# The compact operators on $c_{0}$ as a Calkin algebra

Pavlos Motakis  Department of Mathematics and Statistics, York University,
4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada<EMAIL_ADDRESS>and
Daniele Puglisi  Department of Mathematics and Computer Sciences, University
of Catania, Catania, 95125, Italy<EMAIL_ADDRESS>
###### Abstract.

For a Banach space $X$, let $\mathcal{L}(X)$ denote the algebra of all bounded
linear operators on $X$ and let $\mathcal{K}(X)$ denote the compact operator
ideal in $\mathcal{L}(X)$. The quotient algebra
$\mathcal{L}(X)/\mathcal{K}(X)$ is called the Calkin algebra of $X$, and it is
denoted $\mathpzc{Cal}(X)$. We prove that the unitization of
$\mathcal{K}(c_{0})$ is isomorphic as a Banach algebra to the Calkin algebra
of some Banach space $\mathcal{Z}_{\mathcal{K}(c_{0})}$. This Banach space is
an Argyros-Haydon sum $(\oplus_{n=1}^{\infty}X_{n})_{\mathrm{AH}}$ of a
sequence of copies $X_{n}$ of a single Argyros-Haydon space
$\mathfrak{X}_{\mathrm{AH}}$, and the external versus the internal Argyros-
Haydon construction parameters are chosen from disjoint sets.

###### 2020 Mathematics Subject Classification:

46B07, 46B25, 46B28.

The first author was supported by NSERC Grant RGPIN-2021-03639.

###### Contents

  1. 1 Introduction
  2. 2 Horizontally approximable operators
  3. 3 Bourgain-Delbaen-$\mathscr{L}_{\infty}$ sums of Banach spaces
  4. 4 Argyros-Haydon sums of a sequence of Banach spaces
    1. 4.1 Definition of the sets $(\Delta_{n})_{n=1}^{\infty}$
    2. 4.2 Definition of the space $\mathcal{Z}=\mathcal{Z}\big{(}(m_{j},n_{j})_{j=1}^{\infty},(X_{n})_{n=1}^{\infty}\big{)}$
    3. 4.3 Evaluation analysis of $e_{\gamma}^{*}$
  5. 5 Operators on Argyros-Haydon sums of Banach spaces
  6. 6 The space $\mathcal{Z}_{\mathcal{K}(c_{0})}$
  7. 7 Open problems and directions

##  1\. Introduction

The Calkin algebra of a Banach space $X$ (see, e.g., [4], [20], or [5]) is
defined as $\mathpzc{Cal}(X)=\mathcal{L}(X)/\mathcal{K}(X)$, where
$\mathcal{L}(X)$ denotes the algebra of bounded linear opetors on $X$ and
$\mathcal{K}(X)$ denotes its ideal of compact operators. Given a unital Banach
algebra $\mathcal{B}$, showing that it is a Calkin algebra means finding a
Banach space $X$ such that $\mathpzc{Cal}(X)$ is isomorphic to $\mathcal{B}$
as a Banach algebra. This is a challenging problem; solving it demands
designing $X$ having in mind the algebraic structure of the indirectly defined
object $\mathcal{L}(X)$ and the position of $\mathcal{K}(X)$ in it. The
starting point of this field of study is the Argyros-Haydon space
$\mathfrak{X}_{\mathrm{AH}}$ (see [2]) with
$\mathpzc{Cal}(\mathfrak{X}_{\mathrm{AH}})$ one-dimensional, i.e., it has the
scalar-plus-compact property. The construction of $\mathfrak{X}_{\mathrm{AH}}$
relies on immensely influential works by Tsirelson ([19]), Bourgain and
Delbaen ([3]), Schlumprecht ([17]), and Gowers and Maurey ([9]). Following
[2], several examples of explicit unital Banach algebras $\mathcal{B}$ have
been shown to be Calkin algebras. Some of them are the convolution algebra
$\ell_{1}(\mathbb{N}_{0})$ ([18]) and all separable commutative
$C^{*}$-algebras ([15] and [12]). Other examples include the Hilbert space
$\ell_{2}$ with coordinate-wise multiplication induced by an orthonormal basis
(this is the first known reflexive infinite dimensional Calkin algebra) ([13],
see also [16]) and the quasireflexive James space with coordinate-wise
multiplication induced by its standard shrinking basis ([14]). The
aforementioned examples are commutative; non-commutative ones also exist, but
are confined to spaces of the type
$\mathcal{B}=\oplus_{i=1}^{n}M_{k_{i}}(\mathcal{B}_{i})$ for some
$\mathcal{B}_{i}$ from the previous lists ([10, Page 1022, Note added in
proof] and [12, Corollary 8.8]). The endeavour of finding more involved
explicit non-commutative Calkin algebras was sparked by a set of questions of
N. C. Phillips focusing primarily on $C^{*}$-algebras (see [12, Section 9]).
We contribute to this by proving the following.

######  Theorem.

There exists a Banach space $\mathcal{Z}_{\mathcal{K}(c_{0})}$ with Calkin
algebra isomorphic to the unitization of $\mathcal{K}(c_{0})$ as a Banach
algebra.

The Banach space $\mathcal{Z}_{\mathcal{K}(c_{0})}$ is a Bourgain-
Delbaen-$\mathscr{L}_{\infty}$ sum of Bourgain-
Delbaen-$\mathscr{L}_{\infty}$-spaces. Bourgain-
Delbaen-$\mathscr{L}_{\infty}$-spaces (introduced in [3], see also [1])
comprise an important class that has played a major role in Banach space
theory. Prominent examples are the original Bourgain-Delbaen space from [3],
and the Argyros-Haydon space from [2], but there are many others. For example,
in [11], such spaces were used to negatively solve Godefroy’s problem from [7]
on whether the isomorphism class of $c_{0}$ is Borel. Another application is
from [6] where it was proved that such a space from [1] is a counterexample to
a conjecture of Godefroy, Kalton, and Lancien from [8] stating an
$\ell_{1}$-predual with summable Slenk should be isomorphic to $c_{0}$. In
this paper, we use Bourgain-Delbaen-$\mathscr{L}_{\infty}$ sums from [21]
similarly to [15]. The Banach space $\mathcal{Z}_{\mathcal{K}(c_{0})}$ is a
special type of Bourgain-Delbaen-$\mathscr{L}_{\infty}$-sum: it is an Argyros-
Haydon-sum $(\oplus_{n=1}^{\infty}X_{n})_{\mathrm{AH}}$, and each $X_{n}$ is a
copy of a single Argyros-Haydon space $\mathfrak{X}_{\mathrm{AH}}$. The
classical Argyros-Haydon construction uses as parameter a sequence of pairs or
natural numbers $(m_{j},n_{j})_{j=1}^{\infty}$ satisfying certain growth
conditions. Here, the space $\mathfrak{X}_{\mathrm{AH}}$ has parameter a
sequence of pairs $(m_{j},n_{j})_{j\in L_{1}}$ whereas the external Argyros-
Haydon has parameter a sequence of pairs $(m_{j},n_{j})_{j\in L_{2}}$, and the
sets $L_{1}$ and $L_{2}$ are disjoint infinite subsets of $\mathbb{N}$. As
such, our construction relies on directly applying known techniques and does
not involve radical modifications of heavy machinery.

We postpone the details of the construction of
$\mathcal{Z}_{\mathcal{K}(c_{0})}$ and proof of its main properties until
Sections 4, 5, and 6. We initially focus on the principles that yield that its
Calkin algebra is the unitization of $\mathcal{K}(c_{0})$. Recall that a
Schauder decomposition of a Banach space $X$ is a sequence of closed subspaces
$(X_{n})_{n=1}^{\infty}$ of $X$ such that each $x\in X$ admits a unique
representation $x=\sum_{n=1}^{\infty}x_{n}$, where $x_{n}\in X_{n}$, for all
$n\in\mathbb{N}$. This induces a sequence of bounded linear projections
$P_{n}:X\to X$, $n\in\mathbb{N}$, where $P_{n}x=\sum_{k=1}^{n}x_{k}$ and
$C=\sup_{n}\|P_{n}\|$ is finite. If, in particular, $C=1$, then the Schauder
decomposition is called monotone. If $\cup_{n=1}^{\infty}P_{n}^{*}(X^{*})$ is
dense in $X^{*}$ then we call $(X_{n})_{n=1}^{\infty}$ shrinking. For every
interval $I$ of $\mathbb{N}$ we also let $P_{I}x=\sum_{k\in I}x_{k}$.

Given a Banach space $X$ with a Schauder decomposition
$(X_{n})_{n=1}^{\infty}$, every bounded linear operator $T:X\to X$ admits a
matrix representation $(T_{m,n})_{m,n=1}^{\infty}$, where $T_{m,n}:X_{n}\to
X_{m}$, $m,n\in\mathbb{N}$. This can be used to approximate $T$ in the strong
operator topology. We define the algebra $\mathcal{HA}(X)$ of horizontally
approximable operators on $X$ comprising the $T:X\to X$ for which this
approximation is valid in the operator norm topology. When
$(X_{n})_{n=1}^{\infty}$ is shrinking, $\mathcal{K}(X)\subset\mathcal{HA}(X)$,
and thus, $\mathcal{HA}(X)/\mathcal{K}(X)$ is a subalgebra of
$\mathpzc{Cal}(X)$. We observe that if $X$ has the scalar-plus-compact
property, then $\mathcal{HA}(c_{0}(X))/\mathcal{K}(c_{0}(X))$ coincides with
$\mathcal{K}(c_{0})$. We exploit this fact as follows. We introduce an
equivalence condition between Schauder decompositions $(X_{n})_{n=1}^{\infty}$
and $(Y_{n})_{n=1}^{\infty}$ of Banach spaces $X$ and $Y$ respectively, called
finite Fredholm equivalence. If it is satisfied, then
$\mathcal{HA}(X)/\mathcal{K}(X)$ and $\mathcal{HA}(Y)/\mathcal{K}(Y)$ are
isomorphic Banach algebras, even if $X$ and $Y$ are not isomorphic Banach
spaces. We prove that in particular the standard Schauder decomposition of a
Bourgain-Delbaen-$\mathscr{L}_{\infty}$-sum of a sequence of Banach spaces
$(X_{n})_{n=1}^{\infty}$ is finitely Fredholm equivalent to the standard
Schauder decomposition of $(\oplus_{n=1}^{\infty}X_{n})_{0}$. This is directly
applicable to the following distinguishing features of
$\mathcal{Z}_{\mathcal{K}(c_{0})}$, equipped with its standard Schauder
decomposition:

  1. (i)

It is a Bourgain-Delbaen-$\mathscr{L}_{\infty}$ sum of countably many copies
of a space $\mathfrak{X}_{\mathrm{AH}}$ with the scalar-plus compact property.

  2. (ii)

Every bounded linear operator on $\mathcal{Z}_{\mathcal{K}(c_{0})}$ is a
scalar multiple of the identity plus a horizontally approximable operator.

By the preceding discussion,
$\mathcal{HA}(\mathcal{Z}_{\mathcal{K}(c_{0})})/\mathcal{K}(\mathcal{Z}_{\mathcal{K}(c_{0})})$
is isomorphic as a Banach algebra to
$\mathcal{HA}(c_{0}(\mathfrak{X}_{\mathrm{AH}}))/\mathcal{K}(c_{0}(\mathfrak{X}_{\mathrm{AH}}))$,
which, by the first item, coincides with $\mathcal{K}(c_{0})$. The second
property yields that
$\mathcal{HA}(\mathcal{Z}_{\mathcal{K}(c_{0})})/\mathcal{K}(\mathcal{Z}_{\mathcal{K}(c_{0})})$
is of codimension-one in $\mathpzc{Cal}(X)$.

##  2\. Horizontally approximable operators

In this section, we study the algebra $\mathcal{HA}(X)$ of horizontally
approximable operators on a Banach space $X$ with a Schauder decomposition
$(X_{n})_{n=1}^{\infty}$. This is a subcollection of the algebra of
horizontally compact operators, introduced in [21, Definition 7.1] (see also
Definition 5.1). An operator $T:X\to X$ is horizontally approximable if the
finite initial submatrices of its standard representation as an infinite
matrix $(T_{m,n})_{m,n=1}^{\infty}$ with respect to $(X_{n})_{n=1}^{\infty}$
can be used to approximate $T$ in operator norm. When $(X_{n})_{n=1}^{\infty}$
is shrinking, $\mathcal{K}(X)\subset\mathcal{HA}(X)$, and thus,
$\mathcal{HA}(X)/\mathcal{K}(X)$ is a subalgebra of the Calkin algebra of $X$.
For example, if $X$ is an infinite dimensional Banach space with the scalar-
plus-compact property, we prove that
$\mathcal{HA}(c_{0}(X))/\mathcal{K}(c_{0}(X))$ coincides with
$\mathcal{K}(c_{0})$. We then provide an equivalence condition between
Schauder decompositions $(X_{n})_{n=1}^{\infty}$ and $(Y_{n})_{n=1}^{\infty}$
of Banach spaces $X$ and $Y$ respectively. When it is satisfied,
$\mathcal{HA}(X)/\mathcal{K}(X)$ and $\mathcal{HA}(Y)/\mathcal{K}(Y)$ are
isomorphic as Banach algebras, even if $X$ and $Y$ are not isomorphic.

######  Definition 2.1.

Let $X$ be a Banach space with a Schauder decomposition
$(X_{n})_{n=1}^{\infty}$ and associated sequence of projections
$(P_{n})_{n=1}^{\infty}$.

  1. (a)

A bounded linear operator $A$ on $X$ is called horizontally approximable if
any of the following equivalent conditions is satisfied.

    1. ($\alpha$)

$A=\lim_{n}AP_{n}$ and $A=\lim_{n}P_{n}A$,

    2. ($\beta$)

$A=\lim_{m}\lim_{n}P_{m}AP_{n}$,

    3. ($\gamma$)

$A=\lim_{n}\lim_{m}P_{m}AP_{n}$,

    4. ($\delta$)

$A=\lim_{n}P_{n}AP_{n}$.

where all limits are with respect to the operator norm topology.

  2. (b)

The collection of horizontally approximable bounded linear operators on $X$ is
denoted $\mathcal{HA}(X)$.

The definitions of horizontally approximable operators and $\mathcal{HA}(X)$
depend on the Schauder decomposition $(X_{n})_{n=1}^{\infty}$, but this will
always be fixed, and thus, we suppress it in the notation.

######  Remark 2.2.

Let $X$ be a Banach space with a Schauder decomposition
$(X_{n})_{n=1}^{\infty}$. We will identify a $T\in\mathcal{L}(X)$ with its
standard matrix representation $(T_{m,n})_{m,n=1}^{\infty}$ with respect to
$(X_{n})_{n=1}^{\infty}$, i.e., for $m,n\in\mathbb{N}$,
$T_{m,n}=P_{\\{m\\}}TP_{\\{n\\}}:X\to X$. Letting “$\mathrm{sot}$” denote the
strong operator topology of $\mathcal{L}(X)$, the following hold.

  1. (i)

If $T=(T_{m,n})_{m,n=1}^{\infty}$ then

$T=\mathrm{sot}\text{-}\sum_{m=1}^{\infty}\mathrm{sot}\text{-}\sum_{n=1}^{\infty}T_{m,n}=\mathrm{sot}\text{-}\sum_{n=1}^{\infty}\mathrm{sot}\text{-}\sum_{m=1}^{\infty}T_{m,n}.$

  2. (ii)

If $T=(T_{m,n})_{m,n=1}^{\infty}$ and $S=(S_{m,n})_{n=1}^{\infty}$ then

$TS=\Big{(}\mathrm{sot}\text{-}\sum_{k=1}^{\infty}T_{m,k}S_{k,n}\Big{)}_{m,n=1}^{\infty}.$

We also sometimes naturally identify each $T_{m,n}$ with a map
$T_{m,n}:X_{n}\to X_{m}$ by restricting its domain and codomain. It will
always be clear from context which map the notation $T_{m,n}$ refers to.

######  Remarks.

Let $X$ be a Banach space with a Schauder decomposition
$(X_{n})_{n=1}^{\infty}$.

  1. (i)

$\mathcal{HA}(X)$ is a closed subalgebra of $\mathcal{L}(X)$ and

$\mathcal{HA}_{00}(X)=\cup_{n=1}^{\infty}\big{\\{}A\in\mathcal{L}(X):A=P_{n}AP_{n}\big{\\}}$

is dense in $\mathcal{HA}(X)$. An operator
$T=(T_{m,n})_{m,n=1}^{\infty}\in\mathcal{L}(X)$ is in $\mathcal{HA}_{00}(X)$
if and only if all but finitely many of its entries are zero.

  2. (ii)

An operator $T=(T_{m,n})_{m,n=1}^{\infty}\in\mathcal{L}(X)$ is in
$\mathcal{HA}(X)$ if and only if

$T=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}T_{m,n}=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}T_{m,n},$

where the convergence is in the operator norm topology.

  3. (iii)

If $A=(A_{m,n})_{m,n=1}^{\infty}$, $B=(B_{m,n})_{m,n=1}^{\infty}$ are in
$\mathcal{HA}(X)$ then

$AB=\Big{(}\sum_{k=1}^{\infty}A_{m,k}B_{k,n}\Big{)}_{m,n=1}^{\infty},$

where the convergence is in the operator norm topology.

  4. (iv)

If we additionally assume that the Schauder decomposition
$(X_{n})_{n=1}^{\infty}$ is shrinking, $\mathcal{K}(X)\subset\mathcal{HA}(X)$.
In particular, $\mathcal{HA}(X)/\mathcal{K}(X)$ is a closed subalgebra of
$\mathpzc{Cal}(X)$.

######  Notation 2.3.

For a sequence of Banach spaces $(X_{n})_{n=1}^{\infty}$ we denote

$(\oplus_{n=1}^{\infty}X_{n})_{0}=\Big{\\{}(x_{n})_{n=1}^{\infty}\in\prod_{n=1}^{\infty}X_{n}:\lim_{n}\|x_{n}\|=0\Big{\\}}$

with $\|(x_{n})_{n=1}^{\infty}\|=\max_{n}\|x_{n}\|$. If $X$ is a Banach space
and, for all $n\in\mathbb{N}$, $X_{n}=X$, then we denote this space
$c_{0}(X)$.

######  Remark 2.4.

For a sequence of Banach spaces $(X_{n})_{n=1}^{\infty}$, the space
$Y=(\oplus_{n=1}^{\infty}X_{n})_{0}$ admits a standard shrinking Schauder
decomposition $(\hat{X}_{n})_{n=1}^{\infty}$ where, for $n\in\mathbb{N}$,

$\hat{X}_{n}=\Big{\\{}(x_{m})_{m=1}^{\infty}\in Y:x_{m}=0\text{ for }m\neq
n\Big{\\}}.$

######  Remark 2.5.

We identify the Banach space $c_{0}(\ell_{1})$ with all infinite scalar
matrices $A=(a_{m,n})_{m,n=1}^{\infty}$ such that

(2.1) $\displaystyle\|A\|=\sup_{m}\sum_{n=1}^{\infty}|a_{m,n}|<\infty\text{
and}$ (2.2)
$\displaystyle\lim_{m}\Big{(}\sum_{n=1}^{\infty}|a_{m,n}|\Big{)}=0.$

This is then naturally isometrically identified with $\mathcal{K}(c_{0})$ by
letting for $A=(a_{m,n})_{m,n=1}^{\infty}$,
$Ae_{n}=\sum_{m=1}^{\infty}a_{m,n}e_{m}$. With this identification, usual
matrix multiplication corresponds to composition of operators, and the
collection of matrices $A$ with finitely many non-zero entries is dense in
$\mathcal{K}(c_{0})$.

######  Proposition 2.6.

Let $X$ be an infinite dimensional Banach space with the scalar-plus-compact
property and consider $c_{0}(X)$ with its standard shrinking Schauder
decomposition. Then, $\mathcal{HA}(c_{0}(X))/\mathcal{K}(c_{0}(X))$ is
isometrically isomorphic to $\mathcal{K}(c_{0})$ as a Banach algebra.

###### Proof.

Denote $Y=c_{0}(X)$. For $A=(a_{m,n})_{m,n=1}^{\infty}$ in
$\mathcal{K}(c_{0})$ with all but finitely many zero entries we define
$\hat{A}=(a_{m,n}I_{X})_{m,n=1}^{\infty}\in\mathcal{HA}_{00}(Y)$, i.e., for
$(x_{n})_{n=1}^{\infty}\in Y$,

$\hat{A}(x_{n})_{n=1}^{\infty}=\Big{(}\sum_{n=1}^{\infty}a_{m,n}x_{n}\Big{)}_{m=1}^{\infty}.$

Let us verify that $\|\hat{A}\|\leq\|A\|$. Indeed, if $(x_{n})_{n=1}^{\infty}$
is in the unit ball of $Y$, then

$\big{\|}\hat{A}\big{(}(x_{n})_{n=1}^{\infty}\big{)}\big{\|}=\max_{m}\Big{\|}\sum_{n=1}^{\infty}a_{m,n}x_{n}\Big{\|}\leq\max_{m}\sum_{n=1}^{\infty}|a_{m,n}|=\|A\|.$

It is then entirely straightforward to check that $A\mapsto\hat{A}$ extends to
a non-expansive linear homomorphism from $\mathcal{K}(c_{0})$ to
$\mathcal{L}(Y)$. If we denote
$\mathcal{Q}:\mathcal{HA}(Y)\to\mathcal{HA}(Y)/\mathcal{K}(Y)$ the quotient
map, then $A\mapsto\mathcal{Q}\hat{A}$ is such a map as well. We will use the
infinite-dimensionality of $X$ to also easily obtain
$\|A\|\leq\|\mathcal{Q}\hat{A}\|$. Indeed, let
$A=(a_{m,n})_{m,n=1}^{\infty}\in\mathcal{K}(c_{0})$ with all but finitely many
zero entries and choose $M,N\in\mathbb{N}$ such that, for all
$(x_{n})_{n=1}^{\infty}\in Y$,

$\hat{A}\big{(}(x_{n})_{n=1}^{\infty}\big{)}=\Big{(}\sum_{n=1}^{N}a_{1,n}x_{n},\ldots,\sum_{n=1}^{N}a_{M,n}x_{n},0,0,\ldots\Big{)}.$

Let $K:Y\to Y$ be a compact operator and fix $\varepsilon>0$. For every
subspace $Z$ of $X$ and $n\in\mathbb{N}$ let

$\hat{Z}(n)=\big{\\{}(0,\ldots 0,x_{n},0,\ldots):x_{n}\in Z\big{\\}},$

which is a subspace of $Y$. By the compactness of $K$, we may inductively
choose infinite-dimensional subspaces $Z_{1}\supset Z_{2}\supset\cdots\supset
Z_{N}=Z$ of $X$ such that, for $n=1,\ldots,N$,
$\|K|_{\hat{Z}(n)}\|<\varepsilon/N$. Then, for every $x\in Z$ of norm one and
signs $(\varepsilon_{n})_{n=1}^{N}$,

$\Big{\|}\big{(}\hat{A}-K\big{)}\big{(}(\varepsilon_{1}x,\varepsilon_{2}x,\ldots,\varepsilon_{N}x,0,\ldots)\big{)}\Big{\|}\geq\max_{1\leq
m\leq M}\Big{|}\sum_{n=1}^{N}\varepsilon_{n}a_{m,n}\Big{|}-\varepsilon.$

In particular, $\|\hat{A}-K\|\geq\|A\|-\varepsilon$. Thus
$A\mapsto\mathcal{Q}\hat{A}$ is an isometric linear homomorphism.

To show that $A\mapsto\mathcal{Q}\hat{A}$ is onto, let
$B=(S_{m,n})_{m,n=1}^{\infty}\in\mathcal{HA}_{00}(Y)$. Because $X$ has the
scalar-plus-compact property, for $m,n\in\mathbb{N}$, there are a scalar
$a_{m,n}$ and a compact operator $K_{m,n}:X\to X$ such that
$S_{m,n}=a_{m,n}I_{X}+K_{m,n}$ and $a_{m,n}$, $K_{m,n}$ are non-zero for for
only finitely many $m,n\in\mathbb{N}$. Consider
$A=(a_{m,n})_{m,n=1}^{\infty}\in\mathcal{K}(c_{0})$ and note that
$\hat{A}-B\in\mathcal{K}(Y)$, i.e., $\mathcal{Q}\hat{A}=\mathcal{Q}B$. By the
density of $\mathcal{HA}_{00}(Y)$ in $\mathcal{HA}(Y)$,
$A\mapsto\mathcal{Q}\hat{A}$ is onto. ∎

######  Notation 2.7.

For Banach spaces $X$ and $Y$ and bounded linear operators $T,S:X\to Y$ we
will write $T\sim S$ to mean $T-S\in\mathcal{K}(X,Y)$.

######  Definition 2.8.

Let $X$ and $Y$ be Banach spaces with Schauder decompositions
$(X_{n})_{n=1}^{\infty}$, $(Y_{n})_{n=1}^{\infty}$ respectively and associated
projections $(P_{n})_{n=1}^{\infty}$, $(Q_{n})_{n=1}^{\infty}$ respectively.
We call $(X_{n})_{n=1}^{\infty}$ and $(Y_{n})_{n=1}^{\infty}$ $C$-finitely
Fredholm equivalent, for some $C\geq 1$, if there exist bounded linear
operators $L_{n}:X\to Y$ and $R_{n}:Y\to X$, $n\in\mathbb{N}$, that satisfy

  1. ($\alpha$)

the uniform boundedness condition

$\big{(}\sup_{n}\|L_{n}\|\big{)}\big{(}\sup_{n}\|R_{n}\|\big{)}\leq C,$

  2. ($\beta$)

the compatibility condition

$\displaystyle L_{m}\sim L_{n}P_{m}\sim Q_{m}L_{n}\text{ and }R_{m}\sim
R_{n}Q_{m}\sim P_{m}R_{n},$

for $m\leq n\in\mathbb{N}$,

  3. ($\gamma$)

and the initial Fredholm invertibility condition

$\displaystyle R_{n}L_{n}\sim P_{n}\text{ and }L_{n}R_{n}\sim Q_{n},$

for $n\in\mathbb{N}$.

######  Proposition 2.9.

Let $X$ and $Y$ be Banach spaces with $C$-finitely Fredholm equivalent
shrinking Schauder decompositions, for some $C\geq 1$. Then,
$\mathcal{HA}(X)/\mathcal{K}(X)$ and $\mathcal{HA}(Y)/\mathcal{K}(Y)$ are
$C^{2}$-isomorphic as Banach algebras.

###### Proof.

Adopt the notation of Definition 2.8 and let
$\mathcal{Q}_{Y}:\mathcal{L}(Y)\to\mathcal{L}(Y)/\mathcal{K}(Y)$ and
$\mathcal{Q}_{X}:\mathcal{L}(X)\to\mathcal{L}(X)/\mathcal{K}(X)$ denote the
quotient maps. For $n\in\mathbb{N}$ and $A\in\mathcal{HA}(X)$ we define

$\phi_{n}(A)=L_{n}AR_{n}.$

We note that, by item ($\beta$), $\phi_{n}(A)\sim\phi_{n}(P_{n}AP_{n})\sim
Q_{n}\phi_{n}(A)Q_{n}$, and thus,

$\phi_{n}(A)\in\\{B\in\mathcal{L}(Y):B=Q_{n}BQ_{n}\\}+\mathcal{K}(Y)\subset\mathcal{HA}(Y).$

Note that, for
$A\in\mathcal{HA}_{00}(X)=\cup_{n=1}^{\infty}\\{A\in\mathcal{L}(X):A=P_{n}AP_{n}\\}$,
the sequence $(\mathcal{Q}_{Y}\phi_{n}(A))_{n=1}^{\infty}$ is eventually
constant. Indeed, if $m\in\mathbb{N}$ is such that $A=P_{m}AP_{m}$ then, for
$n\geq m$, by item ($\beta$),

$\phi_{n}(A)=L_{n}AR_{n}=L_{n}P_{m}AP_{m}R_{n}\sim L_{m}AR_{m}=\phi_{m}(A).$

Additionally, for $A,B\in\mathcal{HA}_{00}(X)$ and $n\in\mathbb{N}$
sufficiently large,
$\mathcal{Q}_{Y}\phi_{n}(AB)=\mathcal{Q}_{Y}\phi_{n}(A)\mathcal{Q}_{Y}\phi_{n}(B)$.
Indeed, if $n\in\mathbb{N}$ is such that $A=P_{n}AP_{n}$ and $B=P_{n}BP_{n}$
then, by item ($\gamma$),

$\phi_{n}(AB)=L_{n}ABR_{n}=L_{n}AP_{n}BR_{n}\sim
L_{n}AR_{n}L_{n}BP_{n}=\phi_{n}(A)\phi_{n}(B).$

We conclude that, by item ($\alpha$), that the formula
$A\mapsto\lim_{n}\mathcal{Q}_{Y}(L_{n}AR_{n})$ defines a bounded homomorphism
$\Phi:\mathcal{HA}(X)\to\mathcal{HA}(Y)/\mathcal{K}(Y)$ of norm at most $C$,
and $\mathcal{K}(X)\subset\mathrm{ker}(\Phi)$. Let
$\widehat{\Phi}:\mathcal{HA}(X)/\mathcal{K}(X)\to\mathcal{HA}(Y)/\mathcal{K}(Y)$
be the unique bounded homomorphism such that, for $A\in\mathcal{HA}(X)$,
$\widehat{\Phi}(\mathcal{Q}_{X}A)=\Phi(A)$.

By interchanging the roles of $X$ and $Y$, the formula
$B\mapsto\lim_{n}\mathcal{Q}_{X}(R_{n}BL_{n})$ defines a bounded homomorphism
$\Psi:\mathcal{HA}(Y)\to\mathcal{HA}(X)/\mathcal{K}(X)$ of norm at most $C$,
and $\mathcal{K}(Y)\subset\mathrm{ker}(\Psi)$. Let
$\widehat{\Psi}:\mathcal{HA}(Y)/\mathcal{K}(Y)\to\mathcal{HA}(X)/\mathcal{K}(X)$
be the unique bounded homomorphism such that, for $B\in\mathcal{HA}(Y)$,
$\widehat{\Psi}(\mathcal{Q}_{Y}B)=\Psi(B)$. We claim that $\hat{\Phi}$ and
$\hat{\Psi}$ are each other’s inverse, and to that end, by density, it
suffices to prove

$\begin{split}\hat{\Psi}\big{(}\hat{\Phi}(\mathcal{Q}_{X}A)\big{)}&=\mathcal{Q}_{X}A,\text{
for }A\in\mathcal{HA}_{00}(X)\text{ and}\\\
\hat{\Phi}\big{(}\hat{\Psi}(\mathcal{Q}_{Y}B)\big{)}&=\mathcal{Q}_{Y}B,\text{
for }B\in\mathcal{HA}_{00}(Y).\end{split}$

We only prove the former because the arguments to show each of these are
identical. Let $A\in\mathcal{HA}_{00}(X)$, such that
$\Phi(A)=\lim_{n}\mathcal{Q}_{Y}\phi_{n}(A)$. Recall that, if $m\in\mathbb{N}$
is such that $A=P_{m}AP_{m}$, then, for $n\geq m$,
$\mathcal{Q}_{Y}\phi_{n}(A)=\mathcal{Q}_{Y}\phi_{m}(A)$. Therefore,
$\Phi(A)=\mathcal{Q}_{Y}\phi_{m}(A)$ and

$\begin{split}\hat{\Psi}\big{(}\hat{\Phi}(\mathcal{Q}_{X}A)\big{)}&=\hat{\Psi}\big{(}\Phi(A)\big{)}=\Psi\big{(}\phi_{m}(A)\big{)}=\Psi(L_{m}AR_{m})\\\
&=\lim_{n}\mathcal{Q}_{X}(R_{n}L_{m}AR_{m}L_{n}).\end{split}$

For $n\geq m$, by ($\beta$) and ($\gamma$), $R_{n}L_{m}\sim
R_{n}Q_{m}L_{m}\sim R_{m}L_{m}\sim P_{m}$. Similarly, $R_{m}L_{n}\sim P_{m}$,
and thus,
$\hat{\Psi}\big{(}\hat{\Phi}(\mathcal{Q}_{X}A)\big{)}=\mathcal{Q}_{X}(P_{m}AP_{m})=\mathcal{Q}_{X}(A)$.
∎

We remark that, under additional regularity assumptions on $Y$, $C$-finite
Fredholm equivalence can be improved to 1-finite Fredholm equivalence, after
renorming $X$.

######  Proposition 2.10.

Let $X$ and $Y$ be Banach spaces with $C$-finitely Fredholm equivalent
Schauder decompositions $(X_{n})_{n=1}^{\infty}$, $(Y_{n})_{n=1}^{\infty}$
respectively, for some $C\geq 1$. Adopting the notation of Definition 2.8,
assume that $(Y_{n})_{n=1}^{\infty}$ is monotone and, for every
$m,n\in\mathbb{N}$, $L_{m}R_{n}=Q_{\min\\{m,n\\}}$. Then, after passing to a
$C$-equivalent norm of $X$ but maintaining the original norm of $Y$,
$(X_{n})_{n=1}^{\infty}$, $(Y_{n})_{n=1}^{\infty}$ are 1-finitely Fredholm
equivalent.

###### Proof.

Let $\|\cdot\|_{X}$, $\|\cdot\|_{Y}$ denote the original norms of $X$ and $Y$
respectively and put
$\alpha=\sup_{n}\|R_{n}:(X,\|\cdot\|_{X})\to(Y,\|\cdot\|_{Y})\|$. Define the
equivalent norm on $X$ given by

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|x\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{X}=\max\big{\\{}\alpha^{-1}\|x\|_{X},\sup_{n}\|L_{n}x\|_{Y}\big{\\}}.$

It is easy to check that
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{X}$
is $C$-equivalent to $\|\cdot\|_{X}$. Then, by definition, for every
$n\in\mathbb{N}$,
$\|L_{n}:(X,{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{X})\to(Y,\|\cdot\|_{Y})\|\leq
1$. Also, for $m\in\mathbb{N}$ and $y\in Y$ such that $\|y\|_{Y}\leq 1$ we
will show
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|R_{m}y\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{X}\leq
1$. Indeed,

$\alpha^{-1}\|R_{m}y\|_{X}\leq\alpha^{-1}\|R_{m}:(Y,\|\cdot\|_{Y})\to(X,\|\cdot\|_{X})\|\|y\|_{Y}\leq
1$

and, for $n\in\mathbb{N}$,

$\|L_{n}(R_{m}y)\|_{Y}=\|Q_{\min\\{n,m\\}}y\|_{Y}\leq 1.$

This means that the Schauder decompositions of
$(X,{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{X})$
and $(Y,\|\cdot\|_{Y})$ are 1-finitely Fredholm equivalent. ∎

##  3\. Bourgain-Delbaen-$\mathscr{L}_{\infty}$ sums of Banach spaces

We recall the notion of an abstract $C$-Bourgain-
Delbaen-$\mathscr{L}_{\infty}$ sum $\mathcal{Z}$ of of a sequence of Banach
spaces $(X_{n})_{n=1}^{\infty}$, introduced in [21, Definition 2.3]. This
space $\mathcal{Z}$ admits a Schauder decomposition
$(\mathcal{Z}_{n})_{n=1}^{\infty}$ that we prove is $C$-finitely Fredholm
equivalent to the standard Schauder decomposition of
$(\oplus_{n=1}^{\infty}X_{n})_{0}$. The easy consequence of this is that if
each $X_{n}$ is a copy of a common Banach space $X$ with the scalar-plus-
compact property and $\mathcal{Z}$ satisfies the scalar-plus-horizontally
approximable property, then the Calkin algebra of such a $\mathcal{Z}$ is
isomorphic, as a Banach algebra, to the unitization of $\mathcal{K}(c_{0})$.
The goal of the subsequent Sections 4, 5, and 6 is to gather all the necessary
details from [2] and [21], and to prove that such $\mathcal{Z}$ exists.

######  Definition 3.1.

Let $(X_{n})_{n=1}^{\infty}$ be a sequence of Banach spaces and
$(\Delta_{n})_{n=1}^{\infty}$ be a sequence of disjoint finite sets. Let

$\mathcal{W}=\big{(}\oplus_{n=1}^{\infty}\big{(}X_{n}\oplus\ell_{\infty}(\Delta_{n})\big{)}_{\infty}\big{)}_{\infty}$

denote the Banach space of all sequences of pairs
$(x_{n},y_{n})_{n=1}^{\infty}$ such that, for $n\in\mathbb{N}$,
$(x_{n},y_{n})\in X_{n}\times\ell_{\infty}(\Delta_{n})$, with

$\|(x_{n},y_{n})_{n=1}^{\infty}\|_{\mathcal{W}}=\sup_{n\in\mathbb{N}}\max\\{\|x_{n}\|,\|y_{n}\|\\}<\infty.$

For $n\in\mathbb{N}$, we canonically identify

$(X_{n}\oplus\ell_{\infty}(\Delta_{n}))_{\infty}\text{ and
}\mathcal{W}_{n}=\big{(}\oplus_{k=1}^{n}\big{(}X_{k}\oplus\ell_{\infty}(\Delta_{k})\big{)}_{\infty}\big{)}_{\infty}$

with subspaces of $\mathcal{W}$ and let

$r_{n}:\mathcal{W}\to\mathcal{W}_{n}$

denote the canonical restriction map. For $n\in\mathbb{N}$, let
$\pi_{X_{n}}:\mathcal{W}\to X_{n}$ denote the canonical quotient map.

Assume that $C\geq 1$ and $i_{n}:\mathcal{W}_{n}\to\mathcal{W}$,
$n\in\mathbb{N}$ are linear operators that satisfy the following.

  1. ($\alpha$)

For all $n\in\mathbb{N}$, $\|i_{n}\|\leq C$.

  2. ($\beta$)

For all $n\in\mathbb{N}$, $r_{n}i_{n}:\mathcal{W}_{n}\to\mathcal{W}_{n}$ is
the identity map.

  3. ($\gamma$)

For all $m\leq n\in\mathbb{N}$,
$i_{n}r_{n}i_{m}=i_{m}:\mathcal{W}_{m}\to\mathcal{W}$.

  4. ($\delta$)

For all $m<n\in\mathbb{N}$, $\pi_{X_{n}}i_{m}:\mathcal{W}_{m}\to\mathcal{W}$
is the zero map.

For $n\in\mathbb{N}$, define
$\mathcal{Z}_{n}=i_{n}(X_{n}\oplus\ell_{\infty}(\Delta_{n}))_{\infty}$ and
$\mathcal{Z}=\overline{\langle\cup_{n=1}^{\infty}\mathcal{Z}_{n}\rangle}$. The
space $\mathcal{Z}$ is called a $C$-Bourgain-Delbaen-$\mathscr{L}_{\infty}$
sum of $(X_{n})_{n=1}^{\infty}$.

######  Proposition 3.2.

For a $C$-Bourgain-Delbaen-$\mathscr{L}_{\infty}$ sum $\mathcal{Z}$ of a
sequence of Banach spaces $(X_{n})_{n=1}^{\infty}$ the following hold.

  1. (i)

$(\mathcal{Z}_{n})_{n=1}^{\infty}$ is a Schauder decomposition of
$\mathcal{Z}$ with associated projections
$P_{n}=i_{n}r_{n}|_{\mathcal{Z}}:\mathcal{Z}\to\mathcal{Z}$, $n\in\mathbb{N}$.

  2. (ii)

For all $n\in\mathbb{N}$, $P_{n}(\mathcal{Z})=i_{n}(\mathcal{W}_{n})$.

###### Proof.

It is clear, by item ($\alpha$) of Definition 3.1, that $\sup\|P_{n}\|\leq C$
and, by items ($\beta$) and ($\gamma$) of Definition 3.1, that
$P_{m}P_{n}=P_{\min\\{m,n\\}}$. We will show:

  1. ($\mathfrak{a}$)

For $n\leq m\in\mathbb{N}$, $P_{m}(\mathcal{Z}_{n})=\mathcal{Z}_{n}$.

  2. ($\mathfrak{b}$)

For $m<n\in\mathbb{N}$, $P_{m}(\mathcal{Z}_{n})=\\{0\\}$.

$z=i_{n}\big{(}(0,0),\ldots,(0,0),(x_{n},y_{n})\big{)}.$

These easily yield item (i). For the former, by the definition of
$\mathcal{Z}_{n}$, $P_{n}(\mathcal{Z}_{n})=\mathcal{Z}_{n}$, and thus, for
$m\geq n$,

$P_{m}(\mathcal{Z}_{n})=P_{m}\big{(}P_{n}(\mathcal{Z}_{n})\big{)}=(P_{m}P_{n})(\mathcal{Z}_{n})=P_{n}(\mathcal{Z}_{n})=\mathcal{Z}_{n}.$

For the latter, let $z\in\mathcal{Z}_{n}$, i.e, for some $x_{n}\in X_{n}$ and
$y_{n}\in\ell_{\infty}(\Delta_{n})$. Then, for $m<n$,

$\begin{split}P_{m}z&=i_{m}r_{m}i_{n}\big{(}(0,0),\ldots,(0,0),(x_{n},y_{n})\big{)}\big{)}\\\
&=i_{m}r_{m}\big{(}(0,0),\ldots,(0,0),(x_{n},y_{n})\big{)}\big{)}\\\
&=i_{m}\big{(}(0,0),\ldots,(0,0)\big{)}=0.\end{split}$

For the proof of item (ii), the inclusion $P_{n}(\mathcal{Z})\subset
i_{n}(\mathcal{W}_{n})$ is obvious. We prove, by induction on
$n\in\mathbb{N}$, that $i_{n}(\mathcal{W}_{n})\subset P_{n}(\mathcal{Z}_{n})$.
For $n=1$, this is obvious. Let now $n\geq 2$ such that
$i_{n-1}(\mathcal{W}_{n})\subset P_{n-1}(\mathcal{Z}_{n})$ and let

$z=i_{n}\big{(}(x_{1},y_{1}),\ldots,(x_{n-1},y_{n-1}),(x_{n},y_{n})\big{)}\in
i_{n}(\mathcal{W}_{n})$

Put

$w=i_{n-1}\big{(}(x_{1},y_{1}),\ldots,(x_{n-1},y_{n-1})\big{)},$

which, by the inductive hypothesis, is in
$P_{n-1}(\mathcal{Z})=\mathcal{Z}_{1}+\cdots+\mathcal{Z}_{n-1}$. By items
($\gamma$) and ($\delta$) of Definition 3.1 we may write

$w=i_{n}\big{(}(x_{1},y_{1}),\ldots,(x_{n-1},y_{n-1}),(0,w_{n})\big{)},$

thus,

$z-w=i_{n}\big{(}(0,0),\ldots,(0,0),(x_{n},y_{n}-w_{n})\big{)}\in\mathcal{Z}_{n}.$

In conclusion,
$z\in\mathcal{Z}_{1}+\cdots+\mathcal{Z}_{n-1}+\mathcal{Z}_{n}=P_{n}(\mathcal{Z})$.
∎

######  Proposition 3.3.

Let $\mathcal{Z}$ be a $C$-Bourgain-Delbaen-$\mathscr{L}_{\infty}$ sum of a
sequence of Banach spaces $(X_{n})_{n=1}^{\infty}$. The following hold.

  1. (i)

$(\mathcal{Z}_{n})_{n=1}^{\infty}$ is $C$-finitely Fredholm equivalent to the
standard Schauder decomposition of $(\oplus_{n=1}^{\infty}X_{n})_{c_{0}}$.

  2. (ii)

$\mathcal{Z}$ admits a $C$-equivalent norm such that
$(\mathcal{Z}_{n})_{n=1}^{\infty}$ is $1$-finitely Fredholm equivalent to the
standard Schauder decomposition of $(\oplus_{n=1}^{\infty}X_{n})_{c_{0}}$.

###### Proof.

Denote $Y=(\oplus_{n=1}^{\infty}X_{n})_{c_{0}}$ and, for $n\in\mathbb{N}$,
define the following $L_{n}:\mathcal{Z}\to Y$ and $R_{n}:Y\to\mathcal{Z}$.

$\displaystyle
L_{n}\big{(}(x_{k},y_{k})_{k=1}^{\infty}\big{)}=(x_{1},x_{2},\ldots,x_{n},0,0,\ldots)$
$\displaystyle
R_{n}\big{(}(x_{k})_{k=1}^{\infty}\big{)}=i_{n}\big{(}(x_{1},0),(x_{2},0),\ldots,(x_{n},0)\big{)}.$

and note that $\sup_{n\in\mathbb{N}}\|L_{n}\|\|R_{n}\|\leq C$, i.e., the
uniform boundedness condition is satisfied. We will also show that the
compatibility and initial Fredholm invertibility conditions are satisfied. But
first, let us note that $L_{m}R_{n}=Q_{\min\\{m,n\\}}$ holds trivially. Thus,
item (ii) will follow immediately from item (i).

To show the compatibility condition, fix $m\leq n\in\mathbb{N}$. It is obvious
that $L_{m}=Q_{m}L_{n}$. To show $L_{n}P_{m}=L_{m}$, let
$(x_{k},y_{k})_{k=1}^{\infty}\in\mathcal{Z}$ and note

$\begin{split}P_{m}\big{(}(x_{k},y_{k})_{k=1}^{\infty}\big{)}&=i_{m}\big{(}(x_{1},y_{1}),\ldots,(x_{m},y_{m})\big{)}\\\
&=\big{(}(x_{1},y_{1}),\ldots,(x_{m},y_{m}),(0,w_{m}+1),(0,w_{m+2}),\ldots\big{)},\end{split}$

by items ($\beta$) and ($\delta$), of Definition 3.1. This easily yields

$L_{n}P_{m}\big{(}(x_{k},y_{k})_{k=1}^{\infty}\big{)}=L_{m}\big{(}(x_{k},y_{k})_{k=1}^{\infty}\big{)}.$

To show $R_{n}=P_{m}R_{n}$, let $(x_{k})_{k=1}^{\infty}\in Y$. Then

$\begin{split}P_{m}R_{n}\big{(}(x_{k})_{k=1}^{\infty}\big{)}&=i_{m}r_{m}\big{(}i_{n}\big{(}(x_{1},0),\ldots,(x_{m},0),\ldots,(x_{n},0)\big{)}\big{)}\\\
&=i_{m}\big{(}\big{(}(x_{1},0),\ldots,(x_{m},0)\big{)}\big{)}\text{ (by
\ref{BD assumption b} of \lx@cref{creftypecap~refnum}{abstract BD sum})}\\\
&=R_{m}\big{(}(x_{k})_{k=1}^{\infty}\big{)}.\end{split}$

To prove $R_{n}Q_{m}\sim R_{m}$ we define the auxiliary bounded linear
operator $A_{m,n}:\mathcal{Z}\to\mathcal{Z}$ such that, for
$(x_{k},y_{k})_{k=1}^{\infty}\in\mathcal{Z}$.

$A_{m,n}\big{(}(x_{k},y_{k})_{k=1}^{\infty}\big{)}=i_{n}\big{(}(0,0),\ldots,(0,0),(0,y_{m+1}),\ldots,(0,y_{n})\big{)},$

which is finite rank because of factoring through the finite-dimensional space
$\ell_{\infty}(\Delta_{m+1})\oplus\cdots\oplus\ell_{\infty}(\Delta_{n})$. We
will show $R_{m}-A_{m,n}R_{m}=R_{n}Q_{m}$. Indeed, for
$(x_{k})_{k=1}^{\infty}\in Y$, note that, by items ($\delta$) and ($\gamma$)
of Definition 3.1,

$\begin{split}R_{m}\big{(}(x_{k})_{k=1}^{\infty}\big{)}&=i_{m}\big{(}(x_{1},0),\ldots,(x_{m},0)\big{)}\\\
&=\big{(}(x_{1},0),\ldots,(x_{m},0),(0,y_{m+1}),(0,y_{m+2}),\ldots\big{)}\\\
&=i_{n}\big{(}(x_{1},0),\ldots,(x_{m},0),(0,y_{m+1}),\ldots,(0,y_{n})\big{)}.\end{split}$

Therefore,

$\begin{split}\big{(}R_{m}-A_{m,n}R_{m}\big{)}\big{(}(x_{k})_{k=1}^{\infty}\big{)}&=i_{n}\big{(}(x_{1},0),\ldots,(x_{m},0),(0,0),\ldots,(0,0)\big{)}\\\
&=R_{n}Q_{m}\big{(}(x_{k})_{k=1}^{\infty}\big{)},\end{split}$

and thus, $R_{n}Q_{m}\sim R_{m}$.

We prove the initial Fredholm invertibility condition and fix
$n\in\mathbb{N}$. Note that $R_{n}L_{n}=Q_{n}$ is rather obvious and that, for
$(x_{k},y_{k})_{k=1}^{\infty}\in\mathcal{Z}$,

$L_{n}R_{n}\big{(}(x_{k},y_{k})_{k=1}^{\infty}\big{)}=i_{n}\big{(}(x_{1},0),\ldots,(x_{n},0)\big{)},$

and thus,

$\big{(}P_{n}-L_{n}R_{n}\big{)}\big{(}(x_{k},y_{k})_{k=1}^{\infty}\big{)}=i_{n}\big{(}(0,y_{1}),\ldots,(0,y_{n})\big{)}.$

Therefore $P_{n}-L_{n}R_{n}$ is a finite rank operator because it factors
through the finite-dimensional space
$\ell_{1}(\Delta_{1})\oplus\cdots\oplus\ell_{\infty}(\Delta_{n})$. ∎

We formalize the scalar-plus-horizontally approximable property of a Banach
space with a Schauder decomposition.

######  Definition 3.4.

A Banach space $X$ with a Schauder decomposition $(X_{n})_{n=1}^{\infty}$ is
said to have the scalar-plus-horizontally approximable property if every
bounded linear operator $T$ on $X$ is of the form $\lambda I_{X}+A$, for some
scalar $\lambda$ and $A\in\mathcal{HA}(X)$, where $I_{X}:X\to X$ is the
identity operator.

Note that if $X$ has the scalar-plus-horizontally approximable property, then
the Schauder decomposition $(X_{n})_{n=1}^{\infty}$ is shrinking.

The following is the culmination of Sections 4, 5, and 6. Specifically, its
statement is a combination of Proposition 6.1 and Proposition 6.2.

######  Theorem 3.5.

There exists a Banach space $\mathfrak{X}_{\mathrm{AH}}$ with the scalar-plus-
compact property and a 2-Bourgain-Delbaen-$\mathscr{L}_{\infty}$-sum
$\mathcal{Z}_{\mathcal{K}(c_{0})}$ of countably many copies of
$\mathfrak{X}_{\mathrm{AH}}$ that has the scalar-plus-horizontally
approximable property.

The following is the main result of this paper. Given the existence of the
space from Theorem 3.5, its proof is a direct consequence of the tools and
language developped so far.

######  Theorem 3.6.

The Calkin algebra of $\mathcal{Z}_{\mathcal{K}(c_{0})}$ is isomorphic to the
unitization of $\mathcal{K}(c_{0})$ as a Banach algebra. More precisely, after
passing to a $2$-equivalent norm of $\mathcal{Z}_{\mathcal{K}(c_{0})}$,
$\mathpzc{Cal}(\mathcal{Z}_{\mathcal{K}(c_{0})})$ contains a two-sided ideal
of codimension one that is isometrically isomorphic to $\mathcal{K}(c_{0})$ as
a Banach algebra.

###### Proof.

Because $\mathcal{Z}_{\mathcal{K}(c_{0})}$ has the scalar-plus-horizontally
approximable property,
$\mathcal{HA}(\mathcal{Z}_{\mathcal{K}(c_{0})})/\mathcal{K}(\mathcal{Z}_{\mathcal{K}(c_{0})})$
is a two-sided ideal of codimension one in the space
$\mathpzc{Cal}(\mathcal{Z}_{\mathcal{K}(c_{0})})$. By Proposition 3.3 and
Proposition 2.9, after passing to a 2-equivalent norm of
$\mathcal{Z}_{\mathcal{K}(c_{0})}$,
$\mathcal{HA}(\mathcal{Z}_{\mathcal{K}(c_{0})})/\mathcal{K}(\mathcal{Z}_{\mathcal{K}(c_{0})})$
is isometrically isomorphic to $\mathcal{HA}(c_{0}(X))/\mathcal{K}(c_{0}(X))$
which, by Proposition 2.6, is isometrically isomorphic to
$\mathcal{K}(c_{0})$. ∎

######  Remark 3.7.

The above theorem does not state that, after passing to an equivalent norm,
the Calkin algebra of $\mathcal{Z}_{\mathcal{K}(c_{0})}$ coincides
isometrically with the natural unitization of $\mathcal{K}(c_{0})$ inside
$\mathcal{L}(c_{0})$, i.e., $\mathbb{C}I_{c_{0}}\oplus\mathcal{K}(c_{0})$.

##  4\. Argyros-Haydon sums of a sequence of Banach spaces

In this section we repeat the definition of an Argyros-Haydon sum of a
sequence of Banach space $(X_{n})_{n=1}^{\infty}$ from [21, Sections 4 and 5].
Although the presented definition is essentially the same, there are some
mild, mostly notational, differences that we briefly elaborate on in Remark
4.2 and in the paragraph before Theorem 5.2. Specifically, instead of
considering a sequence of Banach spaces $X_{n}$, $n\in\mathbb{N}$, each with a
norming sequence $(f_{i}^{n})_{i=1}^{\infty}$, we consider a sequence of
closed subspaces $(X_{n})_{n=1}^{\infty}$ of $\ell_{\infty}$, each with the
standard norming sequence of the coordinate functionals on $\ell_{\infty}$.

Let $(m_{j})_{j=1}^{\infty}$ and $(n_{j})_{j=1}^{\infty}$ be increasing
sequences of positive integers.

######  Assumption 4.1.

We assume that $(m_{j})_{j=1}^{\infty}$, $(n_{j})_{j=1}^{\infty}$ satisfy the
following:

  1. ($\alpha$)

$m_{1}\geq 4$.

  2. ($\beta$)

$m_{j+1}\geq m_{j}^{2}$.

  3. ($\gamma$)

$n_{1}\geq m_{1}^{2}$.

  4. ($\delta$)

$n_{j+1}\geq(16n_{j})^{\log_{2}(m_{j+1})}$.

The sequence of pairs $(m_{j},n_{j})_{j=1}^{\infty}$ will be referred to as a
set of parameters because it is used to define a sequence of disjoint finite
sets $(\Delta_{n})_{n=1}^{\infty}$ that depend on it. This definition also
includes additional information encoded into a finite collection of partial
functions from $\Gamma=\cup_{n=1}^{\infty}\Delta_{n}$ to some sets. All this
information is then used to define, for a sequence $(X_{n})_{n=1}^{\infty}$ of
closed subspaces of $\ell_{\infty}$, a specific Bourgain-
Delbaen-$\mathscr{L}_{\infty}$-sum

$\mathcal{Z}=\mathcal{Z}\big{(}(m_{j},n_{j})_{j=1}^{\infty},(X_{n})_{n=1}^{\infty}\big{)}\subset\mathcal{W}=\big{(}\oplus_{n=1}^{\infty}(X_{n}\oplus\ell_{\infty}(\Delta_{n}))_{\infty}\big{)}_{\infty}$

of $(X_{n})_{n=1}^{\infty}$. We will refer to such a space $\mathcal{Z}$ as an
Argyros-Haydon sum of $(X_{n})_{n=1}^{\infty}$ with parameters
$(m_{j},n_{j})_{j=1}^{\infty}$.

###  4.1. Definition of the sets $(\Delta_{n})_{n=1}^{\infty}$

We construct disjoint finite sets $\Delta_{n}$, $n\in\mathbb{N}$, depending on
the pair of sequences $(m_{j})_{j=1}^{\infty}$ and $(n_{j})_{j=1}^{\infty}$
alongside a finite collection of partial functions from
$\Gamma=\cup_{n=1}^{\infty}\Delta_{n}$ to some sets.

Let $\Delta_{1}^{0}=\emptyset$, $\Delta_{1}^{1}=\\{(1)\\}$, and
$\Delta_{1}=\Delta_{1}^{0}\cup\Delta_{1}^{1}$. For $\gamma=(1)$ we let
$\mathrm{rank}(\gamma)=1$ and $\sigma(\gamma)=2$. Assume we have defined
disjoint sets $\Delta_{1},\ldots,\Delta_{n}$ such that, for $1\leq k\leq n$,
$\Delta_{k}$ is the disjoint union of two sets $\Delta_{k}^{0}$ and
$\Delta_{k}^{1}$. Denote $\Gamma_{n}=\cup_{k=1}^{n}\Delta_{n}$ and
$\Gamma^{1}_{n}=\cup_{k=1}^{n}\Delta^{1}_{n}$. Assume that, for
$\gamma\in\Gamma_{1}^{1}$ we have defined $\mathrm{rank}(\gamma)$,
$\mathrm{cut}(\gamma)$, and $\mathrm{age}(\gamma)$ in $\\{1,\ldots,n\\}$,
$\mathrm{weight}(\gamma)\in\\{m_{1}^{-1},\ldots,m_{2(n-1)}^{-1}\\}$, and
$\mathrm{base}(\gamma)$, $\mathrm{top}(\gamma)$ in sets that will be specified
in the inductive step. Assume we have also defined an injection
$\sigma:\Gamma^{1}_{n}\to\mathbb{N}$ such that, for $\gamma\in\Gamma_{n}^{1}$,
$\sigma(\gamma)>\mathrm{rank}(\gamma)$.

We fix $N(n)\in\mathbb{N}$ sufficiently large, to be determined soon. Define

$\displaystyle G_{n}$ $\displaystyle=\big{\\{}pe^{i2\pi
q}:p,q\in[0,1]\cap\mathbb{Q},\;p(N(n)!)\in\mathbb{Z},\text{ and
}q(N(n)!)\in\mathbb{Z}\big{\\}}$ $\displaystyle K_{n}$
$\displaystyle=\Big{\\{}f=\big{(}(f_{k}(i))_{i=1}^{n}\big{)}_{k=1}^{n}\in\prod_{k=1}^{n}G_{n}^{n}:\sum_{k=1}^{n}\sum_{i=1}^{n}|f_{k}(i)|\leq
1\Big{\\}},\text{ and}$ $\displaystyle B_{n}$
$\displaystyle=\Big{\\{}b^{*}=\big{(}(b_{k}^{*}(\gamma))_{\gamma\in\Delta_{k}}\big{)}_{k=1}^{n}\in\prod_{k=1}^{n}G_{n}^{\Delta_{k}}:\sum_{k=1}^{n}\sum_{\gamma\in\Delta_{k}}|b_{k}^{*}(\gamma)|\leq
1\Big{\\}}.$

For $1\leq k\leq n$ and $\xi\in\Delta_{k}$, we let $e_{\xi}^{*}$ denote the
member $\big{(}(b_{k}^{*}(\gamma))_{\gamma\in\Delta_{k}}\big{)}_{k=1}^{n}$ of
$B_{n}$ such that $b^{*}_{k}(\xi)=1$ and it has he value zero in all other
entries.

Note that $G_{n}$ is a $4/(N(n)!)$-net of the complex unit disk, and
therefore, $B_{n}$ is a $(7|\Gamma_{n}|)/(N(n)!)$-net of the unit ball of
$(\oplus_{r=1}^{n}\ell_{1}(\Delta_{r}))_{1}$. We demand that $N(n)$ has been
chosen sufficiently large such that $B_{n}$ is a $1/2^{n}$-net in the unit
ball of $(\oplus_{k=1}^{n}\ell_{1}(\Delta_{k}))_{1}$. Similarly, we demand
that $K_{n}$ is a $1/2^{n}$ net in the unit ball of
$(\oplus_{k=1}^{n}\ell_{1}^{n})_{1}$. We also demand $N(n)\geq N(n-1)$, and
thus $G_{n}\supset G_{n-1}$, (which is the sole purpose of the factorial).

We define

$\Delta_{n+1}^{0}=\\{n+1\\}\times K_{n}.$

For $\gamma\in\Delta_{n+1}^{0}$, we use the following notation.

  1. (1)

Let $\mathrm{rank}(\gamma)=n+1$ denote the first coordinate of $\gamma$.

  2. (2)

Let $\mathrm{top}(\gamma)=f\in K_{n}$ denote the second coordinate of
$\gamma$.

To define $\Delta_{n+1}^{1}$, we first consider the set

$\Delta_{n+1}^{1,\mathrm{aux}}=\\{n+1\\}\times\big{(}\\{\emptyset\\}\cup\Gamma^{1}_{n}\setminus\Gamma_{1}^{1}\big{)}\times\\{m^{-1}_{1},\ldots,m^{-1}_{2n}\\}\times\\{1,\dots,n-1\\}\times
B_{n}.$

For $\gamma\in\Delta_{n+1}^{1,\mathrm{aux}}$, we use the following notation.

  1. (3)

Let $\mathrm{rank}(\gamma)=n+1$ denote the first coordinate of $\gamma$.

  2. (4)

Let $\mathrm{base}(\gamma)$ denote the second coordinate of $\gamma$ and note

     * •

either $\mathrm{base}(\gamma)=\xi\in\Gamma^{1}_{n}\setminus\Gamma_{1}^{1}$

     * •

or $\mathrm{base}(\gamma)=\emptyset$, in which case we say it is undefined.

  3. (5)

Let $\mathrm{weight}(\gamma)\in\\{m^{-1}_{1},\ldots,m^{-1}_{2n}\\}$ denote the
third coordinate of $\gamma$.

  4. (6)

Let $\mathrm{cut}(\gamma)=p\in\\{1\,\ldots,n-1\\}$ denote the fourth
coordinate of $\gamma$.

  5. (7)

Let $\mathrm{top}(\gamma)=b^{*}\in B_{n}$ denote the last coordinate of
$\gamma$.

  6. (8)

Let $\mathrm{age}(\gamma)$ be defined as follows.

     * •

If $\mathrm{base}(\gamma)$ is undefined let $\mathrm{age}(\gamma)=1$.

     * •

If $\mathrm{base}(\gamma)=\xi$ let $\mathrm{age}(\gamma)=\mathrm{age}(\xi)+1$.

We define $\Delta_{n+1}^{1,\mathrm{even}}$ as all
$\gamma\in\Delta_{n+1}^{1,\mathrm{aux}}$ such that, for some $1\leq j\leq n$,
$\mathrm{weight}(\gamma)=m_{2j}^{-1}$ and

  1. ($\alpha$)

either $\mathrm{base}(\gamma)$ is undefined or,

  2. ($\beta$)

if $\mathrm{base}(\gamma)=\xi\in\Gamma_{n}^{1}\setminus\Delta_{1}^{1}$, then
$\mathrm{weight}(\xi)=m_{2j}^{-1}$ and $\mathrm{age}(\gamma)<n_{2j}$.

We define $\Delta_{n+1}^{1,\mathrm{odd}}$ as all
$\gamma\in\Delta_{n+1}^{\mathrm{aux}}$ such that, for some $1\leq j\leq n$,
$\mathrm{weight}(\gamma)=m_{2j-1}^{-1}$ and

  1. ($\gamma$)

either $\mathrm{base}(\gamma)$ is undefined and
$\mathrm{top}(\gamma)=e_{\eta}^{*}$, for some $\eta\in\Gamma^{1}$ satisfying
$\mathrm{weight}(\eta)=m^{-1}_{4i-2}<m_{2j-1}^{-2}$ or,

  2. ($\delta$)

if $\mathrm{base}(\gamma)=\xi\in\Gamma_{n}^{1}\setminus\Gamma_{1}^{1}$, then
$\mathrm{weight}(\xi)=m_{2j-1}^{-1}$ and $\mathrm{age}(\gamma)<n_{2j}$, and
$\mathrm{top}(\gamma)=e_{\eta}^{*}$, for some $\eta\in\Gamma^{1}$ satisfying
$\mathrm{weight}(\eta)=m_{4\sigma(\xi)}$.

Define
$\Delta_{n+1}^{1}=\Delta_{n+1}^{1,\mathrm{even}}\cup\Delta_{n+1}^{1,\mathrm{odd}}$,
$\Delta_{n+1}=\Delta_{n+1}^{0}\cup\Delta_{n+1}^{1}$ and, letting
$\Gamma_{n+1}^{1}=\cup_{k=1}^{n+1}\Gamma_{k}^{1}$, extend $\sigma$ to an
injective function $\sigma:\Gamma^{1}_{n+1}\to\mathbb{N}$ such that, for
$\gamma\in\Gamma_{n+1}^{1}$, $\sigma(\gamma)>\mathrm{rank}(\gamma)$.

We let $\Gamma=\cup_{k=1}^{\infty}\Delta_{k}$ and note the dependence on
$(m_{j})_{j=1}^{\infty}$ and $(n_{j})_{j=1}^{\infty}$. We also denote
$\Gamma^{0}=\cup_{k=1}^{\infty}\Delta_{k}^{0}$ and
$\Gamma^{1}=\cup_{k=1}^{\infty}\Delta_{k}^{1}$.

######  Remark 4.2.

In [21], the set $\Gamma$ does not only depend on
$(m_{j},n_{j})_{j=1}^{\infty}$, but also on a sequence of Banach spaces
$(X_{n})_{n=1}^{\infty}$ and it incorporates in its Definition some
information about them, via some norming functionals on them. We have replaced
this with the sets $K_{n}$, $n\in\mathbb{N}$, which encode the usual action of
absolute convex combinations of the usual norming functionals
$(e_{i}^{*})_{i=1}^{\infty}$ on $\ell_{\infty}$, and thus on all its
subspaces.

###  4.2. Definition of the space
$\mathcal{Z}=\mathcal{Z}\big{(}(m_{j},n_{j})_{j=1}^{\infty},(X_{n})_{n=1}^{\infty}\big{)}$

For $n\in\mathbb{N}$, let $X_{n}$ be a closed subspace of $\ell_{\infty}$.
Denote

$\mathcal{W}=\big{(}\oplus_{n=1}^{\infty}\big{(}X_{n}\oplus\ell_{\infty}(\Delta_{n})\big{)}_{\infty}\big{)}_{\infty}.$

We will inductively define linear extension maps
$i_{n,n+1}:\mathcal{W}_{n}\to\mathcal{W}_{n+1}$ (i.e.,
$r_{n}i_{n,n+1}:\mathcal{W}_{n}\to\mathcal{W}_{n}$ is the identity map), such
that additionally $\pi_{X_{n+1}}i_{n,n+1}=0$. Assuming that we have defined
$i_{1,2}$,…,$i_{n-1,n}$, we will define
$i_{n,n+1}:\mathcal{W}_{n}\to\mathcal{W}_{n+1}$ by first specifying linear
functionals $c_{\gamma}^{*}:\mathcal{W}\to\mathbb{C}$,
$\gamma\in\Delta_{n+1}$, and, for
$w=\big{(}(x_{1},y_{1}),\ldots,(x_{n},y_{n})\big{)}\in\mathcal{W}_{n}\subset\mathcal{W}$
put

$i_{n,n+1}(w)=\Big{(}(x_{1},y_{1}),\ldots,(x_{n},y_{n}),\big{(}(0,(c_{\gamma}^{*}(w))_{\gamma\in\Delta_{n+1}}\big{)}\Big{)}.$

Denote, for $m<n$, $i_{m,n}=i_{n-1,n}\circ i_{n-2,n-1}\circ\cdots\circ
i_{m,m+1}:\mathcal{W}_{m}\to\mathcal{W}_{n}$ and
$i_{n,n}:\mathcal{W}_{n}\to\mathcal{W}_{n}$ the identity map. We canonically
identify each $b^{*}=((b_{k}^{*}(\gamma))_{\gamma\in\Delta_{k}})_{k=1}^{n}$ in
$B_{n}$ with with a linear functional $b^{*}:\mathcal{W}\to\mathbb{C}$ given
by

(4.1) $b^{*}\big{(}(x_{k},y_{k})_{k=1}^{\infty}\big{)}=\sum_{k=1}^{n}\langle
b_{k}^{*},y_{k}\rangle$

and each $f=\big{(}(f_{k}(i))_{i=1}^{n}\big{)}_{k=1}^{n}\in K_{n}$, is
identified with a linear functional $f:\mathcal{W}\to\mathbb{C}$ given by

(4.2)
$f\big{(}(x_{k},y_{k})_{k=1}^{\infty}\big{)}=\sum_{k=1}^{n}\sum_{i=1}^{n}f_{k}(i)e_{i}^{*}(x_{k}),$

where $(e_{i}^{*})_{i=1}^{\infty}$ are the standard coordinate functionals of
$\ell_{\infty}$.

Let $\gamma\in\Delta_{n+1}$.

  1. (i)

If $\gamma\in\Delta_{n+1}^{0}$ with $\mathrm{top}(\gamma)=f\in K_{n}$, using
the identification (4.2), define

$c_{\gamma}^{*}=f.$

  2. (ii)

If $\gamma\in\Delta_{n+1}^{1}$ is as in ($\alpha$) or ($\gamma$), with
$\mathrm{weight}(\gamma)=m_{j}^{-1}$,
$\mathrm{cut}(\gamma)=p\in\\{1,\ldots,n-1\\}$, and
$\mathrm{top}(\gamma)=b^{*}\in B_{n}$, using (4.1), define

$c_{\gamma}^{*}=\frac{1}{m_{j}}\big{(}b^{*}-b^{*}i_{p,n}r_{p}\big{)}.$

  3. (iii)

If $\gamma\in\Delta_{n+1}^{1}$ is as in ($\beta$) or ($\delta$), with
$\mathrm{base}(\gamma)=\xi\in\Gamma_{n}^{1}$,
$\mathrm{weight}(\gamma)=m_{j}^{-1}$,
$\mathrm{cut}(\gamma)=p\in\\{1,\ldots,n-1\\}$, and
$\mathrm{top}(\gamma)=b^{*}\in B_{n}$, using (4.1), define

$c_{\gamma}^{*}=e_{\xi}^{*}+\frac{1}{m_{j}}\big{(}b^{*}-b^{*}i_{p,n}r_{p}\big{)}.$

By [21, Proposition 5.1], for every $m\leq n$, $\|i_{m,n}\|\leq 2$. We may
therefore, for $n\in\mathbb{N}$, define $i_{n}:\mathcal{W}_{n}\to\mathcal{W}$
such that, for
$z=\big{(}(x_{1},y_{1}),\ldots,(x_{n},y_{n})\big{)}\in\mathcal{W}_{n}$,

$i_{n}(z)=\big{(}(x_{1},y_{1}),\ldots,(x_{n},y_{n}),(0,z_{n+1}),(0,z_{n+2}),\ldots\big{)},$

where, for $k\geq n$,

$r_{k+1}i_{n}(z)=i_{k,k+1}r_{k}(z)=i_{k,k+1}\big{(}(x_{1},y_{1}),\ldots,(x_{n},y_{n}),(0,z_{n+1}),\ldots,(0,z_{k})\big{)}.$

Given this, it is not hard to see that the assumptions of Definition 3.1 are
satisfied, and thus, letting, for $n\in\mathbb{N}$,
$\mathcal{Z}_{n}=i_{n}(X_{n}\oplus\ell_{\infty}(\Delta_{n}))_{\infty}$, the
space

$\mathcal{Z}\big{(}(m_{j},n_{j})_{j=1}^{\infty},(X_{n})_{n=1}^{\infty}\big{)}=\overline{\langle\cup_{n=1}^{\infty}\mathcal{Z}_{n}\rangle}$

is a 2-Bourgain-Delbaen-$\mathscr{L}_{\infty}$ sum of
$(X_{n})_{n=1}^{\infty}$. We refer to this space as the Argyros-Haydon sum of
$(X_{n})_{n=1}^{\infty}$ with parameters $(m_{j},n_{j})_{j=1}^{\infty}$.

The role of the set $\Gamma^{0}$ is the following.

######  Remark 4.3.

For every $x\in\mathcal{Z}$ and $\varepsilon>0$ there exists $\gamma\in\Gamma$
such that $|e_{\gamma}^{*}(x)|\geq\|x\|-\varepsilon$. This was, essentially,
shown in [21, Lemma 5.7], but we repeat the brief argument. Let
$z=(x_{k},y_{k})_{k=1}^{\infty}\in\mathcal{Z}$ and note
$\|z\|=\max\\{\sup_{k}\|x_{k}\|,\sup_{k}\|y_{k}\|\\}$. If this maximum is
obtained by the second term, then the conclusion is obvious. Otherwise assume
that, for some $k_{0}\in\mathbb{N}$, $\|z\|<\|x_{k_{0}}\|+\varepsilon/2$. Pick
$i_{0}\in\mathbb{N}$ such that
$|e_{i_{0}}^{*}(x_{k_{0}})|>\|x_{k_{0}}\|-\varepsilon/2$. Let
$n=\max\\{k_{0},i_{0}\\}$ and $f=((f_{k}(i))_{i=1}^{n})_{k=1}^{n}$ such that
$f_{k_{0}}(i_{0})=1$, and all other entries are zero. Then, for
$\gamma=(n+1,f)\in\Delta_{n+1}^{0}$,
$|e_{\gamma}^{*}(z)|=|e_{i_{0}}^{*}(x_{k_{0}})|>\|z\|-\varepsilon$.

The following is proved in [21, Corollary 5.15].

######  Theorem 4.4.

The Schauder decomposition $(\mathcal{Z}_{n})_{n=1}^{\infty}$ of the space
$\mathcal{Z}=\mathcal{Z}\big{(}(m_{j},n_{j})_{j=1}^{\infty},(X_{n})_{n=1}^{\infty}\big{)}$
is shrinking. In particular, $\mathcal{Z}^{*}$ is $2$-isomorphic to
$\big{(}\oplus_{n=1}^{\infty}(X_{n}^{*}\oplus\ell_{1}(\Delta_{n}))_{1}\big{)}_{1}$.

###  4.3. Evaluation analysis of $e_{\gamma}^{*}$

The evaluation analysis of a coordinate functional $e_{\gamma}^{*}$ is a
central concept invented in [2], and it has been used in all Argyros-Haydon
constructions.

For $\gamma\in\Delta_{1}$ let $c_{\gamma}^{*}=0$, and for $\gamma\in\Gamma$,
let $d_{\gamma}^{*}=e_{\gamma}^{*}-c^{*}_{\gamma}$. For $\gamma\in\Delta_{1}$,
$e_{\gamma}^{*}=d_{\gamma}^{*}$ while, for $\gamma\in\Gamma^{0}$ with
$\mathrm{top}(\gamma)=f\in K_{n}$, it directly follows
$e_{\gamma}^{*}=d_{\gamma}^{*}+f$.

The following can be proved easily by induction on the age of a $\gamma$ (see,
e.g., [21, Proposition 5.3] or [2, Proposition 4.5]).

######  Proposition 4.5 (Evaluation analysis).

For $\gamma\in\Gamma^{1}$ with $\mathrm{rank}(\gamma)>1$,

$e_{\gamma}^{*}=\sum_{r=1}^{a}d_{\xi_{r}}^{*}+\frac{1}{m_{j}}\sum_{r=1}^{a}b_{r}^{*}\circ
P_{(p_{r},q_{r}]},$

where $a=\mathrm{age}(\gamma)$, $m_{j}^{-1}=\mathrm{weigh}(\gamma)$,
$\xi_{a}=\gamma$, for $1\leq r<a$, $\xi_{r}=\mathrm{base}(\xi_{r+1})$, and,
for $1\leq r\leq a$, $b_{r}^{*}=\mathrm{top}(\xi_{r})$,
$p_{r}=\mathrm{cut}(\xi_{r})$, $q_{r}=\mathrm{rank}(\xi_{r})-1$. Furthermore,
for $1\leq r\leq a$, $p_{r}<q_{r}$ and, for $1\leq r<a$, $q_{r}+1<p_{r+1}$.

It is crucial to have an abundance of available coordinate functionals to
achieve lower bounds for linear combinations of certain vectors. This is the
purpose of the following, and it was proved in [2, Proposition 4.7].

######  Proposition 4.6.

Let $j\in\mathbb{N}$, $1\leq a\leq n_{2j}$, and $(p_{r},q_{r})_{r=1}^{a}$ be
pairs of positive integers such that $2j\leq p_{r}<q_{r}$, $1\leq r\leq a$,
and $p_{r-1}+1<q_{r}$, $2\leq r\leq a$. Let also $b_{r}^{*}\in B_{q_{r}}$,
$1\leq r\leq a$. Then, there exist $\xi_{r}\in\Delta^{1}_{q_{r}+1}$, $1\leq
r\leq a$, and $\gamma\in\Gamma^{1}_{q_{a}+1}$ that has evaluation analysis

$e_{\gamma}^{*}=\sum_{r=1}^{a}d_{\xi_{r}}^{*}+\frac{1}{m_{2j}}\sum_{r=1}^{a}b_{r}^{*}\circ
P_{(p_{r},q_{r}]}.$

##  5\. Operators on Argyros-Haydon sums of Banach spaces

Here we recall one of the main results from [21], namely that under
appropriate assumptions a space
$\mathcal{Z}=\mathcal{Z}\big{(}(m_{j},n_{j})_{j=1}^{\infty},(X_{n})_{n=1}^{\infty}\big{)}$
satisfies the scalar-plus-horizontally compact property. We then study
conditions under which an operator $T:\mathcal{Z}^{(1)}\to\mathcal{Z}^{(2)}$,
where $\mathcal{Z}^{(1)}$, $\mathcal{Z}^{(2)}$ are two different Argyros-
Haydon sums, is “horizontally small”.

######  Definition 5.1.

Let $X$ be a Banach space with a Schauder decomposition
$(X_{n})_{n=1}^{\infty}$ and associated projections $(P_{n})_{n=1}^{\infty}$.

  1. (a)

A sequence $(x_{k})_{k=1}^{\infty}$ is called horizontally block if there
exist successive intervals $I_{k}$, $k\in\mathbb{N}$, of $\mathbb{N}$ such
that, for all $k\in\mathbb{N}$, $x_{k}=P_{I_{k}}x_{k}$.

  2. (b)

For a Banach space $Y$, a bounded linear operator $T:X\to Y$ is called
horizontally compact if any of the following equivalent conditions holds:

    1. ($\alpha$)

Whenever $(x_{k})_{k=1}^{\infty}$ is a bounded horizontally block sequence in
$X$, $\lim_{k}\|Tx_{k}\|=0$.

    2. ($\beta$)

$T=\lim_{n}TP_{n}$, in operator norm.

The following, although not stated in this more general form, is proved in
[21, Proposition 7.8]. There, the assumption that every $T:\mathcal{Z}\to
X_{n}$ is horizontally compact is realized by demanding that each $X_{n}$
either has the Schur property, or $X_{n}^{*}$ contains no isomorphic copy of
$\ell_{1}$ (see [21, Proposition 3.6]). We note that in [21, page 21], the
author made additional assumptions about the norming functionals
$(f_{i}^{n})_{i=1}^{\infty}$ associated to each space $X_{n}$,
$n\in\mathbb{N}$ (see the introduction of Section 4 and Remark 4.2). These
additional assumptions are not a requirement in the proof and they were only
used to simplify a perturbation argument as in [2].

######  Theorem 5.2 ([21]).

Let $(m_{j},n_{j})_{j=1}^{\infty}$ satisfy 4.1, $(X_{n})_{n=1}^{\infty}$ be a
sequence of closed subspaces of $\ell_{\infty}$, and denote
$\mathcal{Z}=\mathcal{Z}\big{(}(m_{j},n_{j})_{j=1}^{\infty},(X_{n})_{n=1}^{\infty}\big{)}$.
Assume that, for every $n\in\mathbb{N}$, every bounded linear operator
$T:\mathcal{Z}\to X_{n}$ is horizontally compact. Then, for every bounded
linear operator $T:\mathcal{Z}\to\mathcal{Z}$, there exists
$\lambda\in\mathbb{C}$ such that $T-\lambda I_{\mathcal{Z}}$ is horizontally
compact.

######  Notation 5.3.

Let $(m_{j},n_{j})_{j=1}^{\infty}$ satisfy 4.1, $L$ be an infinite subset of
$\mathbb{N}$, and $(X_{n})_{n=1}^{\infty}$ be a sequence of closed subspaces
of $\ell_{\infty}$. When we write

$\mathcal{Z}\big{(}(m_{j},n_{j})_{j\in L},(X_{n})_{n=1}^{\infty}\big{)},$

we will mean
$\mathcal{Z}\big{(}(m^{\prime}_{j},n^{\prime}_{j})_{j=1}^{\infty},(X_{n})_{n=1}^{\infty}\big{)}$,
where $(m_{j}^{\prime},n_{j}^{\prime})_{j=1}^{\infty}$ is relabeling of
$(m_{j},n_{j})_{j\in L}$ given by the unique order preserving bijection
$\mathbb{N}\mapsto L$.

The following statement is a combination of [21, Corollary 5.12 and
Proposition 5.14]. The “in particular” part of item (ii) follows from [2,
Proposition 2.5].

######  Proposition 5.4.

Let $(m_{j},n_{j})_{j=1}^{\infty}$ satisfy 4.1, $L$ be an infinite subset of
$\mathbb{N}$, $(X_{n})_{n=1}^{\infty}$ be a sequence of closed subspaces of
$\ell_{\infty}$, and let $\mathcal{Z}=\mathcal{Z}\big{(}(m_{j},n_{j})_{j\in
L},(X_{n})_{n=1}^{\infty}\big{)}$. Then, there exists a class
$\mathcal{C}_{\mathrm{RIS}}$ of bounded horizontally block sequences
satisfying the following.

  1. (i)

For every bounded linear operator $T:\mathcal{Z}\to Y$, where $Y$ is a Banach
space, $T$ is horizontally compact if and only if for every
$(y_{k})_{k=1}^{\infty}$ in $\mathcal{C}_{\mathrm{RIS}}$,
$\lim_{k}\|Ty_{k}\|=0$.

  2. (ii)

For every $(y_{k})_{k=1}^{\infty}$ in $\mathcal{C}_{\mathrm{RIS}}$ there
exists $C>0$ such that the map $e_{k}\mapsto y_{k}$ extends to a bounded
linear operator of norm at most $C$ from the mixed-Tsirelson space
$T[(\mathscr{A}_{3n_{j}},m^{-1}_{j})_{j\in L}]$ to $\mathcal{Z}$.

In particular, for every $j\in\mathbb{N}\setminus L$ and positive integers
$k_{1}<\cdots<k_{n_{j}}$,

$\big{\|}\sum_{i=1}^{n_{j}}y_{k_{i}}\big{\|}\leq C\frac{n_{j}}{m_{j}^{2}}.$

The following is proved with essentially the same argument as in [2, Lemma
10.3].

######  Proposition 5.5.

Let $(m_{j},n_{j})_{j=1}^{\infty}$ satisfy 4.1 and let $L_{1}$, $L_{2}$ be
subsets of $\mathbb{N}$ such that $L_{1}\setminus L_{2}$ and $L_{2}\setminus
L_{1}$ are infinite. Let $(X^{(1)}_{n})_{n=1}^{\infty}$,
$(X^{(2)}_{n})_{n=1}^{\infty}$ be sequences of closed subspaces of
$\ell_{\infty}$. For $i=1,2$, let
$\mathcal{Z}^{(i)}=\mathcal{Z}\big{(}(m_{j},n_{j})_{j\in
L_{i}},(X_{n}^{(i)})_{n=1}^{\infty}\big{)}$ and denote its associated sequence
of projections $(P_{n}^{(i)})_{n=1}^{\infty}$. For a bounded linear operator
$T:\mathcal{Z}^{(1)}\to\mathcal{Z}^{(2)}$, the following hold.

  1. (i)

If, for every $n\in\mathbb{N}$, $TP_{\\{n\\}}^{(1)}$ is compact then
$\lim_{n}\|T-P_{n}^{(2)}T\|=0$.

  2. (ii)

If, for every $n\in\mathbb{N}$, $P_{\\{n\\}}^{(2)}T$ is compact then
$\lim_{n}\|T-TP_{n}^{(1)}\|=0$, i.e., $T$ is horizontally compact.

###### Proof.

We will only prove the first assertion; the second one is similar and somewhat
simpler. Denote $\mathcal{C}^{(1)}_{\mathrm{RIS}}$ the class of bounded
horizontally block sequences in $\mathcal{Z}^{(1)}$ given by Proposition 5.4.

######  Claim.

For every $(y_{k})_{k=1}^{\infty}$ in $\mathcal{C}^{(1)}_{\mathrm{RIS}}$,
$\lim_{n}\sup_{k,m\geq n}\|P^{(2)}_{(m,\infty)}Ty_{k}\|=0$.

We will first use the claim to deduce $\lim_{n}\|P^{(2)}_{(n,\infty)}T\|=0$,
as desired. Assume this is false. Because, for each $n\in\mathbb{N}$,
$TP_{\\{n\\}}^{(1)}$ is compact it easily follows that, for $n\in\mathbb{N}$,
$\lim_{k}\|P_{(k,\infty)}^{(2)}TP_{n}^{(1)}\|=0$. Therefore, for some
$\varepsilon>0$ and for all $n\in\mathbb{N}$,
$\limsup_{k}\|P^{(2)}_{(k,\infty)}TP^{(1)}_{(n,\infty)}\|>\varepsilon$. It is
thus easy find a bounded horizontally block sequence $(z_{n})_{n=1}^{\infty}$
and a strictly increasing sequence of positive integers
$(k_{n})_{n=1}^{\infty}$ such that, for all $n\in\mathbb{N}$,
$\|P^{(2)}_{(k_{n},\infty)}Tz_{n}\|>\varepsilon/2$. For every
$n\in\mathbb{N}$, fix $f_{n}$ in the unit ball of $\mathcal{Z}^{(2)}$ such
that $|f_{n}(P^{(2)}_{(k_{n},\infty)}Tz_{n})|>\varepsilon/2$ and define the
bounded linear operator $S:\mathcal{Z}^{(1)}\to c_{0}$ given by
$Sx=(f_{n}(P^{(2)}_{(k_{n},\infty)}Tx))_{n=1}^{\infty}$. This is well defined
because $P_{(k_{n},\infty)}T$, $n\in\mathbb{N}$, converges to zero in the
strong operator topology. The sequence $(z_{n})_{n=1}^{\infty}$ witnesses the
non-horizontal compactness of $S$, therefore, there exists
$(y_{m})_{m=1}^{\infty}$ in $\mathcal{C}^{(1)}_{\mathrm{RIS}}$ such that
$\limsup_{m}\|Sy_{m}\|>0$. Because the Schauder decomposition of
$\mathcal{Z}^{(1)}$ is shrinking, $(y_{m})_{m=1}^{\infty}$ is weakly null.
Therefore, we deduce $\lim_{n}\sup_{\ell,m\geq
n}|f_{\ell}(P_{(k_{\ell},\infty)}^{(2)}Ty_{m})|>0$, and in particular,

$\lim_{n}\sup_{k,m\geq n}\|P_{(k,\infty)}^{(2)}Ty_{m}\|>0,$

contradicting the claim.

We proceed to prove the claim now. Fix a sequence $(y_{k})_{k=1}^{\infty}$ in
$\mathcal{C}^{(1)}_{\mathrm{RIS}}$ and let $C$ be the constant given by
Proposition 5.4 (ii). Assume, towards contradiction, that, for some
$\varepsilon>0$, $\lim_{n}\sup_{m,k\geq
n}\|P^{(2)}_{(m,\infty)}y_{k}\|>\varepsilon$. We will prove that, for
arbitrary $j\in L_{2}\setminus L_{1}$, that has even position in the
increasing order of $L_{2}$, $\|T\|\geq\varepsilon m_{j}/(4C)$, which, by 4.1
would be absurd. We denote $\Gamma$ the set associated with the construction
of $\mathcal{Z}^{(2)}$. After passing to a subsequence of
$(y_{k})_{k=1}^{\infty}$ and relabeling, there are $\eta_{k}\in\Gamma$, $j\leq
p_{k}<q_{k}<p_{k+1}-1\in\mathbb{N}$, $\lambda_{k}\in G_{q_{k}}$,
$k\in\mathbb{N}$, such that

$\mathfrak{Re}\Big{(}\lambda_{k}e_{\eta_{k}}^{*}\big{(}P^{(2)}_{(p_{k},q_{k}]}Ty_{k}\big{)}\Big{)}>\varepsilon\text{
and }\big{\|}P_{(q_{k},\infty)}Ty_{k}\big{\|}<\frac{\varepsilon}{6n^{2}_{j}}.$

Because $(y_{k})_{k=1}^{\infty}$ is weakly null, we may further assume that,
for every $k\in\mathbb{N}$,

$\max_{1\leq
i<k}\Big{|}e_{\eta_{i}}^{*}\big{(}P^{(2)}_{(q_{i},p_{i}]}Ty_{k}\big{)}\Big{|}<\frac{\varepsilon
m_{j}}{6n^{2}_{j}}\text{ and
}\max_{\xi\in\Gamma_{q_{k-1}+1}}|d_{\xi}^{*}(Ty_{k})|<\frac{\varepsilon}{6n^{2}_{j}}.$

Because $j$ has even position in $L_{2}$, by Proposition 4.6 there exists
$\gamma\in\Gamma$ with evaluation analysis

$e_{\gamma}^{*}=\frac{1}{m_{j}}\sum_{k=1}^{n_{j}}\lambda_{k}e_{\eta_{k}}^{*}\circ
P_{(p_{k},q_{k}]}+\sum_{k=1}^{n_{j}}d_{\xi_{k}}^{*},$

for some $\xi_{k}\in\Delta_{q_{k}+1}^{1}$, $1\leq k\leq n_{j}$. We use this
$\gamma$ to bound from below the norm of $T$.

(5.1)
$\begin{split}\|T\|C\frac{n_{j}}{m^{2}_{j}}&\geq\|T\|\Big{\|}\sum_{k=1}^{n_{j}}y_{k}\Big{\|}\geq\Big{|}e_{\gamma}^{*}\Big{(}\sum_{k=1}^{n_{j}}Ty_{k}\Big{)}\Big{|}\\\
&\geq\underbrace{\Big{|}\sum_{k=1}^{n_{j}}e_{\gamma}^{*}\Big{(}P^{(2)}_{(p_{k},q_{k}]}Ty_{k}\Big{)}\Big{|}}_{=:\alpha}-\sum_{k=1}^{n_{j}}\underbrace{\Big{|}e_{\gamma}^{*}\Big{(}P^{(2)}_{[1,q_{k-1}+1]}Ty_{k}\Big{)}\Big{|}}_{=:\beta_{k}}\\\
&\phantom{geq}-\sum_{k=1}^{n_{j}}\underbrace{\Big{|}e_{\gamma}^{*}\Big{(}P^{(2)}_{(q_{k-1}+1,p_{k}]}Ty_{k}\Big{)}\Big{|}}_{=0}-\sum_{k=1}^{n_{j}}\underbrace{\Big{|}e_{\gamma}^{*}\Big{(}P^{(2)}_{(q_{k},\infty)}Ty_{k}\Big{)}\Big{|}}_{\leq\varepsilon/(6n_{2j})}.\end{split}$

We estimate $\alpha$ and the $\beta_{k}$ as follows:

(5.2)
$\alpha\geq\mathfrak{Re}\Big{(}\frac{1}{m_{j}}\sum_{k=1}^{n_{j}}\lambda_{k}e_{\eta_{k}}^{*}\big{(}P^{(2)}_{(p_{k},q_{k}]}Ty_{k})\Big{)}>\varepsilon\frac{n_{j}}{m_{j}},$

and, for $1\leq k\leq n_{j}$,

(5.3)
$\beta_{k}=\Big{|}\frac{1}{m_{j}}\sum_{i=1}^{k-1}\lambda_{i}e_{\eta_{i}}^{*}\Big{(}P_{(q_{i},p_{i}]}Ty_{k}\Big{)}+\sum_{i=1}^{k-1}d_{\xi_{i}}^{*}(Ty_{k})\Big{|}<\frac{\varepsilon}{6n_{j}}+\frac{\varepsilon}{6n_{j}}.$

Combining (5.1), (5.2), and (5.3), we deduce $\|T\|\geq\varepsilon
m_{j}/(4C)$. ∎

##  6\. The space $\mathcal{Z}_{\mathcal{K}(c_{0})}$

In this brief penultimate section we define the space
$\mathcal{Z}_{\mathcal{K}(c_{0})}$ and prove Theorem 3.5. Fix a sequence of
pairs $(m_{j},n_{j})_{j=1}^{\infty}$ satisfying 4.1 and let $L_{1}$, $L_{2}$
be disjoint infinite subsets of $\mathbb{N}$. Define

$\mathfrak{X}_{\mathrm{AH}}=\mathcal{Z}\big{(}(m_{j},n_{j})_{j\in
L_{1}},(\\{0\\})_{n=1}^{\infty}\big{)},$

i.e., the Argyros-Haydon sum of infinitely many copies of the zero subspace
with parameters $(m_{j},n_{j})_{j\in L_{1}}$. This Banach space is separable,
because its standard Schauder decomposition $(\mathcal{Z}_{n})_{n=1}^{\infty}$
consists of finite dimensional spaces. This space is practically the same as
the original Argyros-Haydon space from [2], but for completeness, we justify
its main property.

######  Proposition 6.1.

The space $\mathfrak{X}_{\mathrm{AH}}$ has the scalar-plus-compact property.

###### Proof.

For each $n\in\mathbb{N}$, every $T:\mathcal{Z}\to X_{n}=\\{0\\}$ is,
trivially, horizontally compact. Therefore, by Theorem 5.2, every bounded
linear operator is a scalar multiple of the identity plus a horizontally
compact operator. But, because the standard Schauder decomposition is finite
dimensional, horizontally compact operators are in fact compact. ∎

We fix, for each $n\in\mathbb{N}$, an isometric copy $X_{n}$ of
$\mathfrak{X}_{AH}$ in $\ell_{\infty}$ and define

$\mathcal{Z}_{\mathcal{K}(c_{0})}=\mathcal{Z}\big{(}(m_{j},n_{j})_{j\in
L_{2}},(X_{n})_{n=1}^{\infty}\big{)}.$

######  Proposition 6.2.

The space $\mathcal{Z}_{\mathcal{K}(c_{0})}$ has the scalar-plus-horizontally
approximable property.

###### Proof.

We first argue that $\mathcal{Z}_{\mathcal{K}(c_{0})}$ has the scalar-plus-
horizontally compact property. By the finite-dimensionality of the standard
Schauder decomposition of $\mathfrak{X}_{\mathrm{AH}}$, for every bounded
linear operator
$T:\mathcal{Z}_{\mathcal{K}(c_{0})}\to\mathfrak{X}_{\mathrm{AH}}$ the
assumption of Proposition 5.5 (ii) is satisfied, and therefore, $T$ is
horizontally compact. By Theorem 5.2, every bounded linear operator
$T:\mathcal{Z}_{\mathcal{K}(c_{0})}\to\mathcal{Z}_{\mathcal{K}(c_{0})}$ is a
scalar multiple of the identity plus a horizontally compact operator.

We will next show that horizontally compact operators on
$\mathcal{Z}_{\mathcal{K}(c_{0})}$ are horizontally approximable. Let
$T:\mathcal{Z}_{\mathcal{K}(c_{0})}\to\mathcal{Z}_{\mathcal{K}(c_{0})}$ be
horizontally compact; we will show $\lim_{n}\|P_{(m,\infty)}T\|=0$. If this is
false, because $T=\lim_{n}TP_{n}$, there is $n_{0}\in\mathbb{N}$ such that
$\limsup_{m}\|P_{(m,\infty)}TP_{n_{0}}\|>0$. Then, for some $1\leq n\leq
n_{0}$, $\limsup_{m}\|P_{(m,\infty)}TP_{\\{n\\}}\|>0$. Because
$\mathcal{Z}_{n}$ is isomorphich to
$\mathfrak{X}_{\mathrm{AH}}\oplus\ell_{\infty}(\Delta_{n})$, and $\Delta_{n}$
is finite, there exists a bounded linear operator
$S:\mathfrak{X}_{\mathrm{AH}}\to\mathcal{Z}_{\mathcal{K}(c_{0})}$ such that
$\liminf_{m}\|P_{(m,\infty)}S\|>0$. But, by the finite-dimensionality of the
Schauder decomposition of $\mathfrak{X}_{\mathrm{AH}}$, $S$ satisfies the
assumption of Proposition 5.5 (ii) and, therefore,
$\liminf_{m}\|P_{(m,\infty)}S\|=0$. ∎

######  Remark 6.3.

By Theorem 4.4, the dual of $\mathcal{Z}_{\mathcal{K}(c_{0})}$ is 4-isomorphic
to $\ell_{1}$. In particular, $\mathcal{Z}_{\mathcal{K}(c_{0})}$ has the
approximation property, and thus,
$\mathcal{K}(\mathcal{Z}_{\mathcal{K}(c_{0})})$ is the minimum non-trivial
closed ideal of $\mathcal{L}(\mathcal{Z}_{\mathcal{K}(c_{0})})$. Because
$\mathcal{K}(c_{0})$ is simple, it follows that the only non-trivial closed
ideals of $\mathcal{L}(\mathcal{Z}_{\mathcal{K}(c_{0})})$ are
$\mathcal{K}(\mathcal{Z}_{\mathcal{K}(c_{0})})$ and
$\mathcal{HA}(\mathcal{Z}_{\mathcal{K}(c_{0})})$.

##  7\. Open problems and directions

We share some open problems and directions that arise naturally from this
work. For additional questions of a similar nature we refer the reader to [14,
Section 8], [12, Section 9], and [13, Section 3].

A combination and refinement of techniques from [15] and this paper has
potential to answer the following.

######  Problem 1.

Let $K$ be a countable compact metric space. Is the unitization of
$\mathcal{K}(C(K))\equiv C(K;\ell_{1}(K))$ isomorphic as a Banach algebra to
the Calkin algebra of some Banach space?

Let $1<p<\infty$ and let $q$ denote its conjugate exponent. In the definition
of a Bourgain-Delbaen-$\mathscr{L}_{\infty}$-sum, one can replace the space
$\mathcal{W}$ with the space
$\big{(}\big{(}\oplus_{k=1}^{\infty}X_{n}\big{)}_{p}\oplus\big{(}\oplus_{n=1}^{\infty}\ell_{\infty}(\Delta_{n})\big{)}_{\infty}\big{)}_{\infty}$
to achieve a mixed Bourgain-Delbaen-$(p,\infty)$-sum of a sequence of Banach
spaces $(X_{n})_{n=1}^{\infty}$ (see [14] for a similar mixed sum). The
endeavour of studying the properties of such a sum may be useful in studying
the following. Let $X$ be a infinite dimensional Banach space with trivial
type (such as $c_{0}$ or $\ell_{1}$), and let $Y=\ell_{p}(X)$, considered with
its standard Schauder decomposition. Consider the subalgebra of
$\mathcal{HA}(Y)$ of all horizontally approximable operators with matrix
representation $A=(a_{m,n}I_{X})_{m,n=1}^{\infty}$. Then,

$\begin{split}\|A\|_{\mathcal{L}(Y)}&=\sup\Big{\\{}\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}|x_{n}y_{m}a_{m,n}|:\|(x_{n})_{n=1}^{\infty}\|_{p}\|(y_{m})_{m=1}^{\infty}\|_{q}\leq
1\Big{\\}}\\\
&=\|(|a_{m,n}|)_{m,n=1}^{\infty}\|_{\mathcal{L}(\ell_{p})}.\end{split}$

Put differently, this space is the completion of the space of finitely
supported scalar matrices in $\mathcal{K}(\ell_{p})$ with the smallest norm
dominating $\|\cdot\|_{\mathcal{L}(\ell_{p})}$ making the collection
$E_{i,j}=(\delta_{i}(m)\delta_{j}(n))_{m,n=1}^{\infty}$, $i,j\in\mathbb{N}$, a
1-unconditional basis. Let us denote this Banach algebra
$\mathcal{K}(\ell_{p})_{\mathrm{u}}$.

######  Problem 2.

For $1<p<\infty$, is $\mathcal{K}(\ell_{p})_{\mathrm{u}}$ isomorphic as a
Banach algebra to the Calkin algebra of some Banach space?

This can also be asked for the analogously defined space
$\mathcal{K}(V)_{\mathrm{u}}$, for any Banach space $V$ with an unconditional
basis. A mixed Bourgain-Delbaen-$(V,\infty)$ is the natural candidate for
solving this if $V$ does not contain $\ell_{1}$. Other types of mixed
Bourgain-Delbaen sums based on those in [14] may be able to address the case
when $\ell_{1}$ is a subspace of $V$.

2 can be seen as a variation of the more challenging goal of providing a
representation of an explicit infinite dimensional non-commutative
$C^{*}$-algebra, such as $\mathcal{K}(\ell_{2})$, as a Calkin algebra (see
[12, Section 9] for a detailed discussion on related problems posed by N. C.
Phillips). Some far simpler known examples are those in [12, Corollary 8.8],
and it was shown in [10] that every finite-dimensional semi-simple complex
algebra is a Calkin algebra.

######  Problem 3.

Is the unitization of $\mathcal{K}(\ell_{2})$ isomorphic as a Banach algebra
to the Calkin algebra of some Banach space?

More generally, if $X$ is a Banach space with a shrinking Schauder basis, is
$\mathcal{K}(X)$ isomorphic to the Calkin algebra of some Banach space?

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