# Banach algebra mappings preserving the invertibility of linear pencils

Francois Schulz  Department of Mathematics and Applied Mathematics, Faculty of
Science, University of Johannesburg, P.O. Box 524, Auckland Park, 2006, South
Africa<EMAIL_ADDRESS>
###### Abstract.

Let $A$ and $B$ be complex unital Banach algebras, and let $\varphi,\psi:A\to
B$ be surjective mappings. If $A$ is semisimple with an essential socle and
$\varphi$ and $\psi$ preserves the invertibility of linear pencils in both
directions, that is, for any $x,y\in A$ and $\lambda\in\mathbb{C}$, $\lambda
x+y$ is invertible in $A$ if and only if $\lambda\varphi(x)+\psi(y)$ is
invertible in $B$, then we show that there exists an invertible element $u$ in
$B$ and a Jordan isomorphism $J:A\to B$ such that $\varphi(x)=\psi(x)=uJ(x)$
for all $x\in A$.

###### Key words and phrases:

Banach algebra, rank, trace, invertibility preserving mappings, Jordan
isomorphism

###### 2020 Mathematics Subject Classification:

46H05; 46H10; 47A05; 47A10

##  1\. Introduction

Throughout $A$ and $B$ shall denote complex unital Banach algebras, where we
will always use $\mathbf{1}$ to denote the identity of an algebra under
consideration. The group of invertible elements of a Banach algebra $A$ shall
be denoted by $G(A)$. For any element $x$ in a unital Banach algebra $A$,
recall that its spectrum is given by

$\sigma(x)=\\{\lambda\in\mathbb{C}:\lambda\mathbf{1}-x\notin G(A)\\}.$

Moreover, we use $\sigma^{\prime}(x)=\sigma(x)\setminus\\{0\\}$ and
$\rho(x)=\sup_{\lambda\in\sigma(x)}|\lambda|$ to denote its nonzero spectrum
and spectral radius, respectively. For the Jacobson radical of a Banach
algebra $A$ we shall write $\operatorname{rad}(A)$. We say that $A$ is
semisimple if $\operatorname{rad}(A)=\\{0\\}$. If $A$ is semisimple, then it
has a smallest two-sided ideal containing all minimal left and right ideals,
called the socle of $A$ which we denote by $\operatorname{soc}(A)$. Lastly we
mention that if $A$ is semisimple, then $\operatorname{soc}(A)$ is essential
if and only if its left annihilator is trivial.

The study of linear mappings preserving the invertibility or eigenvalues of
matrices can be traced back to Dieudonné in [9]. Let $M_{n}(\mathbb{C})$
denote the Banach algebra of all complex $n\times n$ matrices. Building on the
work of Dieudonné, Marcus and Purves showed in [12] that if a linear map
$\varphi:M_{n}(\mathbb{C})\to M_{n}(\mathbb{C})$ preserves invertibility (that
is, $x\in G(M_{n}(\mathbb{C}))\implies\varphi(x)\in G(M_{n}(\mathbb{C}))$),
then there exist $u,v\in G(M_{n}(\mathbb{C}))$ such that

$\varphi(x)=uxv\mbox{ for all }x\in M_{n}(\mathbb{C})\;\;\mbox{ or
}\;\;\varphi(x)=ux^{t}v\mbox{ for all }x\in M_{n}(\mathbb{C}),$ (1.1)

where $x^{t}$ denotes the transpose of $x$. Of course, if the mapping
$\varphi$ above is unital, that is, $\varphi(\mathbf{1})=\mathbf{1}$, then
$u=v^{-1}$. Notice also that one does not need to assume here that $\varphi$
is surjective (since injectivity can be established from the hypotheses).
Motivated by the preceding result of Marcus and Purves, and by Kaplansky’s
famous problem, Sourour classified in [15] the form of any bijective linear
invertibility preserving map between $\mathcal{L}(X)$ and $\mathcal{L}(Y)$,
where $\mathcal{L}(X)$ and $\mathcal{L}(Y)$ are the Banach algebras of all
bounded linear operators acting on the Banach spaces $X$ and $Y$,
respectively. This was further generalized by Brešar, Fošner, and Šemrl in
[7], where it is shown that if $\varphi:A\to B$ is a unital bijective linear
mapping that preserves invertibility, with $A$ and $B$ semisimple and
$\operatorname{soc}(A)$ essential, it follows that $\varphi$ is a Jordan
isomorphism; that is, a linear bijective map satisfying
$\varphi(x^{2})=\varphi(x)^{2}$ for all $x\in A$. In more recent years there
has been much interest in general preserver problems, where the condition of
linearity is removed from the hypotheses and the preserving condition is
replaced by something which can potentially be used to recover linearity (or
semi-linearity) in the conclusion. For example, it is easy to see that a
linear spectrum preserving map $\varphi:A\to B$ has the property that
$\sigma(\lambda x+y)=\sigma(\lambda\varphi(x)+\varphi(y))$ for any $x,y\in A$
and $\lambda\in\mathbb{C}$. However, if $\varphi$ is not assumed to be linear,
can one recover linearity from the preserving property alone? For recent work
in this direction see [1] and [4].

In this note we shall study pairs of surjective mappings $\varphi,\psi:A\to B$
preserving the invertibility of linear pencils in both directions, that is,
which satisfy the property that

$\lambda x+y\in G(A)\iff\lambda\varphi(x)+\psi(y)\in G(B)\mbox{ for all
}x,y\in A\mbox{ and }\lambda\in\mathbb{C}.$ (1.2)

Our work is motivated by a recent paper of Costara, [8], which generalizes a
result on determinant preserving mappings by Dolinar and Šemrl in [10], and
deals with our situation in the setting of complex matrices, i.e. the case
where $\varphi,\psi$ satisfy (1.2) and $A=B=M_{n}(\mathbb{C})$. Remarkably, if
only one of the mappings in this case is either surjective or continuous, then
[8, Theorem 1] says that $\varphi=\psi$ and $\varphi$ takes one of the forms
in (1.1). Our aim here is to show that if a pair of surjective mappings
$\varphi,\psi:A\to B$ satisfy (1.2), where $A$ is semisimple with
$\operatorname{soc}(A)$ essential, then $\varphi=\psi$ and $\varphi:x\mapsto
uJ(x)$, where $u\in G(B)$ and $J:A\to B$ is a Jordan isomorphism. Of course,
since our consideration includes infinite-dimensional Banach algebras (in
particular, $\mathcal{L}(X)$ with $X$ of any dimension), and since we do not
place any restrictions on $B$, our assumption that both mappings are
surjective seems reasonable.

##  2\. Variation of elements in terms of invertibility

We firstly prove a few preliminary results concerning the radical and the
variation of elements in terms of invertibility in a semisimple Banach
algebra. These are similar to some of the results in [8] and [11], which were
obtained for complex matrices and bounded linear operators on a Hilbert space.
However, our proofs use contemporary techniques from spectral theory in
abstract Banach algebras and, in this generality, some of our results seem to
be new. The first result is probably well-known, but it will be useful in
sequel.

######  Proposition 2.1.

For any $x\in A$ it follows that

$x\in\operatorname{rad}(A)\iff x+y\in G(A)\mbox{ for all }y\in G(A).$

###### Proof.

The forward implication follows from [2, Theorem 3.1.3], which also implies
that

$x\in\operatorname{rad}(A)\iff\sigma(yx)=\\{0\\}\mbox{ for all }y\in A.$

Conversely, if $y+x\in G(A)$ for all $y\in G(A)$, then $\sigma(yx)=\\{0\\}$
for all $y\in G(A)$. Let $w\in A$ be given. Then
$\sigma((\lambda\mathbf{1}+w)x)=\\{0\\}$ for all $\lambda\in\mathbb{C}$ with
$|\lambda|>\rho(w)$. If we now take $f:\mathbb{C}\to A$ to be the entire
function defined by $f(\lambda)=(\lambda\mathbf{1}+w)x$ for each
$\lambda\in\mathbb{C}$, then by Aupetit’s Scarcity Theorem [2, Theorem 3.4.25]
and [2, Corollary 3.4.18] we infer that $\sigma(f(\lambda))=\\{0\\}$ for all
$\lambda\in\mathbb{C}$. Taking $\lambda=0$ allows us to conclude that
$\sigma(wx)=\\{0\\}$ for all $w\in A$. Hence, $x\in\operatorname{rad}(A)$. ∎

The next result is reminiscent of [8, Lemma 1] for matrices.

######  Proposition 2.2.

Suppose that $A$ is semisimple. For any $a,b\in A$, if

$x+a\in G(A)\iff x+b\in G(A)\mbox{ for all }x\in A,$

then $a=b$.

###### Proof.

By replacing $x$ by $-\lambda\mathbf{1}+x$, the hypothesis implies that

$\sigma(x+a)=\sigma(x+b)\mbox{ for all }x\in A.$

Thus, since $A$ is semisimple, we can apply either [5, Theorem 2.2] or [13,
Theorem 2.1] to infer that $a=b$. ∎

It turns out that it is enough to restrict $x$ in Proposition 2.2 to the set
$G(A)$. For operators, a similar result can be found in [11, Lemma 3.2]. To
prove the result for semisimple Banach algebras in general, we closely follow
an argument which appears in the proof of [5, Theorem 2.6].

######  Theorem 2.3.

Let $A$ be semisimple with $a,b\in A$. Assume that for every $x\in G(A)$ we
have that $x+a\in G(A)\iff x+b\in G(A)$. Then $a=b$.

###### Proof.

Let $x\in A$ and $\lambda\in\mathbb{C}$ with $|\lambda|>\rho(x)$ be given.
Then $-\lambda\mathbf{1}+x\in G(A)$. Thus, the hypothesis implies that

$\lambda\mathbf{1}-(x+a)\in G(A)\iff\lambda\mathbf{1}-(x+b)\in G(A).$

Consequently, for any $x\in A$ and $\lambda\in\mathbb{C}$, we have that

$|\lambda|>\rho(x)\implies\left[\lambda\in\sigma(x+a)\iff\lambda\in\sigma(x+b)\right].$
(2.1)

Take any quasinilpotent $q\in A$ (i.e. $q\in A$ with $\rho(q)=0$). We claim
that

$\max\\{\rho(n(b-a)+q),\rho(nb-(n-1)a+q)\\}\leq\|b\|+\|q\|$ (2.2)

for all $n\in\mathbb{N}$. We proceed by induction on $n$. Certainly,

$\rho(b+q)\leq\|b+q\|\leq\|b\|+\|q\|.$

Moreover, by (2.1) with $x=-b-q$ we have

$\rho(b-a+q)=\rho(a-b-q)\leq\rho(-b-q)=\rho(b+q),$

which establishes the base case. Assume now that (2.2) holds for $n=k$. Taking
$x=k(b-a)+q$ in (2.1) then yields

$\rho((k+1)b-ka+q)\leq\|b\|+\|q\|.$

Moreover, taking $x=-\left((k+1)b-ka+q\right)$ in (2.1) now gives

$\rho((k+1)(b-a)+q)=\rho((k+1)(a-b)-q)\leq\|b\|+\|q\|.$

By induction, this then proves our claim that (2.2) holds for all
$n\in\mathbb{N}$. For any $\lambda\in\mathbb{C}$, the hypothesis allows us to
replace $a$ and $b$ in (2.1) by $\lambda a$ and $\lambda b$. Moreover, recall
that $q\in A$ was an arbitrary quasinilpotent element, and notice that $nq$ is
quasinilpotent for all $n\in\mathbb{N}$. Consequently, we conclude that

$\rho(n\lambda(b-a)+nq)\leq\|\lambda b\|+\|nq\|\mbox{ for all
}n\in\mathbb{N}\mbox{ and }\lambda\in\mathbb{C}.$ (2.3)

If we divide (2.3) throughout by $n$ and then let $n\rightarrow\infty$, we
obtain that

$\rho(\lambda(b-a)+q)\leq\|q\|\mbox{ for all }\lambda\in\mathbb{C}.$ (2.4)

By Vesentini’s Theorem [2, Theorem 3.4.7] it follows that the map
$\lambda\mapsto\rho(\lambda(b-a)+q)$ is subharmonic on $\mathbb{C}$. Since
this map is bounded by (2.4), it is constant by Liouville’s Spectral Theorem
[2, Theorem 3.4.14]. Taking $\lambda=0$ and $\lambda=1$ then yields

$\rho((b-a)+q)=\rho(q)=0.$

Since the above holds for any quasinilpotent element $q\in A$, we can now use
Zemánek’s characterization of the radical [2, Theorem 5.3.1] to conclude that
$b-a\in\operatorname{rad}(A)=\\{0\\}$. ∎

If we assume in addition that $a,b\in G(A)$ and that $\operatorname{soc}(A)$
is an essential ideal, then it is possible to restrict $x$ in Proposition 2.2
to the set of (spectrally) rank one elements of $A$, which we will denote by
$\mathscr{F}_{1}(A)$. Firstly, however, we digress to give a brief description
of the rank and trace in the setting of a semisimple Banach algebra.

The _spectral rank_ of an element $a\in A$, is given by

$\mathrm{rank}^{\sigma}(a)=\sup_{x\in A}\\#\sigma^{\prime}(xa)\leq\infty,$

where $\\#K$ denotes the number of distinct elements in the set $K$. From the
definition it is clear that
$\mathrm{rank}^{\sigma}(ua)=\mathrm{rank}^{\sigma}(a)$ for each $u\in G(A)$.
Moreover, observe that $x\in\operatorname{rad}(A)$ if and only if
$\mathrm{rank}^{\sigma}(x)=0$. We also note that this definition of rank
coincides with the classical operator rank when $A=\mathcal{L}(X)$. In
particular, if $A$ is semisimple, Aupetit and Mouton have shown that
$\operatorname{soc}(A)$ is equal to the set of all finite-rank elements of
$A$; in fact, $\operatorname{soc}(A)=\operatorname{span}(\mathscr{F}_{1}(A))$
(see [3, Corollary 2.9] and its proof). If $a\in\operatorname{soc}(A)$, then
[3, Theorem 2.2] says that the set

$E(a):=\left\\{x\in
A:\\#\sigma^{\prime}(xa)=\mathrm{rank}^{\sigma}(a)\right\\}$

is dense and open in $A$. We say that $a\in\operatorname{soc}(A)$ is a
_maximal finite-rank_ element if and only if $\mathbf{1}\in E(a)$. If $a$ is a
maximal finite-rank element, then by [3, Theorem 2.8] it can be expressed as a
linear combination of mutually orthogonal rank one idempotents, where the
scalar coefficients are distinct and exhaust all the values in
$\sigma^{\prime}(a)$. The notion of the _trace_ in the general setting of a
Banach algebra was firstly studied together with the rank in [3]. It turns out
that the trace will be a useful tool in our work. For
$a\in\operatorname{soc}(A)$, we define the trace of $a$ by

$\operatorname{tr}(a)=\sum_{\lambda\in\sigma(a)}\lambda
m\left(\lambda,a\right),$

where $m(\lambda,a)$ is the _multiplicity_ of $a$ at $\lambda$. For more
details on the multiplicity, see [3, Theorem 2.4] and the accompanying
discussion. Importantly, we mention here that if $a\in\mathscr{F}_{1}(A)$,
then

$\operatorname{tr}(a)=\left\\{\begin{array}[]{ll}0&\mbox{if
}\sigma(a)=\\{0\\},\\\ \lambda&\mbox{if
}\sigma^{\prime}(a)=\\{\lambda\\}.\end{array}\right.$ (2.5)

The (generalized) trace retains many of its classical properties; for example,
$a\mapsto\operatorname{tr}(a)$ is a linear functional on
$\operatorname{soc}(A)$ ([3, Theorem 3.3] and [14, Lemma 2.1]) and it
satisfies the well-known _trace property_ , that is,
$\operatorname{tr}(ab)=\operatorname{tr}(ba)$ for any
$a\in\operatorname{soc}(A)$ and $b\in A$ ([6, Theorem 2.4]). From the
linearity of the trace and the preceding discussion, if
$a\in\operatorname{soc}(A)$ is a maximal finite-rank element, then

$\operatorname{tr}(a)=\sum_{\lambda\in\sigma^{\prime}(a)}\lambda.$ (2.6)

Although the trace is not continuous in general, it does follow from [3,
Theorem 3.3] that for any fixed $n\in\mathbb{N}$, the trace is continuous on
the set of elements with rank at most $n$. Finally, we highlight the following
important property of the trace which will be useful in our work:

######  Theorem 2.4 ([3, Corollary 3.6]).

For any $x\in A$, if $\operatorname{tr}(xa)=0$ for each
$a\in\operatorname{soc}(A)$, then $x\operatorname{soc}(A)=\\{0\\}$. Moreover,
if $x\in\operatorname{soc}(A)$, then $\operatorname{tr}(xa)=0$ for each
$a\in\operatorname{soc}(A)$ implies that $x=0$.

We are now in a position to extend [11, Lemma 3.3] for Hilbert space operators
to semisimple Banach algebras with essential socles.

######  Proposition 2.5.

Let $A$ be semisimple and suppose that $\operatorname{soc}(A)$ is an essential
ideal. If $a,b\in G(A)$ and

$x+a\in G(A)\iff x+b\in G(A)\mbox{ for all }x\in\mathscr{F}_{1}(A),$

then $a=b$.

###### Proof.

For any $x\in\mathscr{F}_{1}(A)$ and $\alpha\in\mathbb{C}$, the assumption
gives

$\mathbf{1}-\alpha a^{-1}x\in G(A)\iff\mathbf{1}-\alpha b^{-1}x\in G(A).$

Hence, $\sigma^{\prime}(a^{-1}x)=\sigma^{\prime}(b^{-1}x)$ for all
$x\in\mathscr{F}_{1}(A)$. Since $x\in\mathscr{F}_{1}(A)$, it follows from
(2.5) that

$\operatorname{tr}(a^{-1}x)=\operatorname{tr}(b^{-1}x)\mbox{ for each
}x\in\mathscr{F}_{1}(A).$

By the linearity of the trace and the fact that
$\operatorname{soc}(A)=\operatorname{span}(\mathscr{F}_{1}(A))$, we conclude
that

$\operatorname{tr}((a^{-1}-b^{-1})y)=0\mbox{ for all
}y\in\operatorname{soc}(A).$

Hence, by Theorem 2.4, we have $(a^{-1}-b^{-1})\operatorname{soc}(A)=\\{0\\}$.
Since $\operatorname{soc}(A)$ is essential, we therefore obtain that $a=b$ as
desired. ∎

##  3\. Classification of mappings preserving the invertibility of linear
pencils

In this section we prove our main result which we state next.

######  Theorem 3.1.

Let $\varphi,\psi:A\to B$ be two surjective mappings, where $A$ is semisimple
and $\operatorname{soc}(A)$ is essential. If $\varphi$ and $\psi$ satisfy
(1.2), that is,

$\lambda x+y\in G(A)\iff\lambda\varphi(x)+\psi(y)\in G(B)\mbox{ for all
}x,y\in A\mbox{ and }\lambda\in\mathbb{C},$

then there exist some $u\in G(B)$ and a Jordan isomorphism $J:A\to B$ such
that

$\varphi(x)=\psi(x)=uJ(x)\mbox{ for all }x\in A.$

Before we proceed to give a proof of our main theorem, we firstly establish a
few lemmas which reveal some useful properties of the preservers under
consideration.

######  Lemma 3.2.

Let $\varphi,\psi:A\to B$ be two mappings, where $A$ is semisimple. If
$\varphi$ and $\psi$ satisfy (1.2), then $\varphi$ and $\psi$ are injective.

###### Proof.

Let $a,b\in A$ and suppose that $\varphi(a)=\varphi(b)$. Then

$\varphi(a)+\psi(x)\in G(B)\iff\varphi(b)+\psi(x)\in G(B)\mbox{ for all }x\in
A.$

Consequently, by (1.2), we deduce that

$a+x\in G(A)\iff b+x\in G(A)\mbox{ for all }x\in A.$

Since $A$ is semisimple, Proposition 2.2 yields that $a=b$. Similarly one
shows that $\psi(a)=\psi(b)\implies a=b$. ∎

######  Lemma 3.3.

Let $\varphi,\psi:A\to B$ be two surjective mappings, where $A$ is semisimple.
If $\varphi$ and $\psi$ satisfy (1.2), then:

  * (i)

$\psi(G(A))=G(B)$.

  * (ii)

$B$ is semisimple.

  * (iii)

$\varphi(0)=\psi(0)=0$.

  * (iv)

$\varphi(G(A))=G(B)$.

###### Proof.

  * (i)

This follows immediately from condition (1.2) with $\lambda=0$ and the
surjectivity of $\psi$.

  * (ii)

Let $q\in\operatorname{rad}(B)$ be given. Since $\varphi$ is surjective, it
follows that $q=\varphi(x)$ for some $x\in A$. From Proposition 2.1 and part
(i) above it follows that

$\varphi(x)+\psi(y)\in G(B)\mbox{ for each }y\in G(A).$

Consequently, by condition (1.2), we have

$x+y\in G(A)\mbox{ for each }y\in A.$

Thus, if we apply Proposition 2.1 once again, we can now conclude that
$x\in\operatorname{rad}(A)=\\{0\\}$. This shows that
$\operatorname{rad}(B)=\\{\varphi(0)\\}$, and so, $B$ is semisimple.

  * (iii)

From $0\in\operatorname{rad}(B)=\\{\varphi(0)\\}$, it readily follows that
$\varphi(0)=0$. To see that $\psi(0)=0$, notice first that
$G(B)\subseteq\varphi(G(A))$. Indeed, if $y\in G(B)$, then by the surjectivity
of $\varphi$ it follows that $y=\varphi(a)$ for some $a\in A$. If we assume
that $a\notin G(A)$, then by condition (1.2) we would have for all
$\lambda\neq 0$ that

$\displaystyle\lambda a+0\notin G(A)$ $\displaystyle\implies$
$\displaystyle\lambda\varphi(a)+\psi(0)\notin G(B)$ $\displaystyle\implies$
$\displaystyle\lambda\mathbf{1}+\varphi(a)^{-1}\psi(0)\notin G(B).$

However, then $\sigma(\varphi(a)^{-1}\psi(0))$ would be unbounded which is
impossible. From this contradiction we conclude that $a\in G(A)$, and so,
$G(B)\subseteq\varphi(G(A))$. With this observation, we can now proceed to
show that $\psi(0)\in\operatorname{rad}(B)$. For any $y=\varphi(a)\in G(B)$,
we have that $a\in G(A)$, and so, by condition (1.2)

$a+0\in G(A)\implies\varphi(a)+\psi(0)\in G(B).$

Hence, $y+\psi(0)\in G(B)$ for all $y\in G(B)$. By Proposition 2.1 we
therefore have $\psi(0)\in\operatorname{rad}(B)=\\{0\\}$ as desired.

  * (iv)

This follows easily from $\psi(0)=0$, condition (1.2), and the surjectivity of
$\varphi$. ∎

Our key strategy is to establish the linearity of $\varphi$ and $\psi$. This
will eventually be achieved by showing that $\varphi^{-1}$ and $\psi^{-1}$ are
linear under the hypotheses in Theorem 3.1. However, for the time being, we
focus on $\varphi$ and $\psi$. It is straightforward to show, using condition
(1.2), that we can factor out scalars.

######  Lemma 3.4.

Let $\varphi,\psi:A\to B$ be two surjective mappings, where $A$ is semisimple.
If $\varphi$ and $\psi$ satisfy (1.2), then $\varphi$ and $\psi$ are
homogeneous.

###### Proof.

From condition (1.2) it readily follows that

$\varphi(\lambda x)+\psi(y)\in G(B)\iff\lambda x+y\in
G(A)\iff\lambda\varphi(x)+\psi(y)\in G(B)$

for all $x,y\in A$ and $\lambda\in\mathbb{C}$. Thus, since $\psi$ is
surjective and $B$ is semisimple by Lemma 3.3, we conclude from Proposition
2.2 that $\varphi(\lambda x)=\lambda\varphi(x)$ for all $x\in A$ and
$\lambda\in\mathbb{C}$. From Lemma 3.3 we recall that $\psi(0)=0$. Thus, if we
combine this with condition (1.2) and use the fact that $G(A)$ is a
multiplicative group, we can conclude that

$x+\lambda y\in G(A)\iff\varphi(x)+\lambda\psi(y)\in G(B)\mbox{ for all
}x,y\in A\mbox{ and }\lambda\in\mathbb{C}.$ (3.1)

Hence, since $\varphi$ is surjective and $B$ is semisimple by Lemma 3.3, the
same argument as before also yields that $\psi(\lambda y)=\lambda\psi(y)$ for
all $y\in A$ and $\lambda\in\mathbb{C}$. ∎

The next step is to show that the restrictions of the mappings $\varphi$ and
$\psi$ to $\operatorname{soc}(A)$ are additive (linear) mappings onto
$\operatorname{soc}(B)$. For this we will require that $\varphi$ and $\psi$
both preserve rank. The latter readily follows from an equivalent description
of rank which appears in [1] (compare also [3, Theorem 2.1]).

######  Theorem 3.5 ([1, Theorem 2.2]).

Suppose that $A$ is semisimple. Let $a\in A$, let $m\in\mathbb{N}$, and let
$K$ be any subset of $\mathbb{C}$ with at least $m+1$ nonzero elements. Then
the following are equivalent:

  * (a)

$\mathrm{rank}^{\sigma}(a)=m$.

  * (b)

$\sup_{y\in G(A)}\\#\left\\{t\in K:y+ta\notin G(A)\right\\}=m$.

In light of Theorem 3.5 and Lemma 3.3, we now deduce the following:

######  Lemma 3.6.

Let $\varphi,\psi:A\to B$ be two surjective mappings, where $A$ is semisimple.
If $\varphi$ and $\psi$ satisfy (1.2), then
$\mathrm{rank}^{\sigma}(a)=\mathrm{rank}^{\sigma}(\varphi(a))=\mathrm{rank}^{\sigma}(\psi(a))$
for all $a\in A$.

######  Lemma 3.7.

Let $\varphi,\psi:A\to B$ be two surjective mappings, where $A$ is semisimple.
If $\varphi$ and $\psi$ satisfy (1.2), then the restrictions
$\varphi:\operatorname{soc}(A)\to\operatorname{soc}(B)$ and
$\psi:\operatorname{soc}(A)\to\operatorname{soc}(B)$ are well-defined linear
bijective mappings.

###### Proof.

From Lemma 3.4 it follows that $B$ is semisimple. Moreover, Lemma 3.6 says
that $\varphi$ and $\psi$ both preserve rank. Thus, since $\varphi$ and $\psi$
are bijections by Lemma 3.2 and the hypotheses, we can therefore conclude that
the restrictions are well-defined bijective mappings. According to Lemma 3.4
we also already have the homogeneity of $\varphi$ and $\psi$. It therefore
remains to show that $\varphi$ and $\psi$ are additive on
$\operatorname{soc}(A)$. Since
$\operatorname{soc}(A)=\operatorname{span}(\mathscr{F}_{1}(A))$, it suffices
to show that

$\psi(x_{1}+\cdots+x_{n})=\psi(x_{1})+\cdots+\psi(x_{n})\mbox{ for any
}x_{1},\ldots,x_{n}\in\mathscr{F}_{1}(A),$

and likewise for $\varphi$. To this end, fix any
$x_{1},\ldots,x_{n}\in\mathscr{F}_{1}(A)$ and let $x=x_{1}+\cdots+x_{n}$. Next
let $u\in E(\psi(x))\cap G(B)$ be arbitrary. From Lemma 3.3 we have that
$\varphi(G(A))=G(B)$, and so, $u=\varphi(y)^{-1}$ for some $y\in G(A)$. Using
condition (1.2) we now observe that

$\displaystyle\lambda\mathbf{1}+y^{-1}w\in G(A)$ $\displaystyle\iff$
$\displaystyle\lambda y+w\in G(A)$ $\displaystyle\iff$
$\displaystyle\lambda\varphi(y)+\psi(w)\in G(B)$ $\displaystyle\iff$
$\displaystyle\lambda\mathbf{1}+\varphi(y)^{-1}\psi(w)\in G(B)$

for any $w\in A$ and $\lambda\in\mathbb{C}$. Consequently,

$\sigma(y^{-1}w)=\sigma(\varphi(y)^{-1}\psi(w))\mbox{ for all }w\in A.$

Since $\sigma(y^{-1}x)=\sigma(\varphi(y)^{-1}\psi(x))$, it follows from the
definition of $E(\psi(x))$ and Lemma 3.6 that

$\\#\sigma^{\prime}(y^{-1}x)=\\#\sigma^{\prime}(\varphi(y)^{-1}\psi(x))=\mathrm{rank}^{\sigma}(\psi(x))=\mathrm{rank}^{\sigma}(x).$

Since $\mathrm{rank}^{\sigma}(x)=\mathrm{rank}^{\sigma}(y^{-1}x)$ and
$\mathrm{rank}^{\sigma}(\psi(x))=\mathrm{rank}^{\sigma}(\varphi(y)^{-1}\psi(x))$,
we can now use (2.6) to conclude that

$\operatorname{tr}(\varphi(y)^{-1}\psi(x))=\operatorname{tr}(y^{-1}x).$ (3.2)

Moreover, since $\sigma(y^{-1}x_{j})=\sigma(\varphi(y)^{-1}\psi(x_{j}))$ and

$\mathrm{rank}^{\sigma}(y^{-1}x_{j})=\mathrm{rank}^{\sigma}(\varphi(y)^{-1}\psi(x_{j}))=1$

for each $j\in\left\\{1,\ldots,n\right\\}$, it follows from (2.5) that

$\operatorname{tr}(y^{-1}x_{j})=\operatorname{tr}(\varphi(y)^{-1}\psi(x_{j}))\mbox{
for each }j\in\left\\{1,\ldots,n\right\\}.$

Thus, using the linearity of the trace, we have

$\displaystyle\operatorname{tr}(y^{-1}x)$ $\displaystyle=$
$\displaystyle\operatorname{tr}(y^{-1}x_{1})+\cdots+\operatorname{tr}(y^{-1}x_{n})$
(3.3) $\displaystyle=$
$\displaystyle\operatorname{tr}(\varphi(y)^{-1}\psi(x_{1}))+\cdots+\operatorname{tr}(\varphi(y)^{-1}\psi(x_{n})).$

Combining (3.2) with (3.3) now allows us to conclude that

$\operatorname{tr}(u\psi(x))=\operatorname{tr}(u\psi(x_{1}))+\cdots+\operatorname{tr}(u\psi(x_{n}))$
(3.4)

for all $u\in E(\psi(x))\cap G(B)$. Since the set $E(\psi(x))$ is dense in
$B$, and the trace is continuous on the set of all elements with rank at most
$\max\\{\mathrm{rank}^{\sigma}(\psi(x)),1\\}$, we infer that (3.4) holds for
all $u\in G(B)$. Now let $v\in B$ be given, and let

$b=\psi(x)-(\psi(x_{1})+\cdots+\psi(x_{n})).$

For all nonzero $\lambda\in\mathbb{C}$ with $1/|\lambda|>\rho(v)$ it now
follows from (3.4) (which holds for all $u\in G(B)$) that
$\operatorname{tr}((\mathbf{1}-\lambda v)^{-1}b)=0$. Thus, by the continuity
property and linearity of the trace, we have

$0=\operatorname{tr}\left(\sum_{j=0}^{\infty}\lambda^{j}v^{j}b\right)=\sum_{j=0}^{\infty}\lambda^{j}\operatorname{tr}(v^{j}b).$

Consequently, $\operatorname{tr}(v^{j}b)=0$ for all $j\in\mathbb{N}$. In
particular, $\operatorname{tr}(bv)=\operatorname{tr}(vb)=0$ for all
$v\in\operatorname{soc}(A)$. Thus, since $b\in\operatorname{soc}(A)$ we
conclude that $b=0$ by Theorem 2.4. Hence,

$\psi(x_{1}+\cdots+x_{n})=\psi(x_{1})+\cdots+\psi(x_{n}),$

and the additivity of $\psi$ on $\operatorname{soc}(A)$ follows. In a similar
way one establishes the additivity of $\varphi$ on $\operatorname{soc}(A)$.∎

We are now ready to give a proof of our main result.

###### Proof of Theorem 3.1.

From Lemma 3.2 it follows that $\varphi,\psi:A\to B$ are bijective mappings.
Moreover, $B$ is semisimple by Lemma 3.3. Notice now that
$\varphi^{-1},\psi^{-1}:B\to A$ are surjective mappings satisfying

$\lambda a+b\in G(B)\iff\lambda\varphi^{-1}(a)+\psi^{-1}(b)\in G(A)\mbox{ for
all }a,b\in B\mbox{ and }\lambda\in\mathbb{C},$ (3.5)

but with the added advantage that the domain is semisimple and the codomain is
semisimple with an essential socle. From Lemma 3.7 it follows that the
restrictions $\varphi^{-1}:\operatorname{soc}(B)\to\operatorname{soc}(A)$ and
$\psi^{-1}:\operatorname{soc}(B)\to\operatorname{soc}(A)$ are well-defined
linear mappings. Since we already know that $\varphi^{-1}$ and $\psi^{-1}$ are
homogeneous by Lemma 3.4, we now proceed to systematically extend the
additivity of $\varphi^{-1}$ on $\operatorname{soc}(B)$ to all of $B$. In the
process we also show that $\varphi^{-1}=\psi^{-1}$. The details are as
follows.

Let $u\in G(B)$ and $a\in\mathscr{F}_{1}(B)$ be given. Notice that
$\varphi^{-1}(u)$ and $\varphi^{-1}(u+a)-\psi^{-1}(a)$ are both invertible in
$A$ by Lemma 3.3 and (3.5). Moreover, for any $b\in\mathscr{F}_{1}(B)$, by
using (3.5) and the linearity of $\psi^{-1}$ on $\operatorname{soc}(B)$ we
have

$\displaystyle\varphi^{-1}(u)+\psi^{-1}(b)\in G(A)$ $\displaystyle\iff$
$\displaystyle u+b\in G(B)$ $\displaystyle\iff$ $\displaystyle(u+a)-a+b\in
G(B)$ $\displaystyle\iff$ $\displaystyle\varphi^{-1}(u+a)+\psi^{-1}(-a+b)\in
G(A)$ $\displaystyle\iff$
$\displaystyle\varphi^{-1}(u+a)-\psi^{-1}(a)+\psi^{-1}(b)\in G(A).$

Thus, since $\psi^{-1}(\mathscr{F}_{1}(B))=\mathscr{F}_{1}(A)$ and
$\operatorname{soc}(A)$ is essential, it follows from Proposition 2.5 that

$\varphi^{-1}(u+a)=\varphi^{-1}(u)+\psi^{-1}(a)\mbox{ for each }u\in
G(B)\mbox{ and }a\in\mathscr{F}_{1}(B).$

Now let $u,v\in G(B)$ be given. As before we have that $\varphi^{-1}(u)$ and
$\psi^{-1}(u+v)-\varphi^{-1}(v)$ are both invertible in $A$. Moreover, for any
$b\in\mathscr{F}_{1}(B)$, we can now use (3.5) and the homogeneity of
$\varphi^{-1}$ to find that

$\displaystyle\psi^{-1}(u+v)-\varphi^{-1}(v)+\psi^{-1}(b)\in G(A)$
$\displaystyle\iff$ $\displaystyle\psi^{-1}(u+v)+\varphi^{-1}(-v+b)\in G(B)$
$\displaystyle\iff$ $\displaystyle(u+v)+(-v+b)\in G(B)$ $\displaystyle\iff$
$\displaystyle u+b\in G(A)$ $\displaystyle\iff$
$\displaystyle\varphi^{-1}(u)+\psi^{-1}(b)\in G(A).$

Consequently, Proposition 2.5 yields that

$\psi^{-1}(u+v)=\varphi^{-1}(u)+\varphi^{-1}(v)\mbox{ for all }u,v\in G(B).$
(3.6)

If we interchange the roles of $\varphi^{-1}$ and $\psi^{-1}$ in both parts of
the preceding argument, then we also obtain

$\varphi^{-1}(u+v)=\psi^{-1}(u)+\psi^{-1}(v)\mbox{ for all }u,v\in G(B).$
(3.7)

Taking $u=v$ in (3.7) and using the homogeneity of $\varphi^{-1}$, we infer
that

$\varphi^{-1}(u)=\psi^{-1}(u)\mbox{ for all }u\in G(B).$ (3.8)

Now let $x\in B$ be arbitrary, and fix any $\lambda\in\mathbb{C}$ with
$|\lambda|>\rho(x)$. Then $-\lambda\mathbf{1},\lambda\mathbf{1}+x\in G(B)$.
Thus, by (3.6), (3.7), and (3.8), we have

$\displaystyle\psi^{-1}(x)$ $\displaystyle=$
$\displaystyle\psi^{-1}(-\lambda\mathbf{1}+\lambda\mathbf{1}+x)$
$\displaystyle=$
$\displaystyle\varphi^{-1}(-\lambda\mathbf{1})+\varphi^{-1}(\lambda\mathbf{1}+x)$
$\displaystyle=$
$\displaystyle\psi^{-1}(-\lambda\mathbf{1})+\psi^{-1}(\lambda\mathbf{1}+x)$
$\displaystyle=$
$\displaystyle\varphi^{-1}(-\lambda\mathbf{1}+\lambda\mathbf{1}+x)$
$\displaystyle=$ $\displaystyle\varphi^{-1}(x).$

Hence, $\varphi^{-1}=\psi^{-1}$. We are now in a position to show that
$\varphi^{-1}$ is additive and hence linear. Let $v\in G(B)$ and $x\in B$ be
arbitrary. Keeping in mind that $\varphi^{-1}=\psi^{-1}$ is an homogeneous map
which satisfies (3.6), for any $u\in G(B)$, we have from (3.5) that

$\displaystyle\varphi^{-1}(x+v)-\varphi^{-1}(v)+\varphi^{-1}(u)\in G(A)$
$\displaystyle\iff$ $\displaystyle\varphi^{-1}(x+v)+\varphi^{-1}(-v+u)\in
G(A)$ $\displaystyle\iff$ $\displaystyle x+v-v+u\in G(B)$ $\displaystyle\iff$
$\displaystyle x+u\in G(B)$ $\displaystyle\iff$
$\displaystyle\varphi^{-1}(x)+\varphi^{-1}(u)\in G(A).$

Hence, since $\varphi^{-1}(G(B))=G(A)$ by Lemma 3.3, it follows from Theorem
2.3 that

$\varphi^{-1}(x+v)=\varphi^{-1}(x)+\varphi^{-1}(v)\mbox{ for all }v\in
G(B)\mbox{ and }x\in B.$

In light of the preceding identity and the homogeneity of $\varphi^{-1}$, to
obtain the additivity of $\varphi^{-1}$, we can now simply repeat the
preceding argument with $v\in G(B)$ replaced by $v\in B$. This then
establishes the linearity of $\varphi^{-1}=\psi^{-1}$. Consequently, we have
that $\varphi=\psi$ is linear.

If we now let $J:A\to B$ be defined by
$J(x)=\varphi(\mathbf{1})^{-1}\varphi(x)$ for each $x\in A$, then $J$ is a
unital linear bijection which preserves invertibility. Thus, since $A$ and $B$
are semisimple and $\operatorname{soc}(A)$ is essential, it follows from [7,
Theorem 1.1] that $J$ is a Jordan isomorphism. Hence, with
$u=\varphi(\mathbf{1})$, we have $\varphi(x)=\psi(x)=uJ(x)$ for all $x\in A$.
∎

Using Sourour’s results [15, Theorem 1.1 and Corollary 1.2], we obtain, as a
consequence of Theorem 3.1, the following classification for pairs of
surjective mappings which preserve the invertibility of linear operator
pencils in both directions. Here we use $X^{\prime}$ to denote the dual space
of $X$, and $S^{\ast}$ to denote the adjoint of an operator $S$.

######  Corollary 3.8.

Let $\varphi,\psi:\mathcal{L}(X)\to\mathcal{L}(Y)$ be two surjective mappings
which satisfy

$\lambda S+T\in G(\mathcal{L}(X))\iff\lambda\varphi(S)+\psi(T)\in
G(\mathcal{L}(Y))\mbox{ for all }S,T\in\mathcal{L}(X)\mbox{ and
}\lambda\in\mathbb{C}.$

Then either

$\varphi(S)=\psi(S)=TSU\mbox{ for every }S\in\mathcal{L}(X)$

or

$\varphi(S)=\psi(S)=VS^{\ast}W\mbox{ for every }S\in\mathcal{L}(X),$

where $T:X\to Y$, $U:Y\to X$, $V:X^{\prime}\to Y$ and $W:Y\to X^{\prime}$ are
bounded invertible operators.

## References

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# Combining Counting Processes and Classification Improves a Stopping Rule for
Technology Assisted Review

Reem Bin-Hezam${}^{\alpha}{{}^{,}}^{\beta}$ Mark Stevensonα
αDepartment of Computer Science, University of Sheffield, United Kingdom
βInformation Systems Department, Princess Nourah bint Abdulrahman University,
Saudi Arabia
<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Technology Assisted Review (TAR) stopping rules aim to reduce the cost of
manually assessing documents for relevance by minimising the number of
documents that need to be examined to ensure a desired level of recall. This
paper extends an effective stopping rule using information derived from a text
classifier that can be trained without the need for any additional annotation.
Experiments on multiple data sets (CLEF e-Health, TREC Total Recall, TREC
Legal and RCV1) showed that the proposed approach consistently improves
performance and outperforms several alternative methods.111Code to reported
results available from https://github.com/ReemBinHezam/TAR_Stopping_CP_CLF

##  1 Background

Information Retrieval (IR) systems often return large numbers of documents in
response to user queries and screening them for relevance represents a
significant effort. This problem is particularly acute in applications where
it is important to identify most (or all) of the relevant documents, for
example, systematic reviews of scientific literature Higgins et al. (2019) and
legal eDiscovery Oard et al. (2018). Technology Assisted Review (TAR) develops
methods that aim to reduce the effort required to screen a collection of
documents for relevance, such as stopping rules that inform reviewers that a
desired level of recall has been reached (and no further documents need to be
examined) Li and Kanoulas (2020); Yang et al. (2021a).

One of the most commonly applied approaches to developing stopping rules is to
identify when a particular level of recall has been reached by estimating the
total number of documents in the collection (either directly or indirectly).

The majority of algorithms using this approach operate by obtaining manual
judgements for a sample of the unobserved documents and inferring the total
number of relevant documents, e.g. Shemilt et al. (2014); Howard et al.
(2020); Callaghan and Müller-Hansen (2020). However, these approaches may not
account for the fact that most relevant documents appear early in the ranking,
information shown to be useful for stopping algorithms Cormack and Grossman
(2016a); Li and Kanoulas (2020), so they may examine more documents than
necessary.

Counting Processes Counting processes, stochastic models of the number of
occurrences of an event within some interval Cox and Isham (1980), can
naturally model changes to the frequency of relevant document occurances
within a ranking. They have been used to develop stopping rules that ensure
that a desired level of recall is reached while also minimising the number of
documents that need to be reviewed and found to outperform a range of
alternative approaches Sneyd and Stevenson (2019, 2021); Stevenson and Bin-
Hezam (to appear). However, in these methods the estimate of the total number
of relevant documents is only based on the examined initial portion of the
ranking and no information is used from the remaining documents.

Classification Another approach has been to use a text classifier to estimate
the total number of relevant documents Yu and Menzies (2019); Yang et al.
(2021a). This method has the advantage of using information about all
documents in the ranking by observing the classifier’s predictions for the
documents that have not yet been examined. However, the classifier alone does
not consider or model the occurrence rate at which relevant documents have
already been observed. These methods were found to be effective, although each
was only tested on a single dataset.

This paper proposes an extension to stopping algorithms based on counting
processes by using a text classifier to inform the estimate of the total
number of relevant documents. This approach makes use of information about the
relevance of documents from the entire ranking without increasing the number
of documents that need to be examined for relevance, since the text classifier
is trained using information already available.

##  2 Approach

We begin by describing the existing counting process approach and then explain
how the classifier is integrated.

Counting Process Stopping Rule Sneyd and Stevenson (2019, 2021) The approach
starts by obtaining relevance judgements for the $n$ highest ranked documents.
A rate function is then fitted to model the probability of relevant documents
being encountered in the (as yet) unexamined documents later in the ranking.
The counting process uses this rate function to estimate the total number of
relevant documents remaining. Based on this estimate, the algorithm stops if
enough relevant documents have been found to reach a desired recall. If not
then more documents are examined and the process repeated until either the
stopping point is reached or all documents have been examined. Fitting an
appropriate rate function is an important factor in the success of this
approach. A set of sample points are extracted from the documents for which
the relevance judgements are available (i.e. top $n$), and the probability of
a document being relevant at each point is estimated. This set of points is
then provided to a non-linear least squares curve fitting algorithm to produce
the fitted rate function. For each sample point, the probability of a document
being relevant is computed by simply examining a window of documents around it
and computing the proportion that is relevant, i.e.

$\frac{1}{n}\sum_{d\in W}\mathbbm{1}(d)$ (1)

where $W$ is the set of documents in the window and $\mathbbm{1}(d)$ returns 1
if $d$ has been labelled as relevant.

Integrating Text Classification Since the existing counting process approach
only examines the top-ranked documents, it relies on the assumption that the
examined portion of documents provides a reliable indication of the rate at
which relevant documents occur within the ranking. However, this may not
always be the case, particularly when only a small number of documents have
been manually examined or when relevant documents unexpectedly appear late in
the ranking. This problem can be avoided by applying a text classifier to the
unexamined documents at each stage. The information it provides helps to
ensure that the rate function is appropriate and that the estimate of the
total number of relevant documents provided by the counting process is
accurate.

The text classifier is integrated into the stopping algorithm in a
straightforward way. We begin by assuming that the ranking contains a total of
$N$ documents. As before, relevance judgements are obtained for the first $n$.
These judgements are used to train a text classifier which is applied to all
remaining documents without relevance judgements (i.e. $n+1\;\ldots\;N$). The
rate function is now fitted by examining all $N$ documents in the ranking
(rather than only the first $n$), using manual relevance judgements for the
first $n$ and the classifier’s predictions for the remainder (i.e.
$n+1\;\ldots\;N$). As before, the algorithm stops if it has been estimated
that enough relevant documents are contained within the first $n$ to achieve
the desired recall. Otherwise, the value of $n$ is increased and the process
repeated. See Figure 1.

Figure 1: Counting Process with Classifier Approach

It is possible to use information from the classifier in various ways when
computing the probability of document relevance during the curve fitting
process. Two were explored:

ClassLabel: the class labels output by the classifier are used directly, i.e.
using eq. 1 with relevance determined by the classifier.

ClassScore: Alternatively, the scores output by the classifier is used,
subject to a threshold of 0.5, i.e. eq. 1 is replaced by

$\frac{1}{n}\sum_{d\in W}\begin{cases}score(d)&\textrm{if}\;score(d)\geq
0.5\\\ 0&\textrm{otherwise}\\\ \end{cases}$ (2)

where $score(d)$ is the score the classifier assigned to $d$. (Using the class
score without thresholding was found to overestimate the number of relevant
documents.)

##  3 Experiments

Experiments compared three approaches. CP: Baseline counting process approach
without text classification, i.e. Sneyd and Stevenson (2021). CP+ClassLabel
and CP+ClassScore: Counting process combined with the text classifier (see
§2).

The counting process uses the most effective approach previously reported
Sneyd and Stevenson (2021). A power law Zobel (1998) was used as the rate
function with an inhomogeneous Poisson process since its performance is
comparable with a Cox process while being more computationally efficient.
Parameters were set as follows: the confidence level was set to 0.95 while the
sample size was initialised to 2.5% and incremented by 2.5% for each
subsequent iteration. An existing reference implementation was used Sneyd and
Stevenson (2021).

The text classifier uses logistic regression which has been shown to be
effective for TAR problems Yang et al. (2021a); Li and Kanoulas (2020). The
classifier was based on scikit-learn using TF-IDF scores of each document’s
content as features, the content is title and abstract for CLEF datasets,
email messages for TREC datasets (including title and message body), and news
articles content for the RCV1 dataset. The classification threshold was set to
0.5 (model default) by optimising F-measure for relevant documents using a
held-out portion of the dataset. Text classifiers based on BioBERT Lee et al.
(2020) and PubMedBERT Gu et al. (2020) were also developed. However, neither
outperformed the linear logistic regression model on CLEF datasets, a result
consistent with previous findings for TAR models Yang et al. (2022); Sadri and
Cormack (2022).

TAR problems are often highly imbalanced, with very few relevant documents.
However, TAR methods successfully place relevant documents early in the
ranking Cormack and Grossman (2015, 2016b); Li and Kanoulas (2020).
Consequently, the prevalence of relevant documents in the training data is
unlikely to be representative of the entire collection, particularly early in
the ranking. To account for this cost-sensitive learning was used during
classifier training. The weight of the minority class within the training data
was set to 1, while the weight of the majority class was set to the imbalance
ratio (i.e. count of the minority over the majority class in training data)
Ling and Sheng (2008).

###  3.1 Datasets

Performance was evaluated on six diverse datasets widely used in previous work
on TAR. CLEF e-Health (CLEF2017, CLEF2018, CLEF2019) Kanoulas et al. (2017,
2018, 2019): A collection of systematic reviews from the Conference and Labs
of the Evaluation Forum (CLEF) 2017, 2018, and 2019 e-Health lab Task 2:
Technology-Assisted Reviews in Empirical Medicine. The datasets contain 42,
30, and 31 reviews respectively. TREC Total Recall (TR) Grossman et al. (2016)
A collection of 290,099 emails with 34 topics related to Jeb Bush’s eight-year
tenure as Governor of Florida (athome4). TREC Legal (Legal) Cormack et al.
(2010) A collection of 685,592 Enron emails made available by the Federal
Energy Review Commission during their investigation into the company’s
collapse. RCV1 Lewis et al. (2004) A collection of Reuters news articles
labelled with categories. Following previous work Yang et al. (2021b, a), 45
categories were used to represent a range of topics, and the collections
downsampled to 20%.

The ranking used by the stopping model needs to be the same for all datasets
used in the experiments to ensure fair and meaningful comparison. The rankings
produced by AutoTAR Cormack and Grossman (2015) were used since they allow
comparison with a range of alternative approaches Li and Kanoulas (2020) and
can be generated using their reference implementation.

Table 1:  Results for Counting Process and Classification Stopping Methods. (*
indicate scores are Not statistically significantly different from CP, +
indicate CP+ClassLabel and CP+ClassScore are statistically significantly
different)

| Desired recall = 0.9 | Desired recall = 0.8 | Desired recall = 0.7
---|---|---|---
Dataset | Model | Recall | Rel. | Cost | Excess | Recall | Rel. | Cost | Excess | Recall | Rel. | Cost | Excess
CLEF2017 | CP | 1.000 | 1.000 | 0.281 | 0.238 | 1.000 | 1.000 | 0.265 | 0.232 | 1.000 | 1.000 | 0.255 | 0.229
CP+ClassLabel | 0.989 | 1.000 | 0.153 | 0.102 | 0.989 | 1.000 | 0.152 | 0.114 | 0.988 | 1.000 | 0.150 | 0.120
CP+ClassScore | 0.989 | 1.000 | 0.152 | 0.101 | 0.989 | 1.000 | 0.152 | 0.114 | 0.988 | 1.000 | 0.147 | 0.117
CLEF2018 | CP | 1.000 | 1.000 | 0.293 | 0.242 | 1.000 | 1.000 | 0.287 | 0.249 | 1.000 | 1.000 | 0.277 | 0.245
CP+ClassLabel | 0.983 | 1.000 | 0.137 | 0.075 | 0.983 | 1.000 | 0.137 | 0.091 | 0.982 | 1.000 | 0.135 | 0.097
CP+ClassScore | 0.983 | 1.000 | 0.137 | 0.075 | 0.982 | 1.000 | 0.136 | 0.090 | 0.981 | 1.000 | 0.134 | 0.096
CLEF2019 | CP | 0.999 | 1.000 | 0.283 | 0.228 | 0.999 | 1.000 | 0.279 | 0.235 | 0.999 | 1.000 | 0.276 | 0.240
CP+ClassLabel | 0.996 | 1.000 | 0.221 | 0.161 | 0.996 | 1.000 | 0.216 | 0.169 | 0.996 | 1.000 | 0.212 | 0.173
CP+ClassScore | 0.996 | 1.000 | 0.221 | 0.161 | 0.996 | 1.000 | 0.213+ | 0.165+ | 0.994 | 1.000 | 0.207+ | 0.168+
Legal | CP | 1.000 | 1.000 | 0.425 | 0.401 | 1.000 | 1.000 | 0.338 | 0.320 | 1.000 | 1.000 | 0.287 | 0.273
CP+ClassLabel | 0.972∗ | 1.000 | 0.088 | 0.050 | 0.972∗ | 1.000 | 0.088 | 0.064 | 0.972∗ | 1.000 | 0.088 | 0.070
CP+ClassScore | 0.972∗ | 1.000 | 0.088 | 0.050 | 0.972∗ | 1.000 | 0.088 | 0.064 | 0.963∗ | 1.000 | 0.075 | 0.057
TR | CP | 1.000 | 1.000 | 0.059 | 0.054 | 1.000 | 1.000 | 0.056 | 0.052 | 1.000 | 1.000 | 0.052 | 0.049
CP+ClassLabel | 0.999 | 1.000 | 0.028 | 0.023 | 0.999∗ | 1.000 | 0.027 | 0.023 | 0.999∗ | 1.000 | 0.027 | 0.024
CP+ClassScore | 0.999 | 1.000 | 0.028 | 0.023 | 0.999∗ | 1.000 | 0.027 | 0.023 | 0.999∗ | 1.000 | 0.027 | 0.024
RCV1 | CP | 0.999 | 1.000 | 0.193 | 0.180 | 0.998 | 1.000 | 0.154 | 0.145 | 0.998 | 1.000 | 0.134 | 0.127
CP+ClassLabel | 0.972 | 0.956∗ | 0.038 | 0.022 | 0.969 | 1.000 | 0.036 | 0.026 | 0.969 | 1.000 | 0.036 | 0.028
CP+ClassScore | 0.969 | 0.933∗ | 0.036 | 0.020 | 0.969 | 1.000 | 0.036 | 0.026 | 0.969 | 1.000 | 0.036 | 0.028

###  3.2 Evaluation Metrics

Approaches were evaluated using the metrics commonly used in previous work on
TAR stopping criteria, e.g. Yang et al. (2021a); Li and Kanoulas (2020), and
calculated using the tar_eval open-source evaluation
script.222https://github.com/CLEF-TAR/tar Average scores across all topics in
each collection are reported.

Recall: proportion of relevant documents identified by the method. Following
Li and Kanoulas (2020), results closest to the desired recall are considered
best, rather than the highest overall recall.

Reliability (Rel.): percentage of topics where the desired recall was reached
(or exceeded). For each topic, reliability is 1 if the desired recall was
reached, and 0 otherwise.

Cost: percentage of documents examined.

Excess cost (Excess): we introduce this measure which quantifies the
additional documents that have to be examined compared with optimal stopping
(i.e. stopping immediately when the desired recall has been reached). It is
computed as follows:

$excess\;cost=\frac{cost(method)-cost(optimal)}{1-cost(optimal)}$ (3)

where $cost(method)$ and $cost(optimal)$ are the cost of the method being
evaluated and the cost of stopping at the optimal point, respectively. This
metric indicates the proportion of the documents that need to be examined
after the desired recall has been reached.

##  4 Results and Analysis

Experiments were carried out using a range of desired recalls {0.9, 0.8, 0.7}
with results shown in Table 1. Results show that combining the classifier with
the counting process (CP+ClassLabel and CP+ClassScore) is more effective than
using the counting process alone. The improvement in the performance of both
approaches compared with CP is statistically significant across topics
($p<0.05$, paired t-test with Bonferroni correction). There is a significant
reduction in cost which is often substantial (e.g. Legal collection with a
desired recall of 0.9). This is achieved with a minimal reduction to the
number of relevant documents identified, although the reliability remains
unaffected in the majority of cases, and when it is affected (RCV1 collection
with a desired recall of 0.9), the reduction is minimal and not statistically
significant.

Table 2: Proposed Method vs. Baselines (TR collection)

| Desired recall = 0.9 | Desired recall = 0.8
---|---|---
Model | Recall | Rel. | Cost | Excess | Recall | Rel. | Cost | Excess
CP+ClassScore | 0.999 | 1.000 | 0.028 | 0.023 | 0.999 | 1.000 | 0.027 | 0.023
SCAL Cormack and Grossman (2016b) | 0.903 | 0.647 | 0.144 | 0.140 | 0.761 | 0.676 | 0.107 | 0.103
SD-training Hollmann and Eickhoff (2017) | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000
SD-sampling Hollmann and Eickhoff (2017) | 0.936 | 0.794 | 0.779 | 0.778 | 0.896 | 0.735 | 0.690 | 0.689
AutoStop Li and Kanoulas (2020) | 0.953 | 0.941 | 0.766 | 0.765 | 0.885 | 0.912 | 0.754 | 0.753
Target Cormack and Grossman (2016a) | 0.900 | 0.706 | 0.069 | 0.064 | 0.844 | 0.882 | 0.069 | 0.065

The performance of CP+ClassScore and CP+ClassLabel are comparable. The scores
for CP+ClassScore may be marginally better, although the difference is only
significant in limited circumstances. (Differences in the cost and excess
scores were significant for the CLEF 2019 collection with desired recalls of
0.8 and 0.7.) Overall, the way in which the classifier output is integrated
into the approach seems less important than the fact it is used at all. The
average recall for all approaches exceeds the desired recall, indicating a
tendency to be somewhat conservative in proposing the stopping point. However,
the classifier allows the algorithm to identify this point earlier, presumably
because it has indicated a low probability of relevant documents being found
later in the ranking.

###  4.1 Effect of Cost-sensitive Learning

The effect of using cost-sensitive learning during classifier training was
explored by comparing it with the performance obtained when it was not used,
see Table 3.

These results demonstrate the importance of accounting for class imbalance
when training the classifier. The low prevalence of relevant documents can
lead to the classifier being unable to identify them resulting in the approach
stopping too early, particularly when the desired recall is high.

Table 3: Effect of Cost-Sensitive Learning (CSL) for CP+ClassLabel Model
applied to Legal collection

Desired recall | CSL | Recall | Rel. | Cost | Excess
---|---|---|---|---|---
0.9 |  2613 | 0.793 | 0.000 | 0.025 | -0.016
$\checkmark$ | 0.972 | 1.000 | 0.088 | 0.050
0.8 |  2613 | 0.793 | 0.500 | 0.025 | -0.001
$\checkmark$ | 0.972 | 1.000 | 0.088 | 0.064
0.7 |  2613 | 0.793 | 1.000 | 0.025 | 0.006
$\checkmark$ | 0.972 | 1.000 | 0.088 | 0.070

###  4.2 Baseline Comparison

Direct comparison of stopping methods is made difficult by the range of
rankings and evaluation metrics used in previous work. However, Li and
Kanoulas (2020) reported the performance of several approaches using the same
rankings as our experiments, allowing benchmarking against several methods.
SCAL Cormack and Grossman (2016b), and AutoStop Li and Kanoulas (2020) sample
documents from the entire ranking to estimate the number of relevant
documents, SD-training/SD-sampling Hollmann and Eickhoff (2017) use the scores
assigned by the ranking. Results for the target method Cormack and Grossman
(2016a) are also reported. Table 2 compares the proposed approach against
these baselines for the TR collection for the desired recall levels for which
results are available. (Results for other datasets were similar and available
in Li and Kanoulas (2020).) CP+ClassScore has a lower cost than the baselines,
requiring only a fraction of the documents to be examined, while achieving a
higher recall. Although the average recall of some baselines is closer to the
desired recall than the proposed approach, their reliability is also lower
which indicates that they failed to identify enough relevant documents to
achieve the desired recall for multiple topics.

##  5 Conclusion

This paper explored the integration of a text classifier into an existing TAR
stopping algorithm. Experiments on six collections indicated that the proposed
approach was able to achieve the desired recall level with a statistically
significant lower cost than the existing method based on counting processes
alone. Integrating the classifier output in the form of labels or scores made
little difference, with improvements observed using either approach. The text
classifier provides the stopping algorithm with information about the likely
relevance of the documents that have not yet been manually examined, allowing
it to better estimate the number remaining.

## Limitations

We presented a stopping rule for ranked lists. Comparison against multiple
datasets requires the same ranking model for all datasets, which may not
always be available. In addition, reported results have been limited to a
single rate function (power law) and confidence level (0.95). However, the
pattern of results for other hyperparameters is similar to those reported.

## Ethics Statement

This work presents a TAR method designed to reduce the amount of effort
required to identify information of interest within large document
collections. The most common applications of this technology are literature
reviews and identification of documents for legal disclosure. We do not
foresee any ethical issues associated with these uses.

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# Three-manifolds with bounded curvature and uniformly positive scalar
curvature

Conghan Dong  Mathematics Department, Stony Brook University, NY 11790, United
States<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

In this note, we prove that for a complete noncompact three dimensional
Riemannian manifold with bounded sectional curvature, if it has uniformly
positive scalar curvature, then there is a uniform lower bound on the
injectivity radius.

###### Key words and phrases:

$3$-manifolds, positive scalar curvature, collapsing, injectivity radius

##  1\. Introduction

A well-known problem posed by Yau (see Problem Section in [Yau82]) is to
classify $3$-manifolds admitting complete Riemannian metrics of positive
scalar curvature up to diffeomorphism. In the case of closed $3$-manifolds, it
has been resolved by Schoen-Yau [SY79], Gromov-Lawson [GL83], Hamilton [Ham82]
and Perelman [Per02, Per03b, Per03a]. But for open $3$-manifolds, this problem
remains wide open. Some recent progress were made under the assumption of
uniformly positive scalar curvature. Particularly, Chang-Weinberger-Yu [CWY10]
classified complete $3$-manifolds with uniformly positive scalar curvature and
finitely generated fundamental group. Bessieres-Besson-Maillot [BBM11]
classified complete $3$-manifolds with uniformly positive scalar curvature and
bounded geometry. Here the bounded geometry means the sectional curvature is
bounded and the injectivity radius is bounded away from zero.

In this note, we make further progress towards Yau’s problem and show that
bounded curvature and uniformly positive scalar curvature on complete open
$3$-manifolds are enough to derive a lower bound on the injectivity radius.

######  Theorem 1.1.

Assume $(M^{3},g)$ is a smooth three dimensional complete noncompact
Riemannian manifold. If the sectional curvature is bounded by
$|K_{g}|\leq\Lambda$ and the scalar curvature is bounded below by $R_{g}\geq
1$, then the injectivity radius has a uniform lower bound. In other words,
there exists a uniform positive number $r_{0}=r_{0}(\Lambda)>0$, depending
only on $\Lambda$, such that for any $x\in M$, the injectivity radius at $x$
is bounded from below by $r_{0}$.

This theorem together with the main result in [BBM11] gives the following
corollary.

######  Corollary 1.2.

Let $M$ be a connected orientable $3$-manifold which carries a complete metric
of bounded sectional curvature and uniformly positive scalar curvature, then
there is a collection $\mathcal{F}$ of spherical space forms with finitely
many diffeomorphim type such that $M$ is a (possibly infinite) connected sum
of copies of $S^{2}\times S^{1}$ or members of $\mathcal{F}$.

######  Remark 1.1.

Jian Wang recently claimed a result on classification of open three manifolds
carrying uniformly positive scalar curvature in his preprint.

Also, we have the following improvement of the main result in [BBMM20].

######  Corollary 1.3.

The moduli space of complete Riemannian metrics of bounded curvature and
uniformly positive scalar curvature on an orientable $3$-manifold is path-
connected or empty.

By regularity results in [And90], the main theorem implies the following
compactness result.

######  Corollary 1.4.

For any sequence of pointed complete noncompact Riemannian $3$-manifolds
$(M_{i}^{3},g_{i},p_{i})$ with uniformly bounded curvature and uniformly
positive scalar curvature, up to a subsequence, they converge locally in
$C^{1,\alpha}$-topology for any $0<\alpha<1.$

Another corollary is that $3$-manifolds with uniformly positive scalar
curvature and bounded curvature have at least linear volume growth.

######  Corollary 1.5.

Assume $(M^{3},g)$ is a complete noncompact $3$-manifold satisfying that the
sectional curvature is bounded by $|K_{g}|\leq\Lambda$ and the scalar
curvature is bounded below by $R_{g}\geq 1$. Then there is a positive constant
$c=c(\Lambda)>0$, depending only on $\Lambda$, such that for any point $p\in
M$,

$\liminf_{r\to\infty}\frac{\text{Vol}_{g}B(p,r)}{r}\geq c.$

###### Proof.

For any $r\gg 1$, and $q\in\partial B(p,r)$, choose a geodesic segment
$\gamma:[0,r]\to M$ between $p$ and $q$. Let $r_{0}$ be the lower bound on the
injectivity radius given in Theorem 1.1. Choose a partition of the interval
$[0,r]$ by $0=t_{0}<t_{1}<\cdots<t_{k+1}\leq r$ with $t_{i+1}=t_{i}+3r_{0}$
for $0\leq i\leq k$ and $r<t_{k+1}+3r_{0}$. Then the geodesic balls
$\\{B(\gamma(t_{i}),r_{0})\\}_{i=0}^{k}$ are disjoint and all included in
$B(p,r)$. Note that $k\geq\frac{r}{3r_{0}}$, and $\text{Vol}_{g}B(q,r_{0})\geq
v_{\Lambda}(r_{0})$ for any $q\in M$, where $v_{\Lambda}(r_{0})$ is the volume
of a geodesic ball with radius $r_{0}$ in the $3$-sphere with constant
curvature $\Lambda$. So

$\text{Vol}_{g}B(p,r)\geq\sum_{i=0}^{k}\text{Vol}_{g}B(\gamma(t_{i}),r_{0})\geq
r\cdot\frac{v_{\Lambda}(r_{0})}{3r_{0}}.$

It’s done by taking $c=\frac{v_{\Lambda}(r_{0})}{3r_{0}}$. ∎

######  Remark 1.2.

Yau [Yau76] proved that any complete noncompact manifolds with nonnegative
Ricci curvature have at least linear volume growth. In general, the volume
growth is not uniform. Munteanu-Wang [MW22] proved that for a complete
$3$-manifold with nonnegative Ricci curvature and uniformly positive scalar
curvature, the volume has at most uniform linear growth. So in the case of a
complete noncompact $3$-manifold $(M^{3},g)$ with $0\leq Ric_{g}\leq\Lambda$
and $R_{g}\geq 1$, the volume of a geodesic ball has uniformly linear growth.

Now we explain the ideas in the proof of the main theorem and give some
remarks. The proof is by contradiction, and involves two main ingredients: the
structure of collapsing manifolds with bounded curvature, which has been
intensively studied, see among others the work of [CG86, Fuk87, Fuk88, CFG92];
and Gromov’s width inequality [Gro18, Gro19].

Let us assume by contradiction that, up to a double covering of orientations,
there is a sequence of three dimensional complete noncompact orientable
Riemannian manifolds $(M_{i}^{3},g_{i},p_{i})$ with uniformly bounded
curvature and uniformly positive scalar curvature, which, up to a subsequence,
converges to a complete Alexandrov space $(X,d,o)$ in the pointed Gromov-
Hausdorff topology, and the injectivity radius $\text{inj}(p_{i})$ at $p_{i}$
converges to $0$.

By our assumptions, the sequence is collapsing and $X$ can’t be a point, i.e.
$\text{dim}X=1$ or $2$. The theory of collapsing with bounded curvature gives
a symmetric structure around sufficiently collapsed part. Roughly speaking, if
$\text{dim}X=2$, there is a wide Seifert fibered space with boundary around
$p_{i}$ in $M_{i}$, see Proposition 2.2; if $\text{dim}X=1$, there is a long
torical band in $M_{i}$, see Proposition 2.3. The noncompact assumption
ensures that we can find an arbitrarily wide Seifert fibered space or
arbitrarily long torical band.

But Gromov’s width inequality tells us that uniformly positive scalar
curvature on a Riemannian band imposes a uniform upper bound on the width,
except that there are spheres separating this band, see Propositions 3.2 and
3.3. When $\text{dim}X=1$, this immediately gives a contradiction. When
$\text{dim}X=2$, note that a Seifert fibered space with boundary is
$S^{2}$-irreducible, so spheres can not separate the wide Seifert fibered
space in the sufficiently collapsed part, which also gives a contradiction
(see Section 3 for more details).

######  Remark 1.3.

The following example shows that the assumption on the upper bound of
sectional curvature is necessary.

######  Example 1.1.

Consider the warped product metric $dr^{2}+\rho(r)^{2}ds_{2}^{2}$ on
$\mathbb{R}\times S^{2}$. Then the scalar curvature is given by

$R=-\frac{4\ddot{\rho}}{\rho}+\frac{2-2\dot{\rho}^{2}}{\rho^{2}},$

and the sectional curvature $K$ is between the values
$-\frac{\ddot{\rho}}{\rho},\frac{1-\dot{\rho}^{2}}{\rho^{2}}$. See [Pet06] for
computations. Define

$\rho(r)=\begin{cases}f(r),&|r|\leq R_{0}\\\ \frac{1}{|r|},&|r|\geq
R_{0},\end{cases}$

where

$f(r)=\frac{3}{8R_{0}^{5}}r^{4}-\frac{5}{4R_{0}^{3}}r^{2}+\frac{15}{8R_{0}}.$

Then $\rho$ is a positive $C^{2}$-function, and for a fixed $R_{0}\geq 100$,
it’s easy to see that $R>1$, and $K\geq-\frac{C}{R_{0}^{2}}$. A smooth
modification of $\rho$ gives a metric with sectional curvature bounded from
below and uniformly positive scalar curvature on $\mathbb{R}\times S^{2}$. But
the injectivity radius converges to $0$ and the curvature upper bound blows up
as $|r|\to\infty$.

######  Remark 1.4.

The noncompact assumption is important. There exists closed $3$-manifolds with
uniformly bounded sectional curvature and uniformly positive scalar curvature,
but arbitrarily small injectivity radius. Some examples are given by product
metrics $ds_{2}^{2}+\varepsilon^{2}ds_{1}^{2}$ on $S^{2}\times S^{1}$ with
small $\varepsilon>0$, and collapsing Berger $3$-spheres.

######  Remark 1.5.

The main result does not hold in higher dimensions. For example, we can
consider $S^{2}\times\mathbb{R}^{2}$ with product metric
$g=g_{S^{2}}+g_{\mathbb{R}^{2}}$, where $(\mathbb{R}^{2},g_{\mathbb{R}^{2}})$
is a complete metric with bounded curvature, and the injectivity radius goes
to zero at infinity in $\mathbb{R}^{2}$, and $(S^{2},g_{S^{2}})$ is a constant
curvature metric. We can choose $g_{S^{2}}$ with big enough curvature such
that $g$ has uniformly positive scalar curvature and bounded sectional
curvature, but the injectivity radius does not have a uniform lower bound.

Acknowledgements. The author would like to thank his advisor Prof. Xiuxiong
Chen for the encouragements. He also thanks Jian Wang and Shaosai Huang for
careful reading and helpful comments, and Jian Wang for helpful discussions on
positive scalar curvature and informing the author some corollaries.

##  2\. Collapsing with bounded curvature

In this section, following [Fuk87, Fuk88], we talk about the structure around
sufficiently collapsing part of three manifolds with bounded curvature.

Assume there is a sequence of complete noncompact orientable three manifolds
$(M_{i},g_{i},p_{i})$ with sectional curvature uniformly bounded
$|K_{g_{i}}|\leq\Lambda$ and injectivity radius $\text{inj}(p_{i})$ satisfying

$\lim_{i\to\infty}\text{inj}(p_{i})=0.$

By Cheeger-Gromov compactness theorem [Che70, Gro07], we can take a convergent
subsequence in the pointed Gromov-Hausdorff topology. For simplicity, we abuse
the notation and write $(M_{i},d_{i},p_{i})$ for a convergent subsequence,
that is

$(M_{i},d_{i},p_{i})\to(X,d,o),$

where $(X,d)$ is a complete Alexandrov space with curvature bounded below and
$0\leq\text{dim}X\leq 3$. By results in [Che70, CC97], the assumption that
$\text{inj}(p_{i})\to 0$ implies the sequence is collapsing, i.e.
$\text{dim}X<3$; the noncompactness assumption implies that $\text{dim}X>0$.
Namely, $\text{dim}X=1$ or $2$.

######  Proposition 2.1.

If $\text{dim}X=2$, then $(X,d)$ is a Riemannian orbifold without boundary.

######  Remark 2.1.

If $X$ has bounded diameter and comes from codimension one collapsing, then
the fact that it is a Riemannian orbifold was known in the literature, see
Proposition 11.5 in [Fuk90], and Proposition 8.1 in [SY00]. See also [NT08]
for the general collapsing case. For reader’s convenience, we reproduce the
proof and add some details here.

###### Proof.

For any $q_{0}\in X$, take $q_{i}\in M_{i}$ with $q_{i}\to q_{0}$. For a small
ball $B(0,\varepsilon)\subset\mathbb{R}^{3}\cong T_{q_{i}}M_{i}$ with center
$0$ and a very small radius $\varepsilon\ll\frac{\pi}{\sqrt{\Lambda}}$,
consider the pull back metrics $\exp_{q_{i}}^{*}g_{i}$, which we still denote
by $g_{i}$, then $(B(0,\varepsilon),g_{i})$ has a uniform lower bound on
injectivity radius. By Cheeger-Gromov compactness theorem, up to a subsequence
we have a metric $g_{0}$ on $B(0,\varepsilon)$ such that

$(B(0,\varepsilon),g_{i})\to(B(0,\varepsilon),g_{0})$

in $C^{1,\alpha}$-topology. Take $G_{i}$ to be the local fundamental group

$G_{i}=G(q_{i},\varepsilon):=\\{\gamma:\gamma\text{ is a geodesic loop at
}q_{i}\text{ with length }<\varepsilon\\}.$

Then $G_{i}$ is a finite set and acts on $B(0,\varepsilon)$ by free local
isometries and $B(0,\varepsilon)/G_{i}=B(q_{i},\varepsilon)$. The local group
structure on $G_{i}$ is defined by Gromov’s product, see [BK81] for more
details. Put

$L=\\{\gamma\in
C(B(0,\varepsilon),B(0,2\varepsilon)):\frac{1}{2}\leq\frac{d_{0}(\gamma(x),\gamma(y))}{d_{0}(x,y)}\leq
2,\ \forall x,y\in B(0,\varepsilon)\\},$

where $d_{0}$ is the metric associated to $g_{0}$. Note $L$ is a compact set
and for large enough $i$, we have $G_{i}\subset L$. By taking a subsequence,
there exists a closed subset $G\subset L$ such that $G_{i}\to G$, and $G$ acts
isometrically on $(B(0,\varepsilon),g_{0})$. It was proved in Section 3 of
[Fuk88] that $G$ is a Lie group germ.

Then passing to a subsequence, there is an equivariant convergent sequence

$(B(0,\varepsilon),g_{i},G_{i})\to(B(0,\varepsilon),g_{0},G),$

and

$(B(0,\varepsilon),g_{0})/G=B(q_{0},\varepsilon).$

In the case $\text{dim}X=2$, we know $\text{dim}G=1$.

Let $H_{0}$ be the isotropy group of $G$ at $0$ and
$H_{0}^{\prime},G^{\prime}$ the identity components of $H_{0},G$ respectively.
From Section 5 of [Fuk88], we know for $\varepsilon$ small enough,
$B(q_{0},\varepsilon)$ is isometric to $B(0,\varepsilon)/H_{0}G^{\prime}$.

######  Claim.

$H_{0}$ is a discrete subgroup.

###### Proof of the claim.

We argue it by contradiction and assume that $H_{0}$ is not discrete. Then
$H_{0}$ contains a one dimensional Lie group germ. Since $\text{dim}G=1$,
$H_{0}$ and $G$ has the same Lie group germ at identity. Hence there exists a
small neighbourhood $U$ of the identity $e$ in $G$ such that $\forall\alpha\in
U$, $\alpha(0)=0$.

There is a small $\delta_{0}>0$ such that $\\{\alpha\in
G:d_{0}(0,\alpha(0))\leq\delta_{0}\\}\subset H_{0}$. If not, then there exist
$\delta_{j}\to 0$ and a sequence $\alpha_{j}\in G$ such that
$d_{0}(0,\alpha_{j}(0))\leq\delta_{j}$ but $\alpha_{j}(0)\neq 0$. Then up to a
subsequence, assume $\alpha_{j}\to\alpha_{\infty}\in G$. So
$\alpha_{\infty}(0)=0$. Note $\alpha_{\infty}^{-1}\cdot\alpha_{j}\in U$ for
large $j$ implies $\alpha_{\infty}^{-1}\cdot\alpha_{j}(0)=0$. So
$\alpha_{j}(0)=\alpha_{\infty}(\alpha_{\infty}^{-1}\cdot\alpha_{j}(0))=\alpha_{\infty}(0)=0$,
a contradiction.

Similarly, for any small $0<\delta_{1}<\delta_{0}$, we have that for any
$\gamma_{i}\in G_{i}$, if $d_{i}(0,\gamma_{i}(0))<\delta_{0}$, then
$d_{i}(0,\gamma_{i}(0))<\delta_{1}$ for all large $i$. If not, then there is a
sequence $\gamma_{i}$ satisfying $\delta_{1}\leq
d_{i}(0,\gamma_{i}(0))<\delta_{0}$. Up to a subsequence, assume
$\gamma_{i}\to\gamma_{0}\in G$. Then $\delta_{1}\leq
d_{0}(0,\gamma_{0}(0))\leq\delta_{0}$. But we just proved that
$d_{0}(0,\gamma_{0}(0))\leq\delta_{0}$ implies $\gamma_{0}(0)=0$, which is a
contradiction.

Now consider the local group

$G_{i}(\delta_{0}):=\\{\gamma\in G_{i}:d_{i}(0,\gamma(0))\leq\delta_{0}\\}.$

There are two cases:

(1) $G_{i}(\delta_{0})=\\{e\\}$. In this case,
$B(q_{0},\delta_{0})=(B(0,\delta_{0}),g_{0})$ has dimension three and is thus
noncollapsing, a contradiction.

(2) $G_{i}(\delta_{0})\neq\\{e\\}$. Assume there exists $e\neq\gamma_{i}\in
G_{i}(\delta_{0})$. Denote the uniform lower bound of injectivity radius on
$(B(0,\varepsilon),g_{i})$ by $r_{0}$, and choose
$\delta_{1}<\min\\{\frac{1}{2}\delta_{0},r_{0},\frac{\pi}{4\sqrt{\Lambda}}\\}$.
Then for large enough $i$, from previous paragraph, for any
$n\in\mathbb{Z}_{+}$, inductively we know
$d_{i}(0,\gamma_{i}^{n}(0))<\delta_{1}$. Note $G_{i}(\delta_{0})$ is a finite
set, so there exists an $n_{i}$ such that $\gamma_{i}^{n_{i}}=e$.

Then we can use the technique of center of mass to show that $\gamma_{i}=e$.
By our choice of $\delta_{1}$, from Section 8 of [BK81], there is a unique
minimum point $q_{c}$ of
$P(x)=\frac{1}{2}\sum_{j=1}^{n_{i}}d_{i}(x,\gamma_{i}^{j}(0))$ in
$B(0,\delta_{1})$. Since $\gamma_{i}$ is an isometry and
$\gamma_{i}^{n_{i}}=e$, $P(q_{c})=P(\gamma_{i}(q_{c}))$. From the uniqueness
of $q_{c}$, we know $\gamma_{i}(q_{c})=q_{c}$. Note $\gamma_{i}$ behaves like
deck transformation, so $\gamma_{i}$ has a fixed point only when
$\gamma_{i}=e$. This gives a contradiction and completes the proof of the
claim. ∎

From the claim, together with the fact that $H_{0}$ is compact, we know
$H_{0}$ is a finite group. Then the orbit $G\cdot 0\cong G/H_{0}$ is one
dimensional. By standard slice theorem, we know $B(q_{0},\varepsilon)$ is
isometric to $S_{0}/H_{0}$, where $S_{0}=\exp_{0}(V)$ for some small ball
$V\subset\mathbb{R}^{2}\cong(T_{0}(G\cdot 0))^{\perp}\subset
T_{0}B(0,\varepsilon)$, and via $\exp_{0}$, $H_{0}$ acts isometrically on $V$.
Note that $M_{i}$ are orientable and $G_{i}$ preserve orientations, so $G$
also preserves orientation, which implies that $H_{0}\subset SO(2)$, i.e.
$H_{0}$ is a finite cyclic group. This implies that $q_{0}$ is either a
regular point or an interior Riemannian orbifold point. ∎

######  Proposition 2.2.

If $\text{dim}X=2$, then for any fixed number $r>0$, for all sufficiently
large $i$, there exists a Seifert fibered space $\Omega_{i}$ with boundary
such that $B(p_{i},r)\subset\Omega_{i}\subset B(p_{i},2r)$.

###### Proof.

Under the same notations as above, if $\text{dim}X=2$, then from Proposition
2.1 we have a decomposition $S_{1}(X)\subset X$, where $S_{1}(X)$ consists of
orbifold singularities and $X\setminus S_{1}(X)$ is a smooth surface.
Perturbing $p_{i}$ if necessary, we can assume $o\in X\setminus S_{1}(X)$.
Choose $\varepsilon>0$ small, $r^{\prime}<2r$ and $r^{\prime}-2\varepsilon>r$.
By Theorem 0.12 and Theorem 10.1 in [Fuk88], there is a continuous map

$f_{i}:B(p_{i},r^{\prime})\to X$

such that

$B(p_{i},r^{\prime}-2\varepsilon)\subset
f_{i}^{-1}(B(o,r^{\prime}-\varepsilon))\subset B(p_{i},r^{\prime}),$

the restriction on $f_{i}^{-1}(X\setminus S_{1}(X))$ is a smooth fiber bundle
with fiber diffeomorphic to $S^{1}$, and $\forall p\in S_{1}(X)\cap
B(o,r^{\prime}-\varepsilon)$, $f_{i}^{-1}(p)$ is diffeomorphic to
$S^{1}/H_{p}$ for some cyclic group $H_{p}=\mathbb{Z}_{m}$. Then for any small
disk $D\subset B(o,r^{\prime}-\varepsilon)$ with $D\cap S_{1}(X)=\\{p\\}$, a
$m$-fold covering of $f_{i}^{-1}(D)$ is diffeomorphic to $D\times S^{1}$. So

$\Omega_{i}:=f_{i}^{-1}(B(o,r^{\prime}-\varepsilon))$

is a desired Seifert fibered space with boundary.

∎

######  Proposition 2.3.

If $\text{dim}X=1$, then for any fixed number $r>0$, there is a submanifold
$T_{i}\times(-1,1)\subset M_{i}$ with
$d_{i}(T_{i}\times\\{-1\\},T_{i}\times\\{1\\})\geq r$, where $T_{i}$ is
diffeomorphic to a torus $T^{2}$ or a Klein bottle $K^{2}$.

###### Proof.

If $\text{dim}X=1$, then $X=\mathbb{R}$ or $[0,\infty)$. In both cases, we can
choose $q\in X$ with $d(o,q)=r+1$ and $B(q,r)\subset X$ consists of regular
points. Then by Theorem 0.12 and Theorem 10.1 in [Fuk88], for large enough
$i$, there is a map

$f_{i}:B(p_{i},2r+2)\to X$

such that the restriction on $f_{i}^{-1}(B(q,r))$ is a fiber bundle, whose
fiber is diffeomorphic to a torus or Klein bottle with arbitrary small
diameter and $f_{i}$ is an almost Riemannian submersion. So

$\Omega_{i}=f_{i}^{-1}(B(q,r))\cong T_{i}\times(-1,1)$

is a desired submanifold. Note
$\lim_{i\to\infty}d_{i}(T_{i}\times\\{-1\\},T_{i}\times\\{1\\})=2r.$

∎

##  3\. Proof of the theorem

Recall that a Riemannian band is a Riemannian manifold $(X,\partial_{\pm})$
with two distinguished disjoint non-empty subsets in the boundary $\partial
X$, denoted by $\partial_{-}$ and $\partial_{+}$. A band is called proper if
$\partial_{\pm}$ are unions of connected components of $\partial X$ and
$\partial_{-}\cup\partial_{+}=\partial X$. The width of a Riemannian band
$(X,\partial_{\pm})$ is defined as $d(\partial_{-},\partial_{+}).$ Gromov
proved the following width inequalities in [Gro19] by using $\mu$-bubbles.

######  Proposition 3.1 ($\frac{2\pi}{n}$-Inequality).

Let $(X^{n},\partial_{\pm})$ be a proper compact Riemannian band of dimension
$n\leq 7$ with scalar curvature $R\geq 1$. If no closed hypersurface in $X$
which separates $\partial_{-}$ from $\partial_{+}$ admits a metric with
positive scalar curvature, then

$d(\partial_{-},\partial_{+})\leq\frac{2\pi}{n}\cdot\sqrt{n(n-1)}.$

As a corollary, width inequalities hold for torical bands, see also [Gro18].

######  Proposition 3.2.

For a Riemannian band $X^{n}=T^{n-1}\times[-1,1]$ with scalar curvature $R\geq
1$, $n\leq 7,$

$d(T^{n-1}\times\\{-1\\},T^{n-1}\times\\{1\\})\leq\frac{2\pi}{n}\cdot\sqrt{n(n-1)}.$

Note that a closed orientable surface which admits a metric with positive
scalar curvature is diffeomorphic to a $2$-sphere $S^{2}$. Taking $n=3$, we
have the following corollary.

######  Proposition 3.3.

Let $X$ be an orientable three manifold with the same assumptions in
Proposition 3.1. There exists a positve number $C_{0}=\frac{2\sqrt{6}\pi}{3}$
such that if $d(\partial_{-},\partial_{+})\geq C_{0}$, then there exist
$2$-spheres separating $\partial_{-}$ from $\partial_{+}$.

Now we can prove the main theorem.

###### Proof of Theorem 1.1.

Assume by contradiction that there exists a sequence of complete noncompact
three manifolds $(M_{i},g_{i},p_{i})$, satisfying $|K_{g_{i}}|\leq\Lambda$ and
$R_{g_{i}}\geq 1$, such that $\text{inj}(p_{i})\to 0$. Taking a double
covering if necessary, we assume $M_{i}$ are orientable. As in Section 2, up
to a subsequence, we assume $(M_{i},g_{i},p_{i})\to(X,d,o)$ in the pointed
Gromov-Hausdorff topology, with $\text{dim}X=1$ or $2$.

If $\text{dim}X=2$, from Proposition 2.2, we know for a big fixed radius
$r>C_{0}+100$, where $C_{0}$ is the constant in Proposition 3.3, there exists
a Seifert fibered space $\Omega_{i}$ with boundary such that
$B(p_{i},2r)\subset\Omega_{i}$ for all sufficiently large $i$. In the
following, we will abuse the notation and write $(M,g,p)$ for
$(M_{i},g_{i},p_{i})$ when $i$ is large enough.

By modifying the distance function, we can find a smooth function $\varphi$ on
$M$ such that

$d(p,x)+1\leq\varphi(x)\leq d(p,x)+2.$

For suitable small numbers $\varepsilon_{1},\varepsilon_{2}>0$, we can assume
$r_{1}:=r-\varepsilon_{1}$ and $r_{2}:=2r+\varepsilon_{2}$ are regular values
of $\varphi$ and take

$X:=\\{x\in M:r_{1}<\varphi(x)<r_{2}\\},\ \partial_{-}=\varphi^{-1}(r_{1}),\
\partial_{+}=\varphi^{-1}(r_{2}).$

Then $X\subset\Omega$ and $d(\partial_{-},\partial_{+})>C_{0}$ by our choice
of $r$.

From Proposition 3.3, we know there are $2$-spheres
$\\{S_{k}^{2}\\}_{k=1}^{N}$ in $X$ separating $\partial_{-}$ from
$\partial_{+}$. Note that a Seifert fibered space with boundary is
$S^{2}$-irreducible, see [Sco83], so each sphere $S_{k}^{2}$ bounds a $3$-ball
$B^{3}_{k}$ in $\Omega$, and

$M\setminus\cup_{k=1}^{N}S^{2}_{k}=(\sqcup_{k=1}^{N}B^{3}_{k})\sqcup
M^{\prime},$

where $M^{\prime}$ is the unbounded connected component. Then
$\partial_{+}\subset M^{\prime}$, otherwise $M$ has bounded diameter. So we
can choose one ball $B^{3}\subset\Omega$ such that $p\in B^{3}$, then

$B(p,\frac{r}{2})\subset\\{x\in M:\varphi(x)<r_{1}\\}\subset B^{3}.$

From the proof of Proposition 2.2, we know the restriction of $\Omega$ to
smaller domain gives a Seifert fibered space $\Omega^{\prime}$ with boundary
such that

$\Omega^{\prime}\subset B(p,\frac{r}{2})\subset B^{3}.$

From the exact sequence of homotopy groups of Seifert fibered space with
boundary, the homotopy class of a fiber in $\Omega^{\prime}$ is not trivial in
$\Omega$, but it is trivial in $B^{3}\subset\Omega$, which is a contradiction.

If $\text{dim}X=1$, from Proposition 2.3, we know there is a band
$T_{i}\times(-1,1)$ with width bigger than $r$ admitting a positive scalar
curvature $R_{g_{i}}\geq 1$, where $T_{i}$ is either a torus or Klein bottle.
Up to a double covering, we can assume $T_{i}$ is a torus. Choosing $r$ big
enough gives a contradiction to Proposition 3.2.

∎

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