# Uniqueness of algebra norm on quotients of the algebra of bounded operators
on a Banach space

Max Arnott  Department of Mathematics and Statistics, Fylde College, Lancaster
University, Lancaster, LA1 4YF, United Kingdom<EMAIL_ADDRESS>and
Niels Jakob Laustsen  Department of Mathematics and Statistics, Fylde College,
Lancaster University, Lancaster, LA1 4YF, United Kingdom
<EMAIL_ADDRESS>
###### Abstract.

We show that for each of the following Banach spaces $X$, the quotient algebra
$\mathscr{B}(X)/\mathscr{I}$ has a unique algebra norm for every closed ideal
$\mathscr{I}$ of $\mathscr{B}(X)\colon$

  * •

$X=\bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\bigr{)}_{c_{0}}$ and its
dual, $X=\bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\bigr{)}_{\ell_{1}}$,

  * •

$X=\bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\bigr{)}_{c_{0}}\oplus
c_{0}(\Gamma)$ and its dual,
$X=\bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\bigr{)}_{\ell_{1}}\oplus\ell_{1}(\Gamma)$,
for an uncountable cardinal number $\Gamma$,

  * •

$X=C_{0}(K_{\mathcal{A}})$, the Banach space of continuous functions vanishing
at infinity on the locally compact Mrówka space $K_{\mathcal{A}}$ induced by
an uncountable, almost disjoint family $\mathcal{A}$ of infinite subsets of
$\mathbb{N}$, constructed such that $C_{0}(K_{\mathcal{A}})$ admits “few
operators”.

Equivalently, this result states that every homomorphism from $\mathscr{B}(X)$
into a Banach algebra is continuous and has closed range. The key step in our
proof is to show that the identity operator on a suitably chosen Banach space
factors through every operator in $\mathscr{B}(X)\setminus\mathscr{I}$ with
control over the norms of the operators used in the factorization. These
quantitative factorization results may be of independent interest.

###### Key words and phrases:

Banach space, bounded operator, closed operator ideal, quotient algebra,
uniqueness of algebra norm, long sequence space, Mrówka space, quantitative
factorization of operators

###### 2010 Mathematics Subject Classification:

46H10, 46H40, 47L10 (primary); 46B26, 46B45, 46E15, 47B01, 47L20 (secondary)

##  1\. Introduction and statement of main results

By an _algebra norm_ on a (real or complex) algebra $\mathscr{A}$, we
understand a norm $\lVert\,\cdot\,\rVert$ on $\mathscr{A}$ that is
submultiplicative in the sense that $\lVert ab\rVert\leqslant\lVert
a\rVert\,\lVert b\rVert$ for every $a,b\in\mathscr{A}$. A  _normed algebra_ is
an algebra equipped with an algebra norm. We say that a normed algebra
$\mathscr{A}$ has a unique algebra norm if every algebra norm on $\mathscr{A}$
is equivalent to the given norm.

In the case where $\mathscr{A}$ is a Banach algebra (which will be the case
for us), this notion should not be confused with the weaker notion of having a
unique _complete_ algebra norm; see Remark 2.3 for a more detailed discussion
of the difference between these two notions.

We shall focus on the Banach algebra $\mathscr{B}(X)$ of bounded operators on
a Banach space $X$ and its quotients by closed ideals. One of the earliest
results about this Banach algebra (predating even the term “Banach algebra”)
is due to Eidelheit [13, Theorem 1], who showed that $\mathscr{B}(X)$ has a
unique complete algebra norm for every Banach space $X$. In fact, building on
Eidelheit’s methods, one can deduce a stronger conclusion, which we shall
state in Theorem 2.7 below, once we have established the necessary
terminology.

Many related results have subsequently been obtained; [50, §0] contains an
excellent survey of what was known around the turn of the millenium. We would
like to highlight the following results, which provide the most important
context for our work:

  1. (i)

Meyer [37] proved that the Calkin algebra $\mathscr{B}(X)/\mathscr{K}(X)$ has
a unique algebra norm for each of the classical sequence spaces $X=c_{0}$ and
$X=\ell_{p}$, where $1\leqslant p<\infty$. For later reference, we recall the
classical result of Gohberg, Markus and Feldman [17] that the ideal
$\mathscr{K}(X)$ of compact operators is the unique non-trivial closed ideal
of $\mathscr{B}(X)$ for each of these Banach spaces $X$.

  2. (ii)

In his dissertation [51], Ware launched a systematic attack on the uniqueness-
of-algebra-norm question for Calkin algebras, generalizing the aforementioned
results of Meyer by showing that the Calkin algebra
$\mathscr{B}(X)/\mathscr{K}(X)$ has a unique algebra norm for a wide range of
Banach spaces $X$, including the following [51, §§5.2–5.5]:

     * •

every finite direct sum of spaces from the family
$\\{c_{0}\\}\cup\\{\ell_{p}:1\leqslant p<\infty\\};$

     * •

the infinite direct sums
$\bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{p}^{n}\bigr{)}_{c_{0}}$ and
$\bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{p}^{n}\bigr{)}_{\ell_{q}}$ for
$1\leqslant p\leqslant\infty$ and $1\leqslant q<\infty;$

     * •

the Tsirelson space $T$ and its dual $T^{*}$;

     * •

the quasi-reflexive James spaces $J_{p}$ for $1<p<\infty$.

  3. (iii)

In a somewhat different direction, Ware [51, Section 6] generalized Meyer’s
results to the non-separable setting as follows. Take an uncountable index set
$\Gamma$, and set $X=c_{0}(\Gamma)$ or $X=\ell_{p}(\Gamma)$ for some
$1\leqslant p<\infty$. Then the quotient $\mathscr{B}(X)/\mathscr{I}$ has a
unique algebra norm for every closed ideal $\mathscr{I}$ of $\mathscr{B}(X)$.
Daws’ classification [10] of the closed ideals of $\mathscr{B}(X)$ for these
Banach spaces $X$ plays a key role in this work.

  4. (iv)

More recently, Johnson and Phillips [26] have answered one of the main
questions left open in Ware’s dissertation [51] by showing that the Calkin
algebra $\mathscr{B}(L_{p}[0,1])/\mathscr{K}(L_{p}[0,1])$ has a unique algebra
norm whenever $1<p<\infty$. Furthermore, for $p\neq 2$, they have shown that
$\mathscr{B}(L_{p}[0,1])$ contains a closed ideal $\mathscr{I}$ for which the
quotient $\mathscr{B}(L_{p}[0,1])/\mathscr{I}$ admits at least two
inequivalent algebra norms.

Motivated by these results, especially (i), (iii) and (iv), we asked ourselves
for which Banach spaces $X$ other than $c_{0}(\Gamma)$ and $\ell_{p}(\Gamma)$
for $1\leqslant p<\infty$ is it true that every quotient of $\mathscr{B}(X)$
by a closed ideal has a unique algebra norm? Our focus has been on the
(relatively few) Banach spaces $X$ for which all of the closed ideals of
$\mathscr{B}(X)$ have been classified. The following theorem summarizes our
main findings.

######  Theorem 1.1.

Every quotient of $\mathscr{B}(X)$ by one of its closed ideals has a unique
algebra norm for each of the following Banach spaces $X\colon$

  1. (i)

$X=\bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\bigr{)}_{c_{0}}$ and
$X=\bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\bigr{)}_{\ell_{1}};$

  2. (ii)

$X=\bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\bigr{)}_{c_{0}}\oplus
c_{0}(\Gamma)$ and
$X=\bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\bigr{)}_{\ell_{1}}\oplus\ell_{1}(\Gamma)$
for an uncountable index set $\Gamma;$

  3. (iii)

$X=C_{0}(K_{\mathcal{A}})$, the Banach space of continuous functions vanishing
at infinity on the Mrówka space $K_{\mathcal{A}}$ induced by an uncountable,
almost disjoint family $\mathcal{A}$ of infinite subsets of $\mathbb{N}$,
chosen such that $C_{0}(K_{\mathcal{A}})$ admits “few operators”.

We refer to Section 6 for details of the terminology used in (iii). Note that
(i) can be viewed as a special case of (ii) by allowing the index set $\Gamma$
to be countable because

$\Bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\Bigr{)}_{c_{0}}\cong\Bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\Bigr{)}_{c_{0}}\oplus
c_{0}\qquad\text{and}\qquad\Bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\Bigr{)}_{\ell_{1}}\cong\Bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\Bigr{)}_{\ell_{1}}\oplus\ell_{1}.$

This manuscript is organized as follows. In Section 2, we introduce some
background material and associated terminology, notably concerning maximality,
minimality and incompressibility of algebra norms, and we explain the roles
that automatic continuity of homomorphisms and quantitative factorizations of
idempotents play in their study. We also give a “skeleton proof” of Theorem
1.1 and some illustrative examples for context.

Section 3 contains a quantitative factorization theorem for the identity
operator on $c_{0}$ that we shall use in the proofs of Theorem 1.1(ii) and
(iii). It is based on theorems of Rosenthal and
Dowling–Randrianantoanina–Turett.

With these preparations under our belt, we complete the proofs of the three
parts of Theorem 1.1 in Sections 4–6. We believe that our approach via
quantitative factorizations of the identity operators on suitably chosen
Banach spaces may be of interest to researchers studying topics quite
different from uniqueness of algebra norm and refer to Theorems 4.1, 5.4, 6.1
and Corollary 5.6 for our main conclusions stated in terms of such
factorizations.

Finally, in Section 7 we show that the quantitative factorization route we
took to proving Theorem 6.1 is not only sufficient, but also necessary.

##  2\. Preliminaries, including a skeleton proof of Theorem 1.1

All normed spaces and algebras are over the same scalar field $\mathbb{K}$,
either the real or the complex numbers. As usual, for a locally compact
Hausdorff space $K$, $C_{0}(K)$ denotes the Banach space of continuous
functions $f\colon K\to\mathbb{K}$ which “vanish at infinity” in the sense
that the set $\\{k\in K:\lvert f(k)\rvert\leqslant\varepsilon\\}$ is compact
for every $\varepsilon>0$.

The term “operator” means a bounded linear map between normed spaces. We write
$B_{X}$ for the closed unit ball of a normed space $X$, while $I_{X}$ (or
simply $I$ when $X$ is clear from the context) denotes the identity operator
on $X$. For a Banach space with a basis and $n\in\mathbb{N}$, $P_{n}$ will
usually denote the $n^{\text{th}}$ basis projection. Other notation will be
introduced as and when required.

Equivalence of norms plays a central role in this investigation. The
definition splits naturally in two parts, an upper and a lower bound. This is
the motivation behind the following terminology.

######  Definition 2.1.

Let $(\mathscr{A},\lVert\,\cdot\,\rVert)$ be a normed algebra. We say that the
given norm $\lVert\,\cdot\,\rVert$ on $\mathscr{A}$ is maximal if, for every
algebra norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\,\cdot\,\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
on $\mathscr{A}$, there is a constant $C_{1}>0$ such that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|a\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\leqslant
C_{1}\|a\|$ for every $a\in\mathscr{A}$. Analogously, we say that
$\lVert\,\cdot\,\rVert$ is minimal if, for every algebra norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\,\cdot\,\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
on $\mathscr{A}$, there is a constant $C_{2}>0$ such that $\|a\|\leqslant
C_{2}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|a\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
for every $a\in\mathscr{A}$. We say that $\mathscr{A}$ has a unique algebra
norm if the given norm is both maximal and minimal, or in other words if every
algebra norm on $\mathscr{A}$ is equivalent to the given norm.

######  Remark 2.2.

One usually requires that an algebra norm on a unital algebra $\mathscr{A}$
must take the value $1$ at the multiplicative identity $1_{\mathscr{A}}$. We
shall not adhere to this convention. This will not cause any problems because,
given any algebra norm $\lVert\,\cdot\,\rVert$ on $\mathscr{A}$,

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|a\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}=\sup\bigl{\\{}\lVert
ab\rVert:b\in\mathscr{A},\,\lVert b\rVert\leqslant
1\bigr{\\}}\qquad(a\in\mathscr{A})$

defines an equivalent algebra norm on $\mathscr{A}$ which satisfies
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|1_{\mathscr{A}}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}=1$.

######  Remark 2.3.

For a Banach algebra $\mathscr{A}$, there is a related notion of having a
unique complete algebra norm, which asserts that every complete algebra norm
on $\mathscr{A}$ is equivalent to the given norm. The Banach Isomorphism
Theorem implies that whenever the given norm on a Banach algebra $\mathscr{A}$
is either minimal or maximal, then it is automatically equivalent to any other
complete norm on $\mathscr{A}$, so $\mathscr{A}$ has unique complete algebra
norm.

It is well-known that maximality of the norm can be rephrased in terms of
automatic continuity of homomorphisms, which is one of the oldest topics in
the theory of Banach algebras. Here, and elsewhere, the term “homomorphism”
means a linear and multiplicative map between two algebras. There are many
ways to express this relationship. We have chosen the following, which
reflects the applications we have in mind. A much more comprehensive result
can be found in [42, Theorem 6.1.5].

######  Proposition 2.4.

The following three conditions are equivalent for a normed algebra
$(\mathscr{A},\lVert\,\cdot\,\rVert)\colon$

  1. (a)

The given norm $\lVert\,\cdot\,\rVert$ on $\mathscr{A}$ is maximal.

  2. (b)

Every homomorphism from $\mathscr{A}$ into a normed algebra is continuous.

  3. (c)

Every injective homomorphism from $\mathscr{A}$ into a Banach algebra is
continuous.

Now suppose that $(\mathscr{A},\lVert\,\cdot\,\rVert)$ is a Banach algebra.
Then conditions (a)–(c) are equivalent to the following two conditions:

  1. (d)

The quotient norm on $\mathscr{A}/\mathscr{I}$ is maximal for every closed
ideal $\mathscr{I}$ of $\mathscr{A}$.

  2. (e)

Every homomorphism from $\mathscr{A}/\mathscr{I}$ into a normed algebra is
continuous for every closed ideal $\mathscr{I}$ of $\mathscr{A}$.

###### Proof.

This is standard. For example, it is easy to adapt the proof of [8,
Proposition 2.1.7] to verify that conditions (a)–(c) are equivalent, and
therefore conditions (d) and (e) are also equivalent. This part of the proof
does not require completeness of $\mathscr{A}$. Completeness (in the shape of
the Open Mapping Theorem) is, however, required to prove that (b) implies (e);
see [8, Proposition 2.1.5] for details. Finally, the implication
(e)$\Rightarrow$(b) is trivial. ∎

B. E. Johnson [24] proved what has become the classical automatic continuity
result for homomorphisms from the Banach algebra $\mathscr{B}(X)$. We state it
in a simplified non-technical form, as it will suffice for our purposes.

######  Theorem 2.5 (Johnson).

Let $X$ be a Banach space which is isomorphic to its square $X\oplus X$. Then
every homomorphism from $\mathscr{B}(X)$ into a Banach algebra is continuous.

Proposition 2.4 has a counterpart for minimality. Its statement involves the
following notion: a linear map $T\colon X\to Y$ between normed spaces $X$ and
$Y$ is _bounded below_ if there is a constant $\eta>0$ such that $\lVert
Tx\rVert\geqslant\eta\lVert x\rVert$ for every $x\in\ X$. If we wish to
specify the constant, we say that $T$ is _bounded below by_ $\eta$.

######  Lemma 2.6.

Let $(\mathscr{A},\lVert\,\cdot\,\rVert)$ be a normed algebra. Then its norm
$\lVert\,\cdot\,\rVert$ is minimal if and only if every injective homomorphism
from $\mathscr{A}$ into a normed algebra is bounded below.

###### Proof.

Straightforward; we refer to the proof of [42, Theorem 6.1.5(a)–(b)] for
details. ∎

The following result originates in the work of Eidelheit [13, Lemma 1 and
Theorem 1], as already mentioned in the Introduction. It can for instance be
found in [8, Theorem 5.1.14] or [42, Theorem, page 107].

######  Theorem 2.7.

Let $X$ be a Banach space, and let $\mathscr{A}$ be a subalgebra of
$\mathscr{B}(X)$ which contains the ideal of finite-rank operators. Then the
operator norm on $\mathscr{A}$ is minimal.

In a forthcoming paper [26], W. B. Johnson and N. C. Phillips introduce and
explore the following two notions.

######  Definition 2.8.

A normed algebra $\mathscr{A}$ is:

  * •

incompressible if every continuous, injective homomorphism from $\mathscr{A}$
into a normed algebra is bounded below;

  * •

uniformly incompressible if there is a function
$f\colon(0,\infty)\to(0,\infty)$ such that every continuous, injective
homomorphism $\varphi$ from $\mathscr{A}$ into a normed algebra is bounded
below by $f(\lVert\varphi\rVert)$.

Lemma 2.6 shows that minimality of the norm implies incompressibility and that
the two notions are equivalent if every homomorphism from $\mathscr{A}$ into a
normed algebra is continuous. More precisely, we have the following result.

######  Lemma 2.9.

A normed algebra $(\mathscr{A},\lVert\,\cdot\,\rVert)$ has a unique algebra
norm if and only if $\mathscr{A}$ is incompressible and the given norm
$\lVert\,\cdot\,\rVert$ is maximal.

###### Proof.

This is immediate from Lemma 2.6 because, by Proposition 2.4, maximality of
the norm $\lVert\,\cdot\,\rVert$ means that every homomorphism from
$\mathscr{A}$ into a normed algebra is continuous. ∎

######  Corollary 2.10.

The following three conditions are equivalent for a Banach algebra
$\mathscr{A}\colon$

  1. (a)

Every quotient algebra of $\mathscr{A}$ by a closed ideal has a unique algebra
norm.

  2. (b)

Every homomorphism from $\mathscr{A}$ into a Banach algebra is continuous and
has closed range.

  3. (c)

Every homomorphism from $\mathscr{A}$ into a Banach algebra is continuous, and
$\mathscr{A}/\mathscr{I}$ is incompressible for every closed ideal
$\mathscr{I}$ of $\mathscr{A}$.

###### Proof.

(a)$\Rightarrow$(b). Let $\varphi\colon\mathscr{A}\to\mathscr{B}$ be a
homomorphism into a Banach algebra $\mathscr{B}$. Proposition 2.4 shows that
$\varphi$ is continuous because the norm on $\mathscr{A}$ is maximal. In
particular, the ideal $\ker\varphi$ is closed, and the quotient norm on
$\mathscr{A}/\ker\varphi$ is minimal by hypothesis. Let
$\widetilde{\varphi}\colon\mathscr{A}/\ker\varphi\to\mathscr{B}$ be the
induced homomorphism given by $\widetilde{\varphi}(a+\ker\varphi)=\varphi(a)$;
it satisfies:

  1. (i)

$\widetilde{\varphi}[\mathscr{A}/\ker\varphi]=\varphi[\mathscr{A}]$,

  2. (ii)

$\widetilde{\varphi}$ is injective and therefore bounded below by Lemma 2.6,

  3. (iii)

$\widetilde{\varphi}$ is continuous, hence it has closed range by (ii), and
therefore so does $\varphi$ by (i).

(b)$\Rightarrow$(c). Let $\mathscr{I}$ be a closed ideal of $\mathscr{A}$, and
consider a continuous, injective homomorphism
$\varphi\colon\mathscr{A}/\mathscr{I}\to\mathscr{B}$ into a normed algebra
$\mathscr{B}$. By hypothesis, the composite homomorphism
$\iota\varphi\pi\colon\mathscr{A}\to\widehat{\mathscr{B}}$ has closed range,
where $\pi\colon\mathscr{A}\to\mathscr{A}/\mathscr{I}$ is the quotient map and
$\iota\colon\mathscr{B}\to\widehat{\mathscr{B}}$ is the isometric embedding of
$\mathscr{B}$ into its completion $\widehat{\mathscr{B}}$. It follows that the
injection $\iota\varphi$ has closed range because $\pi$ is surjective, so
$\iota\varphi$ is bounded below, and therefore the same is true for $\varphi$.

(c)$\Rightarrow$(a). This follows from Proposition 2.4 and Lemma 2.9. ∎

We can now outline how we shall prove Theorem 1.1.

###### Skeleton proof of Theorem 1.1.

According to Corollary 2.10, we must show that each of the Banach spaces $X$
listed in clauses (i)–(iii) of Theorem 1.1 satisfies:

  1. (1)

every homomorphism from $\mathscr{B}(X)$ into a Banach algebra is continuous;

  2. (2)

$\mathscr{B}(X)/\mathscr{I}$ is incompressible for every closed ideal
$\mathscr{I}$ of $\mathscr{B}(X)$.

In each case, (1) is already known to hold true. This follows from Johnson’s
result stated in Theorem 2.5 above for the Banach spaces listed in clauses
(i)–(ii) because they are all isomorphic to their squares, while [30,
Corollary 39] verifies it for the particular Banach space
$X=C_{0}(K_{\mathcal{A}})$ considered in clause (iii), which fails to be
isomorphic to its square.

Considering (2), we may clearly suppose that the ideal $\mathscr{I}$ is
proper, and in view of Theorem 2.7, we may also suppose that $\mathscr{I}$ is
non-zero. Hence, it remains to show that

$\mathscr{B}(X)/\mathscr{I}\ \text{is incompressible for every non-trivial
closed ideal}\ \mathscr{I}\ \text{of}\ \mathscr{B}(X),$ (2.1)

which we shall do in Sections 4–6. ∎

When verifying (2.1), we shall follow the lead of Johnson and Phillips [26]
and use “quantitative factorizations” of idempotent operators to deduce that
the quotient algebra $\mathscr{B}(X)/\mathscr{I}$ is uniformly incompressible.
This is a natural tool to bring to bear on the problem at hand because
factorizations of idempotent operators already play a key role in the
classifications of the closed ideals of $\mathscr{B}(X)$ for each of the
Banach spaces $X$ we consider. The information we need to add to these results
is that the factorizations can be carried out with uniform norm bounds on the
auxiliary operators. We believe that the results we obtain in this regard will
be of interest to researchers working on problems very different from
uniqueness of algebra norm. The following notion is at the heart of our
approach.

######  Definition 2.11.

Let $C\geqslant 1$. A normed algebra $\mathscr{A}$ has the _idempotent
factorization property with constant_ $C$ (abbreviated $C$-IFP) if, for every
$a\in\mathscr{A}$ of norm $1$, there are $b,c\in\mathscr{A}$ with $\lVert
b\rVert\,\lVert c\rVert\leqslant C$ such that $bac$ is a non-zero idempotent.

It is sometimes more convenient to use the following equivalent formulation of
this definition: $\mathscr{A}$ has the $C$-IFP if, for every
$a\in\mathscr{A}\setminus\\{0\\}$, there are $b,c\in\mathscr{A}$ with $\lVert
b\rVert\,\lVert c\rVert\leqslant C/\lVert a\rVert$ such that $bac$ is a non-
zero idempotent.

Using this terminology, we can express the basic observation of Johnson and
Phillips [26] as follows; see also [38] for an earlier related, more abstract
result. We include a short proof for ease of reference.

######  Proposition 2.12.

Every normed algebra which has the $C$-IFP for some constant $C\geqslant 1$ is
uniformly incompressible.

###### Proof.

Suppose that $\mathscr{A}$ is a normed algebra with the $C$-IFP for some
$C\geqslant 1$, and let $\varphi\colon\mathscr{A}\to\mathscr{B}$ be a
continuous, injective homomorphism into some normed algebra $\mathscr{B}$. For
every $a\in\mathscr{A}\setminus\\{0\\}$, we can find $b,c\in\mathscr{A}$ with
$\lVert b\rVert\,\lVert c\rVert\leqslant C/\lVert a\rVert$ such that $p=bac$
is a non-zero idempotent. Then $\varphi(p)$ is a non-zero idempotent in
$\mathscr{B}$, so

$1\leqslant\lVert\varphi(p)\rVert\leqslant\lVert\varphi(b)\rVert\,\lVert\varphi(a)\rVert\,\lVert\varphi(c)\rVert\leqslant\lVert\varphi\rVert^{2}\,\frac{C}{\lVert
a\rVert}\,\lVert\varphi(a)\rVert.$

Rearranging this inequality, we see that $\varphi$ is bounded below by
$1/(C\lVert\varphi\rVert^{2})$, and consequently the definition of uniform
incompressibility is satisfied with respect to the function
$f\colon(0,\infty)\to(0,\infty)$ given by $f(t)=1/(Ct^{2})$. ∎

We shall usually apply a variant of Proposition 2.12, adapted to the context
of operator ideals. According to Pietsch [44], an operator ideal is an
assignment $\mathscr{J}$ which designates to every pair $(X,Y)$ of Banach
spaces a subspace $\mathscr{J}(X,Y)$ of $\mathscr{B}(X,Y)$ satisfying:

  * •

$\mathscr{J}(X,Y)\neq\\{0\\}$ for some Banach spaces $X$ and $Y$;

  * •

$UTS\in\mathscr{J}(W,Z)$ whenever $W$, $X$, $Y$ and $Z$ are Banach spaces and
$S\in\mathscr{B}(W,X)$, $T\in\mathscr{J}(X,Y)$ and $U\in\mathscr{B}(Y,Z)$ are
operators.

An operator ideal $\mathscr{J}$ is _closed_ if the subspace $\mathscr{J}(X,Y)$
is closed in the operator norm for every pair of Banach spaces $(X,Y)$. As
usual, we abbreviate $\mathscr{J}(X,X)$ to $\mathscr{J}(X)$.

######  Lemma 2.13.

Let $X$ be a Banach space and $\mathscr{J}$ a closed operator ideal. Suppose
that there exists a constant $C\geqslant 1$ such that, for every
$T\in\mathscr{B}(X)\setminus\mathscr{J}(X)$ with $\lVert
T+\mathscr{J}(X)\rVert=1$, there exist a Banach space $Y$ and operators
$R\in\mathscr{B}(X,Y)$ and $S\in\mathscr{B}(Y,X)$ with $\lVert R\rVert\,\lVert
S\rVert\leqslant C$ such that $RTS+\mathscr{J}(Y)$ is a non-zero idempotent in
$\mathscr{B}(Y)/\mathscr{J}(Y)$. Then $\mathscr{B}(X)/\mathscr{J}(X)$ has the
$C^{2}$-IFP and is therefore uniformly incompressible.

###### Proof.

If $\mathscr{J}(X)=\mathscr{B}(X)$, there is nothing to prove, so suppose that
the closed ideal $\mathscr{J}(X)$ is proper, and choose $C\geqslant 1$ as
specified. Then, for every
$t=T+\mathscr{J}(X)\in\mathscr{B}(X)/\mathscr{J}(X)$ with $\lVert t\rVert=1$,
we can find a Banach space $Y$ and operators $R\in\mathscr{B}(X,Y)$ and
$S\in\mathscr{B}(Y,X)$ with $\lVert R\rVert\,\lVert S\rVert\leqslant C$ such
that $P+\mathscr{J}(Y)$ is a non-zero idempotent, where
$P=RTS\in\mathscr{B}(Y)$. Then $P\notin\mathscr{J}(Y)$, and we have
$P^{3}-P=(I_{Y}+P)(P^{2}-P)\in\mathscr{J}(Y)$, so the operator
$Q=(TSR)^{2}\in\mathscr{B}(X)$ satisfies

$RQTS=P^{3}\notin\mathscr{J}(Y)\qquad\text{and}\qquad
Q^{2}-Q=TS(P^{3}-P)R\in\mathscr{J}(X).$

It follows that $q=Q+\mathscr{J}(X)$ is a non-zero idempotent in
$\mathscr{B}(X)/\mathscr{J}(X)$. Therefore its image under the injective
homomorphism $\varphi$ is also a non-zero idempotent, which implies that
$\lVert\varphi(q)\rVert\geqslant 1$. Set
$u=SR+\mathscr{J}(X)\in\mathscr{B}(X)/\mathscr{J}(X)$, and observe that
$\lVert u\rVert\leqslant\lVert S\rVert\,\lVert R\rVert\leqslant C$ and
$q=(tu)^{2}$. Combining these facts, we conclude that
$\mathscr{B}(X)/\mathscr{J}(X)$ has the $C^{2}$-IFP. ∎

######  Remark 2.14.

In our applications of Lemma 2.13, a stronger hypothesis will be satisfied: we
can always take the Banach space $Y$ and the operators $R$ and $S$ with
$\lVert R\rVert\,\lVert S\rVert\leqslant C$ such that the non-zero idempotent
$RTS+\mathscr{J}(Y)$ is equal to $I_{Y}+\mathscr{J}(Y)$. In this case we can
draw the stronger conclusion that $\mathscr{B}(X)/\mathscr{J}(X)$ has the
$C$-IFP. The reason is that, in the notation already introduced in the proof,
the hypothesis that $I_{Y}-RTS\in\mathscr{J}(Y)$ implies that

$q=(tu)^{2}=TSRTSR+\mathscr{J}(X)=TSI_{Y}R+\mathscr{J}(X)=tu,$

and $\lVert u\rVert\leqslant C$ as before.

Suppose that the operator ideal $\mathscr{J}$ is the ideal of compact
operators, so that we consider the Calkin algebra
$\mathscr{B}(X)/\mathscr{K}(X)$ of $X$. A result of Barnes [4, Lemma 1,
including the remark following it] implies that idempotents lift from the
Calkin algebra to $\mathscr{B}(X)$. Our next lemma will allow us to establish
quantitative versions of this result in certain cases. More precisely, under
the hypothesis of Remark 2.14 that we can factor $I_{Y}+\mathscr{K}(Y)$
through $T+\mathscr{K}(X)$ with control over the norms of the factors, it
turns out that we can lift the factorization to obtain an analogous
quantitative factorization for the operator $T$ itself, provided that the
Banach space $Y$ is sufficiently “nice” (which it will be in our applications,
where $Y=c_{0}$ or $Y=\ell_{1}$; see also Remark 2.16 for how to handle some
other cases). As well as formally strengthening our conclusions, we believe
that presenting them in this way will make them more accessible to researchers
from other fields.

######  Lemma 2.15.

Let $Y$ be a Banach space with a normalized, bimonotone basis
$(b_{n})_{n\in\mathbb{N}}$, and suppose that

$\biggl{\lVert}\sum_{n=1}^{m}\alpha_{n}b_{n}\biggr{\rVert}=\biggl{\lVert}\sum_{n=1}^{m}\alpha_{n}b_{n+1}\biggr{\rVert}\qquad(m\in\mathbb{N},\,\alpha_{1},\ldots,\alpha_{m}\in\mathbb{K}).$
(2.2)

Then, for every $C>1$ and every operator $T\in\mathscr{B}(Y)$ such that
$I_{Y}-T$ is compact, there are operators $U,V\in\mathscr{B}(Y)$ with $\lVert
U\rVert\,\lVert V\rVert\leqslant C$ such that $UTV=I_{Y}$.

###### Proof.

Equation (2.2) shows that the linear map $R$ given by $Rb_{n}=b_{n+1}$ for
$n\in\mathbb{N}$ is an isometry. Now consider the linear map $L$ given by
$Lb_{1}=0$ and $Lb_{n+1}=b_{n}$ for $n\in\mathbb{N}$. It satisfies

$\displaystyle\biggl{\lVert}L\Bigl{(}\sum_{n=1}^{m+1}\alpha_{n}b_{n}\Bigr{)}\biggr{\rVert}=\biggl{\lVert}\sum_{n=1}^{m}\alpha_{n+1}b_{n}\biggr{\rVert}$
$\displaystyle=\biggl{\lVert}\sum_{n=1}^{m}\alpha_{n+1}b_{n+1}\biggr{\rVert}=\biggl{\lVert}(I_{Y}-P_{1})\Bigl{(}\sum_{n=1}^{m+1}\alpha_{n}b_{n}\Bigr{)}\biggr{\rVert}$
$\displaystyle\leqslant\biggl{\lVert}\sum_{n=1}^{m+1}\alpha_{n}b_{n}\biggr{\rVert}\qquad(m\in\mathbb{N},\,\alpha_{1},\ldots,\alpha_{m+1}\in\mathbb{K})$

by (2.2) and bimonotonicity. Hence $Y$ admits shift operators
$R,L\in\mathscr{B}(Y)$, both having norm $1$, and

$L^{n}R^{n}=I_{Y}\qquad\text{and}\qquad
R^{n}L^{n}=I_{Y}-P_{n}\qquad(n\in\mathbb{N}).$ (2.3)

Let $C>1$, and suppose that $S=I_{Y}-T$ is compact. Then $P_{n}S\to S$ as
$n\to\infty$, so we can find $n\in\mathbb{N}$ such that
$\lVert(I_{Y}-P_{n})S\rVert\leqslant 1-\frac{1}{C}$. Combining this with
(2.3), we obtain

$\displaystyle\lVert I_{Y}-L^{n}TR^{n}\rVert$ $\displaystyle=\lVert
L^{n}(I_{Y}-T)R^{n}\rVert=\lVert L^{n}SR^{n}\rVert$
$\displaystyle\leqslant\lVert L^{n}S\rVert=\lVert
R^{n}L^{n}S\rVert=\lVert(I_{Y}-P_{n})S\rVert\leqslant 1-\frac{1}{C}.$

This implies that the operator $L^{n}TR^{n}$ is invertible by the Neumann
series, and its inverse has norm at most $C$. Consequently we can define
operators $U=(L^{n}TR^{n})^{-1}L^{n}\in\mathscr{B}(Y)$ and
$V=R^{n}\in\mathscr{B}(Y)$ with $\lVert U\rVert\leqslant C$ and $\lVert
V\rVert=1$, and $UTV=I_{Y}$ by definition. ∎

######  Remark 2.16.

We shall also require the counterpart of Lemma 2.15 for
$Y=\bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\bigr{)}_{E}$, where
$E=c_{0}$ or $E=\ell_{1}$. Clearly $Y$ has a bimonotone basis, but it does not
satisfy (2.2). One could set up a more abstract, unified framework which
covers both these spaces and those in Lemma 2.15. However, we find it more
instructive to explain directly how to modify the proof of Lemma 2.15 to
obtain the desired conclusion for
$Y=\bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\bigr{)}_{E}$.

For $n\in\mathbb{N}$, define
$L_{n}\in\mathscr{B}(\ell_{2}^{n+1},\ell_{2}^{n})$ and
$R_{n}\in\mathscr{B}(\ell_{2}^{n},\ell_{2}^{n+1})$ by

$L_{n}(t_{1},\ldots,t_{n+1})=(t_{1},\ldots,t_{n}),\qquad
R_{n}(t_{1},\ldots,t_{n})=(t_{1},\ldots,t_{n},0)\qquad(t_{1},\ldots,t_{n+1}\in\mathbb{K}),$

and then define $L,R\in\mathscr{B}(Y)$ by

$Ly=(L_{n}y_{n+1})_{n\in\mathbb{N}},\qquad
Ry=(0,R_{1}y_{1},R_{2}y_{2},\ldots)\qquad(y=(y_{n})_{n\in\mathbb{N}}\in Y).$

As in the proof of Lemma 2.15, these operators satisfy that $R$ is an
isometry, $L$ has norm $1$, $L^{n}R^{n}=I_{Y}$ for each $n\in\mathbb{N}$, and
$L^{n}S\to 0$ as $n\to\infty$ for every $S\in\mathscr{K}(Y)$. Using these
facts, we can now repeat the final part of the argument verbatim: Let $C>1$,
and suppose that $T\in\mathscr{B}(Y)$ is an operator for which $S=I_{Y}-T$ is
compact. Then, choosing $n\in\mathbb{N}$ such that $\lVert
L^{n}S\rVert\leqslant 1-\frac{1}{C}$, we have

$\lVert I_{Y}-L^{n}TR^{n}\rVert=\lVert L^{n}SR^{n}\rVert\leqslant\lVert
L^{n}S\rVert\,\lVert R\rVert^{n}\leqslant 1-\frac{1}{C},$

so the operator $L^{n}TR^{n}$ is invertible, and
$\lVert(L^{n}TR^{n})^{-1}\rVert\leqslant C$. Hence the operators
$U=(L^{n}TR^{n})^{-1}L^{n}\in\mathscr{B}(Y)$ and $V=R^{n}\in\mathscr{B}(Y)$
satisfy $UTV=I_{Y}$ and $\lVert U\rVert\,\lVert V\rVert\leqslant C$, as
desired.

######  Remark 2.17.

The closed ideals of $\mathscr{B}(X)$ have been classified for several other
Banach spaces $X$ than those specified in Theorem 1.1. All of these Banach
spaces originate from the famous Argyros–Haydon space $X_{{\text{AH}}}$, which
was created to solve the scalar-plus-compact problem [1]. We do not know
whether $\mathscr{B}(X_{{\text{AH}}})$ has a unique algebra norm because we do
not know whether every homomorphism from $\mathscr{B}(X_{{\text{AH}}})$ into a
Banach algebra is continuous. This question is also open for the variants of
$X_{{\text{AH}}}$ whose closed operator ideals have been classified in [48,
Theorem 2.1] and [28, Theorem 1.4], respectively. (We shall consider the
latter Banach space in more detail in Example 2.20 below.)

The situation is even more intriguing for the family of Banach spaces $X_{K}$,
where $K$ is an infinite compact metric space, having the property that the
Calkin algebra $\mathscr{B}(X_{K})/\mathscr{K}(X_{K})$ is isometrically
isomorphic to the Banach algebra $C(K)$ of continuous, scalar-valued functions
on $K$. A Banach space $X_{K}$ with this property was originally constructed
by Motakis, Puglisi and Zizimopoulou [40] in the case where $K$ is countably
infinite, while Motakis [39] has subsequently found a new approach which
handles the general case.

It is undecidable in ZFC whether every homomorphism from $C(K)$ into a Banach
algebra is continuous; more precisely, Dales [7] and Esterle [14]
independently proved that discontinuous homomorphisms from $C(K)$ exist under
the Continuum Hypothesis, whereas Solovay and Woodin constructed a different
model of ZFC in which all such homomorphisms are continuous; see [9] for an
exposition of this result. Consequently it is also undecidable in ZFC whether
every homomorphism from $\mathscr{B}(X_{K})$ into a Banach algebra is
continuous. Note that we have a complete classification of the closed ideals
of $\mathscr{B}(X_{K})$, as described in [28, Remark 1.5(vi)] and [39,
Proposition 8.6].

We conclude this section with three examples which use results from the
literature to illustrate how quotient algebras of $\mathscr{B}(X)$ may fail to
possess certain combinations of the properties we consider. The first two
examples concern the _weak Calkin algebra,_ that is, the quotient
$\mathscr{B}(X)/\mathscr{W}(X)$, where $\mathscr{W}(X)$ denotes the ideal of
weakly compact operators on $X$.

######  Example 2.18.

Let $X$ be a Banach space, identified with its canonical image in the bidual
$X^{**}$. Since an operator $T\in\mathscr{B}(X)$ is weakly compact if and only
if $T^{**}[X^{**}]\subseteq X$, the formula

$\varphi(T+\mathscr{W}(X))(x^{**}+X)=T^{**}(x^{**})+X\qquad(x^{**}\in
X^{**},\,T\in\mathscr{B}(X))$

defines an injective and contractive homomorphism
$\varphi\colon\mathscr{B}(X)/\mathscr{W}(X)\to\mathscr{B}(X^{**}/X)$, which is
non-zero if and only if $X$ is non-reflexive. González, Saksman and Tylli
studied this homomorphism in [18], showing in particular that when
$X=(J_{2}\oplus J_{2}\oplus\cdots)_{\ell_{2}}$ is the $\ell_{2}$-sum of
countably many copies of James’ quasi-reflexive Banach space $J_{2}$, the
range of $\varphi$ is not closed [18, Theorem 2.6]. Hence the weak Calkin
algebra $\mathscr{B}(X)/\mathscr{W}(X)$ fails to be incompressible in this
case. Clearly $X\cong X\oplus X$, so Proposition 2.4 and Theorem 2.5 imply
that the quotient norm on $\mathscr{B}(X)/\mathscr{W}(X)$ is maximal.

Our second example involves the following piece of notation. For a Banach
space $E$, we write $E^{\sim}$ for the Banach algebra which is the unitization
of $E$ equipped with the trivial product; that is,
$E^{\sim}=E\times\mathbb{K}$ as a vector space, with the norm and product
defined by

$\lVert(x,s)\rVert=\lVert x\rVert+\lvert
s\rvert\qquad\text{and}\qquad(x,s)(y,t)=(sy+tx,st)\qquad(x,y\in
E,\,s,t\in\mathbb{K}),$

respectively. Given a linear map $\theta\colon E\to F$ into some Banach space
$F$, we define its unitization $\widetilde{\theta}\colon E^{\sim}\to F^{\sim}$
by $\widetilde{\theta}(x,s)=(\theta(x),s)$ for $x\in E$ and $s\in\mathbb{K}$.
This is a unital homomorphism which is continuous if and only if $\theta$ is
continuous.

######  Example 2.19.

Let $X_{\text{R}}$ denote the Banach space which Read constructed in [45] with
the following remarkable property: the weak Calkin algebra of $X_{\text{R}}$
is isomorphic as a Banach algebra to the unitization $\ell_{2}^{\sim}$ of the
Hilbert space $\ell_{2}$ equipped with the trivial product. (In fact,
$X_{{\text{R}}}$ has a slightly stronger property, namely: there exists a
continuous, surjective homomorphism
$\psi\colon\mathscr{B}(X_{{\text{R}}})\to\ell_{2}^{\sim}$ with
$\ker\psi=\mathscr{W}(X_{{\text{R}}})$ such that the extension

$\textstyle{\\{0\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{W}(X_{{\text{R}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{B}(X_{{\text{R}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\displaystyle{\psi}}$$\textstyle{\ell_{2}^{\sim}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{0\\}}$

splits in the category of Banach algebras, as shown in [33, Theorem 1.2].)

We claim that the norm on $\ell_{2}^{\sim}$ is neither maximal nor
incompressible, and hence the same is true for the weak Calkin algebra
$\mathscr{B}(X_{{\text{R}}})/\mathscr{W}(X_{{\text{R}}})$.

To see that the norm on $\ell_{2}^{\sim}$ fails to be maximal, we must show
that there exists a discontinuous homomorphism from $\ell_{2}^{\sim}$ into a
Banach algebra. This is well known and not difficult to verify. Indeed, since
$\ell_{2}$ is infinite-dimensional, it admits a discontinuous linear
functional $\theta\colon\ell_{2}\to\mathbb{K}$, so its unitization
$\widetilde{\theta}\colon\ell_{2}^{\sim}\to\mathbb{K}^{\sim}$ is a
discontinuous homomorphism.

To verify that $\ell_{2}^{\sim}$ fails to be incompressible, take
$p\in(2,\infty)$ and consider the unitization
$\widetilde{\varphi}\colon\ell_{2}^{\sim}\to\ell_{p}^{\sim}$ of the formal
inclusion map $\varphi$ from $\ell_{2}$ into $\ell_{p}$. This is a continuous,
injective homomorphism with dense range, but it is not surjective, so it
cannot be bounded below. (In fact, it is strictly singular, as observed in
[34, page 76], for instance.)

Our third and final example displays a Calkin algebra which is incompressible
without being uniformly incompressible.

######  Example 2.20.

This example is based on the Banach space $Z$ studied in [28]. It is given by

$Z=X_{{\text{AH}}}\oplus_{\infty}Y,$ (2.4)

where $X_{{\text{AH}}}$ denotes the Banach space of Argyros and Haydon already
mentioned in Remark 2.17, $Y$ is a certain closed, infinite-dimensional
subspace of $X_{{\text{AH}}}$ constructed in [28, Theorem 1.2], and the
subscript $\infty$ indicates that we equip $Z$ with the norm

$\lVert(x,y)\rVert=\max\\{\lVert x\rVert,\lVert y\rVert\\}\qquad(x\in
X_{{\text{AH}}},\,y\in Y).$

The key property of $Z$ that we require is that, according to [28, Equation
(1.2)], every operator $T\in\mathscr{B}(Z)$ has the form

$T=\begin{pmatrix}\alpha_{1,1}I_{X_{{\text{AH}}}}&\alpha_{1,2}J\\\
0&\alpha_{2,2}I_{Y}\end{pmatrix}+S,$ (2.5)

where $\alpha_{1,1},\alpha_{1,2},\alpha_{2,2}\in\mathbb{K}$, $J\colon Y\to
X_{{\text{AH}}}$ denotes the inclusion map and $S\in\mathscr{K}(Z)$. (We note
in passing that $Z$ has a basis, so the ideal of finite-rank operators is
dense in $\mathscr{K}(Z)$.)

Johnson and Phillips [26] observed that the subalgebra

$\left\\{\begin{pmatrix}\alpha&\beta\\\
0&\alpha\end{pmatrix}:\alpha,\beta\in\mathbb{K}\right\\}$

of the algebra $M_{2}(\mathbb{K})$ of scalar-valued $2\times 2$ matrices,
equipped with the spectral norm, is incompressible, but not uniformly
incompressible. Building on this example, we shall show that the same
conclusion holds true for the Calkin algebra $\mathscr{B}(Z)/\mathscr{K}(Z)$.

First, (2.5) shows that $\mathscr{B}(Z)/\mathscr{K}(Z)$ is finite-dimensional
and therefore incompressible.

Second, we observe that the scalars $\alpha_{1,1}$, $\alpha_{1,2}$ and
$\alpha_{2,2}$ in (2.5) are uniquely determined by $T$ because the operators
$I_{X_{{\text{AH}}}}$, $J$ and $I_{Y}$ are non-compact, so for each
$\delta\in(0,1)$, we can define a map $\varphi_{\delta}\colon\mathscr{B}(Z)\to
M_{2}(\mathbb{K})$ by

$\varphi_{\delta}(T)=\begin{pmatrix}\alpha_{1,1}&\delta\alpha_{1,2}\\\
0&\alpha_{2,2}\end{pmatrix}.$

Straightforward calculations show that $\varphi_{\delta}$ is a unital
homomorphism. Furthermore, $\varphi_{\delta}$ is continuous with norm $1$
provided that we equip $M_{2}(\mathbb{K})$ with the norm induced by
identifying it with the Banach algebra $\mathscr{B}(\ell_{\infty}^{2})$, that
is,

$\left\lVert\begin{pmatrix}\alpha_{1,1}&\alpha_{1,2}\\\
\alpha_{2,1}&\alpha_{2,2}\end{pmatrix}\right\rVert=\max\bigl{\\{}\lvert\alpha_{1,1}\rvert+\lvert\alpha_{1,2}\rvert,\lvert\alpha_{2,1}\rvert+\lvert\alpha_{2,2}\rvert\bigr{\\}}.$

Since $\ker\varphi_{\delta}=\mathscr{K}(Z)$, the Fundamental Isomorphism
Theorem implies that $\varphi_{\delta}$ induces a continuous, injective,
unital homomorphism
$\widetilde{\varphi}_{\delta}\colon\mathscr{B}(Z)/\mathscr{K}(Z)\to
M_{2}(\mathbb{K})$, also of norm $1$, by the formula
$\widetilde{\varphi}_{\delta}(T+\mathscr{K}(Z))=\varphi_{\delta}(T)$.

Consequently, if $\mathscr{B}(Z)/\mathscr{K}(Z)$ were uniformly
incompressible, there would be a constant $\eta>0$ such that
$\widetilde{\varphi}_{\delta}$ is bounded below by $\eta$ for every
$\delta\in(0,1)$. However, we have

$\left\lVert\begin{pmatrix}0&J\\\
0&0\end{pmatrix}+\mathscr{K}(Z)\right\rVert=\lVert
J+\mathscr{K}(Y,X_{{\text{AH}}})\rVert=1$

because $Y$ is infinite-dimensional, and therefore

$\eta\leqslant\left\lVert\widetilde{\varphi}_{\delta}\left(\begin{pmatrix}0&J\\\
0&0\end{pmatrix}+\mathscr{K}(Z)\right)\right\rVert=\left\lVert\begin{pmatrix}0&\delta\\\
0&0\end{pmatrix}\right\rVert=\delta\qquad(\delta\in(0,1)),$

contradicting that the same constant $\eta>0$ should work for every
$\delta\in(0,1)$. This contradiction proves that the Calkin algebra
$\mathscr{B}(Z)/\mathscr{K}(Z)$ fails to be uniformly incompressible.

Proposition 2.12 implies that $\mathscr{B}(Z)/\mathscr{K}(Z)$ cannot have the
$C$-IFP for any $C\geqslant 1$. This is also clear from the fact that the
inclusion operator $J$ is inessential, as observed in [28], because every
inessential, idempotent operator has finite rank.

A similar conclusion can be drawn for the Banach space
$X=\ell_{p}\oplus\ell_{q}$ for $1\leqslant p<q<\infty$. As already mentioned
on page (ii), Ware [51, Corollary 5.2.3] showed that the Calkin algebra of $X$
has a unique algebra norm and is therefore incompressible. However,
$\mathscr{B}(X)$ contains many non-compact operators which no idempotent
operator of infinite rank factors through because [15, Corollaries 9 and 12]
shows that there are $2^{\mathfrak{c}}$ many closed ideals between the ideals
of compact and strictly singular operators on $X$, and every strictly
singular, idempotent operator has finite rank. Hence the Calkin algebra of $X$
cannot have the $C$-IFP for any $C\geqslant 1$.

##  3\. A quantitative factorization result for the identity on $c_{0}$

The aim of this section is to establish the following characterization of when
the identity operator on $c_{0}$ factors through an operator defined on a
Banach space of the form $C_{0}(K)$ for a locally compact Hausdorff space $K$.

######  Theorem 3.1.

Let $T\in\mathscr{B}(C_{0}(K),Y)$, where $K$ is a locally compact Hausdorff
space and $Y$ is a Banach space for which the unit ball of $Y^{*}$ is weak*
sequentially compact, and let $C>0$. Then $C_{0}(K)$ contains a sequence
$(f_{n})_{n\in\mathbb{N}}$ such that

$\sup_{k\in K}\sum_{n=1}^{\infty}\lvert f_{n}(k)\rvert\leqslant
1\qquad\text{and}\qquad\inf_{n\in\mathbb{N}}\lVert Tf_{n}\rVert>\frac{1}{C}$
(3.1)

if and only if there are operators $U\in\mathscr{B}(Y,c_{0})$ and
$V\in\mathscr{B}(c_{0},C_{0}(K))$ such that

$UTV=I_{c_{0}}\qquad\text{and}\qquad\lVert U\rVert\,\lVert V\rVert<C.$ (3.2)

Our main application of Theorem 3.1 will be in Section 6, where it will
provide the foundation for the proof of Theorem 1.1(iii). However, a simple
special case of it will also help us establish Theorem 1.1(ii), which is why
we state and prove it at this point.

To aid the presentation of the proof of Theorem 3.1, we have split it into a
number of simpler statements, beginning with one which explains the
significance of the left-hand part of (3.1). For an index set $\Gamma$, we
write $(e_{\gamma})_{\gamma\in\Gamma}$ for the unit vector basis of
$c_{0}(\Gamma)$.

######  Lemma 3.2.

Let $(f_{n})_{n\in\mathbb{N}}$ be a sequence in $C_{0}(K)$ for some locally
compact Hausdorff space $K$. There is an operator
$V\in\mathscr{B}(c_{0},C_{0}(K))$ such that $Ve_{n}=f_{n}$ for every
$n\in\mathbb{N}$ if and only if

$\sup_{k\in K}\sum_{n=1}^{\infty}\lvert f_{n}(k)\rvert<\infty.$ (3.3)

If one, and hence both, of these conditions are satisfied, the norm of the
operator $V$ is equal to the supremum (3.3).

###### Proof.

Let $C\in[0,\infty]$ denote the supremum in (3.3), and observe that

$C=\sup\biggl{\\{}\sum_{n=1}^{m}\lvert f_{n}(k)\rvert:k\in
K,\,m\in\mathbb{N}\biggr{\\}}.$

$\Rightarrow$. Suppose that $V\in\mathscr{B}(c_{0},C_{0}(K))$ is an operator
with $Ve_{n}=f_{n}$ for every $n\in\mathbb{N}$, and take $m\in\mathbb{N}$ and
$k\in K$. For each $n\in\\{1,\ldots,m\\}$, choose $\sigma_{n}\in\mathbb{K}$
such that $\lvert\sigma_{n}\rvert=1$ and $\sigma_{n}f_{n}(k)\geqslant 0$. Then
$x=\sum_{n=1}^{m}\sigma_{n}e_{n}\in c_{0}$ has norm $1$, so

$\lVert V\rVert\geqslant\lVert
Vx\rVert_{\infty}=\biggl{\lVert}\sum_{n=1}^{m}\sigma_{n}f_{n}\biggr{\rVert}_{\infty}\geqslant\biggl{\lvert}\sum_{n=1}^{m}\sigma_{n}f_{n}(k)\biggr{\rvert}=\sum_{n=1}^{m}\sigma_{n}f_{n}(k)=\sum_{n=1}^{m}\lvert
f_{n}(k)\rvert.$

This proves that $C\leqslant\lVert V\rVert<\infty$.

$\Leftarrow$. Suppose that the supremum $C$ is finite. Since $c_{00}$ is dense
in $c_{0}$, it suffices to show that the linear map $V\colon c_{00}\to
C_{0}(K)$ given by $Ve_{n}=f_{n}$ for every $n\in\mathbb{N}$ is bounded. To
verify this, we observe that, for $m\in\mathbb{N}$ and
$\alpha_{1},\ldots,\alpha_{m}\in\mathbb{K}$ with $\max_{1\leqslant n\leqslant
m}\lvert\alpha_{n}\rvert\leqslant 1$,

$\biggl{\lVert}V\Bigl{(}\sum_{n=1}^{m}\alpha_{n}e_{n}\Bigr{)}\biggr{\rVert}_{\infty}=\biggl{\lVert}\sum_{n=1}^{m}\alpha_{n}f_{n}\biggr{\rVert}_{\infty}=\sup_{k\in
K}\biggl{\lvert}\sum_{n=1}^{m}\alpha_{n}f_{n}(k)\biggr{\rvert}\leqslant\sup_{k\in
K}\sum_{n=1}^{m}\lvert\alpha_{n}\rvert\,\lvert f_{n}(k)\rvert\leqslant C.$

Hence $V$ is bounded with $\lVert V\rVert\leqslant C$. The final clause
follows by combining the estimates obtained in the two parts of the proof. ∎

We now come to the two main ingredients in the proof of Theorem 3.1. Both are
generalizations of known results and follow from careful examination of the
published proofs, as we shall explain in more detail below. The first is a
quantitative version of a celebrated theorem of Rosenthal, originally stated
as the first remark following [46, Theorem 3.4].

######  Theorem 3.3 (Rosenthal).

Let $T\in\mathscr{B}(c_{0}(\Gamma),X)$, where $\Gamma$ is an infinite set and
$X$ is a Banach space, and suppose that
$\delta:=\inf_{\gamma\in\Gamma}\|Te_{\gamma}\|>0$. Then, for every
$\eta\in(0,\delta)$, there is a subset $\Gamma^{\prime}$ of $\Gamma$ of the
same cardinality as $\Gamma$ for which the restriction of $T$ to the subspace
$\overline{\operatorname{span}}\\{e_{\gamma}:\gamma\in\Gamma^{\prime}\\}$ is
bounded below by $\eta$.

###### Proof.

Rosenthal stated the result without specifying for which values of $\eta>0$ we
can find a subset $\Gamma^{\prime}$ of the same cardinality as $\Gamma$ such
that the restriction of $T$ to
$\overline{\operatorname{span}}\\{e_{\gamma}:\gamma\in\Gamma^{\prime}\\}$ is
bounded below by $\eta$. However, inspection of his proof shows that this is
possible for every $\eta<\delta$. ∎

The second result generalizes a theorem of Dowling, Randrianantoanina and
Turett [12, Theorem 6]. To state it concisely, we require the following
notion.

######  Definition 3.4.

Let $C\geqslant 1$. Two Banach spaces $X$ and $Y$ are $C$_-isomorphic_ if
there exists an isomorphism $T\in\mathscr{B}(X,Y)$ with
$\|T\|\,\|T^{-1}\|\leqslant C$.

######  Theorem 3.5 (Dowling, Randrianantoanina and Turett).

Let $Z$ be a closed subspace of a Banach space $Y$ for which the unit ball of
$Y^{*}$ is weak* sequentially compact, and suppose that $Z$ contains a closed
subspace which is isomorphic to $c_{0}$. Then, for every $C>1$, there is a
projection $P\in\mathscr{B}(Y)$ with $\|P\|\leqslant C$ such that $P[Y]$ is
contained in $Z$ and $C$-isomorphic to $c_{0}$.

###### Proof.

For $Z=Y$, this is precisely the result which Dowling, Randrianantoanina and
Turett proved in [12, Theorem 6]. In order to adapt their proof to the setting
where $Z\subsetneq Y$, it suffices to observe that they define the projection
$P$ using Hahn–Banach extensions of the coordinate functionals corresponding
to a $(1-\delta)^{-1}$-isomorphic copy of $c_{0}$ inside $Z$, for a suitably
defined $\delta\in(0,1)$. By extending these coordinate functionals to all of
$Y$, rather than only to $Z$, we can follow the rest of their proof verbatim
to obtain the stated result. ∎

######  Remark 3.6.

In the case of real scalars, Galego and Plichko [16, Theorem 4.3] have proved
a result similar to Theorem 3.5 under the weaker hypothesis that $Y$ does not
contain a copy of $\ell_{1}$. However, it is not clear to us whether this
result carries over to complex scalars. Therefore we have opted for the above
version, which will suffice for our purposes.

We note in passing that Galego and Plichko do not state explicitly that the
$C$-complemented copy of $c_{0}$ they construct inside $Z$ is $C$-isomorphic
to $c_{0}$. However, a close inspection of their proof reveals that it is.

######  Lemma 3.7.

Let $X$ and $Y$ be Banach spaces, where $X$ contains a subspace which is
isomorphic to $c_{0}$ and the unit ball of $Y^{*}$ is weak* sequentially
compact, and let $R\in\mathscr{B}(X,Y)$ be an operator which is bounded below
by $\eta>0$. Then, for every $C>1$, $R[X]$ contains a closed subspace $Z$
which is $C$-complemented in $Y$ and $C$-isomorphic to $c_{0}$, and there are
operators $U\in\mathscr{B}(Y,c_{0})$ and $V\in\mathscr{B}(c_{0},X)$ such that
$URV=I_{c_{0}}$ and $\lVert U\rVert\,\lVert V\rVert\leqslant C^{2}/\eta$.

###### Proof.

Since $R$ is bounded below and $X$ contains a subspace which is isomorphic to
$c_{0}$, we can apply Theorem 3.5 to the closed subspace $Z=R[X]$ of $Y$ to
find a projection $P$ of $Y$ onto a closed subspace $Y_{0}$ of $R[X]$ such
that $\lVert P\rVert\leqslant C$ and there is an isomorphism
$S\in\mathscr{B}(c_{0},Y_{0})$ with $\lVert S\rVert\,\lVert
S^{-1}\rVert\leqslant C$. Set $X_{0}=R^{-1}[Y_{0}]$, and let
$R_{0}\in\mathscr{B}(X_{0},Y_{0})$ be the restriction of $R$, which is an
isomorphism with $\lVert R_{0}^{-1}\rVert\leqslant 1/{\eta}$ because $R$ is
bounded below by $\eta$. Then we have a commutative diagram

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where $J\colon X_{0}\to X$ denotes the inclusion map. Hence the operators
$U=S^{-1}P\in\mathscr{B}(Y,c_{0})$ and $V=JR_{0}^{-1}S\in\mathscr{B}(c_{0},X)$
satisfy $URV=I_{c_{0}}$, and

$\lVert U\rVert\,\lVert V\rVert\leqslant\lVert S^{-1}\rVert\,\lVert
P\rVert\,\lVert J\rVert\,\lVert R_{0}^{-1}\rVert\,\lVert
S\rVert\leqslant\frac{C^{2}}{\eta}.\qed$

###### Proof of Theorem 3.1.

$\Rightarrow$. Suppose that $(f_{n})_{n\in\mathbb{N}}$ is a sequence in
$C_{0}(K)$ which satisfies (3.1). Lemma 3.2 shows that we can define an
operator $V_{1}\in\mathscr{B}(c_{0},C_{0}(K))$ with $\lVert
V_{1}\rVert\leqslant 1$ by $V_{1}e_{n}=f_{n}$ for every $n\in\mathbb{N}$.
Choose $\eta\in(C^{-1},\inf_{n\in\mathbb{N}}\lVert Tf_{n}\rVert)$. Then
Theorem 3.3 implies that $\mathbb{N}$ contains an infinite subset $N$ such
that the restriction $R$ of the operator $TV_{1}$ to the subspace
$X=\overline{\operatorname{span}}\\{e_{n}:n\in N\\}$ is bounded below by
$\eta$.

Take $\xi\in(1,\sqrt{C\eta})$. Since $X$ is (isometrically) isomorphic to
$c_{0}$, we can apply Lemma 3.7 to obtain operators $U\in\mathscr{B}(Y,c_{0})$
and $V_{2}\in\mathscr{B}(c_{0},X)$ such that

$URV_{2}=I_{c_{0}}\qquad\text{and}\qquad\lVert U\rVert\,\lVert
V_{2}\rVert\leqslant\frac{\xi^{2}}{\eta}<C.$

It follows that (3.2) is satisfied provided that we define
$V=V_{1}JV_{2}\in\mathscr{B}(c_{0},C_{0}(K))$, where $J\colon X\to c_{0}$
denotes the inclusion map.

Conversely, suppose that $U\in\mathscr{B}(Y,c_{0})$ and
$V\in\mathscr{B}(c_{0},C_{0}(K))$ are operators satisfying (3.2), and set
$f_{n}=\frac{1}{\lVert V\rVert}Ve_{n}\in C_{0}(K)$ for every $n\in\mathbb{N}$.
Lemma 3.2 shows that $\sup_{k\in K}\sum_{n=1}^{\infty}\lvert f_{n}(k)\rvert=1$
because the operator $V/\lVert V\rVert$ has norm $1$. Furthermore, since
$UTf_{n}=\frac{1}{\lVert V\rVert}e_{n}$ for every $n\in\mathbb{N}$, we have

$\inf_{n\in\mathbb{N}}\lVert Tf_{n}\rVert\geqslant\frac{1}{\lVert
U\rVert\,\lVert V\rVert}>\frac{1}{C},$

so the sequence $(f_{n})$ satisfies (3.1). ∎

##  4\. The proof of Theorem 1.1(i)

For Banach spaces $E$, $X$ and $Y$, let $\overline{\mathscr{G}}_{E}(X,Y)$
denote the closure of the set of operators $T\in\mathscr{B}(X,Y)$ which factor
through $E$ in the sense that $T=UV$ for some operators $U\in\mathscr{B}(E,Y)$
and $V\in\mathscr{B}(X,E)$. This set is closed under addition provided that
$E$ contains a complemented subspace isomorphic to $E\oplus E$, and therefore
$\overline{\mathscr{G}}_{E}$ defines a closed operator ideal in this case.

In the remainder of this section, set $E=c_{0}$ or $E=\ell_{1}$ (so that
$\overline{\mathscr{G}}_{E}$ is indeed a closed operator ideal), and let $X$
denote either of the Banach spaces
$X=\bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\bigr{)}_{E}$, the latter
being the dual space of the former. The lattice of closed ideals of
$\mathscr{B}(X)$, originally given as [31, Corollary 5.6] for $E=c_{0}$ and as
[32, Theorem 2.12] for $E=\ell_{1}$, is

$\\{0\\}\subsetneq\mathscr{K}(X)\subsetneq\overline{\mathscr{G}}_{E}(X)\subsetneq\mathscr{B}(X).$
(4.1)

A result of Ware [51, Theorem 5.3.1], already mentioned in the second bullet
point of clause (ii) on page (ii), tells us that the Calkin algebra
$\mathscr{B}(X)/\mathscr{K}(X)$ has a unique algebra norm for both of these
Banach spaces $X$. (Alternatively, applying Corollary 5.6 below with
$\Gamma=\emptyset$, we obtain the stronger conclusion that
$\mathscr{B}(X)/\mathscr{K}(X)$ has the $C$-IFP for every $C>1$.) Hence, in
view of (2.1), Theorem 1.1(i) will be established once we prove the following
result.

######  Theorem 4.1.

Let $X=\bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\bigr{)}_{E}$, where
$E=c_{0}$ or $E=\ell_{1}$. For every constant $C>1$ and every operator
$T\in\mathscr{B}(X)\setminus\overline{\mathscr{G}}_{E}(X)$ with $\lVert
T+\overline{\mathscr{G}}_{E}(X)\rVert=1$, there are operators
$U,V\in\mathscr{B}(X)$ such that $UTV=I_{X}$ and $\lVert U\rVert\,\lVert
V\rVert\leqslant C$.

Consequently the quotient algebra
$\mathscr{B}(X)\big{/}\,\overline{\mathscr{G}}_{E}(X)$ has the $C$-IFP and is
uniformly incompressible.

######  Notation 4.2.

Direct sums and operators between them will play a key role in this section
and the next, so to avoid any possible confusion, let us explain the notation
we use, all of which is fairly standard.

Throughout this paragraph, $\Gamma$ denotes a non-empty set,
$(X_{\gamma})_{\gamma\in\Gamma}$ is a family of Banach spaces indexed by
$\Gamma$, and $E=c_{0}$ or $E=\ell_{1}$ as above. We write
$\bigl{(}\bigoplus_{\gamma\in\Gamma}X_{\gamma}\bigr{)}_{E}$ for the _$E$
-direct sum_ of the family $(X_{\gamma})_{\gamma\in\Gamma}$; that is,
$\bigl{(}\bigoplus_{\gamma\in\Gamma}X_{\gamma}\bigr{)}_{E}$ is the Banach
space consisting of those functions
$x\colon\Gamma\to\bigcup_{\gamma\in\Gamma}X_{\gamma}$ such that $x(\gamma)\in
X_{\gamma}$ for each $\gamma\in\Gamma$ and

$\begin{cases}\text{the set}\ \\{\gamma\in\Gamma:\lvert
x(\gamma)\rvert\geqslant\varepsilon\\}\ \text{is finite for each}\
\varepsilon>0\ &\text{for}\quad E=c_{0},\\\ \sum_{\gamma\in\Gamma}\lVert
x(\gamma)\rVert<\infty\quad&\text{for}\quad E=\ell_{1}.\end{cases}$ (4.2)

The vector-space operations are defined pointwise, and the norm is given by

$\lVert x\rVert=\begin{cases}\sup_{\gamma\in\Gamma}\lVert
x(\gamma)\rVert\quad&\text{for}\quad E=c_{0},\\\ \sum_{\gamma\in\Gamma}\lVert
x(\gamma)\rVert\quad&\text{for}\quad E=\ell_{1}.\end{cases}$

(Of course, (4.2) is trivially satisfied when $\Gamma$ is finite; in this case
we simply equip the full Cartesian product of $(X_{\gamma})_{\gamma\in\Gamma}$
with the norm specified above.) We often write elements of
$\bigl{(}\bigoplus_{\gamma\in\Gamma}X_{\gamma}\bigr{)}_{E}$ in the form
$(x_{\gamma})_{\gamma\in\Gamma}$, where $x_{\gamma}=x(\gamma)$. Throughout
this section and the next, for $\beta\in\Gamma$, we reserve the symbols
$J_{\beta}\colon
X_{\beta}\to\bigl{(}\bigoplus_{\gamma\in\Gamma}X_{\gamma}\bigr{)}_{E}$ and
$Q_{\beta}\colon\bigl{(}\bigoplus_{\gamma\in\Gamma}X_{\gamma}\bigr{)}_{E}\to
X_{\beta}$ for the canonical $\beta^{\text{th}}$ coordinate embedding and
projection, respectively.

Given an operator
$T\in\mathscr{B}\bigl{(}\bigl{(}\bigoplus_{\gamma\in\Gamma}X_{\gamma}\bigr{)}_{E},\bigl{(}\bigoplus_{\gamma\in\Gamma}Y_{\gamma}\bigr{)}_{E}\bigr{)}$,
for some family $(Y_{\gamma})_{\gamma\in\Gamma}$ of Banach spaces, we
associate with it the $\Gamma\times\Gamma$ matrix
$(T_{\beta,\gamma})_{\beta,\gamma\in\Gamma}$ whose
$(\beta,\gamma)^{\text{th}}$ coefficient is the operator defined by
$T_{\beta,\gamma}=Q_{\beta}TJ_{\gamma}\in\mathscr{B}(X_{\gamma},Y_{\beta})$.
Suppose that $\Gamma$ is infinite. Following [31], we say that the operator
$T$ has _finite rows_ if the set $\\{\gamma\in\Gamma:T_{\beta,\gamma}\neq
0\\}$ is finite for each $\beta\in\Gamma$; and analogously $T$ has _finite
columns_ if the set $\\{\beta\in\Gamma:T_{\beta,\gamma}\neq 0\\}$ is finite
for each $\gamma\in\Gamma$. An operator which has both finite rows and finite
columns has _locally finite matrix._

Given a family
$\\{T_{\gamma}\in\mathscr{B}(X_{\gamma},Y_{\gamma}):\gamma\in\Gamma\\}$ of
operators such that $\sup_{\gamma\in\Gamma}\lVert T_{\gamma}\rVert<\infty$, we
define the associated _diagonal operator_ by

$\bigoplus_{\gamma\in\Gamma}T_{\gamma}\colon\
(x_{\gamma})_{\gamma\in\Gamma}\mapsto(T_{\gamma}x_{\gamma})_{\gamma\in\Gamma},\quad\Bigl{(}\bigoplus_{\gamma\in\Gamma}X_{\gamma}\Bigr{)}_{E}\to\Bigl{(}\bigoplus_{\gamma\in\Gamma}Y_{\gamma}\Bigr{)}_{E}.$

This is clearly a bounded operator with norm $\sup_{\gamma\in\Gamma}\lVert
T_{\gamma}\rVert$.

### The case $E=c_{0}$.

Throughout this subsection, we consider the Banach space

$X=\Bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\Bigr{)}_{c_{0}}.$

The following index $m_{\varepsilon}(T)$, which was originally introduced in
[31, Definition 5.2(ii)], played a key role in the classification (4.1) of the
closed ideals of $\mathscr{B}(X)$, and it will also be an essential ingredient
in our proof of Theorem 4.1.

######  Definition 4.3.

For $\varepsilon>0$ and an operator
$T\in\mathscr{B}\bigl{(}\bigl{(}\bigoplus_{n\in N}H_{n}\bigr{)}_{c_{0}},Y)$,
where the index set $N$ is finite, $H_{n}$ is a Hilbert space for each $n\in
N$ and $Y$ can be any Banach space, we define

$m_{\varepsilon}(T)=\sup\biggl{\\{}m\in\mathbb{N}_{0}:\Bigl{\lVert}T\circ\bigoplus_{n\in
N}(I_{H_{n}}-P_{G_{n}})\Bigr{\rVert}>\varepsilon\ \text{for every subspace}\
G_{n}\ \text{of}\ H_{n}\biggr{.}\\\\[-11.38109pt] \biggl{.}\text{with}\ \dim
G_{n}\leqslant m\ \text{for each}\ n\in
N\biggr{\\}}\in\mathbb{N}_{0}\cup\\{\pm\infty\\},$

where $P_{G_{n}}$ denotes the orthogonal projection of $H_{n}$ onto the
subspace $G_{n}$.

Loosely speaking, $m_{\varepsilon}(T)$ is the largest number of dimensions
that we can remove from each of the Hilbert spaces in the domain of $T$ and
still obtain an operator with norm greater than $\varepsilon$.

Suppose that $T\in\mathscr{B}(X)$ is an operator with finite rows. As
described in [31, Remark 5.4], the above definition of $m_{\varepsilon}$ can
be applied to define $m_{\varepsilon}(Q_{j}T)$ for every $j\in\mathbb{N}$ by
ignoring the cofinite number of Hilbert spaces in the domain of $Q_{j}T$ on
which it acts trivially.

The following lemma explains why the quantities $m_{\varepsilon}(Q_{j}T)$ for
$j\in\mathbb{N}$ are relevant in our investigation. It is proved in [31,
Theorem 5.5(iii)]. (The bounds on the norms of the operators $U$ and $V$ are
not stated explicitly in [31], but follow from the proof of the result.)

######  Lemma 4.4.

Let $T\in\mathscr{B}(X)$ be an operator with locally finite matrix, and
suppose that the set $\\{m_{\varepsilon}(Q_{j}T):j\in\mathbb{N}\\}$ is
unbounded for some $\varepsilon>0$. Then there are operators
$U,V\in\mathscr{B}(X)$ with $\lVert U\rVert\leqslant 1/\varepsilon$ and
$\lVert V\rVert\leqslant 1$ such that $UTV=I_{X}$.

In order to apply this result, we require specific values of $\varepsilon>0$
for which the set $\\{m_{\varepsilon}(Q_{j}T):j\in\mathbb{N}\\}$ is unbounded.
Our next lemma addresses this point.

######  Lemma 4.5.

Let $T\in\mathscr{B}(X)\setminus\overline{\mathscr{G}}_{c_{0}}(X)$ be an
operator with locally finite matrix, and suppose that the set
$\\{m_{\varepsilon}(Q_{j}T):j\in\mathbb{N}\\}$ is bounded for some
$\varepsilon>0$. Then $\varepsilon\geqslant\lVert
T+\overline{\mathscr{G}}_{c_{0}}(X)\rVert$.

Consequently the set $\\{m_{\varepsilon}(Q_{j}T):j\in\mathbb{N}\\}$ is
unbounded whenever $0<\varepsilon<\lVert
T+\overline{\mathscr{G}}_{c_{0}}(X)\rVert$.

###### Proof.

Set $m=\sup\\{m_{\varepsilon}(Q_{j}T):j\in\mathbb{N}\\}\in\mathbb{N}_{0}$. Our
proof uses the machinery from [31, Construction 4.2], so we begin by
introducing the necessary notation. For every $j\in\mathbb{N}$, define
$N_{j}=\\{k\in\mathbb{N}:T_{j,k}\neq 0\\}$, which is finite, and
$B_{j}=\bigl{(}\bigoplus_{k\in N_{j}}\ell_{2}^{k}\bigr{)}_{c_{0}}$, and let
$L_{j}\colon B_{j}\to X$ be the natural inclusion map. Furthermore, set
$B=\bigl{(}\bigoplus_{j\in\mathbb{N}}B_{j}\bigr{)}_{c_{0}}$ and
$\widetilde{T}=\bigoplus_{j\in\mathbb{N}}Q_{j}TL_{j}\in\mathscr{B}(B,X)$.
Then, as shown in [31, Construction 4.2], there exists an operator
$V\in\mathscr{B}(X,B)$ with $\lVert V\rVert=1$ such that $T=\widetilde{T}V$.

The convention described after Definition 4.3 means that
$m_{\varepsilon}(Q_{j}TL_{j})=m_{\varepsilon}(Q_{j}T)\leqslant m$ for every
$j\in\mathbb{N}$. Therefore we can find orthogonal projections
$P_{j,k}\in\mathscr{B}(\ell_{2}^{k})$ for $k\in N_{j}$, each having rank at
most $m+1$, such that

$\Bigl{\lVert}Q_{j}TL_{j}\circ\bigoplus_{k\in
N_{j}}(I_{\ell_{2}^{k}}-P_{j,k})\Bigr{\rVert}\leqslant\varepsilon.$ (4.3)

Set $R_{j}=\bigoplus_{k\in N_{j}}P_{j,k}\in\mathscr{B}(B_{j})$ and
$R=\bigoplus_{j\in\mathbb{N}}R_{j}\in\mathscr{B}(B)$. Since each of the
projections $P_{j,k}$ has rank at most $m+1$, the image of $R$ embeds
isometrically into a direct sum of the form
$(\ell_{2}^{m+1}\oplus\ell_{2}^{m+1}\oplus\cdots)_{c_{0}}$, which in turn is
isomorphic to
$(\ell_{\infty}^{m+1}\oplus\ell_{\infty}^{m+1}\oplus\cdots)_{c_{0}}\equiv
c_{0}$. This shows that $R$ factors through $c_{0}$, and therefore the same is
true for the composite operator $\widetilde{T}RV$. Hence we have

$\displaystyle\lVert T+\overline{\mathscr{G}}_{c_{0}}(X)\lVert$
$\displaystyle\leqslant\lVert
T-\widetilde{T}RV\rVert=\lVert\widetilde{T}(I_{B}-R)V\rVert\leqslant\Bigl{\lVert}\bigoplus_{j\in\mathbb{N}}Q_{j}TL_{j}(I_{B_{j}}-R_{j})\Bigr{\rVert}\,\lVert
V\rVert\leqslant\varepsilon,$

where the final estimate follows because $\lVert V\rVert=1$ and $\lVert
Q_{j}TL_{j}(I_{B_{j}}-R_{j})\rVert\leqslant\varepsilon$ for every
$j\in\mathbb{N}$ by (4.3). The final clause is immediate. ∎

###### Proof of Theorem 4.1 for $E=c_{0}$.

Take a constant $C>1$ and an operator $T\in\mathscr{B}(X)$ with $\lVert
T+\overline{\mathscr{G}}_{c_{0}}(X)\rVert=1$. By [31, Lemma 2.7(iii)], we can
find $S\in\mathscr{K}(X)$ such that $T-S$ has locally finite matrix. Since

$C^{-\frac{1}{2}}<1=\lVert T+\overline{\mathscr{G}}_{c_{0}}(X)\rVert=\lVert
T-S+\overline{\mathscr{G}}_{c_{0}}(X)\rVert,$

Lemma 4.5 shows that the set
$\\{m_{C^{-\frac{1}{2}}}(Q_{j}(T-S)):j\in\mathbb{N}\\}$ is unbounded.
Consequently, applying Lemma 4.4, we can find operators
$U_{1},V_{1}\in\mathscr{B}(X)$ with $\lVert U_{1}\rVert\leqslant
C^{\frac{1}{2}}$ and $\lVert V_{1}\rVert\leqslant 1$ such that
$U_{1}(T-S)V_{1}=I_{X}$. Since $U_{1}SV_{1}$ is compact, Remark 2.16 implies
that there are operators $U_{2},V_{2}\in\mathscr{B}(X)$ with $\lVert
U_{2}\rVert\,\lVert V_{2}\rVert\leqslant C^{\frac{1}{2}}$ such that
$U_{2}(U_{1}TV_{1})V_{2}=I_{X}$. Now the conclusion follows by taking
$U=U_{2}U_{1}$ and $V=V_{1}V_{2}$. ∎

### The case $E=\ell_{1}$.

We now turn our attention to the Banach space
$X=\bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\bigr{)}_{\ell_{1}}$. It
identifies naturally with the dual space of the Banach space
$\bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\bigr{)}_{c_{0}}$ that we
considered in the previous subsection, and our proof is essentially a dual
version of the proof we have just completed. As observed in [32, Remark 2.13],
the dualization is not entirely straightforward because we can no longer
perturb operators to arrange that they have locally finite matrix; only finite
columns can be achieved. Fortunately, that will suffice to carry out the
necessary steps, beginning with the following index originally introduced in
[32, Definition 2.4], which is the dual version of Definition 4.3 above.

######  Definition 4.6.

For $\varepsilon>0$ and an operator
$T\in\mathscr{B}\bigl{(}Y,\bigl{(}\bigoplus_{j\in
M}H_{j}\bigr{)}_{\ell_{1}})$, where $Y$ can be any Banach space, the index set
$M$ is finite and $H_{j}$ is a Hilbert space for each $j\in M$, we define

$n_{\varepsilon}(T)=\sup\biggl{\\{}n\in\mathbb{N}_{0}:\biggl{\lVert}\Bigl{(}\bigoplus_{j\in
M}(I_{H_{j}}-P_{G_{j}})\Bigr{)}T\biggr{\rVert}>\varepsilon\ \text{for every
subspace}\ G_{j}\ \text{of}\ H_{j}\biggr{.}\\\\[-11.38109pt]
\biggl{.}\text{with}\ \dim G_{j}\leqslant n\ \text{for each}\ j\in
M\biggr{\\}}\in\mathbb{N}_{0}\cup\\{\pm\infty\\},$

where $P_{G_{j}}$ denotes the orthogonal projection of $H_{j}$ onto the
subspace $G_{j}$, as before.

Hence $n_{\varepsilon}(T)$ is the largest number of dimensions that we can
remove from each of the Hilbert spaces in the codomain of $T$ and still obtain
an operator with norm greater than $\varepsilon$.

When $T\in\mathscr{B}(X)$ has finite columns, we can apply this definition to
the operator $TJ_{k}$ for every $k\in\mathbb{N}$ by simply ignoring the
Hilbert spaces $\ell_{2}^{j}$ in the codomain of $TJ_{k}$ corresponding to the
cofinite set of indices $j\in\mathbb{N}$ for which $Q_{j}TJ_{k}=0$. Then, as
one would hope, the quantities $n_{\varepsilon}(TJ_{k})$ for $k\in\mathbb{N}$
measure how close the identity operator on $X$ is to factoring through $T$.
More precisely, we have the following counterpart of Lemma 4.4, originally
proved as part of [32, Proposition 2.10]. (The norm bounds on $U$ and $V$ are
not stated explicitly in [32], but follow easily from the proof of the
implication (ii)$\Rightarrow$(iii).)

######  Lemma 4.7.

Let $T\in\mathscr{B}(X)$ be an operator with finite columns, and suppose that
the set $\\{n_{\varepsilon}(TJ_{k}):k\in\mathbb{N}\\}$ is unbounded for some
$\varepsilon>0$. Then there are operators $U,V\in\mathscr{B}(X)$ with $\lVert
U\rVert\leqslant 1/\varepsilon$ and $\lVert V\rVert\leqslant 1$ such that
$VTU=I_{X}$.

Our next lemma serves the same purpose as Lemma 4.5, and its proof is similar.

######  Lemma 4.8.

Let $T\in\mathscr{B}(X)\setminus\overline{\mathscr{G}}_{\ell_{1}}(X)$ be an
operator with finite columns, and suppose that
$\sup\\{n_{\varepsilon}(TJ_{k}):k\in\mathbb{N}\\}$ is finite for some
$\varepsilon>0$. Then $\varepsilon\geqslant\lVert
T+\overline{\mathscr{G}}_{\ell_{1}}(X)\rVert$.

Consequently the set $\\{n_{\varepsilon}(TJ_{k}):k\in\mathbb{N}\\}$ is
unbounded whenever $0<\varepsilon<\lVert
T+\overline{\mathscr{G}}_{\ell_{1}}(X)\rVert$.

###### Proof.

Set $n=\sup\\{n_{\varepsilon}(TJ_{k}):k\in\mathbb{N}\\}\in\mathbb{N}_{0}$. The
set $M_{k}=\\{j\in\mathbb{N}:T_{j,k}\neq 0\\}$ is finite for each
$k\in\mathbb{N}$ because $T$ has finite columns. Define
$Y_{k}=\bigl{(}\bigoplus_{j\in M_{k}}\ell_{2}^{j}\bigr{)}_{\ell_{1}}$, and let
$L_{k}\colon Y_{k}\to X$ and $S_{k}\colon X\to Y_{k}$ denote the natural
inclusion map and projection, respectively. Furthermore, define
$Y=\bigl{(}\bigoplus_{k\in\mathbb{N}}Y_{k}\bigr{)}_{\ell_{1}}$ and
$\widetilde{T}=\bigoplus_{k\in\mathbb{N}}S_{k}TJ_{k}\in\mathscr{B}(X,Y)$.
Then, for each element $y=(y_{k})_{k\in\mathbb{N}}\in Y$, the series
$\sum_{k\in\mathbb{N}}L_{k}y_{k}$ converges absolutely in $X$, and the
operator $W\in\mathscr{B}(Y,X)$ given by $Wy=\sum_{k\in\mathbb{N}}L_{k}y_{k}$
satisfies $\lVert W\rVert=1$ and $T=W\widetilde{T}$.

For every $k\in\mathbb{N}$, we have $n\geqslant
n_{\varepsilon}(TJ_{k})=n_{\varepsilon}(S_{k}TJ_{k})$, so we can find
orthogonal projections $P_{j,k}\in\mathscr{B}(\ell_{2}^{j})$ for $j\in M_{k}$,
each having rank at most $n+1$, such that

$\biggl{\lVert}\Bigl{(}\bigoplus_{j\in
M_{k}}(I_{\ell_{2}^{j}}-P_{j,k})\Bigr{)}S_{k}TJ_{k}\biggr{\rVert}\leqslant\varepsilon.$
(4.4)

Set $R_{k}=\bigoplus_{j\in M_{k}}P_{j,k}\in\mathscr{B}(Y_{k})$ and
$R=\bigoplus_{k\in\mathbb{N}}R_{k}\in\mathscr{B}(Y)$, and observe that $R$
factors through $\ell_{1}$ because the image of $R$ embeds isometrically into
$(\ell_{2}^{n+1}\oplus\ell_{2}^{n+1}\oplus\cdots)_{\ell_{1}}$, which is
isomorphic to
$(\ell_{1}^{n+1}\oplus\ell_{1}^{n+1}\oplus\cdots)_{\ell_{1}}\equiv\ell_{1}$.
Consequently we have

$\lVert T+\overline{\mathscr{G}}_{\ell_{1}}(X)\rVert\leqslant\lVert
T-WR\widetilde{T}\rVert=\lVert W(I_{Y}-R)\widetilde{T}\rVert\leqslant\lVert
W\rVert\,\Bigl{\lVert}\bigoplus_{k\in\mathbb{N}}(I_{Y_{k}}-R_{k})S_{k}TJ_{k}\Bigr{\rVert}\leqslant\varepsilon$

by (4.4), as required. The final clause is immediate. ∎

###### Proof of Theorem 4.1 for $E=\ell_{1}$.

Take $C>1$ and $T\in\mathscr{B}(X)$ with $\lVert
T+\overline{\mathscr{G}}_{\ell_{1}}(X)\rVert=1$, and apply [31, Lemma 2.7(i)]
to find $S\in\mathscr{K}(X)$ such that $T-S$ has finite columns. Then
$C^{-\frac{1}{2}}<1=\lVert T-S+\overline{\mathscr{G}}_{\ell_{1}}(X)\rVert$,
and using Lemmas 4.8 and 4.7 as well as Remark 2.16, we can complete the proof
in the same way as we did for $E=c_{0}$ above. ∎

##  5\. The proof of Theorem 1.1(ii)

The aim of this section is to show that every quotient of $\mathscr{B}(X)$ by
one of its closed ideals has a unique algebra norm for each of the Banach
spaces

$X=\Bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\Bigr{)}_{c_{0}}\oplus
c_{0}(\Gamma)\qquad\text{and}\qquad
X=\Bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\Bigr{)}_{\ell_{1}}\oplus\ell_{1}(\Gamma),$
(5.1)

where the index set $\Gamma$ is an uncountable cardinal number. As before, it
suffices to show that the quotient of $\mathscr{B}(X)$ by each of its non-
trivial closed ideals is incompressible, and in line with our previous
approach, we shall in fact prove the stronger result that each of these
quotients has the $C$-IFP for some $C>1$; see Theorem 5.4 and Corollary 5.6
for details.

The closed ideals of $\mathscr{B}(X)$ were classified in [2]. Unsurprisingly,
this result relies heavily on the ideal classifications for each of the two
summands of $X$. The first of these played the main role in Section 4 and was
stated explicitly in (4.1). Let us now review the second. The following
definition is at the heart of it.

######  Definition 5.1.

Let $\kappa$ be an infinite cardinal number. An operator
$T\in\mathscr{B}(X,Y)$ between Banach spaces $X$ and $Y$ is _$\kappa$
-compact_ if, for every $\varepsilon>0$, the closed unit ball $B_{X}$ of $X$
contains a subset $X_{\varepsilon}$ with $\lvert X_{\varepsilon}\rvert<\kappa$
such that

$\inf\bigl{\\{}\lVert T(x-y)\rVert:y\in
X_{\varepsilon}\bigr{\\}}\leqslant\varepsilon\qquad(x\in B_{X}).$

We write $\mathscr{K}_{\kappa}(X,Y)$ for the subset of $\mathscr{B}(X,Y)$
consisting of $\kappa$-compact operators. This is a closed operator ideal. The
name “$\kappa$-compact operator” refers to the fact that it extends the usual
notion of compactness for operators (which corresponds to $\kappa=\aleph_{0}$)
to uncountable cardinal numbers. We call the quotient algebra
$\mathscr{B}(X)/\mathscr{K}_{\kappa}(X)$ the _$\kappa$ -Calkin algebra_.

Generalizing a theorem of Gramsch [19] and Luft [36] in the non-separable
Hilbertian case, Daws [10, Theorem 7.4] showed that the lattice of closed
ideals of $\mathscr{B}(X)$ for each of the long sequence spaces
$X=c_{0}(\Gamma)$ and $X=\ell_{p}(\Gamma)$, where $1\leqslant p<\infty$ and
the index set $\Gamma$ is an uncountable cardinal number, is given by

$\begin{split}\\{0\\}\subsetneq\mathscr{K}(X)=\mathscr{K}_{\aleph_{0}}(X)\subsetneq\mathscr{K}_{\aleph_{1}}(X)\subsetneq&\cdots\subsetneq\mathscr{K}_{\kappa}(X)\subsetneq\mathscr{K}_{\kappa^{+}}(X)\subsetneq\cdots\\\
&\cdots\subsetneq\mathscr{K}_{\Gamma}(X)\subsetneq\mathscr{K}_{\Gamma^{+}}(X)=\mathscr{B}(X),\end{split}$
(5.2)

where $\kappa^{+}$ denotes the cardinal successor of a cardinal number
$\kappa$. Johnson, Kania and Schechtman [25, Theorem 1.5] gave an alternative
statement and proof of this result, with additional details provided in [22,
Theorem 3.7].

Let $X_{1}$ and $X_{2}$ be Banach spaces. As in Notation 4.2, we equip their
direct sum $X_{1}\oplus X_{2}$ with the $E$-norm, where $E=c_{0}$ or
$E=\ell_{1}$, and for $m\in\\{1,2\\}$, $J_{m}\colon X_{m}\to X_{1}\oplus
X_{2}$ and $Q_{m}\colon X_{1}\oplus X_{2}\to X_{m}$ denote the $m^{\text{th}}$
coordinate embedding and projection, respectively.

For a subset $\mathscr{I}$ of $\mathscr{B}(X_{1}\oplus X_{2})$ and
$j,k\in\\{1,2\\}$, we define the $(j,k)^{\text{th}}$ quadrant of $\mathscr{I}$
by

$\mathscr{I}_{j,k}=\\{Q_{j}TJ_{k}:T\in\mathscr{I}\\}\subseteq\mathscr{B}(X_{k},X_{j}).$

Conversely, given subsets $\mathscr{I}_{j,k}$ of $\mathscr{B}(X_{k},X_{j})$
for $j,k\in\\{1,2\\}$, we set

$\begin{pmatrix}\mathscr{I}_{1,1}&\mathscr{I}_{1,2}\\\
\mathscr{I}_{2,1}&\mathscr{I}_{2,2}\end{pmatrix}=\left\\{\begin{pmatrix}T_{1,1}&T_{1,2}\\\
T_{2,1}&T_{2,2}\end{pmatrix}:T_{j,k}\in\mathscr{I}_{j,k}\
\bigl{(}j,k\in\\{1,2\\}\bigr{)}\right\\}\subseteq\mathscr{B}(X_{1}\oplus
X_{2}).$

The significance of this notation is due to [2, Lemma 2.1], which states that

$\mathscr{I}=\begin{pmatrix}\mathscr{I}_{1,1}&\mathscr{I}_{1,2}\\\
\mathscr{I}_{2,1}&\mathscr{I}_{2,2}\end{pmatrix}$ (5.3)

for every ideal $\mathscr{I}$ of $\mathscr{B}(X_{1}\oplus X_{2})$, and
$\mathscr{I}_{j,j}$ is an ideal of $\mathscr{B}(X_{j})$ for each
$j\in\\{1,2\\}$. Furthermore, $\mathscr{I}_{j,k}$ is closed in
$\mathscr{B}(X_{k},X_{j})$ for each $j,k\in\\{1,2\\}$ if and only if
$\mathscr{I}$ is closed in $\mathscr{B}(X_{1}\oplus X_{2})$.

Now, to state the classification of the closed ideals of $\mathscr{B}(X)$ for
each of the Banach spaces $X$ given by (5.1) in a unified way and align our
notation with that used in the previous paragraph, set
$(E,E_{\Gamma})=(c_{0},c_{0}(\Gamma))$ or
$(E,E_{\Gamma})=(\ell_{1},\ell_{1}(\Gamma))$, where $\Gamma$ is an uncountable
cardinal number, and observe that

$X=X_{1}\oplus X_{2}\qquad\text{for}\qquad
X_{1}=\Bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\Bigr{)}_{E}\qquad\text{and}\qquad
X_{2}=E_{\Gamma}.$ (5.4)

In this notation, the main result of [2] states that the non-trivial closed
ideals of $\mathscr{B}(X)$ are

$\mathscr{K}_{\kappa}(X)\quad(\aleph_{0}\leqslant\kappa\leqslant\Gamma)\quad\text{and}\quad\mathscr{J}_{\kappa}(X)=\begin{pmatrix}\overline{\mathscr{G}}_{E}(X_{1})&\mathscr{B}(X_{2},X_{1})\\\
\mathscr{B}(X_{1},X_{2})&\mathscr{K}_{\kappa}(X_{2})\end{pmatrix}\quad(\aleph_{1}\leqslant\kappa\leqslant\Gamma^{+}).$
(5.5)

We split the proof that the quotient of $\mathscr{B}(X)$ by each of these
ideals has the $C$-IFP for some $C>1$ in two cases, beginning with those
ideals that are strictly larger than the ideal of compact operators (which we
recall is labelled $\mathscr{K}_{\aleph_{0}}(X)$ in the list above). This
relies on the following elementary general result.

######  Lemma 5.2.

Let $X_{1}$ and $X_{2}$ be Banach spaces, and suppose that $\mathscr{I}$ is a
closed ideal of $\mathscr{B}(X_{1}\oplus X_{2})$ which satisfies:

  1. (i)

There are constants $C_{1},C_{2}\geqslant 1$ such that
$\mathscr{B}(X_{j})/\mathscr{I}_{j,j}$ has the $C_{j}$-IFP for
$j\in\\{1,2\\}$.

  2. (ii)

$\mathscr{I}_{j,k}=\mathscr{B}(X_{k},X_{j})$ whenever $j,k\in\\{1,2\\}$ are
distinct.

Then $\mathscr{B}(X_{1}\oplus X_{2})/\mathscr{I}$ has the $C$-IFP for
$C=\max\\{C_{1},C_{2}\\}$.

###### Proof.

Take $T=(T_{j,k})_{j,k=1}^{2}\in\mathscr{B}(X_{1}\oplus X_{2})$ with $\lVert
T+\mathscr{I}\rVert=1$. Condition (ii) implies that $\lVert
T+\mathscr{I}\rVert=\max\bigl{\\{}\lVert
T_{1,1}+\mathscr{I}_{1,1}\rVert,\lVert
T_{2,2}+\mathscr{I}_{2,2}\rVert\bigr{\\}}$, so we can choose $j\in\\{1,2\\}$
such that $\lVert T_{j,j}+\mathscr{I}_{j,j}\rVert=1$. Now condition (i) shows
that there are operators $R,S\in\mathscr{B}(X_{j})$ with $\lVert
R+\mathscr{I}_{j,j}\rVert\,\lVert S+\mathscr{I}_{j,j}\rVert\leqslant C_{j}$
such that $RT_{j,j}S+\mathscr{I}_{j,j}$ is a non-zero idempotent.

Set $U=J_{j}RQ_{j}$ and $V=J_{j}SQ_{j}$. Then $U,V\in\mathscr{B}(X_{1}\oplus
X_{2})$ satisfy

$\lVert U+\mathscr{I}\rVert\,\lVert V+\mathscr{I}\rVert=\lVert
R+\mathscr{I}_{j,j}\rVert\,\lVert S+\mathscr{I}_{j,j}\rVert\leqslant
C_{j}\leqslant\max\\{C_{1},C_{2}\\},$

and
$(U+\mathscr{I})(T+\mathscr{I})(V+\mathscr{I})=J_{j}RT_{j,j}SQ_{j}+\mathscr{I}$
is an idempotent, which is non-zero because
$RT_{j,j}S\notin\mathscr{I}_{j,j}$. ∎

In order to apply this lemma, we require the following result, which Ware [51,
Propositions 6.3.1–6.3.2] observed was established in Daws’ proofs of [10,
Theorems 6.2 and 7.3]. (We note in passing that we do not know whether the
constant $4$ in (i) is optimal.)

######  Theorem 5.3.

Let $\Gamma$ and $\kappa$ be uncountable cardinal numbers with
$\kappa\leqslant\Gamma$. Then:

  1. (i)

The $\kappa$-Calkin algebras of $c_{0}(\Gamma)$ and $\ell_{p}(\Gamma)$ for
$1<p<\infty$ have the $4$-IFP.

  2. (ii)

The $\kappa$-Calkin algebra of $\ell_{1}(\Gamma)$ has the $C$-IFP for every
constant $C>1$.

######  Theorem 5.4.

Let $X$ be given by (5.1). Then, for every proper closed ideal $\mathscr{I}$
of $\mathscr{B}(X)$ such that $\mathscr{K}(X)\subsetneq\mathscr{I}$, the
quotient algebra $\mathscr{B}(X)/\mathscr{I}$ has the $4$-IFP and is therefore
uniformly incompressible.

###### Proof.

Inspecting the list (5.5) of proper closed ideals of $\mathscr{B}(X)$ which
are strictly larger than $\mathscr{K}(X)$, we see that

$\displaystyle\mathscr{I}_{1,1}$
$\displaystyle=\begin{cases}\overline{\mathscr{G}}_{E}(X_{1})\ &\text{for}\
\mathscr{I}=\mathscr{J}_{\kappa}(X),\ \text{where}\
\aleph_{1}\leqslant\kappa\leqslant\Gamma^{+},\\\ \mathscr{B}(X_{1})\
&\text{for}\ \mathscr{I}=\mathscr{K}_{\kappa}(X),\ \text{where}\
\aleph_{1}\leqslant\kappa\leqslant\Gamma,\end{cases}$
$\displaystyle\mathscr{I}_{2,2}$
$\displaystyle=\begin{cases}\mathscr{K}_{\kappa}(X_{2})\ &\text{for}\
\mathscr{I}=\mathscr{K}_{\kappa}(X)\ \text{or}\
\mathscr{I}=\mathscr{J}_{\kappa}(X),\ \text{where}\
\aleph_{1}\leqslant\kappa\leqslant\Gamma,\\\ \mathscr{B}(X_{2})\ &\text{for}\
\mathscr{I}=\mathscr{J}_{\Gamma^{+}}(X),\end{cases}$ and
$\displaystyle\mathscr{I}_{j,k}$
$\displaystyle=\mathscr{B}(X_{k},X_{j})\qquad(j,k\in\\{1,2\\},\,j\neq k).$
(5.6)

Indeed, this is clear from the definition if
$\mathscr{I}=\mathscr{J}_{\kappa}(X)$ for some
$\aleph_{1}\leqslant\kappa\leqslant\Gamma^{+}$. Otherwise
$\mathscr{I}=\mathscr{K}_{\kappa}(X)$ for some
$\aleph_{1}\leqslant\kappa\leqslant\Gamma$, so
$\mathscr{I}_{j,k}=\mathscr{K}_{\kappa}(X_{k},X_{j})$, and the above
identities follow from the fact that every operator with separable range is
$\aleph_{1}$-compact.

Hence Theorems 4.1 and 5.3 imply that the quotient algebras
$\mathscr{B}(X_{j})/\mathscr{I}_{j,j}$ for $j\in\\{1,2\\}$ have the 4-IFP
whenever they are non-zero, and (5.6) ensures that Lemma 5.2 applies. It
follows that $\mathscr{B}(X)/\mathscr{I}$ has the 4-IFP if
$\mathscr{B}(X_{1})/\mathscr{I}_{1,1}$ and
$\mathscr{B}(X_{2})/\mathscr{I}_{2,2}$ are both non-zero. Otherwise we have
$\mathscr{I}_{j,j}=\mathscr{B}(X_{j})$ for $j=1$ or $j=2$, and
$\mathscr{B}(X)/\mathscr{I}\cong\mathscr{B}(X_{k})/\mathscr{I}_{k,k}$, where
$k=2$ if $j=1$ and $k=1$ if $j=2$, so $\mathscr{B}(X)/\mathscr{I}$ also has
the 4-IFP in this case. ∎

It remains to tackle the Calkin algebra of $X$. For brevity, we write $\lVert
T\rVert_{e}$ for the essential norm of an operator $T\in\mathscr{B}(X,Y)$;
that is,

$\lVert T\rVert_{e}=\lVert T+\mathscr{K}(X,Y)\rVert.$ (5.7)

Furthermore, for a set $\Gamma$, $[\Gamma]^{<\omega}$ denotes the collection
of finite subsets of $\Gamma$.

######  Lemma 5.5.

Let $X=\bigl{(}\bigoplus_{\gamma\in\Gamma}F_{\gamma}\bigr{)}_{E}$ and
$Y=\bigl{(}\bigoplus_{\xi\in\Xi}G_{\xi}\bigr{)}_{E}$, where $E=c_{0}$ or
$E=\ell_{1}$, the index sets $\Gamma$ and $\Xi$ are infinite and $F_{\gamma}$
and $G_{\xi}$ are non-zero, finite-dimensional Banach spaces for every
$\gamma\in\Gamma$ and $\xi\in\Xi$. Then, for every non-compact operator
$T\in\mathscr{B}(X,Y)$ with $\lVert T\rVert_{e}=1$ and every constant $C>1$,
there are operators $U\in\mathscr{B}(Y,E)$ and $V\in\mathscr{B}(E,X)$ such
that

$UTV=I_{E}\qquad\text{and}\qquad\lVert U\rVert\,\lVert V\rVert\leqslant C.$
(5.8)

###### Proof.

For $A\in[\Gamma]^{<\omega}$, set $R_{A}=\sum_{\gamma\in
A}J_{\gamma}Q_{\gamma}\in\mathscr{B}(X)$. This is a finite-rank projection,
and both $R_{A}$ and $I_{X}-R_{A}$ have norm $1$ (provided $A\neq\emptyset$).
(In the second part of the proof, we shall also use this notation for the
Banach space $Y$; this should not cause any confusion.)

In the case where $E=c_{0}$, take $\eta\in(C^{-1},1)$. Combining the density
of the subspace
$\operatorname{span}\\{J_{\gamma}(F_{\gamma}):\gamma\in\Gamma\\}$ in $X$ with
the fact that $\lVert T(I_{X}-R_{A})\rVert\geqslant\lVert T\rVert_{e}=1$ for
each $A\in[\Gamma]^{<\omega}$, we can recursively construct a sequence
$(A_{n})_{n\in\mathbb{N}}$ of non-empty disjoint sets in $[\Gamma]^{<\omega}$
such that, for each $n\in\mathbb{N}$, there is a unit vector
$x_{n}\in\operatorname{span}\\{J_{\gamma}(F_{\gamma}):\gamma\in A_{n}\\}$ for
which $\lVert Tx_{n}\rVert>\eta$. Since the sets $A_{n}$ are disjoint and
$\lVert x_{n}\rVert=1$, we can define a linear isometry
$V_{1}\in\mathscr{B}(c_{0},X)$ by $V_{1}e_{n}=x_{n}$ for each
$n\in\mathbb{N}$.

We can now apply Theorem 3.1 to the operator $TV_{1}\in\mathscr{B}(c_{0},Y)$,
noting that the unit vector basis $(e_{n})_{n\in\mathbb{N}}$ of $c_{0}$ is a
sequence which satisfies (3.1) because $\inf_{n\in\mathbb{N}}\lVert
TV_{1}e_{n}\rVert\geqslant\eta>1/C$, and that the unit ball of
$Y^{*}\equiv\bigl{(}\bigoplus_{\xi\in\Xi}G_{\xi}^{*}\bigr{)}_{\ell_{1}}$ is
weak* sequentially compact, for instance because its elements have countable
support, so every sequence is contained in a subspace of the form
$\bigl{(}\bigoplus_{\xi\in\Xi_{0}}G_{\xi}^{*}\bigr{)}_{\ell_{1}}$ for some
countable subset $\Xi_{0}$ of $\Xi$. It follows that there are operators
$U\in\mathscr{B}(Y,c_{0})$ and $V_{2}\in\mathscr{B}(c_{0})$ such that
$UTV_{1}V_{2}=I_{c_{0}}$ and $\lVert U\rVert\,\lVert V_{2}\rVert<C$, so (5.8)
is satisfied for the operator $V=V_{1}V_{2}$.

In the case where $E=\ell_{1}$, we no longer have a Rosenthal-style result
such as Theorem 3.1, so we build the factorization from scratch. Set
$\eta=\frac{C-1}{3C}\in(0,\frac{1}{3})$. We shall require two elementary facts
about operators $S\in\mathscr{B}(X,Y)$ between $\ell_{1}$-direct sums of
finite-dimensional spaces:

$\displaystyle\lVert S\rVert_{e}$
$\displaystyle=\inf\bigl{\\{}\lVert(I_{Y}-R_{A})S\rVert:A\in[\Xi]^{<\omega}\bigr{\\}},$
(5.9) $\displaystyle\lVert S\rVert$ $\displaystyle=\sup\bigl{\\{}\lVert
SJ_{\gamma}w\rVert:\gamma\in\Gamma,\,w\in B_{F_{\gamma}}\bigr{\\}}.$ (5.10)

(In fact, (5.9) holds true no matter what form the domain $X$ of $S$ has,
while (5.10) holds true irrespective of the codomain $Y$.) Applying (5.9) to
the given operator $T$ with essential norm $1$, we can find
$A_{0}\in[\Xi]^{<\omega}$ such that

$\lVert(I_{Y}-R_{A_{0}})T\rVert\leqslant 1+\eta.$ (5.11)

Now, using (5.10) together with the fact that
$\lVert(I_{Y}-R_{A})T(I_{X}-R_{B})\rVert\geqslant\lVert T\rVert_{e}>1-\eta$
whenever $A\in[\Xi]^{<\omega}$ and $B\in[\Gamma]^{<\omega}$, we can
recursively choose a sequence $(A_{n})_{n\in\mathbb{N}}$ of disjoint sets in
$[\Xi\setminus A_{0}]^{<\omega}$ and a sequence
$(\gamma_{n})_{n\in\mathbb{N}}$ of distinct elements in $\Gamma$ such that,
for each $n\in\mathbb{N}$, there is a unit vector $w_{n}\in F_{\gamma_{n}}$
for which the vector $y_{n}=R_{A_{n}}TJ_{\gamma_{n}}w_{n}\in Y$ satisfies

$\lVert y_{n}\rVert\geqslant 1-\eta.$ (5.12)

Then $(J_{\gamma_{n}}w_{n})$ is a sequence of disjointly supported unit
vectors in $X$, so we can define a linear isometry
$V\in\mathscr{B}(\ell_{1},X)$ by $Ve_{n}=J_{\gamma_{n}}w_{n}$ for each
$n\in\mathbb{N}$.

Take $n\in\mathbb{N}$, and choose $f_{n}\in Y^{*}$ of norm $1$ such that
$\langle y_{n},f_{n}\rangle=\lVert y_{n}\rVert$. Since $y_{n}=R_{A_{n}}y_{n}$
and $\lVert R_{A_{n}}\rVert=1$, we may suppose that
$R_{A_{n}}^{*}f_{n}=f_{n}$. This implies that $\langle y_{m},f_{n}\rangle=0$
for $m\neq n$, and we can define an operator $U_{1}\in\mathscr{B}(Y,\ell_{1})$
by $U_{1}y=\bigl{(}\langle y,f_{n}\rangle/\lVert
y_{n}\rVert\bigr{)}_{n\in\mathbb{N}}$; it satisfies $U_{1}y_{n}=e_{n}$ for
$n\in\mathbb{N}$, and $\lVert U_{1}\rVert\leqslant\frac{1}{1-\eta}$ by (5.12).

The definition of $y_{n}$ together with the fact that
$A_{n}\subseteq\Xi\setminus A_{0}$ means that $y_{n}$ is the component of the
vector $(I_{Y}-R_{A_{0}})TJ_{\gamma_{n}}w_{n}$ supported on the set $A_{n}$,
so by the definition of the $\ell_{1}$-norm and (5.11)–(5.12), we have

$\lVert(I_{Y}-R_{A_{0}})TJ_{\gamma_{n}}w_{n}-y_{n}\rVert=\lVert(I_{Y}-R_{A_{0}})TJ_{\gamma_{n}}w_{n}\rVert-\lVert
y_{n}\rVert\leqslant 1+\eta-(1-\eta)=2\eta.$

Consequently the operator
$U_{2}=U_{1}(I_{Y}-R_{A_{0}})TV\in\mathscr{B}(\ell_{1})$ satisfies

$\lVert(U_{2}-I_{\ell_{1}})e_{n}\rVert=\lVert
U_{1}((I_{Y}-R_{A_{0}})TJ_{\gamma_{n}}w_{n}-y_{n})\rVert\leqslant\frac{2\eta}{1-\eta}\qquad(n\in\mathbb{N}).$

This implies that $\lVert U_{2}-I_{\ell_{1}}\rVert\leqslant 2\eta/(1-\eta)<1$
by the well-known counterpart of (5.10) for operators defined on the space
$\ell_{1}$ itself. Therefore $U_{2}$ is invertible, and

$\lVert
U_{2}^{-1}\rVert\leqslant\Bigl{(}1-\frac{2\eta}{1-\eta}\Bigr{)}^{-1}=\frac{1-\eta}{1-3\eta}.$

Set $U=U_{2}^{-1}U_{1}(I_{Y}-R_{A_{0}})\in\mathscr{B}(Y,\ell_{1})$. Then
$UTV=I_{\ell_{1}}$ by the definition of $U_{2}$,

$\lVert
U\rVert\leqslant\frac{1-\eta}{1-3\eta}\frac{1}{1-\eta}=C\qquad\text{and}\qquad\lVert
V\rVert=1.\qed$

######  Corollary 5.6.

Let $X=\bigl{(}\bigoplus_{n\in\mathbb{N}}\ell_{2}^{n}\bigr{)}_{E}\oplus
E_{\Gamma}$, where $(E,E_{\Gamma})=(c_{0},c_{0}(\Gamma))$ or
$(E,E_{\Gamma})=(\ell_{1},\ell_{1}(\Gamma))$ for some index set $\Gamma$. For
every non-compact operator $T\in\mathscr{B}(X)$ with $\lVert T\rVert_{e}=1$
and every constant $C>1$, there are operators $U\in\mathscr{B}(X,E)$ and
$V\in\mathscr{B}(E,X)$ such that

$UTV=I_{E}\qquad\text{and}\qquad\lVert U\rVert\,\lVert V\rVert\leqslant C.$

Consequently the Calkin algebra of $X$ has the $C$-IFP and is uniformly
incompressible.

###### Proof.

This follows immediately from LABEL:{L:Calkin_c0_ell1_FinDim} because we can
write $X=\bigl{(}\bigoplus_{\xi\in\Xi}F_{\xi}\bigr{)}_{E}$, where
$\Xi=(\mathbb{N}\times\\{1\\})\cup(\Gamma\times\\{2\\})$ and
$F_{\xi}=\ell_{2}^{n}$ for $\xi=(n,1)$ and $F_{\xi}=\mathbb{K}$ for
$\xi=(\gamma,2)$. ∎

##  6\. The proof of Theorem 1.1(iii)

Let $[\mathbb{N}]^{\omega}$ be the set of all infinite subsets of
$\mathbb{N}$, recall that $[\Gamma]^{<\omega}$ denotes the set of all finite
subsets of a general set $\Gamma$, and consider a family
$\mathcal{A}\subseteq[\mathbb{N}]^{\omega}$ which is _almost disjoint_ in the
sense that $A\cap B\in[\mathbb{N}]^{<\omega}$ whenever $A,B\in\mathcal{A}$ are
distinct. Then the closed subspace $X_{\mathcal{A}}$ of $\ell_{\infty}$
spanned by the indicator functions $\mathbbm{1}_{A}$ for $A\in\mathcal{A}$ and
$\mathbbm{1}_{\\{n\\}}$ for $n\in\mathbb{N}$ is a self-adjoint subalgebra.
Hence, assuming complex scalars, we see that $X_{\mathcal{A}}$ is
isometrically isomorphic to $C_{0}(K_{\mathcal{A}})$ for a locally compact
Hausdorff space $K_{\mathcal{A}}$, which is the _Mrówka space_ induced by the
almost disjoint family $\mathcal{A}$ (also known as the Alexandroff–Urysohn
space, the $\Psi$-space and the Isbell–Mrówka space). Note that
$X_{\mathcal{A}}$ is non-unital whenever $\mathcal{A}$ is infinite, so
$K_{\mathcal{A}}$ is not compact. Furthermore, $K_{\mathcal{A}}$ is metrizable
if and only if $C_{0}(K_{\mathcal{A}})$ is separable, if and only if
$\mathcal{A}$ is countable.

Alternatively, one can define $K_{\mathcal{A}}$ explicitly as follows. As a
set,

$K_{\mathcal{A}}=\\{x_{n}:n\in\mathbb{N}\\}\cup\\{y_{A}:A\in\mathcal{A}\\},$
(6.1)

where $x_{n}$ is an isolated point for each $n\in\mathbb{N}$, while a
neighbourhood basis of $y_{A}$, for each $A\in\mathcal{A}$, is given by the
collection of sets of the form $U(A,F)=\\{y_{A}\\}\cup\\{x\in A\setminus
F\\}$, where $F\in[\mathbb{N}]^{<\omega}$. It follows easily from this
description that $K_{\mathcal{A}}$ is _scattered_ in the sense that every non-
empty subset of it contains a relatively isolated point.

Our goal is to prove the following theorem.

######  Theorem 6.1.

Let $K_{\mathcal{A}}$ be the Mrówka space induced by an uncountable, almost
disjoint family $\mathcal{A}\subseteq[\mathbb{N}]^{\omega}$. For every $C>4$
and every non-compact operator $T\in\mathscr{B}(C_{0}(K_{\mathcal{A}}))$ with
$\lVert T\rVert_{e}=1$, there are operators
$U\in\mathscr{B}(C_{0}(K_{\mathcal{A}}),c_{0})$ and
$V\in\mathscr{B}(c_{0},C_{0}(K_{\mathcal{A}}))$ such that

$UTV=I_{c_{0}}\qquad\text{and}\qquad\lVert U\rVert\,\lVert V\rVert<C.$

Consequently the Calkin algebra of $C_{0}(K_{\mathcal{A}})$ has the $C$-IFP
and is uniformly incompressible.

Before embarking on the proof of Theorem 6.1, let us explain how Theorem
1.1(iii) follows from it.

The setting is as follows: for an uncountable, almost disjoint family
$\mathcal{A}\subseteq[\mathbb{N}]^{\omega}$, we say that
$C_{0}(K_{\mathcal{A}})$ admits _few operators_ if every operator on
$C_{0}(K_{\mathcal{A}})$ is the sum of a scalar multiple of the identity
operator and an operator which factors through $c_{0}$, so in other words the
ideal $\mathscr{G}_{c_{0}}(C_{0}(K_{\mathcal{A}}))$ of operators factoring
through $c_{0}$ has codimension $1$ in $\mathscr{B}(C_{0}(K_{\mathcal{A}}))$.
We remark that the ideal $\mathscr{G}_{c_{0}}(C_{0}(K_{\mathcal{A}}))$ is
closed for every Mrówka space $K_{\mathcal{A}}$ because it coincides with the
ideal of operators with separable range, which is a closed operator ideal.

Assuming the Continuum Hypothesis, Koszmider [29] constructed an uncountable,
almost disjoint family $\mathcal{A}\subseteq[\mathbb{N}]^{\omega}$ for which
$C_{0}(K_{\mathcal{A}})$ admits few operators; a construction within ZFC is
given in [30]. Kania and Kochanek [27, Theorem 5.5] proved that when
$C_{0}(K_{\mathcal{A}})$ admits few operators,
$\mathscr{B}(C_{0}(K_{\mathcal{A}}))$ contains only four closed ideals, namely

$\\{0\\}\subsetneq\mathscr{K}(C_{0}(K_{\mathcal{A}}))\subsetneq\mathscr{G}_{c_{0}}(C_{0}(K_{\mathcal{A}}))\subsetneq\mathscr{B}(C_{0}(K_{\mathcal{A}})).$
(6.2)

This result was also obtained independently by Brooker (unpublished).

In view of (2.1) and (6.2), to prove Theorem 1.1(iii), we need only establish
incompressibility of the quotients
$\mathscr{B}(C_{0}(K_{\mathcal{A}}))/\mathscr{G}_{c_{0}}(C_{0}(K_{\mathcal{A}}))\cong\mathbb{K}$
and $\mathscr{B}(C_{0}(K_{\mathcal{A}}))/\mathscr{K}(C_{0}(K_{\mathcal{A}}))$.
The former is finite-dimensional, so trivially has a unique algebra norm,
while Theorem 6.1 gives the conclusion for the latter.

It remains to prove Theorem 6.1. Our strategy is to verify that
$C_{0}(K_{\mathcal{A}})$ contains a sequence $(f_{n})_{n\in\mathbb{N}}$ which
satisfies (3.1) and then apply Theorem 3.1. We begin by explaining why the
latter result applies to the space $Y=C_{0}(K_{\mathcal{A}})$.

######  Theorem 6.2.

The unit ball of $C_{0}(K)^{*}$ is weak* sequentially compact for every
scattered, locally compact Hausdorff space $K$.

The following well-known observation will be useful in the proof.

######  Remark 6.3.

Let $K$ be a scattered, compact Hausdorff space. Then $C(K)$ contains a
complemented subspace which is isomorphic to $c_{0}$, so $C(K)$ is isomorphic
to its hyperplanes.

As an application of this fact, suppose that $K$ is only _locally_ compact (as
well as scattered and Hausdorff). Then its one-point compactification $\alpha
K$ is scattered. Since $C_{0}(K)$ is a hyperplane in $C(\alpha K)$, these two
Banach spaces are isomorphic.

###### Proof of Theorem 6.2.

In view of Remark 6.3 and the isomorphic nature of this result, we may suppose
that $K$ is compact by replacing it with its one-point compactification if
necessary. Now the conclusion follows by combining two famous results: on the
one hand, Hagler and Johnson [20, Theorem 1(b)] have shown that the unit ball
of $X^{*}$ is weak* sequentially compact for every Asplund space $X$, and on
the other Namioka and Phelps [41, Theorem 18] have shown that $C(K)$ is an
Asplund space whenever $K$ is a scattered, compact Hausdorff space. ∎

To facilitate the presentation of our proof of Theorem 6.1, we introduce some
more notation. Throughout the following list,
$\mathcal{A}\subseteq[\mathbb{N}]^{\omega}$ denotes an almost disjoint family.

  * •

A (finite or infinite) sequence $(f_{n})_{n\in N}$ of scalar-valued functions,
all defined on the same set $K$, is _disjoint_ if, for every $k\in K$, there
is at most one $n\in N$ such that $f_{n}(k)\neq 0$. This is a slight weakening
of the usual notion of disjointly supported functions in the case where $K$ is
a topological space.

  * •

For a function $f\in C_{0}(K_{\mathcal{A}})$, define

$r(f)=\\{n\in\mathbb{N}:f(x_{n})\neq 0\\}\qquad\text{and}\qquad
s(f)=\\{A\in\mathcal{A}:f(y_{\mathcal{A}})\neq 0\\}.$

In view of (6.1), these sets correspond to a partition of $\\{k\in
K_{\mathcal{A}}:f(k)\neq 0\\}$, and consequently a sequence $(f_{n})$ in
$C_{0}(K_{\mathcal{A}})$ is disjoint in the sense of the preceding bullet
point if and only if

$r(f_{m})\cap r(f_{n})=\emptyset\quad\text{and}\quad s(f_{m})\cap
s(f_{n})=\emptyset\quad\text{whenever}\quad m\neq n.$

  * •

For $\mathcal{A}^{\prime}\in[\mathcal{A}]^{<\omega}$, define

$D(\mathcal{A}^{\prime})=\bigcup\\{A\cap B:A,B\in\mathcal{A}^{\prime},\,A\neq
B\\}\in[\mathbb{N}]^{<\omega}.$

In words, $D(\mathcal{A}^{\prime})$ is the set of those natural numbers which
belong to more than one of the sets in $\mathcal{A}^{\prime}$. The fact that
$D(\mathcal{A}^{\prime})$ is finite follows from the fact that the family
$\mathcal{A}$ is almost disjoint and the set $\mathcal{A}^{\prime}$ is finite.

  * •

For $F\in[\mathbb{N}]^{<\omega}$ and
$\mathcal{A}^{\prime}\in[\mathcal{A}]^{<\omega}$, we define a finite-rank
operator by

$\operatorname{Proj}(F,\mathcal{A}^{\prime})=\sum_{n\in
F}\delta_{x_{n}}\otimes\mathbbm{1}_{\\{x_{n}\\}}+\sum_{A\in\mathcal{A}^{\prime}}\delta_{y_{A}}\otimes\mathbbm{1}_{U(A,F)}\in\mathscr{B}(C_{0}(K_{\mathcal{A}})),$

where $\delta_{k}\in C_{0}(K_{\mathcal{A}})^{*}$ denotes the point evaluation
at a point $k\in K_{\mathcal{A}}$, that is,

$\langle f,\delta_{k}\rangle=f(k)\qquad(f\in C_{0}(K_{\mathcal{A}}));$

$\mathbbm{1}_{L}\colon K_{\mathcal{A}}\to\\{0,1\\}$ denotes the indicator
function of a subset $L$ of $K_{\mathcal{A}}$, and we recall that the sets
$U(A,F)$ were defined in the text below (6.1); and as usual,
$\delta_{k}\otimes\mathbbm{1}_{L}$ denotes the rank-one operator given by
$f\mapsto f(k)\mathbbm{1}_{L}$.

The following two lemmas explain the properties of the operators
$\operatorname{Proj}(F,\mathcal{A}^{\prime})$ that we require in our
construction. The first is verified by direct computation, so we omit the
details.

######  Lemma 6.4.

Let $F\in[\mathbb{N}]^{<\omega}$ and
$\mathcal{A}^{\prime}\in[\mathcal{A}]^{<\omega}$. Then:

  1. (i)

$\operatorname{Proj}(F,\mathcal{A}^{\prime})$ is a projection.

  2. (ii)

The adjoint $\operatorname{Proj}(F,\mathcal{A}^{\prime})^{*}$ of
$\operatorname{Proj}(F,\mathcal{A}^{\prime})$ satisfies

$\operatorname{Proj}(F,\mathcal{A}^{\prime})^{*}\delta_{y_{A}}=\begin{cases}\delta_{y_{A}}&\quad\text{for}\
A\in\mathcal{A}^{\prime},\\\ 0&\quad\text{for}\
A\in\mathcal{A}\setminus\mathcal{A}^{\prime}.\end{cases}$

  3. (iii)

Suppose that $D(\mathcal{A}^{\prime})\subseteq F$. Then
$\lVert\operatorname{Proj}(F,\mathcal{A}^{\prime})\rVert=1$ (except in the
trivial case where $\mathcal{A}^{\prime}$ and $F$ are both empty).

  4. (iv)

$\lVert I-\operatorname{Proj}(F,\emptyset)\rVert=1$.

######  Lemma 6.5.

Let
$h\in\operatorname{span}\\{\mathbbm{1}_{\\{x_{n}\\}},\,\mathbbm{1}_{U(A,\emptyset)}:n\in\mathbb{N},\,A\in\mathcal{A}\\}$,
where $\mathcal{A}\subseteq[\mathbb{N}]^{\omega}$ is an almost disjoint
family. Then:

  1. (i)

There exists a set $H\in[\mathbb{N}]^{<\omega}$ such that

$D(s(h))\subseteq H\qquad\text{and}\qquad
h\in\operatorname{span}\\{\mathbbm{1}_{\\{x_{n}\\}},\,\mathbbm{1}_{U(A,H)}:n\in
H,\,A\in s(h)\\}.$

  2. (ii)

Suppose that $H\in[\mathbb{N}]^{<\omega}$ is chosen as in (i). Then, for every
$G\in[\mathbb{N}]^{<\omega}$ with $H\subseteq G$, the function
$g=(I-\operatorname{Proj}(G,\emptyset))h\in C_{0}(K_{\mathcal{A}})$ satisfies

$s(g)=s(h)\qquad\text{and}\qquad g=\sum_{A\in
s(g)}g(y_{A})\mathbbm{1}_{U(A,G)}.$

###### Proof.

(i). Since $s(h)$ is a finite subset of $\mathcal{A}$, we may express $h$ in
the form

$h=\sum_{n\in N}\lambda_{n}\mathbbm{1}_{\\{x_{n}\\}}+\sum_{A\in
s(h)}h(y_{A})\mathbbm{1}_{U(A,\emptyset)}$

for some $N\in[\mathbb{N}]^{<\omega}$ and some scalars $\lambda_{n}$ for $n\in
N$. Then the set $H=D(s(h))\cup N\in[\mathbb{N}]^{<\omega}$ satisfies

$h=\sum_{n\in H}h(x_{n})\mathbbm{1}_{\\{x_{n}\\}}+\sum_{A\in
s(h)}h(y_{A})\mathbbm{1}_{U(A,H)}.$

(ii). First, we note that $g(y_{A})=h(y_{A})$ for every $A\in\mathcal{A}$
because $s(\operatorname{Proj}(G,\emptyset)h)=\emptyset$, so $s(g)=s(h)$. To
help the readability of the proof of the second part, we define

$f=\sum_{A\in s(g)}g(y_{A})\mathbbm{1}_{U(A,G)}.$

Then, what we seek to verify is that $g=f$. It is clear by inspection that
$g(y_{A})=f(y_{A})$ for every $A\in\mathcal{A}$. So, it remains to check that
$g(x_{n})=f(x_{n})$ for every $n\in\mathbb{N}$. There are three cases to
consider:

  * •

If $n\in G$, then $f(x_{n})=0$ and
$(\operatorname{Proj}(G,\emptyset)h)(x_{n})=h(x_{n})$, so $g(x_{n})=0$ as
well.

  * •

If $n\in A\setminus G$ for some $A\in s(g)$, then $n\notin B$ for every $B\in
s(g)\setminus\\{A\\}$ because $A\cap B\subseteq D(s(g))=D(s(h))\subseteq
H\subseteq G$. Therefore we have

$f(x_{n})=g(y_{A})=h(y_{A})=h(x_{n})=g(x_{n}).$

  * •

The final case is that $n\notin G$ and $n\notin A$ for every $A\in s(g)$. This
implies that $f(x_{n})=0$, $h(x_{n})=0$ and
$(\operatorname{Proj}(G,\emptyset)h)(x_{n})=0$, so we also have $g(x_{n})=0$.
∎

###### Proof of Theorem 6.1.

As previously mentioned, our strategy is to show that, for every $C>4$ and
every non-compact operator $T\in\mathscr{B}(C_{0}(K_{\mathcal{A}}))$ with
$\lVert T\rVert_{e}=1$, $C_{0}(K_{\mathcal{A}})$ contains a sequence
$(f_{n})_{n\in\mathbb{N}}$ which satisfies condition (3.1) in Theorem 3.1.
This will prove the result because Theorem 6.2 ensures that Theorem 3.1
applies.

Take $\delta\in(\frac{1}{C},\frac{1}{4})$. By recursion, we shall construct a
disjoint sequence $(f_{n})_{n\in\mathbb{N}}$ in the unit ball of
${C_{0}(K_{\mathcal{A}})}$ such that

$\lVert Tf_{n}\rVert\geqslant\delta\qquad(n\in\mathbb{N}).$ (6.3)

We observe that a sequence $(f_{n})$ with these properties will satisfy (3.1)
because, for each $k\in K_{\mathcal{A}}$, there is at most one
$n\in\mathbb{N}$ such that $f_{n}(k)\neq 0$, and therefore
$\sum_{n=1}^{\infty}\lvert f_{n}(k)\rvert\leqslant 1$. We carry out the
recursive construction in two distinct cases, depending on the norms of the
restrictions of a certain family of operators to the subspace
$X_{00}=\operatorname{span}\\{\mathbbm{1}_{\\{x_{n}\\}}:n\in\mathbb{N}\\}$ of
$C_{0}(K_{\mathcal{A}})$.

Case 1. Suppose that

$\lVert
T(I-\operatorname{Proj}(F,\emptyset))|_{X_{00}}\rVert>\delta\qquad(F\in[\mathbb{N}]^{<\omega}).$
(6.4)

In this case, it is quite easy to construct a disjoint sequence
$(f_{n})_{n\in\mathbb{N}}$ satisfying (6.3) inside the unit ball of $X_{00}$.

Indeed, to start the recursion, we apply (6.4) with $F_{0}=\emptyset$ to see
that $\|T|_{X_{00}}\|>\delta$, so we can select $f_{1}\in B_{X_{00}}$ such
that $\|Tf_{1}\|>\delta$.

Now assume recursively that, for some $n\in\mathbb{N}$, we have chosen a
disjoint sequence of functions $(f_{j})_{j=1}^{n}$ in $B_{X_{00}}$ for which
$\|Tf_{j}\|>\delta$ for each $j\in\\{1,\dots,n\\}$. Set
$F_{n}=\bigcup^{n}_{j=1}r(f_{j})$, and observe that
$F_{n}\in[\mathbb{N}]^{<\omega}$ because $f_{1},\ldots,f_{n}\in X_{00}$. Hence
we may apply (6.4) to find a function $g_{n+1}\in B_{X_{00}}$ for which

$\lVert T(I-\operatorname{Proj}(F_{n},\emptyset))g_{n+1}\rVert>\delta.$

Define $f_{n+1}=(I-\operatorname{Proj}(F_{n},\emptyset))g_{n+1}\in X_{00}$.
Then we have $\lVert Tf_{n+1}\rVert>\delta$,
$r(f_{n+1})\subseteq\mathbb{N}\setminus F_{n}$, and
$\|f_{n+1}\|\leqslant\|g_{n+1}\|\leqslant 1$ by Lemma 6.4(iv), so the
recursion continues.

Case 2. Suppose that (6.4) fails, so that $\lVert
T(I-\operatorname{Proj}(F,\emptyset))|_{X_{00}}\rVert\leqslant\delta$ for some
$F\in[\mathbb{N}]^{<\omega}$, and set
$S=T(I-\operatorname{Proj}(F,\emptyset))\in\mathscr{B}(C_{0}(K_{\mathcal{A}}))$.
Since $\operatorname{Proj}(F,\emptyset)$ is a finite-rank operator, we have
$\|S\|_{e}=\|T\|_{e}=1$. In this case, we shall construct the disjoint
sequence $(f_{n})_{n\in\mathbb{N}}$ satisfying (6.3) within the unit ball of
the subspace

$Y_{00}=\operatorname{span}\\{\mathbbm{1}_{\\{x_{n}\\}},\,\mathbbm{1}_{U(A,\emptyset)}:n\in\mathbb{N},\,A\in\mathcal{A}\\},$

which is dense in $C_{0}(K_{\mathcal{A}})$. This argument is somewhat more
involved, with the main part consisting of a recursive construction of a
disjoint sequence of functions $(g_{n})_{n\in\mathbb{N}}$ in $B_{Y_{00}}$ and
a sequence $(G_{n})_{n\in\mathbb{N}}$ in $[\mathbb{N}]^{<\omega}$ for which

$\|Sg_{n}\|>\delta,\quad\ D\Bigl{(}\bigcup_{j=1}^{n}s(g_{j})\Bigr{)}\subseteq
G_{n}\quad\ \text{and}\quad\ g_{n}=\sum_{A\in
s(g_{n})}g_{n}(y_{A})\mathbbm{1}_{U(A,G_{n})}\qquad(n\in\mathbb{N}).$ (6.5)

To begin this recursion, observe that there is a function $h_{1}\in
B_{Y_{00}}$ with $\|Sh_{1}\|>2\delta$ because
$\|S\|\geqslant\|S\|_{e}=1>2\delta$ and $Y_{00}$ is dense in
$C_{0}(K_{\mathcal{A}})$. Applying Lemma 6.5(i), we can find a set
$H_{1}\in[\mathbb{N}]^{<\omega}$ such that

$D(s(h_{1}))\subseteq H_{1}\qquad\text{and}\qquad
h_{1}\in\operatorname{span}\\{\mathbbm{1}_{\\{x_{m}\\}},\,\mathbbm{1}_{U(A,H_{1})}:m\in
H_{1},\,A\in s(h_{1})\\}.$

Define $G_{1}=H_{1}$, $k_{1}=\operatorname{Proj}(H_{1},\emptyset)h_{1}\in
B_{X_{00}}$ and
$g_{1}=h_{1}-k_{1}=(I-\operatorname{Proj}(H_{1},\emptyset))h_{1}\in
B_{Y_{00}}$. Then $\|Sk_{1}\|\leqslant\delta$ by the choice of $S$, so that

$\|Sg_{1}\|\geqslant\|Sh_{1}\|-\|Sk_{1}\|>2\delta-\delta=\delta,$

while the second and third identities in (6.5) both follow from Lemma 6.5(ii).
This completes the base step.

Now assume that, for some $n\in\mathbb{N}$, we have selected disjointly
supported functions $g_{1},\ldots,g_{n}\in B_{Y_{00}}$ and sets
$G_{1},\ldots,G_{n}\in[\mathbb{N}]^{<\omega}$ satisfying (6.5). The definition
of $Y_{00}$ implies that the set $\mathcal{A}_{n}=\bigcup_{j=1}^{n}s(g_{j})$
is finite, so $Q_{n}=\operatorname{Proj}(G_{n},\mathcal{A}_{n})$ is a finite-
rank projection, and $\lVert S(I-Q_{n})\rVert\geqslant\lVert
S\rVert_{e}=1>4\delta$. Hence we can find a function $\zeta_{n+1}\in
B_{Y_{00}}$ for which $\|S(I-Q_{n})\zeta_{n+1}\|>4\delta$.

By hypothesis, $D(\mathcal{A}_{n})\subseteq G_{n}$, so Lemma 6.4(iii) implies
that $\lVert Q_{n}\rVert=1$. Consequently the function
$h_{n+1}=\frac{1}{2}(I-Q_{n})\zeta_{n+1}$ belongs to $B_{Y_{00}}$. With
another application of Lemma 6.5(i), we produce a set
$H_{n+1}\in[\mathbb{N}]^{<\omega}$ for which $D(s(h_{n+1}))\subseteq H_{n+1}$
and

$h_{n+1}\in\operatorname{span}\\{\mathbbm{1}_{\\{x_{m}\\}},\,\mathbbm{1}_{U(A,H_{n+1})}:m\in
H_{n+1},\,A\in s(h_{n+1})\\}.$

For every $A\in\mathcal{A}_{n}$, we have

$h_{n+1}(y_{A})=\frac{1}{2}\langle\zeta_{n+1},(I-Q_{n})^{*}\delta_{y_{A}}\rangle=0$

because $Q_{n}^{*}\delta_{y_{A}}=\delta_{y_{A}}$ by Lemma 6.4(ii), and
therefore $s(h_{n+1})\cap\mathcal{A}_{n}=\emptyset$. Define

$\displaystyle G_{n+1}=H_{n+1}\cup D(\mathcal{A}_{n}\cup
s(h_{n+1}))\in[\mathbb{N}]^{<\omega},\qquad
k_{n+1}=\operatorname{Proj}(G_{n+1},\emptyset)h_{n+1}\in B_{X_{00}},$
$\displaystyle
g_{n+1}=h_{n+1}-k_{n+1}=(I-\operatorname{Proj}(G_{n+1},\emptyset))h_{n+1}\in
B_{Y_{00}}.$

Then we have $\lVert Sk_{n+1}\rVert\leqslant\delta$, which implies that

$\lVert Sg_{n+1}\rVert\geqslant\lVert Sh_{n+1}\rVert-\delta=\frac{1}{2}\lVert
S(I-Q_{n})\zeta_{n+1}\rVert-\delta>\frac{4\delta}{2}-\delta=\delta.$

Furthermore, Lemma 6.5(ii) shows that

$s(g_{n+1})=s(h_{n+1})\qquad\text{and}\qquad g_{n+1}=\sum_{A\in
s(g_{n+1})}g_{n+1}(y_{A})\mathbbm{1}_{U(A,G_{n+1})}.$ (6.6)

In particular, we see that
$D\bigl{(}\bigcup^{n+1}_{j=1}s(g_{j})\bigr{)}\subseteq G_{n+1}$, bearing the
definition of $\mathcal{A}_{n}$ in mind. Thus (6.5) is satisfied for $n+1$.

It remains to verify that $g_{n+1}$ is disjoint from $g_{j}$ for each
$j\in\\{1,\ldots,n\\}$. Since

$s(g_{n+1})\cap s(g_{j})\subseteq s(h_{n+1})\cap\mathcal{A}_{n}=\emptyset,$

we need only check that $r(g_{n+1})\cap r(g_{j})=\emptyset$. Take $m\in
r(g_{n+1})$, so that $m\in A\setminus G_{n+1}$ for some $A\in s(g_{n+1})$ by
(6.6). Then, for each $B\in s(g_{j})$, we see that $m\notin B$ because

$A\cap B\subseteq D\biggl{(}\bigcup^{n+1}_{i=1}s(g_{i})\biggr{)}\subseteq
G_{n+1}.$

Hence the final identity in (6.5) for the function $g_{j}$ implies that

$g_{j}(x_{m})=\sum_{B\in
s(g_{j})}g_{j}(y_{B})\mathbbm{1}_{U(B,G_{j})}(x_{m})=0,$

and therefore $m\notin r(g_{j})$. This completes the recursive construction of
the sequences $(g_{n})_{n\in\mathbb{N}}$ and $(G_{n})_{n\in\mathbb{N}}$.

We conclude the proof by defining
$f_{n}=(I-\operatorname{Proj}(F,\emptyset))g_{n}$ for every $n\in\mathbb{N}$.
Then $\lVert f_{n}\rVert\leqslant\lVert g_{n}\rVert\leqslant 1$ by Lemma
6.4(iv), $\lVert Tf_{n}\rVert=\lVert Sg_{n}\rVert>\delta$ by the left-hand
part of (6.5), and the sequence $(f_{n})_{n\in\mathbb{N}}$ is disjoint because
$s(f_{n})=s(g_{n})$ and $r(f_{n})\subseteq r(g_{n})$ for every
$n\in\mathbb{N}$, where the sequence $(g_{n})_{n\in\mathbb{N}}$ is disjoint. ∎

##  7\. Incompressibility of the Calkin algebra of $C_{0}(K)$ for scattered
$K$

In each of the cases we encountered in the proof of Theorem 1.1, we deduced
incompressibility from quantitative factorizations of idempotent operators,
which in view of Example 2.20 would appear to be a much stronger property than
necessary. Notably, in the previous section we used Theorem 3.1 to show that
the Calkin algebra of $C_{0}(K_{\mathcal{A}})$ is (uniformly) incompressible
for every uncountable, almost disjoint family
$\mathcal{A}\subseteq[\mathbb{N}]^{\omega}$.

The aim of this section is to prove that for a scattered, locally compact
Hausdorff space $K$, the idempotent factorization property is not only a
sufficient, but also a necessary condition for incompressibility of the Calkin
algebra of $C_{0}(K)$. The significance of this result is that when somebody
seeks to determine whether the Calkin algebra of $C_{0}(K)$ is incompressible,
they will know that verifying condition (a) below (as we did in the previous
section for $K=K_{\mathcal{A}}$) is most likely the optimal route. We consider
the case where $K$ equals the ordinal interval $[0,\omega^{\omega})$
particularly interesting.

Recall from (5.7) that $\lVert\,\cdot\,\rVert_{e}$ denotes the essential norm.

######  Theorem 7.1.

The following conditions are equivalent for a scattered, locally compact
Hausdorff space $K\colon$

  1. (a)

There is a constant $\delta\in(0,1)$ such that, for every non-compact operator
$T\in\mathscr{B}(C_{0}(K))$ with $\lVert T\rVert_{e}=1$, $C_{0}(K)$ contains a
sequence $(f_{n})_{n\in\mathbb{N}}$ with

$\sup_{k\in K}\sum_{n=1}^{\infty}\lvert f_{n}(k)\rvert\leqslant
1\qquad\text{and}\qquad\inf_{n\in\mathbb{N}}\lVert Tf_{n}\rVert>\delta.$ (7.1)

  2. (b)

There is a constant $C>1$ such that, for every non-compact operator
$T\in\mathscr{B}(C_{0}(K))$ with $\lVert T\rVert_{e}=1$, there are operators
$U\in\mathscr{B}(C_{0}(K),c_{0})$ and $V\in\mathscr{B}(c_{0},C_{0}(K))$ with
$\lVert U\rVert\,\lVert V\rVert<C$ such that $UTV=I_{c_{0}}$.

  3. (c)

The Calkin algebra $\mathscr{B}(C_{0}(K))/\mathscr{K}(C_{0}(K))$ is uniformly
incompressible.

  4. (d)

The Calkin algebra $\mathscr{B}(C_{0}(K))/\mathscr{K}(C_{0}(K))$ is
incompressible.

  5. (e)

The algebra norm $\nu$ given by (7.2) below dominates the essential norm in
the sense that there is a constant $\eta>0$ such that

$\nu(T+\mathscr{K}(C_{0}(K)))\geqslant\eta\lVert
T\rVert_{e}\qquad(T\in\mathscr{B}(C_{0}(K))).$

The proof of Theorem 7.1 has two main ingredients. The first is Theorem 3.1,
which was already established in Section 3 and has the equivalence of
conditions (a) and (b) as an immediate consequence. The other is the algebra
norm $\nu$ on the Calkin algebra of $C_{0}(K)$ that appears in condition (e).

To define it, we note that according to the proof of [27, Proposition
5.4(ii)], the identity operator on $c_{0}$ factors through every non-compact
operator on $C_{0}(K)$ whenever $K$ is a scattered, locally compact Hausdorff
space. (This will also follow from our results, as we shall explain in detail
at the beginning of the proof of Proposition 7.2 below.) Therefore we can
define a map
$\nu\colon\mathscr{B}(C_{0}(K))/\mathscr{K}(C_{0}(K))\to[0,\infty)$ by

$\nu(T+\mathscr{K}(C_{0}(K)))=\begin{cases}0\quad&\text{if}\
T\in\mathscr{K}(C_{0}(K)),\\\ \sup\biggl{\\{}\displaystyle{\frac{1}{\lVert
U\rVert\,\lVert V\rVert}}:I_{c_{0}}-UTV\in\mathscr{K}(c_{0})\biggr{\\}}\
&\text{otherwise.}\end{cases}$ (7.2)

As already indicated, our aim is to establish the following result.

######  Proposition 7.2.

Let $K$ be a scattered, locally compact Hausdorff space. Then $\nu$ defined by
(7.2) is an algebra norm on $\mathscr{B}(C_{0}(K))/\mathscr{K}(C_{0}(K))$, and

$\nu(T+\mathscr{K}(C_{0}(K))\leqslant\lVert
T\rVert_{e}\qquad(T\in\mathscr{B}(C_{0}(K))).$ (7.3)

######  Remark 7.3.

Recall from Remark 6.3 that $C_{0}(K)$ is isomorphic to $C(\alpha K)$ whenever
$K$ is a scattered, locally compact Hausdorff space and $\alpha K$ denotes its
one-point compactification, which is also scattered. Consequently, in view of
the of the isomorphic invariance of Theorem 7.1, we could have stated it only
for compact $K$ without formally losing any generality. We have nevertheless
chosen to state it in the locally compact case for the following reasons:

  1. (i)

The isomorphism between $C_{0}(K)$ and $C(\alpha K)$ is not isometric (as the
example of $c_{0}$ and $c$ illustrates; see [5] for details), and the
constants $\delta$, $C$ and $\eta$ in conditions (a), (b) and (e) may
therefore change when passing from one space to the other. We consider the
values of these constants to be of some interest in the context of
quantitative factorizations.

  2. (ii)

The present version includes the case $K=\mathbb{N}$ (or in other words
$C_{0}(K)=c_{0}$) explicitly, which seems natural given the role $c_{0}$ plays
in the result. Perhaps more importantly, when proving Theorem 7.1
(specifically, in the proof of Proposition 7.2), we shall apply various lemmas
to $c_{0}$ as well as $C_{0}(K)$. If the latter class did not contain $c_{0}$,
some statements would become more complicated.

  3. (iii)

When we applied a precursor of Theorem 7.1 in Section 6, it was for a locally
compact space; the “vanishing at infinity” property was very helpful in that
proof.

Before initiating the preparations for the proof of Proposition 7.2, let us
point out that the definition of $\nu$ and the proof that it is an algebra
norm are heavily inspired by a similar result in [26] concerning operators
that the identity operator on $\ell_{2}$ factors through. Our version, which
involves replacing $\ell_{2}$ with $c_{0}$, is somewhat harder to prove
because $c_{0}$ has a much richer subspace structure than $\ell_{2}$.

We begin with a variant of a standard result about non-compact operators
defined on $c_{0}$, tailored to our particular needs. Combining it with Lemma
3.7, we arrive at Corollary 7.6, which we shall apply in a number of slightly
different contexts in the proof of Proposition 7.2. To ensure that it covers
all these applications, we have stated the exact details we require in each
case, even though this means there is some redundancy in the statement.

######  Lemma 7.4.

Let $R\in\mathscr{B}(X,Y)$ be a non-compact operator between Banach spaces $X$
and $Y$, where $X$ is $C$-isomorphic to $c_{0}$ for some constant $C\geqslant
1$. Then, for each $\eta\in(0,1)$, $X$ contains a $C$-complemented subspace
$W$ which is $C$-isomorphic to $c_{0}$ and satisfies

$(1+\eta)C\lVert R\rVert_{e}\lVert w\rVert\geqslant\lVert
Rw\rVert\geqslant\frac{1-\eta}{C}\lVert R\rVert_{e}\lVert w\rVert\qquad(w\in
W).$ (7.4)

###### Proof.

Take an isomorphism $U\in\mathscr{B}(c_{0},X)$ with $\lVert U\rVert\,\lVert
U^{-1}\rVert\leqslant C$, and set $S=RU\in\mathscr{B}(c_{0},Y)$. Choose
$T\in\mathscr{K}(c_{0},Y)$ such that $\lVert
S+T\rVert\leqslant(1+\frac{\eta}{2})\lVert S\rVert_{e}$. We have $TP_{n}\to T$
as $n\to\infty$ because the basis of $c_{0}$ is shrinking, so $\lVert
T(I_{c_{0}}-P_{m_{0}})\rVert\leqslant\frac{\eta}{2}\lVert S\rVert_{e}$ for
some $m_{0}\in\mathbb{N}$.

On the other hand, $\lVert S(I_{c_{0}}-P_{m})\rVert>(1-\frac{\eta}{2})\lVert
S\rVert_{e}$ for each $m\in\mathbb{N}$, and therefore we can find an integer
$m^{\prime}>m$ and a unit vector $w\in\operatorname{span}\\{e_{j}:m<j\leqslant
m^{\prime}\\}$ such that $\lVert Sw\rVert>(1-\frac{\eta}{2})\lVert
S\rVert_{e}$. Using this, we can recursively choose integers
$m_{1}<m_{2}<\cdots$ with $m_{1}>m_{0}$ and unit vectors
$w_{n}\in\operatorname{span}\\{e_{j}:m_{n-1}<j\leqslant m_{n}\\}$ such that

$\lVert Sw_{n}\rVert>\Bigl{(}1-\frac{\eta}{2}\Bigr{)}\lVert
S\rVert_{e}\qquad(n\in\mathbb{N}).$

Since the linear map on $c_{0}$ determined by $e_{n}\mapsto w_{n}$ for each
$n\in\mathbb{N}$ is an isometry, Theorem 3.3 implies that we can find a
subsequence $(w_{n_{j}})$ of $(w_{n})$ such that the restriction of $S$ to the
subspace $W_{0}=\overline{\operatorname{span}}\\{w_{n_{j}}:j\in\mathbb{N}\\}$
is bounded below by $(1-\eta)\lVert S\rVert_{e}$. Furthermore, $(w_{n_{j}})$
is a normalized block basic sequence of $(e_{n})$, so $W_{0}$ is isometrically
isomorphic to $c_{0}$, and we can take a projection $Q$ of $c_{0}$ onto
$W_{0}$ with $\lVert Q\rVert=1$. Consequently the subspace $W=U[W_{0}]$ is
$C$-isomorphic to $c_{0}$ and complemented in $X$ via the projection
$UQU^{-1}$, which has norm at most $C$.

Finally, to verify (7.4), take $w\in W$, and set $w_{0}=U^{-1}w\in W_{0}$.
Then $Rw=Sw_{0}$ and $w_{0}=(I_{c_{0}}-P_{m_{0}})w_{0}$, so

$\displaystyle\lVert Rw\rVert$
$\displaystyle\leqslant\lVert(S+T)w_{0}\rVert+\lVert
T(I_{c_{0}}-P_{m_{0}})w_{0}\rVert\leqslant(1+\eta)\lVert S\rVert_{e}\lVert
w_{0}\rVert$ $\displaystyle\leqslant(1+\eta)\lVert R\rVert_{e}\lVert
U\rVert\,\lVert U^{-1}\rVert\,\lVert w\rVert\leqslant(1+\eta)C\lVert
R\rVert_{e}\lVert w\rVert$

and

$\displaystyle\lVert Rw\rVert$ $\displaystyle\geqslant(1-\eta)\lVert
S\rVert_{e}\lVert w_{0}\rVert\geqslant(1-\eta)\frac{\lVert
SU^{-1}\rVert_{e}}{\lVert U^{-1}\rVert}\frac{\lVert Uw_{0}\rVert}{\lVert
U\rVert}\geqslant\frac{1-\eta}{C}\lVert R\rVert_{e}\lVert w\rVert.\qed$

######  Definition 7.5.

Let $X$, $Y$ and $Z$ be Banach spaces. An operator $R\in\mathscr{B}(X,Y)$
_fixes a copy of $Z$_ if there is an operator $V\in\mathscr{B}(Z,X)$ such that
the composite operator $RV$ is bounded below.

######  Corollary 7.6.

Let $X$ and $Y$ be Banach spaces for which the unit ball of $Y^{*}$ is weak*
sequentially compact, and let $R\in\mathscr{B}(X,Y)$ be an operator which
fixes a copy of $c_{0}$. Then, for every $\varepsilon\in(0,1)$, there exist a
constant $\xi>0$ and a closed subspace $W$ of $X$ such that $W$ is isomorphic
to $c_{0}$,

$(1+\varepsilon)\xi\lVert w\rVert\geqslant\lVert
Rw\rVert\geqslant(1-\varepsilon)\xi\lVert w\rVert\qquad(w\in W),$ (7.5)

the subspace $R[W]$ is $(1+\varepsilon)$-complemented in $Y$ and
$(1+\varepsilon)$-isomorphic to $c_{0}$, and there are operators
$U\in\mathscr{B}(Y,c_{0})$ and $V\in\mathscr{B}(c_{0},X)$ such that

$URV=I_{c_{0}}\qquad\text{and}\qquad\lVert U\rVert\,\lVert
V\rVert\leqslant\frac{(1+\varepsilon)^{2}}{(1-\varepsilon)\xi}.$ (7.6)

###### Proof.

Given $\varepsilon\in(0,1)$, choose $\eta>0$ such that
$\eta^{2}+2\eta\leqslant\varepsilon$. Since $R$ fixes a copy of $c_{0}$, we
can take a closed subspace $X_{1}$ of $X$ such that $X_{1}$ is isomorphic to
$c_{0}$ and the restriction of $R$ to $X_{1}$ is bounded below. By James’
Distortion Theorem [23], $X_{1}$ contains a closed subspace $X_{2}$ which is
$(1+\eta)$-isomorphic to $c_{0}$. The restriction of $R$ to $X_{2}$ is non-
compact because it is bounded below, so in view of Lemma 7.4, we can find a
closed subspace $X_{3}$ of $X_{2}$ such that $X_{3}$ is isomorphic to $c_{0}$
and

$(1+\eta)^{2}\lVert R|_{X_{2}}\rVert_{e}\lVert w\rVert\geqslant\lVert
Rw\rVert\geqslant\frac{1-\eta}{1+\eta}\lVert R|_{X_{2}}\rVert_{e}\lVert
w\rVert\qquad(w\in X_{3}).$ (7.7)

Now, applying Lemma 3.7 to the restriction of $R$ to $X_{3}$ and with the
constant $C=1+\varepsilon$, we obtain a closed subspace $Z$ of $R[X_{3}]$ such
that $Z$ is $(1+\varepsilon)$-complemented in $Y$ and
$(1+\varepsilon)$-isomorphic to $c_{0}$. Set $W=R^{-1}[Z]\cap X_{3}$ and
$\xi=\lVert R|_{X_{2}}\rVert_{e}$. Then $R$ maps $W$ isomorphically onto $Z$,
so in particular $W$ is isomorphic to $c_{0}$, and (7.5) follows from (7.7)
because the choice of $\eta$ implies that $1+\varepsilon\geqslant(1+\eta)^{2}$
and $1-\varepsilon\leqslant(1-\eta)/(1+\eta)$. Finally the existence of the
operators $U$ and $V$ satisfying (7.6) also follows from Lemma 3.7 because the
restriction of $R$ to $X_{3}$ is bounded below by $(1-\varepsilon)\xi$. ∎

When applying this result to $C_{0}(K)$ for $K$ scattered, the following
variant of a classical theorem of Pełczyński will be helpful.

######  Theorem 7.7.

Let $K$ be a scattered, locally compact Hausdorff space. An operator from
$C_{0}(K)$ into a Banach space is compact if and only if it is weakly compact,
if and only if it does not fix a copy of $c_{0}$.

###### Proof.

In view of Remark 6.3 and the isomorphic nature of this result, we may suppose
that $K$ is compact by replacing it with its one-point compactification if
necessary. Pełczyński [43] showed that an operator defined on a $C(K)$-space
is weakly compact if and only if it does not fix a copy of $c_{0}$, and of
course every compact operator is weakly compact.

To complete the proof, suppose that $R$ is a weakly compact operator defined
on $C(K)$. By Gantmacher’s Theorem, its adjoint $R^{*}$ is also weakly
compact. This implies that $R^{*}$ is compact because a famous result of Rudin
[47] states that its codomain $C(K)^{*}$ is isomorphic to $\ell_{1}(K)$, which
has the Schur property. Hence $R$ is compact by Schauder’s Theorem. ∎

Finally, before proving Proposition 7.2, we establish an elementary lemma that
will help us shorten a couple of steps in the proof.

######  Lemma 7.8.

Let $X$, $Y$ and $Z$ be infinite-dimensional Banach spaces, and suppose that
the operators $R\in\mathscr{B}(X,Y)$, $T\in\mathscr{B}(Y,Z)$ and
$U\in\mathscr{B}(Z,X)$ satisfy

$I_{X}-UTR\in\mathscr{K}(X).$ (7.8)

Then $R[X]$ is closed and isomorphic to a closed subspace of finite
codimension in $X$, and the restriction of $T$ to any closed, infinite-
dimensional subspace of $R[X]$ is non-compact.

###### Proof.

Equation (7.8) implies that $UTR$ is a Fredholm operator. Therefore $R$ is an
upper semi-Fredholm operator by [6, Corollary 1.3.4], so $R$ has closed range
and finite-dimensional kernel. Let $W$ be a closed, complementary subspace of
$\ker R$; that is, $W+\ker R=X$ and $W\cap\ker R=\\{0\\}$. Then the
restriction of $R$ to $W$ is an isomorphism onto $R[X]$.

Suppose that $Y_{0}$ is a closed subspace of $R[X]$ such that $T|_{Y_{0}}$ is
compact, set $X_{0}=R^{-1}[Y_{0}]$, and regard the restriction $\widetilde{R}$
of $R$ to $X_{0}$ as an operator into $Y_{0}$; it satisfies
$UTR|_{X_{0}}=UT|_{Y_{0}}\widetilde{R}$, which implies that $UTR|_{X_{0}}$ is
compact. Hence $X_{0}$ is finite-dimensional by (7.8), so $R[X_{0}]=Y_{0}$ is
also finite-dimensional. ∎

######  Remark 7.9.

The same proof as above will work if one replaces $\mathscr{K}(X)$ with the
larger ideal of inessential operators in (7.8).

###### Proof of Proposition 7.2.

We shall apply Corollary 7.6 several times in this proof, in each case to a
non-compact operator $R$ whose domain and codomain are (isomorphic to) either
$c_{0}$ or $C_{0}(K)$. To avoid repetition, let us once and for all state that
Theorems 7.7 and 6.2 ensure that the hypotheses of Corollary 7.6 on $R$ and
its codomain are satisfied in this case.

This observation shows in particular that the identity operator on $c_{0}$
factors through every non-compact operator on $C_{0}(K)$, so the definition of
$\nu$ makes sense. Furthermore, it is clear that $\nu$ is faithful and
absolutely homogeneous.

In the remainder of the proof, suppose that
$T_{1},T_{2}\in\mathscr{B}(C_{0}(K))$.

Subadditivity. Let us begin by observing that the inequality

$\nu(T_{1}+T_{2}+\mathscr{K}(C_{0}(K)))\leqslant\nu(T_{1}+\mathscr{K}(C_{0}(K)))+\nu(T_{2}+\mathscr{K}(C_{0}(K)))$