# The Dyson equation for $2$-positive maps and Hölder bounds for the Lévy
distance of densities of states

Tobias Mai  Saarland University, Department of Mathematics, D-66123
Saarbrücken, Germany<EMAIL_ADDRESS>
###### Abstract.

The so-called density of states is a Borel probability measure on the real
line associated with the solution of the Dyson equation which we set up, on
any fixed $C^{\ast}$-probability space, for a selfadjoint offset and a
$2$-positive linear map. Using techniques from free noncommutative function
theory, we prove explicit Hölder bounds for the Lévy distance of two such
measures when any of the two parameters varies. As the main tools for the
proof, which are also of independent interest, we show that solutions of the
Dyson equation have strong analytic properties and evolve along any
$C^{1}$-path of $2$-positive linear maps according to an operator-valued
version of the inviscid Burgers equation.

###### Key words and phrases:

operator-valued semicircular elements, Dyson equation, density of states,
Burgers equation, free noncommutative function theory

###### 2010 Mathematics Subject Classification:

46T20, 46G05, 46G20, 35R15, 46L54, 60B20

The author wants to thank Johannes Hoffmann and Roland Speicher for the
interesting discussions which have inspired this work.

##  1\. Introduction and presentation of the main results

Voiculescu’s groundbreaking discovery [59] of the phenomenon of asymptotic
freeness has started an enormously fruitful exchange of techniques between the
area of operator algebras and random matrix theory through free probability
theory. Undoubtedly among the most prominent of such connections which were
facilitated by free probability theory are operator-valued semicircular
elements. Let us recall that those are noncommutative random variables which
constitute the canonical generalization of semicircular elements to the
framework of operator-valued free probability theory, while the latter are the
free probability counterpart of normally distributed random variables in
classical probability; we refer to [54, 42, 41] for an introduction to
(operator-valued) free probability theory and to operator-valued semicircular
elements in particular. These are fundamental objects in free probability [60,
54, 50, 51, 15, 53], which fully reveal their rich structure at the frontiers
of that field. On the one hand, the study of random matrix models for
operator-valued semicircular elements has led to spectacular developments on
the operator-algebraic side like in [22, 46, 50] and [28, 27]. On the other
hand, random matrix models for operator-valued semicircular elements are of
great interest beyond operator-algebraic applications and this elegant
description of their asymptotic eigenvalue distribution enabled a detailed
study thereof like in [49, 31, 45, 2, 4, 3]; see also [52, 39, 40, 12, 20].

Operator-valued semicircular elements often do not show up explicitly in such
contexts, but rather in the guise of their operator-valued Cauchy transforms
or by equations determining the latter. To build our intuition, let us take a
closer look at the model considered in [28, 27]. Voiculescu’s multivariate
generalization [59] of Wigner’s semicircle law [64] (in its almost sure
version [32, 57] which generalizes [7] to the multivariate case) teaches us
that tuples $(X^{N}_{1},\dots,X^{N}_{n})$ of independent standard Gaussian
random matrices of size $N\times N$ converge in distribution almost surely as
$N\to\infty$ towards a tuple $(S_{1},\dots,S_{n})$ of freely independent
standard semicircular elements. Explicitly, with respect to the normalized
trace $\operatorname{tr}_{N}$ on $M_{N}(\mathbb{C})$, we have for every
$k\in\mathbb{N}$ and each choice of indices $1\leq i_{1},\dots,i_{k}\leq n$
that

(1.1) $\lim_{N\to\infty}\operatorname{tr}_{N}(X_{i_{1}}^{N}X_{i_{2}}^{N}\cdots
X_{i_{k}}^{N})=\tau(S_{i_{1}}S_{i_{2}}\cdots S_{i_{k}})\qquad\text{almost
surely},$

where we suppose that $(S_{1},\dots,S_{n})$ is realized in some
$C^{\ast}$-probability space $(M,\tau)$. For fixed selfadjoint matrices
$b_{0},b_{1},\dots,b_{n}\in M_{m}(\mathbb{C})$, we aim at understanding the
asymptotic behavior of the selfadjoint random matrices
$X^{N}:=b_{0}\otimes\mathbf{1}_{N}+b_{1}\otimes X_{1}^{N}+\dots+b_{n}\otimes
X_{n}^{N}$. It follows from (1.1) that the empirical eigenvalue distribution
$\mu_{X^{N}}$ of $X^{N}$ converges in distribution almost surely as
$N\to\infty$ to the analytic distribution $\mu_{S}$ of the noncommutative
random variable $S:=b_{0}\otimes\mathbf{1}+b_{1}\otimes
S_{1}+\dots+b_{n}\otimes S_{n}$ in the $C^{\ast}$-probability space
$(M_{m}(\mathbb{C})\otimes M,\operatorname{tr}_{m}\otimes\tau)$. In fact, by
means of operator-valued free probability, we can even compute $\mu_{S}$. The
crucial observation is that $S$, in the operator-valued $C^{\ast}$-probability
space $(M_{m}(\mathbb{C})\otimes M,\mathbb{E},M_{m}(\mathbb{C}))$ with
conditional expectation
$\mathbb{E}:=\operatorname{id}_{M_{m}(\mathbb{C})}\otimes\tau$, forms an
operator-valued semicircular element with mean $b_{0}$ and the covariance
$\eta:M_{m}(\mathbb{C})\to M_{m}(\mathbb{C})$ given by
$\eta(b):=\sum^{n}_{j=1}b_{j}bb_{j}$ for $b\in M_{m}(\mathbb{C})$. This
implies that the operator-valued Cauchy transform
$G_{S}:\mathbb{H}^{+}(M_{m}(\mathbb{C}))\to\mathbb{H}^{-}(M_{m}(\mathbb{C}))$
defined by $G_{S}(b):=\mathbb{E}[(b-S)^{-1}]$ solves the equation

$(b-b_{0})G_{S}(b)=\mathbf{1}+\eta(G_{S}(b))G_{S}(b)\qquad\text{for
$b\in\mathbb{H}^{+}(M_{m}(\mathbb{C}))$}.$

In fact, $G_{S}$ is hereby uniquely determined and can be computed efficiently
by some fixed point iteration; see [31, 34]. This change of perspective thus
settles the problem of determining $\mu_{S}$, since $G_{S}$ recovers the
scalar-valued Cauchy transform $\mathcal{G}_{\mu_{S}}$ of $\mu_{S}$ via the
relation $\mathcal{G}_{\mu_{S}}(z)=\operatorname{tr}_{m}(G_{S}(z\mathbf{1}))$
for all $z\in\mathbb{C}^{+}$ and since $\mu_{S}$ can be obtained from
$\mathcal{G}_{\mu_{S}}$ via Stieltjes inversion.

In this paper, we are concerned with a significant generalization of this
construction which originates in [2, 3] and which we briefly sketch now; a
more detailed presentation will be given in Section 3, in particular in
Definition 3.4. Let $(B,\phi)$ be a $C^{\ast}$-probability space. We put
$B_{\mathrm{sa}}:=\\{b\in B\mid b^{\ast}=b\\}$ and denote by $\mathcal{P}(B)$
the set of all $\mathbb{C}$-linear maps $\eta:B\to B$ which are positive; see
Subsection 2.1 for more details. To any $\rho=(b_{0},\eta)\in
B_{\mathrm{sa}}\times\mathcal{P}(B)$, called _data pair_ , we associate the
holomorphic function $G_{\eta}:\mathbb{H}^{+}(B)\to\mathbb{H}^{-}(B)$ which is
uniquely determined by the _Dyson equation_

(1.2) $bG_{\eta}(b)=\mathbf{1}+\eta(G_{\eta}(b))G_{\eta}(b)\qquad\text{for
$b\in\mathbb{H}^{+}(B)$}$

and consider in turn the holomorphic function
$\mathcal{G}_{\rho}:\mathbb{C}^{+}\to\mathbb{C}^{-}$ which is defined by
$\mathcal{G}_{\rho}(z):=\phi(G_{\eta}(z\mathbf{1}-b_{0}))$ for all
$z\in\mathbb{C}^{+}$. Since $\mathcal{G}_{\rho}$ satisfies
$\lim_{y\to\infty}iy\mathcal{G}_{\rho}(iy)=1$, it must be the Cauchy transform
of a hereby uniquely determined Borel probability measure $\mu_{\rho}$ on
$\mathbb{R}$, to which we shall refer as the _density of states_ associated
with $\rho$. In this way, we obtain a mapping

$\mu:\
B_{\mathrm{sa}}\times\mathcal{P}(B)\to\operatorname{Prob}(\mathbb{R}),\quad\rho\mapsto\mu_{\rho}$

taking values in the set $\operatorname{Prob}(\mathbb{R})$ of all Borel
probability measures on the real line $\mathbb{R}$.

For the operator-valued semicircular element $S$ with mean $b_{0}$ and
covariance $\eta$ which we introduced above, this procedure recovers $\mu_{S}$
as $\mu_{(b_{0},\eta)}$; in fact, it works for any operator-valued
semicircular element $S$ in an arbitrary operator-valued
$C^{\ast}$-probability space $(A,\mathbb{E},B)$ and with respect to some
distinguished state $\phi$ on $B$. This highlights the fundamental fact that,
if we keep $(B,\phi)$ fixed, then $\mu_{S}$ only depends on $b_{0}$ and the
completely positive linear map $\eta$, but not on the concrete realization $S$
or the corresponding space $(A,\mathbb{E},B)$. We point out that with [44,
Proposition 2.2], it is possible to construct for every data pair
$(b_{0},\eta)$ with $\eta$ being completely positive an operator-valued
$C^{\ast}$-probability space $(A,\mathbb{E},B)$ and an operator-valued
semicircular element $S$ with mean $b_{0}$ and covariance $\eta$. The map
$\mu$ introduced above not only circumvents this construction, but even works
in situations where no such realization of $(b_{0},\eta)$ can exist.

Our aim is to quantify the proximity of $\mu_{\rho_{0}}$ and $\mu_{\rho_{1}}$
in terms of the data pairs $\rho_{0},\rho_{1}\in
B_{\mathrm{sa}}\times\mathcal{P}(B)$. To this end, we endow

  * •

$\operatorname{Prob}(\mathbb{R})$ with the Lévy metric
$L:\operatorname{Prob}(\mathbb{R})\times\operatorname{Prob}(\mathbb{R})\to[0,1]$,
which is known to provide a metrization of weak convergence;

  * •

$B_{\mathrm{sa}}$ with the metric $d_{B_{\mathrm{sa}}}$ induced by the norm
$\|\cdot\|$ on the $C^{\ast}$-algebra $B$;

  * •

$\mathcal{P}(B)$ with the metric $d_{\mathcal{P}(B)}$ induced by the norm
$\|\eta\|:=\sup_{\|b\|\leq 1}\|\eta(b)\|$ on the space of all bounded linear
operators $\eta:B\to B$.

More precisely, we will derive Hölder bounds for
$L(\mu_{\rho_{1}},\mu_{\rho_{0}})$ in terms of
$d_{B_{\mathrm{sa}}}(b_{0,1},b_{0,0})$ and
$d_{\mathcal{P}(B)}(\eta_{1},\eta_{0})$, depending on the varying argument,
but uniformly in the respective other argument; for that purpose, however, we
need to restrict $\mu$ to the subset $B_{\mathrm{sa}}\times\mathcal{P}_{2}(B)$
of $B_{\mathrm{sa}}\times\mathcal{P}(B)$, where $\mathcal{P}_{2}(B)$ stands
for the set of all $\eta\in\mathcal{P}(B)$ which are $2$-positive. The
necessity of restricting $\mu$ arose also in [3], but there the authors
required $\eta$ to satisfy some strict version of positivity called
_flatness_. This point and that we drop any symmetry condition for $\eta$ are
the crucial differences between our approach and that of [3], aside from the
technical variation that we work in the more general setting of
$C^{\ast}$-probability spaces which are not necessarily tracial, whereas [3]
is formulated for tracial $W^{\ast}$-probability spaces.

For the sake of simplicity, we will denote the restriction of $\mu$ to
$B_{\mathrm{sa}}\times\mathcal{P}_{2}(B)$ again by $\mu$. The precise
statements, which constitute the first main result of this paper, read as
follows.

######  Theorem 1.1.

Let $(B,\phi)$ be a $C^{\ast}$-probability space and consider the map
$\mu:B_{\mathrm{sa}}\times\mathcal{P}_{2}(B)\to\operatorname{Prob}(\mathbb{R})$
defined above. Then the following bounds hold true:

  1. (i)

With $c_{1}:=3(\frac{1}{\pi})^{1/3}<2.0484$, we have for each choice of
$\eta\in\mathcal{P}_{2}(B)$ that

$L(\mu_{(b_{0,1},\eta)},\mu_{(b_{0,0},\eta)})\leq
c_{1}\,\|b_{0,1}-b_{0,0}\|^{1/3}\qquad\text{for all $b_{0,0},b_{0,1}\in
B_{\mathrm{sa}}$.}$

  2. (ii)

With $c_{2}:=5(\frac{1}{4\pi})^{2/5}<1.8168$, we have for each choice of
$b_{0}\in B_{\mathrm{sa}}$ that

$L(\mu_{(b_{0},\eta_{1})},\mu_{(b_{0},\eta_{0})})\leq
c_{2}\,\|\eta_{1}-\eta_{0}\|^{1/5}\qquad\text{for all
$\eta_{0},\eta_{1}\in\mathcal{P}_{2}(B)$.}$

In particular, if the Cartesian product
$B_{\mathrm{sa}}\times\mathcal{P}_{2}(B)$ is endowed with any product metric
$d$, then the map
$\mu:(B_{\mathrm{sa}}\times\mathcal{P}_{2}(B),d)\to(\operatorname{Prob}(\mathbb{R}),L)$
is uniformly continuous.

The proof of Theorem 1.1 will be given in Section 5. For both of the Hölder
bounds (i) and (ii), we will consider an affine linear interpolation
$(\rho_{t})_{t\in[0,1]}$ between $\rho_{0}$ and $\rho_{1}$ for which the “time
derivative” of $\mathcal{G}_{\rho_{t}}$ can be controlled. Along the way, we
obtain estimates for the integrals
$\frac{1}{\pi}\int^{\infty}_{-\infty}|\mathcal{G}_{\mu_{\rho_{1}}}(s+i\varepsilon)-\mathcal{G}_{\mu_{\rho_{0}}}(s+i\varepsilon)|\,ds$
for $\varepsilon>0$ which we record in Corollary 5.2.

Theorem 1.1 can be seen as an extension of [3, Proposition 2.3] to the less
restrictive setting of covariance maps that are $2$-positive, where the
proximity of two densities of states is controlled “globally” in terms of the
Lévy distance. Due to its relaxed conditions, our result applies readily to
the limit laws of Kronecker random matrices as discussed in [3, Section 9];
see also [4].

The bounds in Theorem 1.1 (ii) and Corollary 5.2 (ii) appeared already in [12,
Theorem 3.5], but only in the centered case and under much stronger
assumptions regarding $\eta_{0}$ and $\eta_{1}$. While the proof given in [12]
used some “discrete interpolation” stemming from the (noncommutative)
Lindeberg method, we will obtain these results via some “continuous
interpolation” as explained above. For that purpose, it is crucial to
understand how solutions $G_{\eta}$ of the Dyson equation (1.2) depend on
$\eta\in\mathcal{P}_{2}(B)$; this is the content of the next theorem, which is
our second main result. Notice that
$\overline{\mathbb{H}^{+}_{\gamma}(B)}=\\{b\in
B\mid\operatorname{Im}(b)\geq\gamma\mathbf{1}\\}$ for $\gamma>0$.

######  Theorem 1.2.

Let
$G:\mathcal{P}_{2}(B)\times\mathbb{H}^{+}(B)\to\mathbb{H}^{-}(B),(\eta,b)\mapsto
G_{\eta}(b)$ be the function determined by the Dyson equation (1.2). We endow
$\mathcal{P}_{2}(B)\times\mathbb{H}^{+}(B)$ with the product topology. Then:

  1. (i)

For every fixed $\gamma>0$, the family
$G_{\boldsymbol{\cdot}}(b):\mathcal{P}_{2}(B)\to\mathbb{H}^{-}(B),\eta\mapsto
G_{\eta}(b)$ with $b\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$ is
equicontinuous, i.e., we have

(1.3)
$\lim_{\eta\to\eta_{0}}\sup_{b\in\overline{\mathbb{H}^{+}_{\gamma}(B)}}\|G_{\eta}(b)-G_{\eta_{0}}(b)\|=0\qquad\text{for
each $\eta_{0}\in\mathcal{P}_{2}(B)$}.$

In particular,
$G:\mathcal{P}_{2}(B)\times\mathbb{H}^{+}(B)\to\mathbb{H}^{-}(B),(\eta,b)\mapsto
G_{\eta}(b)$ is continuous.

  2. (ii)

For every fixed $\gamma>0$, the family
$(DG_{\boldsymbol{\cdot}})(b):\mathcal{P}_{2}(B)\to\mathcal{L}(B),\eta\mapsto(DG_{\eta})(b)$
with $b\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$ is equicontinuous, i.e., we
have

(1.4)
$\lim_{\eta\to\eta_{0}}\sup_{b\in\overline{\mathbb{H}^{+}_{\gamma}(B)}}\|(DG_{\eta})(b)-(DG_{\eta_{0}})(b)\|=0\qquad\text{for
each $\eta_{0}\in\mathcal{P}_{2}(B)$}.$

In particular,
$DG:\mathcal{P}_{2}(B)\times\mathbb{H}^{+}(B)\to\mathcal{L}(B),(\eta,b)\mapsto(DG_{\eta})(b)$
is continuous.

The proof of Theorem 1.2 will be given in Section 4.

Our third main result given in the following theorem says that the evolution
of solutions of the Dyson equation (1.2) along $C^{1}$-paths
$\eta:[0,T]\to\mathcal{P}_{2}(B)$ can be described by an operator-valued
version of the inviscid Burgers equation (1.6).

######  Theorem 1.3.

Let $\eta:[0,T]\to\mathcal{P}_{2}(B),t\mapsto\eta_{t}$ be a $C^{1}$-path. Let
$G:[0,T]\times\mathbb{H}^{+}(B)\to\mathbb{H}^{-}(B),(t,b)\mapsto G_{t}(b)$ be
the function defined by $G_{t}(b):=G_{\eta_{t}}(b)$ for all $t\in[0,T]$ and
$b\in\mathbb{H}^{+}(B)$. We endow $[0,T]\times\mathbb{H}^{+}(B)$ with the
product topology. Then, for every fixed $\gamma>0$, the family
$G_{\boldsymbol{\cdot}}(b):[0,T]\to\mathbb{H}^{-}(B)$ with
$b\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$ is equidifferentiable, i.e., we
have

(1.5) $\lim_{t\to
t_{0}}\sup_{b\in\overline{\mathbb{H}^{+}_{\gamma}(B)}}\Big{\|}\frac{1}{t-t_{0}}\big{(}G_{t}(b)-G_{t_{0}}(b)\big{)}-\overset{\,{}_{\mbox{\bf.}}}{G}_{t_{0}}(b)\Big{\|}=0\qquad\text{for
each $t_{0}\in[0,T]$}.$

Moreover, $G$ satisfies the partial differential equation

(1.6)
$\overset{\,{}_{\mbox{\bf.}}}{G}_{t}(b)=-(DG_{t})(b)\big{(}\overset{\,{}_{\mbox{\bf.}}}{\eta}_{t}(G_{t}(b))\big{)}\qquad\text{for
$(t,b)\in[0,T]\times\mathbb{H}^{+}(B)$}.$

In particular,
$\overset{\,{}_{\mbox{\bf.}}}{G}:[0,T]\times\mathbb{H}^{+}(B)\to
B,(t,b)\mapsto\overset{\,{}_{\mbox{\bf.}}}{G}_{t}(b)$ is continuous and
$\overset{\,{}_{\mbox{\bf.}}}{G}_{t}:\mathbb{H}^{+}(B)\to B$ is holomorphic
for every fixed $t\in[0,T]$.

In the scalar-valued framework of [58], Voiculescu derived the inviscid
Burgers equation as the free probability counterpart of the heat equation. An
operator-valued generalization of his result (see also [60]) was obtained by
Anshelevich, Belinschi, Fevrier, and Nica in [5], but under more restrictive
assumptions. We prove Theorem 1.3 in Section 4.

The rest of this paper is organized as follows.

In Section 2, we set up the notation and we recall some fundamental results
from different areas that will be used in the subsequent parts; concretely,
Section 2.1 is about positive and completely positive maps on
$C^{\ast}$-algebras, Section 2.2 gives a brief survey on holomorphic functions
between (subsets of) Banach spaces, Section 2.3 focuses on Cauchy transforms
and their characterization, and Section 2.4 recalls the definition of the Lévy
distance and explains how this quantity can be controlled in terms of the
respective Cauchy transforms.

Section 3 has two purposes. On the one hand, we want to describe the
construction of the density of states
$\mu:B_{\mathrm{sa}}\times\mathcal{P}(B)\to\operatorname{Prob}(\mathbb{R})$,
which is the main object of this paper. In doing so, we follow [3] but we
generalize their construction to $C^{\ast}$-probability spaces $(B,\phi)$,
where $\phi$ does not need to be tracial, and to positive maps $\eta$ defined
on $B$, which are not necessarily selfadjoint with respect to the
distinguished state $\phi$. The definition of the density of states
$\mu:B_{\mathrm{sa}}\times\mathcal{P}(B)\to\operatorname{Prob}(\mathbb{R})$
will be given in Section 3.2, while Section 3.1 provides the results about the
Dyson equation which justify this construction; as for the latter, we build on
[31] but we add some explanation why solutions of the Dyson equation are
holomorphic (and in fact uniformly Fréchet differentiable on
$\overline{\mathbb{H}^{+}_{\gamma}(B)}$ for every $\gamma>0$ as shown in Lemma
3.3). On the other hand, we undertake in Section 3 a detailed study of
properties of solutions of the Dyson equation, especially in the case of
$2$-positive maps $\eta$. To this end, we consider in Section 3.3
amplifications of the Dyson equation. More precisely, we show in Proposition
3.7 that for an $n$-positive map $\eta:B\to B$ the solutions
$(G_{\eta^{(k)}})_{k=1}^{n}$ for the amplifications $\eta^{(k)}:M_{k}(B)\to
M_{k}(B)$ for $k=1,\dots,n$ glue to a noncommutative function of height $n$;
this terminology is outlined in the Appendix A. In Section 3.4, we focus on
$2$-positive linear maps $\eta$ and derive in Lemma 3.8 with the help of
Proposition 3.7 bounds for the Fréchet derivative $DG_{\eta}$ of $G_{\eta}$,
which enable us to establish Lipschitz continuity of
$G_{\eta}:\mathbb{H}^{+}(B)\to\mathbb{H}^{-}(B)$ and of
$DG_{\eta}:\mathbb{H}^{+}(B)\to\mathcal{L}(B)$ on each
$\overline{\mathbb{H}^{+}_{\gamma}(B)}$ for $\gamma>0$; see Lemma 3.9 and
Lemma 3.10.

Section 4 is devoted to the proofs of Theorem 1.2 and of Theorem 1.3. Finally,
with the help of these results, the proof of Theorem 1.1 will be given in
Section 5.

##  2\. Preliminaries

###  2.1. Positive and completely positive maps

For complex Banach spaces $(E,\|\cdot\|_{E})$ and $(F,\|\cdot\|_{F})$, we
denote by $\mathcal{L}(E,F)$ the Banach space of all bounded linear operators
$T$ from $E$ to $F$ with the norm given by $\|T\|:=\sup_{\|x\|_{E}\leq
1}\|Tx\|_{F}$. We abbreviate $\mathcal{L}(E):=\mathcal{L}(E,E)$.

For a unital $C^{\ast}$-algebra $B$, whose unit we denote by $\mathbf{1}$, we
let $B_{\mathrm{sa}}:=\\{b\in B\mid b^{\ast}=b\\}$ be the real subspace of $B$
consisting of all selfadjoint elements and we let $B_{+}:=\\{b\in B\mid b\geq
0\\}$ be the convex cone of all positive elements in $B$. For every
$k\in\mathbb{N}$, we denote by $M_{k}(B)$ the unital $C^{\ast}$-algebra of all
$k\times k$ matrices over $B$; the unit in $M_{k}(B)$ will be denoted by
$\mathbf{1}_{k}$.

A map $\eta:B\to B$ is said to be _positive_ if it is $\mathbb{C}$-linear and
maps $B_{+}$ into itself. Note that every such $\eta$ is automatically
bounded, i.e., it belongs to $\mathcal{L}(B)$; in fact, we have
$\|\eta\|=\|\eta(\mathbf{1})\|$ (see, for instance, Corollary 2.9 in [43]).
The set of all positive maps $\eta:B\to B$, which will be denoted by
$\mathcal{P}(B)$, thus forms a convex subset of $\mathcal{L}(B)$. Further, for
every $k\in\mathbb{N}$, we denote by $\mathcal{P}_{k}(B)$ the subset of
$\mathcal{P}(B)$ consisting of those $\eta$ which are _$k$ -positive_, i.e.,
for which the amplification $\eta^{(k)}:M_{k}(B)\to M_{k}(B)$, given by
$\eta^{(k)}(b):=(\eta(b_{ij}))_{i,j=1}^{k}$ for every matrix
$b=(b_{ij})_{i,j=1}^{k}\in M_{k}(B)$, are positive in the aforementioned
sense. Clearly,

$\mathcal{P}(B)=\mathcal{P}_{1}(B)\supseteq\dots\supseteq\mathcal{P}_{k}(B)\supseteq\mathcal{P}_{k+1}(B)\supseteq\dots\supseteq\mathcal{P}_{\infty}(B),$

where $\mathcal{P}_{\infty}(B)$ denotes the subset of $\mathcal{P}(B)$
consisting of those $\eta\in\mathcal{P}(B)$ which are _completely positive_ ,
i.e., which belong to $\mathcal{P}_{k}(B)$ for every $k\in\mathbb{N}$; in
other words, $\mathcal{P}_{\infty}(B):=\bigcap_{k\geq 1}\mathcal{P}_{k}(B)$.
Notice that $\mathcal{P}_{k}(B)$, for every $k\in\mathbb{N}$, and likewise
$\mathcal{P}_{\infty}(B)$ form convex subsets of $\mathcal{L}(B)$.

As shown in [18], each of the inclusions in the above chain can be strict
depending on $B$; more precisely, [18, Theorem 1] says that we have
$\mathcal{P}_{n-1}(M_{n}(\mathbb{C}))\supsetneq\mathcal{P}_{n}(M_{n}(\mathbb{C}))$.
On the other hand, [18, Theorem 5] tells us that
$\mathcal{P}_{n}(M_{n}(\mathbb{C}))=\mathcal{P}_{\infty}(M_{n}(\mathbb{C}))$.
In particular, already on $M_{3}(\mathbb{C})$ there are $2$-positive maps
which are not completely positive; a concrete example that was given in [18]
is $\eta:M_{3}(\mathbb{C})\to M_{3}(\mathbb{C})$ defined by
$\eta(b):=2\operatorname{Tr}_{3}(b)\mathbf{1}_{3}-b$. In view of these facts,
our results on the operator-valued inviscid Burgers equation provide a
significant extension of the previous work done in this direction.

###  2.2. Holomorphic functions between Banach spaces

In this section, we give a survey on notions of holomorphy for functions
between subsets of complex Banach spaces and we briefly recall some of their
properties which will be used in the sequel. For more details, we refer the
interested readers to the monographs [33, 21].

In the following, let $(E,\|\cdot\|_{E})$ and $(F,\|\cdot\|_{F})$ be complex
Banach spaces.

Recall that a function $f:U\to F$ on an open subset $\emptyset\neq U\subseteq
E$ is said to be _Gâteaux holomorphic on $U$_ if the limit, called the
_Gâteaux derivative_ of $f$ at $x$ in the direction $h$,

$\delta f(x;h):=\lim_{\begin{subarray}{c}z\rightarrow 0\\\ z\in
U(x;h)\backslash\\{0\\}\end{subarray}}\frac{1}{z}(f(x+zh)-f(x))$

exists for all $x\in U$ and $h\in E$, where $U(x;h):=\\{z\in\mathbb{C}\mid
x+zh\in U\\}$. According to [21, Lemma 3.3], a function $f:U\to F$ is Gâteaux
holomorphic if and only if, for every choice of $x\in U$ and $h\in E$ and each
bounded linear functional $\phi:F\to\mathbb{C}$, the $\mathbb{C}$-valued
function $U(x;h)\ni z\mapsto\phi(f(x+zh))$ is holomorphic in the sense of one-
variable complex analysis.

Further, we recall that a function $f:U\to F$ is said to be _Fréchet
holomorphic on $U$_, if

  1. (i)

it is Gâteaux holomorphic on $U$,

  2. (ii)

its Gâteaux derivative $\delta f(x;\cdot)$ at any point $x\in U$ is a bounded
linear operator, and

  3. (iii)

$\displaystyle\lim_{h\rightarrow 0}\frac{1}{\|h\|_{E}}\|f(x+h)-f(x)-\delta
f(x;h)\|_{F}=0$.

In this case, we write $Df(x)$ for the bounded linear operator given by the
Gâteaux derivative $\delta f(x;\cdot)$ and call it the _Fréchet derivative_ of
$f$ at the point $x\in U$. It can be shown (see (2.3) in [66]) that $\delta
f(x;\cdot)$ is always linear; therefore, condition (ii) above reduces to the
requirement that $\delta f(x;\cdot)$ is bounded. A result of Zorn [67] says
that condition (iii) is redundant.

It is an immediate consequence of the definitions that every Fréchet
holomorphic function $f:U\to F$ is Gâteaux holomorphic and continuous. In
particular, every Fréchet holomorphic function is holomorphic in the following
sense (see [29]): a function $f:U\to F$ is said to be _holomorphic_ if it is
Gâteaux holomorphic on $U$ and _locally bounded on $U$_, i.e., for every
$x_{0}\in U$ there exists $r=r(x_{0})>0$ such that

$\sup_{x\in U\colon\|x-x_{0}\|_{E}<r}\|f(x)\|_{F}<\infty.$

Remarkably, it turns out that conversely every holomorphic function $f:U\to F$
necessarily is Fréchet holomorphic; see [33, Theorem 3.17.1]. In other words,
holomorphy and Fréchet holomorphy are equivalent concepts. The following lemma
is a slight generalization of some key arguments in the proof of the
aforementioned fact, which we record for later use. To shorten the notation,
we set $\mathbb{B}_{E}(x_{0},r):=\\{x\in E\mid\|x-x_{0}\|_{E}<r\\}$ for
$x_{0}\in E$ and $r>0$.

######  Lemma 2.1.

Let $f:U\to F$ be a Gâteaux holomorphic function. Suppose that $x_{0}\in U$
and $r>0$ are such that $\mathbb{B}_{E}(x_{0},r)\subseteq U$ and
$M_{f}(x_{0},r):=\sup_{x\in\mathbb{B}_{E}(x_{0},r)}\|f(x)\|_{F}<\infty$. Then
the following statements hold true:

  1. (i)

For every $h\in E$, we have that $\|\delta
f(x_{0};h)\|_{F}\leq\frac{1}{r}M_{f}(x_{0},r)\|h\|_{E}$.

  2. (ii)

For $h\in\mathbb{B}_{E}(0,r)$, we have $\|f(x_{0}+h)-f(x_{0})-\delta
f(x_{0};h)\|_{F}\leq\frac{M_{f}(x_{0},r)}{r(r-\|h\|_{E})}\|h\|_{E}^{2}.$

The different characterizations of holomorphy are useful to carry over results
from one-variable complex analysis. The next lemma, for instance, is a
consequence of Vitali’s theorem; it is contained in [33, Theorem 3.18.1], but
since the proof simplifies significantly in the particular situation that we
are considering here, we include the argument for the sake of clarity. Recall
that a family $\mathcal{F}$ of functions $f:U\to F$ is called _locally
uniformly bounded_ if for every $x_{0}\in U$ there exists $r=r(x_{0})>0$ such
that

$\sup_{f\in\mathcal{F}}\ \sup_{x\in
U\colon\|x-x_{0}\|_{E}<r}\|f(x)\|_{F}<\infty.$

######  Lemma 2.2.

Let $(f_{n})_{n=1}^{\infty}$ be a sequence of Gâteaux holomorphic functions
$f_{n}:U\to F$ which is locally uniformly bounded and pointwise convergent to
a function $f:U\to F$. Then $f$ is (Fréchet) holomorphic.

###### Proof.

Let us fix $x\in U$, $h\in E$, as well as some bounded linear functional
$\phi:F\to\mathbb{C}$. Since each $f_{n}$ is Gâteaux holomorphic on $U$, we
obtain holomorphic functions $g_{n}:U(x;h)\to\mathbb{C}$ by
$g_{n}(z):=\phi(f_{n}(x+zh))$; since $(f_{n})_{n=1}^{\infty}$ is locally
uniformly bounded on $U$ and converges pointwise on $U$ to $f$, we infer that
$(g_{n})_{n=1}^{\infty}$ is locally uniformly bounded on $U(x;h)$ and
pointwise convergent on $U(x;h)$ to the function $g:U(x;h)\to\mathbb{C}$ which
is given by $g(z):=\phi(f(x+zh))$. By Vitali’s theorem, it follows that
$(g_{n})_{n=1}^{\infty}$ converges locally uniformly to $g$, so that $g$ must
be holomorphic. This verifies that $f$ is Gâteaux holomorphic on $U$. Because
$(f_{n})_{n=1}^{\infty}$ is locally uniformly bounded and pointwise convergent
to $f$ by assumption, it follows that $f$ is locally bounded. In summary, we
have that $f$ is (Fréchet) holomorphic. ∎

###  2.3. Cauchy transforms

We denote by $\operatorname{Prob}(\mathbb{R})$ the set of all Borel
probability measures on the real line $\mathbb{R}$. For every
$\mu\in\operatorname{Prob}(\mathbb{R})$, let $\mathcal{G}_{\mu}$ be the
_Cauchy transform_ of $\mu$, i.e., the holomorphic function

$\mathcal{G}_{\mu}:\ \mathbb{C}^{+}\to\mathbb{C}^{-},\quad
z\mapsto\int_{\mathbb{R}}\frac{1}{z-t}\,d\mu(t)$

which is defined on the complex upper half-plane
$\mathbb{C}^{+}:=\\{z\in\mathbb{C}\mid\operatorname{Im}(z)>0\\}$ and takes its
values in the complex lower half-plane
$\mathbb{C}^{-}:=\\{z\in\mathbb{C}\mid\operatorname{Im}(z)<0\\}$.

For later use, we recall the following well-known characterization of Cauchy
transforms. We refer the readers to [25] for an excellent survey on this
subject; in particular, the statement given below is [25, Lemma 2], except
that the authors work with the _Stieltjes transform_
$\mathcal{S}_{\mu}:\mathbb{C}^{+}\to\mathbb{C}^{+}$ instead of the Cauchy
transform $\mathcal{G}_{\mu}:\mathbb{C}^{+}\to\mathbb{C}^{-}$, which differ
only by sign.

######  Theorem 2.3.

Let $\mathcal{G}:\mathbb{C}^{+}\to\mathbb{C}$ be holomorphic. The following
statements are equivalent:

  1. (i)

$\mathcal{G}$ is the Cauchy transform of a Borel probability measure on
$\mathbb{R}$, i.e., there exists some $\mu\in\operatorname{Prob}(\mathbb{R})$
such that $\mathcal{G}(z)=\mathcal{G}_{\mu}(z)$ for all $z\in\mathbb{C}^{+}$.

  2. (ii)

$\mathcal{G}$ satisfies the conditions $\operatorname{Im}(\mathcal{G}(z))>0$
for all $z\in\mathbb{C}^{+}$ and $\lim_{y\to\infty}iy\mathcal{G}(iy)=1$.

We notice that $\mu\in\operatorname{Prob}(\mathbb{R})$ is in fact uniquely
determined by its Cauchy transform $\mathcal{G}_{\mu}$; in other words (see
[25, Lemma 1]), for $\mu_{1},\mu_{2}\in\operatorname{Prob}(\mathbb{R})$, we
have that $\mathcal{G}_{\mu_{1}}(z)=\mathcal{G}_{\mu_{2}}(z)$ for all
$z\in\mathbb{C}^{+}$ if and only if $\mu_{1}=\mu_{2}$. In particular, we see
that the measure $\mu$ in Item (i) of Theorem 2.3 is unique.

###  2.4. Lévy and Kolmogorov distance

The Lévy and the Kolmogorov distance yield metrics on
$\operatorname{Prob}(\mathbb{R})$ which are among the most prominent ones
which are usually considered on that space; see, e.g., [17]. In this paper, we
provide bounds for the Lévy distance of densities of states, but we point out
that such bounds often go over to the Kolmogorov distance.

Before giving the definitions of the Lévy and the Kolmogorov distance, we
stipulate that $\mathcal{F}_{\mu}:\mathbb{R}\to[0,1]$ will stand for the
_cumulative distribution function_ of any
$\mu\in\operatorname{Prob}(\mathbb{R})$; recall that $\mathcal{F}_{\mu}$ is
defined by $\mathcal{F}_{\mu}(t):=\mu((-\infty,t])$ for every
$t\in\mathbb{R}$.

If $\mu,\nu\in\operatorname{Prob}(\mathbb{R})$ are given, then their _Lévy
distance_ is defined by

$L(\mu,\nu):=\inf\\{\varepsilon>0\mid\forall t\in\mathbb{R}:\
\mathcal{F}_{\mu}(t-\varepsilon)-\varepsilon\leq\mathcal{F}_{\nu}(t)\leq\mathcal{F}_{\mu}(t+\varepsilon)+\varepsilon\\}$

and their _Kolmogorov distance_ is defined by

$\Delta(\mu,\nu):=\sup_{t\in\mathbb{R}}|\mathcal{F}_{\mu}(t)-\mathcal{F}_{\nu}(t)|.$

It is well-known that the Lévy distance defines a metric on
$\operatorname{Prob}(\mathbb{R})$ which provides a metrization of weak
convergence in the sense that $L(\mu_{n},\mu)\to 0$ if and only if
$(\mu_{n})_{n=1}^{\infty}$ converges weakly to $\mu$.

The Lévy distance is always upper-bounded by the Kolmogorov distance, i.e., we
have $L(\mu,\nu)\leq\Delta(\mu,\nu)$ for all
$\mu,\nu\in\operatorname{Prob}(\mathbb{R})$. If we suppose in addition that
$\nu$ has a cumulative distribution function $\mathcal{F}_{\nu}$ which is
Hölder continuous with exponent $\beta\in(0,1]$ and Hölder constant $C>0$,
i.e., $|\mathcal{F}_{\nu}(t)-\mathcal{F}_{\nu}(s)|\leq C|t-s|^{\beta}$ holds
for all $s,t\in\mathbb{R}$, then [10, Lemma B.18] says that we also have the
complementary bound $\Delta(\mu,\nu)\leq(C+1)L(\mu,\nu)^{\beta}$.

Both the Lévy and the Kolmogorov distance can be controlled in terms of the
Cauchy transforms of the involved measures. For the Kolmogorov distance, such
bounds are given in [8, Theorem 2.2]; see also [9, 10]. These bounds, however,
require some regularity of the cumulative distribution function of at least
one of the considered measures; see [26, 11]. For the Lévy distance, we have
the following statement, by which this quantity becomes accessible via Cauchy
transforms and which remarkably works without any constraints.

######  Theorem 2.4.

Let $\mu,\nu\in\operatorname{Prob}(\mathbb{R})$. Then, for any choice of
$\varepsilon>0$, we have the bound

(2.1) $L(\mu,\nu)\leq
2\sqrt{\frac{\varepsilon}{\pi}}+\frac{1}{\pi}\int^{\infty}_{-\infty}\big{|}\operatorname{Im}(\mathcal{G}_{\mu}(t+i\varepsilon))-\operatorname{Im}(\mathcal{G}_{\nu}(t+i\varepsilon))\big{|}\,dt.$

While this result seems to be well-known, it is surprisingly challenging to
localize it in the literature. The reason is that bounds for $L(\mu,\nu)$ are
mostly given in terms of the characteristic functions of $\mu$ and $\nu$
rather than in terms of their Cauchy transforms; see [17, Section 3]. A
slightly weaker version of Theorem 2.4, namely with an additional factor
$\sqrt{2}$ in front of the term $2\sqrt{\frac{\varepsilon}{\pi}}$, was
obtained recently in [47]. A direct proof of Theorem 2.4 is given in the
appendix of [12]. For convenience of the readers, we include here another
derivation based on the so-called _smoothing inequality_

(2.2) $L(\mu,\nu)\leq
L(\mu\ast\gamma,\nu\ast\gamma)+\max\\{2\delta,1-\mathcal{F}_{\gamma}(\delta)+\mathcal{F}_{\gamma}(-\delta)\\}$

holding for arbitrary $\mu,\nu,\gamma\in\operatorname{Prob}(\mathbb{R})$ and
each $\delta>0$; a proof of (2.2) can be found in [65].

###### Proof of Theorem 2.4.

For $\mu\in\operatorname{Prob}(\mathbb{R})$ and $\varepsilon>0$, we define
$\mu_{\varepsilon}\in\operatorname{Prob}(\mathbb{R})$ by
$d\mu_{\varepsilon}(t)=-\frac{1}{\pi}\operatorname{Im}(\mathcal{G}_{\mu}(t+i\varepsilon))\,dt$.
Note that if we let $\gamma_{\varepsilon}\in\operatorname{Prob}(\mathbb{R})$
be given by
$d\gamma_{\varepsilon}(t):=\frac{1}{\pi}\frac{\varepsilon}{\varepsilon^{2}+t^{2}}\,dt$,
then $\mu_{\varepsilon}=\mu\ast\gamma_{\varepsilon}$ for each $\varepsilon>0$.
Now, since we have

$\mathcal{F}_{\gamma_{\varepsilon}}(-\delta)=\gamma_{\varepsilon}((-\infty,-\delta])=\frac{1}{\pi}\int^{-\delta}_{-\infty}\frac{\varepsilon}{\varepsilon^{2}+t^{2}}\,dt\leq\frac{\varepsilon}{\pi\delta}$

and similarly
$1-\mathcal{F}_{\gamma_{\varepsilon}}(\delta)=\gamma_{\varepsilon}((\delta,\infty))\leq\frac{\varepsilon}{\pi\delta}$,
the smoothing inequality (2.2) yields $L(\mu,\nu)\leq
L(\mu_{\varepsilon},\nu_{\varepsilon})+2\max\\{\delta,\frac{\varepsilon}{\pi\delta}\\}$.
By choosing $\delta=\sqrt{\frac{\varepsilon}{\pi}}$, we obtain that
$L(\mu,\nu)\leq
L(\mu_{\varepsilon},\nu_{\varepsilon})+2\sqrt{\frac{\varepsilon}{\pi}}$. In
combination with the fact that

$L(\mu_{\varepsilon},\nu_{\varepsilon})\leq\frac{1}{\pi}\int^{\infty}_{-\infty}\big{|}\operatorname{Im}(\mathcal{G}_{\mu}(t+i\varepsilon))-\operatorname{Im}(\mathcal{G}_{\nu}(t+i\varepsilon))\big{|}\,dt,$

which is easily verified by straightforward estimates, this yields the
asserted bound (2.1). ∎

##  3\. The Dyson equation and density of states

Throughout the following, let $B$ be a unital $C^{\ast}$-algebra; the unit of
$B$ will be denoted by $\mathbf{1}$. We define $\mathbb{H}^{+}(B)$ and
$\mathbb{H}^{-}(B)$ to be the upper and lower half-plane in $B$, respectively,
that is, $\mathbb{H}^{\pm}(B):=\\{b\in
B\mid\exists\varepsilon>0:\pm\operatorname{Im}(b)\geq\varepsilon\mathbf{1}\\}$
where $\operatorname{Im}(b):=\frac{1}{2i}(b-b^{\ast})$. Notice that $b\in B$
belongs to $\mathbb{H}^{\pm}(B)$ if and only if $\pm\operatorname{Im}(b)$ is
an invertible positive element in $B$; recall that
$b\geq\|b^{-1}\|^{-1}\mathbf{1}$ for every positive $b\in B$ which is
invertible. Further, for every $\gamma>0$, we define
$\mathbb{H}^{+}_{\gamma}(B):=\\{b+i\gamma\mathbf{1}\mid
b\in\mathbb{H}^{+}(B)\\}$. Notice that
$\overline{\mathbb{H}^{+}_{\gamma}(B)}=\\{b\in\mathbb{H}^{+}(B)\mid\operatorname{Im}(b)\geq\gamma\mathbf{1}\\}$.

###  3.1. The Dyson equation

This subsection is devoted to the first step in the construction of the
mapping $\mu$ that we outlined in the introduction. It consists in showing
that there exists for every $\eta\in\mathcal{P}(B)$ a holomorphic function
$G_{\eta}:\mathbb{H}^{+}(B)\to\mathbb{H}^{-}(B)$ uniquely determined among all
functions defined on $\mathbb{H}^{+}(B)$ and taking values in
$\mathbb{H}^{-}(B)$ by the Dyson equation (1.2); note that in the sequel, we
shall refer to (3.1) instead of (1.2). The next theorem, a von Neumann algebra
version of which was used also in [3], gives the precise statement.

######  Theorem 3.1.

Let $\eta:B\to B$ be a positive linear map. Then there exists a unique
function $G_{\eta}:\mathbb{H}^{+}(B)\to\mathbb{H}^{-}(B)$ solving the _Dyson
equation_

(3.1) $bG_{\eta}(b)=\mathbf{1}+\eta(G_{\eta}(b))G_{\eta}(b)\qquad\text{for all
$b\in\mathbb{H}^{+}(B)$}.$

This function $G_{\eta}$ is holomorphic and enjoys the property that

(3.2) $\|G_{\eta}(b)\|\leq\|\operatorname{Im}(b)^{-1}\|\qquad\text{for all
$b\in\mathbb{H}^{+}(B)$}.$

Essentially, this result is proven in [31], except that the authors have not
discussed that the solution of the Dyson equation is holomorphic, because this
observation was irrelevant for their purpose. Here, we supplement that
argument. To this end, we first recall another result from [31] and we take a
closer look at its proof.

######  Theorem 3.2 (Proposition 3.2, [31]).

Let $\eta:B\to B$ be a positive linear map. For every fixed
$b\in\mathbb{H}^{+}(B)$, the equation

(3.3) $bw=\mathbf{1}+\eta(w)w$

has a unique solution $w\in\mathbb{H}^{-}(B)$; this $w$ is the limit of the
sequence $(h_{b}^{\circ n}(w_{0}))_{n=1}^{\infty}$ of iterates of the
holomorphic function

$h_{b}:\ \mathbb{H}^{-}(B)\to\mathbb{H}^{-}(B),\quad w\mapsto(b-\eta(w))^{-1}$

for every choice of an initial point $w_{0}\in\mathbb{H}^{-}(B)$. Moreover, we
have that $\|w\|\leq\|\operatorname{Im}(b)^{-1}\|$.

Notice that our formulation of Theorem 3.2 differs from the original one in
[31] as we prefer to perform the iteration in $\mathbb{H}^{-}(B)$ instead of
the right half-plane of $B$, which would be $\\{b\in
B\mid\exists\varepsilon>0:\operatorname{Re}(b)\geq\varepsilon\mathbf{1}\\}$,
where $\operatorname{Re}(b):=\frac{1}{2}(b+b^{\ast})$.

The proof of Theorem 3.2 given in [31] crucially relies on the Earle-Hamilton
fixed point theorem [23], an important result about fixed points of
holomorphic functions between subsets of complex Banach spaces; we refer the
readers who are interested in this topic also to the excellent exposition
given in [29]. For our purpose, it suffices to recap what the authors of [31]
have proved (in the translation to our setting) in order to apply the Earle-
Hamilton fixed point theorem: not only have they shown that
$h_{b}(w)=(b-\eta(w))^{-1}$ defines a holomorphic function
$h_{b}:\mathbb{H}^{-}(B)\to\mathbb{H}^{-}(B)$ for every
$b\in\mathbb{H}^{+}(B)$, but they also proved that each $h_{b}$ is uniformly
bounded on $\mathbb{H}^{-}(B)$ with
$\sup_{w\in\mathbb{H}^{-}(B)}\|h_{b}(w)\|\leq\|\operatorname{Im}(b)^{-1}\|$.
With these insights, we are ready to return to Theorem 3.1.

###### Proof of Theorem 3.1.

By assigning to $b\in\mathbb{H}^{+}(B)$ the unique solution
$G_{\eta}(b):=w\in\mathbb{H}^{-}(B)$ of (3.3), we obtain according to Theorem
3.2 a unique function $G_{\eta}:\mathbb{H}^{+}(B)\to\mathbb{H}^{-}(B)$ which
solves the Dyson equation (3.1) and we infer that $G_{\eta}$ satisfies in
addition the bound (3.2). It remains to prove that $G_{\eta}$ is holomorphic.

To this end, we fix any point $w_{0}\in\mathbb{H}^{-}(B)$ and use it to define
a sequence $(G_{n})_{n=1}^{\infty}$ of functions
$G_{n}:\mathbb{H}^{+}(B)\to\mathbb{H}^{-}(B)$ by $G_{n}(b):=h_{b}^{\circ
n}(w_{0})$ for all $b\in\mathbb{H}^{+}(B)$; inductively, one easily sees that
each $G_{n}$ is Gâteaux holomorphic. Theorem 3.2 tells us that the sequence
$(G_{n})_{n=1}^{\infty}$ converges pointwise on $\mathbb{H}^{+}(B)$ to
$G_{\eta}$. Therefore, Lemma 2.2 will imply that $G_{\eta}$ is holomorphic
once we have verified that $(G_{n})_{n=1}^{\infty}$ is locally uniformly
bounded.

This can be done as follows. Thanks to the uniform boundedness of $h_{b}$, we
have $\|G_{n}(b)\|=\|h_{b}^{\circ
n}(w_{0})\|\leq\|\operatorname{Im}(b)^{-1}\|$ for all $n\in\mathbb{N}$ and
each $b\in\mathbb{H}^{+}(B)$. This shows that the sequence
$(G_{n})_{n=1}^{\infty}$ is uniformly bounded on
$\overline{\mathbb{H}^{+}_{\gamma}(B)}$ for every $\gamma>0$ and hence locally
uniformly bounded on $\mathbb{H}^{+}(B)$ since we have
$\mathbb{H}^{+}(B)=\bigcup_{\gamma>0}\overline{\mathbb{H}^{+}_{\gamma}(B)}$. ∎

Since holomorphy entails Fréchet differentiability, we have the Fréchet
derivative $DG_{\eta}:\mathbb{H}^{+}(B)\to\mathcal{L}(B)$ of $G_{\eta}$ at our
disposal. By definition, it satisfies for every $b\in\mathbb{H}^{+}(B)$

$\lim_{h\rightarrow
0}\frac{1}{\|h\|}\|G_{\eta}(b+h)-G_{\eta}(b)-(DG_{\eta})(b)h\|=0.$

For later use, we record with the following lemma some strengthening of the
latter condition.

######  Lemma 3.3.

Let $\eta:B\to B$ be a positive linear map and let
$G_{\eta}:\mathbb{H}^{+}(B)\to\mathbb{H}^{-}(B)$ be the function uniquely
determined by the Dyson equation (3.1) according to Theorem 3.1. Then the
Fréchet derivative $DG_{\eta}:\mathbb{H}^{+}(B)\to\mathcal{L}(B)$ of
$G_{\eta}$ satisfies for each $\gamma>0$

$\lim_{h\rightarrow
0}\frac{1}{\|h\|}\sup_{b\in\overline{\mathbb{H}^{+}_{\gamma}(B)}}\|G_{\eta}(b+h)-G_{\eta}(b)-(DG_{\eta})(b)h\|=0.$

Before giving the proof of Lemma 3.3, we first introduce some notation. For
$b_{0}\in\mathbb{H}^{+}(B)$ and $\sigma\in(0,1)$, we set
$\mathbb{D}(b_{0},\sigma):=\\{b\in
B\mid\|b-b_{0}\|<\sigma\|\operatorname{Im}(b_{0})^{-1}\|^{-1}\\}$. Notice that

(3.4)
$\mathbb{D}(b_{0},\sigma)\subset\mathbb{H}^{+}_{\gamma}(B)\qquad\text{for}\qquad\gamma:=(1-\sigma)\|\operatorname{Im}(b_{0})^{-1}\|^{-1}.$

Indeed, if any $b\in\mathbb{D}(b_{0},\sigma)$ is given, we may check for every
state $\phi$ on $B$ that

$\phi(\operatorname{Im}(b))=\phi(\operatorname{Im}(b_{0}))+\phi(\operatorname{Im}(b)-\operatorname{Im}(b_{0}))\\\
\geq\|\operatorname{Im}(b_{0})^{-1}\|^{-1}-\|b-b_{0}\|>(1-\sigma)\|\operatorname{Im}(b_{0})^{-1}\|^{-1}=\gamma,$

where we used that
$\operatorname{Im}(b_{0})\geq\|\operatorname{Im}(b_{0})^{-1}\|^{-1}\mathbf{1}$;
this shows $b\in\mathbb{H}^{+}_{\gamma}(B)$, as claimed.

###### Proof of Lemma 3.3.

Let $\gamma>0$ be given and choose $\sigma\in(0,1)$. Take any
$b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$. Since
$\|\operatorname{Im}(b_{0})^{-1}\|^{-1}\geq\gamma$, we get from (3.4) that
$\mathbb{B}_{B}(b_{0},\sigma\gamma)\subseteq\mathbb{D}(b_{0},\sigma)\subset\mathbb{H}^{+}_{(1-\sigma)\gamma}(B)$,
and (3.2) yields
$\sup_{b\in\overline{\mathbb{B}_{B}(b_{0},\sigma\gamma)}}\|G_{\eta}(b)\|\leq\frac{1}{(1-\sigma)\gamma}$.
Using this fact, we deduce from Lemma 2.1 (ii) that

$\|G_{\eta}(b_{0}+h)-G_{\eta}(b_{0})-(DG_{\eta})(b_{0})h\|\leq\frac{1}{(1-\sigma)\sigma\gamma^{2}(\sigma\gamma-\|h\|)}\|h\|^{2}$

for each $h\in B$ which satisfies $\|h\|<\sigma\gamma$. Consequently, as
$b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$ was arbitrary,

$\frac{1}{\|h\|}\sup_{b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}}\|G_{\eta}(b_{0}+h)-G_{\eta}(b_{0})-(DG_{\eta})(b_{0})h\|\leq\frac{1}{(1-\sigma)\sigma\gamma^{2}(\sigma\gamma-\|h\|)}\|h\|$

whenever $h\in B$ satisfies $0<\|h\|<\sigma\gamma$. This yields the assertion.
∎

###  3.2. Density of states

Based on the results about the Dyson equation obtained in the previous
section, we finally can construct the map $\mu$ which we announced in the
introduction.

We suppose that $(B,\phi)$ is a $C^{\ast}$-probability space, i.e., a
$C^{\ast}$-algebra $B$ with unit $\mathbf{1}$ which is endowed with a
distinguished state $\phi$.

######  Definition 3.4.

Let any data pair $\rho=(b_{0},\eta)\in B_{\mathrm{sa}}\times\mathcal{P}(B)$
be given. Consider the holomorphic function
$G_{\eta}:\mathbb{H}^{+}(B)\to\mathbb{H}^{-}(B)$ associated with $\eta$ by
Theorem 3.1. Then $\mathcal{G}_{\rho}(z):=\phi(G_{\eta}(z\mathbf{1}-b_{0}))$
for $z\in\mathbb{C}^{+}$ defines a holomorphic function
$\mathcal{G}_{\rho}:\mathbb{C}^{+}\to\mathbb{C}^{-}$ with the property
$\lim_{y\to\infty}iy\mathcal{G}_{\rho}(iy)=1$ and is thus, by Theorem 2.3, the
Cauchy transform of a Borel probability measure $\mu_{\rho}$ on the real line
$\mathbb{R}$; more explicitly, we have that

$\phi(G_{\eta}(z\mathbf{1}-b_{0}))=\int_{\mathbb{R}}\frac{1}{z-t}\,d\mu_{\rho}(t)\qquad\text{for
all $z\in\mathbb{C}^{+}$},$

which determines $\mu_{\rho}$ uniquely. We call $\mu_{\rho}$ the _density of
states_ associated with the data pair $\rho$. In this way, we obtain a mapping

$\mu:\
B_{\mathrm{sa}}\times\mathcal{P}(B)\to\operatorname{Prob}(\mathbb{R}),\quad\rho\mapsto\mu_{\rho}.$

Notice that $\lim_{y\to\infty}iy\mathcal{G}_{\rho}(iy)=1$ follows easily from
(3.1) with the help of (3.2).

Definition 3.4 extends [3, Definition 2.2] to the setting of
$C^{\ast}$-probability spaces which are not necessarily tracial and removes
any additional assumptions on the positive map.

We emphasize that $\mu_{\rho}$, despite its name, may very well possess an
atomic part and hence can fail to have a density with respect to the Lebesgue
measure on $\mathbb{R}$. To ensure that $\mu_{\rho}$ enjoys regularity
properties of this kind, one needs to impose further conditions on $\rho$;
see, for instance, [3, 39].

###  3.3. Matricial amplifications of the Dyson equation

Theorem 3.1 characterizes $G_{\eta}$ as the unique function on
$\mathbb{H}^{+}(B)$ with values in $\mathbb{H}^{-}(B)$ which solves the Dyson
equation (3.1). In order to further analyze $G_{\eta}$, it is helpful to take
also matricial amplifications $\eta^{(n)}$ of $\eta$ into consideration,
especially for $n=2$. For this purpose, we need a better understanding of the
structure of $\mathbb{H}^{+}(M_{2}(B))$; this goal is achieved by the next
lemma.

######  Lemma 3.5.

Let $b_{1},b_{2}\in\mathbb{H}^{+}(B)$ and $w\in B$ be given. Then
$b:=\begin{bmatrix}b_{1}&w\\\ 0&b_{2}\end{bmatrix}\in M_{2}(B)$ belongs to
$\mathbb{H}^{+}(M_{2}(B))$ if and only if

(3.5)
$\|\operatorname{Im}(b_{2})^{-1/2}w^{\ast}\operatorname{Im}(b_{1})^{-1}w\operatorname{Im}(b_{2})^{-1/2}\|<4.$

Consequently, we have the following implications:

  1. (i)

If
$\|w\|^{2}<4\|\operatorname{Im}(b_{1})^{-1}\|^{-1}\|\operatorname{Im}(b_{2})^{-1}\|^{-1}$,
then $b\in\mathbb{H}^{+}(M_{2}(B))$.

  2. (ii)

If $b\in\mathbb{H}^{+}(M_{2}(B))$, then
$\|w\|^{2}<4\|\operatorname{Im}(b_{1})\|\|\operatorname{Im}(b_{2})\|$.

The proof of Lemma 3.5 relies substantially on the following general facts.

######  Lemma 3.6.

Let $B$ be a $C^{\ast}$-algebra with unit $\mathbf{1}$. For each $v\in B$, we
have that

  1. (i)

$\begin{bmatrix}\mathbf{1}&v\\\ v^{\ast}&\mathbf{1}\end{bmatrix}\in M_{2}(B)$
is positive if and only if $\|v\|\leq 1$;

  2. (ii)

$\begin{bmatrix}\mathbf{1}&v\\\ v^{\ast}&\mathbf{1}\end{bmatrix}\in M_{2}(B)$
is positive and invertible if and only if $\|v\|<1$.

###### Proof.

The first equivalence (i) is [43, Lemma 3.1 (i)]; a direct proof of (ii) was
given in [14, Section 2.1]. We note that (ii) can also be deduced from (i);
indeed, for any $\varepsilon\in(0,1)$, it gives that
$\begin{bmatrix}\mathbf{1}&v\\\
v^{\ast}&\mathbf{1}\end{bmatrix}\geq\varepsilon\mathbf{1}_{2},$ or
equivalently $\begin{bmatrix}\mathbf{1}&\frac{1}{1-\varepsilon}v\\\
\frac{1}{1-\varepsilon}v^{\ast}&\mathbf{1}\end{bmatrix}\geq 0,$ if and only if
$\|v\|\leq 1-\varepsilon$. ∎

###### Proof of Lemma 3.5.

The proof of the first assertion is basically given in the discussion at the
end of Section 2.1 in [14]; for the readers’ convenience, we include the
argument, adapted to our situation. First of all, we notice that
$b\in\mathbb{H}^{+}(M_{2}(B))$ if and only if the matrix

$\operatorname{Im}(b)=\operatorname{Im}\bigg{(}\begin{bmatrix}b_{1}&w\\\
0&b_{2}\end{bmatrix}\bigg{)}=\begin{bmatrix}\operatorname{Im}(b_{1})&\frac{1}{2i}w\\\
-\frac{1}{2i}w^{\ast}&\operatorname{Im}(b_{2})\end{bmatrix}$

is positive and invertible. Since $b_{1},b_{2}\in\mathbb{H}^{+}(B)$ by
assumption, both $\operatorname{Im}(b_{1})$ and $\operatorname{Im}(b_{2})$ are
positive and invertible; thus, the operator $\operatorname{Im}(b)$ is positive
and invertible if and only if

$\begin{bmatrix}\operatorname{Im}(b_{1})^{-1/2}&0\\\
0&\operatorname{Im}(b_{2})^{-1/2}\end{bmatrix}\operatorname{Im}(b)\begin{bmatrix}\operatorname{Im}(b_{1})^{-1/2}&0\\\
0&\operatorname{Im}(b_{2})^{-1/2}\end{bmatrix}=\begin{bmatrix}\mathbf{1}&v\\\
v^{\ast}&\mathbf{1}\end{bmatrix},$

with
$v:=\frac{1}{2i}\operatorname{Im}(b_{1})^{-1/2}w\operatorname{Im}(b_{2})^{-1/2}$,
is positive and invertible. With the aid of Lemma 3.6 (ii), we infer that
$\operatorname{Im}(b)$ is positive and invertible if and only if

$\frac{1}{4}\|\operatorname{Im}(b_{2})^{-1/2}w^{\ast}\operatorname{Im}(b_{1})^{-1}w\operatorname{Im}(b_{2})^{-1/2}\|=\|v^{\ast}v\|=\|v\|^{2}<1,$

which is the statement we wanted to prove.

Having established that $b\in\mathbb{H}^{+}(M_{2}(B))$ holds true if and only
if condition (3.5) is satisfied, we can deduce the additional assertions:

Suppose that
$\|w\|^{2}<4\|\operatorname{Im}(b_{1})^{-1}\|^{-1}\|\operatorname{Im}(b_{2})^{-1}\|^{-1}$
is satisfied. Since
$\operatorname{Im}(b_{i})^{-1}\leq\|\operatorname{Im}(b_{i})^{-1}\|\mathbf{1}$
for $i=1,2$, we get in this case that

$\|\operatorname{Im}(b_{2})^{-1/2}w^{\ast}\operatorname{Im}(b_{1})^{-1}w\operatorname{Im}(b_{2})^{-1/2}\|\leq\|w\|^{2}\|\operatorname{Im}(b_{1})^{-1}\|\|\operatorname{Im}(b_{2})^{-1}\|<4,$

Thus, (3.5) is satisfied and hence we have $b\in\mathbb{H}^{+}(M_{2}(B))$;
this proves (i).

In order to prove (ii), we suppose that $b\in\mathbb{H}^{+}(M_{2}(B))$. First,
we notice that
$\|w\|\leq\|w\operatorname{Im}(b_{2})^{-1/2}\|\|\operatorname{Im}(b_{2})\|^{1/2}$
since $\|\operatorname{Im}(b_{2})^{1/2}\|=\|\operatorname{Im}(b_{2})\|^{1/2}$
by the $C^{\ast}$-identity. Then, we use
$\mathbf{1}\leq\|\operatorname{Im}(b_{1})\|\operatorname{Im}(b_{1})^{-1}$ to
infer from the latter that

$\displaystyle\|w\|^{2}$
$\displaystyle\leq\|w\operatorname{Im}(b_{2})^{-1/2}\|^{2}\|\operatorname{Im}(b_{2})\|$
$\displaystyle=\|\operatorname{Im}(b_{2})^{-1/2}w^{\ast}w\operatorname{Im}(b_{2})^{-1/2}\|\|\operatorname{Im}(b_{2})\|$
$\displaystyle\leq\|\operatorname{Im}(b_{1})\|\|\operatorname{Im}(b_{2})\|\|\operatorname{Im}(b_{2})^{-1/2}w^{\ast}\operatorname{Im}(b_{1})^{-1}w\operatorname{Im}(b_{2})^{-1/2}\|.$

Involving (3.5), which holds since we have $b\in\mathbb{H}^{+}(M_{2}(B))$ by
assumption, we conclude that
$\|w\|^{2}<4\|\operatorname{Im}(b_{1})\|\|\operatorname{Im}(b_{2})\|$, as
desired. ∎

The next proposition provides a strengthening of Theorem 3.1 for $n$-positive
maps $\eta:B\to B$. In such cases, the Dyson equation uniquely determines a
noncommutative function of height $n$, in the terminology that we explain in
the Appendix A; see also Proposition A.2.

######  Proposition 3.7.

Let $\eta:B\to B$ be an $n$-positive linear map. Consider the noncommutative
sets of height $n$ which are given by $\mathbb{H}^{\pm}(M_{\leq
n}(B)):=\coprod_{k=1}^{n}\mathbb{H}^{\pm}(M_{k}(B))$. Then there exists a
unique noncommutative function $G^{(\leq n)}_{\eta}:\mathbb{H}^{+}(M_{\leq
n}(B))\to\mathbb{H}^{-}(M_{\leq n}(B))$ of height $n$ such that, for each
$k=1,\dots,n$, the $k$-th level function
$G_{\eta}^{(k)}:\mathbb{H}^{+}(M_{k}(B))\to\mathbb{H}^{-}(M_{k}(B))$ satisfies

(3.6)
$bG_{\eta}^{(k)}(b)=\mathbf{1}_{k}+\eta^{(k)}(G_{\eta}^{(k)}(b))G_{\eta}^{(k)}(b)\qquad\text{for
all $b\in\mathbb{H}^{+}(M_{k}(B))$}.$

In particular, if $n\geq 2$, then for every $b\in\mathbb{H}^{+}(B)$ and each
$w\in B$ satisfying $\begin{bmatrix}b&w\\\
0&b\end{bmatrix}\in\mathbb{H}^{+}(M_{2}(B))$, we have that

(3.7) $G^{(2)}_{\eta}\bigg{(}\begin{bmatrix}b&w\\\
0&b\end{bmatrix}\bigg{)}=\begin{bmatrix}G_{\eta}(b)&(DG_{\eta})(b)w\\\
0&G_{\eta}(b)\end{bmatrix}.$

###### Proof.

For $k=1,\dots,n$, because $\eta^{(k)}:M_{k}(B)\to M_{k}(B)$ is positive,
Theorem 3.1 guarantees the existence of a unique function
$G_{\eta}^{(k)}:\mathbb{H}^{+}(M_{k}(B))\to\mathbb{H}^{-}(M_{k}(B))$
satisfying (3.6); in fact, we have that $G_{\eta}^{(k)}=G_{\eta^{(k)}}$. Thus,
in order to establish the existence of a unique noncommutative function
$G^{(\leq n)}_{\eta}:\mathbb{H}^{+}(M_{\leq n}(B))\to\mathbb{H}^{-}(M_{\leq
n}(B))$ of height $n$ with the properties required in the proposition, it
suffices to show that the aforementioned functions $G_{\eta}^{(k)}$ combine to
a noncommutative function of height $n$. To this end, we check the following:

Whenever $1\leq k,l\leq n$ are such that $k+l\leq n$ and if
$b_{1}\in\mathbb{H}^{+}(M_{k}(B))$ and $b_{2}\in\mathbb{H}^{+}(M_{l}(B))$,
then both $G_{\eta}^{(k+l)}(b_{1}\oplus b_{2})$ and
$G_{\eta}^{(k)}(b_{1})\oplus G_{\eta}^{(l)}(b_{2})$ are in
$\mathbb{H}^{-}(M_{k+l}(B))$ and solve (3.3) for
$\eta^{(k+l)}\in\mathcal{P}(M_{k+l}(B))$ at $b_{1}\oplus
b_{2}\in\mathbb{H}^{+}(M_{k+l}(B))$. By uniqueness,
$G_{\eta}^{(k+l)}(b_{1}\oplus b_{2})=G_{\eta}^{(k)}(b_{1})\oplus
G_{\eta}^{(l)}(b_{2})$.

For $1\leq k\leq n$, if $b\in\mathbb{H}^{+}(M_{k}(B))$ and an invertible
matrix $S\in M_{k}(\mathbb{C})$ are given such that
$SbS^{-1}\in\mathbb{H}^{+}(M_{k}(B))$ holds, then Theorem 3.2 tells us that

$G^{(k)}_{\eta}(b)=\lim_{n\to\infty}h^{\circ
n}_{b}(-i\mathbf{1}_{k})\qquad\text{and}\qquad
G^{(k)}_{\eta}(SbS^{-1})=\lim_{n\to\infty}h^{\circ
n}_{SbS^{-1}}(-i\mathbf{1}_{k}).$

Notice that the functions $h_{b}$ and $h_{SbS^{-1}}$ are defined like in
Theorem 3.2, but with respect to $\eta^{(k)}$ instead of $\eta$; both of them
map $\mathbb{H}^{-}(M_{k}(B))$ into itself. Since for all
$w\in\mathbb{H}^{-}(M_{k}(B))$

$h_{SbS^{-1}}(w)=\big{(}SbS^{-1}-\eta^{(k)}(w)\big{)}^{-1}=S\big{(}b-S^{-1}\eta^{(k)}(w)S\big{)}^{-1}S^{-1}\\\
=S\big{(}b-\eta^{(k)}(S^{-1}wS)\big{)}^{-1}S^{-1}=Sh_{b}(S^{-1}wS)S^{-1},$

we see that $h^{\circ n}_{SbS^{-1}}(-i\mathbf{1}_{k})=S^{-1}h_{b}^{\circ
n}(-i\mathbf{1}_{k})S$ for all $n\in\mathbb{N}$ and so
$G_{\eta}^{(k)}(SbS^{-1})=SG_{\eta}^{(k)}(b)S^{-1}$.

The asserted formula (3.7) follows immediately from Theorem A.1. ∎

###  3.4. Properties of solutions of the Dyson equation for $2$-positive maps

With the aid of Proposition 3.7, we now derive bounds for the Fréchet
derivative of $G_{\eta}$ for $\eta\in\mathcal{P}_{2}(B)$, which are in the
spirit of [14, Proposition 3.1].

######  Lemma 3.8.

Suppose that $\eta\in\mathcal{P}_{2}(B)$. Then, for every
$b\in\mathbb{H}^{+}(B)$,

(3.8) $\|(DG_{\eta})(b)\|=\sup_{w\in B\colon\|w\|\leq
1}\|(DG_{\eta})(b)w\|\leq\|\operatorname{Im}(b)^{-1}\|^{2},$

and moreover, if $\phi$ is any state on $B$, then

(3.9) $\sup_{w\in B\colon\|w\|\leq
1}\big{|}\phi((DG_{\eta})(b)w)\big{|}\leq-\operatorname{Im}(\phi(G_{\eta}(b)))\|\operatorname{Im}(b)^{-1}\|.$

###### Proof.

We take $b\in\mathbb{H}^{+}(B)$ and $w\in B$ such that the condition
$\|w\|\leq 2(1-\varepsilon)\|\operatorname{Im}(b)^{-1}\|^{-1}$ is satisfied
for some $\varepsilon\in(0,1)$. In this situation, Lemma 3.5 (i) tells us that

$\begin{bmatrix}b&w\\\ 0&b\end{bmatrix}\in\mathbb{H}^{+}(M_{2}(B)).$

Since $\eta$ is supposed to be $2$-positive, we may apply Proposition 3.7; the
formula (3.7) yields

(3.10) $G_{\eta}^{(2)}\bigg{(}\begin{bmatrix}b&w\\\
0&b\end{bmatrix}\bigg{)}=\begin{bmatrix}G_{\eta}(b)&(DG_{\eta})(b)w\\\
0&G_{\eta}(b)\end{bmatrix}.$

Since $G_{\eta}^{(2)}$ takes its values in $\mathbb{H}^{-}(M_{2}(B))$, we
conclude from the latter with the help of Lemma 3.5 (ii) that
$\|(DG_{\eta})(b)w\|<2\|\operatorname{Im}(G_{\eta}(b))\|$. By rescaling and
using (3.2), we infer that every $w\in B$ with $\|w\|\leq 1$ satisfies

$\|(DG_{\eta})(b)w\|<\frac{1}{1-\varepsilon}\|\operatorname{Im}(G_{\eta}(b))\|\|\operatorname{Im}(b)^{-1}\|\leq\frac{1}{1-\varepsilon}\|\operatorname{Im}(b)^{-1}\|^{2}.$

By letting $\varepsilon\searrow 0$, we obtain (3.8), the first of the asserted
formulas.

In order to prove (3.9), we proceed as follows. First of all, we deduce from
(3.10) that

$\begin{bmatrix}\operatorname{Im}(G_{\eta}(b))&\frac{1}{2i}\big{(}(DG_{\eta})(b)w\big{)}\\\
-\frac{1}{2i}\big{(}(DG_{\eta})(b)w\big{)}^{\ast}&\operatorname{Im}(G_{\eta}(b))\end{bmatrix}=\operatorname{Im}\bigg{(}\begin{bmatrix}G_{\eta}(b)&(DG_{\eta})(b)w\\\
0&G_{\eta}(b)\end{bmatrix}\bigg{)}\leq 0.$

By applying the amplification $\phi^{(2)}$ of $\phi$ to the latter, we
conclude that the scalar matrix

$\begin{bmatrix}\phi\big{(}\operatorname{Im}(G_{\eta}(b))\big{)}&\frac{1}{2i}\phi\big{(}(DG_{\eta})(b)w\big{)}\\\
-\frac{1}{2i}\overline{\phi\big{(}(DG_{\eta})(b)w\big{)}}&\phi\big{(}\operatorname{Im}(G_{\eta}(b))\big{)}\end{bmatrix}$

is negative semidefinite; hence, we get that
$\frac{1}{2}|\phi((DG_{\eta})(b)w)|\leq-\phi(\operatorname{Im}(G_{\eta}(b)))$.
By rescaling, we infer from the latter that if $w\in B$ is such that
$\|w\|\leq 1$, then

$\big{|}\phi\big{(}(DG_{\eta})(b)w\big{)}\big{|}\leq-\frac{1}{1-\varepsilon}\phi(\operatorname{Im}(G_{\eta}(b)))\|\operatorname{Im}(b)^{-1}\|.$

By letting $\varepsilon\searrow 0$, we get (3.9), the second of the asserted
formulas. ∎

The bound (3.8) can be used to derive, for every $\gamma>0$, Lipschitz
continuity of $G_{\eta}$ on $\overline{\mathbb{H}^{+}_{\gamma}(B)}$ with
Lipschitz constant $\frac{1}{\gamma^{2}}$. We prove a more general result; cf.
[14, Proposition 3.1].

######  Lemma 3.9.

Suppose that $\eta\in\mathcal{P}_{2}(B)$. Then, for all
$b_{0},b_{1}\in\mathbb{H}^{+}(B)$, we have that

(3.11)
$\|G_{\eta}(b_{1})-G_{\eta}(b_{0})\|\leq\|\operatorname{Im}(b_{0})^{-1}\|\|\operatorname{Im}(b_{1})^{-1}\|\|b_{1}-b_{0}\|.$

###### Proof.

Fix $b_{0},b_{1}\in\mathbb{H}^{+}(B)$. We consider the function
$f:[0,1]\to\mathbb{H}^{-}(B)$ which is defined by
$f(t):=G_{\eta}(tb_{1}+(1-t)b_{0})$ for $t\in[0,1]$. Then $f$ is
differentiable as a $B$-valued function with
$f^{\prime}(t)=(DG_{\eta})(tb_{1}+(1-t)b_{0})(b_{1}-b_{0})$. By using the
bound (3.8) provided by Lemma 3.8, we observe that

$\|f^{\prime}(t)\|\leq\big{\|}\big{(}t\operatorname{Im}(b_{1})+(1-t)\operatorname{Im}(b_{0})\big{)}^{-1}\big{\|}^{2}\|b_{1}-b_{0}\|.$

Since
$t\operatorname{Im}(b_{1})+(1-t)\operatorname{Im}(b_{0})\geq(t\|\operatorname{Im}(b_{1})^{-1}\|^{-1}+(1-t)\|\operatorname{Im}(b_{0})^{-1}\|^{-1})\,\mathbf{1}$,
we have

$\big{\|}\big{(}t\operatorname{Im}(b_{1})+(1-t)\operatorname{Im}(b_{0})\big{)}^{-1}\big{\|}\leq\big{(}t\|\operatorname{Im}(b_{1})^{-1}\|^{-1}+(1-t)\|\operatorname{Im}(b_{0})^{-1}\|^{-1}\big{)}^{-1}.$

By using
$\int^{1}_{0}(\gamma_{1}t+\gamma_{0}(1-t))^{-2}\,dt=(\gamma_{0}\gamma_{1})^{-1}$
for every $\gamma_{0},\gamma_{1}>0$, we finally obtain that

$\|G_{\eta}(b_{1})-G_{\eta}(b_{0})\|\leq\int^{1}_{0}\|f^{\prime}(t)\|\,dt\leq\|\operatorname{Im}(b_{0})^{-1}\|\|\operatorname{Im}(b_{1})^{-1}\|\|b_{1}-b_{0}\|,$

which is the asserted bound (3.11). ∎

Let us point out that an alternative proof of (3.11) can be given by
evaluating $G^{(2)}_{\eta}$ at the point
$\begin{bmatrix}b_{0}&r(b_{1}-b_{0})\\\ 0&b_{1}\end{bmatrix}$ for sufficiently
small $r>0$ (using the identity (A.3) together with the fact that $G^{(\leq
2)}_{\eta}$ is a noncommutative function of height $2$ by Proposition 3.7) and
examining the result with Lemma 3.5 in analogy to the proof of Lemma 3.8; see
also Remark A.3.

From Lemma 3.9, we can derive Lipschitz continuity of $DG_{\eta}$ on
$\overline{\mathbb{H}^{+}_{\gamma}(B)}$ for every $\gamma>0$.

######  Lemma 3.10.

There is a universal constant $c>0$ such that, for all
$\eta\in\mathcal{P}_{2}(B)$ and $\gamma>0$,

(3.12)
$\|(DG_{\eta})(b_{1})-(DG_{\eta})(b_{0})\|\leq\frac{c}{\gamma^{3}}\,\|b_{1}-b_{0}\|\qquad\text{for
all $b_{0},b_{1}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$}.$

###### Proof.

Choose $\sigma\in(0,1)$. Like in the proof of Lemma 3.3, we infer from (3.4)
that
$\mathbb{B}_{B}(b,\sigma\gamma)\subseteq\mathbb{D}(b,\sigma)\subset\mathbb{H}^{+}_{(1-\sigma)\gamma}(B)$
for every $b\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$. Thus,
$f(b):=G_{\eta}(b_{1}+b)-G_{\eta}(b_{0}+b)$ yields a well-defined holomorphic
function $f:\mathbb{B}_{B}(0,\sigma\gamma)\to B$ satisfying
$(Df)(0)=(DG_{\eta})(b_{1})-(DG_{\eta})(b_{0})$ and moreover
$M_{f}(0,\sigma\gamma)\leq\frac{1}{(1-\sigma)^{2}\gamma^{2}}\|b_{1}-b_{0}\|$
according (3.11). With the help of the Cauchy estimate stated in Lemma 2.1
(i), we derive that
$\|(DG_{\eta})(b_{1})-(DG_{\eta})(b_{0})\|\leq\frac{1}{\sigma(1-\sigma)^{2}}\frac{1}{\gamma^{3}}\|b_{1}-b_{0}\|$,
from which (3.12) follows with
$c=\min_{\sigma\in(0,1)}\frac{1}{\sigma(1-\sigma)^{2}}=\frac{27}{4}$. ∎

##  4\. The operator-valued inviscid Burgers equation for paths in
$\mathcal{P}_{2}(B)$

This section is about an operator-valued version of the inviscid Burgers
equation for paths in $\mathcal{P}_{2}(B)$. In the scalar-valued framework,
the fundamental insight that the inviscid Burgers equation constitutes the
free probability counterpart of the heat equation is due to Voiculescu; see
[58], especially the comments following the proof of Theorem 4.3 therein. In
[5, Theorem 8.3], Anshelevich, Belinschi, Fevrier, and Nica generalized the
aforementioned result of Voiculescu by an operator-valued version of the
inviscid Burgers equation. On the level of formal series, a variant thereof
has already appeared in [60, Proposition 4.5].

In the literature on noncommutative probability and on free probability in
particular, there are numerous other places where such a dynamical perspective
turned out to be extremely successful; we refer the interested readers for
instance to [16, 13, 19, 35, 6, 48, 24, 36, 37] and the references listed
therein.

With Theorem 1.3, we generalize [5, Theorem 8.3] from invertible completely
positive maps to arbitrary $2$-positive maps, to the relatively small price
that we are limited to initial conditions which are themselves solutions of
some Dyson equation; this, however, is entirely sufficient for the intended
application of Theorem 1.3 in the proof of Theorem 1.1.

Theorem 1.3 builds upon Theorem 1.2, the proof of which will also be given in
this section. Both proofs will make use of the following fact, which we borrow
from [34]; because it is stated there in slightly different form and under the
much more restrictive assumption of complete positivity instead of
$2$-positivity, we include here also the proof.

######  Proposition 4.1.

Let $B$ be a unital $C^{\ast}$-algebra and let $\eta\in\mathcal{P}_{2}(B)$ be
given. Define holomorphic functions $\Psi_{\eta}:\mathbb{H}^{-}(B)\to B$ by
$\Psi_{\eta}(w):=w^{-1}+\eta(w)$ and $\Delta_{b,\eta}:\mathbb{H}^{-}(B)\to B$
by $\Delta_{b,\eta}(w):=b-\Psi_{\eta}(w)$ for some fixed
$b\in\mathbb{H}^{+}(B)$. Furthermore, we set lCll U_b,η &:= {w ∈H^-(B)
∣∥Δ_b,η(w)∥ ¡ ∥Im(b)^-1∥^-1} and
U^σ_b,η := {w ∈H^-(B) ∣∥Δ_b,η(w)∥ ≤σ∥Im(b)^-1∥^-1} for $\sigma\in(0,1)$. Then
the following statements hold true:

  1. (i)

$\Psi_{\eta}$ restricts to a holomorphic function
$\Psi_{\eta}:\mathbb{U}_{b,\eta}\to\mathbb{H}^{+}(B)$ which satisfies
$G_{\eta}(\Psi_{\eta}(w))=w$ for all $w\in\mathbb{U}_{b,\eta}$. Further, for
every $\sigma\in(0,1)$, we have that

$\Psi_{\eta}(\mathbb{U}^{\sigma}_{b,\eta})\subseteq\overline{\mathbb{D}(b,\sigma)}\subset\overline{\mathbb{H}^{+}_{\gamma}(B)}\qquad\text{for}\qquad\gamma:=(1-\sigma)\|\operatorname{Im}(b)^{-1}\|^{-1}.$

  2. (ii)

Put $w_{0}:=G_{\eta}(b)\in\mathbb{H}^{-}(B)$. Then, for every $\sigma\in(0,1)$
and each $w\in\mathbb{U}^{\sigma}_{b,\eta}$, we have

(4.1)
$\|w-w_{0}\|\leq\frac{1}{1-\sigma}\|\operatorname{Im}(b)^{-1}\|^{2}\|\Delta_{b,\eta}(w)\|\leq\frac{\sigma}{1-\sigma}\|\operatorname{Im}(b)^{-1}\|.$

Note that $\mathbb{U}_{b,\eta}$ is an open set containing $w_{0}$, which is
the unique solution of the equation (3.3) in $\mathbb{H}^{-}(B)$; indeed, we
have $\Delta_{b,\eta}(w_{0})=0$ so that in fact
$w_{0}\in\mathbb{U}^{\sigma}_{b,\eta}$ for every $\sigma\in(0,1)$.
Accordingly, we regard points in $\mathbb{U}_{b,\eta}^{\sigma}$ as approximate
solutions of (3.3), which by (4.1) are getting closer to the accurate solution
$w_{0}$ when $\sigma$ tends to $0$.

###### Proof of Proposition 4.1.

(i) It is obvious that $\Psi_{\eta}$ and hence $\Delta_{b,\eta}$ define
$B$-valued holomorphic functions on $\mathbb{H}^{-}(B)$. Take any
$\sigma\in(0,1)$. Notice that
$\|\Psi_{\eta}(w)-b\|=\|\Delta_{b,\eta}(w)\|\leq\sigma\|\operatorname{Im}(b)^{-1}\|^{-1}$
and hence $\Psi_{\eta}(w)\in\overline{\mathbb{D}(b,\sigma)}$ for
$w\in\mathbb{U}^{\sigma}_{b,\eta}$; this proves that
$\Psi_{\eta}(\mathbb{U}^{\sigma}_{b,\eta})\subseteq\overline{\mathbb{D}(b,\sigma)}\subset\overline{\mathbb{H}^{+}_{\gamma}(B)}$
for $\gamma=(1-\sigma)\|\operatorname{Im}(b)^{-1}\|^{-1}$, where the second
inclusion comes from (3.4). Because
$\mathbb{U}_{b,\eta}=\bigcup_{\sigma\in(0,1)}\mathbb{U}^{\sigma}_{b,\eta}$, it
follows that the restriction of $\Psi_{\eta}$ to $\mathbb{U}_{b,\eta}$ takes
its values in $\mathbb{H}^{+}(B)$.

Next, we verify that $G_{\eta}(\Psi_{\eta}(w))=w$ for all
$w\in\mathbb{U}_{b,\eta}$. To this end, we take any $w\in\mathbb{U}_{b,\eta}$.
First, we observe that $\Psi_{\eta}(w)w=\mathbf{1}+\eta(w)w$. This tells us
that $w$ is the unique solution of (3.3) in $\mathbb{H}^{-}(B)$ at the point
$\Psi_{\eta}(w)\in\mathbb{H}^{+}(B)$, which implies
$G_{\eta}(\Psi_{\eta}(w))=w$, as we wished to show.

(ii) Take any $w\in\mathbb{U}^{\sigma}_{b,\eta}$. From (i), we know
$\Psi_{\eta}(w)\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$ for
$\gamma=(1-\sigma)\|\operatorname{Im}(b)^{-1}\|^{-1}$; consequently,
$\|\operatorname{Im}(\Psi_{\eta}(w))^{-1}\|\leq\frac{1}{1-\sigma}\|\operatorname{Im}(b)^{-1}\|$.
With the help of the bound (3.11), we get

$\|w-w_{0}\|=\|G_{\eta}(\Psi_{\eta}(w))-G_{\eta}(b)\|\leq\|\operatorname{Im}(\Psi_{\eta}(w))^{-1}\|\|\operatorname{Im}(b)^{-1}\|\|\Delta_{b,\eta}(w)\|\\\
\leq\frac{1}{1-\sigma}\|\operatorname{Im}(b)^{-1}\|^{2}\|\Delta_{b,\eta}(w)\|\leq\frac{\sigma}{1-\sigma}\|\operatorname{Im}(b)^{-1}\|$

for every $w\in\mathbb{U}^{\sigma}_{b,\eta}$, which are the bounds asserted in
(4.1). ∎

We continue these investigations with the following two propositions which
allow a local comparison of solutions $G_{\eta_{0}}$ and $G_{\eta_{1}}$ of the
Dyson equation (3.1) for $\eta_{0},\eta_{1}\in\mathcal{P}_{2}(B)$. With the
first one, which builds on Proposition 4.1 (i), we show that $G_{\eta_{0}}$
and $G_{\eta_{1}}$ are locally subordinated (to each other) whenever
$\eta_{0}$ and $\eta_{1}$ in $\mathcal{P}_{2}(B)$ are sufficiently close and
we control in terms of $\|\eta_{1}-\eta_{0}\|$ the deviation of the
subordination function from the identity.

######  Proposition 4.2.

Fix $\gamma>0$ and $0<\sigma^{\prime}<\sigma<1$. Suppose that
$\eta_{0},\eta_{1}\in\mathcal{P}_{2}(B)$ are such that the condition
$\|\eta_{1}-\eta_{0}\|\leq(1-\sigma^{\prime})(\sigma-\sigma^{\prime})\gamma^{2}$
is satisfied. Then, for each $b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$,
we have the inclusion
$G_{\eta_{1}}(\overline{\mathbb{D}(b_{0},\sigma^{\prime})})\subseteq\mathbb{U}^{\sigma}_{b_{0},\eta_{0}}$,
so that we obtain a well-defined function by

$\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}:\
\overline{\mathbb{D}(b_{0},\sigma^{\prime})}\to\overline{\mathbb{D}(b_{0},\sigma)},\quad
b\mapsto\Psi_{\eta_{0}}(G_{\eta_{1}}(b)),$

which is holomorphic on $\mathbb{D}(b_{0},\sigma^{\prime})$ and furthermore
has the property that
$G_{\eta_{0}}(\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}(b))=G_{\eta_{1}}(b)$ holds
for all $b\in\overline{\mathbb{D}(b_{0},\sigma^{\prime})}$. Moreover, we have
that

(4.2) $\displaystyle\sup_{b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}}\
\sup_{b\in\overline{\mathbb{D}(b_{0},\sigma^{\prime})}}\|\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}(b)-b\|$
$\displaystyle\leq\frac{1}{(1-\sigma^{\prime})\gamma}\|\eta_{1}-\eta_{0}\|\qquad\text{and}$
(4.3)
$\displaystyle\sup_{b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}}\|(D\omega^{b_{0}}_{\eta_{0}\to\eta_{1}})(b_{0})-\operatorname{id}_{B}\|$
$\displaystyle\leq\frac{1}{\sigma^{\prime}(1-\sigma^{\prime})\gamma^{2}}\|\eta_{1}-\eta_{0}\|.$

Before we proceed to the proof of Proposition 4.2, we collect in the following
remark some useful identities to which we will come back in the sequel.

######  Remark 4.3.

Let $\eta_{0},\eta_{1}\in\mathcal{P}_{2}(B)$, $\gamma>0$, and $\sigma\in(0,1)$
be given. For $b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$ and
$b\in\overline{\mathbb{D}(b_{0},\sigma)}$, we may compute using (3.1) for
$G_{\eta_{1}}$ that

(4.4)
$\Delta_{b_{0},\eta_{0}}(G_{\eta_{1}}(b))=b_{0}-G_{\eta_{1}}(b)^{-1}-\eta_{0}(G_{\eta_{1}}(b))=b_{0}-b+(\eta_{1}-\eta_{0})(G_{\eta_{1}}(b))$

and in particular
$\Delta_{b_{0},\eta_{0}}(G_{\eta_{1}}(b_{0}))=(\eta_{1}-\eta_{0})(G_{\eta_{1}}(b_{0}))$.
From (3.4), we know that
$\overline{\mathbb{D}(b_{0},\sigma)}\subset\overline{\mathbb{H}^{+}_{(1-\sigma)\gamma}(B)}$;
thanks to (3.2), we thus have that
$\|G_{\eta_{1}}(b)\|\leq\frac{1}{(1-\sigma)\gamma}$. It follows from (4.4)
that

(4.5)
$\|\Delta_{b_{0},\eta_{0}}(G_{\eta_{1}}(b))\|\leq\|b-b_{0}\|+\|\eta_{1}-\eta_{0}\|\|G_{\eta_{1}}(b)\|\leq\sigma\|\operatorname{Im}(b_{0})^{-1}\|^{-1}+\frac{1}{(1-\sigma)\gamma}\|\eta_{1}-\eta_{0}\|$

and in particular
$\|\Delta_{b_{0},\eta_{0}}(G_{\eta_{1}}(b_{0}))\|\leq\frac{1}{\gamma}\|\eta_{1}-\eta_{0}\|$.

###### Proof of Proposition 4.2.

Suppose that $\eta_{0},\eta_{1}\in\mathcal{P}_{2}(B)$ satisfy the condition
$\|\eta_{1}-\eta_{0}\|\leq(1-\sigma^{\prime})(\sigma-\sigma^{\prime})\gamma^{2}$.
We take $b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$ and any
$b\in\overline{\mathbb{D}(b_{0},\sigma^{\prime})}$. Using that
$\gamma\leq\|\operatorname{Im}(b_{0})^{-1}\|^{-1}$, we obtain with the aid of
the bound (4.5) from Remark 4.3 that

$\|\Delta_{b_{0},\eta_{0}}(G_{\eta_{1}}(b))\|\leq\sigma^{\prime}\|\operatorname{Im}(b_{0})^{-1}\|^{-1}+(\sigma-\sigma^{\prime})\gamma\leq\sigma\|\operatorname{Im}(b_{0})^{-1}\|^{-1},$

which means that $G_{\eta_{1}}(b)\in\mathbb{U}_{b_{0},\eta_{0}}^{\sigma}$.
This verifies the asserted inclusion
$G_{\eta_{1}}(\overline{\mathbb{D}(b_{0},\sigma^{\prime})})\subseteq\mathbb{U}^{\sigma}_{b_{0},\eta_{0}}$.

From Proposition 4.1 (i), we learn that
$\Psi_{\eta_{0}}(\mathbb{U}^{\sigma}_{b_{0},\eta_{0}})\subseteq\overline{\mathbb{D}(b_{0},\sigma)}$
and $G_{\eta_{0}}(\Psi_{\eta_{0}}(w))=w$ for all
$w\in\mathbb{U}^{\sigma}_{b_{0},\eta_{0}}$; hence,
$\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}:\overline{\mathbb{D}(b_{0},\sigma^{\prime})}\to\overline{\mathbb{D}(b_{0},\sigma)}$
is well-defined and satisfies
$G_{\eta_{0}}(\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}(b))=G_{\eta_{1}}(b)$ for
all $b\in\overline{\mathbb{D}(b_{0},\sigma^{\prime})}$. The holomorphy of
$\Psi_{\eta_{0}}$ and $G_{\eta_{1}}$ established in Proposition 4.1
respectively in Theorem 3.1 ensures that
$\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}$ is holomorphic on
$\mathbb{D}(b_{0},\sigma^{\prime})$, as asserted.

To prove (4.2), we take $b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$ and
$b\in\overline{\mathbb{D}(b_{0},\sigma^{\prime})}$ and check with the help of
Remark 4.3 that
$b-\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}(b)=\Delta_{b,\eta_{0}}(G_{\eta_{1}}(b))=(\eta_{1}-\eta_{0})(G_{\eta_{1}}(b))$
and in turn
$\|\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}(b)-b\|\leq\frac{1}{(1-\sigma^{\prime})\gamma}\|\eta_{1}-\eta_{0}\|$.
By taking suprema over $b\in\overline{\mathbb{D}(b_{0},\sigma^{\prime})}$ and
$b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$, we get (4.2).

To prove (4.3), we proceed as follows. Take an arbitrary
$b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$. Because
$\mathbb{B}_{B}(b_{0},\sigma^{\prime}\gamma)\subseteq\mathbb{D}(b_{0},\sigma^{\prime})$,
we get by $f:\mathbb{B}_{B}(b_{0},\sigma^{\prime}\gamma)\to
B,b\mapsto\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}(b)-b$ a well-defined
holomorphic function. From Lemma 2.1 (i), we infer that

$\|(D\omega^{b_{0}}_{\eta_{0}\to\eta_{1}})(b_{0})-\operatorname{id}_{B}\|=\|(Df)(b_{0})\|\leq\frac{1}{\sigma^{\prime}\gamma}M_{f}(b_{0},\sigma^{\prime}\gamma)\leq\frac{1}{\sigma^{\prime}\gamma}\sup_{b\in\overline{\mathbb{D}(b_{0},\sigma^{\prime})}}\|\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}(b)-b\|.$

By passing to the supremum over all
$b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$ and employing (4.2), we arrive
at (4.3). ∎

The second proposition exploits Proposition 4.1 (ii).

######  Proposition 4.4.

Fix $\gamma>0$ and $\sigma_{0}\in(0,1)$. Suppose that
$\eta_{0},\eta_{1}\in\mathcal{P}_{2}(B)$ are such that the condition
$\|\eta_{1}-\eta_{0}\|\leq\sigma_{0}\gamma^{2}$ is satisfied. Then, we have

(4.6)
$\sup_{b\in\overline{\mathbb{H}^{+}_{\gamma}(B)}}\|G_{\eta_{1}}(b)-G_{\eta_{0}}(b)\|\leq\frac{1}{(1-\sigma_{0})\gamma^{3}}\|\eta_{1}-\eta_{0}\|$

and moreover, with the constant $c>0$ which appeared in Lemma 3.10,

(4.7)
$\sup_{b\in\overline{\mathbb{H}^{+}_{\gamma}(B)}}\|(DG_{\eta_{1}})(b)-(DG_{\eta_{0}})(b)\|\leq\frac{1}{\gamma^{4}}\Bigg{[}\min_{\begin{subarray}{c}0<\sigma^{\prime}<\sigma<1\colon\\\
\sigma_{0}=(1-\sigma^{\prime})(\sigma-\sigma^{\prime})\end{subarray}}\frac{1-\sigma+c\sigma^{\prime}}{\sigma^{\prime}(1-\sigma^{\prime})(1-\sigma)^{3}}\Bigg{]}\|\eta_{1}-\eta_{0}\|.$

###### Proof.

Let us consider arbitrary $0<\sigma^{\prime}<\sigma<1$ such that
$\sigma_{0}=(1-\sigma^{\prime})(\sigma-\sigma^{\prime})$.

Take any $b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$. Thanks to
Proposition 4.2, we thus have that
$G_{\eta_{1}}(\overline{\mathbb{D}(b_{0},\sigma^{\prime})})\subseteq\mathbb{U}^{\sigma}_{b_{0},\eta_{0}}$
and in particular
$G_{\eta_{1}}(b_{0})\in\mathbb{U}^{\sigma}_{b_{0},\eta_{0}}$. Hence, we may
apply Proposition 4.1 (ii), which gives us
$\|G_{\eta_{1}}(b_{0})-G_{\eta_{0}}(b_{0})\|\leq\frac{1}{(1-\sigma)\gamma^{2}}\|\Delta_{b_{0},\eta_{0}}(G_{\eta_{1}}(b_{0}))\|$.
Further, due to Remark 4.3, we have that
$\|\Delta_{b_{0},\eta_{0}}(G_{\eta_{1}}(b_{0}))\|\leq\frac{1}{\gamma}\|\eta_{1}-\eta_{0}\|$.
In combination, we get
$\|G_{\eta_{1}}(b_{0})-G_{\eta_{0}}(b_{0})\|\leq\frac{1}{(1-\sigma)\gamma^{3}}\|\eta_{1}-\eta_{0}\|$,
which yields (4.6) by taking the supremum over all
$b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$ and letting
$\sigma^{\prime}\searrow 0$.

Take again any $b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$. From
Proposition 4.2, we know that
$\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}:\mathbb{D}(b_{0},\sigma^{\prime})\to\overline{\mathbb{D}(b_{0},\sigma)}$
is holomorphic and satisfies
$G_{\eta_{0}}(\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}(b))=G_{\eta_{1}}(b)$ for
all $b\in\mathbb{D}(b_{0},\sigma^{\prime})$. This identity allows us to
compute the Fréchet derivative of $G_{\eta_{1}}$ at the point $b_{0}$ as
$(DG_{\eta_{1}})(b_{0})=(DG_{\eta_{0}})(\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}(b_{0}))(D\omega^{b_{0}}_{\eta_{0}\to\eta_{1}})(b_{0})$.
Therefore, we can estimate

(4.8)
$\big{\|}(DG_{\eta_{1}})(b_{0})-(DG_{\eta_{0}})(b_{0})\big{\|}\leq\big{\|}(DG_{\eta_{0}})(\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}(b_{0}))\big{(}(D\omega^{b_{0}}_{\eta_{0}\to\eta_{1}})(b_{0})-\operatorname{id}_{B}\big{)}\big{\|}\\\
+\big{\|}(DG_{\eta_{0}})(\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}(b_{0}))-(DG_{\eta_{0}})(b_{0})\big{\|}.$

Keeping in mind that both $b_{0}$ and, by (3.4),
$\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}(b_{0})$ belong to
$\overline{\mathbb{H}^{+}_{(1-\sigma)\gamma}(B)}$, we obtain

$\big{\|}(DG_{\eta_{0}})(\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}(b_{0}))\big{(}(D\omega^{b_{0}}_{\eta_{0}\to\eta_{1}})(b_{0})-\operatorname{id}_{B}\big{)}\big{\|}\leq\frac{1}{(1-\sigma)^{2}\gamma^{2}}\|(D\omega^{b_{0}}_{\eta_{0}\to\eta_{1}})(b_{0})-\operatorname{id}_{B}\|$

from the bound (3.8) provided by Lemma 3.8, and from Lemma 3.10 we deduce that

$\|(DG_{\eta_{0}})(\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}(b_{0}))-(DG_{\eta_{0}})(b_{0})\|\leq\frac{c}{(1-\sigma)^{3}\gamma^{3}}\|\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}(b_{0})-b_{0}\|.$

By involving the bounds (4.3) and (4.2), respectively, the latter estimates
lead us to

$\big{\|}(DG_{\eta_{0}})(\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}(b_{0}))\big{(}(D\omega^{b_{0}}_{\eta_{0}\to\eta_{1}})(b_{0})-\operatorname{id}_{B}\big{)}\big{\|}\leq\frac{1}{\sigma^{\prime}(1-\sigma^{\prime})(1-\sigma)^{2}\gamma^{4}}\|\eta_{1}-\eta_{0}\|\qquad\text{and}\\\
\|(DG_{\eta_{0}})(\omega^{b_{0}}_{\eta_{0}\to\eta_{1}}(b_{0}))-(DG_{\eta_{0}})(b_{0})\|\leq\frac{c}{(1-\sigma^{\prime})(1-\sigma)^{3}\gamma^{4}}\|\eta_{1}-\eta_{0}\|,$

respectively. By combining these insights with (4.8), passing to the supremum
over $b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$, and optimizing over
$0<\sigma^{\prime}<\sigma<1$ with
$\sigma_{0}=(1-\sigma^{\prime})(\sigma-\sigma^{\prime})$, we obtain (4.7). ∎

With the help of Proposition 4.4, we can prove Theorem 1.2.

###### Proof of Theorem 1.2.

First, we notice that the assertions in Item (i) and Item (ii) concerning the
continuity of
$G:\mathcal{P}_{2}(B)\times\mathbb{H}^{+}(B)\to\mathbb{H}^{-}(B)$ and
$DG:\mathcal{P}_{2}(B)\times\mathbb{H}^{+}(B)\to\mathcal{L}(B)$ immediately
follow once the equicontinuity in the respective case is established. The
general mechanism behind (albeit not in its most general version but tailored
to our situation) is the following: if
$f:\mathcal{P}_{2}(B)\times\mathbb{H}^{+}(B)\to F$, for some Banach space
$(F,\|\cdot\|)$, is such that

  * •

for all $\gamma>0$, the family $f(\cdot,b):\mathcal{P}_{2}(B)\to F$ with
$b\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$ is equicontinuous, and

  * •

for every fixed $\eta\in\mathcal{P}_{2}(B)$, the function
$f(\eta,\cdot):\mathbb{H}^{+}(B)\to F$ is continuous,

then $f$ must be continuous. To see this, we take any sequence
$((\eta_{n},b_{n}))_{n=1}^{\infty}$ in
$\mathcal{P}_{2}(B)\times\mathbb{H}^{+}(B)$ converging to
$(\eta,b)\in\mathcal{P}_{2}(B)\times\mathbb{H}^{+}(B)$. Note that
$(b_{n})_{n=1}^{\infty}$ lies in $\overline{\mathbb{H}^{+}_{\gamma}(B)}$ for
some $\gamma>0$; thus,

$\|f(\eta_{n},b_{n})-f(\eta,b)\|\leq\sup_{b^{\prime}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}}\|f(\eta_{n},b^{\prime})-f(\eta,b^{\prime})\|+\|f(\eta,b_{n})-f(\eta,b)\|$

for all $n\in\mathbb{N}$. With the help of the latter estimate, we easily
infer from the assumptions on the separate continuity of $f$ that
$f(\eta_{n},b_{n})\to f(\eta,b)$ as $n\to\infty$. Thus, $f$ is continuous.

The statements on the equicontinuity follow immediately from Proposition 4.4.
∎

With these preparations done, we may get down to prove Theorem 1.3.

###### Proof of Theorem 1.3.

First of all, we notice that the functions
$G:[0,T]\times\mathbb{H}^{+}(B)\to\mathbb{H}^{-}(B)$ and
$DG:[0,T]\times\mathbb{H}^{+}(B)\to\mathcal{L}(B)$ are continuous thanks to
Theorem 1.2 as $\eta:[0,T]\to\mathcal{P}_{2}(B)$ is continuous. Therefore and
because $\overset{\,{}_{\mbox{\bf.}}}{\eta}:[0,T]\to\mathcal{P}_{2}(B)$ is
continuous, the continuity of
$\overset{\,{}_{\mbox{\bf.}}}{G}:[0,T]\times\mathbb{H}^{+}(B)\to B$ follows
once the Burgers equation (1.6) is shown. Moreover, once (1.5) is established,
Lemma 2.2 yields that the function
$\overset{\,{}_{\mbox{\bf.}}}{G}_{t}:\mathbb{H}^{+}(B)\to B$ is holomorphic
for every fixed $t\in[0,T]$. For the proof of Theorem 1.3 it thus suffices to
prove (1.5) and (1.6).

We fix $\gamma>0$ and $t_{0}\in[0,T]$. Let $\varepsilon>0$ be given. By Lemma
3.3, we find $r>0$ such that

(4.9)
$\Big{\|}G_{t_{0}}(b)-G_{t_{0}}(b_{0})-(DG_{t_{0}})(b_{0})\big{(}b-b_{0}\big{)}\Big{\|}\leq\frac{\gamma\varepsilon}{3(1+\|\overset{\,{}_{\mbox{\bf.}}}{\eta}_{t_{0}}\|)}\,\|b-b_{0}\|$

whenever $b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$ and
$b\in\mathbb{B}_{B}(b_{0},r)\cap\mathbb{H}^{+}(B)$. Next, we choose some
$\sigma\in(0,1)$ such that

(4.10)
$\frac{\sigma}{(1-\sigma)\gamma^{3}}\|\overset{\,{}_{\mbox{\bf.}}}{\eta}_{t_{0}}\|<\frac{\varepsilon}{3}\qquad\text{and}\qquad\sigma\gamma<r.$

By continuity and differentiability of $\eta$ at $t_{0}$, there exists
$\delta>0$ such that the conditions

(4.11) $\|\eta_{t}-\eta_{t_{0}}\|\leq\sigma\gamma^{2}\quad\text{for all $t\in
I$}\qquad\text{and}\\\
\Big{\|}\frac{1}{t-t_{0}}(\eta_{t}-\eta_{t_{0}})-\overset{\,{}_{\mbox{\bf.}}}{\eta}_{t_{0}}\Big{\|}<\min\Big{\\{}1,\frac{\gamma^{3}\varepsilon}{3}\Big{\\}}\quad\text{for
all $t\in I\setminus\\{t_{0}\\}$}$

are satisfied with $I:=[0,T]\cap(t_{0}-\delta,t_{0}+\delta)$. Let us consider
an arbitrary $b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$. Thanks to the
first condition in (4.11), we may proceed like in the proof of Proposition 4.2
and deduce from Remark 4.3 that
$G_{t}(b_{0})\in\mathbb{U}_{b_{0},\eta_{t_{0}}}^{\sigma}$ for all $t\in I$.
Using the bound (3.8) from Lemma 3.8 and Proposition 4.1 (ii), we infer from
the first condition in (4.10) that

(4.12)
$\|(DG_{t_{0}})(b_{0})\|\|\overset{\,{}_{\mbox{\bf.}}}{\eta}_{t_{0}}\|\|G_{t}(b_{0})-G_{t_{0}}(b_{0})\|<\frac{\varepsilon}{3}\qquad\text{for
all $t\in I$}.$

By involving Proposition 4.2, we see that
$b_{0,t}:=\omega^{b_{0}}_{\eta_{t_{0}}\to\eta_{t}}(b_{0})$ lies in
$\mathbb{H}^{+}(B)$ and satisfies $G_{t}(b_{0})=G_{t_{0}}(b_{0,t})$ for all
$t\in I$. From Remark 4.3 and the first condition in (4.11), we get

(4.13)
$b_{0}-b_{0,t}=\Delta_{b_{0},\eta_{t_{0}}}(G_{t}(b_{0}))=(\eta_{t}-\eta_{t_{0}})(G_{t}(b_{0}))$

and in turn $\|b_{0,t}-b_{0}\|\leq\sigma\gamma$ for all $t\in I$; by the
second condition imposed in (4.10), this ensures that
$b_{0,t}\in\mathbb{B}_{B}(b_{0},r)\cap\mathbb{H}^{+}(B)$ for all $t\in I$.
Thus, we can apply (4.9), which gives us

(4.14)
$\frac{1}{|t-t_{0}|}\Big{\|}G_{t}(b_{0})-G_{t_{0}}(b_{0})-(DG_{t_{0}})(b_{0})\big{(}b_{0,t}-b_{0}\big{)}\Big{\|}\leq\frac{\varepsilon}{3}\qquad\text{for
all $t\in I\setminus\\{t_{0}\\}$},$

where we have used that (4.13) and the second condition in (4.11) imply

$\frac{1}{|t-t_{0}|}\|b_{0,t}-b_{0}\|\leq\frac{1}{\gamma}\bigg{(}\Big{\|}\frac{1}{t-t_{0}}(\eta_{t}-\eta_{t_{0}})-\overset{\,{}_{\mbox{\bf.}}}{\eta}_{t_{0}}\Big{\|}+\|\overset{\,{}_{\mbox{\bf.}}}{\eta}_{t_{0}}\|\bigg{)}\leq\frac{1+\|\overset{\,{}_{\mbox{\bf.}}}{\eta}_{t_{0}}\|}{\gamma}.$

From (4.13) and the second condition in (4.11), we further obtain by using
(3.8) from Lemma 3.8 as well as one more time the bound
$\|G_{t}(b_{0})\|\leq\frac{1}{\gamma}$ deduced from (3.2) that

(4.15)
$\|(DG_{t_{0}})(b_{0})\|\Big{\|}\frac{1}{t-t_{0}}(b_{0}-b_{0,t})-\overset{\,{}_{\mbox{\bf.}}}{\eta}_{t_{0}}(G_{t}(b_{0}))\Big{\|}<\frac{\varepsilon}{3}\qquad\text{for
all $t\in I\setminus\\{t_{0}\\}$}.$

Putting together the bounds (4.12), (4.14), and (4.15) which we just
established, we obtain

$\Big{\|}\frac{1}{t-t_{0}}\big{(}G_{t}(b_{0})-G_{t_{0}}(b_{0})\big{)}+(DG_{t_{0}})(b_{0})\big{(}\overset{\,{}_{\mbox{\bf.}}}{\eta}_{t_{0}}(G_{t_{0}}(b_{0}))\big{)}\Big{\|}\\\
\leq\frac{1}{|t-t_{0}|}\Big{\|}G_{t}(b_{0})-G_{t_{0}}(b_{0})-(DG_{t_{0}})(b_{0})\big{(}b_{0,t}-b_{0}\big{)}\Big{\|}\\\
+\|(DG_{t_{0}})(b_{0})\|\Big{\|}\frac{1}{t-t_{0}}(b_{0}-b_{0,t})-\overset{\,{}_{\mbox{\bf.}}}{\eta}_{t_{0}}(G_{t}(b_{0}))\Big{\|}+\|(DG_{t_{0}})(b_{0})\|\|\overset{\,{}_{\mbox{\bf.}}}{\eta}_{t_{0}}\|\|G_{t}(b_{0})-G_{t_{0}}(b_{0})\|<\varepsilon$

for all $t\in I\setminus\\{t_{0}\\}$. Because
$b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$ was arbitrary, it follows that

$\sup_{b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}}\Big{\|}\frac{1}{t-t_{0}}\big{(}G_{t}(b_{0})-G_{t_{0}}(b_{0})\big{)}+(DG_{t_{0}})(b_{0})\big{(}\overset{\,{}_{\mbox{\bf.}}}{\eta}_{t_{0}}(G_{t_{0}}(b_{0}))\big{)}\Big{\|}\leq\varepsilon.$

Thus, in summary, we see that

$\lim_{t\to
t_{0}}\sup_{b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}}\Big{\|}\frac{1}{t-t_{0}}\big{(}G_{t}(b_{0})-G_{t_{0}}(b_{0})\big{)}+(DG_{t_{0}})(b_{0})\big{(}\overset{\,{}_{\mbox{\bf.}}}{\eta}_{t_{0}}(G_{t_{0}}(b_{0}))\big{)}\Big{\|}=0.$

This shows that the family
$G_{\boldsymbol{\cdot}}(b_{0}):[0,T]\to\mathbb{H}^{-}(B)$ for
$b_{0}\in\overline{\mathbb{H}^{+}_{\gamma}(B)}$ is equidifferentiable at
$t_{0}$ with
$\overset{\,{}_{\mbox{\bf.}}}{G}_{t_{0}}(b_{0})=-(DG_{t_{0}})(b_{0})(\overset{\,{}_{\mbox{\bf.}}}{\eta}_{t_{0}}(G_{t_{0}}(b_{0})))$,
which establishes at once (1.5) and (1.6). ∎

##  5\. Hölder continuity of the density of states

In this section, we give the proof of Theorem 1.1. For each of the two Hölder
bounds stated therein, the proof follows the same strategy based on the next
lemma.

######  Lemma 5.1.

Let $(\rho_{t})_{t\in[0,T]}$ be a family of data pairs in
$B_{\mathrm{sa}}\times\mathcal{P}(B)$. Suppose that the function
$\mathcal{G}:[0,T]\times\mathbb{C}^{+}\to\mathbb{C}^{-},(t,z)\mapsto\mathcal{G}_{t}(z)$
defined by $\mathcal{G}_{t}(z):=\mathcal{G}_{\rho_{t}}(z)$ for
$(t,z)\in[0,T]\times\mathbb{C}^{+}$ satisfies the following conditions:

  1. (i)

The function $\mathcal{G}:[0,T]\times\mathbb{C}^{+}\to\mathbb{C}^{-}$ is
continuous with respect to the product topology on
$[0,T]\times\mathbb{C}^{+}$.

  2. (ii)

For each $z\in\mathbb{C}^{+}$, the function
$\mathcal{G}_{\boldsymbol{\cdot}}(z):[0,T]\to\mathbb{C}^{-},t\mapsto\mathcal{G}_{t}(z)$
is continuously differentiable; we denote its derivative by
$\overset{\,{}_{\mbox{\bf.}}}{\mathcal{G}}_{\boldsymbol{\cdot}}(z):[0,T]\to\mathbb{C}^{-}$.

  3. (iii)

There are $k\in\mathbb{N}$ and $\theta\geq 0$ such that

$\big{|}\overset{\,{}_{\mbox{\bf.}}}{\mathcal{G}}_{t}(z)\big{|}\leq-\frac{\operatorname{Im}(\mathcal{G}_{t}(z))}{\operatorname{Im}(z)^{k}}\,\theta\qquad\text{for
all $(t,z)\in[0,T]\times\mathbb{C}^{+}$}.$

Then, for every $\varepsilon>0$, it holds true that

(5.1)
$\frac{1}{\pi}\int^{\infty}_{-\infty}\big{|}\mathcal{G}_{\rho_{T}}(s+i\varepsilon)-\mathcal{G}_{\rho_{0}}(s+i\varepsilon)\big{|}\,ds\leq\frac{T\theta}{\varepsilon^{k}},$

and with the absolute constant
$c_{k}:=(2k+1)(\frac{1}{k^{2}\pi})^{\frac{k}{2k+1}}$, we have that

(5.2) $L(\mu_{\rho_{T}},\mu_{\rho_{0}})\leq
c_{k}T^{\frac{1}{2k+1}}\theta^{\frac{1}{2k+1}}.$

###### Proof.

Take any $\varepsilon>0$. Thanks to condition (ii), the fundamental theorem of
calculus allows us to write
$\mathcal{G}_{\rho_{T}}(s+i\varepsilon)-\mathcal{G}_{\rho_{0}}(s+i\varepsilon)=\int^{T}_{0}\overset{\,{}_{\mbox{\bf.}}}{\mathcal{G}}_{t}(s+i\varepsilon)\,dt$
for every $s\in\mathbb{R}$. In combination with the bound required in
condition (iii), this yields that

$\big{|}\mathcal{G}_{\rho_{T}}(s+i\varepsilon)-\mathcal{G}_{\rho_{0}}(s+i\varepsilon)\big{|}\leq-\frac{\theta}{\varepsilon^{k}}\int^{T}_{0}\operatorname{Im}(\mathcal{G}_{t}(s+i\varepsilon))\,dt$

for all $s\in\mathbb{R}$. By integrating the latter inequality over
$s\in\mathbb{R}$ and using the Fubini-Tonelli theorem, which can be applied
thanks to condition (i), we conclude that

$\frac{1}{\pi}\int^{\infty}_{-\infty}\big{|}\mathcal{G}_{\rho_{T}}(s+i\varepsilon)-\mathcal{G}_{\rho_{0}}(s+i\varepsilon)\big{|}\,ds\leq\frac{\theta}{\varepsilon^{k}}\int^{T}_{0}\int^{\infty}_{-\infty}\Big{(}-\frac{1}{\pi}\operatorname{Im}(\mathcal{G}_{t}(s+i\varepsilon))\Big{)}\,ds\,dt=\frac{T\theta}{\varepsilon^{k}}.$

Notice that the last step relies on the fact that
$-\frac{1}{\pi}\operatorname{Im}(\mathcal{G}_{t}(s+i\varepsilon))\,ds$ is a
probability measure on $\mathbb{R}$ for each $t\in[0,T]$. This verifies the
bound asserted in (5.1). Finally, we involve the estimate for the Lévy
distance given in Theorem 2.4. The already established bound (5.1) gives us
that $L(\mu_{\rho_{T}},\mu_{\rho_{0}})\leq
2\sqrt{\frac{\varepsilon}{\pi}}+\frac{T\theta}{\varepsilon^{k}}$ for all
$\varepsilon>0$. Minimizing the right hand side over
$\varepsilon\in(0,\infty)$ yields the asserted bound (5.2). ∎

With the help of Lemma 5.1, we can finally prove Theorem 1.1. Notice that the
uniform continuity of
$\mu:(B_{\mathrm{sa}}\times\mathcal{P}_{2}(B),d)\to(\operatorname{Prob}(\mathbb{R}),L)$
with respect to any metric $d$ on $B_{\mathrm{sa}}\times\mathcal{P}_{2}(B)$
which yields a metrization of the product topology is an immediate consequence
of the bounds stated in the items (i) and (ii) of that theorem. To see this,
we only need to notice that, for any two data pairs
$\rho_{0}=(b_{0,0},\eta_{0})$ and $\rho_{1}=(b_{0,1},\eta_{1})$ in
$B_{\mathrm{sa}}\times\mathcal{P}_{2}(B)$, we have that

$L(\mu_{\rho_{1}},\mu_{\rho_{0}})\leq
L(\mu_{(b_{0,1},\eta_{1})},\mu_{(b_{0,0},\eta_{1})})+L(\mu_{(b_{0,0},\eta_{1})},\mu_{(b_{0,0},\eta_{0})})\leq
c_{1}\,\|b_{0,1}-b_{0,0}\|^{1/3}+c_{2}\,\|\eta_{1}-\eta_{0}\|^{1/5}.$

Thus, it suffices to prove (i) and (ii) in Theorem 1.1; we treat these two
parts separately.

###### Proof of Theorem 1.1 (i).

For fixed $b_{0}\in B_{\mathrm{sa}}$ and
$\eta_{0},\eta_{1}\in\mathcal{P}_{2}(B)$, we consider the $C^{1}$-path

$\eta:\ [0,1]\to\mathcal{P}_{2}(B),\quad
t\mapsto\eta_{t}:=(1-t)\eta_{0}+t\eta_{1}$

and the family of data pairs $(\rho_{t})_{t\in[0,1]}$ given by
$\rho_{t}:=(b_{0},\eta_{t})$ for $t\in[0,1]$. In this case, the function
$\mathcal{G}:[0,1]\times\mathbb{C}^{+}\to\mathbb{C}^{-},(t,z)\mapsto\mathcal{G}_{t}(z)$
defined by $\mathcal{G}_{t}(z):=\mathcal{G}_{\rho_{t}}(z)$ for
$(t,z)\in[0,1]\times\mathbb{C}^{+}$ is related with the function
$G:[0,1]\times\mathbb{H}^{+}(B)\to\mathbb{H}^{-}(B),(t,b)\mapsto G_{t}(b)$
defined by $G_{t}(b):=G_{\eta_{t}}(b)$ for all
$(t,z)\in[0,1]\times\mathbb{H}^{+}(B)$ via
$\mathcal{G}_{t}(z)=\phi(G_{t}(z\mathbf{1}-b_{0}))$ for all
$(t,z)\in[0,1]\times\mathbb{C}^{+}$.

From Theorem 1.2, we derive that
$G:[0,1]\times\mathbb{H}^{+}(B)\to\mathbb{H}^{-}(B)$ is continuous, and in
Theorem 1.3, we have seen that
$G_{\boldsymbol{\cdot}}(b):[0,1]\to\mathbb{H}^{-}(B)$ is continuously
differentiable for every fixed $b\in\mathbb{H}^{+}(B)$; we infer that
$\mathcal{G}$ satisfies the conditions formulated in Item (i) and Item (ii) of
Lemma 5.1. To verify also the condition formulated in Item (iii), we proceed
as follows. First, we deduce from Theorem 1.3 that $G$ solves the partial
differential equation

$\overset{\,{}_{\mbox{\bf.}}}{G}_{t}(b)=-(DG_{t})(b)\big{(}(\eta_{1}-\eta_{0})(G_{t}(b))\big{)}\qquad\text{for
$(t,b)\in[0,1]\times\mathbb{H}^{+}(B)$}.$

By applying $\phi$ to both sides, we derive with the bound (3.9) provided by
Lemma 3.8 that

$\big{|}\phi(\overset{\,{}_{\mbox{\bf.}}}{G}_{t}(b))\big{|}\leq-\operatorname{Im}(\phi(G_{t}(b)))\|\operatorname{Im}(b)^{-1}\|\|(\eta_{1}-\eta_{0})\big{(}G_{t}(b)\big{)}\|\\\
\leq-\operatorname{Im}(\phi(G_{t}(b)))\|\operatorname{Im}(b)^{-1}\|^{2}\|\eta_{1}-\eta_{0}\|.$

When applied for $b=z\mathbf{1}-b_{0}$, the latter says that for $\mathcal{G}$
the condition (iii) is satisfied with

(5.3)
$\big{|}\overset{\,{}_{\mbox{\bf.}}}{\mathcal{G}}_{t}(z)\big{|}\leq-\frac{\operatorname{Im}(\mathcal{G}_{t}(z))}{\operatorname{Im}(z)^{2}}\,\|\eta_{1}-\eta_{0}\|\qquad\text{for
$(t,z)\in[0,1]\times\mathbb{C}^{+}$}.$

Thus, (5.2) in Lemma 5.1 yields the asserted bound
$L(\mu_{\rho_{1}},\mu_{\rho_{0}})\leq c_{2}\|\eta_{1}-\eta_{0}\|^{1/5}$. ∎

###### Proof of Theorem 1.1 (ii).

Suppose that $b_{0,0},b_{0,1}\in B_{\mathrm{sa}}$ and
$\eta\in\mathcal{P}_{2}(B)$ are given. We set

$b_{0}:\ [0,1]\to B_{\mathrm{sa}},\quad t\mapsto
b_{0,t}:=(1-t)b_{0,0}+tb_{0,1}$

and consider the family of data pairs $(\rho_{t})_{t\in[0,1]}$ given by
$\rho_{t}:=(b_{0,t},\eta)$ for $t\in[0,1]$. We define
$G:[0,1]\times\mathbb{H}^{+}(B)\to\mathbb{H}^{-}(B),(t,b)\mapsto G_{t}(b)$ by
$G_{t}(b):=G_{\eta}(b-b_{0,t})$ for all
$(t,b)\in[0,1]\times\mathbb{H}^{+}(B)$. The function
$\mathcal{G}:[0,1]\times\mathbb{C}^{+}\to\mathbb{C}^{-},(t,z)\mapsto\mathcal{G}_{t}(z)$
defined by $\mathcal{G}_{t}(z):=\mathcal{G}_{\rho_{t}}(z)$ for
$(t,z)\in[0,1]\times\mathbb{C}^{+}$ is then related with $G$ via
$\mathcal{G}_{t}(z)=\phi(G_{t}(z\mathbf{1}))$ for all
$(t,z)\in[0,1]\times\mathbb{C}^{+}$.

Clearly, $G:[0,1]\times\mathbb{H}^{+}(B)\to\mathbb{H}^{-}(B)$ is continuous
and $G_{\boldsymbol{\cdot}}(b):[0,1]\to\mathbb{H}^{-}(B)$ is continuously
differentiable with derivative given by
$\overset{\,{}_{\mbox{\bf.}}}{G}_{t}(b)=-(DG_{\eta})(b-b_{0,t})(b_{0,1}-b_{0,0})$
for every fixed $b\in\mathbb{H}^{+}(B)$; hence, $\mathcal{G}$ satisfies the
conditions formulated in Item (i) and Item (ii) of Lemma 5.1. To verify
condition (iii), we use the bound (3.9) provided by Lemma 3.8,

$\big{|}\phi(\overset{\,{}_{\mbox{\bf.}}}{G}_{t}(b))\big{|}\leq-\operatorname{Im}(\phi(G_{t}(b)))\|\operatorname{Im}(b)^{-1}\|\|b_{0,1}-b_{0,0}\|,$

which yields, when applied for $b=z\mathbf{1}$, that

(5.4)
$\big{|}\overset{\,{}_{\mbox{\bf.}}}{\mathcal{G}}_{t}(z)\big{|}\leq-\frac{\operatorname{Im}(\mathcal{G}_{t}(z))}{\operatorname{Im}(z)}\,\|b_{0,1}-b_{0,0}\|\qquad\text{for
$(t,z)\in[0,1]\times\mathbb{C}^{+}$}.$

Thus, (5.2) in Lemma 5.1 yields the asserted bound
$L(\mu_{\rho_{1}},\mu_{\rho_{0}})\leq c_{1}\|b_{0,1}-b_{0,0}\|^{1/3}$. ∎

We conclude by the following consequence of the proof of Theorem 1.1.

######  Corollary 5.2.

In the situation of Theorem 1.1, the following bounds hold true:

  1. (i)

For each choice of $\eta\in\mathcal{P}_{2}(B)$, we have for all
$b_{0,0},b_{0,1}\in B_{\mathrm{sa}}$ and $\varepsilon>0$ that

$\frac{1}{\pi}\int^{\infty}_{-\infty}\big{|}\mathcal{G}_{\mu_{(b_{0,1},\eta)}}(s+i\varepsilon)-\mathcal{G}_{\mu_{(b_{0,0},\eta)}}(s+i\varepsilon)\big{|}\,ds\leq\frac{1}{\varepsilon}\,\|b_{0,1}-b_{0,0}\|.$

  2. (ii)

For each choice of $b_{0}\in B_{\mathrm{sa}}$, we have for all
$\eta_{0},\eta_{1}\in\mathcal{P}_{2}(B)$ and $\varepsilon>0$ that

$\frac{1}{\pi}\int^{\infty}_{-\infty}\big{|}\mathcal{G}_{\mu_{(b_{0},\eta_{1})}}(s+i\varepsilon)-\mathcal{G}_{\mu_{(b_{0},\eta_{0})}}(s+i\varepsilon)\big{|}\,ds\leq\frac{1}{\varepsilon^{2}}\,\|\eta_{1}-\eta_{0}\|.$

###### Proof.

Using (5.1) from Lemma 5.1, we obtain the bounds asserted in the corollary
from the bounds (5.3) and (5.4) which we derived in the proofs of (i) and (ii)
in Theorem 1.1. ∎

##  Appendix A Free noncommutative function theory on the second level

Free noncommutative function theory has its origins in the work of Taylor [55,
56], whose ideas were taken up subsequently by numerous authors for various
applications; we mention [61, 62, 63] for their relevance on free probability
theory and since Proposition 3.7 fits nicely into that context. A unified and
very general approach was presented by Kaliuzhnyi-Verbovetskyi and Vinnikov in
[38], building on the notion of noncommutative sets and functions. With an eye
towards applications in the concrete situation of this paper, we introduce
here the terminology of noncommutative sets and functions in some truncated
version. For the sake of simplicity, we restrict ourselves to the case where
those are built on $C^{\ast}$-algebras.

Let $A$ be a unital $C^{\ast}$-algebra. For any $n\in\mathbb{N}$, we refer to
$M_{\leq n}(A):=\coprod_{k=1}^{n}M_{k}(A)$ as the _noncommutative space over
$A$ of height $n$_. For a subset $\Omega\subseteq M_{\leq n}(A)$, we call
$\Omega_{k}:=M_{k}(A)\cap\Omega$ for $1\leq k\leq n$ the _$k$ -th level of
$\Omega$_. We say that $\Omega$ is a _noncommutative set over $A$ of height
$n$_ if $\Omega$ is closed under direct sums of appropriate size, i.e., we
require that whenever $1\leq k,l\leq n$ are such that $k+l\leq n$, then

$X\oplus Y:=\begin{bmatrix}X&0\\\ 0&Y\end{bmatrix}\in\Omega_{k+l}$

for all $X\in\Omega_{k}$ and $Y\in\Omega_{l}$. We say that $\Omega$ is _open_
if each level $\Omega_{k}$ is open in the operator-norm topology of the
respective $C^{\ast}$-algebra $M_{k}(A)$.

Now, consider two unital $C^{\ast}$-algebras $A$ and $B$ and let
$\Omega\subseteq M_{\leq n}(A)$ be a noncommutative set of height $n$.
Consider a function $f^{(\leq n)}:\Omega\to M_{\leq n}(B)$. We shall denote by
$f^{(k)}:\Omega_{k}\to M_{\leq n}(B)$ the restriction of $f^{(\leq n)}$ to the
$k$-th level $\Omega_{k}$ of $\Omega$. We call $f^{(\leq n)}$ a
_noncommutative function of height $n$_ if $f^{(k)}(\Omega_{k})\subseteq
M_{k}(B)$ for $k=1,\dots,n$ and the following two conditions are satisfied:

  * •

_$f^{(\leq n)}$ respects direct sums_, i.e., whenever $1\leq k,l\leq n$ are
such that $k+l\leq n$, then $f^{(k+l)}(X\oplus Y)=f^{(k)}(X)\oplus f^{(l)}(Y)$
holds true for all $X\in\Omega_{k}$ and $Y\in\Omega_{l}$.

  * •

_$f^{(\leq n)}$ respects similarities_, i.e., if $X\in\Omega_{k}$ with $1\leq
k\leq n$ and an invertible matrix $S\in M_{k}(\mathbb{C})$ are given such that
$SXS^{-1}\in\Omega_{k}$, then $f^{(k)}(SXS^{-1})=Sf^{(k)}(X)S^{-1}$.

We say that $f$ is _continuous_ if $f^{(k)}:\Omega_{k}\to M_{k}(B)$ is
continuous for $k=1,\dots,n$ in the respective operator-norm topologies.

A fundamental fact in free noncommutative function theory is that
noncommutative functions, due to their algebraic structure, show strong
analytic properties under rather weak assumptions; see [38, Theorem 7.4]. To
some extent, this remains true in the truncated setting. The following
theorem, which is a variant of [30, Proposition 2.5] in the spirit of [1,
Lemma 3.1], makes this claim precise. In essence, it says that continuous
functions between subsets of $C^{\ast}$-algebras are automatically (Fréchet)
holomorphic if they admit an extension to a noncommutative function of height
$2$.

######  Theorem A.1.

Let $A$ and $B$ be unital $C^{\ast}$-algebras and let
$\emptyset\neq\Omega_{1}\subseteq A$ be an open subset. Consider a continuous
function $f^{(1)}:\Omega_{1}\to B$ with the following property: there exists
an open noncommutative set $\Omega$ over $A$ of height $2$ and a continuous
noncommutative function $f^{(\leq 2)}:\Omega\to M_{\leq 2}(B)$ of height $2$
which extends $f^{(1)}:\Omega_{1}\to B$ in the sense that $\Omega_{1}$ is the
first level set of $\Omega$ and $f^{(1)}$ is the restriction of $f^{(\leq 2)}$
to $\Omega_{1}$. Then $f^{(1)}$ is holomorphic and, for all $a\in\Omega_{1}$
and $h\in A$ satisfying the condition

(A.1) $\begin{bmatrix}a&h\\\ 0&a\end{bmatrix}\in\Omega_{2},$

it holds true that

(A.2) $f^{(2)}\bigg{(}\begin{bmatrix}a&h\\\
0&a\end{bmatrix}\bigg{)}=\begin{bmatrix}f^{(1)}(a)&(Df^{(1)})(a)h\\\
0&f^{(1)}(a)\end{bmatrix}.$

Before we give the proof, we notice that for $a_{1},a_{2}\in A$ and
$z\in\mathbb{C}$ the matrix identity

(A.3) $\begin{bmatrix}a_{1}&z(a_{2}-a_{1})\\\
0&a_{2}\end{bmatrix}\begin{bmatrix}1&z\\\
0&1\end{bmatrix}=\begin{bmatrix}1&z\\\
0&1\end{bmatrix}\begin{bmatrix}a_{1}&0\\\ 0&a_{2}\end{bmatrix}$

holds; this provides similarity relations on $M_{2}(A)$, which will be used in
the sequel.

###### Proof of Theorem A.1.

Take $a\in\Omega_{1}$ and $h\in A$ such that (A.1) is satisfied. Since
$\Omega_{1}$ and $\Omega_{2}$ are assumed to be open, we find $r>0$ such that

$a+zh\in\Omega_{1}\qquad\text{and}\qquad\begin{bmatrix}a&h\\\
0&a+zh\end{bmatrix}\in\Omega_{2}$

for all $z\in\mathbb{C}$ satisfying $0<|z|<r$. For every such $z$, notice that
(A.3) gives

$\begin{bmatrix}a&h\\\ 0&a+zh\end{bmatrix}\begin{bmatrix}1&\frac{1}{z}\\\
0&1\end{bmatrix}=\begin{bmatrix}1&\frac{1}{z}\\\
0&1\end{bmatrix}\begin{bmatrix}a&0\\\ 0&a+zh\end{bmatrix},$

from which we infer that

$f^{(2)}\bigg{(}\begin{bmatrix}a&h\\\
0&a+zh\end{bmatrix}\bigg{)}=\begin{bmatrix}1&\frac{1}{z}\\\
0&1\end{bmatrix}\begin{bmatrix}f^{(1)}(a)&0\\\
0&f^{(1)}(a+zh)\end{bmatrix}\begin{bmatrix}1&-\frac{1}{z}\\\
0&1\end{bmatrix}.$

By another application of (A.3), we deduce from the latter that

$f^{(2)}\bigg{(}\begin{bmatrix}a&h\\\
0&a+zh\end{bmatrix}\bigg{)}=\begin{bmatrix}f^{(1)}(a)&\frac{1}{z}(f^{(1)}(a+zh)-f^{(1)}(a))\\\
0&f^{(1)}(a+zh)\end{bmatrix}.$

As both $f^{(1)}$ and $f^{(2)}$ are continuous, the previous identity tells us
when letting $z\to 0$ that the limit $(\delta f^{(1)})(a;h)=\lim_{z\to
0}\frac{1}{z}(f^{(1)}(a+zh)-f^{(1)}(a))$ exists and that

(A.4) $f^{(2)}\bigg{(}\begin{bmatrix}a&h\\\
0&a\end{bmatrix}\bigg{)}=\begin{bmatrix}f^{(1)}(a)&(\delta f^{(1)})(a;h)\\\
0&f^{(1)}(a)\end{bmatrix}.$

Because both $\Omega_{1}$ and $\Omega_{2}$ are supposed to be open, there
exists for every given $a\in\Omega_{1}$ some $\varepsilon>0$ such that (A.1)
is satisfied for all $h\in A$ with $\|h\|<\varepsilon$. Thus, it follows that
$f^{(1)}$ is Gâteaux holomorphic, and because it is continuous by assumption
and so in particular locally bounded, we conclude that $f^{(1)}$ is
holomorphic; see the discussion in Section 2.2. In particular, we may replace
in (A.4) the Gâteaux derivative $\delta f^{(1)}(a;h)$ by the Fréchet
derivative $(Df^{(1)})(a)h$ which leads us to the formula (A.2). ∎

For a function $f^{(\leq n)}:\Omega\to M_{\leq n}(B)$ with
$f^{(k)}(\Omega_{k})\subseteq M_{k}(B)$ for $k=1,\dots,n$, we say that
_$f^{(\leq n)}$ respects intertwinings_ if it satisfies the following
condition: whenever $X\in\Omega_{k}$, $Y\in\Omega_{l}$, and $T\in M_{k\times
l}(\mathbb{C})$ for $1\leq k,l\leq n$ are such that $XT=TY$, then
$f^{(k)}(X)T=Tf^{(l)}(Y)$. Like in [38, Proposition 2.1], one sees that if
$f^{(\leq n)}$ respects intertwinings, then it respects direct sums and
similarities, and is thus a noncommutative function of height $n$; the proof
of the converse implication, however, does not work in the truncated setting
due to the limitation of height. It is therefore interesting to note that
Proposition 3.7 can be strengthened:

######  Proposition A.2.

Let $\eta:B\to B$ be an $n$-positive linear map on the unital
$C^{\ast}$-algebra $B$. Then the noncommutative function $G^{(\leq
n)}_{\eta}:\mathbb{H}^{+}(M_{\leq n}(B))\to\mathbb{H}^{-}(M_{\leq n}(B))$ of
height $n$ introduced in Proposition 3.7 respects intertwinings.

The proof of Proposition A.2 is similar to that of Proposition 3.7 and is thus
omitted.

To some extent, one can imitate in such cases the construction of the left and
right difference-differential operators $\Delta_{L}$ and $\Delta_{R}$ from
[38, Section 2.2]. We focus on $\Delta_{R}$ since $\Delta_{L}$ can be treated
analogously. Let $f^{(\leq n)}:\Omega\to M_{\leq n}(B)$ be a noncommutative
function of height $n$ on an open noncommutative set $\Omega\subseteq M_{\leq
n}(A)$ of height $n$ which respects intertwinings. In the same way as [38,
Proposition 2.4], one proves that if $X\in\Omega_{k}$ and $Y\in\Omega_{l}$ are
given for $1\leq k,l\leq n$ with $k+l\leq n$, then there exists a
$\mathbb{C}$-homogeneous map $\Delta_{R}f^{(\leq n)}(X,Y):M_{k\times l}(A)\to
M_{k\times l}(B)$ such that

(A.5) $f^{(k+l)}\bigg{(}\begin{bmatrix}X&Z\\\
0&Y\end{bmatrix}\bigg{)}=\begin{bmatrix}f^{(k)}(X)&\Delta_{R}f^{(\leq
n)}(X,Y)(Z)\\\
0&f^{(l)}(Y)\end{bmatrix}\quad\text{whenever}\quad\begin{bmatrix}X&Z\\\
0&Y\end{bmatrix}\in\Omega_{k+l}.$

While the proof of [38, Proposition 2.6] does not work without imposing
further constraints on $k$ and $l$ due to the limitation of height, the first
order difference formula from [38, Theorem 2.11] extends to the truncated
setting with exactly the same proof: if $X\in\Omega_{k}$, $Y\in\Omega_{l}$,
and $S\in M_{k\times l}(\mathbb{C})$ are given for $1\leq k,l\leq n$ with
$k+l\leq n$, then

(A.6) $Sf^{(k)}(Y)-f^{(l)}(X)S=\Delta_{R}f^{(\leq n)}(X,Y)(SY-XS).$

In the situation of Theorem A.1 and under the additional assumption that
$f^{(\leq 2)}:\Omega\to M_{\leq 2}(B)$ respects intertwinings, one can easily
deduce from (A.6) that $\Delta_{R}f^{(\leq 2)}(a,a)(h)=Df^{(1)}(a)h$ for all
$a\in\Omega_{1}$ and $h\in A$; thus, (A.1) is recovered by applying (A.5).

######  Remark A.3.

It is worthwhile to note that with the aforementioned facts, the alternative
derivation of Lemma 3.9 which we sketched after the detailed proof becomes
more transparent; in fact, we see that (A.3) was a shortcut avoiding the
difference-differential operator. For $\eta\in\mathcal{P}_{2}(B)$, we can
derive from (A.5) with the help of Lemma 3.5 like in the proof of Lemma 3.8
that $\|\Delta_{R}G_{\eta}^{(\leq
2)}(b_{0},b_{1})(w)\|\leq\|\operatorname{Im}(b_{0})^{-1}\|\|\operatorname{Im}(b_{1})^{-1}\|\|w\|$
for all $b_{0},b_{1}\in\mathbb{H}^{+}(B)$ and every $w\in B$; this result is
in accordance with [14, Proposition 3.1]. Further, we infer from the first
order difference formula (A.6) that
$G_{\eta}(b_{1})-G_{\eta}(b_{0})=\Delta_{R}G_{\eta}^{(\leq
2)}(b_{0},b_{1})(b_{1}-b_{0})$ for all $b_{0},b_{1}\in\mathbb{H}^{+}(B)$. By
combining both observations, we arrive again at the bound (3.11); we remark
that this strategy is similar to the proof of [14, Proposition 3.1].

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