# Continuity of inner-outer factorization and cross sections from invariant
subspaces to inner functions

Bingzhe Hou  Bingzhe Hou, School of Mathematics, Jilin University, 130012,
Changchun, P. R. China<EMAIL_ADDRESS>and  Yue Xin  Yue Xin, School of
Mathematics, Jilin University, 130012, Changchun, P. R. China
<EMAIL_ADDRESS>
###### Abstract.

Let $H^{\infty}$ be the Banach algebra of bounded analytic functions on the
unit open disc $\mathbb{D}$ equipped with the supremum norm. As well known,
inner functions play an important role of in the study of bounded analytic
functions. In this paper, we are interested in the study of inner functions.
Following by the canonical inner-outer factorization decomposition, define
$Q_{inn}$ and $Q_{out}$ the maps from $H^{\infty}$ to $\mathfrak{I}$ the set
of inner functions and $\mathfrak{F}$ the set of outer functions,
respectively. In this paper, we study the $H^{2}$-norm continuity and
$H^{\infty}$-norm discontinuity of $Q_{inn}$ and $Q_{out}$ on some subsets of
$H^{\infty}$. On the other hand, the Beurling theorem connects invariant
subspaces of the multiplication operator $M_{z}$ and inner functions. We show
the nonexistence of continuous cross section from some certain invariant
subspaces to inner functions in the supremum norm. The continuity problem of
$Q_{inn}$ and $Q_{out}$ on $\textrm{Hol}(\overline{\mathbb{D}})$, the set of
all analytic functions in the closed unit disk, are also considered.

###### Key words and phrases:

Inner-outer factorization; bounded analytic functions; cross sections; inner
functions; essential supremum norm.

###### 2010 Mathematics Subject Classification:

Primary 30J05, 30J10; Secondary 15A60, 15B05.

##  1\. Introduction

Let $\mathbb{D}$ be the unit open disc and $\mathbb{T}$ be the unit circle.
Let $H^{\infty}$ be the Banach algebra of bounded analytic functions on
$\mathbb{D}$ equipped with the supremum norm
${\|}f{\|}_{\infty}=\sup\\{{|}f(z){|};z\in\mathbb{D}\\}$. Moreover, denoted by
$(H^{\infty})^{-1}$ the set of invertible functions in $H^{\infty}$. Let
$L^{\infty}(\mathbb{T})$ (or $L^{\infty}$ in brief) be the collection of all
essentially bounded measurable functions on the unit circle $\mathbb{T}$ with
regard to the normalized Lebesgue measure of $\mathbb{T}$. $L^{\infty}$ is
also a Banach algebra equipped with the the essential supremum norm
${\|}g{\|}_{L^{\infty}}=\sup_{\xi\in\mathbb{T}}{\rm ess}{|}g(z){|}$. Then
there is an inclusion $i:H^{\infty}\rightarrow L^{\infty}$ defined by

$f\rightarrow\widetilde{f}(\mathrm{e}^{\mathbf{i}\theta})=\lim\limits_{r\to
1^{-}}f(r\mathrm{e}^{\mathbf{i}\theta}).$

Notice that this inclusion is an isometry, i.e., for any $f\in H^{\infty}$,

$\|f\|_{\infty}=\|\widetilde{f}\|_{L^{\infty}}.$

Moreover, denoted by $(L^{\infty})^{-1}$ the set of invertible functions in
$L^{\infty}$.

A bounded analytic function $u$ on $\mathbb{D}$ is called an inner function if
it has unimodular radial limits almost everywhere on $\mathbb{T}$, and denote
by $\mathfrak{I}$ the set of all inner functions. As well known, it plays an
important role of inner functions in the study of bounded analytic functions,
for instance, the Beurling theorem tells us that each invariant subspace $M$
of the classical Hardy space $H^{2}$ under the multiplication operator $M_{z}$
there exists an inner function $u$ such that $M=uH^{2}$. In addition, a
bounded analytic function $F$ on $\mathbb{D}$ is called an outer function if
$F$ is a cyclic vector for multiplication operator $M_{z}$, i.e.,

$\bigvee\\{z^{n}F(z);\ n\in\mathbb{N}\\}=H^{2},$

and denote by $\mathfrak{F}$ the set of all outer functions.

For any $f\in H^{\infty}$, there is a canonical inner-outer factorization
decomposition $f=uF$, where $F$ is an outer function and $u$ is an inner
function which is unique up to a scalar of modulus $1$. Through the present
paper, we assume that $u$ is an inner function with $u^{(n_{0})}(0)>0$ where
$n_{0}$ is the smallest nonnegative integer such that $u^{(n_{0})}(0)$ is non-
vanishing, which fixes the choice of the inner function in the inner-outer
factorization decomposition and is called a normalized inner function.
Furthermore, let $Q_{inn}$ be the mapping

$Q_{inn}:H^{\infty}\rightarrow\mathfrak{I},\ \ \ \ Q_{inn}(f)=u,$

and let $Q_{out}$ be the mapping

$Q_{out}:H^{\infty}\rightarrow\mathfrak{F},\ \ \ \ Q_{out}(f)=F.$

A natural question is to ask whether the mapping $Q_{inn}$ or $Q_{out}$ is
continuous in some certain norm? Unfortunately, V. Kabaila [8] showed that
neither $Q_{inn}$ nor $Q_{out}$ is continuous in $H^{p}$-norm, $1\leq
p<\infty$.

In [3], R. Douglas and C. Pearcy made a study on the topology of the invariant
subspaces of certain bounded linear operators, where the distance of two
invariant subspaces is the norm of the difference of the corresponding
orthogonal projections. If $p$ is a nontrivial projection such that $p(H^{2})$
is invariant under $M_{z}$ multiplication by the coordinate function, then the
Beurling theorem gives an inner function $\varphi$ in $H^{\infty}$ such that
$p(H^{2})=\varphi H^{2}$. This $\varphi$ is also unique up to a scalar of
modulus $1$, and $p=T_{\varphi}T^{*}_{\varphi}$, where $T_{\varphi}$ is the
Toeplitz operator induced by $\varphi$. Let $p_{t}$, $t\in[0,1]$, be a
(operator) norm continuous family of nontrivial projections such that
$p_{t}(H^{2})$ is invariant under $M_{z}$ for each $t$. In this paper, we
denote $\varphi_{t}$ the inner function each $t$ such that
$p_{t}(H^{2})=\varphi_{t}H^{2}$, and call $\varphi_{t}$ a cross section of
$p_{t}$ in $\mathfrak{I}$. In addition, we always denote $u_{t}$ the inner
function chosen for each $t$ such that $p_{t}(H^{2})=u_{t}H^{2}$ and
$u_{t}^{(n_{t})}(0)>0$, where $n_{t}$ is the smallest nonnegative integer such
that $u_{t}^{(n_{t})}(0)$ is non-vanishing, and call $u_{t}$ the normalized
cross section of $p_{t}$ in $\mathfrak{I}$.

The components of the set of inner functions has been considered by Herrero in
[6] and [7], and by Nestoridis in [14] and [15]. Let $\mathfrak{I}^{*}$
($CN^{*}$) be the open set in $H^{\infty}$ of functions of the form $f=uh$,
where $u$ is an inner function (Carleson-Newman Blaschke product) and
$f\in(H^{\infty})^{-1}$. Notice that a function $f$ belongs to
$\mathfrak{I}^{*}$ if and only if $h\in H^{\infty}$ and
$\inf\limits_{\xi\in\mathbb{T}}|\widetilde{f}(\xi)|>0$, that is $f\in
H^{\infty}$ and $\widetilde{f}\in(L^{\infty})^{-1}$. A result of Laroco [12]
asserts that the set $\mathfrak{I}^{*}$ is dense in $H^{\infty}$. A. Nicolau
and D. Suárez [16] studied the connected components of $\mathfrak{I}^{*}$ and
$CN^{*}$.

In this paper, we will consider the subsets of $\mathfrak{I}^{*}$ with the
same multiplicity of the zero point at $0$. For any $f\in H^{\infty}$, denote
by $\textrm{Mul}_{0}(f)$ be the multiplicity of the zero point of $f$ at $0$,
more precisely,

$\textrm{Mul}_{0}(f)=\inf\\{n;f^{(n)}(0)\neq 0\\}.$

Furthermore, we denote

$\mathfrak{I}^{*}_{n}=\\{f\in\mathfrak{I}^{*};\textrm{Mul}_{0}(f)=n\\},\ \ \
\text{for any}\ n=0,1,2,\ldots.$

Notice that

$\mathfrak{I}^{*}_{n}=\\{z^{n}f;f\in\mathfrak{I}^{*}_{0}\\}.$

In the present paper, we study the $H^{2}$-norm continuity of $Q_{inn}$ and
$Q_{out}$ on $\mathfrak{I}^{*}_{n}$ and $\mathfrak{I}^{*}$ in section $2$. In
section $3$, we study the the $H^{\infty}$-norm discontinuity of $Q_{inn}$ and
$Q_{out}$ on $\mathfrak{I}^{*}_{n}$, and show the nonexistence of continuous
cross section from invariant subspaces to inner functions under essential
supremum norm. In the final section $4$, we also consider the continuity
problem of $Q_{inn}$ and $Q_{out}$ on $\textrm{Hol}(\overline{\mathbb{D}})$,
the set of all analytic functions in the closed unit disk.

##  2\. Continuity of inner-outer factorization

Following from the proof of Theorem $7.1$ in [5], one can see that if
$p\mathbf{1}\neq 0$, where $\mathbf{1}$ is the constant function $1$ in
$H^{2}$, then $u=\frac{p\mathbf{1}}{\|p\mathbf{1}\|}$ is the normalized inner
function of an orthogonal projection $p$ on an invariant subspace of $M_{z}$.
Consequently, $u$ is $H^{2}$-norm continuous with respect to $p$ for
$p\mathbf{1}(0)\neq 0$.

######  Lemma 2.1 ([5], p. 80).

Suppose that $p_{t}\mathbf{1}(0)\neq 0$ for all $t$. The normalized cross
section $u_{t}$ of $p_{t}$ in $\mathfrak{I}$ is continuous in $H^{2}$-norm.

Let $\mathfrak{M}$ and $\mathfrak{N}$ be two nontrivial subspaces in a Hilbert
space $\mathcal{H}$, $p_{\mathfrak{M}}$ and $p_{\mathfrak{N}}$ be the
orthogonal projections on $\mathfrak{M}$ and $\mathfrak{N}$, respectively. The
gap (aperture) between subspaces $\mathfrak{M}$ and $\mathfrak{N}$ defined as,
e.g., [3] and [9],

${\rm
gap}(\mathfrak{M},\mathfrak{N})=\|p_{\mathfrak{M}}-p_{\mathfrak{N}}\|=\max\\{\|p_{\mathfrak{M}}p_{\mathfrak{N}^{\bot}}\|,\|p_{\mathfrak{N}}p_{\mathfrak{M}^{\bot}}\|\\}$

is used to measure the distance between subspaces.

The maximal angle $\theta_{\rm max}(\mathfrak{M},\mathfrak{N})$ between
$\mathfrak{M}$ and $\mathfrak{N}$ was introduced in [11] and is defined as the
angle in $[0,\pi/2]$ given by

$\sin\theta_{\rm
max}(\mathfrak{M},\mathfrak{N})=\sup\limits_{x\in\mathfrak{M},\|x\|=1}{\rm
dist}(x,\mathfrak{N})=\sup\limits_{x\in\mathfrak{M},\|x\|=1}\sqrt{1-\|p_{\mathfrak{N}}(x)\|^{2}}.$

######  Lemma 2.2 ([1] or [10]).

$\|p_{\mathfrak{M}}-p_{\mathfrak{N}}\|=\max\\{\sin\theta_{\rm
max}(\mathfrak{M},\mathfrak{N}),\sin\theta_{\rm
max}(\mathfrak{N},\mathfrak{M})\\}.$

######  Theorem 2.3.

For any nonnegative integer $n$, the maps
$Q_{inn}:(\mathfrak{I}^{*}_{n},\|\cdot\|_{\infty})\rightarrow(\mathfrak{I},\|\cdot\|_{2})$
and
$Q_{out}:(\mathfrak{I}^{*}_{n},\|\cdot\|_{\infty})\rightarrow(\mathfrak{F},\|\cdot\|_{2})$
are continuous in the $H^{2}$-norm.

###### Proof.

Notice that for any nonnegative integer $n$,

$\mathfrak{I}^{*}_{n}=\\{z^{n}f;f\in\mathfrak{I}^{*}_{0}\\}.$

It suffices to consider the case of $n=0$.

Given any element $f$ in $\mathfrak{I}^{*}_{0}$. There is a canonical
decomposition $f=uF$, where $u=Q_{inn}(f)$ is a normalized inner function and
$F=Q_{out}(f)$ is an outer function. Since $F$ is an invertible element in
$H^{\infty}$, there exist two positive constants $c_{1}$ and $C_{1}$, such
that

$0<c_{1}\leq\|F\|_{\infty}\leq C_{1}<+\infty.$

Similarly, for an element $g$ in $\mathfrak{I}^{*}_{0}$, we may write $g=vG$,
where $v=Q_{inn}(g)$ and $G=Q_{out}(g)$. Then, there are two positive
constants $c_{2}$ and $C_{2}$, such that

$0<c_{2}\leq\|G\|_{\infty}\leq C_{2}<+\infty.$

Now denote

$c=\min\\{c_{1},c_{2}\\}\ \ \ \ \text{and}\ \ \ \ C=\max\\{C_{1},C_{2}\\}.$

We will prove the continuity of
$Q_{inn}:(\mathfrak{I}^{*}_{0},\|\cdot\|_{\infty})\rightarrow(\mathfrak{I},\|\cdot\|_{2})$
and
$Q_{out}:(\mathfrak{I}^{*}_{0},\|\cdot\|_{\infty})\rightarrow(\mathfrak{F},\|\cdot\|_{2})$,
respectively.

Part (1) The $H^{2}$-norm continuity of $Q_{inn}$ on $\mathfrak{I}^{*}_{0}$.

Denote $\mathfrak{M}$ the nontrivial invariant subspace $u(H^{2})$ of
multiplication operator $M_{z}$ and denote $v(H^{2})$ by $\mathfrak{N}$. Let
$p_{\mathfrak{M}}$ and $p_{\mathfrak{N}}$ be the corresponding nontrivial
orthogonal projections from $H^{2}$ onto $\mathfrak{M}$ and $\mathfrak{N}$,
respectively. Since $F$ and $G$ are invertible in $H^{\infty}$, we have

$p_{\mathfrak{M}}(H^{2})=uH^{2}=fH^{2}\ \ \ \ \text{and}\ \ \ \
p_{\mathfrak{N}}(H^{2})=vH^{2}=gH^{2}.$

For any element $\varphi\in\mathfrak{M}$ with $\|\varphi\|_{2}=1$, there is an
element $h\in H^{2}$ such that $\varphi=uh$. Moreover, $\|h\|_{2}=1$ and
$\varphi=uF\cdot\frac{h}{F}=f\cdot\frac{h}{F}$. Since $g\cdot\frac{h}{F}$ is
an element in the subspace $\mathfrak{N}$, we have

$\displaystyle\textrm{dist}(\varphi,\mathfrak{N})\triangleq$
$\displaystyle\inf\\{\|\varphi-w\|_{2};~{}w\in\mathfrak{N}\\}$
$\displaystyle\leq$ $\displaystyle\|\varphi-g\cdot\frac{h}{F}\|_{2}$
$\displaystyle=$ $\displaystyle\|f\cdot\frac{h}{F}-g\cdot\frac{h}{F}\|_{2}$
$\displaystyle\leq$
$\displaystyle\|f-g\|_{\infty}\|\frac{1}{F}\|_{\infty}\|h\|_{2}$
$\displaystyle\leq$ $\displaystyle\frac{1}{c}\cdot\|f-g\|_{\infty}.$

By the arbitrariness of $\varphi$, we have

$\displaystyle\sup_{\varphi\in\mathfrak{M},\
\|\varphi\|_{2}=1}\textrm{dist}(\varphi,\mathfrak{N})\leq\frac{1}{c}\cdot\|f-g\|_{\infty}.$

Similarly, we have

$\displaystyle\sup_{\psi\in\mathfrak{N},\
\|\psi\|_{2}=1}\textrm{dist}(\psi,\mathfrak{M})\leq\frac{1}{c}\cdot\|f-g\|_{\infty}.$

By Lemma 2.2, one can see that

(2.1) $\|p_{\mathfrak{M}}-p_{\mathfrak{N}}\|=\max\\{\sin\theta_{\rm
max}(\mathfrak{M},\mathfrak{N}),\sin\theta_{\rm
max}(\mathfrak{N},\mathfrak{M})\\}\leq\frac{1}{c}\cdot\|f-g\|_{\infty}.$

Given any $\epsilon>0$. It follows from Lemma 2.1 that there is a positive
number $\epsilon^{\prime}$ such that when
$\|p_{\mathfrak{M}}-p_{\mathfrak{N}}\|\leq\epsilon^{\prime}$, we have
$\|u-v\|_{2}\leq\epsilon$.

Let $\delta=c\epsilon^{\prime}$. Then, when $\|f-g\|_{\infty}\leq\delta$, we
have

$\displaystyle\|p_{\mathfrak{M}}-p_{\mathfrak{N}}\|\leq\frac{\delta}{c}=\frac{c\epsilon^{\prime}}{c}=\epsilon^{\prime}$

and consequently

$\displaystyle\|u-v\|_{2}\leq\epsilon.$

Therefore, the map
$Q_{inn}:(\mathfrak{I}^{*}_{0},\|\cdot\|_{\infty})\rightarrow(\mathfrak{I},\|\cdot\|_{2})$
is continuous in $H^{2}$-norm.

Part (2) The $H^{2}$-norm continuity of $Q_{out}$ on $\mathfrak{I}^{*}_{0}$.

Since

$\displaystyle f-g=$ $\displaystyle uF-vG$ $\displaystyle=$ $\displaystyle uF-
vF+vF-vG$ $\displaystyle=$ $\displaystyle F(u-v)+v(F-G),$

we have

$\displaystyle\|F-G\|_{2}$ $\displaystyle=\|v(F-G)\|_{2}$
$\displaystyle\leq\|f-g\|_{2}+\|F(u-v)\|_{2}$
$\displaystyle\leq\|f-g\|_{\infty}+\|F\|_{\infty}\|u-v\|_{2}$
$\displaystyle\leq\|f-g\|_{\infty}+C\|u-v\|_{2}.$

Given any $\eta>0$. It follows from Part (1) that there is a positive number
$\delta<\frac{\eta}{2}$ such that when $\|f-g\|_{\infty}\leq\delta$, we have
$\|u-v\|_{2}\leq\frac{\eta}{2C}$. Then, we have

$\|F-G\|_{2}\leq\delta+C\cdot\frac{\eta}{2C}\leq\frac{\eta}{2}+\frac{\eta}{2}=\eta.$

Therefore, the map
$Q_{out}:(\mathfrak{I}^{*}_{0},\|\cdot\|_{\infty})\rightarrow(\mathfrak{F},\|\cdot\|_{2})$
is continuous in $H^{2}$-norm. ∎

In the above theorem, the setting of $\mathfrak{I}^{*}_{n}$ is necessary.
Neither $Q_{inn}:(H^{p},\|\cdot\|_{p})\rightarrow(\mathfrak{I},\|\cdot\|_{p})$
nor $Q_{out}:(H^{p},\|\cdot\|_{p})\rightarrow(\mathfrak{F},\|\cdot\|_{p})$ is
continuous, $1\leq p\leq\infty$. Kabaila [8] proved the discontinuity for the
case of $1\leq p<\infty$. The discontinuity for the case of $p=\infty$ could
be found in Corollary $3$ in [13] by Nakazi. Now, we will give a simple
example to show that neither
$Q_{inn}:(\mathfrak{I}^{*},\|\cdot\|_{\infty})\rightarrow(\mathfrak{I},\|\cdot\|_{p})$
nor
$Q_{out}:(\mathfrak{I}^{*},\|\cdot\|_{\infty})\rightarrow(\mathfrak{F},\|\cdot\|_{p})$
is continuous for any $1\leq p\leq\infty$, which implies that
$\mathfrak{I}^{*}_{n}$ could not be extended to $\mathfrak{I}^{*}$ in the
Theorem 2.3.

######  Example 2.4.

Let $f_{t}(z)=\frac{t-z}{1-tz}$ for $t\in[-1,1]$. Obviously, $f_{t}(z)$ is a
continuous path in $\mathfrak{I}^{*}$. However,

$Q_{inn}(f_{t})=\left\\{\begin{array}[]{cc}-\frac{t-z}{1-tz}&\text{for}\
t\in[-1,0],\\\ \frac{t-z}{1-tz}&\text{for}\ t\in(0,1].\end{array}\right.$

and

$Q_{out}(f_{t})=\left\\{\begin{array}[]{cc}-1&\text{for}\ t\in[-1,0],\\\
1&\text{for}\ t\in(0,1].\end{array}\right.$

Then, neither $Q_{inn}(f_{t})$ nor $Q_{out}(f_{t})$ is continuous at $t=0$ in
$H^{p}$-norm, for any $1\leq p\leq\infty$.

##  3\. Nonexistence of continuous cross sections from invariant subspaces to
inner functions in the supremum norm

In this section, we will show the discontinuity of
$Q_{inn}:(\mathfrak{I}^{*}_{n},\|\cdot\|_{\infty})\rightarrow(\mathfrak{I},\|\cdot\|_{\infty})$
and
$Q_{out}:(\mathfrak{I}^{*}_{n},\|\cdot\|_{\infty})\rightarrow(\mathfrak{F},\|\cdot\|_{\infty})$.
Moreover, we will illustrate the nonexistence of continuous cross section from
invariant subspaces to inner functions in the supremum norm. To prove those,
we need some lemmas as preliminaries.

Firstly, Nicolau and Suárez have obtained a result with regard to path in
$\mathfrak{I}^{*}$ in the supremum norm.

######  Lemma 3.1 (Proposition 4.5 in [16]).

Let $f$, $g\in\mathfrak{I}^{*}$. Then there is a normalized inner function $b$
(in fact, it is a CNBP) such that $bf$ and $bg$ can be joined by a polygonal
path contained in $\mathfrak{I}^{*}$ in the supremum norm. Moreover, if $f$,
$g\in CN^{*}$, then $b$ can be chosen such that $bf$ and $bg$ can be joined by
a polygonal path contained in $CN^{*}$.

For a polygonal path, we have an intuitive observation as follows.

######  Lemma 3.2.

Let $\gamma(t):[0,1]\rightarrow\mathbb{C}$ be a polygonal path. Then for any
$\mathrm{e}^{\mathbf{i}\eta}$ which is not parallel to any segment of the
polygonal path $\gamma$, there exists $\epsilon_{0}>0$ such that for any
$0<\epsilon\leq\epsilon_{0}$, the path
$\gamma(t)+\epsilon\mathrm{e}^{\mathbf{i}\eta}$ does not pass through $0$.

###### Proof.

The proof is simple. Since a polygonal path is composed of finite segments,
one can see the conclusion from the following figure.

Figure 1.

∎

Following from the above simple lemma, we could strengthen the conclusion of
Nicolau and Suárez (Lemma 3.1) from $\mathfrak{I}^{*}$ to
$\mathfrak{I}^{*}_{0}$.

######  Lemma 3.3.

Let $h_{0},h_{1}\in(H^{\infty})^{-1}$. Then there exists a normalized inner
function $\varphi$ such that $\varphi h_{0}$ and $\varphi h_{1}$ can be joined
by a path contained in $\mathfrak{I}^{*}_{0}$ in the supremum norm.

###### Proof.

By Lemma 3.1, there is a normalized inner function $b$ such that $bh_{0}$ and
$bh_{1}$ can be joined by a polygonal path $f_{t}(z)$ in $\mathfrak{I}^{*}$,
$t\in[0,1]$. Write $f_{t}=u_{t}h_{t}$, where $u_{t}$ is the normalized inner
function part of $f_{t}$ and $h_{t}$ is the outer function part of $f_{t}$.

Whenever $b(0)=0$ or $b(0)>0$, for a fixed number $r\in(0,1)$, we have
$b(0)+r>0$. Let $\varphi(z)=\frac{b(z)+r}{1+rb(z)}$. Then, $\varphi(0)>0$ and

$g_{t}(z)=\frac{b(z)+(1-t)r}{1+(1-t)rb(z)},\ \ \ \ t\in[0,1],$

is a continuous path from $\varphi$ to $b$ in $\mathfrak{I}$ in the supremum
norm, and

$\widehat{g}_{t}(z)=\frac{b(z)+tr}{1+trb(z)},\ \ \ \ t\in[0,1],$

is the inverse path of $g_{t}$. Furthermore, let

$\widetilde{f}_{t}(z)=\left\\{\begin{array}[]{cc}g_{4t}(z)h_{0}(z)&\text{for}\
t\in[0,\frac{1}{4}],\\\ f_{4t-1}(z)&\text{for}\
t\in[\frac{1}{4},\frac{1}{2}],\\\ \widehat{g}_{2t-1}(z)h_{1}(z)&\text{for}\
t\in[\frac{1}{2},1].\end{array}\right.$

Then, $\widetilde{f}_{t}(z)$ is a path from $\varphi h_{0}$ to $\varphi h_{1}$
in $\mathfrak{I}^{*}$. Since the unit closed interval is compact, there is a
positive number $\epsilon_{0}$ such that

$0<\epsilon_{0}<\inf\limits_{\xi\in\mathbb{T}}\textrm{ess}\widetilde{f}_{t}(\xi)\
\ \ \ \text{for all}\ t\in[0,1].$

Moreover, let $\gamma(t)=\widetilde{f}_{t}(0)$, $t\in[0,1]$. By Lemma 3.1 and
the construction of $\widetilde{f}_{t}(z)$, one can see that $\gamma(t)$ is a
polygonal path in $\mathbb{C}$. By Lemma 3.2, there exist a number $\eta$ and
a positive number $\epsilon$ with
$0<\epsilon<\min\\{\epsilon_{0},|\varphi(0)h_{0}(0)|,|\varphi(0)h_{1}(0)|\\}$
such that the path $\gamma(t)+\epsilon\mathrm{e}^{\mathbf{i}\eta}$ does not
pass through $0$. Consequently,

$\widetilde{f}_{t}(z)+\epsilon\mathrm{e}^{\mathbf{i}\eta}$

is a path from $\varphi h_{0}+\epsilon\mathrm{e}^{\mathbf{i}\eta}$ to $\varphi
h_{1}+\epsilon\mathrm{e}^{\mathbf{i}\eta}$ in $\mathfrak{I}^{*}_{0}$. Notice
that $\varphi h_{0}+t\epsilon\mathrm{e}^{\mathbf{i}\eta}$, $t\in[0,1]$, is a
path from $\varphi h_{0}$ to $\varphi
h_{0}+\epsilon\mathrm{e}^{\mathbf{i}\eta}$ in $\mathfrak{I}^{*}_{0}$, and
$\varphi h_{1}+(1-t)\epsilon\mathrm{e}^{\mathbf{i}\eta}$, $t\in[0,1]$, is a
path from $\varphi h_{1}+\epsilon\mathrm{e}^{\mathbf{i}\eta}$ to $\varphi
h_{1}$ in $\mathfrak{I}^{*}_{0}$. Then, we could obtain a path from $\varphi
h_{0}$ to $\varphi h_{1}$ in $\mathfrak{I}^{*}_{0}$. ∎

######  Theorem 3.4.

Neither
$Q_{inn}:(\mathfrak{I}^{*}_{n},\|\cdot\|_{\infty})\rightarrow(\mathfrak{I},\|\cdot\|_{\infty})$
nor
$Q_{out}:(\mathfrak{I}^{*}_{n},\|\cdot\|_{\infty})\rightarrow(\mathfrak{F},\|\cdot\|_{\infty})$
are continuous in the supremum norm.

###### Proof.

It suffices to consider the case of $n=0$.

As well known, there exist $h_{0},h_{1}\in(H^{\infty})^{-1}$ such that they
can not be joined by a path in $(H^{\infty})^{-1}$ (see [4] or [2] for
example). More precisely, one can choose

$h_{0}=\mathbf{1}\ \ \ \ \text{and}\ \ \ \
h_{1}(z)=\mathrm{e}^{\frac{2\mathbf{i}}{\pi}\log\frac{1+z}{1-z}}$

as required. By Lemma 3.3, there exists a normalized inner function $\varphi$
such that $\varphi h_{0}$ and $\varphi h_{1}$ can be joined by a path
$f_{t}(z)$, $t\in[0,1]$, contained in $\mathfrak{I}^{*}_{0}$ in the supremum
norm.

Notice that $Q_{out}(f_{t})$ belongs to $(H^{\infty})^{-1}$ for each
$t\in[0,1]$. If
$Q_{out}:(\mathfrak{I}^{*}_{n},\|\cdot\|_{\infty})\rightarrow(\mathfrak{F},\|\cdot\|_{\infty})$
is continuous, $Q_{out}(f_{t})$ is a path from $h_{0}$ to $h_{1}$ in
$(H^{\infty})^{-1}$. That is a contradiction.

Now suppose that
$Q_{inn}:(\mathfrak{I}^{*}_{n},\|\cdot\|_{\infty})\rightarrow(\mathfrak{I},\|\cdot\|_{\infty})$
is continuous. Then, $Q_{inn}(f_{t})$ is a path in $\mathfrak{I}$. Let

$C\triangleq\sup\limits_{t\in[0,1]}\|f_{t}\|_{\infty}=\sup\limits_{t\in[0,1]}\|Q_{out}(f_{t})\|_{\infty}.$

For any $s,t\in[0,1]$, since

$\displaystyle f_{s}-f_{t}=$ $\displaystyle u_{s}F_{s}-u_{t}F_{t}$
$\displaystyle=$ $\displaystyle u_{s}F_{s}-u_{t}F_{s}+u_{t}F_{s}-u_{t}F_{t}$
$\displaystyle=$ $\displaystyle F_{s}(u_{s}-u_{t})+u_{t}(F_{s}-F_{t}),$

we have

(3.1) $\displaystyle\|F_{s}-F_{t}\|_{\infty}$
$\displaystyle=\|u_{t}(F_{s}-F_{t})\|_{\infty}$ (3.2)
$\displaystyle\leq\|f_{s}-f_{t}\|_{\infty}+\|F_{s}\|_{\infty}\|u_{s}-u_{t}\|_{\infty}$
(3.3) $\displaystyle\leq\|f_{s}-f_{t}\|_{\infty}+C\|u_{s}-u_{t}\|_{\infty}.$

Then, the continuity of $f_{t}$ and $u_{t}=Q_{inn}(f_{t})$ implies the
continuity of $F_{t}=Q_{out}(f_{t})$. That means $F_{t}=Q_{out}(f_{t})$ is a
path from $h_{0}$ to $h_{1}$ in $(H^{\infty})^{-1}$, which is a contradiction.
∎

######  Theorem 3.5.

There exists a norm continuous family of nontrivial projections $p_{t}$ such
that $p_{t}(H^{2})$ is invariant for each $t$, $t\in[0,1]$, such that there is
no continuous cross section of $p_{t}$ in $\mathfrak{I}$ in the supremum norm.

###### Proof.

Similar to the proof of the previous theorem, let

$h_{0}=\mathbf{1}\ \ \ \ \text{and}\ \ \ \
h_{1}(z)=\mathrm{e}^{\frac{2\mathbf{i}}{\pi}\log\frac{1+z}{1-z}},$

and let $f_{t}(z)$, $t\in[0,1]$, be a path from $\varphi h_{0}$ to $\varphi
h_{1}$ in $\mathfrak{I}^{*}_{0}$ for some certain normalized inner function
$\varphi$. Furthermore, let $\mathfrak{M}_{t}$ be the subspace
$f_{t}\mathbb{H}^{2}$, and let $p_{t}$ be the orthogonal projection onto
$\mathfrak{M}_{t}$. Following from the inequality (2.1), the continuity of
$f_{t}$ in the supremum norm implies the continuity of $p_{t}$ in the operator
norm.

Suppose that $\varphi_{t}(z)$ is a supremum norm continuous cross section of
$p_{t}$ in $\mathfrak{I}$, which is not necessary to be normalized. Write
$f_{t}(z)=\varphi_{t}(z)G_{t}(z)$ for $t\in[0,1]$. As the same as inequalities
(3.1)-(3.3), one can see for any $s,t\in[0,1]$,

$\|G_{s}-G_{t}\|_{\infty}\leq\|f_{s}-f_{t}\|_{\infty}+C\|\varphi_{s}-\varphi_{t}\|_{\infty},$

where $C=\sup\limits_{t\in[0,1]}\|f_{t}\|_{\infty}$ is a positive constant.
Then, $G_{t}$ is a path from $h_{0}$ to $h_{1}$ in $(H^{\infty})^{-1}$, which
is a contradiction. Therefore, there is no continuous cross section of such
$p_{t}$ in $\mathfrak{I}$ in the supremum norm. ∎

######  Remark 3.6.

The above Theorem 3.5 tells us that the supremum norm is quite different from
$\mathbb{H}^{2}$-norm. Replace $\mathbb{H}^{2}$-norm by the supremum norm in
Lemma 2.1, the conclusion will no longer hold true, even if the normalized
restriction is removed.

##  4\. On analytic functions in the closed disk

In this section, we will consider $\textrm{Hol}(\overline{\mathbb{D}})$
instead of $\mathfrak{I}^{*}$. For any nonnegative integer $n$, denote

$\textrm{Hol}_{n}(\overline{\mathbb{D}})=\\{f\in\textrm{Hol}(\overline{\mathbb{D}});\textrm{Mul}_{0}(f)=n\\}.$

######  Theorem 4.1.

Given any nonnegative integer $n$. The map
$Q_{out}:(\textrm{Hol}_{n}(\overline{\mathbb{D}}),\
\|\cdot\|_{\infty})\rightarrow(\mathfrak{F},\|\cdot\|_{\infty})$ is continuous
in the supremum norm, but not the map
$Q_{inn}:(\textrm{Hol}_{n}(\overline{\mathbb{D}}),\|\cdot\|_{\infty})\rightarrow(\mathfrak{I},\|\cdot\|_{\infty})$.
Both
$Q_{inn}:(\textrm{Hol}_{n}(\overline{\mathbb{D}}),\|\cdot\|_{\infty})\rightarrow(\mathfrak{I},\|\cdot\|_{2})$
and
$Q_{out}:(\textrm{Hol}_{n}(\overline{\mathbb{D}}),\|\cdot\|_{\infty})\rightarrow(\mathfrak{F},\|\cdot\|_{2})$
are continuous in the $H^{2}$-norm.

###### Proof.

Notice that for any nonnegative integer $n$,

$\textrm{Hol}_{n}(\overline{\mathbb{D}})=\\{z^{n}f;f\in\textrm{Hol}_{0}(\overline{\mathbb{D}})\\}.$

It suffices to consider the case of $n=0$.

(1) The continuity of $Q_{out}:(\textrm{Hol}_{0}(\overline{\mathbb{D}}),\
\|\cdot\|_{\infty})\rightarrow(\mathfrak{F},\|\cdot\|_{\infty})$

Given any element $f$ in $\textrm{Hol}_{0}(\overline{\mathbb{D}})$. Then $f$
has finite zero points in $\overline{\mathbb{D}}\setminus\\{0\\}$. More
precisely, $f$ has zero points $\\{\alpha_{k}\\}_{k=1}^{n}$ in $\mathbb{D}$
and $\\{z_{s}\\}_{s=1}^{m}$ in $\partial\mathbb{D}$. Consequently, $f$ has a
canonical decomposition,

$f=B_{f}F_{f}=\prod\limits_{k=1}^{n}\frac{|\alpha_{k}|}{\alpha_{k}}\frac{\alpha_{k}-z}{1-\overline{\alpha_{k}}z}\cdot
F_{0}\cdot\prod\limits_{s=1}^{m}(z_{s}-z),$

in which
$Q_{inn}(f)=B_{f}=\prod\limits_{k=1}^{n}\frac{|\alpha_{k}|}{\alpha_{k}}\frac{\alpha_{k}-z}{1-\overline{\alpha_{k}}z}$
is the normalized inner function part of $f$ and
$Q_{out}(f)=F_{f}=F_{0}\prod\limits_{s=1}^{m}(z_{s}-z)$ is the outer function
part of $f$. Denote $F_{1}=\prod\limits_{s=1}^{m}(z_{s}-z)$ and then
$F_{f}=F_{0}F_{1}$. Since $F_{0}$ is an invertible element in $H^{\infty}$,
there exist two positive numbers $C_{1}$ and $C_{2}>1$ such that

$0<C_{1}\leq\|F_{0}\|_{\infty}\leq C_{2}<+\infty.$

By Rouché’s theorem, if $g\in\textrm{Hol}_{0}(\overline{\mathbb{D}})$ is a
small perturbation of $f$, then $g$ and $f$ have the same number of zeros in a
small neighborhood of $\overline{\mathbb{D}}$. In fact, each zero point of $g$
is close to a zero point of $f$. Furthermore, we may assume that $g$ has zero
points $\\{\widetilde{\alpha_{k}}\\}_{k=1}^{n}$ in $\mathbb{D}$,
$\\{\widetilde{z_{s}}\\}_{s=1}^{m_{1}}$ in $\mathbb{D}$ and
$\\{\widetilde{z_{s}}\\}_{s=m_{1}+1}^{m}$ out of $\mathbb{D}$, which are close
to $\\{\alpha_{k}\\}_{k=1}^{n}$, $\\{z_{s}\\}_{s=1}^{m_{1}}$ and
$\\{z_{s}\\}_{s=m_{1}+1}^{m}$, respectively. Then, denote

$\widetilde{B}_{f}=\prod_{k=1}^{n}\frac{|\widetilde{\alpha_{k}}|}{\widetilde{\alpha_{k}}}\frac{\widetilde{\alpha_{k}}-z}{1-\overline{\widetilde{\alpha_{k}}}z},$
$G^{inn}_{F_{1}}=\prod_{s=1}^{m_{1}}(\widetilde{z_{s}}-z),$

and

$G^{out}_{F_{1}}=\prod_{s=m_{1}+1}^{m}(\widetilde{z_{s}}-z).$

Moreover, denote

$\widehat{G^{inn}_{F_{1}}}=\prod_{s=1}^{m_{1}}\frac{|\widetilde{z_{s}}|}{\overline{\widetilde{z_{s}}}}(1-\overline{\widetilde{z_{s}}}z)$

and

$\widetilde{B}_{F}=\frac{G^{inn}_{F_{1}}}{\widehat{G^{inn}_{F_{1}}}}=\prod_{s=1}^{m_{1}}\frac{\overline{\widetilde{z_{s}}}}{|\widetilde{z_{s}}|}\frac{(\widetilde{z_{s}}-z)}{(1-\overline{\widetilde{z_{s}}}z)}.$

Then, $Q_{inn}(g)=B_{g}=\widetilde{B}_{f}\widetilde{B}_{F}$ is the normalized
inner function part of $g$ and
$Q_{out}(g)=F_{g}=\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}G_{0}$ is the outer
function part of $g$, in which $G_{0}$ is an invertible element in
$H^{\infty}$. So one could write $g$ as follows

$\displaystyle g$ $\displaystyle=B_{g}F_{g}$
$\displaystyle=\widetilde{B}_{f}\widetilde{B}_{F}\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}G_{0}$
$\displaystyle=\prod_{k=1}^{n}\frac{|\widetilde{\alpha_{k}}|}{\widetilde{\alpha_{k}}}\frac{\widetilde{\alpha_{k}}-z}{1-\overline{\widetilde{\alpha_{k}}}z}\prod_{s=1}^{m_{1}}\frac{\overline{\widetilde{z_{s}}}}{|\widetilde{z_{s}}|}\frac{(\widetilde{z_{s}}-z)}{(1-\overline{\widetilde{z_{s}}}z)}\prod_{s=1}^{m_{1}}\frac{|\widetilde{z_{s}}|}{\overline{\widetilde{z_{s}}}}(1-\overline{\widetilde{z_{s}}}z)\prod_{s=m_{1}+1}^{m}(\widetilde{z_{s}}-z)G_{0}.$

Given any $\epsilon>0$. Let

$M=\epsilon+\max\\{\|\prod\limits_{s=1}^{m_{1}}(z_{s}-z)\|_{\infty},\
\|\prod\limits_{s=m_{1}+1}^{m}(z_{s}-z)\|_{\infty}\\}.$

Since the zero points of $g$ tend to the zero points of $f$ when $g$ tends to
$f$ in the supremum norm, there exists a positive number
$\delta<\frac{\epsilon}{2}$ such that if
$g\in\textrm{Hol}_{0}(\overline{\mathbb{D}})$ with $\|f-g\|\leq\delta$, then
each $\widetilde{\alpha_{k}}$ is sufficiently close to $\alpha_{k}$ such that

(4.1)
$\|B_{f}-\widetilde{B}_{f}\|_{\infty}\leq\frac{\epsilon}{2C_{2}(M^{2}+4M)},$

and each $\widetilde{z_{s}}$ is sufficiently close to $z_{s}$ such that

(4.2)
$\displaystyle\|G^{inn}_{F_{1}}-\prod\limits_{s=1}^{m_{1}}(z_{s}-z)\|_{\infty}\leq\frac{\epsilon}{2C_{2}(M^{2}+4M)},$
(4.3)
$\displaystyle\|\widehat{G^{inn}_{F_{1}}}-\prod_{s=1}^{m_{1}}\frac{|z_{s}|}{\overline{z_{s}}}(1-\overline{z_{s}}z)\|_{\infty}\leq\frac{\epsilon}{2C_{2}(M^{2}+4M)},$
(4.4)
$\displaystyle\|G^{out}_{F_{1}}-\prod\limits_{s=m_{1}+1}^{m}(z_{s}-z)\|_{\infty}\leq\frac{\epsilon}{2C_{2}(M^{2}+4M)}.$

Notice that $z_{s}\in\partial\mathbb{D}$, for $s=1,\ldots,m$. Then, it is easy
to see that

$\displaystyle\frac{|z_{s}|}{\overline{z_{s}}}(1-\overline{z_{s}}z)=\frac{1}{\overline{z_{s}}}(\overline{z_{s}}z_{s}-\overline{z_{s}}z)=z_{s}-z,$

and consequently we could rewrite the inequality (4.3) as

(4.5)
$\displaystyle\|\widehat{G^{inn}_{F_{1}}}-\prod_{s=1}^{m_{1}}(z_{s}-z)\|_{\infty}\leq\frac{\epsilon}{2C_{2}(M^{2}+4M)}.$

Following from the inequalities (4.2), (4.5) and (4.4), one can see that

$\|G^{inn}_{F_{1}}\|\leq\|\prod\limits_{s=1}^{m_{1}}(z_{s}-z)\|_{\infty}+\frac{\epsilon}{2C_{2}(M^{2}+4M)}\leq
M,$ $\|\widehat{G^{inn}_{F_{1}}}\|_{\infty}\leq M\ \ \ \text{and}\ \ \
\|G^{out}_{F_{1}}\|_{\infty}\leq M.$

Furthermore, together with inequality (4.1), we have

$\displaystyle\|F_{1}-\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}\|_{\infty}$
$\displaystyle=$
$\displaystyle\|\prod\limits_{s=1}^{m_{1}}(z_{s}-z)\prod\limits_{s=m_{1}+1}^{m}(z_{s}-z)-\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\|\prod\limits_{s=1}^{m_{1}}(z_{s}-z)\prod\limits_{s=m_{1}+1}^{m}(z_{s}-z)-\prod\limits_{s=1}^{m_{1}}(z_{s}-z)G^{out}_{F_{1}}\|_{\infty}+\|\prod\limits_{s=1}^{m_{1}}(z_{s}-z)G^{out}_{F_{1}}-\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\|\prod\limits_{s=1}^{m_{1}}(z_{s}-z)\|_{\infty}\|\prod\limits_{s=m_{1}+1}^{m}(z_{s}-z)-G^{out}_{F_{1}}\|_{\infty}+\|G^{out}_{F_{1}}(z)\|_{\infty}\|\prod\limits_{s=1}^{m_{1}}(z_{s}-z)-\widehat{G^{inn}_{F_{1}}}\|_{\infty}$
$\displaystyle\leq$ $\displaystyle\frac{2M\epsilon}{2C_{2}(M^{2}+4M)}$

and similarly,

$\displaystyle\|F_{1}-{G^{inn}_{F_{1}}}G^{out}_{F_{1}}\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\|\prod\limits_{s=1}^{m_{1}}(z_{s}-z)\|_{\infty}\|\prod\limits_{s=m_{1}+1}^{m}(z_{s}-z)-G^{out}_{F_{1}}\|_{\infty}+\|G^{out}_{F_{1}}(z)\|_{\infty}\|\prod\limits_{s=1}^{m_{1}}(z_{s}-z)-{G^{inn}_{F_{1}}}\|_{\infty}$
$\displaystyle\leq$ $\displaystyle\frac{2M\epsilon}{2C_{2}(M^{2}+4M)}.$

Consequently,

$\displaystyle\|B_{f}F_{1}-\widetilde{B}_{f}G^{inn}_{F_{1}}G^{out}_{F_{1}}\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\|B_{f}F_{1}-B_{f}G^{inn}_{F_{1}}G^{out}_{F_{1}}\|_{\infty}+\|B_{f}G^{inn}_{F_{1}}G^{out}_{F_{1}}-\widetilde{B}_{f}G^{inn}_{F_{1}}G^{out}_{F_{1}}\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\|B_{f}\|_{\infty}\|F_{1}-G^{inn}_{F_{1}}G^{out}_{F_{1}}\|_{\infty}+\|G^{inn}_{F_{1}}G^{out}_{F_{1}}\|_{\infty}\|B_{f}-\widetilde{B}_{f}\|_{\infty}$
$\displaystyle=$
$\displaystyle\|F_{1}-G^{inn}_{F_{1}}G^{out}_{F_{1}}\|_{\infty}+\|G^{inn}_{F_{1}}\|_{\infty}\|G^{out}_{F_{1}}\|_{\infty}\|B_{f}-\widetilde{B}_{f}\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\frac{(M^{2}+2M)\epsilon}{2C_{2}(M^{2}+4M)}.$

In addition, we have

$\displaystyle f-g$
$\displaystyle=B_{f}F_{1}F_{0}-B_{g}\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}G_{0}$
$\displaystyle=(B_{f}F_{1}F_{0}-B_{g}\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}F_{0})+(B_{g}\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}F_{0}-B_{g}\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}G_{0})$

then

$\displaystyle\|\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}F_{0}-\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}G_{0}\|_{\infty}$
$\displaystyle=$
$\displaystyle\|B_{g}(\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}F_{0}-\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}G_{0})\|_{\infty}$
$\displaystyle=$
$\displaystyle\|(f-g)-(B_{f}F_{1}F_{0}-B_{g}\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}F_{0})\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\|f-g\|_{\infty}+\|F_{0}\|_{\infty}\|B_{f}F_{1}-B_{g}\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\delta+C_{2}\cdot\frac{(M^{2}+2M)\epsilon}{2C_{2}(M^{2}+4M)}$
$\displaystyle\leq$
$\displaystyle\frac{\epsilon}{2}+\frac{(M^{2}+2M)\epsilon}{2(M^{2}+4M)}.$

Therefore,

$\displaystyle\|F_{f}-F_{g}\|_{\infty}$ $\displaystyle=$
$\displaystyle\|F_{1}F_{0}-\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}G_{0}\|_{\infty}$
$\displaystyle=$
$\displaystyle\|F_{1}F_{0}-\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}F_{0}+\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}F_{0}-\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}G_{0}\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\|F_{0}\|_{\infty}\|F_{1}-\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}\|_{\infty}+\|\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}F_{0}-\widehat{G^{inn}_{F_{1}}}G^{out}_{F_{1}}G_{0}\|_{\infty}$
$\displaystyle\leq$ $\displaystyle
C_{2}\cdot\frac{2M\epsilon}{2C_{2}(M^{2}+4M)}+\frac{\epsilon}{2}+\frac{(M^{2}+2M)\epsilon}{2(M^{2}+4M)}$
$\displaystyle=$ $\displaystyle\epsilon.$

So $f\to g$ in the supremum norm implies $F_{f}\to F_{g}$ in the supremum
norm, which means the map
$Q_{out}:(\textrm{Hol}_{n}(\overline{\mathbb{D}}),\|\cdot\|_{\infty})\rightarrow(\mathfrak{F},\|\cdot\|_{\infty})$
is continuous.

(2) The discontinuity of $Q_{inn}:(\textrm{Hol}_{0}(\overline{\mathbb{D}}),\
\|\cdot\|_{\infty})\rightarrow(\mathfrak{I},\|\cdot\|_{\infty})$

However, the map
$Q_{inn}:(\textrm{Hol}_{0}(\overline{\mathbb{D}}),\|\cdot\|_{\infty})\rightarrow(\mathfrak{I},\|\cdot\|_{\infty})$
is not continuous in the supremum norm. Here we give an example to show the
discontinuity. Let $f_{t}(z)=t-z$ for $t\in[\frac{1}{2},2]$. Then $f_{t}$ is a
continuous path in $\textrm{Hol}(\overline{\mathbb{D}})$. The inner part of
$f_{1}$ is $Q_{inn}(f_{1})=1$ and the inner part of $f_{t}$, for
$t\in[\frac{1}{2},1)$, is $Q_{inn}(f_{t})=\frac{t-z}{1-tz}$. It is easy to see
that as $t\to 1$,

$\|f_{t}-f_{1}\|_{\infty}=|1-t|\rightarrow 0$

but

$\|Q_{inn}(f_{t})-Q_{inn}(f_{1})\|_{\infty}=\|\frac{t-z}{1-tz}-1\|_{\infty}=2\nrightarrow
0.$

(3) The continuity of
$Q_{inn}:(\textrm{Hol}_{n}(\overline{\mathbb{D}}),\|\cdot\|_{\infty})\rightarrow(\mathfrak{I},\|\cdot\|_{2})$

Next, we want to prove that
$Q_{inn}:(\textrm{Hol}_{n}(\overline{\mathbb{D}}),\|\cdot\|_{\infty})\rightarrow(\mathfrak{I},\|\cdot\|_{2})$
is continuous in the $H^{2}$-norm, which is different from the case in the
supremum norm.

Recall that $Q_{inn}(g)=B_{g}=\widetilde{B}_{f}\widetilde{B}_{F}$ and
$Q_{inn}(f)=B_{f}$. Notice that if $g$ tends to $f$ in the supremum norm, then
each $\widetilde{\alpha_{k}}$ is sufficiently close to $\alpha_{k}$. More
precisely, there exists a positive number $\delta^{\prime}<\frac{\epsilon}{2}$
such that if $g\in\textrm{Hol}_{0}(\overline{\mathbb{D}})$ with
$\|f-g\|\leq\delta^{\prime}$, we have

(4.6) $\|B_{f}-\widetilde{B}_{f}\|_{2}\leq\frac{\epsilon}{2}$

and by $z_{s}\in\mathbb{T}$ for $s=1,2,\ldots,m_{1}$,

(4.7)
$|\prod\limits_{s=1}^{m_{1}}|\widetilde{z_{s}}|-1|=|\prod\limits_{s=1}^{m_{1}}|\widetilde{z_{s}}|-\prod\limits_{s=1}^{m_{1}}|z_{s}||\leq\frac{\epsilon^{2}}{8}.$

Write $\widetilde{B}_{F}$ as follows

$\widetilde{B}_{F}(z)=\sum_{k=0}^{\infty}c_{k}z^{k},$

where $c_{k}$ is the $k$-th Taylor coefficient of $\widetilde{B}_{F}$.

Since
$\widetilde{B}_{F}(z)=\prod_{s=1}^{m_{1}}\frac{\overline{\widetilde{z_{s}}}}{|\widetilde{z_{s}}|}\frac{(\widetilde{z_{s}}-z)}{(1-\overline{\widetilde{z_{s}}}z)}$
is an inner function, it is not difficult to see that

$\displaystyle\|\widetilde{B}_{F}-1\|_{2}^{2}$
$\displaystyle=\|\sum_{k=0}^{\infty}c_{k}z^{k}-1\|_{2}^{2}$
$\displaystyle=\|c_{0}-1+\sum_{k=1}^{\infty}c_{k}z^{k}\|_{2}^{2}$
$\displaystyle\leq|c_{0}-1|^{2}+|\sum_{k=1}^{\infty}c_{k}|^{2}$
$\displaystyle=|c_{0}-1|^{2}+\sum_{k=0}^{\infty}|c_{k}|^{2}-|c_{0}|^{2}$
$\displaystyle=|\prod\limits_{s=1}^{m_{1}}|\widetilde{z_{s}}|-1|^{2}+1-\left(\prod\limits_{s=1}^{m_{1}}|\widetilde{z_{s}}|\right)^{2}$
$\displaystyle\leq\frac{\epsilon^{4}}{64}+1-(1-\frac{\epsilon^{2}}{8})^{2}$
$\displaystyle=\frac{\epsilon^{2}}{4},$

that is

(4.8) $\|\widetilde{B}_{F}-1\|_{2}<\frac{\epsilon}{2}.$

Consequently, it follows from inequalities (4.6) and (4.8) that

$\displaystyle\|B_{f}-B_{g}\|_{2}$
$\displaystyle=\|B_{f}-\widetilde{B}_{f}\widetilde{B}_{F}\|_{2}$
$\displaystyle=\|B_{f}-B_{f}\widetilde{B}_{F}+B_{f}\widetilde{B}_{F}-\widetilde{B}_{f}\widetilde{B}_{F}\|_{2}$
$\displaystyle\leq\|B_{f}-B_{f}\widetilde{B}_{F}\|_{2}+\|B_{f}\widetilde{B}_{F}-\widetilde{B}_{f}\widetilde{B}_{F}\|_{2}$
$\displaystyle=\|\widetilde{B}_{F}-1\|_{2}+\|B_{f}-\widetilde{B}_{f}\|_{2}$
$\displaystyle<\frac{\epsilon}{2}+\frac{\epsilon}{2}$ $\displaystyle=\epsilon$

So $f\to g$ in the $H^{2}$-norm implies $B_{f}\to B_{g}$ in the $H^{2}$-norm,
which means the map
$Q_{inn}:(\textrm{Hol}_{n}(\overline{\mathbb{D}}),\|\cdot\|_{\infty})\rightarrow(\mathfrak{I},\|\cdot\|_{2})$
is continuous.

(4) The continuity of
$Q_{out}:(\textrm{Hol}_{n}(\overline{\mathbb{D}}),\|\cdot\|_{\infty})\rightarrow(\mathfrak{F},\|\cdot\|_{2})$

At the end, since
$Q_{out}:(\textrm{Hol}_{n}(\overline{\mathbb{D}}),\|\cdot\|_{\infty})\rightarrow(\mathfrak{F},\|\cdot\|_{\infty})$
is continuous in the supremum norm, it is obvious that
$Q_{out}:(\textrm{Hol}_{n}(\overline{\mathbb{D}}),\|\cdot\|_{\infty})\rightarrow(\mathfrak{F},\|\cdot\|_{2})$
is also continuous in the $H^{2}$-norm. ∎

## Declarations

Ethics approval

Not applicable.

Competing interests

The author declares that there is no conflict of interest or competing
interest.

Authors’ contributions

All authors contributed equally to this work.

Funding

There is no funding source for this manuscript.

Availability of data and materials

Data sharing is not applicable to this article as no data sets were generated
or analyzed during the current study.

## References

  * [1] A. BöTtcher, I. M Spitkovsky, A gentle guide to the basics of two projections theory, Linear Algebra Appl., 432(6) (2010), 1412-1459.
  * [2] K. R. Davidson, Nest Algebras, Pittman Research Notes in Mathematics Series 191, Longman Scientific $\&$ Technical, Harlow; copublished in the United States with John Wiley $\&$ Sons, Inc., New York, 1988.
  * [3] R. G. Douglas, C. Pearcy, On a Topology for Invariant Subspaces, J. Funct. Anal., 2 (1968), 323-341.
  * [4] T. W. Gamelin, Uniform algebras, Prentice Hall Inc. englewood Cliffs N.j, 1984.
  * [5] J. B. Garnett, Bounded Analytic Functions, revised first edition, GTM 236, Springer-Verlag, New York, 2007.
  * [6] D. A. Herrero, Inner functions under uniform topology, Pacific Z. Math. 51 (1974), 167-175.
  * [7] D. A. Herrero, Inner functions under uniform topology II, Rev. Un. Mat. Argentina, 28 (1976), 23-35.
  * [8] V. Kabaila, Continuous factorization of functions in $H^{p}$, Lith. Math. J., 17(4) (1977), 519-522.
  * [9] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995.
  * [10] A. Knyazev, A. Jujunashvili, M. Argentati, Angles Between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods, J. Funct. Anal., 259(6) (2010), 1323-1345.
  * [11] M. Krein, M. Krasnoselski, D. Milman, On the defect numbers of operators in Banach spaces and on some geometric questions, Trudy Inst. Mat. Akad. Nauk Ukrain. SSR 11 (1948) 97-112. (in Russian)
  * [12] L. A. Laroco, Stable rank and approximation theorems in $H^{\infty}$, T. Am. Math. Soc. 327 (1991), 815-832.
  * [13] T. Nakazi, Factorizations of outer functions and extremal problems, P. Edinburgh Math. Soc., 39(3) (1996), 535-546.
  * [14] V. Nestoridis, Inner functions invariant connected components, Pacific J. Math., 83 (2) (1979), 473-480.
  * [15] V. Nestoridis, Inner functions: noninvariant connected components, Pacific J. Math., 87 (1) (1980), 199-209.
  * [16] A. Nicolau, D. Suárez, Paths of inner-related functions, J. Funct. Anal., 262 (9) 2010, 3749-3774.


# A Multi-Agent Framework for Ethically Translating Data to Knowledge

Anonymous Authors  Email InstituteStreet AddressCityStateCountryPostal Code

(2023)

###### Abstract.

We propose a new knowledge translation framework from which we derive novel
ethical considerations of discovery support systems. While intersecting with
traditional knowledge discovery and decision support systems, discovery
support systems delineate themselves by hypothesizing new knowledge as opposed
to recommending actions or detecting patterns. This nuance allows the human
collaborator to be seeded with new hypothesized knowledge that can be be
further refined, validated, or expounded upon either from within or outside
the system. As the power of generative AI becomes more pervasive in society,
particularly in academia and industry, it will inevitably penetrate human-AI
collaborations fundamental to discovery support use-cases. This calls for a
treatment of ethics in the context of these systems. We will center our
analysis around three principle stages in our knowledge translation model: (1)
the process of collecting data based on prior knowledge, (2) the analysis that
brings forth information from data, and (3) the validation of information and
acceptance of knowledge. These stages yield three primary ethical questions:
(1) what prior knowledge is required for ethical data sharing and how should
that knowledge be communicated, (2) what prior knowledge should be shared
regarding the mining procedure operated on the data, and (3) how do you ensure
those responsible for producing new knowledge consider the implications
surfaced by questions (1) and (2).

Generative AI, KDD, Ethics, Decision Support Systems

††copyright: acmcopyright††journalyear: 2023††doi:
XXXXXXX.XXXXXXX††conference: The 6th AAAI/ACM Conference on AI, Ethics, and
Society; August 9–11 2023; Montreal, CA††ccs: Social and professional topics
Computing / technology policy††ccs: General and reference Metrics††ccs:
Applied computing Law††ccs: Computing methodologies Knowledge representation
and reasoning††ccs: Computing methodologies Artificial intelligence

##  1\. Introduction

The mission of science is the robust and efficient discovery and translation
of knowledge. However, this process has historically been inefficient due to
the inherently disjoint nature of the enterprise formed of multiple agents
acting asynchronously with varied levels of collaboration (if at all). Yet,
this asynchronous and disjoint behavior is natural as it is unrealistic for
any single agent to be competent at all aspects of the knowledge discovery and
translation processes, i.e., the processes by which raw data is collected via
experimentation, mined for patterns to develop hypotheses, and tested to
validate knowledge assertions that are disseminated via publication. However,
with recent advances in the field of AI, particularly in deep learning, it is
becoming increasingly clear that AI will transform these processes and usher
in this new field of AI for Science (AI4Science) (wang_scientific_2023) with
systems capable of being end-to-end experts in knowledge translation and
discovery.

While this transformation is revolutionary, we contend that AI4Science has the
potential to exacerbate issues, both ethical and practical, regarding
effective collaboration and knowledge dissemination within the scientific
community. For instance, self-driving labs, being a natural manifestation of
AI4Science, pose to have the remarkable capacity to automate knowledge
discovery, but what happens when these systems are closed? Further, how will
we mitigate misaligned incentives and motivations that limit the dissemination
of derived findings? In essence, what of openness in AI for Science?

It is critical we address these questions before such systems arrive. To that
end, we present an extension of AI4Science under a collaborative multi-agent
framing of Open Science (ramachandran_open_2021), namely AI for Open Science
(AI4OS). We represent AI4OS as a Multi-agent Discovery Support System (MaDiSS)
with the explicit intent of maximizing discovery for all stakeholders,
transcending the limitations of isolated, self-serving systems that could
arise in AI4Science implementations agnostic to openness.

While definitions of Open Science vary, we adopt that Open Science is “a
collaborative culture facilitated by technology, promoting the open exchange
of _data_ , _information_ , and _knowledge_ among the scientific community and
the general public, ultimately expediting scientific research and enhancing
comprehension” (ramachandran_open_2021). Crystallizing this sentiment with
established definitions of _data_ , _information_ , and _knowledge_ from the
Knowledge Discovery and Data Mining (KDD) (fayyad_kdd_1996;
fayyad_knowledge_1996) community, we develop a formal language to characterize
each stage of the knowledge translation process. Leveraging this language, we
derive a theoretical optimization metric for openness to build an ethical
argument supporting AI4OS. Lastly, we use MaDiSS to offer recommendations for
promoting openness in self-driving labs and other AI-driven systems.

Contributions This work makes three primary contributions:

  1. (1)

Introduce a formal language for discussing issues of openness in AI4Science,

  2. (2)

Develop a corresponding openness metric,

  3. (3)

Construct a robust ethical argument in support of this framing.

##  2\. Background and Related Work

###  2.1. The Knowledge Discovery and Data Mining Process

Figure 1. An Overview of the KDD process.

KDD provides useful and precise definitions of data, information, and
knowledge, central to our definition of Open Science. The KDD literature
(fayyad_knowledge_1996; mitra_data_2002) defines data as a set of examples
collected using experimentation and a pattern as an expression in some
language describing a subset of data exemplars that is shorter than the
enumeration of the entire dataset. We extend these definitions to include
information as a set of these patterns that yield knowledge, a label placed on
this information. The KDD process can be summarized in the following steps and
is depicted in Figure 1.

  1. (1)

Learning the application domain and acquiring prior knowledge necessary for
the goals of the application,

  2. (2)

Creating a target dataset which includes the selection or filtering over a
subset of features or samples,

  3. (3)

Data cleaning and preprocessing,

  4. (4)

Data reduction, projection, and transformation,

  5. (5)

Choosing the function of data mining, i.e., the required task that will be
solved such as summarization, classification, regression, etc.,

  6. (6)

Selecting the data mining algorithm and appropriate parameters required to run
the model on the desired data,

  7. (7)

Data mining or searching for patterns of interest in a particular
representation form such as clustering, dependency analysis, rules, etc.,

  8. (8)

Interpretation of the generated patterns and translating the useful ones into
terms understandable by the user.

As stated by Fayyad et al. (fayyad_kdd_1996), the KDD process is iterative and
may contain loops between any two steps in the process. For our purposes, the
main limitation of KDD is that it assumes a single-agent perspective. However,
we argue that a single-agent perspective is insufficient to model Open Science
as it is inherently collaborative among many disjoint agents. Rather, it is
this multi-agent communication/collaboration that is crucial to a successful
knowledge translation. Therefore, we should consider three additional elements
of the knowledge translation process, which to the best of our knowledge, have
not been formulated into a single coherent framework. These elements
correspond to the successful communication of prior knowledge (1) from the
experimenter to the data miner and/or knowledge interpreter, (2) from the data
miner to the knowledge interpreter, and (3) the decisive review of all
propagated knowledge by the interpreter.

###  2.2. Discovery Support Systems

As our representation of AI4OS is formulated as a MaDiSS, we believe a brief
review of traditional Discovery Support Systems (DiSS) is warranted. The first
mention of DiSS came in 1986 when Don R. Swanson developed a literature-based
procedure for generating new hypotheses focused on biomedical information
(swanson_fish_1986). The first hypothesis emerging from his process proposed
that fish oil could be used to cure Raynaud’s disease and was later tested as
well as experimentally and clinically proved (swanson_fish_1986;
swanson_two_1987; swanson_link_1998). This research launched the field of
Literature-Based Discovery (LBD) (gordon_toward_1996; lindsay_literature-
based_1999; mack_text-based_2002; bareinboim_recovering_2015) to infer new and
useful knowledge by logically connecting information fragments from disparate
textual sources. As LBD has developed into an immense field, we point the
reader to Gopalakrishnan et. al. (gopalakrishnan_survey_2019) for a complete
review of recent LBD approaches focused on the biomedical domain and for more
general applications to Thilakaratne et al. (thilakaratne_systematic_2019) and
Hue et al. (hui_application_2019). While LDB techniques can be leveraged in
the context of our definition of MaDiSS, our framework and discussion are more
general than any individual methodology for discovery. Further, these
approaches do not focus on our main contribution which is the knowledge
communication between and consideration of each agent in the translation
process.

A prime example of a modern DiSS as well as a MaDiSS is the US National
Institute of Health (NIH) National Center for Advancing Translational Science
(NCATS) Translator program
(the_biomedical_data_translator_consortium_biomedical_2019) which was founded
with the mission to create a “system capable of integrating existing
biomedical data sets and “translating” those data into insights to accelerate
translational research, generate new hypotheses, and drive innovations in
clinical care and drug discovery.” Importantly, the NCATS program takes an
Open Science approach where knowledge is shared and reasoned over by many
different independent teams via API calls (fecho_progress_2022). These types
of collaborative efforts come with their own set of ethical considerations as
they pertain to Open Science (peters_philosophy_2010; rubinstein_case_2020;
peters_open_2014; fox_open_2021; duwell_open_2019), Open Data
(stieglitz_when_2020; molloy_open_2011; zuiderwijk_open_2014; murray-
rust_open_2008) and the provenance of such systems (cheney_provenance_2009;
bhardwaj_datahub_2014; fecho_progress_2022) and our framing subsumes these
issues as well.

Figure 2. A knowledge flow diagram between three critical teams in the
knowledge translation process. Each variable $k$ represents a knowledge base
of each respective team in the process such that $\overline{k_{i}}$ denotes
all the knowledge of the experimenting team responsible for collecting a
dataset, $\underline{k_{j}}$ denotes the knowledge of the data mining team
responsible for KDD steps 1-7, and $\widecheck{k}_{l}$ represents the
knowledge of the labeling team. $k_{ijl}$ is the new knowledge that is labeled
by the labeling team and numbers 1-4 label the four potential communication
flows that can occur between the agents, not including potential loops.

##  3\. Enhancing Discovery Support with Data and Information Provenance

We define a Discovery Support System (DiSS) as:

> An AI system that supports the generation of useful hypotheses that aid in
> labeling new knowledge in-line with the set of true knowledge $K$.

In science, the knowledge translation process is inherently disjointed,
involving multiple agent collaborators and knowledge sharing. This is because
the same agent will likely not perform, or even be included, in all steps of
the knowledge translation process, i.e., the process by which raw data is
collected via experimentation, mined for patterns to create information, which
is then interpreted knowledge through appropriate labeling. For instance, you
may have a biomedical research team that decides to collect genomics data on
breast cancer patients. This team designs the experiments, selects the
population, and performs data aggregation. As is now common in the biomedical
and machine learning community, this research team then makes this dataset
open to others. Then, a separate data mining team then consumes this open data
and extracts potential patterns to form a set of information. From this point
a separate labeling team, such as a team of oncologists, are presented with
this mined information and asked to assign labels to any potential new
knowledge derivable from that information. As such, it can be helpful to
consider a DiSS system as a collaborative team effort where each team
(composed of humans, machines or a mix) has a duty to ”aid in the labeling of
new knowledge” by transferring their prior knowledge, used in their
contribution to the knowledge translation process, to downstream teams.
Moreover, while we argue that it is ethically imperative for teams to transfer
their prior knowledge to downstream teams, we also recognize a duty for teams
to ingest prior knowledge from their peers and ensure it is, ”in-line with the
set of true knowledge $K$”, or more likely, their approximation of $K$. We
draw from this intuition when deriving our objective function for DiSS in
Section 4.

Unlike the days of Swanson (swanson_fish_1986), high degrees of domain
specialization exacerbates the need for diverse teams. Each team has different
domain knowledge related to their disciplines, world knowledge based on their
lived experiences, as well as knowledge of assumptions regarding potential
biases introduced into their particular step of the knowledge translation
process. For instance, the experimenting team may have selected patients with
a demographic bias based on the team’s geographical location. The data mining
team may have selected a novel normalization step based on their mining
experience of similar datasets skewing the information obtained. Further
still, the knowledge labeling team may have a particular self-affirming bias
toward research and findings they hope to see or deduce from the mined
information.

In any case, it is clear knowledge sharing between these teams is essential
for appropriate knowledge labeling. Figure 2 depicts the primary channels of
knowledge flow between these various teams. Channel 1 represents the
communication between the experimenting team and data mining team regarding
the data collection and experimental design assumptions present in a dataset.
Naturally, the more knowledge the data mining team has in regards to
$\overline{k_{i}}$, the more contextualized the mined information will become.
Channel 2 corresponds to the knowledge flow of algorithmic bias, model
selection, etc., including domain and world knowledge of the data mining team
to the knowledge labeling team. This allows the labeling team to review
potential bias avoiding knowledge labeling mishaps. Similarly, channel 3
fosters the communication of knowledge from the experimenting team to the
labeling team, allowing the labeling team to account for any biases that yield
the mined information. Lastly, channel 4 (a culmination of channels 1-3)
depicts the ideal knowledge flow such that the labeling team has access to the
knowledge bases of both the experimenting team and the data mining team and
the data mining team has access to the knowledge base of the experimenting
team. Naturally, it is also possible for teams to develop simultaneous nested
knowledge translation processes such that their individual knowledge bases are
also increasing over time. In other words, this knowledge flow need not be
static and may contain loops such that agents can update and refine their own
knowledge bases at will.

These four channels illustrate the need for data and information provenance,
i.e., origination knowledge, that allows each team to make the best decisions
at each point in the process. In the next section, we will take these general
intuitions and form a concrete framework for knowledge translation based on
team communication throughout the knowledge process. This will culminate in a
clear argument validating these ideas in terms of the amount of new knowledge
generation and yield technical and societal solutions recommendations for
implementing these communication channels ethically and robustly.

##  4\. A Formal Language of AI for Open Science

In this section, our aim is to introduce a formal language to characterize
AI4OS. We propose that a useful way to frame AI4OS is through a MaDiSS that
aligns with the principles of Open Science. The primary motivation is that
such a framing should enable researchers to conduct ethical assessments of
AI4Science-inspired systems in a unified manner.

Our formalism begins by considering the three primary roles in a MaDiSS (and
KDD in general), which revolve around dataset curation, information
extraction, and knowledge labeling. Recognizing that multiple agents can
participate in each role, we define the knowledge required by each agent to
fulfill their specific role and represent the transfer of this knowledge to
downstream collaborators promoting an open exchange of insights and expertise
within their respective roles. These knowledge priors and roles are as
follows:

  1. (1)

The knowledge base of the experimenting agent, generating the raw data
associated with KDD Steps 1-4,

  2. (2)

The knowledge base of the data mining agent, in their considerations and
selections of the data mining algorithm of KDD Steps 5-7,

  3. (3)

The knowledge base of the labeling agent, responsible for accepting,
distilling and labeling the information patterns in KDD Step 8.

Figure 3. An overview of MaDiSS. The colored portions of the figure are
illustrative of an arbitrary instantiation of the translation process. (a) the
experimenting agent(s) responsible for constructing any particular dataset
$d_{i}$ via experimentation. All $d_{i}$ represents some sub-population of the
global ground truth data, $\hat{D}$ with some instrumentation error $\Delta$.
(b) The data mining agent(s) responsible for the standard KDD process up until
Step 7. (c) The labeling agent responsible for accepting/labeling new
knowledge as derived from information presented by the data mining agent.

We quickly define some common notation that we will assume and exploit later.
We assume the universe of knowledge consists of a (potentially infinite) set
of true knowledge $K$ and a set of false knowledge (also potentially infinite)
$K^{c}$. We let $A$ denote the set of all agents, human or otherwise, where
$\rho=|A|$. Each agent has an assumed set of prior knowledge which is a subset
of $K\cup K^{c}$. While operating over the three role types (experimenting,
mining and labeling) previously introduced, we allow agents to form teams in
which they can belong to multiple or no teams along each role axis. Therefore,
its possible that the same set of agents forming the same team can perform
each role in the pipeline. This situation is analogous to a self-driving lab.
However, as in the real world, it is natural that an experimenting agent may
have their data mined by many different data mining agents. Further, these
mined patterns can be labeled by many different labeling agents. Since we are
interested in the collaborative aspect of Open Science, MaDiSS provides a
formalism for the translation of knowledge among teams. A visual depiction of
MaDiSS is shown in Figure 3. In the following sections, we will outline each
role more formally.

###  4.1. The Experimenting Agent

In the experimenting process (Figure 3.a), experimenting agents form sets of
experimenting teams, $t_{i}\subseteq A$. Each team forms a team knowledge
base, $\overline{k_{i}}$, assumed to rectify conflicts from $K^{c}$ and $K$
between agents as it pertains to the experimenting process. Data sets,
$d_{i}\in D$, are collected by some means of experimentation which we assume
to be informed by $\overline{k_{i}}$.

Grounding ourselves in KDD formalisms, a team’s knowledge, $\overline{k_{i}}$,
is a precursor to Step 1 in the KDD process, which is to say,
$\overline{k_{i}}$ is a set of prior knowledge on a domain that motivates the
capture of data pertaining to that domain. Edges connecting $\overline{k_{i}}$
to their respective data, $d_{i}$, in Figure 3.a, can be thought of as a
sequence of decisions from steps 2-4 in the KDD processes used to aggregate
and refine data. While $\overline{k_{i}}$ can contain elements of $K^{c}$,
$d_{i}$ should not be considered as right or wrong as data will always
represent some sub-population of true data, $\hat{D}$, up to some measurement
noise $\Delta$. However, since $\overline{k_{i}}$ instantiates, $d_{i}$, it is
possible that the sub-population $d_{i}$ may not be suitable to achieve new
knowledge in-line with $K$.

###  4.2. The Data Mining Agent

In the mining process (Figure 3.b), data mining agents form data mining teams,
$t_{j}\subseteq A$. While the agent pool remains the same, we use different
team indices to denote the data mining teams. In other words, $t_{i}=t_{j}$ if
and only if $i=j$. Like _experimenting teams_ , we assume _data mining teams_
have a rectified set of knowledge $\underline{k_{j}}$. $\underline{k_{j}}$ is
used to inform the data mining approach(s), $F_{j}$, applied over any raw
dataset $d_{i}$ to extract patterns. We call the set of extracted patterns
information and denote it $I_{ij}$.

Under the KDD lens, the edges that connect $d_{i}$ and $\underline{k_{j}}$ to
$I_{ij}$ via $F_{j}$, in Figure 3.b, can be thought of as a sequence of
decisions from steps 5-7 in the KDD process used to extra patterns from data.
This includes (5) choosing the function of mining, (6) selecting appropriate
algorithms and parameters to produce models, and (7) searching for patterns by
leveraging these models, ultimately yielding $I_{ij}$. Not included in the KDD
pipeline is the explicit distinction between $\overline{k_{i}}$ and
$\underline{k_{j}}$. If $i=j$ then crucial team knowledge from the
experimenting processes is maintained during the data mining process. This is
assumed in the KDD pipeline as it is framed from the perspective of a single
agent. However, if $i\neq j$ then $\overline{k_{i}}$ is potentially lost,
resulting in potential misuse or misunderstanding of $d_{i}$. We argue that
this results in an $I_{ij}$ less suitable for achieving new knowledge in-line
with the $K$ and potentially subverts the intended use of $d_{i}$ in regards
to $\overline{k_{i}}$.

###  4.3. The Labeling Agent

In the labeling process (Figure 3.c), labeling agents form labeling teams,
$t_{l}\subseteq A$. Similar to data mining teams, we use different team
indices for labeling agents to capture distinct teams. That is to say,
$t_{i}=t_{j}=t_{l}$ if and only if $i=j=l$. Like the teams that came before,
we assume _labeling teams_ have a rectified set of knowledge
$\widecheck{k_{l}}$. The goal of labeling teams is to interpret patterns in
$I_{ij}$ informed by $\widecheck{k_{l}}$ and label those patterns to form a
set of new knowledge labels, $k_{ijl}$, where ideally, $k_{ijl}\subset K$,
though in practice $k_{ijl}$ likely contains elements in $K$ and $K^{c}$.

In Figure 3.c, the edges that connect $I_{ij}$ and $\widecheck{k_{l}}$ to
$k_{ijl}$ capture step 8 in the KDD pipeline, which produces a set of labeled
knowledge garnered from $I_{ij}$. Similarly to the data mining step, the KDD
pipeline does not formalize instances where knowledge is not propagated
forward into the labeling step. If $i=j=l$ knowledge is maintained (again,
this is assumed in KDD from the perspective of a single agent). However, if
$l\neq j$ then the labeling team runs the risk of misusing or misunderstanding
$I_{ij}$, unless $t_{j}$ translated $\overline{k_{j}}$ onto $t_{l}$. We argue
this potentially results in $k_{ijl}$ containing less elements in $K$ and more
elements in $K^{c}$. This can be further confounded when $j\neq i$ from the
mining processes, resulting in a less adequate $I_{ij}$ unless $t_{i}$
translated their $\underline{k_{i}}$ onto $t_{j}$. On the other hand, even if
$l=j$ or $t_{j}$ translated their $\overline{k_{j}}$ onto $t_{l}$, if $l\neq
i$, the labeling team runs the risk of not understanding the initial bias in
$d_{i}$ resulting from $\underline{k_{i}}$ unless $t_{i}$ translated their
$\underline{k_{i}}$ onto $t_{l}$.

##  5\. Optimizing an Openness Metric in AI for Science

To demonstrate the utility of MaDiSS, we use our language to construct an
argument of knowledge translation among teams asserting that it is imperative
teams pass along all provenance required in fulfilling their role in order to
optimize a novel openness metric. A crucial question that we have not yet
considered is whether an agent or team can truly know if any individual piece
of their knowledge is in $K$ or $K^{c}$. To address this we will make the
following assumption.

###### Assumption 1.

A human or machine cannot know with exact certainty whether any knowledge
element in their knowledge base is a member of $K$ or $K^{c}$.

We believe Assumption 1 is justified because no entity has direct access to
$K$, and thereby, cannot be fully certain on the membership of any individual
piece of knowledge in their knowledge base. Since no upstream team can exactly
know if their knowledge is in $K$ or $K^{c}$, they should always pass all
provenance including accompanying confidence scores, allowing downstream teams
to consider their entire context and rectify on their own. The remaining
portion of this section will provide a brief monotonicity argument to justify
this claim along with potential entry points to aid knowledge translation
between teams in line with the AI4OS mission.

Consider an arbitrary $d_{i}$ constructed by $t_{i}$ informed by
$\overline{k_{i}}$ as well as a data mining approach(s), $F_{j}$, of $t_{j}$
informed by $\underline{k_{j}}$. From the data mining team’s perspective, the
information, i.e., the set of patterns as defined by a set of chosen
languages, yielded by $F_{j}$, denoted $I_{ij}$, can be defined as the
function $F_{j}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{D}\times\mathcal{P}(K\cup
K^{c})\rightarrow\mathcal{P}(\mathbb{P})$ where $\mathbb{D}$ is the set of all
possible datasets and $\mathcal{P}(\mathbb{P})$ is the power set of all
possible patterns, $\mathbb{P}$, such that

(1)
$I_{ij}=F_{j}\bigg{(}d_{i},\underline{k_{j}}\cup\overline{k_{i}}\cup\bigcup_{h\in
H}\underline{k_{h}}\bigg{)}$

where $H$ is the set of other data mining team’s knowledge for which the team
$t_{j}$ may or may not have access. It should also be noted that $t_{j}$ may
or may not have access to $\overline{k_{i}}$. Similarly, the labeling team,
$t_{l}$, also has their interpretation of information, $I_{ijl}$, under their
knowledge base and all the other team’s knowledge bases of which they have
access. This information can be defined as another function
$G_{l}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{P}(\mathbb{P})\times\mathcal{P}(K\cup
K^{c})\rightarrow\mathcal{P}(\mathbb{P})$ such that

(2)
$I_{ijl}=G_{l}\bigg{(}I_{ij},\widecheck{k_{l}}\cup\underline{k_{j}}\cup\overline{k_{i}}\cup\bigcup_{h\in
H}\underline{k_{h}}\bigg{)}$

where $\underline{k_{j}}$ may also be empty if inaccessible. If the data
mining and labeling team are the same, $i=j$, then $I_{ij}=I_{ijl}$ because
both teams have the same set of combined knowledge. However, if these teams
are different, then it may not be the case that $I_{ij}=I_{ijl}$.

Consider a simple example in which the data mining team, $t_{j}$, selects only
one data mining approach. In other words, $||I_{ij}||=||I_{ijl}||=1$. For the
sake of this example, let the form of $I_{ij}$ and $I_{ijl}$ be a Knowledge
Graph (KG) with nodes representing entities in some system and edges
representing the relationships between those entities. For the data mining
team, $I_{ij}$ is a construction algorithm $F_{j}$ that learns the edges
(relationships) from $d_{i}$. Now consider the instance in which
$\widecheck{k_{l}}$ contains a piece of knowledge that is absent in
$\overline{k_{j}}$. This could be informed by some specific domain knowledge
that states that two potential entities cannot not be directly connected in
the KG. If $\overline{k_{j}}$ is missing this knowledge, and they produce
$I_{ij}$ with said edge, the labeling team could simply correct the
information based on their (assumed to be) more complete knowledge base by
removing the erroneous edge. However, this now transforms $I_{ij}$ into the
new set of information, $I_{ijl}$.

With the concept of information from the perspective of team $t_{l}$ in
Equation 2, we lastly define a function $L\mathrel{\mathop{\mathchar
58\relax}}\mathcal{P}(\mathbb{P})\rightarrow\mathcal{P}(K\cup K^{c})$ such
that

(3) $k_{ijl}=L(I_{ijl},\mathbf{K^{+}}\cup\mathbf{K^{-}})$

where

$\displaystyle\mathbf{K^{+}}$
$\displaystyle=\bigg{(}\widecheck{k_{l}}\cup\underline{k_{j}}\cup\overline{k_{i}}\cup\bigcup_{h\in
H}\underline{k_{h}}\bigg{)}\cap K$ $\displaystyle\mathbf{K^{-}}$
$\displaystyle=\bigg{(}\widecheck{k_{l}}\cup\underline{k_{j}}\cup\overline{k_{i}}\cup\bigcup_{h\in
H}\underline{k_{h}}\bigg{)}\cap K^{c}$

and $k_{ijl}\subset K\cup K^{c}$ is all the new knowledge translated from the
dataset, $d_{i}$, in accordance with the knowledge and information flow
between all the teams in MaDiSS. Equation 3 leads us to the following
assumption about $L$ given the following definition of a monotone sequence of
sets.

###### Definition 0.

Let $X$ be a set and $S\subseteq\mathcal{P}(X)$. A monotone sequence of sets
denoted $\langle B_{n}\rangle_{n\in\mathbb{N}}\in S$ is monotonically
increasing if $\forall n\in\mathbb{N}$ $B_{n}\subseteq B_{n+1}$ and is
monotonically decreasing if $\forall n\in\mathbb{N}$ $B_{n}\supseteq B_{n+1}$.

###### Assumption 2.

The quantity $\mathinner{\\!\left\lVert L(\cdot)\cap K\right\rVert}$ is
monotonically increasing with any monotonically increasing sequence of sets,
$\langle
K^{+}_{n}\rangle_{n\in\mathbb{N^{+}}}\subseteq\mathcal{P}(\mathbf{K^{+}})$,
and the quantity $\mathinner{\\!\left\lVert L(\cdot)\cap K^{c}\right\rVert}$
is monotonically increasing with any monotonically increasing sequence of
sets, $\langle
K^{-}_{n}\rangle_{n\in\mathbb{N^{-}}}\subseteq\mathcal{P}(\mathbf{K^{-}})$,
where $\mathbb{N^{+}}=[0,||K||]$ and $\mathbb{N^{-}}=[0,||K^{c}||]$.

In other words, Assumption 2 tells us that the more prior knowledge that is
incorporated from all teams in regard to $\mathbf{K^{+}}$ the greater amount
of new knowledge $k_{ijl}$ aligned with $K$ that is produced. Therefore, if
ever a new piece of knowledge $k\in K$ is included into $\mathbf{K^{+}}$, the
amount of new knowledge produced that is in $K$ will never decrease.
Similarly, the more prior knowledge that is included from all teams in regard
to $\mathbf{K^{-}}$ the greater amount of new knowledge $k_{ijl}$ aligned with
$K^{c}$ that is produced. We believe this is a justified assumption as
included knowledge in $K$ would never lead to a knowledge element $K^{c}$ and
vice versa.

Therefore, if all knowledge is equally important to discover, a natural metric
to gauge the success of a knowledge translation is to measure the quantity:

(4) $\mathinner{\\!\left\lVert\bigcup_{i,j,l}k_{ijl}\cap
K\right\rVert}-\mathinner{\\!\left\lVert\bigcup_{i,j,l}k_{ijl}\cap
K^{c}\right\rVert}$

The goal of knowledge translation, and by extension, a MaDiSS and AI4OS, is to
maximize Equation 4. However, recall that Assumption 1 states that the
membership of elements in $k_{ijl}$ is unknowable, so Equation 4 is not
directly computable. Therefore, it is the job of a MaDiSS to approximate $K$,
and by extension $K^{c}$. In fact, each team’s knowledge resembles their best
approximation of a subset of $K$ already. However, on receiving translated
knowledge from upstream, we conclude that a successful MaDiSS should help
process, filter, reframe, and/or reweight this knowledge to increase the
expectation that the amount of knowledge in $K^{+}$ is maximized and the
amount in $K^{-}$ is minimized. We argue this will maximize Equation 4 under
Assumption 2. Lastly, two extensions to Equation 4 could be to normalize the
metric with respect to the cardinality of $k_{ijl}$ in order to compare across
domains as well as relax our assumption that all knowledge is equally
important.

##  6\. Why Openness in AI for Science

The pursuit of openness in AI4OS can be viewed as an effort to maximize
knowledge dissemination, achieved through the effective conveyance of
essential data and algorithmic provenance to the team responsible for labeling
knowledge as well as its intermediaries (Equation 4). Further, the
optimization of this process, which can better align generated knowledge with
ground truth, can only be achieved by open knowledge sharing among
collaborators. Therefore, we assert that the primary ethical concerns in
AI4Science revolve around open access to and open communication of AI-derived
knowledge discoveries.

Broad communication of knowledge discoveries is not a new ethical
consideration and has been handled extensively by the Open Science community
(peters_philosophy_2010; rubinstein_case_2020; peters_open_2014;
fox_open_2021; duwell_open_2019). However, a critical facet of analyzing the
ethics of these systems involves examining their increased speed and autonomy.
If these systems become entirely self-driven, the rapid acceleration of
discovery may overwhelm our society’s capacity to absorb it, as exemplified by
the challenge of digesting the vast research output from predatory journals
and conferences (grudniewicz_predatory_2019; richtig_problems_2018).
Accelerating knowledge discovery through AI4Science without due consideration
could further burden researchers and hinder scholarship.

Another ethical concern is the risk of restricting open access to AI-derived
scientific discoveries by industry, governments, and the military. These
entities may attempt to silo AI4Science implementations for their benefit,
limiting the knowledge available for labeling exclusively to their
organizational unit. To uphold operationally moral (dignum_responsible_2017)
standards, society should ensure diverse agent participation in knowledge
translation teams and agents through increased regulation, auditing, and
disclosure (chan_gpt-3_2023). Given the current self-regulatory environment of
most AI companies (galaski_ai_2021), AI ethicists must continue to press for
specific enforceable and non-voluntary legislation (hagendorff_ethics_2020;
hagendorff_ai_2022) of AI4Science systems focusing their accessibility,
transparency, and openness.

Additionally, we must consider how denial of access could exacerbate existing
societal inequities via socio-technical forces. For instance, if new knowledge
generated by a DiSS is used to accelerate individual knowledge acquisition
rather than novel scientific discovery, denying access to such systems based
on social class or identity could further marginalize underprivileged
populations, perpetuating systemic disparities. For example, consider an AI-
driven system that aids in childhood education by increasing the speed at
which a child acquires new knowledge. Then denial of access based on class or
identity could further limit a marginalized child’s performance. This effect
would fortify and continually widen systemic inequities among historically
oppressed communities.

Shifting our focus to specific ethical considerations within the MaDiSS
formulation, we emphasize the importance of the experimenting team documenting
and transparently communicating assumptions, biases, and contextual factors
affecting dataset collection, processing, or experimentation. This information
is crucial for both the knowledge labeling process and data mining. Neglecting
to provide this information may lead to mislabeled knowledge and exacerbate
ethical concerns. While initiatives like ”datasheets for datasets” are steps
in the right direction (gebru_datasheets_2021), a robust provenance
architecture for communicating this knowledge remains an open challenge.

The significance of Open Data initiatives (stieglitz_when_2020;
molloy_open_2011; zuiderwijk_open_2014; murray-rust_open_2008) and the role of
provenance in databases (cheney_provenance_2009; bhardwaj_datahub_2014) has
been extensively studied, particularly within the realm of biomedical data.
However, the adoption of these practices within the machine learning community
as part of the knowledge translation process has been notably sluggish and is
poised to face further challenges with the rise of AI-driven systems. An
associated area that has received inadequate attention, evident from a MaDiSS
perspective, is the imperative to communicate data mining or algorithmic
provenance to upstream consumers. The current focus in this domain has
primarily centered on the reproducibility of machine learning algorithms
(heil_reproducibility_2021; pineau_improving_2021). However, MaDiSS
underscores the need to go beyond reproducibility and ensure that downstream
labeling teams consider the underlying algorithmic assumptions, biases, and
contexts. Building on the framework proposed by Gebru et al., this could be
implemented through an ”information sheet for information systems” approach,
fostering community collaboration to identify intents, assumptions, and
contexts that must be captured to enhance the successful knowledge translation
process and supports the adoption of AI4OS.

Our final ethical consideration pertains to the role of the labeling team in
acknowledging and considering the provenance provided by the experimenting and
data mining teams. This aspect relies on human autonomy, as knowledge labelers
can choose to ignore provenance. As human autonomy in the knowledge
translation process becomes curtailed by AI4Science manifestations like self-
driving labs, this becomes increasingly problematic. Consequently, fostering
ethical work cultures becomes imperative, wherein all providential knowledge
is not only valued but also demanded from downstream agents. A pragmatic
societal solution involves enhancing digital literacy among labeling teams,
emphasizing the significance of these communication channels, and aligning
with the contextualist recommendations for AI use as presented by Chan
(chan_gpt-3_2023).

Nevertheless, this requirement to thoughtfully consider provenance encounters
socio-technical resistance. AI professionals and researchers who currently
leverage DiSS and soon AI4Science systems inevitably contend with external
pressures that may hinder their consideration of information or data context.
These pressures could stem from impending deadlines, managerial demands, or
market forces. Therefore, it becomes crucial for researchers to prioritize
knowledge transfer, ensuring that their respective labels are well-informed to
avert potentially detrimental consequences. Most significantly, this
underscores a concluding ethical consideration directed at the developers of
AI4Science.

##  7\. Implementation Recommendations for Future Work

Our KTF is meant to serve as a foundational approach that depicts the
collaborative and interoperable translation processes of data to information
to knowledge. While the KDD process has been considered from a multi-agent
perspective (pop_multi-agent_2006; kadar_automated_2017), to the best of our
knowledge this framework is the first to formalize knowledge discovery from
both a multi-agent perspective as well as accounting for prior knowledge
propagation along each step of agent role hierarchy. While this framework is
designed to be generalizable, there are still gaps we wish to address. In
short, these are as follows:

  1. (1)

Provide technical solutions to the knowledge transfer problem among the
collaborative teams of the knowledge translation processes,

  2. (2)

Relax the assumed rectified team knowledge priors and formalize the process of
how their knowledge is rectified (or not) for a team,

  3. (3)

Formalize qualitative success for the knowledge translation process.

Our first point stems from the issue of knowledge transfer between the
experimenting, data mining and labeling teams. We showed that these prior
knowledge sets form monotonic relationships over resulting information,
ultimately leading to the objective of DiSS in Equation 4. We discussed
societal implications of the knowledge transfer problem (equitable access to
DiSS systems and siloing knowledge within organizations) and even further
societal solutions to the knowledge transfer problem (legislation in support
of Open Science principles, provenance chains maintained by researchers and
educating researchers to rigorously test new knowledge). However, the question
remains, to what extent can knowledge transfer be automated. We discussed
before “datasheets for datasets” (gebru_datasheets_2021) as a way to have
authors produce questionnaires related to intent, composition, collection,
use, distribution and maintenance. What would it look like for a system to
automate this processes and require subsequent teams to respond to these
questionnaires before accessing data. We provide no technical solutions to
this problem, allowing this to be extended in the future.

Our second point stems from our assumption of rectified knowledge. While it is
ideal for teams to agree on all prior knowledge on experimenting, mining, and
labeling, it is likely in practice that among teams there will be conflict
whether or not they are aware of it. Further, not every agent is guaranteed to
influence each aspect of experimenting, mining or labeling. One agent might
employ knowledge from $K^{c}$ on some aspect of the experimenting processes,
while another would have employed knowledge from $K$, but because their jobs
were partitioned, no rectification would have occurred. For this reason it is
relevant to consider instances where teams may or may not be able to rectify
knowledge. We think that framing this under the perspective of measuring
interference (positive or negative) (hoc_towards_2001) would be readily
digestible into the DiSS framework.

Our final point stems from the need to access a DiSS qualitatively. For
example, Cooke et al. (cooke_measuring_2003) measures team knowledge after
applying a knowledge transfer process called cross-training (hands-on practice
of tasks from other team members) and conceptual cross-training (sharing of
mental models from the perspective of team-member roles). However, there is a
need for automated frameworks to accomplish such evaluations of a DiSS.
Further, these assessment strategies should capture some sense of Value-
Sensitive Design so that DiSS aligns with appropriate ethical considerations.
In other words, the quality of a DiSS should measure whether or not human
values are either supported or diminished by a particular design
(cummings_integrating_2006). This suggests the need for evaluating over
benchmark datasets that span ethical considerations like labeling of hate
speech (elsherief_latent_2021) or concepts such as justice, well-being,
duties, virtues, and commonsense morality (hendrycks_aligning_2020).

##  8\. Conclusions and Future Work

In this paper, we introduced AI for Open Science using a Multi-agent Discovery
Support formalism that aligns with Open Science ideals and highlights the need
for openness within the AI for Science community. We demonstrated the power of
this formalism to make assertions that appropriate labeling of new knowledge
is monotonically related to the degree to which the labeling agent understands
the intent, context, and assumptions of the upstream teams. This imposes a
clear ethic on all teams operating within an AI for Open Science paradigm to
propagate their prior knowledge to all downstream teams. Finally, we assessed
ethical considerations of openness in the context of AI for Science justifying
the need for systems to adopt an AI for Open Science framing.

Additionally, we believe there are several fruitful directions for future
work. First, Assumption 2 must be empirically validated, perhaps through human
team-based experimentation. Second, we need to assess as a field the degree to
which current AI-driven systems are effectively closed or open. However, for
this to be accomplished we must develop a proxy for Equation 4 that is
computable and provably optimizes our openness metric.