# Band projections and order idempotents
in Banach lattice algebras

David Muñoz-Lahoz222david.munnozl (at) uam (dot) es Instituto de Ciencias
Matemáticas (CSIC-UAM-UC3M-UCM) Consejo Superior de Investigaciones
Científicas, C/ Nicolás Cabrera, 13–15, Campus de Cantoblanco UAM 28049,
Madrid, Spain.

###### Abstract

Motivated by recent work about band projections on spaces of regular operators
over a Banach lattice, given a Banach lattice algebra $A$, we will say an
element $a\in A_{+}$ is a band projection if the multiplication operator
$L_{a}R_{a}\in\mathcal{L}_{r}(A)$ is a band projection. Our aim in this note
is to explore the relations between this and the notion of order idempotent
(those elements $a$ in a Banach lattice algebra $A$ with identity $e$ such
that $0\leq a\leq e$ and $a^{2}=a$). We also revisit the properties of the
ideal generated by the identity on a Banach lattice algebra, motivated by
those of the centre of a Banach lattice.

Keywords Banach lattice algebra; band projection; order idempotent; regular
operator
MSC2020 46B42; 06F25; 46A32; 47A10; 47B65; 47L10

##  1 Introduction

This is a follow-up of the paper [12], where we studied band projections in
spaces of regular operators. Our aim here is to explore analogous results in
the more general context of Banach lattice algebras. Besides contributing to
the theory of these structures, this serves the purpose of showing that some
properties of spaces of regular operators hold for arbitrary Banach lattice
algebras.

Given a Banach lattice $X$, we can order the space of operators on $X$ using
the cone of positive operators, that is, setting an operator $T\colon X\to X$
to be positive if $T(X_{+})\subseteq X_{+}$. In general, however, not every
operator can be expressed as the difference of two positive operators, so the
positive cone need not be generating. Those operators that are in the span of
the positive cone are called regular operators.

The space of regular operators $\mathcal{L}_{r}(X)$ is an ordered vector
space, but it need not be a lattice. It is a remarkable result by Freudenthal,
Kantorovich and Riesz that when $X$ is order complete, $\mathcal{L}_{r}(X)$ is
again a Banach lattice. Since the composition of positive operators is again
positive, $\mathcal{L}_{r}(X)$ is also a Banach algebra.

So in $\mathcal{L}_{r}(X)$ a lattice and an algebraic structure coexist, with
the compatibility condition that the product of positive elements is positive.
In [12] we used this structure to study band projections on
$\mathcal{L}_{r}(X)$. More precisely, if
$\\{P_{\lambda}\\}_{\lambda\in\Lambda}$ is a family of pairwise disjoint band
projections on $X$, and $\Gamma\subseteq\Lambda\times\Lambda$, we defined for
positive operators $T\in\mathcal{L}_{+}(X)$ the map

$\mathcal{P}_{\Gamma}(T)=\bigvee_{(\alpha,\beta)\in\Gamma}P_{\alpha}TP_{\beta}.$
(1)

We showed that
$\mathcal{P}_{\Gamma}\colon\mathcal{L}_{+}(X)\to\mathcal{L}_{+}(X)$ extends to
a unique band projection on $\mathcal{L}_{r}(X)$, and called inner band
projections the band projections of this form.

Even though the definition of inner band projections uses only the lattice and
algebraic operations of $\mathcal{L}_{r}(X)$, the proof that Eq. 1 actually
defines a band projection relied heavily on the expression of the
Riesz–Kantorovich formulae for the lattice operations in $\mathcal{L}_{r}(X)$.
One of the main motivations for this paper is to determine whether we can
extend inner band projections to more general spaces in which a lattice and an
algebraic operation coexist, in a manner analogous to that of
$\mathcal{L}_{r}(X)$.

This leads to the notion of Banach lattice algebras. Formally, a Banach
lattice algebra is a Banach lattice together with a Banach algebra structure
in which the product of positive elements is positive. One of the most
important open problems in the theory of Banach lattice algebras is the
following:

######  Question 1.1 ([17, Question 5.3]).

Is every Banach lattice algebra isometrically (isomorphic) to a closed
subalgebra and sublattice of some space of regular operators?

It is relevant to this question, and therefore to the theory, to understand
which properties of spaces of regular operators depend only on the Banach
lattice algebra structure, and not on the fact that we are actually dealing
with operators. It is in this line of investigation where our paper fits in.

The first step towards an abstract formulation of inner band projections is to
determine which elements of a Banach lattice algebra $A$ should play the role
of the band projections $\\{P_{\lambda}\\}_{\lambda\in\Lambda}$. It is a
classical result that an operator $P\colon X\to X$ on a Banach lattice $X$ is
a band projection if and only if $P^{2}=P$ and $0\leq P\leq I_{X}$. When $A$
has an identity $e$, it seems then natural to regard the elements $p\in A$
satisfying $p^{2}=p$ and $0\leq p\leq e$ as the analogue of band projections.
These elements, called order idempotents, were already considered by E. A.
Alekhno in the context of ordered Banach algebras (see [4]). For instance,
Alekhno showed that the set of order idempotents forms a Boolean algebra, just
like the set of band projections on a Banach lattice does. We shall give an
alternative proof of this fact, and show that the set of order idempotents is
order closed.

But for this we will need to study the ideal generated by the identity element
in $A$. This ideal can be regarded as the abstract analogue of the center of
an order complete Banach lattice. We will collect results by L. Martignon
[10], C. B. Huijsmans [7] and E. Scheffold [14] to show that this ideal
posseses the most fundamental properties of the center: it is lattice and
algebra isometric to a space of continuous functions on a compact Hausdorff
space, it is a projection band, and it is an inverse closed subalgebra of $A$.

And what happens if we don’t have an identity? Can we still define elements
that play the role of band projections? Recently, it was proved in [12] that
band projections are characterized by their multiplication operators in the
following way: a positive operator $P\colon X\to X$ defined on an order
complete Banach lattice $X$ is a band projection if and only if the
multiplication operator

$\begin{array}[]{cccc}L_{P}R_{P}\colon&\mathcal{L}_{r}(X)&\longrightarrow&\mathcal{L}_{r}(X)\\\
&T&\longmapsto&PTP\\\ \end{array}$

is a band projection. Band projections could be therefore defined as those
elements $p\in A_{+}$ for which the operator

$\begin{array}[]{cccc}L_{p}R_{p}\colon&A&\longrightarrow&A\\\
&a&\longmapsto&pap\\\ \end{array}$

is a band projection. This is analogue to the approach taken by K. Vala in
[16] to define compact elements in normed algebras.

Several properties of band projections, in this sense, are studied. It is
shown that the set of band projections need not be order closed or even
commute; in particular, when we have an identity, band projections and order
idempotents may not be the same. The relation between order idempotents and
band projections appears to be subtle, as it is illustrated with several
finite-dimensional examples. Some conditions guaranteeing that a band
projection is order idempotent are also given.

Much better behaved is the set of left and right band projections, defined as
those positive elements in $A$ for which both the left and right
multiplication operators are band projections. These elements coincide with
order idempotents when an identity is present, and they always commute.
Moreover, they provide the appropiate generalization of band projections that
allows for the abstract construction of inner band projections we were looking
for. Our main result (3.29) is that (1) defines a band projection in any order
complete Banach lattice algebra as long as the
$\\{P_{\lambda}\\}_{\lambda\in\Lambda}$ are taken to be left and right band
projections satisfying $P_{\alpha}P_{\beta}=\delta_{\alpha\beta}P_{\alpha}$
(where $\delta_{\alpha\beta}$ is the Kronecker delta). To prove it, we cannot
resort directly to the Riesz–Kantorovich formulae, as done in [12]; instead,
we have to work with the multiplication operators
$L_{P_{\alpha}}R_{P_{\beta}}$ to translate properties from elements in $A$ to
operators acting on $A$ (i.e., to elements in $\mathcal{L}_{r}(A)$). Even
before reaching inner band projections, many of our results will use this
technique, that allows for the generalization of properties from spaces of
regular operators to Banach lattice algebras. We are not aware of any other
systematic use of this idea in the context of Banach lattice algebras, but one
can find it already mentioned or implicitly used in some parts of the
literature (for instance, in [6, 9, 10, 17]).

The content of this note is organized in two big sections (besides this
introduction): Section 2, which collects some well-known facts about spaces of
regular operators, and Section 3, which generalizes the notions in Section 2
to Banach lattice algebras. The reader familiar with regular operators can
jump directly to Section 3 on page 3; the reader familiar with regular
operators and Banach lattice algebras can skim through Sections 3.1 and 3.2 to
see the notation, and then start reading at Section 3.3 on page 3.3, where
most of the original results are contained.

These two big sections are, in turn, divided into subsections that follow a
parallel structure. A notion introduced for the regular operators in a
subsection of Section 2 is then studied for general Banach lattice algebras in
the corresponding subsection of Section 3. More precisely, we have the
following bijection:

2 Spaces of regular operators | 3 Banach lattice algebras
---|---
2.1 Regular operators | 3.1 Banach lattice algebras
2.2 Band projections | 3.3 Band projections and order idempotents
2.3 The center of a Banach lattice | 3.2 The ideal generated by the unit element
2.4 Complexification and spectra | 3.4 Complexification and spectra
2.5 Inner band projections | 3.5 Inner band projections

##  2 Spaces of regular operators

In this section we review some properties of the space of regular operators
over a Banach lattice that will motivate our study of Banach lattice algebras.
The reader familiar with the topic can therefore skip it, except maybe for
2.4, 2.5 and 2.5. With these exceptions, all the contents are classical, and
may be found in [1, 5, 11].

###  2.1 Regular operators

Let $X$ and $Y$ be Banach lattices, and let $\mathcal{L}(X,Y)$ denote the set
of (linear and bounded) operators from $X$ to $Y$. We say that an operator
$T\colon X\to Y$ is _positive_ if $T(X_{+})\subseteq Y_{+}$. The set of
positive operators $\mathcal{L}_{+}(X,Y)$ forms a cone in $\mathcal{L}(X,Y)$.
With this cone, $\mathcal{L}(X,Y)$ is an ordered Banach space, but it need not
be a lattice; in fact, it may even be the case that
$\mathcal{L}_{+}(X,Y)-\mathcal{L}_{+}(X,Y)$ is a proper subset of
$\mathcal{L}(X,Y)$. The elements of
$\mathcal{L}_{+}(X,Y)-\mathcal{L}_{+}(X,Y)$ are called _regular operators_ ,
and the set of regular operators is denoted by $\mathcal{L}_{r}(X,Y)$. Regular
operators are those operators that can be written as the difference of two
positive operators; equivalently, an operator $T$ is regular if there exists a
positive operator $S\in\mathcal{L}_{+}(X,Y)$ such that $T\leq S$.

The space of regular operators $\mathcal{L}_{r}(X,Y)$ need not be a Banach
lattice either. However, when $Y$ is _order complete_ (that is, when any
bounded above set has supremum), $\mathcal{L}_{r}(X,Y)$ is always a Banach
lattice. More precisely:

######  Theorem 2.1 (Riesz–Kantorovich).

Let $X$ and $Y$ be Banach lattices. If $Y$ is order complete, then
$\mathcal{L}_{r}(X,Y)$ is a Banach lattice, with supremum and infimum given by

$\displaystyle(S\vee T)x$ $\displaystyle=\sup\\{\,Su+Tv:u,v\in
X_{+},\,u+v=x\,\\},$ (2) $\displaystyle(S\wedge T)x$
$\displaystyle=\inf\\{\,Su+Tv:u,v\in X_{+},\,u+v=x\,\\},$ (3)

where $S,T\in\mathcal{L}_{r}(X,Y)$ and $x\in X_{+}$, and with norm

$\|T\|_{r}=\|\,|T|\,\|.$ (4)

Moreover, $\mathcal{L}_{r}(X,Y)$ is order complete.

Equations Eq. 2 and Eq. 3 and their derivatives (e.g.,
$|T|=\sup\\{\,|Tu|:-x\leq u\leq x\,\\}$) are called the _Riesz–Kantorovich
formulae_. The norm defined in Eq. 4 is called the _regular norm_.

When $X=Y$, we will simplify our notation from $\mathcal{L}(X,X)$ to
$\mathcal{L}(X)$, from $\mathcal{L}_{r}(X,X)$ to $\mathcal{L}_{r}(X)$, and so
on. In this case, the space $\mathcal{L}(X)$ is a Banach algebra with
multiplication given by composition, and norm given by operator norm. It is
clear that the composition of positive operators is again positive. From this,
and the fact that the modulus is submultiplicative (i.e., $|TS|\leq|T||S|$),
it follows that $\mathcal{L}_{r}(X)$ is a Banach algebra with respect to the
regular norm. If $X$ is order complete, we have seen that $\mathcal{L}_{r}(X)$
is a Banach lattice. The simultaneous existence of these two structures will
be our object of study in Section 3.

###  2.2 Band projections

The interaction between the lattice and algebraic structures of
$\mathcal{L}_{r}(X)$ manifests itself clearly when we study band projections.
A subspace $B$ of $X$ that is closed under lattice operations (i.e., if $y\in
B$ then $|y|\in B$) is called a _sublattice_. A sublattice that is solid
(i.e., if $|x|\leq|y|$ and $y\in B$ then $x\in B$) is called an _ideal_. An
ideal that is order closed (i.e., if $(y_{\gamma})_{\gamma}\subseteq B$ is an
increasing net and $\sup_{\gamma}y_{\gamma}$ exists in $X$, then
$\sup_{\gamma}y_{\gamma}\in B$) is called a _band_. A band for which
$X=B\oplus B^{d}$, where

$B^{d}=\\{\,x\in X:|x|\wedge|y|=0\text{ for all }y\in B\,\\},$

is called a _projection band_. In this case we have a positive projection
$P\colon X\to X$ onto $B$, called a _band projection_. It turns out that band
projections can be nicely characterized in terms of the algebraic and order
structure in $\mathcal{L}(X)$.

######  Theorem 2.2.

Let $X$ be a Banach lattice. An operator $P\colon X\to X$ is a band projection
if and only if $P^{2}=P$ and $0\leq P\leq I_{X}$.

In general, not all bands are projection bands. When $X$ is order complete,
however, every band is a projection band. In this case, we will talk
indistinguishably about bands, projection bands, and band projections.

Finite infima and suprema of band projections yield new band projections.
Surprisingly enough, lattice operations between band projections are
determined by the algebraic and linear operations only.

######  Theorem 2.3.

Let $X$ be a Banach lattice. The set of band projections on $X$ forms a
Boolean algebra with operations

$\overline{P}=I_{X}-P,\quad P_{1}\vee P_{2}=P_{1}+P_{2}-P_{1}P_{2},\quad
P_{1}\wedge P_{2}=P_{1}P_{2},$

minimum the zero map, and maximum the identity map. Moreover, suprema and
infima coincide with those in $\mathcal{L}_{r}(X)$.

A direct consequence of previous theorem is that band projections commute.
Note here the interplay between the lattice and algebraic operations: we are
able to express the lattice operations in purely algebraic terms, and from
this we are able to deduce an algebraic property.

It will be central to our discussion that band projections are characterized
among positive operators by the multiplication operators they define. Let
$A,B\in\mathcal{L}_{r}(X)$ be regular operators on an order complete Banach
lattice $X$. The operators

$\begin{array}[]{cccc}L_{A}\colon&\mathcal{L}_{r}(X)&\longrightarrow&\mathcal{L}_{r}(X)\\\
&T&\longmapsto&AT\\\
\end{array}\quad\begin{array}[]{cccc}R_{B}\colon&\mathcal{L}_{r}(X)&\longrightarrow&\mathcal{L}_{r}(X)\\\
&T&\longmapsto&TB\\\ \end{array}$

are called _left_ and _right multiplication operators_ , respectively. More
generally, any operator of the form $L_{A}R_{B}$, for some
$A,B\in\mathcal{L}_{r}(X)$, is called a _multiplication operator_. As
promised, using multiplication operators we can characterize those elements in
$\mathcal{L}_{+}(X)$ that are band projections as follows.

######  Theorem 2.4 ([12, Theorem 4.5]).

Let $X$ be an order complete Banach lattice, and let
$A,B\in\mathcal{L}_{r}(X)$ be nonzero regular operators. The multiplication
operator $L_{A}R_{B}$ is a band projection if and only if there exists
$\lambda\in\mathbb{R}\setminus\\{0\\}$ such that $\lambda A$ and $B/\lambda$
are band projections.

######  Corollary 2.5.

Let $X$ be an order complete Banach lattice, and let $A\in\mathcal{L}_{+}(X)$.
Then the multiplication operator $L_{A}R_{A}$ is a band projection if and only
if $A$ is a band projection.

###### Proof.

Suppose that $L_{A}R_{A}$ is a band projection. If $A=0$ the result follows,
so assume $A\neq 0$. By previous theorem, there exists
$\lambda\in\mathbb{R}\setminus\\{0\\}$ such that $\lambda A$ and $A/\lambda$
are band projections. Since $A$ is positive, $\lambda$ must also be positive.
Using that $\lambda A$ is a projection,

$\left(\frac{\lambda A}{\lambda^{2}}\right)^{2}=\frac{(\lambda
A)^{2}}{\lambda^{4}}=\frac{\lambda A}{\lambda^{4}}.$

But $\lambda A/\lambda^{2}=A/\lambda$ is also a projection, so $(\lambda
A/\lambda^{2})^{2}=A/\lambda$. Join both equations to arrive at
$\lambda^{2}A=A$. Since $A\neq 0$, it must be the case that $\lambda^{2}-1=0$.
But $\lambda>0$ implies $\lambda=1$, and therefore $A$ is a band projection. ∎

###  2.3 The center of a Banach lattice

Central operators and the center of a Banach lattice have been extensively
studied in the literature (see [1, Section 3.3]). Here we only recall some
basic definitions and a couple of results that will be useful later.

######  Definition 2.6.

Let $X$ be a Banach lattice. An operator $T\colon X\to X$ is called _central_
if there exists some scalar $\lambda>0$ such that $|Tx|\leq\lambda|x|$ for all
$x\in X$. The collection of all central operators is denoted $\mathcal{Z}(X)$
and is called the _center of $X$_.

For example, central operators on $C(K)$ are precisely the operators of the
form $M_{f}(g)=fg$, where $f,g\in C(K)$. Note also that band projections are
positive central operators. When $X$ is order complete, $\mathcal{Z}(X)$ is
just the ideal generated by the identity in the Banach lattice
$\mathcal{L}_{r}(X)$.

Central operators have nice properties: they are regular and order continuous,
and positive central operators are also lattice homomorphisms. The following
theorem collects some of the most important properties of the center.

######  Theorem 2.7 (Wickstead [18]).

Let $X$ be a Banach lattice.

  1. (i)

The center $\mathcal{Z}(X)$, equipped with the operator norm, is an AM-space
with unit $I_{X}$. In particular,

$\|T\|=\|\,|T|\,\|=\inf\\{\,\lambda>0:|T|\leq\lambda I_{X}\,\\}$

for every $T\in\mathcal{Z}(X)$, and $\mathcal{Z}(X)$ is lattice isometric to
$C(K)$ for a certain compact Hausdorff space $K$.

  2. (ii)

The center $\mathcal{Z}(X)$ is an inverse closed***A subalgebra $B$ of an
algebra $A$ is said to be _inverse closed_ if for every $b\in B$ invertible in
$A$, the inverse $b^{-1}$ is contained in $B$. subalgebra of
$\mathcal{L}_{r}(X)$.

  3. (iii)

When $X$ is order complete, the center $\mathcal{Z}(X)$ is the band generated
by $I_{X}$ in $\mathcal{L}_{r}(X)$. In particular, it is a projection band.

An operator $T\colon X\to X$ on a Banach lattice $X$ is said to be

  1. (i)

an _orthomorphism_ if $|x|\wedge|y|=0$ implies $|x|\wedge|Ty|=0$ for every
$x,y\in X$;

  2. (ii)

_band preserving_ if $T(B)\subseteq B$ for every band $B$ in $X$.

It is immediate that central operators are orthomorphisms, and that
orthomorphisms are band preserving. Not trivial, but true, is that for Banach
lattices these three classes of operators coincide.

######  Theorem 2.8 (Abramovich–Veksler–Koldunov [2]).

For an operator $T\colon X\to X$ on a Banach lattice $X$ the following are
equivalent.

  1. (i)

$T$ is a central operator.

  2. (ii)

$T$ is an orthomorphism.

  3. (iii)

$T$ is band preserving.

###  2.4 Complexification and spectra

We want to introduce the “complex version” of the spaces of regular operators.
But first we need to recall the notion of a complex Banach lattice, following
[11, Section 2.2]. Let $X$ be a Banach lattice. The _complexification_ of $X$,
as a vector space, is the additive group $X\times X$ with scalar
multiplication

$(\alpha+i\beta)(x,y)=(\alpha x-\beta y,\alpha y+\beta x),$

where $\alpha,\beta\in\mathbb{R}$ and $x,y\in X$. The group $X\times X$ with
this scalar multiplication is a complex vector space, usually denoted by
$X_{\mathbb{C}}$. Identifying $x\in X$ with $(x,0)\in X\times X$ and $ix$ with
$(0,x)$, we write $x+iy$ instead of $(x,y)\in X\times X$.

######  Proposition 2.9.

Let $X$ be a Banach lattice. For $z=x+iy\in X_{\mathbb{C}}$ define

$|z|=\sup\\{\,x\cos\phi+y\sin\phi:0\leq\phi\leq 2\pi\,\\}.$

This supremum exists in $X$, and it satisfies:

  1. (i)

$|z|=0$ if and only if $z=0$,

  2. (ii)

$|\lambda z|=|\lambda||z|$ for all $\lambda\in\mathbb{C}$ and $z\in
X_{\mathbb{C}}$,

  3. (iii)

$|z+w|\leq|z|+|w|$ for all $z,w\in X_{\mathbb{C}}$.

Moreover, the map $z\mapsto\||z|\|$ defines a norm on $X_{\mathbb{C}}$, and
$X_{\mathbb{C}}$ with this norm is a complex Banach space.

######  Definition 2.10.

A _complex Banach lattice_ is a Banach space of the form $X_{\mathbb{C}}$, for
a certain Banach lattice $X$, with the norm and modulus introduced in previous
proposition.

We now introduce regular operators between complex Banach lattices
$X_{\mathbb{C}}$ and $Y_{\mathbb{C}}$. For every (complex) linear operator
$T\colon X_{\mathbb{C}}\to Y_{\mathbb{C}}$ there exist unique (real) linear
operators $T_{1},T_{2}\colon X\to Y$ such that $Tx=T_{1}x+iT_{2}x$ for all
$x\in X$. The operator $T_{1}$ is called the _real part_ of $T$, while the
operator $T_{2}$ is called the _imaginary part_ of $T$. Thus
$\mathcal{L}(X_{\mathbb{C}},Y_{\mathbb{C}})$ is isomorphic to the
complexification of $\mathcal{L}(X,Y)$. An operator $T\colon X_{\mathbb{C}}\to
Y_{\mathbb{C}}$ is said to be _real valued_ if $T(X)\subseteq Y$. If $T$ is
real valued and $T(X_{+})\subseteq Y_{+}$, then $T$ is said to be _positive_.
An operator $T\colon X_{\mathbb{C}}\to Y_{\mathbb{C}}$ is said to be _regular_
if both its real and imaginary parts are regular. When $Y$ is order complete,
the space of regular operators between $X_{\mathbb{C}}$ and $Y_{\mathbb{C}}$
may be identified with the complexification of the Banach lattice
$\mathcal{L}_{r}(X,Y)$.

Given a regular operator $T\colon X_{\mathbb{C}}\to X_{\mathbb{C}}$, denote by
$\sigma(T)$ its spectrum. The following result characterizes the lattice
isomorphisms contained in the center in terms of their spectrum, and will be
needed later in the discussion.

######  Theorem 2.11 (Schaefer–Wolff–Arendt [13]).

Let $T$ be a lattice isomorphism of a complex Banach lattice $X$. Then
$T\in\mathcal{Z}(X)$ if and only if $\sigma(T)\subseteq\mathbb{R}_{+}$.

###  2.5 Inner band projections

For an order complete Banach lattice $X$, we have introduced the Banach
lattice $\mathcal{L}_{r}(X)$ of regular operators over $X$. We can then
consider band projections on this space. Since $\mathcal{L}_{r}(X)$ is again
order complete, any band will have an associated band projection. We have
already encountered an example of such a band in $\mathcal{L}_{r}(X)$, namely
the center. Another example are inner band projections, a particular class of
band projections on spaces of regular operators introduced by the author
together with P. Tradacete in [12]. The construction of these band projections
is summarized in the following result.

######  Theorem 2.12 ([12, Theorem 2.1]).

Let $X$ be an order complete Banach lattice and let
$\\{P_{\lambda}\\}_{\lambda\in\Lambda}$ be a family of pairwise disjoint band
projections on $X$. Let $\Gamma\subseteq\Lambda\times\Lambda$ be any subset.
Then the map

$\
\begin{array}[]{cccc}\mathcal{P}_{\Gamma}\colon&\mathcal{L}_{+}(X)&\longrightarrow&\mathcal{L}_{+}(X)\\\
&T&\longmapsto&\displaystyle\bigvee_{(\alpha,\beta)\in\Gamma}P_{\alpha}TP_{\beta}\\\
\end{array}$

extends to a unique band projection
$\mathcal{P}_{\Gamma}\colon\mathcal{L}_{r}(X)\to\mathcal{L}_{r}(X)$.

######  Definition 2.13.

Let $X$ be an order complete Banach lattice. We say that a band projection
$\mathcal{P}$ on $\mathcal{L}_{r}(X)$ is an _inner band projection_ if there
exists a family of pairwise disjoint band projections
$\\{P_{\lambda}\\}_{\lambda\in\Lambda}$ on $X$ such that
$\mathcal{P}=\mathcal{P}_{\Gamma}$, as defined in 2.12, for a certain
$\Gamma\subseteq\Lambda\times\Lambda$. We will call _inner projection bands_
the bands associated with inner band projections, and we will denote
$\mathcal{B}_{\Gamma}={P}_{\Gamma})$.

In Section 3.5 we will be able to generalize this construction to arbitrary
order complete Banach lattice algebras. Together with the construction, we
will generalize some of the properties of inner band projections. In
particular, we will study the following property, regarding the structure of
the set of inner band projections.

######  Proposition 2.14 ([12, Proposition 2.10]).

Let $X$ be an order complete Banach lattice and let
$\\{P_{\lambda}\\}_{\lambda\in\Lambda}$ be a family of pairwise disjoint band
projections on $X$. The inner band projections
$\\{\mathcal{P}_{\Gamma}:\Gamma\subseteq\Lambda\times\Lambda\\}$ (resp. the
inner projection bands
$\\{\mathcal{B}_{\Gamma}:\Gamma\subseteq\Lambda\times\Lambda\\}$) form a
Boolean algebra with the same order, suprema and infima as in
$\mathfrak{P}(\mathcal{L}_{r}(X))$ (resp. $\mathfrak{B}(\mathcal{L}_{r}(X))$).
Moreover, this Boolean algebra is isomorphic to $2^{\Lambda\times\Lambda}$
through the map $\Gamma\mapsto\mathcal{P}_{\Gamma}$ (resp.
$\Gamma\mapsto\mathcal{B}_{\Gamma}$).

Inner band projections are also used to characterize atomic and order
continuous Banach lattices.

######  Theorem 2.15 ([12, Theorem 3.1]).

Let $X$ be an order complete Banach lattice. All band projections on
$\mathcal{L}_{r}(X)$ are inner if and only if $X$ is both atomic and order
continuous.

##  3 Banach lattice algebras

The goal of this section is to generalize the properties studied in the first
section to Banach lattice algebras. Sections 3.1 and 3.2 introduce Banach
lattice algebras and some known results that will be useful throughout. The
reader familiar with the literature can therefore have a quick look at the
notation in these sections, and then go directly to Section 3.3.

###  3.1 Banach lattice algebras

In the space of regular operators over an order complete Banach lattice, a
Banach lattice structure (given by the cone of positive operators) and a
Banach algebra structure (given by composition) coexist. We want to study more
general spaces in which these two structures coexist, and to extend (when
possible) the properties of the first part of the paper to this abstract
setting. This will elucidate which properties of the space of regular
operators do not really depend on the fact that they are regular operators
over a Banach lattice, but rather on the algebraic and lattice structures of
the space.

But first, we need to define Banach lattice algebras. It seems like there is
no consensus in the literature on the notion of Banach lattice algebra with
identity, so we follow the one suggested by A. W. Wickstead in [17].

######  Definition 3.1.

A _Banach lattice algebra_ is a Banach lattice $A$ together with a product
$\ast\colon A\times A\to A$ such that

  1. (i)

$(A,\ast)$ is a Banach algebra,

  2. (ii)

if $x,y\in A_{+}$, then $x\ast y\in A_{+}$.

When the product $\ast$ has an identity $e$ of norm one, we say that $A$ is a
_Banach lattice algebra with identity $e$_. When the product $\ast$ is
commutative, we say that $A$ is a _commutative Banach lattice algebra_.

It is important to note the following consequence of the definition.

######  Proposition 3.2 ([6, Theorem 2]).

The identity in a Banach lattice algebra is positive.

Let us pause here to present some classical examples of Banach lattice
algebras. We shall resort to them when presenting examples and counterexamples
in our exposition.

######  Example 3.3.

  1. (i)

Let $K$ be a compact Hausdorff space. The space $C(K)$ of real-valued
continuous functions on $K$ is a commutative Banach lattice algebra with
pointwise order and product, and supremum norm. The identity is the constant
one function.

  2. (ii)

Let $X$ be an order complete Banach lattice. We have seen in Section 2.1 that
the space of regular operators $\mathcal{L}_{r}(X)$ is a Banach lattice, and
at the same time a Banach algebra, in which the composition of positive
operators is positive. It is, therefore, a Banach lattice algebra, with
identity $I_{X}$. This Banach lattice algebra need not be commutative.

  3. (iii)

Let $G$ be a locally compact group, let $\lambda$ be a left invariant Haar
measure, and consider the Banach lattice $L_{1}(G)=L_{1}(\lambda)$. The
convolution multiplication on $L_{1}(G)$, defined for $\phi,\psi\in L_{1}(G)$
by

$(\phi\ast\psi)(g)=\int_{G}\phi(gh)\psi(h^{-1})\,d\lambda(h),\quad\text{where
}g\in G,$

makes $L_{1}(G)$ into a Banach algebra. It is easy to check that the product
of positive elements is positive, so $L_{1}(G)$ is Banach lattice algebra.
This Banach lattice algebra is commutative if and only if $G$ is abelian, and
has an identity if and only if $G$ is discrete.

  4. (iv)

Let $M(G)$ be the space of signed regular Borel measures on a locally compact
group $G$. Then $M(G)$ is a Banach lattice if we set $\mu\geq\nu$, $\mu,\nu\in
M(G)$, whenever $\mu(A)\geq\nu(A)$ for every Borel set $A$, and give it the
norm $\|\mu\|=|\mu|(G)$. The convolution multiplication defined by

$(\mu\ast\nu)(A)=\int_{G}\int_{G}\chi_{A}(st)\,d\mu(s)\,d\nu(t),$

where $\chi_{A}$ denotes the characteristic function of the Borel set $A$,
gives $M(G)$ a Banach algebra structure. Again, this space is a Banach lattice
algebra. Its identity is the point mass measure on the identity of $G$.

  5. (v)

Let $\\{A_{i}\\}_{i\in I}$ be a family of Banach lattice algebras idexed by a
set $I$, and let ${1\leq p\leq\infty}$. Then their $\ell_{p}$-sum
$\ell_{p}(A_{i})$, equipped with coordinate-wise order and product, is again a
Banach lattice algebra, for if $x=(x_{i})$ and $y=(y_{i})$ are elements of
$\ell_{p}(A_{i})$, then

$\|xy\|_{p}\leq\|x\|_{p}\|y\|_{\infty}\leq\|x\|_{p}\|y\|_{p}.$

It is clear that $\ell_{p}(A_{i})$ is commutative if and only if $A_{i}$ is
commutative for each $i\in I$. Also, $\ell_{\infty}(A_{i})$ has identity if
and only if $A_{i}$ has identity for each $i\in I$.

###  3.2 The ideal generated by the identity element

As explained in Section 2.3, the ideal generated by the identity element in
$\mathcal{L}_{r}(X)$, when $X$ is an order complete Banach lattice, is the
center of $X$, an ideal of operators extensively studied in the literature. It
turns out that the notion of the ideal generated by the identity element in
Banach lattice algebras has also been studied, and that properties analogous
to those in 2.7 hold in general.

If $A$ is a Banach lattice algebra with identity $e$, we adopt the convention
of [14] and denote by $A_{e}$ the (order) ideal generated by $e$ in $A$. The
main properties of this ideal are collected in the following result.

######  Theorem 3.4.

Let $A$ be a Banach lattice algebra with identity $e$.

  1. (i)

The space $(A_{e},\|{\cdot}\|_{e})$, with the order and product of $A$, is a
Banach lattice algebra, lattice and algebra isometric to $C(K)$, for a certain
compact Hausdorff $K$.†††Note that the analogy with 2.7 is not complete: here
we are endowing the ideal generated by the identity with its associated norm,
whereas in 2.7 one works with the operator norm. As it turns out, assumption
$\|e\|=1$ implies that both norms are equal. Yet we need spectra to show this
and it will have to wait until Section 3.4. For the time being, it suffices to
know that $A_{e}$ is closed in $A$. This apparent inconvenience has an
advantage: all the contents of this section are valid under the weaker
hypothesis $e\geq 0$, which is another common definition of Banach lattice
algebras with identity.

  2. (ii)

The ideal $A_{e}$ is a principal projection band in $A$.

  3. (iii)

The space $A_{e}$ is an inverse closed subalgebra of $A$.

The contents of previous theorem are scattered through the literature: the
first assertion follows from the results by L. Martignon in [10], the second
assertion is due to E. Scheffold [14, Theorem 3], while the third is mentioned
by C. B. Huijsmans in [7] without any further reference. Since this result
will be central in our exposition, we devote the rest of this section to
present a full proof of it.

To prove the first part of the theorem, we need to talk about Banach lattice
algebras that are AM-spaces with unit, in which the order unit coincides with
the algebraic identity.

######  Definition 3.5.

Let $A$ be a Banach lattice algebra with identity $e$. If $A$ is also an AM-
space with unit $e$ (i.e., if $A_{e}=A$ and
$\|x\|=\inf\\{\,\lambda>0:|x|\leq\lambda e\,\\}$ for every $x\in A$), we say
that $A$ is an _AM-algebra with unit $e$_.

The following theorem asserts that AM-algebras with unit are nothing but
algebras of continuous functions.

######  Theorem 3.6.

Every AM-algebra with unit is algebra and lattice isometric to $C(K)$, for a
certain compact Hausdorff $K$.

Previous result is a direct consequence of Kakutani’s representation theorem
for AM-spaces with unit, and the following characterization of the pointwise
product in $C(K)$ due to L. Martignon.

######  Proposition 3.7 ([10, Proposition 1.4]).

Let $K$ be a compact Hausdorff space and let $\star\colon C(K)\times C(K)\to
C(K)$ be a binary operation such that $(C(K),\star)$ with pointwise order is a
Banach lattice algebra. Suppose also that the constant one function is the
identity for $\star$. Then $\star$ is the pointwise multiplication.

The proof of this last result relies heavily on Banach lattice techniques.
Here we want to provide an alternative proof of 3.6 in the style of the
Gelfand transform: working with maximal (order and algebraic) ideals and real-
valued (lattice and algebra) homomorphisms. This argument follows closely the
proof of Kakutani’s representation theorem from [15, Chapter 4].

There are two kinds of ideals present in Banach lattice algebras: the ideals
as a Banach lattice and the ideals as a Banach algebra. There are also
subspaces that are ideals in both senses. There does not seem to be a common
terminology for these spaces in the literature. We will refer to ideals in the
Banach lattice sense as _order ideals_ , and to ideals in the Banach algebra
sense as _algebraic ideals_ (these, in turn, can be left or right ideals, or
both). _Ideals_ will be the subspaces that are at the same time an order ideal
and a (two-sided) algebraic ideal. The following is a simple characterization
of the ideal generated by an element (that is, the intersection of all the
ideals containing the element) in a Banach lattice algebra with identity.

######  Lemma 3.8.

Let $A$ be a Banach lattice algebra with identity and let $a\in A$. Then the
ideal generated by $a$ is

$\\{\,x\in A:|x|\leq r|a|r^{\prime}\text{ for some }r,r^{\prime}\in
A_{+}\,\\}.$

###### Proof.

Call the displayed set $I$. Let us check that this set is indeed an ideal. If
$x,y\in I$, say $|x|\leq rar^{\prime}$ and $|y|\leq sas^{\prime}$ for some
$r,r^{\prime},s,s^{\prime}\in A_{+}$, then

$|x+y|\leq|x|+|y|\leq
rar^{\prime}+sas^{\prime}\leq(r+s)a(r^{\prime}+s^{\prime})$

so $x+y\in I$. Clearly, $-x$ and $|x|$ are elements of $I$, so $I$ is a
sublattice. From the definition it follows that $I$ is an order ideal. To
check that it is also an algebraic ideal, let $z\in A$. Then we have

$|zx|\leq|z||x|\leq(|z|r)ar^{\prime}\quad\text{and}\quad|xz|\leq|x||z|\leq
ra(r^{\prime}|z|)$

so both $zx,xz\in I$, and we have thus shown that $I$ is an ideal.

Since $A$ has an identity, $a\in I$. Let $J$ be any ideal of $A$ containing
$a$. Then, since $J$ is a sublattice and an algebraic ideal,
$r|a|r^{\prime}\in J$ for all $r,r^{\prime}\in A_{+}$. Since $J$ is an order
ideal, $x\in J$ whenever $|x|\leq rar^{\prime}$. Hence $I\subseteq J$, and so
$I$ is the ideal generated by $a$ in $A$. ∎

As usual, we say that an ideal in a Banach lattice algebra is _maximal_ if it
is not contained in any proper ideal. The following is pure routine.

######  Lemma 3.9.

In a Banach lattice algebra with identity, any proper ideal is contained in a
maximal ideal.

###### Proof.

Let $I$ be a proper ideal in a Banach lattice algebra $A$ with identity $e$.
Let $\mathcal{I}$ be the family of proper ideals in $A$ containing $I$,
ordered by inclusion. Let $\\{J_{\gamma}\\}_{\gamma\in\Gamma}$ be a chain in
$\mathcal{I}$. Then it is straightforward to check that
$J=\bigcup_{\gamma\in\Gamma}J_{\gamma}$ is again an ideal containing $I$.
Moreover, if $J=A$, then $e\in J$ and so $e\in J_{\gamma}$ for a certain
$\gamma\in\Gamma$. Being $J_{\gamma}$ an algebraic ideal, this would imply
$J_{\gamma}=A$, contradicting the definition. So $J$ is proper, and therefore
is an upper bound of the chain in $\mathcal{I}$. By Zorn’s lemma, there exists
a maximal element in $\mathcal{I}$, and this element is a maximal ideal
containing $I$. ∎

Let $A$ be a Banach lattice algebra and let $I$ be a closed ideal. Perform the
usual quotient $A/I$. Being $I$ a two-sided closed ideal, $A/I$ is again a
Banach algebra, and the quotient map $Q\colon A\to A/I$ is an algebra
homomorphism. Since $I$ is an order ideal, $Q(A_{+})$ is a cone in $A/I$, and
the order it induces makes $A/I$ into a Banach lattice, and $Q$ into a lattice
homomorphism. It is direct to check that the product of positive elements in
$A/I$ is again positive. Hence $A/I$ admits a natural Banach lattice algebra
structure, making $Q$ a lattice and algebra homomorphism.

Since the closure of an algebraic ideal is again an algebraic ideal, and the
closure of an order ideal is again an order ideal, the closure of an ideal in
a Banach lattice algebra is again an ideal. It is then routine to check that
maximal ideals in Banach lattice algebras with identity are closed. Moreover,
we have the following.

######  Lemma 3.10.

Let $A$ be an AM-algebra with unit $e$ and let $M$ be a maximal ideal in $A$.
Then $A/M$ is lattice and algebra isomorphic to $\mathbb{R}$.

###### Proof.

By the discussion preceding the lemma, $A/M$ is a Banach lattice algebra.
Suppose that $\dim(A/M)>1$. Then there would exist two non-zero disjoint
elements $Qx,Qy\in A/M$. We are going to show that the ideal $I$ generated by
$|Qx|$ in $A/M$ must be proper. Indeed, if $Qy\in I$, by 3.8 there would exist
$r,r^{\prime}\in A_{+}$ such that $|Qy|\leq Q(r)|Q(x)|Q(r^{\prime})$, and then

$|Qy|=(Q(r)|Q(x)|Q(r^{\prime}))\wedge|Qy|\leq\|r\|\|r^{\prime}\||Q(x)|\wedge|Q(y)|=0,$

which is a contradiction. Hence $I$ is proper. It is not difficult to check
that $Q^{-1}(I)\subseteq A$ is again an ideal, because $Q$ is both a lattice
and algebra homomorphism. Moreover, $M\subseteq Q^{-1}(I)\subsetneq A$, and
since $x\in Q^{-1}(I)$, the first inclusion is proper. This contradicts that
$M$ is maximal. So $\dim(A/M)$ must be $1$. Then $A/M$ is spanned by $Qe$ and
the map

$\begin{array}[]{cccc}&\mathbb{R}&\longrightarrow&A/M\\\
&\lambda&\longmapsto&\lambda Qe\\\ \end{array}$

is a lattice and algebra isomorphism. ∎

######  Lemma 3.11.

Let $A$ be an AM-algebra with unit $e$ and let $x_{0}\in A_{+}$. If
$x_{0}\not\leq e$, there exists a lattice and algebra homomorphism $\phi\colon
A\to\mathbb{R}$ such that $\phi(x_{0})>1$.

###### Proof.

We have $(x_{0}-e)_{+}\neq 0$. Let $I$ be the ideal generated by
$(x_{0}-e)_{-}$. We claim that $(x_{0}-e)_{+}\not\in I$. Indeed, if it were
the case, according to 3.8 there would exist $r,r^{\prime}\in A_{+}$ such that

$(x_{0}-e)_{+}\leq
r(x_{0}-e)_{-}r^{\prime}\leq\|r\|\|r^{\prime}\|(x_{0}-e)_{-}$

which contradicts the fact that $(x_{0}-e)_{+}$ and $(x_{0}-e)_{-}$ are
disjoint. Hence $I$ is proper, and by 3.9 it is contained in a maximal ideal
$M$. The quotient map associated with $M$ is a lattice and algebra
homomorphism $A\to A/M$. Let $\phi\colon A\to\mathbb{R}$ be the composition of
this quotient map with the isomorphism between $A/M$ and $\mathbb{R}$ given in
3.10. This homomorphism satisfies

$\phi(x_{0}-e)=\phi((x_{0}-e)_{+})>0,$

so $\phi(x_{0})>\phi(e)=1$. ∎

With this set of rather standard tools, we can proceed to the proof of 3.6.

###### Proof of 3.6.

Let $A$ be an AM-algebra with unit $e$. Let $K$ be the set of all lattice and
algebra homomorphisms $\phi\colon A\to\mathbb{R}$ with $\phi(e)=1$. We can
view $K$ as a subset of $\mathbb{R}^{A}$. Equip $\mathbb{R}^{A}$ with the
product topology and $K$ with the subspace topology. This topology is
Hausdorff, because the product topology is. Note that, for every $\phi\in K$,
$|\phi(x)|\leq\phi(\|x\|e)=\|x\|$ holds, which means that
$K\subseteq\prod_{x\in A}[-\|x\|,\|x\|]$. By Tychonoff’s theorem, the latter
product is compact. Moreover, $K$ is closed in this subspace, for if
$(\phi_{\lambda})$ is a net in $K$ pointwise convergent to $\phi$, this means
that $\phi(x)=\lim\phi_{\lambda}(x)$ for all $x\in A$, and then by continuity
of the algebraic and lattice operations it is easy to check that $\phi$ is a
again a lattice and algebra homomorphism with $\phi(e)=1$. Therefore $K$ is a
compact Hausdorff space.

It is useful to note that $K$ has sufficiently many elements. Indeed, for
every $x\in A_{+}$ with $\|x\|>1$, we have $x\not\leq e$, and by 3.11 there is
some element $\phi\in K$ such that $\phi(x)>1$.

Now consider the map $T\colon A\to C(K)$ where $Tx(\phi)=\phi(x)$ for $x\in A$
and $\phi\in K$. It is easy to check that $Tx$ is indeed continuous. Since
operations in $C(K)$ are computed pointwise, and $Te(\phi)=\phi(e)=1$ for all
$\phi\in K$, it follows that $T$ is a lattice and algebra homomorphism that
preserves the identity.

We claim $\|Tx\|=\|x\|$ for all $x\in A$. On one hand,
$|Tx(\phi)|=|\phi(x)|\leq\|x\|$, so $\|Tx\|\leq\|x\|$. On the other hand, if
$\|x\|>1$, we have seen that there exists some $\phi\in K$ such that
$\phi(|x|)>1$, so that $\|Tx\|\geq|Tx(\phi)|=\phi(|x|)>1$. If $\|Tx\|<\|x\|$
for some $x\in A$, the element $|x|/\|Tx\|$ would contradict the previous
observation. So $\|Tx\|=\|x\|$ for all $x\in A$.

Finally, note that $T(A)$ is a closed subalgebra of $C(K)$ that contains the
constants and separates points: if $\phi\neq\psi$ in $K$, there is some $x\in
A$ such that $(Tx)(\phi)=\phi(x)\neq\psi(x)=(Tx)(\psi)$. By the
Stone–Weierstrass theorem, $T(A)=C(K)$, and the result is proved. ∎

Some preliminary remarks before presenting the proof of 3.4. For the second
assertion we will provide an argument similar to [9, Theorems 1 and 2] instead
of the original one appearing in [14] because it serves as a first
illustration of what will be our main trick in the coming sections, namely, to
consider multiplication operators on a Banach lattice algebra, and then
exploit the known properties of the space of regular operators. On a Banach
lattice algebra $A$ operators of the form

$\begin{array}[]{cccc}L_{a}\colon&A&\longrightarrow&A\\\ &x&\longmapsto&ax\\\
\end{array},\quad\begin{array}[]{cccc}R_{b}\colon&A&\longrightarrow&A\\\
&x&\longmapsto&xb\\\ \end{array},$

for certain $a,b\in A$, are called _left_ and _right multiplication operators_
, respectively. Operators of the form $L_{a}R_{b}$, for some $a,b\in A$, are
called _multiplication operators_.

###### Proof of 3.4.

  1. (i)

We are going to check that $(A_{e},\|{\cdot}\|_{e})$ is an AM-algebra with
unit $e$. Then the result will follow from 3.6. That $(A_{e},\|{\cdot}\|_{e})$
is an AM-space with unit $e$ is clear. Moreover, it is a subalgebra, for if
$x,y\in A_{e}$:

$|xy|\leq|x||y|\leq\|x\|_{e}\|y\|_{e}e.$

From this also follows that $\|xy\|_{e}\leq\|x\|_{e}\|y\|_{e}$, so it is a
Banach algebra. Its algebraic identity is obviously $e$.

  2. (ii)

We prove this fact in two steps. First, we show that $B_{e}$, the band
generated by $e$, is a projection band. And then we show that $A_{e}=B_{e}$.

To prove that $B_{e}$ is a projection band, it suffices to check that
$\sup_{n}a\wedge ne$ exists for every $a\in A_{+}$ (in fact, we show a little
more, because we will need it later on). Denote $a_{n}=a\wedge ne$, and
observe that for any $m\geq n\geq 1$:

$(a_{m}-a_{n})\wedge(ne-a_{n})=(a_{m}\wedge ne)-a_{n}=a_{n}-a_{n}=0.$

Using the representation of $I_{e}$ as a $C(K)$ space, and seeing $a_{m}$,
$a_{n}$ and $ne$ as continuous functions in $K$, we are saying that
$(a_{m}-a_{n})(t)\wedge(ne-a_{n})(t)=0$ for every $t\in K$. It follows that
$a_{m}-a_{n}$ and $a_{m}-n^{-1}a_{m}a_{n}=n^{-1}a_{m}(ne-a_{n})$ are also
disjoint, so

$(a_{m}-a_{n})\wedge(a_{m}-n^{-1}a_{m}a_{n})=0$

which implies

$a_{m}=a_{n}\vee(n^{-1}a_{m}a_{n}).$

It follows that

$0\leq\|a_{m}-a_{n}\|=\|(n^{-1}a_{m}a_{n}-a_{n})_{+}\|\leq
n^{-1}\|a_{m}a_{n}\|\leq n^{-1}\|a\|^{2},$ (5)

so $(a_{n})$ is a Cauchy sequence. It therefore converges to some $b\in
A_{+}$. Since $(a_{n})$ is increasing, we must have $b=\sup_{n}a\wedge ne$,
and this shows that $B_{e}$ is a projection band.

To prove that $A_{e}$ is a band we need to show that $A_{e}=B_{e}$, and for
this it suffices to check that $a\in A_{e}$ whenever $a\in(B_{e})_{+}$. For
every $a\in(B_{e})_{+}$, inequality $a\wedge ne\leq ne$ implies that
$L_{a\wedge ne}$ is a central operator for every $n\geq 1$. In particular, it
is an orthomorphism. If $x,y\in A$ are such that $|x|\wedge|y|=0$, then
$|x|\wedge|(a\wedge ne)y|=0$ for every $n\geq 1$. But since $a$ is an element
of the band generated by $e$, we have $a=\sup_{n}a\wedge ne$. Then $a\wedge
ne$ converges to $a$ by (5). Since both multiplication and lattice operations
are continuous for the norm, it follows

$0=\lim_{n\to\infty}|x|\wedge|(a\wedge ne)y|=|x|\wedge|ay|.$

This shows that $L_{a}$ is an orthomorphism. By 2.8, $L_{a}$ must be a central
operator, that is, there exists some $\lambda\in\mathbb{R}_{+}$ such that
$-\lambda I_{A}\leq L_{a}\leq\lambda I_{A}$. In particular,
$a=L_{a}(e)\leq\lambda e$, and $a\in A_{e}$ as wanted.

  3. (iii)

Let $a\in A_{e}$ and suppose that $a^{-1}$ exists in $A$. By the previous
assertion we can decompose $a^{-1}=a_{1}+a_{2}$ with $a_{1}\in A_{e}$ and
$a_{2}\in A_{e}^{d}$. Then $e=aa^{-1}=aa_{1}+aa_{2}$. Since $A_{e}$ is a
subalgebra, $aa_{1}\in A_{e}$. Also

$0\leq|aa_{2}|\wedge e\leq(|a||a_{2}|)\wedge e\leq\|a\|_{e}|a_{2}|\wedge e=0,$

so $aa_{2}\in A_{e}^{d}$. But $aa_{2}=e-aa_{1}\in A_{e}$, so it must be
$aa_{2}=0$. Multiplying by $a^{-1}$ we get $a_{2}=0$ and $a^{-1}=a_{1}\in
A_{e}$, as wanted.∎

######  Example 3.12.

Let $A$ be a Banach lattice algebra with identity $e$ that is an AL-space as a
Banach lattice. Then $(A_{e},\|{\cdot}\|_{e})$ is an AM-space, but at the same
time is closed, so that $(A_{e},\|{\cdot}\|_{e})$ is also isomorphic to an AL-
space. It follows that $A_{e}$ must be finite-dimensional. In fact, it will
follow from our study of spectra that, under the assumption $\|e\|=1$,
$\|{\cdot}\|=\|{\cdot}\|_{e}$ (see 3.24). This implies $A_{e}=\mathbb{R}$.
Without the assumption $\|e\|=1$, however, $A_{e}$ need not be one-
dimensional: just take $\ell_{1}^{n}$ with the coordinate-wise order and
product.

###  3.3 Band projections and order idempotents

Band projections play a prominent role in the theory of Banach lattices. These
operators are characterized among regular operators by 2.2. The goal of this
section is to define elements in a general Banach lattice algebra that play
the same role as band projections do in spaces of regular operators. When an
identity is present, this generalization is straightforward from 2.2, and has
already been considered by E. A. Alekhno in the context of ordered Banach
algebras (see [3, 4]).

######  Definition 3.13.

Let $A$ be a Banach lattice algebra with identity $e$. We say that $p\in A$ is
_order idempotent_ if $p^{2}=p$ and $0\leq p\leq e$. We denote by $OI(A)$ the
set of order idempotents in $A$.

What follows is the generalization of 2.3. This result appears, with a
different proof, in [4, Corollary 2.2].

######  Proposition 3.14.

Let $A$ be a Banach lattice algebra with identity $e$. The set of order
idempotents $OI(A)$ forms a Boolean algebra with operations

$\overline{p}=e-p,\ p\vee q=p+q-pq,\ p\wedge q=pq,\quad\text{for }p,q\in
OI(A),$

with zero as the minimum and the identity $e$ as the maximum. Moreover,
suprema and infima coincide with those in $A$.

###### Proof.

By definition, $OI(A)$ is contained in $A_{e}$. Represent $A_{e}$ as a $C(K)$
space using 3.4. Then $p\in OI(A)$, seen as a continuous function in $K$,
satisfies $p^{2}=p$; that is, $p$ is a characteristic function of a clopen set
in $K$. Conversely, any characteristic function of a clopen set is an order
idempotent in $C(K)$, and therefore in $A$. This exhibits an identification of
$OI(A)$ with $\\{\,\chi_{C}:C\subseteq K\text{ clopen}\,\\}.$ The result now
clearly follows. ∎

######  Proposition 3.15.

In a Banach lattice algebra with identity, the set of order idempotents is
norm and order closed.

###### Proof.

Let $A$ be a Banach lattice algebra with identity $e$. Let $(p_{n})$ be a
sequence of order idempotents converging to $p$ in norm. Then
$p_{n}^{2}=p_{n}$ and $0\leq p_{n}\leq e$ for each $n\in\mathbb{N}$, and since
the product is continuous and the positive cone is closed, it follows that $p$
is an order idempotent. This shows that the set of order idempotents is norm
closed.

To show that it is also order closed, let $(p_{\alpha})$ be a net of order
idempotents converging to $p$ in order. The product need not be order
continuous, so one has to proceed with more care now. Since the positive cone
is always order closed, $0\leq p\leq e$ follows from $0\leq p_{\alpha}\leq e$
for each $\alpha$. By 3.4, $p$ and $p_{\alpha}$ commute, so

$|p^{2}-p_{\alpha}^{2}|\leq(p+p_{\alpha})|p-p_{\alpha}|\leq 2|p-p_{\alpha}|,$

where the first inequality follows from the submultiplicativity of the
modulus. This implies that $p_{\alpha}^{2}$ converges in order to $p^{2}$.
Since also $p_{\alpha}^{2}=p_{\alpha}$ for every $\alpha$, and the order limit
is unique, $p^{2}=p$. So $p$ is an order idempotent, and this shows that the
set of order idempotents is also order closed. ∎

The notion of order idempotent seems to abstract some of the most important
properties of band projections, but it relies on the existence of an identity
element. Of course, this poses no limitation in spaces of regular operators,
where the identity operator is always present. However, this is not the case
for general Banach lattice algebras. Can we somehow still abstract the
definition and properties of band projections to Banach lattice algebras
without identity?

Faced with a similar problem, K. Vala gives in [16] a notion of compact and
finite rank elements in arbitrary normed algebras as follows: for a normed
algebra $A$, he defines an element $a$ in $A$ to be _compact_ (resp.  _of
finite rank_) if the operator $L_{a}R_{a}$ is compact (resp. of finite rank).
In view of 2.5, it makes sense to take Vala’s approach and set the following
definition.

######  Definition 3.16.

Let $A$ be a Banach lattice algebra. An element $a\in A_{+}$ is called a _band
projection_ if $L_{a}R_{a}\colon A\to A$ is a band projection. The set of band
projections in $A$ is denoted

$BP(A)=\\{\,a\in A_{+}:L_{a}R_{a}\text{ is a band projection}\,\\}.$

It is also convenient to have the following notions.

######  Definition 3.17.

Let $A$ be a Banach lattice algebra. An element $a\in A_{+}$ is called a _left
band projection_ (resp.  _right band projection_) if $L_{a}$ (resp. $R_{a}$)
is a band projection. The sets of left and right band projections in $A$ are
denoted

$BP_{l}(A)=\\{\,a\in A_{+}:L_{a}\text{ is a band projection}\,\\},$
$BP_{r}(A)=\\{\,a\in A_{+}:R_{a}\text{ is a band projection}\,\\}.$

######  Remark.

Let $A$ be a Banach lattice algebra. According to 2.2, $a\in A_{+}$ is a band
projection if and only if

$0\leq axa\leq x\text{ and }a^{2}xa^{2}=axa\quad\text{for all }x\in A_{+}.$

Similarly, $a\in A_{+}$ is a left band projection if and only if

$0\leq ax\leq x\text{ and }a^{2}x=ax\quad\text{for all }x\in A_{+},$

and a right band projection if and only if

$0\leq xa\leq x\text{ and }xa^{2}=xa\quad\text{for all }x\in A_{+}.$

In particular, the sets $BP(A)$, $BP_{l}(A)$ and $BP_{r}(A)$ are all norm
closed.

If the multiplication in $A$ is jointly order continuous, it follows from
previous remark that $BP(A)$, $BP_{l}(A)$ and $BP_{r}(A)$ are order closed. In
general, however, this will not be true, as the following example shows.

######  Example 3.18.

Consider the Banach lattice $c$ of convergent real sequences, with the product
defined by

$x\star
y=\big{(}\lim_{n\to\infty}x_{n}\big{)}\big{(}\lim_{n\to\infty}y_{n}\big{)}e\quad\text{where
}e=(1,1,1,\ldots).$

Associativity of this product follows from

$\lim_{n\to\infty}(x\star
y)_{n}=\big{(}\lim_{n\to\infty}x_{n}\big{)}\big{(}\lim_{n\to\infty}y_{n}\big{)},$

bilinearity is clear, and it is not difficult to check that, under the usual
supremum norm, $c$ with $\star$ is a Banach algebra. Moreover, the product of
positive elements is positive, so $c$ is actually a Banach lattice algebra
(without identity).

It is clear that $c_{0}=BP(c)=BP_{l}(c)=BP_{r}(c)$. In particular,
$x_{n}=\sum_{i=1}^{n}e_{i}$, where $e_{i}$ denotes the $i$th coordinate
vector, is a (left and right) band projection. Also, $x_{n}\uparrow e$.
However, $e$ is not a (left or right) band projection, because for $x=e-e_{1}$
we have

$e\star x\star e=e\not\leq x.$

It is natural to ask what is the relation between the sets $BP(A)$,
$BP_{l}(A)$, $BP_{r}(A)$ and, in the case with identity, $OI(A)$. From the
definition, it should be clear that $BP_{l}(A)\cap BP_{r}(A)\subseteq BP(A)$,
and that, in the case with identity, $BP_{l}(A)=OI(A)=BP_{r}(A)$ (for $p\in
BP_{l}(A)$, just evaluate $L_{p}$ at the identity element, and similarly for
$p\in BP_{r}(A)$). In particular, $OI(A)=BP_{l}(A)\cap BP_{r}(A)\subseteq
BP(A)$. Is it true that $OI(A)=BP(A)$? This seems to be the case at least for
the classical examples of Banach lattice algebras.

######  Example 3.19.

  1. (i)

Let $X$ be a Dedekind complete Banach lattice. If $P\in
BP(\mathcal{L}_{r}(X))$, by 2.5, $P$ is a band projection, that is, $P\in
OI(\mathcal{L}_{r}(X))$.

  2. (ii)

Let $f\in BP(C(K))$, for a certain compact Hausdorff space $K$, and let
$\mathbb{1}$ be the constant one function in $C(K)$. Then
$L_{f}R_{f}(\mathbb{1})=(L_{f}R_{f})^{2}(\mathbb{1})$ implies $f^{2}=f^{4}$,
so $f$ must be the characteristic function of a clopen set. It follows that
$f$ is an order idempotent.

  3. (iii)

Let $G$ be a locally compact group and consider the Banach lattice algebra
$M(G)$. Since the identity $\delta_{e}$ is an atom, if $\mu\in OI(M(G))$,
inequality $0\leq\mu\leq\delta_{e}$ implies $\mu=\lambda\delta_{e}$ for some
$\lambda\geq 0$. Then $\mu^{2}=\mu$ (we denote $\mu^{2}=\mu\ast\mu$) forces
either $\lambda=1$ or $\lambda=0$. Hence $OI(M(G))=\\{0,\delta_{e}\\}$ is
trivial.

We are going to show that $BP(M(G))$ is also trivial. If $\mu\in BP(M(G))$,
evaluating $L_{\mu}R_{\mu}$ at $\delta_{e}$ yields that $\mu^{2}$ must be an
order idempotent. That is, either $\mu^{2}=0$ or $\mu^{2}=\delta_{e}$. By
direct computation

$\mu^{2}(G)=\iint_{G\times
G}\chi_{G}(st)\,d\mu(s)d\mu(t)=\int_{G}\mu(s^{-1}G)\,d\mu(s)=\mu(G)^{2},$

so if $\mu^{2}=0$, $\mu(G)^{2}=0$, and since $\mu\geq 0$, this implies
$\mu=0$. Now suppose $\mu\neq 0$, that is, $\mu^{2}=\delta_{e}$. We can
decompose $\mu=\mu_{d}+\mu_{c}$, where $\mu_{d}\geq 0$ is discrete (i.e., is
contained in the band generated by the atomic measures, which is precisely the
closed linear span of said measures, because $M(G)$ is order continuous) and
$\mu_{c}\geq 0$ is continuous (i.e., is contained in the disjoint complement).
Since

$\mu_{c}^{2}(\\{g\\})=\int_{G}\mu_{c}(\\{s^{-1}g\\})\,d\mu_{c}(s)=0\quad\text{for
every }g\in G,$

the measure $\mu_{c}^{2}$ cannot have any discrete component. However,

$\delta_{e}=\mu^{2}=\mu_{d}^{2}+\mu_{c}^{2}+\mu_{d}\ast\mu_{c}+\mu_{c}\ast\mu_{d}$

and since every measure involved is positive, we must have $\mu_{c}^{2}=0$.
Equality $\mu_{c}(G)^{2}=\mu_{c}^{2}(G)=0$ implies $\mu_{c}=0$ and so $\mu$ is
discrete. Write

$\mu=\sum_{g\in G}\lambda_{g}\delta_{g}$

for some $\lambda_{g}\in\mathbb{R}_{+}$. By definition,

$\delta_{e}=\mu^{2}=\sum_{g\in G}\bigg{(}\sum_{h\in
G}\lambda_{h}\lambda_{h^{-1}g}\bigg{)}\delta_{g},$

and since the $\lambda_{h}$ are positive, this can only happen if
$\lambda_{h}=0$ for all $h\neq e$. Therefore $\mu=\lambda_{e}\delta_{e}$, and
from $\mu^{2}=\mu^{4}$ it follows that $\lambda_{e}=1$. We have thus shown
that if $\mu\in BP(M(G))$ is different from $0$, it must be $\delta_{e}$, as
wanted.

  4. (iv)

A. W. Wickstead provided in [19] an explicit list of all two-dimensional
Banach lattice algebras. In the Banach lattice algebras with identity from the
list, one can check, by means of direct computation, that the order
idempotents coincide with the band projections.

Contrary to this evidence, it is not true that in Banach lattice algebras with
identity all band projections are order idempotents, and an example already
exists in dimension 3.

######  Example 3.20.

Consider $\mathbb{R}^{2}$ with the usual lattice structure and any lattice
norm, and let $\mathcal{L}_{r}(\mathbb{R}^{2})$ be its space of regular
operators. The space $\mathcal{L}_{r}(\mathbb{R}^{2})$ is a Banach lattice
algebra, which we may identify with $M_{2}(\mathbb{R})$, the space of $2\times
2$ real matrices, where the order is defined entry-wise and the product is the
composition (that is, the usual matrix product). Consider the set of upper
triangular matrices

$\displaystyle A$
$\displaystyle=\operatorname{span}\left\\{\begin{pmatrix}1&0\\\
0&0\end{pmatrix},\begin{pmatrix}0&1\\\ 0&0\end{pmatrix},\begin{pmatrix}0&0\\\
0&1\end{pmatrix}\right\\}$ $\displaystyle=\left\\{\,\begin{pmatrix}a&b\\\
0&c\end{pmatrix}:\,a,b,c\in\mathbb{R}\right\\}.$

Being spanned by disjoint elements, $A$ is a 3-dimensional sublattice of
$M_{2}(\mathbb{R})$. Moreover, one can easily check that the product of upper-
triangular matrices is again upper-triangular. Hence, $A$ is a Banach lattice
algebra with identity when equipped with the order, product, and norm
inherited from $M_{2}(\mathbb{R})$.

Now consider $p=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}$. Clearly $p\geq 0$
and, for all $a,b,c\in\mathbb{R}$,

$L_{p}R_{p}\begin{pmatrix}a&b\\\ 0&c\end{pmatrix}=\begin{pmatrix}0&0\\\
0&0\end{pmatrix}.$

Hence $L_{p}R_{p}=0$; in particular, $p\in BP(A)$. However, $p\not\leq I_{2}$,
so $p\not\in OI(A)$.

Let us compute exactly $BP(A)$ and $OI(A)$. It is direct to check that

$OI(A)=\left\\{\begin{pmatrix}0&0\\\ 0&0\end{pmatrix},\begin{pmatrix}1&0\\\
0&0\end{pmatrix},\begin{pmatrix}0&0\\\ 0&1\end{pmatrix},\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\right\\}.$

Now let $\bar{p}=\begin{pmatrix}p_{1}&p_{2}\\\ 0&p_{3}\end{pmatrix}\in BP(A)$.
For any $a,b,c\in\mathbb{R}_{+}$:

$\begin{pmatrix}p_{1}&p_{2}\\\ 0&p_{3}\end{pmatrix}\begin{pmatrix}a&b\\\
0&c\end{pmatrix}\begin{pmatrix}p_{1}&p_{2}\\\
0&p_{3}\end{pmatrix}=\begin{pmatrix}p_{1}^{2}a&p_{1}p_{2}a+p_{1}p_{3}b+p_{2}p_{3}c\\\
0&cp_{3}^{2}\end{pmatrix}\leq\begin{pmatrix}a&b\\\ 0&c\end{pmatrix}.$

Setting $b=0$ we get the inequality in the upper right entry

$p_{2}(ap_{1}+cp_{3})\leq 0.$

Since $a$ and $c$ can take any positive value, and $p_{1},p_{2},p_{3}$ are
positive, either $p_{2}=0$ or $p_{1}=p_{3}=0$.

  1. (i)

If $p_{2}=0$, equality ${\bar{p}}^{2}={\bar{p}}^{4}$ implies
$p_{1}^{2}=p_{1}^{4}$, so $p_{1}=0$ or $p_{1}=1$, and similarly $p_{3}=0$ or
$p_{3}=1$. The only band projections in this case are the order idempotents.

  2. (ii)

If $p_{2}\neq 0$, then $p_{1}=p_{3}=0$, and $\bar{p}$ is a band projection for
any $p_{2}\geq 0$.

Putting together these computations:

$BP(A)=OI(A)\cup\left\\{\begin{pmatrix}0&p\\\
0&0\end{pmatrix}:p\in\mathbb{R}_{+}\right\\}.$

######  Remark.

This example already illustrates a difference between $BP(A)$ and $OI(A)$:
while order idempotents commute by 3.4, band projections need not commute.
After this observation, it is not surprising that $BP(A)$ need not be a
Boolean algebra: in previous example, $BP(A)$ is not closed under taking
suprema.

By contrast, the elements of $BP_{l}(A)\cap BP_{r}(A)$ always commute. Indeed,
let $a,b\in BP_{l}(A)\cap BP_{r}(A)$. Since $L_{a},L_{b},R_{a},R_{b}$ are band
projections, they commute, so $aba=L_{a}L_{b}(a)=L_{b}L_{a}(a)=ba^{2}$ and
$aba=R_{a}R_{b}(a)=R_{b}R_{a}(a)=a^{2}b$, which gives
$ab=a^{2}b=aba=ba^{2}=ba$.

One could think that previous example is somewhat pathological, because
$p^{2}=0$ and $L_{p}R_{p}=0$. Maybe this is the only reason a band projection
is not an order idempotent. It is not, and a counterexample is coming. But
first an observation.

######  Remark.

Let $\\{A_{i}\\}_{i\in I}$ be a family of Banach lattice algebras indexed on a
set $I$. Consider $A=\ell_{p}(A_{i})$ for some $1\leq p\leq\infty$. We claim
that

$BP(\ell_{p}(A_{i}))=\prod_{i\in I}BP(A_{i}).$

Indeed, a positive element $(p_{i})_{i\in I}$ is a band projection in
$\ell_{p}(A_{i})$ if and only if for all $(x_{i})_{i\in
I}\in\ell_{p}(A_{i})_{+}$:

$0\leq(p_{i})(x_{i})(p_{i})\leq(x_{i})\quad\text{and}\quad(p_{i})(x_{i})(p_{i})=(p_{i})^{2}(x_{i})(p_{i})^{2}.$

This is the same as saying that $0\leq p_{i}x_{i}p_{i}\leq x_{i}$ and
$p_{i}x_{i}p_{i}=p_{i}^{2}x_{i}p_{i}^{2}$ whenever $x_{i}\in(A_{i})_{+}$, for
all $i\in I$. That is, $(p_{i})\in BP(\ell_{p}(A_{i}))$ if and only if
$p_{i}\in BP(A_{i})$.

Similarly, one can check that

$BP_{l}(\ell_{p}(A_{i}))=\prod_{i\in I}BP_{l}(A_{i})\quad\text{and}\quad
BP_{r}(\ell_{p}(A_{i}))=\prod_{i\in I}BP_{r}(A_{i}).$

If each $A_{i}$ has an identity, then also

$OI(\ell_{\infty}(A_{i}))=\prod_{i\in I}OI(A_{i}).$

######  Example 3.21.

Let $A$ be the Banach lattice algebra of Example 3.20, and consider
$\ell_{\infty}(A,A)$. By previous remark, the element

$q=\left(\begin{pmatrix}1&0\\\ 0&0\end{pmatrix},\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\right)$

is a band projection that is not order idempotent. In this case, $q^{2}\neq 0$
and $L_{q}R_{q}\neq 0$.

At this point, it is clear that the relation between order idempotents and
band projections in Banach lattice algebras with identity is not trivial. The
forthcoming results aim at a better understanding of it.

######  Proposition 3.22.

Let $A$ be a Banach lattice algebra with identity $e$, and let $p\in BP(A)$.
Then the following are equivalent:

  1. (i)

$p\in OI(A)$,

  2. (ii)

$p$ is idempotent,

  3. (iii)

$p\in A_{e}$,

  4. (iv)

$(\lambda e+p)^{-1}\geq 0$ for some $\lambda>\|p\|$.

###### Proof.

Condition $(i)$ clearly implies $(ii)$. If $p$ is idempotent,
$L_{p}R_{p}(e)=p^{2}=p$, and since $0\leq L_{p}R_{p}(e)\leq e$, we have $p\in
A_{e}$. So $(ii)$ implies $(iii)$. If $p\in A_{e}$, then, using that $A_{e}$
is a subalgebra and sublattice of $A$, we can consider the restriction
$L_{p}R_{p}\colon A_{e}\to A_{e}$. This operator is a band projection on
$A_{e}$. Represent $A_{e}$ as a $C(K)$, for a certain compact Hausdorff space
$K$. Since $BP(C(K))=OI(C(K))$, it follows that $p\in OI(A_{e})$. But the
order and product in $A_{e}$ are the same as those in $A$, thus $p\in OI(A)$.
This shows that $(ii)$ implies $(iii)$, and also representing $A_{e}$ as
$C(K)$ it is direct from $BP(C(K))=OI(C(K))$ that $(iii)$ implies $(i)$.

It only remains to show the equivalence between $(iii)$ and $(iv)$. Since
$p\in BP(A)$, we have $p^{2}=L_{p}R_{p}(e)=(L_{p}R_{p})^{2}(e)=p^{4}$.
Inductively, $p^{2}=p^{2n}$ and $p^{3}=p^{2n+1}$ for all $n\geq 1$. Using this
fact and the Neumann series, for every $\lambda>\max\\{1,\|p\|\\}$,

$\displaystyle\lambda(\lambda e+p)^{-1}=(e+p/\lambda)^{-1}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(-p)^{n}}{\lambda^{n}}$
$\displaystyle=e-\frac{p}{\lambda}+\frac{p^{2}}{\lambda^{2}}\sum_{n=0}^{\infty}\frac{1}{\lambda^{2n}}-\frac{p^{3}}{\lambda^{3}}\sum_{n=0}^{\infty}\frac{1}{\lambda^{2n}}$
$\displaystyle=e-\frac{p}{\lambda}+\frac{p^{2}}{\lambda^{2}-1}-\frac{p^{3}}{\lambda(\lambda^{2}-1)}$
$\displaystyle=\left(e-\frac{p}{\lambda}\right)\left(e+\frac{p^{2}}{\lambda^{2}-1}\right)$

Suppose that $p\in A_{e}$. Then $p\leq\|p\|_{e}e$ and, by the previous
computation, $(\|p\|_{e}+2)e+p$ is invertible with positive inverse. This
shows that $(iii)$ implies $(iv)$. Conversely, if ${(\lambda e+p)^{-1}}\geq 0$
for some $\lambda>\|p\|$, either $p=0$, in which case the result is trivial,
or $p\neq 0$, in which case $\lambda>\|p\|_{e}\geq 1$‡‡‡Due to the fact that
the spectral radius of $p$ is $1$, see Section 3.4. and then

$(\lambda e+p)(\lambda e-p)=\lambda^{2}e-p^{2}\geq 0.$

Multiplying by $(\lambda e+p)^{-1}$ on the left

$\lambda e-p=(\lambda e+p)^{-1}(\lambda^{2}e-p^{2})\geq 0$

because the product of positive elements is positive. Hence $p\in A_{e}$. ∎

In order to delve deeper into the relation between band projections and order
idempotents, we need to introduce spectra in our study.

###  3.4 Complexification and spectra

To talk about spectra and see their relation with band projections and order
idempotents, we need to work over the complex field. So first we need to
“complexify” some of the notions introduced so far.

Let $A$ be a Banach lattice algebra. Applying the procedure for the
complexification of Banach lattices, as exposed in Section 2.4, one obtains
the complex Banach lattice $A_{\mathbb{C}}$ with norm
$\|z\|_{\mathbb{C}}=\||z|\|$. With the natural product

$(x_{1}+ix_{2})(y_{1}+iy_{2})=(x_{1}y_{1}-x_{2}y_{2})+i(x_{1}y_{2}+x_{2}y_{1})\quad\text{where
}x_{1},x_{2},y_{1},y_{2}\in A,$

the space $A_{\mathbb{C}}$ becomes a complex algebra, that has an identity
precisely when $A$ does. This product is compatible with the complex lattice
structure in the sense that $|z_{1}z_{2}|\leq|z_{1}||z_{2}|$, for all
$z_{1},z_{2}\in A_{\mathbb{C}}$. This fact is not trivial, and a proof may be
found in [8]. The submultiplicativity of the norm on $A$ immediately implies
$\|z_{1}z_{2}\|_{\mathbb{C}}\leq\|z_{1}\|_{\mathbb{C}}\|z_{2}\|_{\mathbb{C}}$
for $z_{1},z_{2}\in A_{\mathbb{C}}$. Hence $A_{\mathbb{C}}$ is also a complex
Banach algebra with norm $\|{\cdot}\|_{\mathbb{C}}$. The space
$A_{\mathbb{C}}$ with its structures of complex Banach lattice and complex
Banach algebra is called a _complex Banach lattice algebra_. When $A$ has an
identity, we emphasize it by saying that $A_{\mathbb{C}}$ is a _complex Banach
lattice algebra with identity_.

######  Example 3.23.

  1. (i)

If $K$ is a compact Hausdorff space, the complexification of $C(K)$ may be
identified with $C(K,\mathbb{C})$, the space of continuous complex-valued
functions on $K$.

  2. (ii)

If $X$ is an order complete Banach lattice, the complexification of
$\mathcal{L}_{r}(X)$ may be identified with the space of regular operators
over $X_{\mathbb{C}}$ (see Section 2.4).

Next we want to present a result analogous to 3.4 in complex Banach lattice
algebras. If $X$ is a Banach lattice, a subspace $I\subseteq X_{\mathbb{C}}$
is called an _ideal_ if $x\in I$ and $|y|\leq|x|$ imply $y\in I$. It is not
difficult to check that $I$ is the complexification of the ideal $I\cap X$ of
$X$, and that the intersection of all the ideals containing $z\in
X_{\mathbb{C}}$ is given by

$(I_{z})_{\mathbb{C}}=\\{\,w\in X_{\mathbb{C}}:|w|\leq\lambda|z|\text{ for
some }\lambda\in\mathbb{R}_{+}\,\\}.$

This is called the _ideal generated by $z$_.

An ideal $B\subseteq X_{\mathbb{C}}$ is called a _band_ (resp.  _projection
band_) if $B\cap X$ is a band (resp. projection band) in $X$. Again, one can
check that an ideal $B\subseteq X$ is a projection band if and only if
$X_{\mathbb{C}}=B\oplus B^{d}$, where

$B^{d}=\\{\,z\in X_{\mathbb{C}}:|w|\wedge|z|=0\text{ for all }w\in B\,\\}.$

Finally, some common notation: for $z\in A_{\mathbb{C}}$ denote its spectrum
by $\sigma(z)$ and its spectral radius by $r(z)$.

######  Theorem 3.24.

Let $A_{\mathbb{C}}$ be a complex Banach lattice algebra with identity $e$ and
let $(A_{\mathbb{C}})_{e}$ be the ideal generated by $e$ in $A_{\mathbb{C}}$.
Then $(A_{\mathbb{C}})_{e}=(A_{e})_{\mathbb{C}}$, i.e., $(A_{\mathbb{C}})_{e}$
is the complexification of the Banach lattice algebra $A_{e}$. Moreover:

  1. (i)

The space $(A_{\mathbb{C}})_{e}$ can be isometrically identified with
$C(K,\mathbb{C})$, for some compact Hausdorff $K$.

  2. (ii)

The ideal $(A_{\mathbb{C}})_{e}$ is a projection band in $A_{\mathbb{C}}$.

  3. (iii)

The space $(A_{\mathbb{C}})_{e}$ is an inverse closed subalgebra of
$A_{\mathbb{C}}$.

###### Proof.

Since $(A_{\mathbb{C}})_{e}\cap A=A_{e}$, it follows that
$(A_{\mathbb{C}})_{e}=(A_{e})_{\mathbb{C}}$. It is then a direct consequence
of 3.4 that $(A_{\mathbb{C}})_{e}$, with the complexification of the norm
$\|{\cdot}\|_{e}$, is isometric to $C(K,\mathbb{C})$. To prove $(i)$ it only
remains to check that the norm $\|{\cdot}\|_{e,\mathbb{C}}$ is equal to
$\|{\cdot}\|_{\mathbb{C}}$. Being $A_{e}$ an inverse closed subalgebra, it is
easy to check that the spectrum of an element in $A_{e}$ (i.e., its range when
viewed as a function in $C(K,\mathbb{C})$) is the same as its spectrum in
$A_{\mathbb{C}}$. Therefore $\|z\|\geq r(z)=\|z\|_{e}$, and since we are
assuming $\|e\|=1$ in our definition of Banach lattice algebras,
$\|z\|\leq\|z\|_{e}$. Joining both inequalities, $\|z\|=\|z\|_{e}=r(z)$ and
$(i)$ is proved. From 3.4 clearly follows $(ii)$, and exactly the same proof
as in 3.4 can be used to show $(iii)$. ∎

Previous result completes the generalization from 2.7 to 3.4. It also
completes Example 3.12. Some remarks regarding the spectrum of order
idempotents and band projections are in order.

######  Remark.

  1. (i)

Let $p$ be an order idempotent. When we represent $(A_{\mathbb{C}})_{e}$ as a
space of continuous functions, $p$ is the characteristic function of a clopen
set. As such, $\sigma(p)=\\{0,1\\}$ whenever $p\neq 0,e$. By previous theorem,
$\|p\|=r(p)=1$.

  2. (ii)

If $p\in BP(A)$, either $p^{2}=0$, in which case $r(p)=0$ and
$\sigma(p)=\\{0\\}$ (the complexification of Example 3.20 serves as an example
of this), or $p^{2}\neq 0$, in which case
$\|p\|=r(p)=\lim_{n}\|p^{2n}\|^{1/2n}=1$ (because $p^{2}$ is order idempotent,
so $\|p^{2}\|=1$, and also $(p^{2})^{n}=p^{2}$). In both cases,
$\sigma(p^{2})\subseteq\\{0,1\\}$ implies $\sigma(p)\subseteq\\{-1,0,1\\}$.

  3. (iii)

For $p\in BP(A)$ to be an order idempotent, it is necessary that
$\sigma(p)=\\{0,1\\}$. However, it is not sufficient: one can easily check
that the element $q$ in Example 3.21 satisfies $\sigma(q)=\\{0,1\\}$, yet it
is not an order idempotent.

The following is an example of a band projection with spectrum $\\{0,1,-1\\}$.

######  Example 3.25.

In the Banach lattice algebra $M_{3}(\mathbb{R})$ (see Example 3.20), consider
the 3-dimensional subspace:

$\displaystyle A$
$\displaystyle=\operatorname{span}\left\\{\begin{pmatrix}1&0&0\\\ 0&0&0\\\
0&0&1\end{pmatrix},\begin{pmatrix}0&0&1\\\ 0&0&0\\\
1&0&0\end{pmatrix},\begin{pmatrix}0&0&0\\\ 0&1&0\\\
0&0&0\end{pmatrix}\right\\}$ $\displaystyle=\left\\{\begin{pmatrix}x&0&y\\\
0&z&0\\\ y&0&x\end{pmatrix}\,:\,x,y,z\in\mathbb{R}\right\\}.$

Being generated by pairwise disjoint elements, $A$ is a sublattice. Moreover,
the identities

$\begin{pmatrix}x&0&y\\\ 0&z&0\\\
y&0&x\end{pmatrix}\begin{pmatrix}x^{\prime}&0&y^{\prime}\\\ 0&z^{\prime}&0\\\
y^{\prime}&0&x^{\prime}\end{pmatrix}=\begin{pmatrix}xx^{\prime}+yy^{\prime}&0&xy^{\prime}+x^{\prime}y\\\
0&zz^{\prime}&0\\\
xy^{\prime}+x^{\prime}y&0&xx^{\prime}+yy^{\prime}\end{pmatrix},$
$\begin{pmatrix}x&0&y\\\ 0&z&0\\\
y&0&x\end{pmatrix}^{-1}=\frac{1}{x^{2}-y^{2}}\begin{pmatrix}x&0&-y\\\
0&z^{-1}(x^{2}-y^{2})&0\\\ -y&0&x\end{pmatrix},\quad\text{when both }x\neq
y\text{ and }z\neq 0,$

prove that $A$ is an inverse closed subalgebra, with identity because the
identity matrix is contained in $A$. Therefore, $A$ is a Banach lattice
algebra with the inherited product, order and norm. Its complexification
consists in taking matrices with complex entries.

Set

$p=\begin{pmatrix}0&0&1\\\ 0&0&0\\\ 1&0&0\end{pmatrix}.$

It is not difficult to check that $pap=a$ for every $a\in
A_{\mathbb{C}}$.§§§Using elementary matrix theory: right multiplication by $p$
swaps the first and third columns of $a\in A_{\mathbb{C}}$, while left
multiplication swaps the first and third rows of $a$; that is, $pap$ is the
reflection of $a$ with respect to both of its diagonals. The elements of $A$,
however, are invariant under this operation. Therefore
$L_{p}R_{p}=I_{A_{\mathbb{C}}}$, and this implies $p\in BP(A)$. The spectrum
of $p$ is the set of roots of its characteristic polynomial

$\det(p-\lambda I_{3})=\begin{vmatrix}-\lambda&0&1\\\ 0&-\lambda&0\\\
1&0&-\lambda\end{vmatrix}=-\lambda(\lambda+1)(\lambda-1),$

which is $\sigma_{A}(p)=\sigma(p)=\\{-1,0,1\\}$.

Next we want to provide a result that relates spectra with order idempotents.
We start with a simple observation.

######  Remark.

In $C(K)$ order idempotents are characterized by their spectrum. More
precisely, $p\in OI(C(K))$ if and only if $\sigma(p)\subseteq\\{0,1\\}$.

We also need the generalization of 2.11 to arbitrary complex Banach lattice
algebras with identity.

######  Corollary 3.26.

Let $A_{\mathbb{C}}$ be a complex Banach lattice algebra with identity. Let
$a\in A_{+}$ be an invertible element with positive inverse. Then $a\in A_{e}$
if and only if $\sigma(a)\subseteq\mathbb{R}_{+}$.

###### Proof.

Let $a\in A_{+}$ be an invertible element such that $a^{-1}\in A_{+}$. Then
$R_{a}\colon A\to A$, being a positive operator with positive inverse
$R_{a^{-1}}$, is a lattice isomorphism. Note that

$(R_{a}-\lambda I_{A_{\mathbb{C}}})(x)=x(a-\lambda e)=R_{a-\lambda
e}(x)\quad\text{for all }x\in A_{\mathbb{C}},$

where $e$ denotes the identity element of $A_{\mathbb{C}}$. It follows that
$\sigma(R_{a})\subseteq\sigma(a)$, since $R_{a-\lambda e}$ is invertible
whenever $a-\lambda e$ is. Suppose $\sigma(a)\subseteq\mathbb{R}_{+}$. Then
$\sigma(R_{a})\subseteq\mathbb{R}_{+}$ and $R_{a}$ is a lattice isomorphism.
By 2.11, $R_{a}\in\mathcal{Z}(A)$, which readily implies $a\in A_{e}$. The
converse is direct if we see $a\in A_{e}$ as a positive continuous function on
a compact space. ∎

######  Proposition 3.27.

Let $A_{\mathbb{C}}$ be a complex Banach lattice algebra with identity. Let
$a\in A_{+}$ be an invertible element such that:

  1. (i)

$a^{-1}\in A_{+}$,

  2. (ii)

$\sigma(a)\subseteq\\{\lambda,\lambda+1\\}$ for some
$\lambda\in\mathbb{R}_{+}$.

Then $a-\lambda e\in OI(A)$, where $e$ is the identity of $A$.

###### Proof.

Since $a$ has positive spectrum, and $a^{-1}\in A_{+}$, from previous
corollary it follows that $a\in A_{e}$. Then $a-\lambda e\in A_{e}$, and its
spectrum satisfies, by hypothesis, $\sigma(a-\lambda e)\subseteq\\{0,1\\}$.
Since $A_{e}$ is essentially a space of continuous functions, it follows that
$a-\lambda e$ is an order idempotent. ∎

###  3.5 Inner band projections

In this section the construction of inner band projections explained in
Section 2.5 is extended to general Banach lattice algebras. Let $A$ be an
order complete Banach lattice algebra and let
$\\{p_{\lambda}\\}_{\lambda\in\Lambda}\subseteq BP_{l}(A)\cap BP_{r}(A)$ be
such that $p_{\alpha}p_{\beta}=\delta_{\alpha\beta}p_{\alpha}$ for all
$\alpha,\beta\in\Lambda$ (in particular, if $A$ has an identity, we can take
the $p_{\lambda}$ to be pairwise disjoint order idempotents). For
$\Gamma\subseteq\Lambda\times\Lambda$ define the map

$\begin{array}[]{cccc}P_{\Gamma}\colon&A_{+}&\longrightarrow&A_{+}\\\
&x&\longmapsto&\displaystyle\bigvee_{(\alpha,\beta)\in\Gamma}p_{\alpha}xp_{\beta}\\\
\end{array}.$ (6)

Note that this is well-defined, because
$p_{\alpha}xp_{\beta}=L_{p_{\alpha}}R_{p_{\beta}}(x)\leq x$ (recall that
$L_{p_{\alpha}}R_{p_{\beta}}$ is a band projection on $A$), and $A$ is order
complete. If we do not assume that $A$ is order complete, the coming results
work just fine, as long as we ask for this supremum to exist. In particular,
if $\Gamma$ is finite, everything works out.

######  Lemma 3.28.

Let $A$ be a Banach lattice algebra, and let
$(p_{\lambda})_{\lambda\in\Lambda}\subseteq BP_{l}(A)\cap BP_{r}(A)$ be such
that $p_{\alpha}p_{\beta}=\delta_{\alpha\beta}p_{\alpha}$. If
$\Phi\subseteq\Lambda\times\Lambda$ is finite, then

$\bigvee_{(\alpha,\beta)\in\Phi}p_{\alpha}xp_{\beta}=\sum_{(\alpha,\beta)\in\Phi}p_{\alpha}xp_{\beta},\quad\text{for
all }x\in A_{+}.$

###### Proof.

For every $(\alpha,\beta)\in\Phi$, $L_{p_{\alpha}}R_{p_{\beta}}$ is a band
projection; to ease the notation, denote it by $L_{\alpha}R_{\beta}$. Whenever
$(\gamma,\delta)\neq(\alpha,\beta)$:

$(L_{\alpha}R_{\beta})(L_{\gamma}R_{\delta})(x)=p_{\alpha}p_{\gamma}xp_{\delta}p_{\beta}=0$

because either $p_{\alpha}p_{\gamma}=0$ or $p_{\delta}p_{\beta}=0$. Therefore
$(L_{\alpha}R_{\beta})(L_{\gamma}R_{\delta})=0$. Being band projections, this
implies that the bands determined by $L_{\alpha}R_{\beta}$ and
$L_{\gamma}R_{\delta}$ are disjoint. Thus

$p_{\alpha}xp_{\beta}\wedge
p_{\gamma}xp_{\delta}=(L_{\alpha}R_{\beta})(x)\wedge(L_{\gamma}R_{\delta})(x)=0,$

or, equivalently, $p_{\alpha}xp_{\beta}\vee
p_{\gamma}xp_{\delta}=p_{\alpha}xp_{\beta}+p_{\gamma}xp_{\delta}$. Since this
is true for any pair of indices, the result follows by induction. ∎

######  Theorem 3.29.

The map $P_{\Gamma}$, as defined in (6), extends to a unique band projection
on $A$.

###### Proof.

Note first that, using previous lemma,

$P_{\Gamma}x=\bigvee_{(\alpha,\beta)\in\Gamma}p_{\alpha}xp_{\beta}=\sup_{\Phi\in\operatorname{Fin}(\Gamma)}\bigvee_{(\alpha,\beta)\in\Phi}p_{\alpha}xp_{\beta}=\sup_{\Phi\in\operatorname{Fin}(\Gamma)}\sum_{(\alpha,\beta)\in\Phi}p_{\alpha}xp_{\beta}.$
(7)

It follows from the definition that $P_{\Gamma}(x+y)\leq
P_{\Gamma}(x)+P_{\Gamma}(y)$. For every
$\Phi,\Psi\in\operatorname{Fin}(\Gamma)$:

$\displaystyle\sum_{(\alpha,\beta)\in\Phi}p_{\alpha}xp_{\beta}+\sum_{(\gamma,\delta)\in\Psi}p_{\gamma}yp_{\delta}$
$\displaystyle\leq\sum_{(\alpha,\beta)\in\Phi\cup\Psi}p_{\alpha}xp_{\beta}+\sum_{(\gamma,\delta)\in\Phi\cup\Psi}p_{\gamma}yp_{\delta}$
$\displaystyle=\sum_{(\alpha,\beta)\in\Phi\cup\Psi}p_{\alpha}(x+y)p_{\beta}\leq
P_{\Gamma}(x+y).$

Taking suprema separately in $\Phi$ and $\Psi$ we conclude that $P_{\Gamma}$
is additive. Hence it extends to a unique positive operator $P_{\Gamma}\colon
A\to A$. It is clear from the definition that $0\leq P_{\Gamma}\leq I_{A}$. To
show that it is a band projection, it only remains to prove that it is
idempotent. By previous inequality, $P_{\Gamma}^{2}\leq P_{\Gamma}$. To see
the other inequality, note that for an arbitrary
$\Phi\in\operatorname{Fin}(\Gamma)$:

$\displaystyle
P_{\Gamma}(P_{\Gamma}(x))=\sum_{(\alpha,\beta)\in\Phi}p_{\alpha}P_{\Gamma}(x)p_{\beta}$
$\displaystyle\geq\sum_{(\alpha,\beta)\in\Phi}p_{\alpha}\bigg{(}\sum_{(\gamma,\delta)\in\Phi}p_{\gamma}xp_{\delta}\bigg{)}p_{\beta}$
$\displaystyle=\sum_{(\alpha,\beta)\in\Phi}\sum_{(\gamma,\delta)\in\Phi}p_{\alpha}p_{\gamma}xp_{\delta}p_{\beta}$
$\displaystyle=\sum_{(\alpha,\beta)\in\Phi}p_{\alpha}xp_{\beta}.$

Inequality $P_{\Gamma}^{2}\geq P_{\Gamma}$ follows from taking supremum on
$\Phi\in\operatorname{Fin}(\Gamma)$. ∎

######  Definition 3.30.

Let $A$ be a Banach lattice algebra and let $P\colon A\to A$ be a band
projection. If there exist $(p_{\lambda})_{\lambda\in\Lambda}\subseteq
BP_{l}(A)\cap BP_{r}(A)$ such that
$p_{\alpha}p_{\beta}=\delta_{\alpha\beta}p_{\alpha}$ for all
$\alpha,\beta\in\Lambda$, and such that $P=P_{\Gamma}$ for some
$\Gamma\subseteq\Lambda\times\Lambda$ as defined in Eq. 6, then we say that
$P$ is an _inner band projection_. We say that the band $P(A)$ is an _inner
projection band_.

######  Example 3.31.

Let $K$ be a compact Hausdorff space. Even though $C(K)$ may not be order
complete, it still makes sense to consider inner band projections, as long as
the supremum in Eq. 6 is well-defined. In fact, in $C(K)$, any band projection
is inner, because it consists in multiplying by the characteristic function of
a clopen set, which is an order idempotent.

######  Remark.

Let $A$ be an order complete Banach lattice algebra. Since in
$\mathcal{L}_{r}(A)$ the supremum of an increasing net is taken pointwise,
equation (7) implies

$P_{\Gamma}=\operatorname*{o-lim}_{\Phi\in\operatorname{Fin}(\Gamma)}\sum_{(\alpha,\beta)\in\Phi}L_{\alpha}R_{\beta},$

where $\operatorname*{o-lim}$ denotes the limit of the net in the order
convergence (see [15, Section 2.4]). In particular, if $A$ is order
continuous, we have

$P_{\Gamma}(x)=\lim_{\Phi\in\operatorname{Fin}(\Gamma)}\sum_{(\alpha,\beta)\in\Phi}p_{\alpha}xp_{\beta}=\sum_{(\alpha,\beta)\in\Gamma}p_{\alpha}xp_{\beta}.$

######  Proposition 3.32.

Let $A$ be an order complete Banach lattice algebra, and let
$(p_{\lambda})_{\lambda\in\Lambda}\subseteq BP_{l}(A)\cap BP_{r}(A)$ be such
that $p_{\alpha}p_{\beta}=\delta_{\alpha\beta}p_{\alpha}$. The set
$\\{\,P_{\Gamma}:\Gamma\subseteq\Lambda\times\Lambda\,\\}$ forms a Boolean
algebra with maximum element $P_{\Lambda\times\Lambda}$, minimum
$P_{\varnothing}$ and operations

$P_{\Gamma}\wedge P_{\Delta}=P_{\Gamma\cap\Delta},\;P_{\Gamma}\vee
P_{\Delta}=P_{\Gamma\cup\Delta},\;\overline{P}_{\Gamma}=P_{\overline{\Gamma}},\quad\text{for
}\Gamma,\Delta\subseteq\Lambda\times\Lambda,$

where $\overline{\Gamma}=\Lambda\times\Lambda\setminus\Gamma$ denotes the
complementary set.

###### Proof.

On the set of inner band projections, consider the usual order of operators.
Let $\Gamma,\Delta\subseteq\Lambda\times\Lambda$. We first want to prove that
$P_{\Gamma}\wedge P_{\Delta}=P_{\Gamma\cap\Delta}$. For every $x\in A_{+}$,
the net
$(\sum_{(\gamma,\delta)\in\Psi}p_{\gamma}xp_{\delta})_{\Psi\in\operatorname{Fin}(\Delta)}$
increases to $P_{\Delta}(x)$. For every $(\alpha,\beta)\in\Gamma$, using that
band projections are order continuous:

$L_{p_{\alpha}}R_{p_{\beta}}(P_{\Delta}(x))=\operatorname*{o-lim}_{\Psi\in\operatorname{Fin}(\Delta)}\sum_{(\gamma,\delta)\in\Psi}L_{p_{\alpha}}R_{p_{\beta}}(p_{\gamma}xp_{\delta})=\begin{cases}p_{\alpha}xp_{\beta}&\text{if
}(\alpha,\beta)\in\Delta,\\\ 0&\text{if
}(\alpha,\beta)\not\in\Delta.\end{cases}$

Therefore

$P_{\Gamma}(P_{\Delta}(x))=\bigvee_{(\alpha,\beta)\in\Gamma}L_{p_{\alpha}}R_{p_{\beta}}(P_{\Delta}(x))=\bigvee_{(\alpha,\beta)\in\Gamma\cap\Delta}p_{\alpha}xp_{\beta}=P_{\Gamma\cap\Delta}(x),$

as wanted

To check the formula for the supremum, first consider the situation
$\Gamma\cap\Delta=\varnothing$. In this case, $P_{\Gamma}P_{\Delta}=0$, and

$P_{\Gamma}+P_{\Delta}=P_{\Gamma}\vee P_{\Delta}\leq P_{\Gamma\cup\Delta}$

because both $P_{\Gamma}\leq P_{\Gamma\cup\Delta}$ and $P_{\Delta}\leq
P_{\Gamma\cup\Delta}$. To see the reverse inequality, take finite sets
$\Phi\in\operatorname{Fin}(\Gamma)$ and $\Psi\in\operatorname{Fin}(\Delta)$.
Using that they are disjoint:

$P_{\Gamma}(x)+P_{\Delta}(x)\geq\sum_{(\alpha,\beta)\in\Phi}p_{\alpha}xp_{\beta}+\sum_{(\gamma,\delta)\in\Psi}p_{\gamma}xp_{\delta}=\sum_{(\alpha,\beta)\in\Phi\cup\Psi}p_{\alpha}xp_{\beta}.$

The right hand side is an arbitrary finite subset of $\Gamma\cup\Delta$;
taking supremum over these yields $P_{\Gamma}+P_{\Delta}\geq
P_{\Gamma\cup\Delta}$. In particular, when $\Delta=\overline{\Gamma}$, we get
the desired formula for the complementary. Finally, for arbitrary $\Gamma$ and
$\Delta$ (not necessarily disjoint), write their union as the three disjoint
unions
$\Gamma\cup\Delta=(\overline{\Gamma}\cap\Delta)\cup(\Gamma\cap\overline{\Delta})\cup(\Gamma\cap\Delta)$
and apply the previous formulae. ∎

######  Remark.

When $A=\mathcal{L}_{r}(X)$, for some order complete Banach lattice $X$, the
Boolean algebra of inner band projections is isomorphic to
$2^{\Lambda\times\Lambda}$, according to 2.14. In the general case, however,
this need not be true. In Example 3.18, the only inner band projection is the
zero projection, even though there are many pairwise disjoint left and right
band projections.

We finish with an illustration of inner band projections in a Banach lattice
algebra without identity. This example also shows that an analogous of 2.15
for this general setup is not straightforward.

######  Example 3.33.

Consider the 3-dimensional Banach lattice algebra

$\displaystyle A$
$\displaystyle=\operatorname{span}\left\\{\begin{pmatrix}1&0&0\\\ 0&0&0\\\
0&0&0\end{pmatrix},\begin{pmatrix}0&0&0\\\ 0&1&0\\\
0&0&0\end{pmatrix},\begin{pmatrix}0&0&0\\\ 0&0&0\\\
1&0&0\end{pmatrix}\right\\}$ $\displaystyle=\left\\{\,\begin{pmatrix}x&0&0\\\
0&y&0\\\ z&0&0\end{pmatrix}:x,y,z\in\mathbb{R}\,\right\\}\subseteq
M_{3}(\mathbb{R}),$

where the order, product and norm are inherited from $M_{3}(\mathbb{R})$ (see
Example 3.20). It is not difficult to check that this Banach lattice algebra
does not have an identity. By direct computation, one can also check that

$BP_{l}(A)\cap BP_{r}(A)=\left\\{\,\begin{pmatrix}0&0&0\\\ 0&0&0\\\
0&0&0\end{pmatrix},p_{1}=\begin{pmatrix}1&0&0\\\ 0&0&0\\\
0&0&0\end{pmatrix},p_{2}=\begin{pmatrix}0&0&0\\\ 0&1&0\\\
0&0&0\end{pmatrix},\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&0\end{pmatrix}\,\right\\}.$

Since $p_{1}ap_{2}=0=p_{2}ap_{1}$ for $a\in A$, the only inner band
projections besides the zero projection are $a\mapsto p_{1}ap_{1}$, $a\mapsto
p_{2}ap_{2}$, and their sum; explicitly, the only inner band projections are

$\begin{pmatrix}x&0&0\\\ 0&y&0\\\
z&0&0\end{pmatrix}\mapsto\begin{pmatrix}x&0&0\\\ 0&0&0\\\
0&0&0\end{pmatrix},\quad\begin{pmatrix}x&0&0\\\ 0&y&0\\\
z&0&0\end{pmatrix}\mapsto\begin{pmatrix}0&0&0\\\ 0&y&0\\\
0&0&0\end{pmatrix},\quad\begin{pmatrix}x&0&0\\\ 0&y&0\\\
z&0&0\end{pmatrix}\mapsto\begin{pmatrix}x&0&0\\\ 0&y&0\\\ 0&0&0\end{pmatrix}.$

It is easy to check that

$\begin{pmatrix}x&0&0\\\ 0&y&0\\\
z&0&0\end{pmatrix}\mapsto\begin{pmatrix}0&0&0\\\ 0&0&0\\\ z&0&0\end{pmatrix}$

defines a band projection that is not inner.

## Acknowledgements

The author is very thankful to Pedro Tradacete for valuable discussions, ideas
and corrections. Research partially supported by an FPI–UAM 2023 contract
funded by Universidad Autónoma de Madrid.

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