may therefore conclude that the t coefficients of the series expansions agree, provided that b ∈ B_{δ}(0) for δ > 0 small enough. Thus,

\frac{dH^{μ^{▷t}}(b)}{dt} = K^{μ}(H^{μ^{▷t}}(b)).

for b ∈ B_{δ}(0). To extend to arbitrary bounded sets in B^{−}, consider the net of difference quotients

D_{h}^{μ}(b, t) = \frac{H^{μ^{▷t+h}}(b) − H^{μ^{▷t}}(b)}{h}

for t > 0. We have just shown that

lim_{h→0} D_{h}^{μ}(b, t) → K^{μ}(H^{μ^{▷t}}(b))

uniformly on B_{δ}(0). By Theorem 2.10 in [BPV12], this implies that the same is true on all bounded sets in B^{−}. Thus, at the level of functions,

\frac{dH^{μ^{▷t}}(b)}{dt} = K^{μ}(H^{μ^{▷t}}(b)),

proving our result. □

Corollary 4.10.

H^{(μ⊗1_{n})^{▷(s+t)}}(b) = H^{(μ⊗1_{n})^{▷s}} (H^{(μ⊗1_{n})^{▷t}}(b)) .

In particular,

F^{μ^{▷(s+t)}}(b) = F^{μ^{▷s}} (F^{μ^{▷t}}(b)) ,

so the combinatorial definition of monotone convolution powers coincides with the complex analytic one in Definition 2.3.

Proof. By Proposition 4.9, H^{μ^{▷s}} (H^{μ^{▷t}}(b)), as a function of s, satisfies

\frac{d}{ds}H^{μ^{▷s}} (H^{μ^{▷t}}(b)) = K^{μ} (H^{μ^{▷s}} (H^{μ^{▷t}}(b))) ,

H^{μ^{▷s}} (H^{μ^{▷t}}(b))|_{s=0} = H^{μ^{▷t}}(b).

Since, by the same proposition, H^{μ^{▷(s+t)}}(b) also satisfies this differential equation with this initial condition, they coincide for all positive s. For the second statement, we observe that

G^{μ^{▷s}} (F^{μ^{▷t}}(b)) = G^{μ^{▷s}} ((G^{μ^{▷t}}(b))^{−1}) = H^{μ^{▷s}} (H^{μ^{▷t}}(b^{−1})) =

= H^{μ^{▷(s+t)}}(b^{−1}) = G^{μ^{▷(s+t)}}(b).

□

Proposition 4.11. If μ, ν ∈ Σ_{0} and μ ▷ μ = ν ▷ ν, then μ = ν. In particular, if the square root with respect to the monotone convolution exists, it is unique.

Proof. Under the given assumption,

K^{μ} = \frac{1}{2}K^{μ▷μ} = K^{ν},

and therefore μ = ν. □

Remark 4.12. Let γ ∈ B be self-adjoint, and σ : B⟨X⟩ → B be a completely positive but not necessarily a B-bimodule map. Define ν_{⊎}^{γ,σ} via its Boolean cumulant functional

B^{ν_{⊎}^{γ,σ}}[b_{0}Xb_{1}] = b_{0}γb_{1}, B^{ν_{⊎}^{γ,σ}}[b_{0}Xb_{1}X … b_{n−1}Xb_{n}] = b_{0}σ[b_{1}X … b_{n−1}]b_{n}.

It is known [BPV12, ABFN13] that ν_{⊎}^{γ,σ} is a completely positive B-bimodule map. Similarly, define ν_{▷}^{γ,σ} via its monotone cumulant functional

K^{ν_{▷}^{γ,σ}}[b_{0}Xb_{1}] = b_{0}γb_{1}, K^{ν_{▷}^{γ,σ}}[b_{0}Xb_{1}X … b_{n−1}Xb_{n}] = b_{0}σ[b_{1}X … b_{n−1}]b_{n}.

We could also define ν_{⊞}^{γ,σ} via its free cumulant functional

R^{ν_{⊞}^{γ,σ}}[b_{0}Xb_{1}] = b_{0}γb_{1}, R^{ν_{⊞}^{γ,σ}}[b_{0}Xb_{1}X … b_{n−1}Xb_{n}] = b_{0}σ[b_{1}X … b_{n−1}]b_{n}.

Lemma 4.13. Let k_{i} → ∞ be a numerical sequence, {μ_{i} : B⟨X⟩ → B}_{i=1}^{∞} a sequence of linear B-bimodule maps, and ρ : B_{0}⟨X⟩ → B a linear B-bimodule map. The following are equivalent.

(a) k_{i}μ_{i}[P (X)] → ρ[P (X)] for all P (X) ∈ B_{0}⟨X⟩.

(b) k_{i}R^{μ_{i}}[P (X)] → ρ[P (X)] for all P (X) ∈ B_{0}⟨X⟩.

(c) k_{i}B^{μ_{i}}[P (X)] → ρ[P (X)] for all P (X) ∈ B_{0}⟨X⟩.

(d) k_{i}K^{μ_{i}}[P (X)] → ρ[P (X)] for all P (X) ∈ B_{0}⟨X⟩. Here in all cases, the convergence is in norm on B.

Proof. We will prove the equivalence between (a) and (d); the rest are similar, and were proved in [BPV12]. Indeed, on B_{0}⟨X⟩,

\begin{aligned}k_{i}μ_{i}[b_{0}Xb_{1}X & … b_{n−1}Xb_{n}] = k_{i}K^{μ_{i}}[b_{0}Xb_{1}X … b_{n−1}Xb_{n}] \\ & + ∑_{\begin{matrix}π∈NC(n) \\ |π|≥2 \end{matrix}} \frac{1}{k_{i}^{|π|−1}} \frac{o(π)}{|π|!} (k_{i}K^{μ_{i}})_{π} [b_{0}Xb_{1}X … b_{n−1}Xb_{n}]. \end{aligned}

It follows immediately that (d) implies (a). The converse implication follows by induction on n. □

Corollary 4.14. For linear B-bimodule maps μ_{i} : B⟨X⟩ → B, the following are equivalent.

(a)

k_{i}μ_{i}[X] → γ, k_{i}μ_{i}[Xb_{1}X … b_{n−1}X] → σ[b_{1}X … b_{n−1}].

(b)

μ_{i}^{⊞k_{i}} → ν_{⊞}^{γ,σ}.

(c)

μ_{i}^{⊎k_{i}} → ν_{⊎}^{γ,σ}.

(d)

μ_{i}^{▷k_{i}} → ν_{▷}^{γ,σ}.

Proof. We will prove the equivalence between (a) and (d); the rest are similar, see Lecture 13 in [NS06]. Indeed, by Lemma 4.6, the statement in part (d) is equivalent to

k_{i}K^{μ_{i}} → K^{ν_{▷}^{γ,σ}},

which by definition of ν_{▷}^{γ,σ} means

k_{i}K^{μ_{i}}[X] → γ, k_{i}K^{μ_{i}}[Xb_{1}X … b_{n−1}X] → σ[b_{1}X … b_{n−1}]

This is equivalent to (a) by the preceding lemma. □

Corollary 4.15. ν_{▷}^{γ,σ} is a completely positive map.

Proof. We can choose completely positive μ_{i} such that μ_{i}^{⊎i} → ν_{⊎}^{γ,σ}, for example by taking μ_{i} = ν_{⊎}^{\frac{1}{i} γ, \frac{1}{i} σ}. Then ν_{▷}^{γ,σ} is the limit of completely positive maps μ_{i}^{▷i}, and as such is completely positive (monotone convolution of two completely positive maps is known to be positive, see Proposition 6.2 of [Pop08] and also [Pop12]). □

Proposition 4.16. Monotone convolution semigroups of completely positive Bbimodule maps are in a one-to-one correspondence with pairs (γ, σ) as above.

Proof. {ν_{▷}^{tγ,tσ} : t ≥ 0} form a one-parameter monotone convolution semigroup of completely positive B-bimodule maps. Conversely, if {μ_{t}} is such a semigroup, define

γ = \frac{d}{dt}|_{t=0} μ_{t}[X] = K^{μ_{1}}[X] ∈ B^{sa},

σ[b_{1}X … b_{n−1}] = \frac{d}{dt}|_{t=0} μ_{t}[Xb_{1}X … b_{n−1}X] = K^{μ_{1}}[Xb_{1}X … b_{n−1}X].

Since for P_{i} ∈ B⟨X⟩ and c_{i} ∈ B,

∑_{i,j=1}^{N} c_{i}^{∗}σ[P_{i}^{∗}P_{j}]c_{j} = \frac{d}{dt}|_{t=0} μ_{t} [ ∑_{i,j=1}^{N} c_{i}^{∗}XP_{i}^{∗}P_{j}Xc_{j}] =

= lim_{t↓0} \frac{1}{t}μ_{t} [ ∑_{i,j=1}^{N} c_{i}^{∗}XP_{i}^{∗}P_{j}Xc_{j}] ≥ 0,

σ is completely positive □

Remark 4.17. A short calculation shows that

Φ(b) = γ + G_{σ}(b).

This, combined with Theorem 2.1, gives an alternative proof of the result in Theorem 3.5 that generators of semigroups arising from ▷-infinitely divisible distributions coincide with the set Λ. One can also use a standard combinatorial argument to show that ▷-infinitely divisible distributions belong to such one-parameter semigroups. At this point, we do not know how to obtain the more general results in Theorem 3.5 by combinatorial methods.

Appendix A. Characterization of general Cauchy transforms

In this appendix, we extend the main result in [Wil13], namely the classification of the Cauchy transforms associated to distributions μ ∈ Σ_{0}, to the Cauchy transforms associated to more general CP maps.

Theorem A.1. The following are equivalent:

(I) The analytic non-commutative function G = (G^{(n)})_{n≥1} : H^{+}(B) → H^{−}(B) has the property that H = (H^{(n)})_{n≥1} defined through the equalities H^{(n)}(b) := G^{(n)}(b^{−1}) for all n ∈ ℕ and b ∈ M_{n}(B) has uniformly analytic extension to a neighborhood of 0 satisfying H^{(n)}(0) = 0.

(II) There exists a ℂ-linear map σ : B⟨X⟩ → B satisfying (1) and (2) such that G^{(n)}(b) = σ((b − X)^{−1}).

Proof. We begin with (II) ⇒ (I). Let σ satisfy (1) and (2). By [PV13], Lemma 5.8, we may conclude that there exists a ⊞-infinitely divisible distribution μ ∈ Σ_{0} such that ρ_{μ}(XP (X)X) = σ(P (X)) for all P (X) ∈ B⟨X⟩ (here, ρ_{μ} denotes the free cumulant function associated to μ). Thus, the Voiculescu transform of μ satisfies the following equality:

(47)  φ_{μ}^{(n)}(b) = −σ((b − X)^{−1})

for all n ∈ ℕ and where the inverse in the equality is considered as a geometric series, so that the right hand side is convergent for ∥b^{−1}∥ small enough dependent on (1). Since μ is ⊞-infinitely divisible, by Proposition 5.1 in [Wil13], we have that the left hand side of (47) extends to

H^{+}(B) ∪ H^{−}(B) ⋃_{n=1}^{∞}{b ∈ M_{n}(B) : ∥b^{−1}∥ < C}

where C is a fixed constant, independent of n. Now, by Proposition 1.2 in [PV13], the fact that μ ∈ Σ_{0} implies that μ is realized as the distribution arising from a non-commutative probability space (A, E, B). That is,

μ(P (X)) = E(P (a))

for a fixed self-adjoint element a ∈ B and all P (X) ∈ B⟨X⟩. Thus, σ((b − X)^{−1}) = ρ_{μ}(a(b − a)^{−1}a) and, since b − a ∈ M_{n}^{+}(B) and ρ_{μ} is a CP map on B⟨X⟩_{0} we may conclude that the σ((b − X)^{−1}) ∈ M_{n}^{−}(B) for all b ∈ M_{n}^{+}(B). Further note that

H(b) = σ((b^{−1} − X)^{−1}) = ∑_{k=0}^{∞} σ((bX)^{k}b)

is convergent in a neighborhood of zero since σ satisfies (1). It is also immediate that H(0) = 0. This completes one direction of our proof. We now prove (I) ⇒ (II). We will follow the proof of Theorem 4.1 in [Wil13] and refer to this paper for the appropriate terminology. We recover our operator σ through the differential structure of H. Indeed, we define the map σ by letting

(σ⊗1_{n})(b_{1}(X ⊗1_{n})b_{2} ⋯ (X ⊗1_{n})b_{ℓ+1}) := Δ_{R}^{ℓ+1}H^{(n)}( \underbrace{0, …, 0}_{ℓ+2 − times})(b_{1}, b_{2}, …, b_{ℓ+1})

for elements b_{1}, b_{2}, ⋯, b_{ℓ+1} ∈ M_{n}(B). It is a consequence of Proposition 3.1 in [Wil13] and [KVV14], Theorem 3.10 that this is a well defined operator. Moreover, the equality

Δ_{R}^{ℓ+1}H^{(n)}( \underbrace{0, …, 0}_{ℓ+2 − times})(b, b, …, b) = \frac{1}{(ℓ + 1)!} \frac{d^{ℓ+1}}{dt^{ℓ+1}} H^{(n)}(0 + tb)|_{t=0}

and the fact that the function is analytic in a neighborhood of 0 implies that

(48)  H^{(n)}(b) = ∑_{k=0}^{∞}(σ ⊗ 1_{n})((bX)^{k}b)

once we show that σ satisfies (1). Continuation will allow us to conclude that

(49)  G^{(n)}(b) = ∑_{k=0}^{∞}(σ ⊗ 1_{n})((b^{−1}X)^{k}b^{−1}) = (σ ⊗ 1_{n})((b − X)^{−1}).

Thus, our theorem will follow when we can show that σ satisfies properties (1) and (2).

To prove (1), we note that this is equivalent to showing that

∥σ(b_{1}Xb_{2} ⋯ Xb_{ℓ+1})∥ ≤ CM^{ℓ+1}

for a fixed C > 0, provided that ∥b_{1}∥ = ⋯ = ∥b_{ℓ+1}∥ = 1. This will follow from uniform analyticity and matches the proof of the same fact in [Wil13]. Indeed, consider the element of M_{ℓ+2}(B)

B = ( \begin{matrix}0 & b_{1} & 0 & 0 & ⋯ & 0 \\ 0 & 0 & b_{2} & 0 & ⋯ & 0 \\ 0 & 0 & 0 & b_{3} & ⋯ & 0 \\ & ⋮ & & & ⋮ \\ 0 & 0 & 0 & 0 & ⋯ & b_{ℓ+1} \\ 0 & 0 & 0 & 0 & ⋯ & 0 \end{matrix} ) .

Note that H^{(ℓ+1)} has a bound of C on a ball of radius r about 0, independent of ℓ since we are assuming that H is uniformly analytic. Thus,

∥σ(b_{1}Xb_{2} ⋯ Xb_{ℓ+1})∥ = \frac{∥δ^{ℓ+1}H^{(ℓ+2)}(0; B)∥}{(ℓ + 1)!}

= ∥Δ_{R}^{ℓ+1}H^{(ℓ+2)}(0, …, 0)(B, …, B)∥

= ∥r^{−(ℓ+1)}Δ_{R}^{ℓ+1}H^{(ℓ+2)}(0, …, 0)(rB, …, rB)∥

= (\frac{1}{r})^{ℓ+1} \frac{∥δ^{ℓ+1}H^{(ℓ+2)}(0; rB)∥}{(ℓ + 1)!}

≤ C (\frac{1}{r})^{ℓ+1}

where the last inequality follows from the Cauchy estimates in Theorem 2.2. We must prove the technical fact that fact that

(50)  σ|_{M_{n}(B)} ≥ 0

Assume that σ(P) < 0 for some P ∈ M_{n}^{+}(B) where we can assume that P > δ1 for some δ > 0. Note that G^{(n)}(zP^{−1}) ∈ M_{n}^{−}(B) for all z ∈ ℂ^{+} by assumption so that λG^{(n)}(iλP^{−1}) ∈ M_{n}^{−}(B) for all λ ∈ ℝ^{+}. Utilizing the series expansion in (49) as well as the exponential bound that we have just proven, we conclude that the

lim_{λ↑∞} λG^{(n)}(iλP^{−1}) = \frac{σ(P)}{i} = −iσ(P) ∉ M_{n}^{−}(B).

This contradiction implies (50). It remains to show (2). Once again, this will closely follow the proof of the analogous fact in Theorem 4.1 in [Wil13]. Indeed, we will first show that

(51)  (σ ⊗ 1_{n})(P (X ⊗ 1_{n} + b_{0})^{∗}P (X ⊗ 1_{n} + b_{0})) ≥ 0

for any monomial P (X) = b_{1}(X ⊗ 1_{n})b_{2} ⋯ X ⊗ 1_{n}b_{ℓ+1} ∈ M_{n}(B)⟨X⟩ and b_{0} ∈ M_{n}(B). We also assume that |b_{ℓ+1}| > ϵ1_{n} and the general case follows by letting ϵ ↓ 0.

Towards this end, we consider elements C, E_{0}, E_{1} ∈ M_{n(ℓ+1)}(B) defined as follows:

C = ( \begin{matrix}0 & c_{1} & 0 & 0 & 0 & ⋯ & 0 \\ c_{1}^{∗} & 0 & c_{2} & 0 & 0 & ⋯ & 0 \\ 0 & c_{2}^{∗} & 0 & c_{3} & 0 & ⋯ & 0 \\ ⋮ & & & ⋮ & & & ⋮ \\ 0 & 0 & ⋯ & 0 & c_{ℓ−1}^{∗} & 0 & c_{ℓ} \\ 0 & 0 & ⋯ & 0 & 0 & c_{ℓ}^{∗} & |c_{ℓ+1}|^{2} \end{matrix} ); E_{0} = \underbrace{1_{n} ⊕ 1_{n} ⊕ ⋯ ⊕ 1_{n}}_{ℓ times} ⊕0_{n}

and E_{1} = 1_{n(ℓ+1)} −E_{0} where c_{i} = δb_{i} for i = 1, …, ℓ and c_{ℓ+1} = b_{ℓ+1}/δ^{ℓ} for δ > 0 to be specified. Note that b_{1}Xb_{2} ⋯ Xb_{ℓ+1} = c_{1}Xc_{2} ⋯ Xc_{ℓ+1}. We define a function

ĝ^{n(ℓ+1)}(b) := G^{n(ℓ+1)}(b − b_{0}) : M_{n(ℓ+1)}^{+}(B) → M_{n(ℓ+1)}^{−}(B)

The following properties are rather trivial and their proof matches those of Theorem 4.1 in [Wil13].

(a) C + ϵE_{0} > γ1_{n} for some γ > 0 provided that δ > 0 is small enough.

(b) The n × n minor in the top left corner of

[(C + ϵE_{0})(X ⊗ 1_{n(ℓ+1)} + b_{0} ⊗ 1_{ℓ+1})]^{2(ℓ−1)}(C + ϵE_{0})

is equal to P (X + b_{0})P^{∗}(X + b_{0}) + O(ϵ).

(c) ĝ^{(n(ℓ+1))}(b) = ∑_{p=0}^{∞} σ([b^{−1}(X ⊗ 1_{n(ℓ+1)} + b_{0} ⊗ 1_{ℓ+1})]^{p}b^{−1}) for b^{−1} in a neighborhood of 0.

(d) We have that zĝ^{(n(ℓ+1))}(zb) → σ(b^{−1}) in norm as |z| ↑ ∞ for b > γ1_{n}.

(e) ĥ^{(n(ℓ+1))}(b) := ĝ^{(n(ℓ+1))}(b^{−1}) has analytic extension to a neighborhood of zero. The only one of these properties that differs from the proof of Theorem 4.1 in [Wil13] is (d). It follows immediately from the series expansion in (48). We now have the pieces in place to prove (51). Note that (a) implies that C + ϵE_{0} is invertible so that the map

z ↦ ĝ^{(n(ℓ+1))}(z(C + ϵE_{0})^{−1})

sends ℂ^{+} into M_{n}(B)^{−}. Let B_{i,j} ∈ M_{n}(B) for i, j = 1, …, ℓ + 1 and consider the element B = (B_{i,j})_{i,j=1}^{ℓ+1} ∈ M_{n(ℓ+1)}(B). Given a state f ∈ M_{n}(B)^{∗} we define a new state

f_{1,1}(B) := f(B_{1,1}) : M_{n(ℓ+1)}(B) → ℂ.

We may define a map

G_{f,C,ϵ}(z) = f_{1,1} ∘ ĝ^{(n(ℓ+1))}(z(C + ϵE_{0})^{−1}) : ℂ^{+} → ℂ^{−}.

Properties (c) and (d) imply the following for z ∈ ℂ^{+}:

lim_{|z|↑∞} zG_{f,C,ϵ}(z) = lim_{|z|↑∞} f_{1,1} [zĝ^{(n(ℓ+1))}(z(C + ϵE_{0})^{−1})]
= f_{1,1}(σ(C + ϵE_{0})) ≥ 0

where the last inequality will follow from the fact that f_{1,1} is a state, property (a) and (50).

Now, observe that the coefficient of z^{−2ℓ+1} in the function G_{f,C,ϵ} is equal to ρ(t^{2(ℓ−1)}) > 0. Furthermore, since

G_{f,C,ϵ}(z) = G_{ρ}(z) = ∑_{ℓ=0}^{∞} \frac{ρ(t^{ℓ})}{z^{ℓ+1}}

= ∑_{ℓ=0}^{∞} \frac{f_{1,1}(σ([(C + ϵE_{0})(X ⊗ 1_{n(ℓ+1)} + b_{0})]^{ℓ}(C + ϵE_{0})))}{z^{ℓ+1}}

we may conclude that

f_{1,1} ∘ σ([(C + ϵE_{0})(X ⊗ 1_{n(ℓ+1)} + b_{0})]^{2(ℓ−1)}(C + ϵE_{0})) = ρ(t^{2(ℓ−1)}) ≥ 0.

Recalling (b), it follows that f ∘ σ([P (X + b_{0})P^{∗}(X + b_{0}) + O(ϵ)]) ≥ 0. Letting ϵ ↓ 0 and noting that f was an arbitrary state, we have proven that

(σ ⊗ 1_{n})(P (X + b_{0})P^{∗}(X + b_{0})) ≥ 0

for any monomial P (X) ∈ M_{n}(B)⟨X⟩. The extension from the case of monomials to general elements in B⟨X⟩ follows the proof in [Wil13] exactly so we will refrain from repeating it. This implies (2) and, therefore, our theorem. □

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Michael Anshelevich Department of Mathematics Texas A&M University College Station TX 77843-3368 manshel@math.tamu.edu John D. Williams Universität des Saarlandes Fachrichtung Mathematik Postfach 151150 66041. Saarbrücken williams@math.uni-sb.de 872

Documenta Mathematica 21 (2016)