# Kernels of operators on certain Banach spaces associated with almost
disjoint families

Bence Horváth  (B. Horváth) Tolstého 19, Prague 10, 101 00, Czech Republic
<EMAIL_ADDRESS>and  Niels Jakob Laustsen  (N. J. Laustsen) Department of
Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1
4YF, United Kingdom<EMAIL_ADDRESS>
###### Abstract.

Given an infinite set $\Gamma$ and an almost disjoint family $\mathcal{A}$ on
$\Gamma$, let $Y_{\mathcal{A}}$ denote the closed subspace of
$\ell_{\infty}(\Gamma)$ spanned by the indicator functions
$1_{\bigcap_{j=1}^{n}A_{j}}$ for $n\in\mathbb{N}$ and
$A_{1},\ldots,A_{n}\in\mathcal{A}$. We show that if $\mathcal{A}$ has
cardinality greater than $\Gamma$, then $Y_{\mathcal{A}}$ contains closed
subspaces which cannot be realised as the kernels of any bounded operators
$Y_{\mathcal{A}}\rightarrow\ell_{\infty}(\Gamma)$. Consequently the spaces
$\ell_{\infty}(\Gamma)$, for any infinite set $\Gamma$, and
$C_{0}(K_{\mathcal{A}})$, where $\mathcal{A}$ is an uncountable almost
disjoint family on $\mathbb{N}$ and $K_{\mathcal{A}}$ the locally compact
Mrówka space associated with $\mathcal{A}$, contain closed subspaces which do
not arise as the kernels of any bounded operators on them.

###### Key words and phrases:

Key words and phrases. Non-separable Banach space, bounded operator, kernel,
Mrówka space, almost disjoint family.

###### 1991 Mathematics Subject Classification:

2020

†† Mathematics Subject Classification. Primary 46B26, 46E15, 47B01; Secondary
18A30 47B38.

##  1\. Introduction and statement of the main result

The problem of classifying all complemented subspaces — that is, closed
subspaces which arise as the kernel of a bounded idempotent operator — of a
given Banach space is of central importance in the theory of Banach spaces.
This problem has a very natural generalisation, first formulated in [7]:

> Which closed subspaces of a Banach space $X$ can be realised as the kernel
> of a bounded operator $X\rightarrow X$?

In the positive direction, it was shown in [7, Proposition 2.1] that whenever
$W$ is a closed subspace of a Banach space $X$ such that $X/W$ is separable,
there is a bounded operator $X\rightarrow X$ whose kernel is $W$.
Consequently, the above question is of interest only for non-separable Banach
spaces $X$. The focus in [7] was on reflexive spaces; its main result [7,
Corollary 2.8] states that the dual space $X^{*}$ of Wark’s non-separable
reflexive Banach space $X$ with “few operators” introduced in [12] contains a
closed subspace which is not the kernel of any bounded operator $X^{*}\to
X^{*}$.

Many years prior to this work, Kalton [4, Proposition 4] showed that $c_{0}$
is not the kernel of any bounded operator $\ell_{\infty}\to\ell_{\infty}$,
thereby generalising Phillips’ Theorem that $c_{0}$ is not complemented in
$\ell_{\infty}$. More precisely, Kalton proved that whenever the kernel of a
bounded operator $T\colon\ell_{\infty}\to\ell_{\infty}$ contains $c_{0}$,
there is an infinite subset $M$ of $\mathbb{N}$ such that
$\ell_{\infty}(M)\subseteq\ker T$, where we have identified $\ell_{\infty}(M)$
with the subspace $\\{x\in\ell_{\infty}:x(n)=0\ (n\in\mathbb{N}\setminus
M)\\}$ of $\ell_{\infty}$.

Our main theorem generalises Kalton’s result in two ways: First, it replaces
the index set $\mathbb{N}$ with an arbitrary infinite set $\Gamma$, and
second, it applies to certain subspaces of $\ell_{\infty}(\Gamma)$. Before we
can state it precisely, we must introduce some notation and terminology.

Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ be the scalar field.
Throughout, we consider a fixed infinite cardinal number $\Gamma$ and use the
notation

$\displaystyle[\Gamma]^{\Gamma}$ $\displaystyle=\\{A\subseteq\Gamma:\lvert
A|=\Gamma\\}\qquad\text{and}\qquad[\Gamma]^{<\Gamma}=\\{A\subseteq\Gamma:\lvert
A\rvert<\Gamma\\},$

where $\lvert A\rvert$ denotes the cardinality of the set $A$. A family
$\mathcal{A}\subset[\Gamma]^{\Gamma}$ is called _almost disjoint_ if $A\cap
B\in[\Gamma]^{<\Gamma}$ whenever $A,B\in\mathcal{A}$ are distinct. A famous
result of Sierpiński [11] states that $[\Gamma]^{\Gamma}$ contains an almost
disjoint family of cardinality greater than $\Gamma$.

Let $1_{A}$ denote the indicator function of a subset $A$ of $\Gamma$,
considered as an element of $\ell_{\infty}(\Gamma)$. The following two closed
subspaces of $\ell_{\infty}(\Gamma)$ associated with a family $\mathcal{A}$ of
subsets of $\Gamma$ are at the heart of our work:

(1.1) $\displaystyle Y_{\mathcal{A}}$
$\displaystyle=\overline{\operatorname{span}}\,\\{1_{\bigcap_{j=1}^{n}A_{j}}:n\in\mathbb{N},\,A_{1},\ldots,A_{n}\in\mathcal{A}\\}$
and $\displaystyle Z_{\mathcal{A}}$
$\displaystyle=\overline{\operatorname{span}}\,\\{1_{\bigcap_{j=1}^{n}A_{j}}:n\in\mathbb{N}\setminus\\{1\\},\,A_{1},\ldots,A_{n}\in\mathcal{A},\,A_{j}\neq
A_{k}\ \text{for}\ j\neq k\\}.$

We can now state our main result as follows.

######  Theorem 1.1.

Let $\mathcal{A}\subset[\Gamma]^{\Gamma}$ be an almost disjoint family of
cardinality greater than $\Gamma$. Then:

  1. (i)

$Z_{\mathcal{A}}\neq Y_{\mathcal{A}^{\prime}}$ for every subset
$\mathcal{A}^{\prime}$ of $\mathcal{A};$ in particular, $Z_{\mathcal{A}}$ is a
proper, closed subspace of $Y_{\mathcal{A}}.$

  2. (ii)

Suppose that $T\colon Y_{\mathcal{A}}\to\ell_{\infty}(\Gamma)$ is a bounded
operator whose kernel contains $Z_{\mathcal{A}}$. Then $\mathcal{A}$ contains
a subset $\mathcal{A}^{\prime}$ of the same cardinality as $\mathcal{A}$ such
that $Y_{\mathcal{A}^{\prime}}\subseteq\ker T$.

The remainder of this paper is organised as follows. We begin by proving 1.1
in Section 2, before giving two applications of it in Section 3: First, we
extend the above-mentioned result of Kalton to kernels of bounded operators
$\ell_{\infty}(\Gamma)\to\ell_{\infty}(\Gamma)$ for any infinite set $\Gamma$
in 3.1, which also strengthens a result of Pełczyński and Sudakov [9, Theorem
1]. Second, we establish an analogue of Kalton’s result for bounded operators
$C_{0}(K_{\mathcal{A}})\to\ell_{\infty}$ in 3.3, where
$\mathcal{A}\subset[\mathbb{N}]^{\omega}$ is any uncountable, almost disjoint
family and $K_{\mathcal{A}}$ denotes the Mrówka space associated with
$\mathcal{A}$. In particular, it follows that $C_{0}(K_{\mathcal{A}})$
contains a closed subspace isomorphic to $c_{0}$ which is not the kernel of
any bounded operator on $C_{0}(K_{\mathcal{A}})$. Finally, we discuss in
Section 4 how our results could potentially be extended to encompass the
subspace $\ell_{\infty}^{c}(\Gamma)$ of $\ell_{\infty}(\Gamma)$ consisting of
countably supported elements (defined whenever $\Gamma$ is an uncountable
set).

##  2\. Proof of 1.1

We require the following elementary combinatorial lemma, which is almost
certainly known. However, we have not been able to find it anywhere in the
literature, so we include a proof.

######  Lemma 2.1.

Let $A_{1},\ldots,A_{m}$ be distinct subsets of $\Gamma$ for some
$m\in\mathbb{N}$, set

(2.1) $\displaystyle A^{d}_{j}:=A_{j}\setminus\underset{k\neq
j}{\bigcup_{k=1}^{m}}A_{k}$

for each $j\in\\{1,\ldots,m\\}$, and let
$\sigma_{1},\ldots,\sigma_{m}\in\mathbb{K}$. Then

(2.2) $\displaystyle\sum_{j=1}^{m}\sigma_{j}1_{A^{d}_{j}}=\sum_{\emptyset\neq
N\subseteq\\{1,\ldots,m\\}}(-1)^{\lvert N\rvert+1}\Bigl{(}\sum_{j\in
N}\sigma_{j}\Bigr{)}1_{\bigcap_{j\in N}A_{j}}.$

###### Proof.

We shall verify (2.2) pointwise, that is, we shall show that
$\operatorname{lhs}(\gamma)=\operatorname{rhs}(\gamma)$ for every
$\gamma\in\Gamma$, where we have introduced the quantities

$\operatorname{lhs}(\gamma)=\sum_{j=1}^{m}\sigma_{j}1_{A^{d}_{j}}(\gamma)\qquad\text{and}\qquad\operatorname{rhs}(\gamma)=\sum_{\emptyset\neq
N\subseteq\\{1,\ldots,m\\}}(-1)^{\lvert N\rvert+1}\Bigl{(}\sum_{j\in
N}\sigma_{j}\Bigr{)}1_{\bigcap_{j\in N}A_{j}}(\gamma)$

for brevity.

Clearly, we have $\operatorname{lhs}(\gamma)=0=\operatorname{rhs}(\gamma)$ for
every $\gamma\notin\bigcup_{j=1}^{m}A_{j}$. It is also easy to see that
$\operatorname{lhs}(\gamma)=\operatorname{rhs}(\gamma)$ if there is a unique
$j\in\\{1,\ldots,m\\}$ such that $\gamma\in A_{j}$. Indeed, in this case
$\operatorname{lhs}(\gamma)=\sigma_{j}$, while the only non-trivial
contribution to $\operatorname{rhs}(\gamma)$ comes from $N=\\{j\\}$, giving
$\operatorname{rhs}(\gamma)=\sigma_{j}$ as well.

Hence it remains to consider the case where $\gamma$ belongs to at least two
of the sets $A_{1},\ldots,A_{m}$. In this case $\gamma\in\bigcup_{k=1,\,k\neq
j}^{m}A_{k}$ for every $j\in\\{1,\ldots,m\\}$, so
$\operatorname{lhs}(\gamma)=0$. Since the order of the sets
$A_{1},\ldots,A_{m}$ does not matter, we may relabel them to arrange that
there is $n\in\\{2,\ldots,m\\}$ such that $\gamma\in\bigcap_{j=1}^{n}A_{j}$
and $\gamma\notin\bigcup_{j=n+1}^{m}A_{j}$. Then we have

$\displaystyle\operatorname{rhs}(\gamma)$ $\displaystyle=\sum_{\emptyset\neq
N\subseteq\\{1,\ldots,n\\}}(-1)^{\lvert N\rvert+1}\Bigl{(}\sum_{j\in
N}\sigma_{j}\Bigr{)}=\sum_{k=1}^{n}(-1)^{k+1}\sum_{\begin{subarray}{c}N\subseteq\\{1,\ldots,n\\}\\\
|N|=k\end{subarray}}\Bigl{(}\sum_{j\in N}\sigma_{j}\Bigr{)}$
$\displaystyle=\sum_{k=1}^{n}(-1)^{k+1}\sum_{j=1}^{n}a_{k,j}\sigma_{j}$

where $a_{k,j}$ denotes the number of subsets $N$ of $\\{1,\ldots,n\\}$ of
size $k$ such that $j\in N$. Since $a_{k,j}=\binom{n-1}{k-1}$, by re-indexing
and applying the binomial formula, we obtain

$\operatorname{rhs}(\gamma)=\sum_{k=1}^{n}(-1)^{k+1}\binom{n-1}{k-1}\sum_{j=1}^{n}\sigma_{j}=\sum_{\ell=0}^{n-1}(-1)^{\ell}\binom{n-1}{\ell}\sum_{j=1}^{n}\sigma_{j}=(-1+1)^{n-1}\sum_{j=1}^{n}\sigma_{j}=0,$

as desired. ∎

As usual, the _support_ of an element $x\in\ell_{\infty}(\Gamma)$ is
$\operatorname{supp}x=\\{\gamma\in\Gamma:x(\gamma)\neq 0\\}$. We shall use the
following elementary fact about the supports of elements of the dense subspace

(2.3)
$Z_{\mathcal{A}}^{0}:=\operatorname{span}\bigl{\\{}1_{\bigcap_{j=1}^{n}A_{j}}:n\in\mathbb{N}\setminus\\{1\\},\,A_{1},\ldots,A_{n}\in\mathcal{A},\,A_{j}\neq
A_{k}\ \text{for}\ j\neq k\bigr{\\}}$

of $Z_{\mathcal{A}}$ several times, so we state it formally for ease of
reference.

######  Lemma 2.2.

Let $\mathcal{A}\subset[\Gamma]^{\Gamma}$ be an almost disjoint family. Then
$\operatorname{supp}z\in[\Gamma]^{<\Gamma}$ for every $z\in
Z_{\mathcal{A}}^{0}$.

###### Proof.

The almost disjointness of $\mathcal{A}$ implies that
$\bigcap_{j=1}^{n}A_{j}\in[\Gamma]^{<\Gamma}$ whenever
$n\in\mathbb{N}\setminus\\{1\\}$ and $A_{1},\ldots,A_{n}\in\mathcal{A}$ are
distinct. Every element $z\in Z_{\mathcal{A}}^{0}$ vanishes outside a finite
union of intersections of this form, from which the conclusion follows. ∎

###### Proof of Theorem 1.1..

(i). The family $\mathcal{A}$ cannot consist of disjoint sets because
$\lvert\mathcal{A}\rvert>\Gamma$, so
$Z_{\mathcal{A}}\neq\\{0\\}=Y_{\emptyset}$. Now suppose that
$\mathcal{A}^{\prime}\subseteq\mathcal{A}$ is non-empty, and take
$A\in\mathcal{A}^{\prime}$. We claim that the indicator function $1_{A}$,
which belongs to $Y_{\mathcal{A}^{\prime}}$ by definition, does not belong to
$Z_{\mathcal{A}}$, and therefore $Y_{\mathcal{A}^{\prime}}\not\subseteq
Z_{\mathcal{A}}$. More precisely, we shall show that $1_{A}$ has distance $1$
to the dense subspace $Z_{\mathcal{A}}^{0}$ of $Z_{\mathcal{A}}$ defined by
(2.3). Indeed, 2.2 shows that $\operatorname{supp}z\in[\Gamma]^{<\Gamma}$ for
every $z\in Z_{\mathcal{A}}^{0}$, whereas $\lvert A\rvert=\Gamma$, so $A$
contains an element $\gamma$ such that $z(\gamma)=0$. Consequently $\lVert
1_{A}-z\rVert_{\infty}\geqslant\lvert(1_{A}-z)(\gamma)\rvert=1$, which proves
the claim, and (i) follows.

(ii). Set $\alpha=\lvert\mathcal{A}\rvert$, and assume that the statement is
false; that is, there exists a bounded operator $T\colon
Y_{\mathcal{A}}\to\ell_{\infty}(\Gamma)$ such that
$Z_{\mathcal{A}}\subseteq\ker T$, but $Y_{\mathcal{A}^{\prime}}\nsubseteq\ker
T$ for every subset $\mathcal{A}^{\prime}$ of $\mathcal{A}$ with
$\lvert\mathcal{A}^{\prime}\rvert=\alpha$. Since $\ker T$ is closed and
contains $Z_{\mathcal{A}}$, this implies that every subset of $\mathcal{A}$ of
cardinality $\alpha$ contains a set $A$ with $T1_{A}\neq 0$. Hence, by
transfinite induction, we can construct a family of distinct sets
$(A(\beta))_{\beta<\alpha}$ in $\mathcal{A}$ such that $T1_{A(\beta)}\neq 0$
for every ordinal $\beta<\alpha$.

Observe that every ordinal $\beta<\alpha$ belongs to the set
$\Lambda_{\gamma,n}:=\bigl{\\{}\beta<\alpha:\lvert(T1_{A(\beta)})(\gamma)\rvert\geqslant\frac{1}{n}\bigr{\\}}$
for some $\gamma\in\Gamma$ and $n\in\mathbb{N}$. Since $\alpha>\Gamma$, it
follows that $\Lambda_{\gamma_{0},n_{0}}$ has cardinality greater than
$\Gamma$ for some $\gamma_{0}\in\Gamma$ and $n_{0}\in\mathbb{N}$. In fact, all
we need is that $\Lambda_{\gamma_{0},n_{0}}$ contains $m$ distinct elements
$\beta_{1},\ldots,\beta_{m}$, where $m\in\mathbb{N}$ satisfies $m>n_{0}\lVert
T\rVert$. For each $j\in\\{1,\ldots,m\\}$, set $A_{j}=A(\beta_{j})$ and choose
a scalar $\sigma_{j}$ of modulus one such that
$\sigma_{j}\cdot(T1_{A_{j}})(\gamma_{0})>0$.

Lemma 2.1 implies that the vectors

$y=\sum_{j=1}^{m}\sigma_{j}1_{A_{j}}\in Y_{\mathcal{A}}\qquad\text{and}\qquad
z=\underset{\lvert N\rvert\geqslant
2}{\sum_{N\subseteq\\{1,\ldots,m\\}}}(-1)^{\lvert N\rvert}\Bigl{(}\sum_{j\in
N}\sigma_{j}\Bigr{)}1_{\bigcap_{j\in N}A_{j}}\in Z_{\mathcal{A}}\subseteq\ker
T$

satisfy

(2.4) $\displaystyle y-z=\sum_{\emptyset\neq
N\subseteq\\{1,\ldots,m\\}}(-1)^{\lvert N\rvert+1}\Bigl{(}\sum_{j\in
N}\sigma_{j}\Bigr{)}1_{\bigcap_{j\in
N}A_{j}}=\sum_{j=1}^{m}\sigma_{j}1_{A_{j}^{d}},$

where the sets $A_{1}^{d},\ldots,A_{m}^{d}$ are defined by (2.1). In
particular, they are disjoint, so $\lVert
y-z\rVert_{\infty}\leqslant\max\\{\lvert\sigma_{1}\rvert,\ldots,\lvert\sigma_{m}\rvert\\}=1$,
and hence we have

$\displaystyle\lVert T\rVert$ $\displaystyle\geqslant\lVert
T(y-z)\rVert_{\infty}=\lVert
Ty\rVert_{\infty}\geqslant\lvert(Ty)(\gamma_{0})\rvert$
$\displaystyle=\Bigl{|}\sum_{j=1}^{m}\sigma_{j}\cdot(T1_{A_{j}})(\gamma_{0})\Bigr{|}=\sum_{j=1}^{m}\lvert(T1_{A_{j}})(\gamma_{0})\rvert\geqslant\frac{m}{n_{0}}>\lVert
T\rVert,$

which is clearly absurd. ∎

We conclude this section by showing how a slightly weaker version of Theorem
1.1 can be proved using perhaps more standard topological methods. It relies
on the following lemma, which can be viewed as a generalisation of the fact
that $c_{0}(\mathbb{R})$ embeds isometrically into $\ell_{\infty}/c_{0}$ (see,
_e.g.,_ [1, Theorem 1.25]). In the remainder of this paper, given a non-empty
set $J$, we write $(\delta_{j})_{j\in J}$ for the standard unit vector basis
of $c_{0}(J)$.

######  Lemma 2.3.

Let $\mathcal{A}\subset[\Gamma]^{\Gamma}$ be an almost disjoint family. Then
$c_{0}(\mathcal{A})$ embeds isometrically into
$Y_{\mathcal{A}}/Z_{\mathcal{A}}$.

###### Proof.

Let $\pi\colon Y_{\mathcal{A}}\to Y_{\mathcal{A}}/Z_{\mathcal{A}}$ denote the
quotient map. Since $(\delta_{A})_{A\in\mathcal{A}}$ spans a dense subspace of
$c_{0}(\mathcal{A})$, it suffices to show that the linear map
$\psi\colon(\operatorname{span}\\{\delta_{A}:A\in\mathcal{A}\\},\|\cdot\|_{\infty})\to
Y_{\mathcal{A}}/Z_{\mathcal{A}}$ defined by $\psi(\delta_{A})=\pi(1_{A})$ for
each $A\in\mathcal{A}$ is an isometry.

Take distinct sets $A_{1},\ldots,A_{m}\in\mathcal{A}$ for some
$m\in\mathbb{N}$, and let $\sigma_{1},\ldots,\sigma_{m}\in\mathbb{K}$. Then we
have

$\psi\Bigl{(}\sum_{j=1}^{m}\sigma_{j}\delta_{A_{j}}\Bigr{)}=\pi\Bigl{(}\sum_{j=1}^{m}\sigma_{j}1_{A_{j}}\Bigr{)}=\pi\Bigl{(}\sum_{j=1}^{m}\sigma_{j}1_{A^{d}_{j}}\Bigr{)}$

because
$\sum_{j=1}^{m}\sigma_{j}1_{A_{j}}-\sum_{j=1}^{m}\sigma_{j}1_{A^{d}_{j}}\in
Z_{\mathcal{A}}$ by (2.4). Therefore, showing that $\psi$ is an isometry is
equivalent to showing that

(2.5)
$\Bigl{\|}\pi\Bigl{(}\sum_{j=1}^{m}\sigma_{j}1_{A^{d}_{j}}\Bigr{)}\Bigr{\|}=\max_{1\leqslant
j\leqslant m}\lvert\sigma_{j}\rvert.$

The inequality $\leqslant$ is clear because $\lVert\pi\rVert\leqslant 1$ and
the sets $A_{1}^{d},\ldots,A_{m}^{d}$ are disjoint.

To verify the opposite inequality, choose $i\in\\{1,\ldots,m\\}$ such that
$\lvert\sigma_{i}\rvert=\max_{1\leqslant j\leqslant m}\lvert\sigma_{j}\rvert$.
Given $\varepsilon>0$ and $z\in Z_{\mathcal{A}}$, take $z_{0}\in
Z^{0}_{\mathcal{A}}$ such that $\|z-z_{0}\|_{\infty}<\varepsilon$, where
$Z^{0}_{\mathcal{A}}$ is the dense subspace of $Z_{\mathcal{A}}$ defined by
(2.3). 2.2 shows that $\operatorname{supp}z_{0}\in[\Gamma]^{<\Gamma}$. Since
$\lvert A_{i}\rvert=\Gamma$ and $\bigcup_{k=1,\,k\neq i}^{m}A_{k}\cap
A_{i}\in[\Gamma]^{<\Gamma}$, it follows that we can find $\gamma\in A_{i}^{d}$
such that $z_{0}(\gamma)=0$. This implies that $\lvert z(\gamma)\rvert=\lvert
z(\gamma)-z_{0}(\gamma)\rvert<\varepsilon$. Combining this with the fact that
$\gamma\notin A_{j}^{d}$ for each $j\in\\{1,\ldots,m\\}\setminus\\{i\\}$, we
obtain

$\displaystyle\Bigl{\|}\sum_{j=1}^{m}\sigma_{j}1_{A_{j}^{d}}-z\Bigr{\|}_{\infty}$
$\displaystyle\geqslant\Bigl{|}\sum_{j=1}^{m}\sigma_{j}1_{A_{j}^{d}}(\gamma)-z(\gamma)\Bigr{|}=\lvert\sigma_{i}-z(\gamma)\rvert\geqslant\lvert\sigma_{i}\rvert-\lvert
z(\gamma)\rvert>\lvert\sigma_{i}\rvert-\varepsilon.$

Since $\varepsilon>0$ and $z\in Z_{\mathcal{A}}$ were arbitrary, we conclude
that
$\bigl{\|}\pi\bigl{(}\sum_{j=1}^{m}\sigma_{j}1_{A_{j}^{d}}\bigr{)}\bigr{\|}\geqslant\lvert\sigma_{i}\rvert$,
and (2.5) follows. ∎

######  Corollary 2.4.

Let $\mathcal{A}\subset[\Gamma]^{\Gamma}$ be an almost disjoint family of
cardinality greater than $\Gamma$. Then the subspace $Z_{\mathcal{A}}$ is not
the kernel of any bounded operator $Y_{\mathcal{A}}\to\ell_{\infty}(\Gamma)$.

###### Proof.

Assume the contrary. Then, by the Fundamental Isomorphism Theorem, there
exists a bounded linear injection
$Y_{\mathcal{A}}/Z_{\mathcal{A}}\to\ell_{\infty}(\Gamma)$. Since
$c_{0}(\mathcal{A})$ embeds isometrically into
$Y_{\mathcal{A}}/Z_{\mathcal{A}}$ by 2.3, we obtain a bounded linear injection
$c_{0}(\mathcal{A})\to\ell_{\infty}(\Gamma)$. On the one hand, in view of [2,
Fact 4.10 (page 135)], this implies that the weak*-density of
$c_{0}(\mathcal{A})^{*}$ is at most $\Gamma$. On the other hand, since
$c_{0}(\mathcal{A})^{*}\cong\ell_{1}(\mathcal{A})$, the weak*-density of
$c_{0}(\mathcal{A})^{*}$ is $\lvert\mathcal{A}\rvert$, which contradicts the
fact that $\lvert\mathcal{A}\rvert>\Gamma$. ∎

##  3\. Applications

In this section, we shall deduce two important consequences of Theorem 1.1.
The first will simultaneously generalise the previously mentioned result of
Kalton [4, Proposition 4] to uncountable index sets and a theorem of
Pełczyński and Sudakov [9, Theorem 1] stating that the closed subspace

(3.1)
$\displaystyle\ell_{\infty}^{<}(\Gamma):=\bigl{\\{}x\in\ell_{\infty}(\Gamma):\\{\gamma\in\Gamma:\lvert
x(\gamma)\rvert>\varepsilon\\}\in[\Gamma]^{<\Gamma}\ \text{for every}\
\varepsilon>0\bigr{\\}}$

is not complemented in $\ell_{\infty}(\Gamma)$ for any infinite set $\Gamma$.

######  Theorem 3.1.

Let $T\colon\ell_{\infty}(\Gamma)\to\ell_{\infty}(\Gamma)$ be a bounded
operator whose kernel contains $\ell_{\infty}^{<}(\Gamma)$. Then there exists
an almost disjoint family $\mathcal{A}\subset[\Gamma]^{\Gamma}$ of cardinality
greater than $\Gamma$ such that $Y_{\mathcal{A}}\subseteq\ker T$.

In particular, $\ell_{\infty}^{<}(\Gamma)$ is a closed subspace of
$\ell_{\infty}(\Gamma)$ which is not the kernel of any bounded operator on
$\ell_{\infty}(\Gamma)$.

###### Proof.

As previously mentioned, Sierpiński [11] has shown that there is an almost
disjoint family $\mathcal{B}\subset[\Gamma]^{\Gamma}$ of cardinality greater
than $\Gamma$. 2.2 implies that
$Z_{\mathcal{B}}^{0}\subseteq\ell_{\infty}^{<}(\Gamma)$, and hence the same is
true for its closure $Z_{\mathcal{B}}$. It follows that $Z_{\mathcal{B}}$ is
contained in the kernel of the restriction $T|_{Y_{\mathcal{B}}}$ of the
operator $T$ to $Y_{\mathcal{B}}$, so 1.1 implies that $\mathcal{B}$ contains
a subset $\mathcal{A}$ of the same cardinality as $\mathcal{B}$ such that
$Y_{\mathcal{A}}\subseteq\ker T|_{Y_{\mathcal{B}}}$. This establishes the
first part of the result. The second part follows because
$1_{A}\notin\ell_{\infty}^{<}(\Gamma)$ whenever $A\in[\Gamma]^{\Gamma}$. ∎

######  Remark 3.2.

We observe that $\ell_{\infty}^{<}(\mathbb{N})=c_{0}$, so 3.1 is a genuine
generalisation of Kalton’s result, which corresponds to $\Gamma=\mathbb{N}$.

Our second application of 1.1 focuses on the particular case
$\Gamma=\mathbb{N}$. Given an almost disjoint family
$\mathcal{A}\subset[\mathbb{N}]^{\omega}$, let $K_{\mathcal{A}}$ denote the
associated _Mrówka space_ (which is also known under other names such as
_$\Psi$ -space,_ _AU-compactum_ and _Isbell–Mrówka space_). As a set,

$\displaystyle
K_{\mathcal{A}}=\\{x_{n}:n\in\mathbb{N}\\}\cup\\{y_{A}:A\in\mathcal{A}\\}.$

Its topology is defined by declaring that the point $x_{n}$ is isolated in
$K_{\mathcal{A}}$ for each $n\in\mathbb{N}$, while the sets

$\displaystyle U(A,F):=\\{x_{n}:n\in A\setminus
F\\}\cup\\{y_{A}\\},\quad\text{where}\quad F\in[\mathbb{N}]^{<\omega},$

form a neighbourhood basis of the point $y_{A}$ for each $A\in\mathcal{A}$.
This turns $K_{\mathcal{A}}$ into a locally compact Hausdorff space.

Let $C_{0}(K_{\mathcal{A}})$ denote the Banach space of continuous functions
$K_{\mathcal{A}}\to\mathbb{K}$ vanishing at infinity. The map $V\colon
C_{0}(K_{\mathcal{A}})\to\ell_{\infty}$ defined by $(Vf)(n)=f(x_{n})$ for each
$f\in C_{0}(K_{\mathcal{A}})$ and $n\in\mathbb{N}$ is a linear isometry onto
the subspace

$\displaystyle
X_{\mathcal{A}}:=\overline{\operatorname{span}}\,\\{1_{A}:A\in\mathcal{A}\cup[\mathbb{N}]^{<\omega}\\}$

of $\ell_{\infty}$ (see, _e.g.,_ [5, Lemma 5]).

######  Theorem 3.3.

Let $\mathcal{A}\subset[\mathbb{N}]^{\omega}$ be an uncountable, almost
disjoint family, and suppose that $T\colon
C_{0}(K_{\mathcal{A}})\to\ell_{\infty}$ is a bounded operator such that
$T1_{\\{x_{n}\\}}=0$ for every $n\in\mathbb{N}$. Then $\mathcal{A}$ contains a
subset $\mathcal{A}^{\prime}$ of the same cardinality as $\mathcal{A}$ such
that $T1_{U(A,F)}=0$ for every $A\in\mathcal{A}^{\prime}$ and
$F\in[\mathbb{N}]^{<\omega}$.

In particular, the closed subspace of $C_{0}(K_{\mathcal{A}})$ spanned by
$\\{1_{\\{x_{n}\\}}:n\in\mathbb{N}\\}$ is not the kernel of any bounded
operator $C_{0}(K_{\mathcal{A}})\to\ell_{\infty}$.

###### Proof.

Consider the composite operator $S:=TW\colon X_{\mathcal{A}}\to\ell_{\infty}$,
where $W\colon X_{\mathcal{A}}\to C_{0}(K_{\mathcal{A}})$ is the inverse of
the linear isometry $V$ defined above. We observe that $X_{\mathcal{A}}$
contains the subspace $Y_{\mathcal{A}}$ defined by (1.1) because $\mathcal{A}$
is almost disjoint, and that $Z_{\mathcal{A}}\subseteq\ker S$ because
$F:=\bigcap_{j=1}^{n}A_{j}\in[\mathbb{N}]^{<\omega}$ whenever
$n\in\mathbb{N}\setminus\\{1\\}$ and $A_{1},\ldots,A_{n}\in\mathcal{A}$ are
distinct, so that

$\displaystyle S1_{F}=\sum_{m\in F}S1_{\\{m\\}}=\sum_{m\in
F}T1_{\\{x_{m}\\}}=0.$

Hence we can apply Theorem 1.1 to the restriction of $S$ to $Y_{\mathcal{A}}$
to obtain a subset $\mathcal{A}^{\prime}$ of $\mathcal{A}$ having the same
cardinality as $\mathcal{A}$ such that $Y_{\mathcal{A}^{\prime}}\subseteq\ker
S\rvert_{Y_{\mathcal{A}}}$. This implies that $0=TW1_{A}=T1_{U(A,\emptyset)}$
for every $A\in\mathcal{A}^{\prime}$, and since

$\displaystyle 1_{U(A,\emptyset)}-1_{U(A,F)}=\sum_{m\in
F}1_{\\{x_{m}\\}}\in\ker T$

for every $F\in[\mathbb{N}]^{<\omega}$, we conclude that $T1_{U(A,F)}=0$ for
every $A\in\mathcal{A}^{\prime}$ and $F\in[\mathbb{N}]^{<\omega}$.

The final clause follows because $1_{U(A,F)}$ has distance one to the subspace
spanned by $\\{1_{\\{x_{n}\\}}:n\in\mathbb{N}\\}$ for every
$A\in\mathcal{A}^{\prime}$ and $F\in[\mathbb{N}]^{<\omega}$. ∎

######  Remark 3.4.

Let $\mathcal{A}\subset[\mathbb{N}]^{\omega}$ be an uncountable, almost
disjoint family. Since $C_{0}(K_{\mathcal{A}})$ embeds into $\ell_{\infty}$,
3.3 implies that the closed subspace of $C_{0}(K_{\mathcal{A}})$ spanned by
$\\{1_{\\{x_{n}\\}}:n\in\mathbb{N}\\}$, which is isomorphic to $c_{0}$, is not
the kernel of any bounded operator $C_{0}(K_{\mathcal{A}})\to
C_{0}(K_{\mathcal{A}})$.

##  4\. Kernels of operators on $\ell_{\infty}^{c}(\Gamma)$

An interesting counterpart of the Banach space $\ell_{\infty}(\Gamma)$, as
well as its subspace $\ell_{\infty}^{<}(\Gamma)$ defined by (3.1), is

$\displaystyle\ell_{\infty}^{c}(\Gamma):=\\{x\in\ell_{\infty}(\Gamma):\operatorname{supp}(x)\
\text{is countable}\\}.$

This is readily seen to be a closed subspace of $\ell_{\infty}(\Gamma)$ and
hence a Banach space in its own right. Since
$\ell_{\infty}^{c}(\Gamma)=\ell_{\infty}(\Gamma)$ when $\Gamma$ is countable,
one can view $\ell_{\infty}^{c}(\Gamma)$ as a “smaller” generalisation of
$\ell_{\infty}$ to uncountable index sets than $\ell_{\infty}(\Gamma)$ itself.

From a certain perspective, this generalisation is nicer than
$\ell_{\infty}(\Gamma)$. Indeed, Johnson, Kania and Schechtman [3, Theorem
1.4] have given a complete classification of the complemented subspaces of
$\ell_{\infty}^{c}(\Gamma)$, something that has been achieved for
$\ell_{\infty}(\Gamma)$ only when $\Gamma$ is countable. However,
$\ell_{\infty}^{c}(\Gamma)$ does not enjoy the same universal properties as
$\ell_{\infty}(\Gamma)$ when $\Gamma$ is uncountable because
$\ell_{\infty}(\Gamma)$ is injective, whereas $\ell_{\infty}^{c}(\Gamma)$ is
not due to a result of Pełczyński and Sudakov [9, Corollary, page 87] stating
that $\ell_{\infty}^{c}(\Gamma)$ is not complemented in
$\ell_{\infty}(\Gamma)$. We remark that $\ell_{\infty}^{c}(\Gamma)$ has the
weaker property of being separably injective (see [1, Definition 2.1 and
Example 2.4] for details).

Our work in this section is motivated by the following question, which asks
whether a natural generalisation of Kalton’s result for $\ell_{\infty}$ holds
for $\ell_{\infty}^{c}(\Gamma)$.

######  Question 4.1.

Let $\Gamma$ be an uncountable cardinal, and suppose that
$T\colon\ell_{\infty}^{c}(\Gamma)\to\ell_{\infty}^{c}(\Gamma)$ is a bounded
operator with $c_{0}(\Gamma)\subseteq\ker T$. Does $\ker T$ contain the
subspace

$\ell_{\infty}^{c}(\Lambda):=\\{x\in\ell_{\infty}^{c}(\Gamma):\operatorname{supp}x\subseteq\Lambda\\}$

for some $\Lambda\in[\Gamma]^{\Gamma}$?

While we cannot answer this question in full generality, the following result
gives a positive answer in certain cases. It relies on the following piece of
terminology: Let $X$, $Y$ and $Z$ be Banach spaces. We say that a bounded
operator $T\colon X\to Y$ _fixes a copy of_ $Z$ if there is a bounded operator
$U\colon Z\to X$ such that $TU$ is an isomorphic embedding.

######  Proposition 4.2.

Let $\Gamma$ be a cardinal with uncountable cofinality and $X$ a Banach space,
and suppose that $T\colon\ell_{\infty}^{c}(\Gamma)\to X$ is a bounded operator
which does not fix a copy of $c_{0}(\Gamma)$. Then
$\ell_{\infty}^{c}(\Lambda)\subseteq\ker T$ for some
$\Lambda\in[\Gamma]^{\Gamma}$.

###### Proof.

We shall prove the contrapositive statement, so suppose that
$\ell_{\infty}^{c}(\Lambda)\nsubseteq\ker T$ for every
$\Lambda\in[\Gamma]^{\Gamma}$. Since $\Gamma$ is infinite, we can take a
family $(\Lambda_{\gamma})_{\gamma\in\Gamma}$ of mutually disjoint sets in
$[\Gamma]^{\Gamma}$. The hypothesis implies that we can choose a unit vector
$x_{\gamma}\in\ell_{\infty}^{c}(\Lambda_{\gamma})\setminus\ker T$ for each
$\gamma\in\Gamma$. Then the transfinite sequence
$(x_{\gamma})_{\gamma\in\Gamma}$ consists of disjointly supported unit vectors
in $\ell_{\infty}^{c}(\Gamma)\subset\ell_{\infty}(\Gamma)$, so we can define a
linear isometry $S\colon c_{0}(\Gamma)\to\ell_{\infty}^{c}(\Gamma)$ by setting
$S\delta_{\gamma}=x_{\gamma}$ for each $\gamma\in\Gamma$.

Combining the fact that $\Gamma$ has uncountable cofinality with the
observation that

$\Gamma=\bigcup_{n\in\mathbb{N}}\Gamma_{n},\quad\text{where}\quad\Gamma_{n}:=\Bigl{\\{}\gamma\in\Gamma:\|Tx_{\gamma}\|_{\infty}\geqslant\frac{1}{n}\Bigr{\\}},$

we conclude that $\lvert\Gamma_{n}\rvert=\Gamma$ for some $n\in\mathbb{N}$.
Then

$\inf\bigl{\\{}\lVert
TS\delta_{\gamma}\rVert_{\infty}:\gamma\in\Gamma_{n}\bigr{\\}}\geqslant\frac{1}{n}>0,$

so a famous result of Rosenthal stated in [10, Remark 1 (page 30)] implies
that we can find $\Gamma^{\prime}\in[\Gamma_{n}]^{\Gamma}$ such that the
restriction of $TS$ to $c_{0}(\Gamma^{\prime})\subseteq c_{0}(\Gamma)$ is an
isomorphic embedding. This shows that $T$ fixes a copy of $c_{0}(\Gamma)$. ∎

4.2 should be compared with the following result of Wojtówicz [13, Theorem 1]:

######  Theorem 4.3 (Wojtówicz).

Let $T\colon\ell_{\infty}(\Gamma)/c_{0}(\Gamma)\to X$ be a bounded operator,
where $\Gamma$ is infinite and $X$ is a Banach space. If $T$ does not fix a
copy of $\ell_{\infty}$, then $\ker T$ contains a closed subspace which is
isomorphic to $c_{0}$.

######  Remark 4.4.

Let $X:=\ell_{\infty}^{c}(\Gamma)$, and write $\mathscr{B}(X)$ for the Banach
algebra of bounded operators $X\to X$. Johnson, Kania and Schechtman [3,
Theorems 1.1 and 3.14] have shown that the set

$\displaystyle\mathscr{S}_{X}(X):=\\{T\in\mathscr{B}(X):T\ \text{does not fix
a copy of}\ X\\}$

is the unique maximal ideal of $\mathscr{B}(X)$ and that
$T\in\mathscr{S}_{X}(X)$ if and only if $T$ does not fix a copy of
$c_{0}(\Gamma)$. Suppose that $\Gamma$ has uncountable cofinality. Then we can
combine this result with 4.2 to deduce that, for every
$T\in\mathscr{S}_{X}(X)$, there exists $\Lambda\in[\Gamma]^{\Gamma}$ such that
$\ell_{\infty}^{c}(\Lambda)\subseteq\ker T$.

4.4 shows that a positive answer to the following question would imply a
positive answer to 4.1 for cardinals of uncountable cofinality.

######  Question 4.5.

Let $X:=\ell_{\infty}^{c}(\Gamma)$ for some cardinal $\Gamma$ with uncountable
cofinality, and let $T\in\mathscr{B}(X)$ be a bounded operator such that
$c_{0}(\Gamma)\subseteq\ker T$. Does $T$ belong to the unique maximal ideal
$\mathscr{S}_{X}(X)$ of $\mathscr{B}(X)$?

Unfortunately, we have no evidence to suggest that the answer to 4.5 is
positive. On the contrary, if we drop the hypothesis that $\Gamma$ has
uncountable cofinality and consider $\Gamma=\mathbb{N}$, then we know that the
answer is negative! This relies on the following result.

######  Lemma 4.6.

There exists a bounded idempotent operator
$P\colon\ell_{\infty}\to\ell_{\infty}$ with infinite-dimensional range such
that $c_{0}\subseteq\ker P$.

###### Proof.

Consider the closed, complementary subspaces
$X_{1}:=\ell_{\infty}(2\mathbb{N}-1)$ and $X_{2}:=\ell_{\infty}(2\mathbb{N})$
of $\ell_{\infty}$, and let $Y:=c_{0}(2\mathbb{N})\subset X_{2}$. Then we can
define a linear isometry $T\colon Y\to\ell_{\infty}$ by setting
$T\delta_{2n}=\delta_{n}$ for each $n\in\mathbb{N}$. We claim that there
exists an automorphism $\tau\colon\ell_{\infty}\to\ell_{\infty}$ which extends
$T$ in the sense that $\tau|_{Y}=T$; this will follow from [8, Theorem
2.f.12(i)], provided that we can show that the quotient spaces
$\ell_{\infty}/T(Y)$ and $\ell_{\infty}/Y$ are both non-reflexive. The first
of these assertions follows immediately from the fact that $T(Y)=c_{0}$, while
the second is a consequence of the fact that the restriction of the quotient
map $\ell_{\infty}\to\ell_{\infty}/Y$ to the non-reflexive subspace $X_{1}$ is
an isometry.

Hence $\tau(X_{1})$ and $\tau(X_{2})$ are closed, complementary subspaces of
$\ell_{\infty}$, so we can take a bounded idempotent operator
$P\colon\ell_{\infty}\to\ell_{\infty}$ with $P(\ell_{\infty})=\tau(X_{1})$ and
$\ker P=\tau(X_{2})$. It has infinite-dimensional range because $X_{1}$ is
infinite-dimensional and $\tau$ is injective, and

$c_{0}=T(Y)=\tau(Y)\subseteq\tau(X_{2})=\ker P.\qed$

######  Remark 4.7.

We can now explain why the answer to 4.5 is negative for $\Gamma=\mathbb{N}$.
This relies on the fact that the ideal of strictly singular operators on
$\ell_{\infty}$ is the unique maximal ideal of $\mathscr{B}(\ell_{\infty})$,
as observed in [6, §3]. By definition, a bounded operator is _strictly
singular_ if it does not fix any infinite-dimensional Banach space. In
particular, a bounded idempotent operator is strictly singular if and only if
it has finite-dimensional range. Therefore the operator $P$ from 4.6 does not
belong to the unique maximal ideal of $\mathscr{B}(\ell_{\infty})$, but
$c_{0}\subseteq\ker P$.

######  Acknowledgement.

The first-named author is grateful to Max Arnott (Lancaster) for bringing this
problem to his attention, and thanks Tommaso Russo (Innsbruck) for some
enlightening conversations regarding the contents of Section 4. He
acknowledges with thanks the financial support provided by the Jagiellonian
Scholarship “Initiative of Excellence — Jagiellonian University” awarded by
the Faculty of Mathematics and Computer Science. Finally, we would like to
thank Jesús M. F. Castillo (Badajoz) for mentioning the original reference [4]
to us.

## References

  * [1] A. Avilés, F. Cabello Sánchez, J. M. F. Castillo, M. González and Y. Moreno, _Separably injective Banach spaces_ , Lect. Notes Math., Springer-Verlag, 2016.
  * [2] P. Hájek, V. Montesinos, J. Vanderwerff, and V. Zizler, _Biorthogonal Systems in Banach Spaces_ , CMS Books in Mathematics, Springer-Verlag, 2008.
  * [3] W. B. Johnson, T. Kania and G. Schechtman, Closed ideals of operators on and complemented subspaces of Banach spaces of functions with countable support, _Proc. Amer. Math. Soc._ 144 (2016), 4471–4485.
  * [4] N. J. Kalton, Spaces of compact operators, _Math. Ann._ 208 (1974), 267–278.
  * [5] P. Koszmider and N. J. Laustsen, A Banach space induced by an almost disjoint family, admitting only few operators and decompositions, _Adv. Math._ 381, Article ID 107613 (2021).
  * [6] N. J. Laustsen and R. J. Loy, Closed ideals in the Banach algebra of operators on a Banach space, in _Proceedings of the Conference on Topological Algebras, their Applications, and Related Topics_ (ed. K. Jarosz and A. Sołtysiak), _Banach Center Publications_ 67 (2005), 245–264.
  * [7] N. J. Laustsen and J. T. White, Subspaces that can and cannot be the kernel of a bounded operator on a Banach space, in _Banach algebras and applications. Proceedings of the 23rd international conference, University of Oulu, Finland, July 3–11, 2017_ (ed. M. Filali), _De Gruyter Proceedings in Mathematics_ (2020), 189–196.
  * [8] J. Lindenstrauss and L. Tzafriri, _Classical Banach Spaces I_ , Springer-Verlag, 1977.
  * [9] A. Pełczyński and V. N. Sudakov, Remark on non-complemented subspaces of the space $m(S)$, _Colloq. Math._ 9 (1962), 85–88.
  * [10] H. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, _Studia Math._ 37 (1970), 13–36.
  * [11] W. Sierpiński, Sur une décomposition d’ensembles, _Monatsh. Math. Phys._ 35 (1928), 239–242.
  * [12] H. M. Wark, A non-separable reflexive Banach space on which there are few operators, _J. London Math. Soc._ 64, No. 2 (2001), 675–689.
  * [13] M. Wojtówicz, Operators preserving $\ell_{\infty}$, _RACSAM_ 108 (2014), 511–517.