# Spectral resolutions in effect algebras

Anna Jenčová<EMAIL_ADDRESS>Sylvia Pulmannová<EMAIL_ADDRESS>Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73
Bratislava, Slovakia

###### Abstract

Effect algebras were introduced as an abstract algebraic model for Hilbert
space effects representing quantum mechanical measurements. We study
additional structures on an effect algebra $E$ that enable us to define
spectrality and spectral resolutions for elements of $E$ akin to those of
self-adjoint operators. These structures, called compression bases, are
special families of maps on $E$, analogous to the set of compressions on
operator algebras, order unit spaces or unital abelian groups. Elements of a
compression base are in one-to-one correspondence with certain elements of
$E$, called projections. An effect algebra is called spectral if it has a
distinguished compression base with two special properties: the projection
cover property (i.e., for every element $a$ in $E$ there is a smallest
projection majorizing $a$), and the so-called b-comparability property, which
is an analogue of general comparability in operator algebras or unital abelian
groups. It is shown that in a spectral archimedean effect algebra $E$, every
$a\in E$ admits a unique rational spectral resolution and its properties are
studied. If in addition $E$ possesses a separating set of states, then every
element $a\in E$ is determined by its spectral resolution. It is also proved
that for some types of interval effect algebras (with RDP, archimedean
divisible), spectrality of $E$ is equivalent to spectrality of its universal
group and the corresponding rational spectral resolutions are the same. In
particular, for convex archimedean effect algebras, spectral resolutions in
$E$ are in agreement with spectral resolutions in the corresponding order unit
space.

##  1 Introduction

In the mathematical description of quantum theory, the yes-no measurements are
represented by Hilbert space effects, that is, operators between 0 and $I$ on
the Hilbert space representing the given quantum system. More generally, any
quantum measurement is described as a measure with values in the set of
effects. The existence of spectral resolutions of self-adjoint operators on a
Hilbert space plays an important role in quantum theory. It provides a
connection between such operators and projection-valued measures, describing
sharp measurements. Moreover, spectrality appears as a crucial property in
operational derivations of quantum theory, see e.g. [3, 7, 27, 40].

Motivated by characterization of state spaces of operator algebras and JB-
algebras, Alfsen and Shultz introduced a notion of spectrality for order unit
spaces, see [1, 2]. A more algebraic approach to spectral order unit spaces
was introduced in [11], see [30] for a comparison of the two approaches. The
latter approach is based on the works by Foulis [12, 15, 16, 18], who studied
a generalization of spectrality for partially ordered unital abelian groups.
In all these works, the basic notion is that of a _compression_ , which
generalizes the map $a\mapsto pap$ for a projection $p$ on a von Neumann
algebra, or the projection onto the ideal generated by a sharp element in an
interpolation group [19]. One of the highlights of these works is the result
that if there exists a suitable set of compressions with specified properties,
each element has a unique spectral resolution, or a _rational_ spectral
resolution with values restricted to $\mathbb{Q}$ in the case of partially
ordered abelian groups, analogous to the spectral resolution of self-adjoint
elements in von Neumann algebras.

As an algebraic abstraction of the set of Hilbert space effects, effect
algebras were introduced by Foulis and Bennett [9]. Besides the Hilbert space
effects, this abstract definition covers a large class of other structures
with no clear notion of a spectral resolution. It is therefore important to
study additional structures on an effect algebra that may ensure the existence
of some type of a spectral resolution. In analogy with the definition in [13],
compressions and compression bases in effect algebras were studied by Gudder
[22, 24]. It is a natural question (remarked upon also in [22]) whether some
form of spectrality can be obtained in this setting and what structural
properties of the effect algebra are required for existence of spectral
resolutions. Note that another approach to spectrality in _convex_ effect
algebras, based on contexts, was studied in [25], see also [32]. There is also
a dual notion of spectrality of compact convex sets pertaining the state space
of the effect algebra (e.g. [2, 3]), this will not be considered here.

Spectral compression bases in effect algebras were studied in [36], building
on the works [24, 18]. For an effect algebra $E$, a compression is an additive
idempotent mapping $J:E\to E[0,p]$ where $p$ is a special element called the
focus of $J$. A compression base is a family $(J_{p})_{p\in P}$ of
compressions satisfying certain properties, parametrized by a specified set
$P$ of focuses. Elements of $P$ are called projections. We say that a
compression base $(J_{p})_{p\in P}$ in $E$ is spectral if it has (1) the
projection cover property, that is, for every $a\in E$ there is a smallest
projection that majorizes $a$; (2) the so-called b-comparability, introduced
in analogy with general comparability in groups of [15] and [19].

In the present paper, we study the properties of effect algebras with spectral
compression bases. We show that in such a case, every element $a\in E$ admits
a unique spectral resolution in terms of spectral projections in $P$,
parametrized by dyadic rationals. If $E$ is also archimedean, the spectral
resolution can be extended to all rational values in $[0,1]$ and characterized
by several properties resembling the characterization of the spectral
resolution for Hilbert space effects. We prove that for any state $s$ on $E$,
$s(a)$ is determined by the values of $s$ on the spectral projections. In
particular, if $E$ has a separating set of states, then every element is
uniquely determined by its spectral resolution.

We also study some special cases of interval effect algebras, that is, effect
algebras that are isomorphic to the unit interval in a unital abelian
partially ordered group $(G,u)$: effect algebras with RDP, divisible and
convex archimedean effect algebras. We show that in these cases, the
compression bases in $E$ are in one-to-one correspondence with compression
bases in $G$ and that spectrality in $E$ is equivalent to spectrality in $G$,
as defined in [18]. Moreover, the spectral resolutions obtained in $E$
coincide with the spectral resolutions in $G$. However, we show that
spectrality of an interval effect algebra is not always inherited from its
universal group.

##  2 Effect algebras

An _effect algebra_ [9] is a system $(E;\oplus,0,1)$ where $E$ is a nonempty
set, $\oplus$ is a partially defined binary operation on $E$, and $0$ and $1$
are constants, such that the following conditions are satisfied:

  1. (E1)

If $a\oplus b$ is defined then $b\oplus a$ is defined and $a\oplus b=b\oplus
a$.

  2. (E2)

If $a\oplus b$ and $(a\oplus b)\oplus c$ are defined then $b\oplus c$ and
$a\oplus(b\oplus c)$ are defined and $(a\oplus b)\oplus c=a\oplus(b\oplus c)$.

  3. (E3)

For every $a\in E$ there is a unique $a^{\prime}\in E$ such that $a\oplus
a^{\prime}=1$.

  4. (E4)

If $a\oplus 1$ is defined then $a=0$.

######  Example 2.1.

Let $\mathcal{H}$ be a Hilbert space and let $E(\mathcal{H})$ be the set of
operators on $\mathcal{H}$ such that $0\leq A\leq I$. For $A,B\in
E(\mathcal{H})$, put $A\oplus B=A+B$ if $A+B\leq I$, otherwise $A\oplus B$ is
not defined. Then $(E(\mathcal{H});\oplus,0,I)$ is an effect algebra. This is
a prototypical example on which the above abstract definition is modelled. The
elements of $E(\mathcal{H})$ are called effects.

######  Example 2.2.

Let $(G,u)$ be a partially ordered abelian group with an order unit $u$. Let
$G[0,u]$ be the unit interval in $G$ (we will often write $[0,u]$ if the group
$G$ is clear). For $a,b\in G[0,u]$, let $a\oplus b$ be defined if $a+b\leq u$
and in this case $a\oplus b=a+b$. It is easily checked that
$(G[0,u],\oplus,0,u)$ is an effect algebra. Effect algebras of this form are
called _interval_ effect algebras. In particular, the real unit interval
$\mathbb{R}[0,1]$ can be given a structure of an effect algebra. Note also
that the Hilbert space effects in Example 2.1 form an interval effect algebra.

We write $a\perp b$ and say that $a$ and $b$ are _orthogonal_ if $a\oplus b$
exists. In what follows, when we write $a\oplus b$, we tacitly assume that
$a\perp b$. A partial order is introduced on $E$ by defining $a\leq b$ if
there is $c\in E$ with $a\oplus c=b$. If such an element $c$ exists, it is
unique, and we define $b\ominus a:=c$. With respect to this partial order we
have $0\leq a\leq 1$ for all $a\in E$. The element $a^{\prime}=1\ominus a$ in
(E3) is called the _orthosupplement_ of $a$. It can be shown that $a\perp b$
iff $a\leq b^{\prime}$ (equivalently, $b\leq a^{\prime}$). Moreover $a\leq b$
iff $b^{\prime}\leq a^{\prime}$, and $a^{\prime\prime}=a$.

An element $a\in E$ is called _sharp_ if $a\wedge a^{\prime}=0$ (i.e., $x\leq
a,a^{\prime}\implies x=0$). We denote the set of all sharp elements of $E$ by
$E_{S}$. An element $a\in E$ is _principal_ if $x,y\leq a$, and $x\perp y$
implies that $x\oplus y\leq a$. It is easy to see that a principal element is
sharp.

By recurrence, the operation $\oplus$ can be extended to finite sums
$a_{1}\oplus a_{2}\oplus\cdots\oplus a_{n}$ of (not necessarily different)
elements $a_{1},a_{2},\ldots a_{n}$ of $E$. If $a_{1}=\dots=a_{n}=a$ and
$\oplus_{i}a_{i}$ exist, we write $\oplus_{i}a_{i}=na$. An effect algebra $E$
is _archimedean_ if for $a\in E$, $na\leq 1$ for all $n\in\mathbb{N}$ implies
that $a=0$.

An infinite family $(a_{i})_{i\in I}$ of elements of $E$ is called
_orthogonal_ if every its finite subfamily has an $\oplus$-sum in $E$. If the
element $\oplus_{i\in I}a_{i}=\bigvee_{F\subseteq I}\oplus_{i\in F}a_{i}$,
where the supremum is taken over all finite subsets of $I$ exists, it is
called the _orthosum_ of the family $(a_{i})_{i\in I}$. An effect algebra $E$
is called _$\sigma$ -orthocomplete_ if the orthosum exists for any orthogonal
countable subfamily of $E$.

An effect algebra $E$ is _monotone $\sigma$-complete_ if every ascending
sequence $(a_{i})_{i\in{\mathbb{N}}}$ has a supremum $a=\bigvee_{i}a_{i}$ in
$E$, equivalently, every descending sequence $(b_{i})_{i\in{\mathbb{N}}}$ has
an infimum $b=\bigwedge_{i}b_{i}$ in $E$. It turns out that an effect algebra
is monotone $\sigma$-complete if and only if it is $\sigma$-orthocomplete
[29]. A subset $F$ of $E$ is _sup/inf -closed in E_ if whenever $M\subseteq F$
and $\wedge M$ ($\vee M$) exists in $E$, then $\wedge M\in F$ ($\vee M\in F$).

If $E$ and $F$ are effect algebras, a mapping $\phi:E\to F$ is _additive_ if
$a\perp b$ implies $\phi(a)\perp\phi(b)$ and $\phi(a\oplus
b)=\phi(a)\oplus\phi(b)$. An additive mapping $\phi$ such that $\phi(1)=1$, is
a _morphism_. If $\phi:E\to F$ is a morphism, and $\phi(a)\perp\phi(b)$
implies $a\perp b$, then $\Phi$ is a _monomorphism_. A surjective monomorphism
is an _isomorphism_.

A _state_ on an effect algebra $E$ is a morphism $s$ from $E$ into the effect
algebra ${\mathbb{R}}[0,1]$ . We denote the set of states on $E$ by $S(E)$. We
say that $S\subset S(E)$ is _separating_ if $s(a)=s(b)$ for every $s\in S$
implies that $a=b$. We say that $S\subset S(E)$ is _ordering_ (or order
determining) if $s(a)\leq s(b)$ for all $s\in S$ implies $a\leq b$. If $S$ is
ordering, then it is separating, the converse does not hold.

A lattice ordered effect algebra $M$, in which $(a\vee b)\ominus
a=a\ominus(a\wedge b)$ holds for all $a,b\in M$, is called an _MV-effect
algebra_. We recall that MV-effect algebras are equivalent with MV-algebras,
which were introduced by [6] as algebraic bases for many-valued logic. It was
proved in [33] that there is an equivalence between MV-algebras and lattice
ordered groups with order unit, in the sense of category theory.

##  3 Compressions on effect algebras

The definitions in this section follow the works of Foulis [12, 13, 15, 18],
who studied spectrality for unital partially ordered abelian groups. Some
details on the constructions in that setting are given in Appendix A. The
definitions were adapted to effect algebras by Gudder [22] and Pulmannová
[36].

######  Definition 3.1.

Let $E$ be an effect algebra.

  1. (i)

An additive map $J:E\to E$ is a _retraction_ if $a\leq J(1)$ implies $J(a)=a$.

  2. (ii)

The element $p:=J(1)$ is called the _focus_ of $J$.

  3. (iii)

A retraction is a _compression_ if $J(a)=0\ \Leftrightarrow\ a\leq
p^{\prime}$.

  4. (iv)

If $I$ and $J$ are retractions we say that $I$ is a _supplement of_ $J$ if
$\ker(J)=I(E)$ and $\ker(I)=J(E)$.

It is easily seen that any retraction is idempotent, in fact, an additive
idempotent map is a retraction if and only if its range is an interval. If a
retraction $J$ has a supplement $I$, then both $I$ and $J$ are compressions
and $I(1)=J(1)^{\prime}$. The element $p:=J(1)$ for a retraction $J$ is called
the focus of $J$. The focus is a principal element and we have $J(E)=[0,p]$,
moreover, $J$ is a compression if and only if
$\mathrm{Ker}(J)=[0,p^{\prime}]$. For these and further properties see [24,
36].

######  Example 3.2.

Let $E(\mathcal{H})$ be the algebra of effects on $\mathcal{H}$ (Example 2.1)
and let $p\in E(\mathcal{H})$ be a projection. Let us define the map
$J_{p}:a\mapsto pap$, then $J_{p}$ is a compression on $E(\mathcal{H})$ and
$J_{p^{\prime}}$ is a supplement of $J_{p}$. By [13], any retraction on
$E(\mathcal{H})$ is of this form for some projection $p$. In particular, any
projection is the focus of a unique retraction $J_{p}$ with a (unique)
supplement $J_{p^{\prime}}$. Effect algebras such that any retraction is
supplemented and uniquely determined by its focus are called compressible,
[22, 12].

Our aim is to introduce a set of compressions analogous to $(J_{p})_{p\in
P(\mathcal{H})}$ in the above example for any effect algebra $E$. For this, we
need to specify a set $P\subseteq E$ of projections and the corresponding
compressions. We will start with the assumption that $P\subseteq E$ is a sub-
effect algebra.

######  Lemma 3.3.

Let $P\subseteq E$ be a sub-effect algebra and let $(J_{p})_{p\in P}$ be a set
of compressions such that $J_{p}(1)=p$. Then $P$ is an orthomodular poset and
$J_{p^{\prime}}$ is a supplement of $J_{p}$.

###### Proof.

We refer the reader to [8, §1.5] for the definition and characterization of
orthomodular posets (OMP). Since the elements of $P$ are principal hence
sharp, $P$ is an orthoalgebra. To show that $P$ is an OMP, it is enough to
show that if $p\oplus q$ exists, then $p\oplus q=p\vee q$. So assume that
$p\oplus q$ exists, then $p,q\leq p\oplus q$. Suppose that $r\in P$ is such
that $p,q\leq r$, then since $r$ is principal, we have $p\oplus q\leq r$. It
follows that $p\oplus q=p\vee q$ and $P$ is an OMP. Further, since any $J_{p}$
is a compression, for $p\in P$ we have
$J_{p^{\prime}}(E)=[0,p^{\prime}]=\mathrm{Ker}(J_{p})$, so that
$J_{p^{\prime}}$ is a supplement of $J_{p}$.

∎

The set of compressions in $E(\mathcal{H})$ has another important property:
$J_{p}$ and $J_{q}$ commute if and only if the projections $p$ and $q$
commute, and in this case, $J_{p}\circ J_{q}=J_{q}\circ J_{p}=J_{pq}$. To
extend this property to general effect algebras, we need the following notion.
Recall that two elements $a,b$ in an effect algebra $E$ are _Mackey
compatible_ if there are elements $a_{1},b_{1},c\in E$ such that $a_{1}\oplus
b_{1}\oplus c$ exists and $a=a_{1}\oplus c,b=b_{1}\oplus c$. We shall write
$a\leftrightarrow b$ if $a,b$ are Mackey compatible. It is well known that for
two projections (or a projection and an effect) in $E(\mathcal{H})$, Mackey
compatibility is equivalent to commutativity. (Note that this is not true for
general pairs of elements.)

We can now introduce the main definition of this section.

######  Definition 3.4.

A family $(J_{p})_{p\in P}$ of compressions on an effect algebra $E$, indexed
by a sub-effect algebra $P\subseteq E$ is called a _compression base_ on $E$
if the following conditions hold:

  1. (C1)

each $p\in P$ is the focus of $J_{p}$,

  2. (C2)

if $p,q\in P$ are Mackey compatible (in $E$), then $J_{p}\circ J_{q}=J_{r}$
for some $r\in P$.

Elements of $P$ are called _projections_.

The above definition is slightly different from the original definitions of a
compression base used in [24]. We next show that these definitions are
equivalent. For this, recall that a sub-effect algebra $F\subseteq E$ is
_normal_ if for all $e,f,d\in E$ such that $e\oplus f\oplus d$ exists in $E$,
we have $e\oplus d,f\oplus d\in F\ \implies\ d\in F$. It was proved in [24]
that in this case, two elements in $F$ are Mackey compatible as elements of
$E$ if and only if they are Mackey compatible in $F$, that is, the elements
$a_{1},b_{1}$ and $c$ can be chosen in $F$.

######  Proposition 3.5.

Let $(J_{p})_{p\in P}$ be a compression base. Then

  1. (i)

$P$ is a normal sub-effect algebra in $E$;

  2. (ii)

if $p,q,r\in P$ are such that $p\oplus q\oplus r$ exists, then $J_{p\oplus
q}\circ J_{q\oplus r}=J_{r}$.

Conversely, if $(J_{p})_{p\in P}$ is a set of compressions such that
$J_{p}(1)=p$, and with properties (i) and (ii), then it is a compression base.

###### Proof.

Let $(J_{p})_{p\in P}$ be a compression base and let $e,f,d\in E$ be such that
$e\oplus f\oplus d\in E$ and $p=e\oplus d,q=f\oplus d\in P$. Then
$p\leftrightarrow q$ (in $E$), so that $J_{p}\circ J_{q}=J_{r}$ for some $r\in
P$. We have

$r=J_{r}(1)=J_{p}\circ J_{q}(1)=J_{p}(q)=J_{p}(f)\oplus J_{p}(d)=d,$

since $f\leq p^{\prime}$ and $d\leq p$. It follows that $d=r\in P$, so that
$P$ is normal. This also proves (ii). The converse is easy.

∎

######  Corollary 3.6.

Let $(J_{p})_{p\in P}$ be a compression base. Then $p\leftrightarrow q$ if and
only if $J_{p}\circ J_{q}=J_{q}\circ J_{p}=J_{p\wedge q}$.

###### Proof.

Assume that $p\leftrightarrow q$ and let $p=p_{1}\oplus r$ and $q=q_{1}\oplus
r$ be such that $p_{1}\oplus q_{1}\oplus r$ exists. Then $p\wedge p=r$ and we
have by the property (ii) in Proposition 3.5 that $J_{p}\circ
J_{q}=J_{r}=J_{q}\circ J_{p}$. The converse follows by [24, Thm. 4.2]: assume
that $J_{p}\circ J_{q}=J_{r}$ for some $r\in P$, then $J_{p}(q)=J_{r}(1)=r\leq
p$ and $J_{r}(q)=J_{p}\circ J_{q}(q)=J_{p}(q)=r$, so that $r\leq q$. Put
$q_{1}=q\ominus r$, $p_{1}=p\ominus r$, then $J_{p}(q_{1})=r\ominus r=0$, so
that $q_{1}\leq p^{\prime}$. It follows that $p_{1}\oplus r\oplus q_{1}$
exists and $p\leftrightarrow q$.

∎

By [22, Cor. 4.5] and [36, Thm. 2.1], the set $P$ as a subalgebra of $E$ is a
regular orthomodular poset (OMP) with the orthocomplementation $p\mapsto
p^{\prime}$. Recall that an OMP $P$ is _regular_ if for all $a,b,c\in P$, if
$a,b$ and $c$ are pairwise Mackey compatible, then $a\leftrightarrow b\vee c$
and $a\leftrightarrow b\wedge c$, [34, 26].

######  Example 3.7.

Let $P(\mathcal{H})$ be the set of all projections on a Hilbert space
$\mathcal{H}$ and let $J_{p}$ for $p\in P(\mathcal{H})$ be as in Example 3.2.
It is easily observed that the set $(J_{p})_{p\in P(\mathcal{H})}$ is a
compression base in $E(\mathcal{H})$. More generally, the set of all
compressions in a compressible effect algebra is a compression base, [24].

In general, there can be many compression bases on an effect algebra $E$, for
example, $(J_{0}=0,J_{1}=id)$ is trivially a compression base. The compression
bases on $E$ can be ordered by inclusion and by standard Zorn Lemma arguments,
there are maximal elements with respect to this order, called maximal
compression bases, [24]. The following example is a compression base on any
effect algebra $E$.

######  Example 3.8.

Let $E$ be an effect algebra and let $\Gamma(E)$ be the set of _central
elements_ [8, Def. 1.9.11, Lemma 1.9.12]. In this case, both $p$ and its
orthosupplement $p^{\prime}$ are principal elements, $a\wedge p$, $a\wedge
p^{\prime}$ exist and we have

$a=a\wedge p\oplus a\wedge p^{\prime}$

for all $a\in E$. It is easily seen that $U_{p}:a\mapsto a\wedge p$ is the
unique compression with focus $p\in\Gamma(E)$ and $(U_{p})_{p\in\Gamma(E)}$ is
a compression base. This compression base will be called _central_.

The next example shows that there may exist different compression bases with
the same set of projections $P$.

######  Example 3.9.

Let $E=E(\mathcal{H})\dot{\cup}E(\mathcal{H})$ be the horizontal sum (or the
0-1 pasting) of two copies of $E(\mathcal{H})$ for a separable Hilbert space
$\mathcal{H}$. That is, the elements of $E$ are of the form $(a,0)$ or $(0,a)$
where $a\in E(\mathcal{H})$ and $(1,0)$ and $(0,1)$ are identified with the
unit element $u$ of $E$. Two elements of $E$ are summable if and only if they
are of the form $(a,0)$, $(b,0)$ or $(0,a)$, $(0,b)$ for some summable $a,b\in
E(\mathcal{H})$ and then $(a,0)\oplus(b,0)=(a\oplus b,0)$, similarly in the
other copy. The sharp elements of $E$ are of the form $(p,0)$ or $(0,p)$ for
some $p\in P(\mathcal{H})$ and all such elements are principal.

Choose any faithful state $\varphi$ on $E(\mathcal{H})$, that is,
$\varphi(a)=0$ for $a\in E(\mathcal{H})$ implies $a=0$. For any $p\in
P(\mathcal{H})$, $p\neq 1$, we define

$J_{(p,0)}((a,0))=(J_{p}(a),0),\quad J_{(p,0)}((0,a))=(\varphi(a)p,0)$

It is easily checked that $J_{(p,0)}$ is a compression on $E$ with focus
$(p,0)$. The compressions $J_{(0,p)}$ can be defined similarly. We also put
$J_{u}=id$.

Let $P=P(\mathcal{H})\dot{\cup}P(\mathcal{H})$, then $P$ is a subalgebra in
$E$ and we can check that $(J_{e})_{e\in P}$ is a compression base. Indeed,
let $e,f\in P$ and $e\leftrightarrow f$, we may clearly assume that none of
the projections is the unit element. Then $e,f$ must belong to the same copy,
hence there are some commuting projections $p,q\in
P(\mathcal{H})\setminus\\{1\\}$, such that $e=(p,0)$ and $f=(q,0)$, say. Then

$\displaystyle J_{e}\circ J_{f}((a,0))=(J_{p}\circ
J_{q}(a),0)=(J_{pq}(a),0),\qquad J_{e}\circ
J_{f}((0,a))=(J_{p}(\varphi(a)q),0)=(\varphi(a)pq,0),$

so that $J_{e}\circ J_{f}=J_{g}$ with $g=(pq,0)$. Notice that the state
$\varphi$ can be chosen arbitrarily, this shows that there can be different
compression bases with the same set of projections $P$. This example will be
continued in Example 4.6.

More generally, if $E$ is the horizontal sum of two effect algebras $E_{1}$
and $E_{2}$, then all principal elements are of the form $(p_{1},0)$ or
$(0,p_{2})$ for some principal $p_{1}\in E_{1}$ or $p_{2}\in E_{2}$. If
$p_{1}\neq 0,1$, then any retraction $J$ on $E$ with focus $(p_{1},0)$ has the
form

$J((a_{1},0))=(J_{1}(a),0),\qquad J((0,a_{2}))=(\phi(a_{2}),0),$

where $J_{1}$ is a retraction on $E_{1}$ with focus $p_{1}$ and
$\phi:E_{2}\to[0,p_{1}]$ is a morphism, see [22]. Note that $J$ is a
compression if and only if $J_{1}$ is a compression and $\phi$ is faithful,
that is, $\phi(a_{2})=0$ implies $a_{2}=0$. Even if both $E_{1}$ and $E_{2}$
have compressions, faithful morphisms $\phi$ might not exist, so in that case
there are no nontrivial compressions on $E$ (take for example $E_{1}=E_{2}$
the Boolean algebra $2^{2}$).

######  Example 3.10.

An effect algebra $E$ is _sequential_ if it is endowed with a binary operation
$\circ$ called a sequential product, see [23] for definition and more
information. In this case, an element $p\in E$ is sharp if and only if $p\circ
p=p$, equivalently, $p\circ p^{\prime}=0$. Moreover, for $a\in E$, $a\leq p$
if and only if $p\circ a=a$, equivalently, $p^{\prime}\circ a=0$. From the
axioms of the sequential product and these properties it immediately follows
that $J_{p}:a\mapsto p\circ a$ is a compression and $(J_{p})_{p\in E_{S}}$ is
a compression base, see [24]. Note that $E(\mathcal{H})$ is sequential, with
$a\circ b=a^{1/2}ba^{1/2}$.

######  Example 3.11.

Let $E$ be an OMP. Notice that all elements are principal, but in general
there is no compression base such that all elements are projections. Indeed,
by [24, Thm. 4.2], for projections $p$ and $q$, $J_{p}(q)$ is a projection if
and only if $p\leftrightarrow q$. It follows that there is a compression base
$(J_{p})_{p\in E}$ if and only if all elements are compatible, that is $E$ is
a Boolean algebra. In that case we have $J_{p}(q)=U_{p}(q)=p\wedge q$.

######  Example 3.12.

We say that an effect algebra $E$ has the Riesz decomposition property (RDP)
if one of the following equivalent conditions is satisfied:

  1. (i)

For any $a,b,c\in E$, if $a\leq b\oplus c$ then there are some $b_{1},c_{1}\in
E$ such that $b_{1}\leq b$, $c_{1}\leq c$ and $a=b_{1}\oplus c_{1}$.

  2. (ii)

For any $a_{1},a_{2},b_{1},b_{2}\in E$, if $a_{1}\oplus a_{2}=b_{1}\oplus
b_{2}$, then there are elements $w_{ij}\in E$, $i,j=1,2$ such that
$a_{1}=w_{11}\oplus w_{12}$, $a_{2}=w_{21}\oplus w_{22}$, $b_{1}=w_{11}\oplus
w_{21}$ and $b_{2}=w_{12}\oplus w_{22}$.

By [37], $E$ is an interval effect algebra. If, in addition, $E$ is lattice-
ordered, then $E$ is an MV-effect algebra, see [8].

From the condition (i) (or (ii)), it follows that all elements in $E$ are
pairwise Mackey compatible. Moreover, every sharp element in $E$ is principal
and the set $E_{S}$ of sharp elements is the center of $E$. By Example 3.8,
$(U_{p})_{p\in E_{S}}$ is a compression base. Since the focus of a retraction
is always sharp, this compression base contains all retractions on $E$. In
this case, we say that the compression base is _total_.

From now on, we will assume that the effect algebra $E$ is endowed with a
fixed compression base $(J_{p})_{p\in P}$ and all further considerations are
made with respect to this base.

###  3.1 Compatibility and commutants

By [36, Lemma 4.1] we have the following.

######  Lemma 3.13.

If $p\in P,a\in E$, then the following statements are equivalent:

  1. (i)

$J_{p}(a)\leq a$,

  2. (ii)

$a=J_{p}(a)\oplus J_{p^{\prime}}(a)$,

  3. (iii)

$a\in E[0,p]\oplus E[0,p^{\prime}]$,

  4. (iv)

$a\leftrightarrow p$,

  5. (v)

$J_{p}(a)=p\wedge a$.

If any of the conditions in the above lemma is satisfied for $a\in E,p\in P$,
we say that $a$ and $p$ _commute_ or that $a$ and $p$ are _compatible_. The
_commutant_ of $p$ in $E$ is defined by

$C(p):=\\{a\in E:a=J_{p}(a)\oplus J_{p^{\prime}}(a)\\}.$

If $Q\subseteq P$, we write $C(Q):=\bigcap_{p\in Q}C(p)$. Similarly, for an
element $a\in E$, and a subset $A\subseteq E$, we write

$PC(a):=\\{p\in P,\ a\in C(p)\\},\ \ PC(A):=\bigcap_{a\in A}PC(a).$

We also define

$\displaystyle CPC(a):=C(PC(a)),\qquad P(a):=CPC(a)\cap
PC(a)=PC(PC(a)\cup\\{a\\}).$

The set $P(a)\subseteq P$ will be called the _bicommutant_ of $a$. For a
subset $Q\subseteq E$, we put

$P(Q):=PC(PC(Q)\cup Q).$

Note that the elements in $P(Q)$ are pairwise Mackey compatible and since $P$
is a regular OMP, this implies that $P(Q)$ is a Boolean subalgebra in $P$.

######  Lemma 3.14.

[36, Lemma 4.2] Let $p,q\in P$, $a\in E$. If $p\perp q$ and either $a\in C(p)$
or $a\in C(q)$, then

$J_{p\vee q}(a)=J_{p\oplus q}(a)=J_{p}(a)\oplus J_{q}(a).$

Recall that a maximal set of pairwise Mackey compatible elements in a regular
OMP $P$ is called a _block_ of $P$ [34, Cor. 1.3.2]. It is well known that
every block $B$ is a Boolean subalgebra of $P$ [34, Thm. 1.3.29]. If $B$ is a
block of $P$, the set $C(B)$ will be called a _C-block_ of $E$.

######  Example 3.15.

Let $E=E(\mathcal{H})$ for a Hilbert space $\mathcal{H}$. It is easily checked
that for any projection $p\in P(\mathcal{H})$,

$C(p)=\\{p\\}^{\prime}\cap E=\\{a\in E,pa=ap\\}$

and for any $a\in E$,

$P(a)=\\{a\\}^{\prime\prime}\cap P(\mathcal{H})$

(here $C^{\prime}$ denotes the usual commutant of a subset of bounded
operators $C\subset B(\mathcal{H})$). The C-blocks are the unit intervals in
maximal abelian von Neumann subalgebras of $B(\mathcal{H})$. In Section 3.3 we
introduce a property under which the C-blocks can be characterized in a
similar way (see Theorem 3.23).

###  3.2 Projection cover property

######  Definition 3.16.

If $a\in E$ and $p\in P$, then $p$ is a _projection cover_ for $a$ if, for all
$q\in P$, $a\leq q\,\Leftrightarrow\,p\leq q$. We say that $E$ has the
_projection cover property_ if every effect $a\in E$ has a (necessarily
unique) projection cover. The projection cover of $a\in E$ will be denoted as
$a^{\text{\rm o}}$.

######  Theorem 3.17 ([24, Thm. 5.2], [36, Thm. 5.1]).

Suppose that $E$ has the projection cover property. Then $P$ is an
orthomodular lattice (OML). Moreover, $\mathcal{P}$ is sup/inf-closed in $E$.

######  Proposition 3.18.

Let $E$ have the projection cover property. Then for any $a\in E$,
$a^{\text{\rm o}}\in P(a)$.

###### Proof.

Since $a\leq a^{\text{\rm o}}$, $a\in C(a^{\text{\rm o}})$ by Lemma 3.13 (iv).
The rest follows by [36, Thm. 5.2 (i)].

∎

###  3.3 b-property and comparability

In this section we recall the notion of the b-comparability property in effect
algebras, introduced in [36] as an analogue of the general comparability
property in unital partially ordered abelian groups [16], which is itself an
extension of the general comparability property in interpolation groups [19].
Some more details on compressions and comparability for partially ordered
abelian groups can be found in Appendix A.

We first introduce a property that allows us to extend the notion of
commutativity to all pairs of elements $a,b\in E$, obtaining the analogue of
commutativity of quantum effects.

######  Definition 3.19 ([36, Definition 6.1]).

We will say that $a\in E$ has the _b-property_ (or is a _b-element_) if there
is a Boolean subalgebra $B(a)\subseteq P$ such that for all $p\in P$, $a\in
C(p)\,\Leftrightarrow\,B(a)\subseteq C(p)$. We say that $E$ has the _b-
property_ if every $a\in E$ is a b-element.

The Boolean subalgebra $B(a)$ in the above definition is in general not
unique. The next lemma shows that the bicommutant of $a$ is the largest such
subalgebra.

######  Lemma 3.20.

Let $a\in E$ be a b-element. Then

  1. (i)

$B(a)\subseteq P(a)$.

  2. (ii)

For $p\in P$, $a\in C(p)$ $\iff$ $P(a)\subseteq C(p)$.

###### Proof.

Let $q\in B(a)$. Since $B(a)$ is a Boolean subalgebra, all elements are
mutually compatible, which means that $B(a)\subseteq C(q)$. By definition,
this implies $a\in C(q)$. Further, for any $p\in PC(a)$ we have $q\in
B(a)\subseteq C(p)$, so that

$q\in PC(a)\cap CPC(a)=P(a).$

This proves (i). To prove (ii), let $p\in P$ be such that $a\in C(p)$, then
$p\in PC(a)$ so that $P(a)\subseteq C(p)$. The converse follows by (i).

∎

######  Lemma 3.21.

[36, Prop. 6.1] (i) If an element $a\in E$ is a b-element, then there is a
block $B$ of $P$ such that $a\in C(B)$. (ii) Every projection $q\in P$ is a
b-element with $B(q)=\\{0,q,q^{\prime},1\\}$.

Note that (ii) of the above Lemma shows that we may have $B(a)\subsetneq
P(a)$. Indeed, if $E$ has RDP and $q\in E_{S}$, then $P(q)$ is all of $E_{S}$,
see Example 3.12, whereas $B(q)$ in (ii) is the minimal Boolean subalgebra
with the required properties.

For $A,B\subseteq E$ we write $A\leftrightarrow B$ if $a\leftrightarrow b$ for
all $a\in A$, $b\in B$.

######  Lemma 3.22.

Let $e,f\in E$ be b-elements. Then $B(e)\leftrightarrow B(f)$ if and only if
$P(e)\leftrightarrow P(f)$.

###### Proof.

By definition of $B(e)$, $B(f)$ and Lemma 3.20, we have the following chain of
equivalences:

$\displaystyle B(e)\leftrightarrow B(f)$ $\displaystyle\iff\ B(e)\subseteq
C(B(f))\iff\ e\in C(B(f))$ $\displaystyle\iff\ P(e)\subseteq C(B(f))\iff\
P(e)\leftrightarrow B(f),$

the proof is finished by symmetry in $e$ and $f$.

∎

Let $e,f\in E$ have the b-property. We say that $e$ and $f$ _commute_ , in
notation $eCf$, if

$\ P(e)\leftrightarrow P(f).$ (1)

Clearly, for $p\in P$ we have $eCp\iff e\leftrightarrow p\iff e\in C(p)$, [36,
Lemma 6.1], so this definition coincides with previously introduced notions if
one of the elements is a projection. Note also that we do not use the word
”compatible” in this case, since in general this notion is different from
Mackey compatibility.

Assume now that $E$ has the b-property, so all elements are b-elements. By
Lemma 3.21, any element of $E$ is contained in some C-block of $E$.

######  Theorem 3.23.

If $E$ has the b-property, then C-blocks in $E$ coincide with maximal sets of
pairwise commuting elements in $E$.

###### Proof.

First observe that $PC(B)=B$ by maximality of $B$. Let $a,b\in C(B)$, then
$P(a),P(b)\subseteq C(p)$ for all $p\in B$, therefore $P(a),P(b)\subseteq
PC(B)=B$. This implies that $P(a)\leftrightarrow P(b)$, i.e. $aCb$. Let $g\in
E$ be such that $gCa$ for all $a\in C(B)$. In particular, $g\leftrightarrow p$
for all $p\in B$, which means that $g\in C(B)$. This implies that $C(B)$ is a
maximal set of mutually commuting elements.

Conversely, let $C\subset E$ be maximal with respect the property $aCb$ for
all $a,b\in C$. This means that $P(a)\leftrightarrow P(b)$ for all $a,b\in C$,
hence there exists a block $B$ of $P$ with $\bigcup_{a\in C}P(a)\subseteq B$.
This entails that $P(a)\leftrightarrow B$, which implies $a\in C(B)$ for all
$a\in C$, hence $C\subseteq C(B)$. By the first part of the proof and
maximality of $C$, we have $C=C(B)$. ∎

######  Example 3.24.

Let $E=E(\mathcal{H})$. Since for $a\in E$,
$\\{a\\}^{\prime}=\\{a\\}^{\prime\prime\prime}=P(a)^{\prime}$ (see Example
3.15), we see that $E$ has the b-property and for $a,b\in E$, we have $aCb$ if
and only if $ab=ba$. Consequently, the C-blocks of $E(\mathcal{H})$ indeed
coincide with unit intervals in maximal abelian subalgebras in
$B(\mathcal{H})$.

######  Definition 3.25.

(Cf. [36, Definition 6.3]) An effect algebra $E$ has the _b-comparability
property_ if

  1. (a)

$E$ has the b-property.

  2. (b)

For all $e,f\in E$ such that $eCf$, the set

$P_{\leq}(e,f):=\\{p\in P(e,f):\ J_{p}(e)\leq J_{p}(f)\\\ \text{ and
}J_{1-p}(f)\leq J_{1-p}(e)\\}$

is nonempty.

To provide some intuition for the definition of b-comparability, we look at
the spectral resolution of a self-adjoint element $a\in B(\mathcal{H})$. We
may restrict to a commutative subalgebra containing $a$, which is isomorphic
to a space of functions. Spectral projections of $a$ correspond to subsets on
which the function representing $a$ is comparable with a multiple of the unit.
To find such projections, we need to be able to find a decomposition of
elements in the subalgebra to a positive and negative part, which can be seen
to be equivalent to the property (b) in Definition 3.25 for pairs of commuting
self-adjoint elements. See also Example 3.15.

The b-comparability property has important consequences on the set of
projections and on the structure of the C-blocks.

######  Theorem 3.26.

[36, Thm. 6.1] Let $E$ have the b-comparability property. Then every sharp
element is a projection: $P=E_{S}$.

Note that by the above theorem, if b-comparability holds, then the compression
base must be maximal.

######  Theorem 3.27.

[36, Thm. 7.1] Let $E$ have the b-comparability property and let $C=C(B)$ for
a block $B$ of $P$. Then

  1. (i)

$C$ is an MV-effect algebra.

  2. (ii)

For $p\in B$, the restriction ${J}_{p}|_{C}$ coincides with $U_{p}$ (recall
Example 3.12) and $(U_{p})_{p\in B}$ is the total compression base in $C$.
Moreover, $C$ with $(U_{p})_{p\in B}$ has the b-comparability property.

  3. (iii)

If $E$ has the projection cover property, then $C$ has the projection cover
property.

  4. (iv)

If $E$ is $\sigma$-orthocomplete, then $C$ is $\sigma$-orthocomplete.

######  Remark 3.28.

It may seem that we could have defined a notion of comparability without
assuming the b-property, requiring existence of some projection such that
$J_{p}(a)\leq J_{p}(b)$ and $J_{p^{\prime}}(b)\leq J_{p^{\prime}}(a)$ for all
pairs of elements $a,b\in E$. As was shown in [36, Prop. 6.2], this would
imply that all of $E$ is just one C-block, hence an MV-effect algebra.

######  Example 3.29.

Let $E$ be an effect algebra with RDP, see Example 3.12. Since all elements
are mutually compatible, all $E$ is one C-block, $E=C(E_{S})$. By Theorem 3.27
(i), we see that if $E$ has the b-comparability property then $E$ must be an
MV-effect algebra. As we will see in the next paragraph, b-comparability in
this case is equivalent to comparability in interpolation groups, see [19].

######  Example 3.30.

Let $E$ be an effect algebra and let $(U_{p})_{p\in\Gamma(E)}$ be the central
compression base (Example 3.8). By Theorem 3.26, if b-comparability holds then
we must have $E_{S}=\Gamma(E)$, that is, every sharp element is central.
Again, under comparability, all of $E$ becomes one C-block $E=C(E_{S})$, which
is an MV-effect algebra. Note that $E_{S}=\Gamma(E)$ e.g. in the case of
_effect monoids_ (see [38]), that is, effect algebras endowed with a binary
operation $\cdot:E\times E\to E$ which is unital, biadditive and associative.
For $p\in E_{S}$ and $a\in E$, we have

$p\cdot a=a\cdot p=p\wedge a=U_{p}(a).$

Some further examples will be treated in Section 4.

###  3.4 b-comparability and RDP

Let $E$ be an effect algebra with RDP. By [37], $E$ is isomorphic to the unit
interval in an abelian interpolation group $(G,u)$ with order unit $u$ [19],
called the _universal group_ of $E$. If moreover $E$ is lattice ordered (i.e.
an MV-effect algebra), then $G$ is an $\ell$-group, [33]. In the rest of this
section, we will identify $E$ with the unit interval $[0,u]$ in its universal
group.

Let us now recall the general comparability property in interpolation groups
with order unit, see [19, Chap. 8] for more details. For any sharp element
$p\in E_{S}$, the convex subgroup $G_{p}$ generated by $p$ is an ideal of $G$
and by RDP, we have the direct sum $G=G_{p}\oplus G_{p^{\prime}}$ (as ordered
groups). Let $\tilde{U}_{p}$ be the projection of $G$ onto $G_{p}$ with kernel
$G_{p^{\prime}}$, then

$\tilde{U}_{p}(x)=x\wedge np,\quad\text{if }0\leq x\leq np\text{ for some
}n\in\mathbb{N}.$

Obviously, $\tilde{U}_{p}$ is an extension of the compression $U_{p}$ on
$E\simeq[0,u]$ (see Example 3.12). Note that the uniqueness of retractions
implies that $(G,u)$ is a _compressible group_ in the sense of [18]. We say
that $(G,u)$ satisfies _general comparability_ if for any $x,y\in G$, there is
some sharp element $p\in[0,u]$ such that
$\tilde{U}_{p}(x)\leq\tilde{U}_{p}(y)$ and
$\tilde{U}_{p^{\prime}}(x)\geq\tilde{U}_{p^{\prime}}(y)$. It is easily seen
that $[0,u]$ has the b-comparability property if $(G,u)$ satisfies general
comparability. The aim of the rest of this section is to show that the
converse is also true.

As noticed in Example 3.29, if $E=[0,u]$ has the b-comparability property,
then it must be an MV-effect algebra. Consequently, the group $G$ is lattice
ordered. Let $a\in G$ be any element, then we have

$a=a_{+}-a_{-},\qquad a_{+},a_{-}\in G^{+},\ a_{+}\wedge a_{-}=0,$ (2)

with $a_{+}=a\vee 0$ and $a_{-}=-a\vee 0$.

######  Lemma 3.31.

Let $(G,u)$ be a lattice ordered abelian group with order unit an let
$n\in\mathbb{N}$. The following are equivalent.

  1. (i)

For any $a,b\in[0,nu]$ there is some sharp element $p\in[0,u]$ such that
$\tilde{U}_{p}(a)\leq\tilde{U}_{p}(b)$ and
$\tilde{U}_{p^{\prime}}(a)\geq\tilde{U}_{p^{\prime}}(b)$.

  2. (ii)

For any $a\in[-nu,nu]$ there is some sharp element $p\in[0,u]$ such that
$\tilde{U}_{p}(a)\leq 0$ and $\tilde{U}_{p^{\prime}}(a)\geq 0$.

###### Proof.

Assume (i) and let $a\in[-nu,nu]$. Let $a=a_{+}-a_{-}$ as in (2), then clearly
$a_{+},a_{-}\in[0,nu]$. By (i), there is some sharp element $p\in[0,u]$ such
that $\tilde{U}_{p}(a_{+})\leq\tilde{U}_{p}(a_{-})$ and
$\tilde{U}_{p^{\prime}}(a_{+})\geq\tilde{U}_{p^{\prime}}(a_{-})$. But then
$\tilde{U}_{p}(a_{+})=a_{+}\wedge np\leq a_{+}$ and also
$\tilde{U}_{p}(a_{+})\leq\tilde{U}_{p}(a_{-})\leq a_{-}$, hence
$\tilde{U}_{p}(a_{+})=0$. Similarly, we obtain
$\tilde{U}_{p^{\prime}}(a_{-})=0$. It follows that

$\displaystyle\tilde{U}_{p}(a)=\tilde{U}_{p}(a_{+}-a_{-})=-\tilde{U}_{p}(a_{-})\leq
0,\qquad\tilde{U}_{p^{\prime}}(a)=\tilde{U}_{p^{\prime}}(a_{+}-a_{-})=\tilde{U}_{p^{\prime}}(a_{+})\geq
0.$

Conversely, if $a,b\in[0,nu]$, then $a-b\in[-nu,nu]$, so that (ii) clearly
implies (i).

∎

######  Theorem 3.32.

Let $(G,u)$ be an abelian interpolation group with order unit $u$. Then
$(G,u)$ satisfies general comparability if and only if the effect algebra
$[0,u]$ has the b-comparability property.

###### Proof.

Assume that $E=[0,u]$ has the b-comparability property, then $E$ is an MV-
effect algebra and $G$ is lattice ordered. For $a,b\in G$, there is some
$m\in\mathbb{N}$ such that $a,b\in[-mu,mu]$. Then $a-b\in[-2mu,2mu]$, so it is
clearly enough to prove that for each $n\in\mathbb{N}$ and $a\in[-nu,nu]$,
there is some sharp $p\in[0,u]$ such that $\tilde{U}_{p}(a)\leq 0$ and
$\tilde{U}_{p^{\prime}}(a)\geq 0$.

We will proceed by induction on $n$. For $n=1$ the statement follows by Lemma
3.31 and the b-comparability property. So assume that the statement holds for
$n$, we will prove it for $n+1$. Using Lemma 3.31, we need to show that for
$a,b\in[0,(n+1)u]$ there is some sharp $p$ with
$\tilde{U}_{p}(a)\leq\tilde{U}_{p}(b)$ and
$\tilde{U}_{p^{\prime}}(a)\geq\tilde{U}_{p^{\prime}}(b)$. Notice that
$a-u,b-u\in[-nu,nu]$, so that by the induction assumption, there are some
sharp elements $q,r$ such that

$\displaystyle\tilde{U}_{q}(a-u)\geq 0,\quad\tilde{U}_{q^{\prime}}(a-u)\leq
0,\qquad\tilde{U}_{r}(b-u)\geq 0,\quad\tilde{U}_{r^{\prime}}(b-u)\leq 0.$

This implies that

$\tilde{U}_{q}(a)\geq q,\ \tilde{U}_{q^{\prime}}(a)\leq
q^{\prime},\qquad\tilde{U}_{r}(b)\geq r,\ \tilde{U}_{r^{\prime}}(b)\leq
r^{\prime}.$

Consider the decompositions

$\displaystyle a=\tilde{U}_{r\wedge q}(a)+\tilde{U}_{r^{\prime}\wedge
q}(a)+\tilde{U}_{r\wedge q^{\prime}}(a)+\tilde{U}_{r^{\prime}\wedge
q^{\prime}}(a),\qquad b=\tilde{U}_{r\wedge q}(b)+\tilde{U}_{r^{\prime}\wedge
q}(b)+\tilde{U}_{r\wedge q^{\prime}}(b)+\tilde{U}_{r^{\prime}\wedge
q^{\prime}}(b).$

We have $\tilde{U}_{r\wedge q}(a)-r\wedge q=\tilde{U}_{r\wedge
q}(a-u)\in[0,nu]$ and similarly $\tilde{U}_{r\wedge q}(b)-r\wedge q\in[0,nu]$.
By the induction hypothesis and Lemma 3.31, there is some sharp $s$ such that

$\displaystyle\tilde{U}_{s}(\tilde{U}_{r\wedge q}(a)-r\wedge
q)\leq\tilde{U}_{s}(\tilde{U}_{r\wedge q}(b)-r\wedge
q),\qquad\tilde{U}_{s^{\prime}}(\tilde{U}_{r\wedge q}(a)-r\wedge
q)\geq\tilde{U}_{s^{\prime}}(\tilde{U}_{r\wedge q}(b)-r\wedge q).$

It follows that $\tilde{U}_{s\wedge r\wedge q}(a)\leq\tilde{U}_{s\wedge
r\wedge q}(b)$ and $\tilde{U}_{s^{\prime}\wedge r\wedge
q}(a)\geq\tilde{U}_{s^{\prime}\wedge r\wedge q}(b)$. Further, we have

$\tilde{U}_{r^{\prime}\wedge
q}(a)=\tilde{U}_{r^{\prime}}(\tilde{U}_{q}(a))\geq r^{\prime}\wedge
q\geq\tilde{U}_{q}(r^{\prime})\geq\tilde{U}_{q}(\tilde{U}_{r^{\prime}}(b))=\tilde{U}_{r^{\prime}\wedge
q}(b)$

and similarly

$\tilde{U}_{r\wedge q^{\prime}}(a)\leq\tilde{U}_{r\wedge q^{\prime}}(b).$

Finally, we have $\tilde{U}_{r^{\prime}\wedge
q^{\prime}}(a)\leq\tilde{U}_{r^{\prime}}(q^{\prime})=r^{\prime}\wedge
q^{\prime}\leq u$ and $\tilde{U}_{r^{\prime}\wedge q^{\prime}}(b)\leq
r^{\prime}\wedge q^{\prime}\leq u$. Using the general comparability property
in $[0,u]$, there is some sharp element $t$ such that

$\tilde{U}_{t\wedge r^{\prime}\wedge q^{\prime}}(a)\leq\tilde{U}_{t\wedge
r^{\prime}\wedge q^{\prime}}(b),\ \ \tilde{U}_{t^{\prime}\wedge
r^{\prime}\wedge q^{\prime}}(a)\geq\tilde{U}_{t^{\prime}\wedge
r^{\prime}\wedge q^{\prime}}(b).$

Now put

$\displaystyle p:=s\wedge r\wedge q+r\wedge q^{\prime}+t\wedge
r^{\prime}\wedge q^{\prime},\qquad p^{\prime}:=s^{\prime}\wedge r\wedge
q+r^{\prime}\wedge q+t^{\prime}\wedge r^{\prime}\wedge q^{\prime}.$

Then we have

$\displaystyle\tilde{U}_{p}(a)=\tilde{U}_{s\wedge r\wedge
q}(a)+\tilde{U}_{r\wedge q^{\prime}}(a)+\tilde{U}_{t\wedge r^{\prime}\wedge
q^{\prime}}(a)\leq\tilde{U}_{s\wedge r\wedge q}(b)+\tilde{U}_{r\wedge
q^{\prime}}(b)+\tilde{U}_{t\wedge r^{\prime}\wedge
q^{\prime}}(b)=\tilde{U}_{p}(b)$

and similarly also $\tilde{U}_{p^{\prime}}(a)\geq\tilde{U}_{p^{\prime}}(b)$.
This proves the ’if’ part. The converse is obvious.

∎

Recall that if the MV-effect algebra $E$ is archimedean, then it is isomorphic
to a subalgebra of continuous functions on a compact Hausdorff space $X$, [8,
Thm. 7.1.3]. If $E$ has b-comparability, then we have seen that the universal
group has general comparability and by Lemma A.1, the group $G$ is archimedean
as well. Using the results of [19, Chap. 8], we obtain more information on the
representing space $X$.

######  Corollary 3.33.

Let $E$ be an archimedean MV-effect algebra with the b-comparability property.
Then $E$ is isomorphic to a subalgebra of continuous functions $X\to[0,1]$ for
a totally disconnected compact Hausdorff space $X$. The space $X$ is the Stone
space of the Boolean subalgebra $P=E_{S}$ of sharp elements in $E$.

###  3.5 The positive part and splitting

Assume that $E$ has the b-comparability property and let $a,b\in E$, $aCb$.
Then there is some C-block $C$ in $E$ such that $a,b\in C$ and therefore also
$P_{\leq}(a,b)\subseteq C$. By Theorem 3.23, $C$ is an MV-effect algebra with
b-comparability property and hence its universal group $G$ has general
comparability, by Theorem 3.32. The group $G$ contains the element $b-a$ and
any $p\in P_{\leq}(a,b)$ is also contained in $P^{G}_{\pm}(b-a)$. By Lemma
A.4, for any $p\in P_{\leq}(a,b)$, the positive part of $b-a$ has the form

$(b-a)_{+}=\tilde{U}_{p}(b)-\tilde{U}_{p}(a)=J_{p}(b)\ominus J_{p}(a)\in C,$

so that $(b-a)_{+}$ is well defined as an element in $E$.

######  Lemma 3.34.

Assume that $E$ has the b-comparability property and let $a,b\in E$, $aCb$.

  1. (i)

The element $(b-a)_{+}$ is contained in any C-block containing $a$ and $b$.

  2. (ii)

For $q\in P_{\leq}(a,b)$, $(a-b)_{+}=J_{q^{\prime}}(a)\ominus
J_{q^{\prime}}(b)$.

  3. (iii)

Let $r\in PC(a,b)$ be such that $J_{r}(a)\leq J_{r}(b)$ and
$J_{r^{\prime}}(b)\leq J_{r^{\prime}}(a)$, then $J_{r}(b)\ominus
J_{r}(a)=(b-a)_{+}$.

  4. (iv)

If $(b-a)_{+}^{\text{\rm o}}$ exists, then $(b-a)_{+}^{\text{\rm o}}\in
P_{\leq}(a,b)$ and it is the smallest projection such that $r\in PC(a,b)$ and
$J_{r}(a)\leq J_{r}(b)$, $J_{r^{\prime}}(b)\leq J_{r^{\prime}}(a)$.

###### Proof.

The statement (i) is obvious from the above considerations, (ii) follows
easily from the fact that $q^{\prime}\in P_{\leq}(b,a)$ if $q\in
P_{\leq}(a,b)$. The statement (iii) follows from the fact that there is some
C-block containing $a,b$ and $r$ and the properties of the orthogonal
decomposition of $a-b$ in its universal group (Lemma A.4). For (iv), note that
since the projection cover $p:=(b-a)_{+}^{\text{\rm o}}\in P((b-a)_{+})$, (i)
implies that $p$ is contained in any C-block containing $a,b$. This implies
$p\in P(a,b)$ and using Lemma A.4 (ii) in the universal group of any such
C-block, $p\in P_{\leq}(a,b)$. The last part of (iv) follows again from Lemma
A.4 (ii) using the fact that there is a C-block containing $a,b$ and $r$.

∎

For any element $a\in E$, we clearly have $aCa^{\prime}$. Let $q\in
P_{\leq}(a,a^{\prime})$, then $q\in P(a)$ and we have

$\displaystyle(a^{\prime}-a)_{+}$ $\displaystyle=J_{q}(a^{\prime})\ominus
J_{q}(a)=q\ominus 2J_{q}(a)$ (3) $\displaystyle(a-a^{\prime})_{+}$
$\displaystyle=J_{q^{\prime}}(a)\ominus
J_{q^{\prime}}(a^{\prime})=q^{\prime}\ominus 2J_{q^{\prime}}(a^{\prime}).$ (4)

In other words, we have a decomposition $a=J_{q}(a)\oplus
J_{q^{\prime}}(a)=a\wedge q\oplus a\wedge q^{\prime}$ where $2(a\wedge q)$ and
$2(a^{\prime}\wedge q^{\prime})$ exist. This may be interpreted as ”$a\wedge
q\leq 1/2$” although of course the constant $1/2$ may not be defined in $E$.
Similarly, ”$a\wedge q^{\prime}\geq 1/2$”. We will call this decomposition the
_splitting_ of $a$ and $q$ a _splitting projection_ for $a$. If $E$ is
archimedean then $C$ is archimedean as well and hence is isomorphic to a
subalgebra of continuous functions on a compact Hausdorff space (Corollary
3.33). In this case the splitting of $a$ has a clear interpretation.

######  Lemma 3.35.

Let $a\in E$ and let $C$ be any C-block containing $a$, with universal group
$G$.

  1. (i)

For any splitting projection $q$ for $a$, all the elements
$(a-a^{\prime})_{+}$, $(a^{\prime}-a)_{+}$, $q$, $2J_{q}(a)$,
$2J_{q^{\prime}}(a^{\prime})$ are contained in $C$ and
$(a-a^{\prime})_{+}=(2a-1)_{+}$, the latter element computed in $G$.

  2. (ii)

If $q$ is a splitting projection for $a$, then $q^{\prime}$ is a splitting
projection for $a^{\prime}$.

  3. (ii)

If $(a-a^{\prime})_{+}^{\text{\rm o}}$ exists, then
$((a-a^{\prime})_{+}^{\text{\rm o}})^{\prime}$ is the largest splitting
projection for $a$.

###### Proof.

Straightforward from Lemma 3.34.

∎

Let $q\in P$, then $J_{q}(E)=[0,q]$ is an effect algebra, with addition from
$E$ and with unit $q$. Note that the orthosupplement of an element $a$ in
$[0,q]$ is $a^{\prime}_{q}:=q\ominus a=q\wedge a^{\prime}$.

As proved in [24, Thm. 4.4], for $p\leq q$, the restriction of $J_{p}$ is a
compression on $[0,q]$ and $(J_{p})_{P\cap[0,q]}$ is a compression base in
$[0,q]$. Below we will always assume this inherited compression base in
$[0,q]$. For $r\in P\cap[0,q]$, we will denote by $C_{q}(r)$ the commutant of
$r$ and by $P_{q}(a)$ the bicommutant of $a$ in $[0,q]$.

######  Lemma 3.36.

Let $q\in P$ and $a,r\in[0,q]\cap P$.

  1. (i)

$C_{q}(r)=C(r)\cap[0,q]$ and $\\{p\wedge q:\ p\in P(a)\\}\subseteq P_{q}(a)$.

  2. (ii)

If $p\leftrightarrow q$, then $a\in C_{q}(p\wedge q)$ if and only if $a\in
C(p)$.

  3. (iii)

If $E$ has b-comparability then $[0,q]$ has b-comparability.

  4. (iv)

Let $[(a-a^{\prime}_{q})_{+}]_{q}$ and $[(a^{\prime}_{q}-a)_{+}]_{q}$ be the
splittings in $[0,q]$, then

$\displaystyle[(a-a^{\prime}_{q})_{+}]_{q}=(a-a^{\prime})_{+}\wedge
q,\qquad[(a^{\prime}_{q}-a)_{+}]_{q}=(a^{\prime}-a)_{+}\wedge q.$

  5. (v)

Let $[a^{\text{\rm o}}]_{q}$ be the projection cover in $[0,q]$, then
$[a^{\text{\rm o}}]_{q}=a^{\text{\rm o}}$. For $b\in C(q)$, $(b\wedge
q)^{\text{\rm o}}=b^{\text{\rm o}}\wedge q$.

###### Proof.

For (i), let $b=J_{q}(b)$, then $J_{r}(b)\oplus
J_{r^{\prime}_{q}}(b)=J_{r}(b)\oplus J_{r^{\prime}\wedge q}(b)=J_{r}(b)\oplus
J_{r^{\prime}}(b)$, this implies the first statement in (i) and also that
$PC_{q}(a):=\\{s\in P\cap[0,q],a\in C_{q}(s)\\}=PC(a)\cap[0,q]$. Let $p\in
P(a)$, then $p$ commutes with $q$ and $p\wedge q$ exists. For the second
statement of (i), we have to prove that $p\wedge q\in C_{q}(PC(a)\cap[0,q])$
which amounts to showing that $p\wedge q\in C(PC(a)\cap[0,q])$. Since clearly
$p,q\in C(PC(a)\cap[0,q])$, we have the same for $p\wedge q$.

For (ii), it is enough to note that if $p\leftrightarrow q$, then $(p\wedge
q)^{\prime}_{q}=q\wedge(p\wedge q)^{\prime}=q\wedge p^{\prime}$ and hence

$J_{p\wedge q}(a)\oplus J_{(p\wedge q)^{\prime}_{q}}(a)=J_{p}(a)\oplus
J_{p^{\prime}}(a).$

For (iii), let us prove the b-property. By (i), it is enough to show that
$P_{q}(a)\subseteq C(r)$ if and only if $a\in C(r)$. So assume that
$P_{q}(a)\subseteq C(r)$, then for any $p\in P(a)$,we have $p\wedge q\in C(r)$
which by (ii) (applied to $a=r$) implies that $p\in C(r)$. By the b-property
in $E$, $a\in C(r)$. The converse is clear by definition of $P_{q}(a)$, this
shows the b-property in $[0,q]$. If $a$ commutes with $b$ in $[0,q]$ then by
(i) we must have $P(a)\wedge q\leftrightarrow P(b)\wedge q$. Using (ii), we
obtain that for all $p\in P(a)$, $s\in P(b)$ we must have $p\in C(s\wedge q)$,
and since $a\in C(s\wedge q^{\prime})$, $p$ commutes with $s\wedge q^{\prime}$
as well. It follows that $P(a)\leftrightarrow P(b)$ and $aCb$ in $E$. Hence
there is some $t\in P_{\leq}(a,b)$, it is easily seen that we may use the
projection $t\wedge q$ to obtain b-comparability in $[0,q]$.

For (iv), it is easily checked that for any $p\in P_{\leq}(a^{\prime},a)$ we
have $p\wedge q\in P_{q,\leq}(a^{\prime}_{q},a)$ so that

$\displaystyle[(a-a^{\prime}_{q})_{+}]_{q}$ $\displaystyle=J_{p\wedge
q}(a)\ominus J_{p\wedge q}(a^{\prime}_{q})=J_{q}(J_{p}(a)\ominus
J_{p}(1-a))=J_{q}((a-a^{\prime})_{+})=(a-a^{\prime})_{+}\wedge q.$

The proof for $[(a^{\prime}_{q}-a)_{+}]_{q}$ is the same.

The projection cover in $[0,q]$ is clearly the same as in $E$, we only have to
prove the second statement of (v). It is clear that $(b\wedge q)^{\text{\rm
o}}\leq b^{\text{\rm o}}\wedge q$. Conversely, since $b^{\text{\rm o}}\in
C(q)$, we have

$b=b\wedge q\oplus b\wedge q^{\prime}\leq(b\wedge q)^{\text{\rm
o}}\oplus(b\wedge q^{\prime})^{\text{\rm o}}\leq b^{\text{\rm o}}\wedge
q\oplus b^{\text{\rm o}}\wedge q^{\prime}=b^{\text{\rm o}}.$

Since $b^{\text{\rm o}}$ is the smallest projection majorizing $b$, the second
inequality must be an equality, this implies $(b\wedge q)^{\text{\rm
o}}=b^{\text{\rm o}}\wedge q$.

∎

##  4 Spectral effect algebras

We are now ready to introduce a notion of spectrality in effect algebras.

######  Definition 4.1.

[36] Let $E$ be an effect algebra with compression base $(J_{p})_{p\in P}$. We
say that $E$ is spectral if $(J_{p})_{p\in P}$ has both the b-comparability
and projection cover property.

Let us remind the reader about some consequences of this definition. First,
$E$ is endowed with a compression base $(J_{p})_{p\in P}$ such that the set of
projections $P=E_{S}$. This implies that the compression base is maximal and
must contain the central compression base $(U_{p})_{p\in\Gamma(E)}$. Another
consequence is that the set of sharp elements $E_{S}$ is a normal subalgebra
which is an OML with the structure inherited from $E$. Every element $a\in E$
is contained in a C-block $C\subseteq E$, consisting of all elements
compatible with all sharp elements in some block $B$ of $P$. Any C-block is an
MV-effect algebra that is spectral with respect to its unique maximal
compression base $(U_{p})_{p\in C_{S}}$. Spectral MV-effect algebras will be
treated (among others) in the examples below. It will be also shown (Example
4.4) that the covering by spectral MV-effect algebras is in general not enough
for the whole effect algebra $E$ to be spectral.

Note that most of the above properties are obtained from b-comparability.
Without the projection cover property we would have a similar structure, but
the subalgebra $P=E_{S}$ might not be a lattice and the MV-effect algebras
might not be spectral. Notice also that our construction of the spectral
resolution below, based on the splitting, still may be done, but with no
unique choice for the spectral projections (cf. Lemma A.2 below or [30, Thm.
3.22] for order unit spaces).

######  Example 4.2.

Let $E=E(\mathcal{H})$ for some Hilbert space $\mathcal{H}$ (or more
generally, we may assume that $E$ is the interval $[0,1]$ in any von Neumann
algebra or in a JBW algebra), with the compression base $(U_{p})_{p\in
P(\mathcal{H})}$ defined as in Example 3.2. For any $a\in E$, let
$a^{\text{\rm o}}$ be the support projection of $a$, then $a^{\circ}$ is a
projection cover of $a$. As we have seen in Example 3.24, $E$ has the
b-property and $aCb$ if and only if $ab=ba$. In that case, taking the positive
part of the self-adjoint operator $a-b$, we have $(a-b)_{+}^{\text{\rm o}}\in
P_{\leq}(b,a)$. We conclude that $E$ has both the projection cover and
b-comparability property, so that $E$ is spectral.

######  Example 4.3.

Let $E$ be an MV-effect algebra and let $(G,u)$ be the universal group. By
Theorem 3.32, $E$ has the b-comparability property if and only if $G$ has the
comparability property. In this case, the projection cover property in $E$ is
equivalent to the _Rickart property_ in $G$, see [15, Thm. 6.5] and Appendix
A. Hence $E$ is spectral if and only if $G$ is spectral. Assume further that
$E$ is monotone $\sigma$-complete, then $(G,u)$ is a Dedekind
$\sigma$-complete lattice ordered group with order unit. By [19, Thm. 9.9],
$(G,u)$ satisfies general comparability, so $E$ has the b-comparability
property. Moreover, it follows by [19, Lemma 9.8] that $E$ has the projection
cover property, so that $G$ and also $E$ is spectral. We will see in Example
5.13 that under additional conditions a spectral MV-algebra must be monotone
$\sigma$-complete. However, this is not always the case: as we will see next,
any Boolean algebra is spectral.

######  Example 4.4.

Let $E$ be a Boolean algebra, with the central compression base $(U_{p})_{p\in
E}$. It is easily seen that $E$ has b-comparability: the b-property is
obtained by setting $B(q)=\\{0,q,q^{\prime},1\\}$, moreover, for $p,q\in E$,
we have $P(p,q)=E$ and

$\displaystyle U_{q}(p)=q\wedge p\leq q=U_{q}(q),\qquad
U_{q^{\prime}}(q)=0\leq q^{\prime}\wedge p=U_{q^{\prime}}(p),$

so that $q\in P_{\leq}(p,q)\neq\emptyset$. It is also clear that any Boolean
algebra has the projection cover property, by setting $p^{\text{\rm o}}=p$,
$p\in E$. Hence any Boolean algebra is spectral.

More generally, let $E$ be an OMP. Since in this case $E=E_{S}$, we see by
Theorem 3.26 that if $E$ has b-comparability, then $P=E_{S}=E$, so that $E$
must be a Boolean algebra (see Example 3.11). Note that any OMP is covered by
blocks which are Boolean algebras, hence spectral MV-effect algebras, but is
itself not spectral unless it is Boolean.

######  Example 4.5.

Let $E_{1}$, $E_{2}$ be effect algebras with compression bases
$(J_{1,p})_{p\in P_{1}}$ and $(J_{2,p})_{p\in P_{2}}$. Let $E=E_{1}\times
E_{2}$ be the direct product and let $P=P_{1}\times P_{2}$. For
$p=(p_{1},p_{2})\in P$, put $J_{p}=(J_{1,p_{1}},J_{2,p_{2}})$. It is easily
checked that each $J_{p}$ is a compression with focus $p$ and $(J_{p})_{p\in
P}$ is a compression base. Since all the operations are taken pointwise, it
can be checked that $E$ is spectral if and only if both $E_{1}$ and $E_{2}$
are spectral.

######  Example 4.6.

Let $E=E(\mathcal{H})\dot{\cup}E(\mathcal{H})$ be the horizontal sum as in
Example 3.9, with the compression base $(J_{p})_{p\in P}$,
$P=P(\mathcal{H})\dot{\cup}P(\mathcal{H})$, constructed from faithful states
on $E(\mathcal{H})$. Since any element of $E$ can be compatible with a
projection only if both belong to the same component, it is easily seen that
$E$ is spectral. More generally, if $E_{1}$ and $E_{2}$ are effect algebras
with compression bases $(J_{1,p})_{p\in P_{1}}$ and $(J_{2,p})_{p\in P_{2}}$
such that we can construct a compression base with $P=P_{1}\dot{\cup}P_{2}$ as
in Example 3.9, then $E_{1}\dot{\cup}E_{2}$ is spectral if and only if $E_{1}$
and $E_{2}$ are spectral.

All the properties involved in the definition of spectrality depend on the
choice of the compression base. As remarked above, such a compression base
must satisfy $P=E_{S}$ and therefore must be maximal. We have seen in Example
3.9 and 4.6 that such a compression base may be not unique. The next result
shows that spectrality does not depend from the choice of such a compression
base.

######  Proposition 4.7.

Let $E$ be spectral and let $\\{\bar{J}_{p}\\}_{p\in P}$ be a compression base
with $P=E_{S}$. Then $E$ is also spectral with respect to $\\{\bar{J}_{p}\\}$.

###### Proof.

By assumption, the compression bases $(J_{p})$ and $(\bar{J}_{p})$ have the
same set of projections. It is easily seen that the projection cover for any
element is the same for both compression bases, since these only depends on
the properties of the projections and not the corresponding compressions.
Hence $E$ with $\\{\bar{J}_{p}\\}$ has the projection cover property. By Lemma
3.14, we also see that the commutants $C(p)$ and bicommutants $P(a)$ are the
same for both compression bases, and that for $a\in C(p)$,
$J_{p}(a)=\bar{J}_{p}(a)=p\wedge a$. This shows that $E$ with
$\\{\bar{J}_{p}\\}$ has the b-comparability property.

∎

It would be interesting to know if this is true for any maximal compression
base, in other words, if for a spectral effect algebra $E$ any maximal
compression base must satisfy $P=E_{S}$.

###  4.1 Spectral resolutions

We will define the _binary spectral resolution_ of an element $a\in E$ as a
family of projections $\\{p_{\lambda}\\}$, indexed by dyadic rationals in
$[0,1]$. A dyadic rational $\lambda\in[0,1)$ has an expansion

$\displaystyle\lambda=\lambda(w):=\sum_{j=1}^{l(w)}w_{j}2^{-j}=k(w)/2^{l(w)},\qquad
k(w):=\sum_{j=1}^{l(w)}w_{j}2^{l(w)-j}$

for a binary string $w=w_{1}\dots w_{l(w)}\in\mathcal{B}:=\bigcup_{n\geq
0}\\{0,1\\}^{n}$, here $l(w)$ is the length of $w$. We will denote the empty
string as $\varepsilon$ and the concatenation of $w_{1},w_{2}\in\mathcal{B}$
as $w_{1}w_{2}$. The lexicographical order on binary strings will be denoted
by $\leq$.

Any dyadic rational $\lambda\in(0,1)$ has a unique expansion of the form
$\lambda=\lambda(w1)$ for some $w\in\mathcal{B}$, in this case

$\lambda=\lambda(w1)=(2k(w)+1)/2^{-(l(w)+1)}.$

Note that $\lambda(w1)$ is the middle point of the interval
$[\lambda(w),\lambda(w+1)]$ where $w+1$ is the consecutive binary string of
the same length as $w$ (if it exists, if $w$ is a (possibly empty) string of
ones $w=1\dots 1$, we put $\lambda(w+1)=1$). Similarly, we denote by $w-1$ the
previous binary string of the same length as $w$ and if $w$ is already the
smallest element in $\\{0,1\\}^{l(w)}$, then $\lambda(w-1)=0$.

We first construct two families $\\{c_{w}\\}$ and $\\{u_{w}\\}$ indexed by
binary strings:

  * •

we put $c_{\varepsilon}=a$, $u_{\varepsilon}:=a^{\text{\rm o}}$,

  * •

for any $w\in\\{0,1\\}^{n}$, $n=0,1,\dots$, we define $c_{w0}$ and $c_{w1}$ by
the splitting of $c_{w}$ in $[0,u_{w}]$:

$\displaystyle u_{w0}:=$ $\displaystyle((c_{w}-c_{w}^{\prime})_{+}^{\text{\rm
o}})^{\prime}\wedge u_{w},\qquad u_{w1}:=u_{w}\wedge u_{w0}^{\prime},$
$\displaystyle c_{w0}:=$ $\displaystyle
u_{w0}\ominus[(c_{w}^{\prime}-c_{w})_{+}\wedge u_{w}]=2J_{u_{w0}}(c_{w}),$
$\displaystyle c_{w1}:=$ $\displaystyle(c_{w}-c_{w}^{\prime})_{+}\wedge
u_{w}=u_{w1}\ominus 2J_{u_{w1}}(u_{w}\ominus c_{w}).$

Note that we have used the properties of splitting in (3), (4) and Lemmas
3.35, 3.36.

######  Remark 4.8.

It might be useful to write down the elements $u_{w}$ and $c_{w}$ explicitly
in a simple example. Let $E=[0,1]^{n}$, then $E$ is a monotone
$\sigma$-complete $MV$-algebra, hence $E$ is spectral. Let
$a=(a_{1},\dots,a_{n})\in E$. Note that an element $u\in E$ is sharp precisely
if $u_{i}\in\\{0,1\\}$ for all $i$, hence can be identified with a subset
$U\subseteq\\{1,\dots,n\\}$, $U=\\{i,u_{i}=1\\}$. Then we have, for any binary
string $w$ of length $n$:

$\displaystyle
U_{w}=\\{i,\frac{k(w)}{2^{n}}<a_{i}\leq\frac{k(w)+1}{2^{n}}\\},\qquad
c_{w}(i)=\begin{dcases}2^{n}a_{i}-k(w),&i\in U_{w}\\\ 0,&i\neq
U_{w}.\end{dcases}$

See also Lemma 4.11 below.

We collect some easily observed properties of these families.

######  Lemma 4.9.

  1. (i)

For all $w$, we have $u_{w}\in P(a)$, $c_{w}\leq u_{w}$ and $c_{w}$ is
contained in all C-blocks containing $a$.

  2. (ii)

For any strings $w$, $\tilde{w}$, we have $u_{w\tilde{w}}\leq u_{w}$.

  3. (iii)

For binary strings of fixed length $l(w)=n$, $u_{w}$ are pairwise orthogonal
and we have

$\bigoplus_{w\in\\{0,1\\}^{n}}u_{w}=a^{\text{\rm o}}.$

###### Proof.

It is clear from the construction that $c_{w}\leq u_{w}$ for all $w$. By Lemma
3.35, all $u_{w0}$, $u_{w1}$, $c_{w0}$, $c_{w1}$ are contained in any C-block
containing $c_{w}$, which implies (i).

The statement (ii) can be proved by induction on the length of $\tilde{w}$:
for any $w$, we have $u_{w0},u_{w1}\leq u_{w}$ by construction. If the
statement holds for any $\tilde{w}$ of length $n$, then $u_{w\tilde{w}i}\leq
u_{w\tilde{w}}\leq u_{w}$ for $i\in\\{0,1\\}$.

We use induction on the length of $w$ to prove (iii): by construction
$u_{w0}\oplus u_{w1}=u_{w}$ for any $w$, this also proves the statement for
$n=1$. Let $w_{1},w_{2}\in\\{0,1\\}^{n}$ and assume that the statement holds
for any $k<n$. To show orthogonality we only have to deal with the situation
when $w_{1}$ and $w_{2}$ have some prefix $\tilde{w}_{1}$ resp.
$\tilde{w}_{2}$, of length $k<n$ and such that
$\tilde{w}_{1}\neq\tilde{w}_{2}$, in which case by (ii), $u_{w_{1}}\leq
u_{\tilde{w}_{1}}$, $u_{w_{2}}\leq u_{\tilde{w}_{2}}$ and
$u_{\tilde{w}_{1}}\perp u_{\tilde{w}_{2}}$ by the induction assumption.
Moreover,

$\displaystyle\bigoplus_{w\in\\{0,1\\}^{n}}u_{w}=\bigoplus_{\tilde{w}\in\\{0,1\\}^{n-1}}(u_{\tilde{w}0}\oplus
u_{\tilde{w}1})=\bigoplus_{\tilde{w}\in\\{0,1\\}^{n-1}}u_{\tilde{w}}=a^{\text{\rm
o}}.$

∎

We now define the binary spectral resolution of $a$:

  * •

$p_{0}=(a^{\text{\rm o}})^{\prime}$,

  * •

for $l(w)=n$, $n=0,1,\dots$, we put

$p_{\lambda(w1)}=(a^{\text{\rm
o}})^{\prime}\oplus\bigoplus_{\tilde{w}\in\\{0,1\\}^{n+1},\tilde{w}\leq
w0}u_{\tilde{w}}$

  * •

$p_{1}=1$.

######  Lemma 4.10.

For any $w\in\\{0,1\\}^{n}$, we have

  1. (i)

$p_{\lambda(w1)}\wedge u_{w}=u_{w0}$,

  2. (ii)

$(a^{\text{\rm
o}})^{\prime}\oplus\bigoplus_{\tilde{w}\in\\{0,1\\}^{l(w)},\tilde{w}\leq
w}u_{\tilde{w}}=p_{\lambda(w+1)}=p_{\lambda(w)}\oplus u_{w}$,

  3. (iii)

if $w_{1},w_{2}$ are two binary strings such that $w_{1}\leq w_{2}$, then
$p_{\lambda(w_{1})}\leq p_{\lambda(w_{2})}$.

###### Proof.

The statement (i) follows easily from the definition and the fact that
$u_{w}=u_{w0}\oplus u_{w1}$. For (ii), notice first that if $w=1\dots 1$ is
the largest element in $\\{0,1\\}^{n}$, then by Lemma 4.9, we have

$(a^{\text{\rm
o}})^{\prime}\oplus\bigoplus_{\begin{subarray}{c}\tilde{w}\in\\{0,1\\}^{n}\\\
\tilde{w}\leq w\end{subarray}}u_{\tilde{w}}=1=p_{1}=p_{\lambda(w+1)},$

so we may assume $w\neq 1\dots 1$. This also shows that the statement holds
for $l(w)=0$. Assume it is true for strings of length $n$ and let $w$ be a
binary string of length $n+1$. If $w=w_{1}0$ for some $w_{1}\in\\{0,1\\}^{n}$,
then we have by definition that the left hand side is equal to
$p_{\lambda(w_{1}1)}=p_{\lambda(w+1)}$. Let now $w=w_{1}1$, then notice that

$\displaystyle\bigoplus_{\begin{subarray}{c}\tilde{w}\in\\{0,1\\}^{n+1}\\\
\tilde{w}\leq
w\end{subarray}}u_{\tilde{w}}=\bigoplus_{\begin{subarray}{c}\tilde{w}\in\\{0,1\\}^{n+1}\\\
\tilde{w}\leq(w_{1}-1)1\end{subarray}}u_{\tilde{w}}\oplus u_{w_{1}0}\oplus
u_{w_{1}1}=\bigoplus_{\begin{subarray}{c}\tilde{w}\in\\{0,1\\}^{n+1}\\\
\tilde{w}\leq(w_{1}-1)1\end{subarray}}u_{\tilde{w}}\oplus u_{w_{1}}$

and repeating the argument, we obtain that

$\displaystyle(a^{\text{\rm
o}})^{\prime}\oplus\bigoplus_{\begin{subarray}{c}\tilde{w}\in\\{0,1\\}^{n+1}\\\
\tilde{w}\leq{w_{1}1}\end{subarray}}u_{\tilde{w}}=(a^{\text{\rm
o}})^{\prime}\oplus\bigoplus_{\begin{subarray}{c}\tilde{w}\in\\{0,1\\}^{n}\\\
\tilde{w}\leq{w_{1}}\end{subarray}}u_{\tilde{w}}=p_{\lambda(w_{1}+1)}=p_{\lambda(w+1)},$

using the induction assumption and $\lambda(w0)=\lambda(w)$. The second
equality is now clear. If $w_{1}$, $w_{2}$ have the same length, the statement
(iii) follows from $p_{\lambda(w)}\leq p_{\lambda(w+1)}$ which is obvious from
(ii). Otherwise we may add a string of zeros at the end of one of $w_{1}$,
$w_{2}$, which does not change the order nor the values of
$p_{\lambda(w_{1})}$ and $p_{\lambda(w_{2})}$.

∎

The element $a$ is contained in some C-block $C=C(B)$ of $E$, which is a
spectral MV-effect algebra with the compression base $(J_{p}|_{C}=U_{p})_{p\in
B}$. As we have seen in Example 4.3, the universal group $G$ of $C$ is a
spectral lattice ordered abelian group $G$ with order unit, with the
compression base $(\tilde{U}_{p})_{p\in B}$, extending the compression base in
$C$. Hence there is a rational spectral resolution of $a$ in $G$, given by

$p^{C}_{a,\lambda}:=((na-mu)_{+})^{*},\qquad\lambda=\frac{m}{n},\ n>0,$ (5)

see Appendix A. For $g\in G$, $g^{*}$ denotes the Rickart mapping in $G$. The
notation highlights the possible dependence on the choice of the C-block $C$.
If the element $a$ is clear, we will skip the index from the notation.

As it will turn out below, the spectral resolution does not depend on $C$. We
will first settle the result for dyadic rationals, relating the spectral
projection $p^{C}_{\lambda}$ to the corresponding projection $p_{\lambda(w1)}$
constructed above.

######  Lemma 4.11.

Let $w=w_{1}\dots w_{l(w)}$ and let $C=C(B)$ be a C-block containing $a$. Then
$u_{w}\in B$ and

$c_{w}=\tilde{U}_{u_{w}}(2^{l(w)}a-k(w)1),$

computed in the universal group $G$ of $C$.

###### Proof.

By Lemma 4.9, we have $u_{w}\in P(a)\subseteq C\cap P=B$ for all $w$. For
$w=\varepsilon$, we have $l(w)=k(\varepsilon)=0$ and
$c_{\varepsilon}=a=U_{a^{\text{\rm
o}}}(a)=U_{u_{\varepsilon}}(a)=\tilde{U}_{u_{\varepsilon}}(a)$. Assume that
the equality holds for $l(w)=n$. We have $k(w0)=2k(w)$ and by construction and
induction assumption,

$\displaystyle c_{w0}$
$\displaystyle=2U_{u_{w0}}(c_{w})=2\tilde{U}_{u_{w0}}(c_{w})=2\tilde{U}_{u_{w0}}(2^{n}a-k(w)1)=\tilde{U}_{u_{w0}}(2^{n+1}a-2k(w)1)$
$\displaystyle=\tilde{U}_{u_{w0}}(2^{l(w0)}a-k(w0)1),$

here we used the fact that $u_{w0}\leq u_{w}$ and that $\tilde{U}_{p}$ is a
group homomorphism on $G$. Similarly, $k(w1)=2k(w)+1$ and

$\displaystyle
c_{w1}=\tilde{U}_{w1}(2c_{w}-1)=\tilde{U}_{w1}(2(2^{l(w)}a-k(w)1)-1)=\tilde{U}_{w1}(2^{l(w1)}a-k(w1)).$

∎

######  Theorem 4.12.

For any C-block $C$ such that $a\in C$ and any dyadic rational
$\lambda\in[0,1]$ we have

$p^{C}_{\lambda}=p_{\lambda}.$

###### Proof.

It is easily seen from (5) that $p^{C}_{0}=1-a^{\text{\rm o}}=p_{0}$ and
$p^{C}_{1}=1=p_{1}$. Let us now prove that
$p^{C}_{\frac{1}{2}}=p_{\lambda(1)}$. By definition, we have

$p_{\lambda(1)}=(a^{\text{\rm o}})^{\prime}\oplus u_{0}=(a^{\text{\rm
o}})^{\prime}\oplus((a-a^{\prime})_{+}^{\text{\rm o}})^{\prime}\wedge
a^{\text{\rm o}}.$

Notice that, for any $p\in P_{\leq}(a^{\prime},a)$, we have

$\displaystyle(a-a^{\prime})_{+}=J_{p}(a)\ominus J_{p}(a^{\prime})\leq
J_{p}(a)\leq J_{p}(a^{\text{\rm o}})=a^{\text{\rm o}}\wedge p\leq a^{\text{\rm
o}},$

so that $(a-a^{\prime})_{+}^{\text{\rm o}}\leq a^{\text{\rm o}}$ and
$((a-a^{\prime})_{+}^{\text{\rm o}})^{\prime}\wedge(a^{\text{\rm
o}})^{\prime}=(a^{\text{\rm o}})^{\prime}$. It follows that

$\displaystyle p_{\lambda(1)}=((a-a^{\prime})_{+}^{\text{\rm
o}})^{\prime}\wedge(a^{\text{\rm
o}})^{\prime}\oplus((a-a^{\prime})_{+}^{\text{\rm o}})^{\prime}\wedge
a^{\text{\rm o}}=((a-a^{\prime})_{+}^{\text{\rm
o}})^{\prime}=p^{C}_{\frac{1}{2}}.$

We will now show the statement by induction on $l(w)$ in the expansion
$\lambda=\lambda(w1)$, the above paragraph shows that it is true for $l(w)=0$.

Assume that the statement holds if $l(w)<n$ and let $w$ be such that $l(w)=n$.
We have $\lambda(w)\leq\lambda\leq\lambda(w+1)$ and by Lemma 4.10,
$p_{\lambda(w)}\leq p_{\lambda}\leq p_{\lambda(w+1)}$ and
$u_{w}=p_{\lambda(w+1)}\wedge p_{\lambda(w)}^{\prime}$. By the induction
assumption we have $p_{\lambda(w)}=p^{C}_{\lambda(w)}$ and
$p_{\lambda(w+1)}=p^{C}_{\lambda(w+1)}$, so that using the properties of
spectral projections in Lemma 4.10, we obtain

$p_{\lambda(w)}\leq p_{\lambda},p^{C}_{\lambda}\leq p_{\lambda(w+1)}.$

This implies $p_{\lambda}\wedge u_{w}^{\prime}=p_{\lambda}^{C}\wedge
u_{w}^{\prime}=p_{\lambda(w)}$, so that what we have to prove is

$u_{w0}=p_{\lambda(w1)}\wedge u_{w}=p^{C}_{\lambda(w1)}\wedge u_{w}.$

Put $g_{w}:=2^{l(w)+1}a-(2k(w)+1)1\in G$, then
$p^{C}_{\lambda}=(g_{w})_{+}^{*}$ by (5). Using Lemmas A.4 (iii) and 4.11, and
the definition of $u_{w0}$, we have

$\displaystyle\tilde{U}_{u_{w0}}((g_{w})_{+})=\tilde{U}_{u_{w0}}\tilde{U}_{u_{w}}((g_{w})_{+})=\tilde{U}_{w0}((\tilde{U}_{u_{w}}(g_{w}))_{+})=J_{u_{w0}}((c_{w}-(u_{w}\ominus
c_{w}))_{+})=0.$

By the properties of the Rickart mapping, this implies that $u_{w0}\leq
p^{C}_{\lambda}$ and since also $u_{w0}\leq u_{w}$, $u_{w0}\leq
p^{C}_{\lambda}\wedge u_{w}$. The converse follows similarly from

$\displaystyle J_{p_{\lambda}^{C}\wedge u_{w}}((c_{w}-(u_{w}\ominus
c_{w}))_{+})=\tilde{U}_{p_{\lambda}^{C}\wedge
u_{w}}((\tilde{U}_{u_{w}}(g_{w}))_{+})=\tilde{U}_{u_{w}}\tilde{U}_{p_{\lambda}^{C}}((g_{w})_{+})=0,$

the definition of $u_{w0}$ and properties of the projection cover.

∎

To obtain the result for values $\lambda\in\mathbb{Q}\cap[0,1]$ other than
dyadic rationals, it will be convenient to use an additional assumption on
$E$, namely that $E$ is archimedean. The next result shows that this
assumption is necessary for an important right continuity property of the
binary spectral resolution.

######  Proposition 4.13.

Let $E$ be spectral and let $\\{p_{a,\lambda}\\}$ be the binary spectral
resolution of $a\in E$. Then

$p_{a,\lambda}=\bigwedge_{\mu>\lambda}p_{a,\mu}$

holds for any $a$ if and only if $E$ is archimedean.

###### Proof.

Let $C$ be any C-block containing $a$ and let $G$ be its universal group. If
$E$ is archimedean, then $C$ is archimedean as well, so that $G$ is
archimedean by Lemma A.1. By Theorem A.5, we have for all
$\lambda\in\mathbb{Q}\cap[0,1]$ that

$p^{C}_{a,\lambda}=\bigwedge_{\mu>\lambda}p^{C}_{a,\mu},$

where the infimum is over all $\mu\in\mathbb{Q}\cap[0,1]$. Restricting to
dyadic rationals, this implies the statement by Theorem 4.12 and the fact that
dyadic rationals are dense in $[0,1]$. Assume conversely that the equality
holds and let $a\in E$ be such that $2^{n}a$ exists for all $n\in\mathbb{N}$.
Then for all $n$, we have

$p_{a,2^{-n}}=p^{C}_{a,2^{-n}}=(2^{n}a-1)_{+}^{*}=1.$

Consequently,

$(a^{\text{\rm o}})^{\prime}=p_{0}=\bigwedge_{n}p_{a,2^{-n}}=1,$

so that $a=0$. Hence $E$ must be archimedean. ∎

######  Theorem 4.14.

Let $E$ be an archimedean spectral effect algebra and $a\in E$. Then
$p^{C}_{a,\lambda}$ does not depend on the C-block $C$ for any
$\lambda\in[0,1]\cap\mathbb{Q}$, in particular, we have

$p^{C}_{a,\lambda}=\bigwedge_{\mu>\lambda}p_{a,\mu}=:p_{a,\lambda},$

here the infimum is over all dyadic rationals $\mu>\lambda$.

###### Proof.

As in the proof of Proposition 4.13, we have for that

$p^{C}_{a,\lambda}=\bigwedge_{\lambda<\mu}p^{C}_{a,\mu}$

for any C-block $C$ containing $a$ and all $\lambda\in\mathbb{Q}\cap[0,1]$.
The statement now follows from Theorem 4.12 and the fact that dyadic rationals
are dense in $[0,1]$.

∎

The family of projections
$\\{p_{a,\lambda}\\}_{\lambda\in\mathbb{Q}\cap[0,1]}$ obtained in the above
theorem will be called the _rational spectral resolution_ of $a$ in $E$.

We introduce partially defined maps $f_{0},f_{1}$ on an effect algebra $E$,
such that

$f_{0}(b)=2b,\qquad f_{1}(b)=(2b^{\prime})^{\prime}$

defined if the respective elements exist, that is, $f_{0}(b)$ exists if $b\leq
b^{\prime}$ and $f_{1}(b)$ exists if $b^{\prime}\leq b$. For a binary string
$w=w_{1}\dots w_{n}$ we define $f_{w}:=f_{w_{n}}\circ\dots\circ f_{w_{1}}$,
with the understanding that $f_{\varepsilon}$ is the identity map and if
$f_{w}(b)$ exists then $f_{\tilde{w}}(b)$ must exist for all prefixes
$\tilde{w}$ of $w$. It is easily seen that for $p\in P$,
$J_{p}(f_{w}(a))=J_{p}(f_{w}(J_{p}(a)))$ if $f_{w}(a)$ exists.

We have the following characterization of the rational spectral resolution.

######  Theorem 4.15.

Let $E$ be an archimedean spectral effect algebra. Then the rational spectral
resolution of $a\in E$ is the unique family
$\\{p_{\lambda}\\}_{\lambda\in\mathbb{Q}\cap[0,1]}$ such that

  1. (i)

$p_{\lambda}\in PC(a)$ for all $\lambda$,

  2. (ii)

$p_{0}\leq a^{\prime}$, $p_{1}=1$ and if $\lambda\leq\mu$ then
$p_{\lambda}\leq p_{\mu}$,

  3. (iii)

$\bigwedge_{\lambda<\mu}p_{\mu}=p_{\lambda}$ for all $\lambda$,

  4. (iv)

for any $w\in\mathcal{B}$ put $u_{w}:=p_{\lambda(w+1)}\wedge
p_{\lambda(w)}^{\prime}$ and let $f_{w}$ be defined as above for the effect
algebra $[0,u_{w}]$, then $f_{w}(J_{u_{w}}(a))$ exists.

###### Proof.

Assume $\\{p_{\lambda}\\}_{\lambda\in\mathbb{Q}\cap[0,1]}$ is a family of
projections satisfying (i)-(iv). By (ii), all $p_{\lambda}$ commute, so that
by (i) there is some C-block $C$ containing $a$ and all $p_{\lambda}$, with
universal group $G$. We will show that (iv) is equivalent to

  1. (iv’)

$2^{l(w)}J_{p_{\lambda(w)}}(a)\leq k(w)p_{\lambda(w)}$ and
$2^{l(w)}J_{p_{\lambda(w)}^{\prime}}(a)\geq k(w)p^{\prime}_{\lambda(w)}$, for
all $w\in\mathcal{B}$, computed in $G$.

To show this, notice that for $w\in\mathcal{B}$, $i\in\\{0,1\\}$ and
$b\in[0,u_{w}]$, $f_{i}(b)$ exists in $[0,u_{w}]$ if and only if $2b-iu_{w}$
computed in $G$ belongs to $C$ and then $f_{i}(b)=2b-iu_{w}$. We have by (ii)
and (iv) that $u_{wi}\leq u_{w}$ and in $[0,u_{wi}]$

$f_{wi}(J_{u_{wi}}(a))=f_{i}(J_{u_{wi}}(f_{w}(J_{u_{w}}(a)))).$

Using the fact that $k(wi)=2k(w)+i$, one can easily prove by induction that
(iv) is equivalent to

$\tilde{U}_{u_{w}}(2^{l(w)}a-k(w)1)\in C,\qquad\forall w\in\mathcal{B},$

here $u_{w}$ is as in (iv) and $\tilde{U}_{u_{w}}$ is the compression
extending $J_{u_{w}}$ in the lattice group $G$. By definition of $u_{w}$, we
have

$p_{\lambda(w)}=p_{0}\oplus\bigoplus_{\tilde{w}<w}u_{\tilde{w}},\quad
p_{\lambda(w)}^{\prime}=\bigoplus_{\tilde{w}\geq w}u_{\tilde{w}},$

where we sum over strings of length $l(\tilde{w})=l(w)$. Since $k(\tilde{w})$
and $k(w)$ are integers such that $k(\tilde{w})<k(w)$ if $\tilde{w}<w$, we
have in this case

$\tilde{U}_{u_{\tilde{w}}}(2^{l(w)}a-k(w)1)\leq\tilde{U}_{u_{\tilde{w}}}(2^{l(\tilde{w})}a-k(\tilde{w})1-1)\leq
0.$

Since also $\tilde{U}_{p_{0}}(2^{l(w)}a-k(w)1)=-k(w)p_{0}\leq 0$ by (ii), we
obtain

$\tilde{U}_{p_{\lambda(w)}}(2^{l(w)}a-k(w)1)=\tilde{U}_{p_{0}}(2^{l(w)}a-k(w)1)+\sum_{\tilde{w}<w}\tilde{U}_{u_{\tilde{w}}}(2^{l(w)}a-k(w)1)\leq
0.$

Similarly, $\tilde{U}_{p_{\lambda(w)}^{\prime}}(2^{l(w)}a-k(w)1)\geq 0$, this
proves (iv’). Conversely, if (iv’) holds, then

$\tilde{U}_{u_{w}}(2^{l(w)}a-k(w)1)=\tilde{U}_{p_{\lambda(w+1)}}\tilde{U}_{p_{\lambda(w)}^{\prime}}(2^{l(w)}a-k(w)1)\geq
0$

and also

$\tilde{U}_{u_{w}}(2^{l(w)}a-k(w)1)-u_{w}=\tilde{U}_{p^{\prime}_{\lambda(w)}}\tilde{U}_{p_{\lambda(w+1)}}(2^{l(w)}a-k(w+1)1)\leq
0,$

so that $\tilde{U}_{u_{w}}(2^{l(w)}a-k(w)1)\in C$ for all $w\in\mathcal{B}$,
this is equivalent to (iv). The proof is now finished similarly as the proof
of Theorem A.5.

∎

###  4.2 Further properties of the rational spectral resolution

We have obtained a well defined binary spectral resolution of elements in a
spectral effect algebra $E$, which can be extended to a rational spectral
resolution if $E$ is archimedean.111In fact, archimedeanity of $E$ is not
needed, as shown in the previous version of this work [31], but the
constructions are more complicated. There are some properties of the spectral
resolution in $E(\mathcal{H})$ that we would like to hold also in this general
case. Specifically, we would expect that

  1. (a)

the element $a$ is uniquely determined by its spectral resolution,

  2. (b)

for any projection $q\in P$, we have $a\in C(q)$ iff $p_{a,\lambda}\in C(q)$
for all $\lambda\in\mathbb{Q}$.

At present, we can show these properties under the assumption of existence of
a separating family of states on $E$.

######  Proposition 4.16.

Let $E$ be spectral, $a\in E$ and let $\\{c_{w}\\}$ and $\\{u_{w}\\}$ be the
families associated with the binary spectral resolution
$\\{p_{a,\lambda(w)}\\}_{w\in\mathcal{B}}$ of $a$. Then for any state $s$ on
$E$:

  1. (i)

$s(a)=\lim_{n\to\infty}\sum_{l(w)=n}\lambda(w)s(u_{w})$ and $s(a)=0$ if and
only if $s(u_{w})=0$ for all $w\in\mathcal{B}$ such that $\lambda(w)\neq 0$.

  2. (ii)

If $E$ is archimedean and $s$ is $\sigma$-additive, then $s(a)=0$ if and only
if $s(a^{\text{\rm o}})=0$.

  3. (iii)

If $b\in E$ is such that $p_{a,\lambda(w)}=p_{b,\lambda(w)}$ for all
$w\in\mathcal{B}$, then $s(a)=s(b)$.

###### Proof.

Let $C$ be any C-block containing $a$ and let $G$ be the universal group.
Since $c_{w}\leq u_{w}$, $c_{w}$ with $w\in\\{0,1\\}^{n}$ are summable in $E$
by Lemma 4.9 and using Lemma 4.11 we obtain in $G$:

$c_{n}:=\bigoplus_{l(w)=n}c_{w}=\sum_{l(w)=n}\tilde{U}_{u_{w}}(2^{n}a-k(w)1)=2^{n}a-\sum_{l(w)=n}k(w)u_{w}.$
(6)

Note that the restriction $s|_{C}$ is a state on $C$ and hence extends to a
state on $G$, so that

$s(c_{n})=2^{n}s(a)-\sum_{l(w)=n}k(w)s(u_{w}).$

The first part of (i) follows from $c_{n}\in E$, so that $0\leq s(c_{n})\leq
1$. This also implies that if $s(a)=0$, then we must have $s(u_{w})=0$ for all
$w$ such that $\lambda(w)\neq 0$. The converse follows from the first part of
(i).

For (ii), assume $s(a)=0$. Let $0^{n}$ denote a string of zeros of length $n$,
then by (i) and Lemma 4.9,

$0=s(\bigoplus_{l(w)=n,w\neq 0^{n}}u_{w})=s(a^{\text{\rm o}}\wedge
u^{\prime}_{0^{n}})=s(a^{\text{\rm o}}\wedge p_{2^{-n}}^{\prime}),$

here we used the fact that $u_{w}=p_{\lambda(w+1)}\wedge
p_{\lambda(w)}^{\prime}$ and $\lambda(0^{n}+1)=\lambda(0^{n-1}1)=2^{-n}$.
Since $E$ is archimedean and $s$ is $\sigma$-additive, we get

$0=s(a^{\text{\rm o}}\wedge(\bigvee_{n}p^{\prime}_{2^{-n}}))=s(a^{\text{\rm
o}}\wedge p^{\prime}_{0})=s(a^{\text{\rm o}}).$

The converse is obvious from $a\leq a^{\text{\rm o}}$. The statement (iii) is
immediate from (i).

∎

######  Proposition 4.17.

Let $q\in P$ be a projection commuting with all binary spectral projections of
$a$. Then for any state $s$ on $E$,

$s(a)=s(J_{q}(a)+J_{q^{\prime}}(a)).$

###### Proof.

Put $J:=J_{q}+J_{q^{\prime}}$, then $J:E\to E$ is a unital additive map
preserving any element $r$ in the Boolean subalgebra generated by the binary
spectral projections of $a$. We also have $J_{r}J=JJ_{r}$ for any such element
and therefore

$J(a)=J(J_{r}(a)+J_{r^{\prime}}(a))=J_{r}(J(a))+J_{r^{\prime}}(J(a)),$

so that $J(a)\in C(r)$. Let $\tilde{C}$ be a C-block containing $J(a)$ and all
$r$ an let $\tilde{G}$ be its universal group. Our aim is to show that for
$c_{n}$ as in (6), $J(c_{n})\in\tilde{C}$ for all $n$ and, as an element of
$\tilde{G}$,

$J(c_{n})=2^{n}J(a)-\sum_{l(w)=n}k(w)u_{w}.$ (7)

Exactly as in the proof of Proposition 4.16, this then implies that

$s(J(a))=\lim_{n\to\infty}\sum_{l(w)=n}\lambda(w)s(u_{w})=s(a).$

Note that if $C$ is a C-block containing $a$, then it is not clear in general
whether $J$ maps $C$ into $\tilde{C}$ (or into any ordered unital group). We
therefore cannot use any extension of $J$ to the universal group of $C$.

Let us first show by induction that $J(c_{w})\in\tilde{C}$ for all $w$. This
is clearly true for $w=\varepsilon$. Assume that $J(c_{w})\in\tilde{C}$ holds
for some $w$. Then $J(c_{w})\leq J(u_{w})=u_{w}$ and it is easily seen that,
since $u_{w0}\in P_{\leq}(c_{w},c_{w}^{\prime}\wedge u_{w})$, we have
$u_{w0}\in P_{\leq}(J(c_{w}),J(c_{w})^{\prime}\wedge u_{w})$ and

$\displaystyle J((c_{w}^{\prime}-c_{w})_{+}\wedge u_{w})$
$\displaystyle=J(J_{u_{w0}}(c_{w}^{\prime})\ominus
J_{u_{w0}}(c_{w}))=J_{u_{w0}}(J(c_{w})^{\prime}\wedge u_{w})\ominus
J_{u_{w0}}(J(c_{w}))$ $\displaystyle=(J(c_{w})^{\prime}\wedge
u_{w}-J(c_{w}))_{+}=(J(c_{w})^{\prime}-J(c_{w}))_{+}\wedge u_{w}\in\tilde{C}.$

Similarly, $J((c_{w}-c_{w}^{\prime}\wedge
u_{w})_{+})=(J(c_{w})-J(c_{w})^{\prime})_{+}\wedge u_{w}\in\tilde{C}$ so that
$J(c_{w0}),J(c_{w1})\in\tilde{C}$.

The elements $J(c_{w})$, $l(w)=n$ are summable and we have

$J(c_{n})=\bigoplus_{l(w)=n}J(c_{w})\in\tilde{C}.$

We next prove (7). Again, we proceed by induction. The statement is clear for
$l(w)=0$: $J(c_{\varepsilon})=J(a)$. Assume that it holds for $l(w)=n$. Then

$\displaystyle\sum_{l(w)=n+1}$ $\displaystyle
J(c_{w})=\sum_{l(w)=n}(J(c_{w0})+J(c_{w1}))$
$\displaystyle=\sum_{l(w)=n}\biggl{(}u_{w0}-\tilde{\tilde{U}}_{u_{w}}\bigl{(}(J(c_{w})^{\prime}-J(c_{w}))_{+}+(J(c_{w})-J(c_{w})^{\prime})_{+}\bigr{)}\biggr{)}$
$\displaystyle=\sum_{l(w)=n}(u_{w0}+\tilde{\tilde{U}}_{u_{w}}(J(c_{w})-J(c_{w})^{\prime}))=2J(c_{n})-\sum_{l(w)=n}u_{w1},$

here $\tilde{\tilde{U}}_{u_{w}}$ is the extension of $J_{u_{w}}|{\tilde{C}}$
to $\tilde{G}$. Using the induction assumption, we get

$\displaystyle J(c_{n+1})$
$\displaystyle=2^{n+1}J(a)-\sum_{l(w)=n}(2k(w)u_{w}+u_{w1})=2^{n+1}J(a)-\sum_{l(w)=n}(2k(w)u_{w0}+(2k(w)+1)u_{w1})$
$\displaystyle=2^{n+1}J(a)-\sum_{l(w)=n+1}k(w)u_{w}.$

∎

######  Theorem 4.18.

Assume that $E$ is spectral and has a separating set of states. Then

  1. (i)

For $q\in P$ and $a\in E$, $a\in C(q)$ if and only if $p_{a,\lambda(w)}\in
C(q)$ for all $w\in\mathcal{B}$.

  2. (ii)

If $a,b\in E$ are such that $p_{a,\lambda(w)}=p_{b,\lambda(w)}$ for all
$w\in\mathcal{B}$, then $a=b$.

###### Proof.

Immediate from Prop. 4.16 and 4.17.

∎

##  5 Interval effect algebras

The notions of compressions, compression bases and their comparability and
spectrality properties were originally introduced in the setting of partially
ordered abelian groups. The necessary definitions and properties are collected
in Appendix A.

Recall that an effect algebra $E$ is an interval effect algebra if it is
isomorphic to the unit interval in a partially ordered unital abelian group
(Example 2.2). By [10], there is a _universal group_ $(G,u)$ of $E$ (unique up
to isomorphisms), such that $E\simeq G[0,u]$ and the following universal
property holds: for any group $H$, any additive mapping $\Phi:E\to H$ uniquely
extends to a group homomorphism $G\to H$.

It follows from the universal property that any compression $J$ on $E$ extends
uniquely to a group homomorphism $\tilde{J}$ on $G$ which is again a
compression on $G$. Further, any compression base $(J_{p})_{p\in P}$ on $E$
uniquely extends to a compression base $(\tilde{J}_{p})_{p\in P}$ on $G$. It
is clear that also conversely any compression base on $G$ restricts to a
compression base on $E$. We obtain a one-to-one correspondence between
compression bases on $E$ and on its universal group. It is a natural question
what is the relation between spectrality on $E$ with a fixed compression base
$(J_{p})_{p\in P}$ and spectrality of the universal group with the extension
$(\tilde{J}_{p})_{p\in P}$.

The defining properties of a spectral group $(G,u)$ with a compression base
are the Rickart property and general comparability property defined in
Appendix A. Since the Rickart property of $(G,u)$ is equivalent to the
projection cover property in $E$ if general comparability holds, the basic
question to be answered is whether the b-comparability property in $E$ is
equivalent to general comparability for $G$. We have already answered this
question in the affirmative in the case of interpolation groups resp. effect
algebras with RDP in Sec. 3.4. The next example shows, however, that it is not
true in general.

######  Example 5.1.

Let $E$ be the effect algebra as in Example 3.9 (the horizontal sum of a
Hilbert space effect algebra with itself). We have introduced a compression
base in $E$ and noticed in Example 4.6 that $E$ is spectral. As the horizontal
sum of two interval effect algebras, $E$ is an interval effect algebra as
well, [14]. Note however that its universal group cannot be spectral: let
$e=(1/2,0)$ and $f=(0,1/2)$, then $e,f\in E$, $e\neq f$ but $2e=2f=u$. It
follows that the universal group is not torsion free, hence it cannot be
spectral by [12, Lemma 4.8].

In the next sections we show that the above equivalence holds for archimedean
divisible effect algebras.

###  5.1 Divisible effect algebras

######  Definition 5.2.

An element $a$ in an effect algebra $E$ is _divisible_ if for any
$n\in\mathbb{N}$, there is a unique $x\in E$ such that $nx=a$, this element
will be denoted by $a/n$. If every element in $E$ is divisible, we say that
$E$ is _divisible_.

By [35], any divisible effect algebra is an interval effect algebra whose
universal group is divisible. In particular, for any
$\lambda\in[0,1]\cap\mathbb{Q}$ and $a\in E$, there is an element $\lambda
a\in E$ and the map $(\lambda,a)\mapsto\lambda a$ has the properties making it
an _effect module_ over $[0,1]\cap\mathbb{Q}$, see [28] for the definition.
Conversely, the unit interval in a divisible group $(G,u)$ is a divisible
effect algebra.

######  Theorem 5.3.

If $E$ has the b-comparability property, then $G$ has general comparability.
If $E$ is archimedean, the converse is also true.

###### Proof.

For any $g\in G$, there are some $K,N\in\mathbb{N}$ such that $K\leq N$ and
$a:=(1/N)(g+Ku)\in E$. Clearly, any projection commutes with $a$ if and only
if it commutes with $g$, this implies that $P(a)=P(g)$. By the b-comparability
property, there is some $p\in P_{\leq}(a,(K/N)u)$, which means that $p\in
P(a)=P(g)$ and $J_{p}(a)\geq J_{p}((K/N)u)$, $J_{p^{\prime}}(a)\leq
J_{p^{\prime}}((K/N)u)$, which clearly implies that $p\in P_{\pm}(g)$.

Conversely, assume that $E$ is archimedean and that $G$ has general
comparability. Then by Lemma A.1, the order unit seminorm $\|\cdot\|$ is a
norm in $G$. By Lemma A.2, we obtain that for every $g\in G$ and
$n\in\mathbb{N}$, there is a rational linear combination
$\sum_{i}\xi_{i}u_{i}$ of elements in $P(g)$ such that
$\|g-\sum_{i}\xi_{i}u_{i}\|\leq 1/n$, so that $g$ is in the norm closure of
the $\mathbb{Q}$-linear span of $P(g)$. Let $a\in E$ and $q\in P$ be such that
$P(a)\subseteq C(q)$. Then also any $\mathbb{Q}$-linear combination of
elements in $P(a)$ is in $C_{G}(q)$ and since $C_{G}(q)$ is closed in
$\|\cdot\|$ by [18, Cor. 3.2], it follows that $a\in C(q)$. This shows that
$E$ has the b-property. Let $a,b\in E$, $aCb$ and let $g=a-b$, $p\in
P_{\pm}(g)$. Clearly, $J_{p}(a)\geq J_{p}(b)$ and $J_{p^{\prime}}(a)\leq
J_{p^{\prime}}(b)$. Further, it is easily seen that $g\in C_{G}(P(a))$, so
that $P(a)\subseteq C(p)$ and hence $a\in C(p)$, similarly $b\in C(p)$. Assume
next that $q\in P$ is such that $a,b\in C(q)$, then $g\in C_{G}(q)$ so that
$p\in C(q)$, it follows that $p\in CP(a,b)$. Putting all together, we obtain
$p\in P_{\leq}(a,b)\neq\emptyset$. Hence $E$ has the b-comparability property.

∎

If $E$ is divisible then its universal group $G$ is archimedean (see Appendix
A) if and only if for $a,b,c\in E$, $a\leq b\oplus\frac{1}{n}c$ for all
$n\in\mathbb{N}$ implies that $a\leq b$. In general this property is stronger
than archimedeanity of $E$, see [21], so let us call a divisible effect
algebra with this property _strongly archimedean_. As a consequence of the
above result, we show that for divisible effect algebras with b-comparability
these two notions coincide.

######  Corollary 5.4.

An archimedean divisible effect algebra with b-comparability is strongly
archimedean.

###### Proof.

Since b-comparability in $E$ implies general comparability for the universal
group, the results follows from Lemma A.1.

∎

######  Theorem 5.5.

Let $E$ be an archimedean divisible effect algebra and let $(G,u)$ be its
universal group. Then $E$ is spectral if and only if $G$ is spectral. In this
case, for $a\in E$:

  1. (i)

the rational spectral resolution obtained in $E$ and in $G$ is the same,

  2. (ii)

$a$ is the norm limit

$a=\lim_{n\to\infty}\sum_{l(w)=n}\lambda(w)u_{w},$

where $u_{w}=p_{a,\lambda(w+1)}\wedge p_{a,\lambda(w)}^{\prime}$,
$w\in\mathcal{B}$.

  3. (iii)

For $b\in E$, $a=b$ if and only if $p_{a,\lambda(w)}=p_{b,\lambda(w)}$,
$w\in\mathcal{B}$.

  4. (iv)

For $a\in E$, $q\in P$, $a\in C(q)$ if and only if $p_{a,\lambda(w)}\in C(q)$
for all $w\in\mathcal{B}$.

###### Proof.

The first statement is clear from Theorem 5.3 and the fact that the projection
cover property of $E$ and $G$ are clearly equivalent.

Assume that $E$ is spectral. For (i), let $C$ be a C-block containing $a$ and
let $\tilde{G}$ be its universal group, which is a subgroup in $G$ and a
spectral lattice ordered group in its own right. By Theorem 4.14, the rational
spectral resolution of $a$ in $E$ coincides with the spectral resolution in
$\tilde{G}$, which in turn is the same in $G$, by the characterization in
Theorem A.5.

To proceed, notice that $G$ is archimedean by Corollary 5.4, so that
$\|\cdot\|$ is a norm and $E$ has a separating set of states, by Lemma A.1.
The statement (ii) now follows from (6), (iii) and (iv) are from Theorem 4.18.

∎

It is easily observed from Theorem A.5 that for divisible effect algebras, the
rational spectral resolution has a characterization resembling the spectral
resolutions in order unit spaces in spectral duality, see [2, Thm. 8.64], in
particular, in von Neumann algebras.

######  Corollary 5.6.

Let $E$ be a divisible archimedean spectral effect algebra. Then the rational
spectral resolution of an element $a\in E$ is the unique parametrized family
$\\{p_{\lambda}\\}_{\lambda\in\mathbb{Q}}$ in $P$ such that

  1. (i)

$p_{\lambda}=0$ for $\lambda<0$ and $p_{\lambda}=1$ for $\lambda\geq 1$,

  2. (ii)

if $\lambda\leq\mu$ then $p_{\lambda}\leq p_{\mu}$,

  3. (iii)

$\bigwedge_{\lambda<\mu}p_{\mu}=p_{\lambda}$,

  4. (iv)

$p_{\lambda}\in PC(a)$,

  5. (v)

$J_{p_{\lambda}}(a)\leq\lambda p_{\lambda}$,
$J_{p^{\prime}_{\lambda}}(a)\geq\lambda p^{\prime}_{\lambda}$.

###  5.2 Convex effect algebras

An effect algebra $E$ is _convex_ [20] if for every $a\in E$ and
$\lambda\in[0,1]\subset{\mathbb{R}}$ there is an element $\lambda a\in E$ such
that for all $a,b\in E$ and all $\lambda,\mu\in[0,1]$ we have

  1. (C1)

$\mu(\lambda a)=(\lambda\mu)a$.

  2. (C2)

If $\lambda+\mu\leq 1$ then $\lambda a\oplus\mu a\in E$ and
$(\lambda+\mu)a=\lambda a\oplus\mu a$.

  3. (C3)

If $a\oplus b\in E$ then $\lambda a\oplus\lambda b\in E$ and $\lambda(a\oplus
b)=\lambda a\oplus\lambda b$.

  4. (C4)

$1a=a$.

A convex effect algebra is convex in the usual sense: for any $a,b\in E$,
$\lambda\in[0,1]$, the element $\lambda a\oplus(1-\lambda)b\in E$. An
important example of a convex effect algebra is the algebra $E(\mathcal{H})$
of Hilbert space effects, Example 2.1. Clearly, any convex effect algebra is
divisible.

Let $V$ be an ordered real linear space with positive cone $V^{+}$. Let $u\in
V^{+}$ and let us form the interval effect algebra $V[0,u]$. A straightforward
verification shows that $(\lambda,x)\mapsto\lambda x$ is a convex structure on
$V[0,u]$, so $V[0,u]$ is a convex effect algebra which we call a _linear
effect algebra_. By [21, Thm. 3.4], the universal group of a convex effect
algebra is the additive group of an ordered real linear space $(V,V^{+})$ with
an order unit $u$, so that any convex effect algebra is isomorphic to the
linear effect algebra $V[0,u]$. Moreover, this isomorphism is _affine_ , which
means that it preserves the convex structures.

The next theorem describes the relations between order unit spaces, ordering
sets of states and strongly archimedean convex effect algebras (cf. the
definition on p. 5.3).

######  Theorem 5.7.

[21, Thm. 3.6] Let $E\simeq V[0,u]$ be a convex effect algebra with universal
group given by the ordered linear space $(V,V^{+})$ with order unit $u$. Then
the following statements are equivalent. (a) $E$ possesses an ordering set of
states. (b) $E$ is strongly archimedean. (c) $(V,V^{+},u)$ is an order unit
space.

Compressions, compression bases and spectrality in order unit spaces were
studied in [11, 30]. In addition to the properties of compressions in
partially ordered unital abelian groups in Appendix A, the maps are also
required to be linear, which is equivalent to the condition that the
restrictions to the unit interval are affine. The next result shows that if
$E$ is strongly archimedean, this property always holds.

######  Theorem 5.8.

Let $\phi:E\to F$ be an additive mapping from a convex effect algebra $E$ into
a strongly archimedean convex effect algebra $F$. Then $\phi$ is affine.

###### Proof.

As $\phi$ is additive, it is easy to show that $\alpha\phi(a)=\phi(\alpha a)$
for any rational $\alpha$. Assume that $\alpha\leq\beta\leq\gamma$, where
$\alpha$ and $\gamma$ are rational numbers and
$\beta\in[0,1]\subset{\mathbb{R}}$. Then

$\alpha\phi(a)=\phi(\alpha a)\leq\phi(\beta a)\leq\phi(\gamma
a)=\gamma\phi(a).$

From this we get

$\phi(\beta a)\leq\beta\phi(a)\oplus(\gamma-\beta)\phi(a).$

For every $n\in{\mathbb{N}}$ we find $\gamma\in{\mathbb{Q}}$ such that
$\gamma-\beta\leq\frac{1}{n}$. Since $E$ is strongly archimedean, this entails
that

$\phi(\beta a)\leq\beta\phi(a).$

Moreover,

$\beta\phi(a)\ominus\phi(\beta
a)\leq\gamma\phi(a)\ominus\alpha\phi(a)=(\gamma-\alpha)\phi(a),$

$\gamma-\alpha=(\gamma-\beta)+(\beta-\alpha)$, and we find $\gamma$ and
$\alpha$ such that $\gamma-\alpha\leq\frac{1}{n}$ for every $n$. This yields

$\phi(\beta a)=\beta\phi(a).$

∎

######  Corollary 5.9.

Every retraction on a convex strongly archimedean effect algebra $E$ is
affine.

From the above results and the fact that convex effect algebras are divisible,
we now easily derive:

######  Theorem 5.10.

Let $E$ be a convex and strongly archimedean effect algebra and let
$(V,V^{+},u)$ be the corresponding order unit space. Then

  1. (i)

Compressions (compression bases) on $E$ uniquely correspond to compressions
(compression bases) on $(V,V^{+},u)$ (in the sense of [11]).

  2. (ii)

$E$ is spectral if and only if $(V,V^{+},u)$ is spectral (in the sense of
[11]).

The spectral resolutions in order unit spaces are indexed by all real numbers:
the spectral projections are defined by

$p_{a,\lambda}=(a-\lambda 1)_{+}^{*},\qquad\lambda\in\mathbb{R}.$

Again by divisibility and the properties of the spectral resolutions in order
unit spaces [11, Thm. 3.5 and Remark 3.1], we obtain using Corollary 5.4:

######  Theorem 5.11.

Let $E$ be a convex archimedean effect algebra with universal group given by
the ordered linear space $(V,V^{+})$ with order unit $u$. Assume that $E$ is
spectral and let $\\{p_{a,\lambda}\\}_{\lambda\in\mathbb{Q}}$ be the rational
spectral resolution of an element $a\in E$. For $\lambda\in\mathbb{R}$, put

$p_{a,\lambda}:=\bigwedge_{\mu>\lambda,\ \mu\in\mathbb{Q}}p_{a,\mu}.$

Then

  1. (i)

$E$ is strongly archimedean, so that $(V,V^{+},u)$ is an order unit space.

  2. (ii)

$(V,V^{+},u)$ with the extended compression base is a spectral order unit
space.

  3. (iii)

$\\{p_{a,\lambda}\\}_{\lambda\in\mathbb{R}}$ coincides with the spectral
resolution of $a$ in $(V,V^{+},u)$.

  4. (iv)

$a$ is given by the Riemann Stieltjes type integral

$a=\int_{0}^{1}\lambda dp_{a,\lambda}.$

######  Example 5.12.

A prominent example of a convex effect algebra is the Hilbert space effect
algebra $E(\mathcal{H})$. We have already checked in Example 4.2 that
$E(\mathcal{H})$ is spectral. The corresponding spectral resolution coincides
with the usual spectral resolution of Hilbert space effects.

######  Example 5.13.

A convex archimedean MV-effect algebra $E$ is isomorphic to a dense subalgebra
in the algebra $C(X,[0,1])$ of continuous functions $X\to[0,1]$ for a compact
Hausdorff space $X$, [8, Thm. 7.3.4]. If $E$ is norm-complete (in the supremum
norm) then $E\simeq C(X,[0,1])$. By the above results, $E$ is spectral if and
only if the space $C(X,\mathbb{R})$ of all continuous real functions, with its
natural order unit space structure, is spectral in the sense of [11]. It was
proved in [17] that this is true if and only if $X$ is basically disconnected,
which is equivalent to the fact that $E$ is monotone $\sigma$-complete, see
[19].

######  Example 5.14.

Let $(X,\|\cdot\|)$ be a (real) Banach space and let

$E=\\{(\lambda,x):\ \lambda\in\mathbb{R},x\in X,\ \|x\|\leq\lambda\leq
1-\|x\|\\}.$

Then $E$ is a convex archimedean effect algebra. The corresponding order unit
space was considered already in [4] and was subsequently called a _generalized
spin factor_ in [5].

By Theorem 5.10 and [30, Thm. 6.5], $E$ is spectral if and only if $X$ is
reflexive and strictly convex. Moreover, it follows from [5, Thm.1], see also
[30, Thm. 6.6], that in addition $X$ is also smooth if and only if $E$ is
spectral in the stronger sense derived from spectral duality due to Alfsen-
Schultz [2].

##  6 Conclusions and future work

We have shown that in a spectral effect algebra in the sense of [36], any
element has a well defined binary spectral resolution in terms of specific
principal elements called projections. In the case that $E$ is archimedean, we
found a characterization of the spectral resolution as the unique family of
projections satisfying certain conditions. There are important properties
expected from a spectral resolution: that it uniquely determines the
corresponding element as well as all projections compatible with it. We were
able to prove these properties only under the condition that a separating
family of states exists. It is a question whether this seemingly rather strong
condition can be dropped, but note that even for the proof of corresponding
properties for groups (see A.6, [18, Thm 4.4]) it was assumed that $G$ is
archimedean, which in this case is equivalent to existence of an _ordering_
set of states.

We also looked at some special cases of interval effect algebras: (effect
algebras with RDP, divisible and convex effect algebras) and proved that
spectrality and spectral resolutions are equivalent to the respective notions
in their universal group. However, we found an example where this is not true.
We conjecture that such equivalence holds for unit intervals in _unperforated_
partially ordered unital abelian groups. Note that under such an equivalence,
for the spectral resolution to be right continuous, the effect algebra $E$
must have an ordering set of states, this follows from Proposition 4.13 and
Theorem 5.7.

In general, it is still not clear what intrinsic properties of $E$ are
necessary and sufficient for existence of a compression base making it
spectral. For this, it is important to characterize spectrality for MV-effect
algebras, which are the basic building blocks for spectral effect algebras. In
this case, some sufficient conditions are known, such as $E$ being monotone
$\sigma$-complete or a boolean algebra, but a general characterization (even
under some additional assumptions such as archimedeanity or norm completeness)
is not known. The question how the different C-blocks are connected and
intertwined by the compressions in the compression base seems much more
challenging.

For convex spectral effect algebras one can consider the possibility of
introducing the functional calculus for its elements. A further possible line
of research is to look at spectrality in some special cases of (convex) effect
algebras. For example, in [39] a spectral theorem was proved for elements in a
normal sequential effect algebra, one can show that this corresponds to our
results. A related question is the characterization of convex sequential
effect algebras, in particular the unit interval in a JB-algebra, in the
context of spectrality.

##  Appendix A Spectrality in ordered groups

The notions considered in this paper for an effect algebra $E$ were inspired
by similar notions for an unital ordered abelian group $(G,u)$, defined and
studied by Foulis in [16, 18]. Here we collect some basic results on
spectrality in this setting.

If $E=G[0,u]$ is the unit interval in $(G,u)$, we can say, in short, that a
compression on $G$ is an order-preserving group endomorphism $J:G\to G$ such
that the restriction of $J$ to $E$ is a compression on $E$. Similarly, a
compression base $(J_{p})_{p\in P}$ in $G$ is a family of compressions in $G$
such that the restrictions $(J_{p}|_{E})_{p\in P}$ form a compression base in
$E$.

We say that $G$ is a _compressible group_ if every retraction is a compression
and each compression is uniquely determined by its focus. In this case, the
set of all compressions is a compression base, see [16] for the definition,
proofs and further results.

Note that in [18] spectrality was introduced in the setting of compressible
groups. We will use a slightly more general assumption that we have a
distinguished compression base $(J_{p})_{p\in P}$ in $G$, that will be fixed
throughout the present section. The proofs of the statements below remain the
same.

The definitions of compatibility and commutants for $G$ are analogous to those
in Sec. 3.1 for effect algebras. For $p\in P$, we will use the notation
$C_{G}(p)$ for the set of all elements $g\in G$ compatible with $p$. The
definitions of $PC(g)$ and the bicommutant $P(g)$ extend straightforwardly to
all $g\in G$.

General comparability. We say that $G$ has _general comparability_ if for any
$g\in G$, the set

$P_{\pm}(g):=\\{p\in P(g):\ J_{p^{\prime}}(g)\leq 0\leq J_{p}(g)\\}$

is nonempty. In this case, $G$ is unperforated (that is, $ng\in G^{+}$ implies
$g\in G^{+}$ for $g\in G$ and $n\in\mathbb{N}$) [12, Lemma 4.8].

Since $u$ is an order unit in $G$, it defines an order unit seminorm in $G$ as

$\|g\|:=\inf\\{\frac{n}{k}:\ k,n\in\mathbb{N},-nu\leq kg\leq nu\\}.$

We say that $G$ is _archimedean_ [19] if $g,h\in G$ and $g\leq nh$ for all
$n\in\mathbb{N}$ implies $g\leq 0$, we remind the reader that this is in
general stronger than archimedeanity of $E$.

The following result was basically proved in [18].

######  Lemma A.1.

Assume that $G$ has general comparability. Then the following are equivalent.

  1. (i)

$E$ is archimedean.

  2. (ii)

$G$ is archimedean.

  3. (iii)

$\|\cdot\|$ is a norm on $G$.

  4. (iv)

$G$ has an order determining set of states.

  5. (v)

$G$ has a separating set of states.

###### Proof.

Assume (i) and let $g,h\in G$ be such that $ng\leq h$ for all
$n\in\mathbb{N}$. Let $p\in P_{\pm}(g)$, then $J_{p}(g)\in G^{+}$ and hence
$0\leq nJ_{p}(g)\leq J_{p}(h)$ for all $n\in\mathbb{N}$. Since $u$ is an order
unit, there is some $m\in\mathbb{N}$ such that $J_{p}(h)\leq mu$, and then
$nJ_{p}(g)\leq mu$ for all $n$. In particular, $mkJ_{p}(g)\leq mu$ for all
$k\in\mathbb{N}$ and since $G$ is unperforated, we get $kJ_{p}(g)\leq u$, for
all $k\in\mathbb{N}$. By (i), this implies that $J_{p}(g)=0$, hence

$g=J_{p^{\prime}}(g)\leq 0,$

so that (ii) holds. The equivalence of (ii)-(iv) was proved in [19, Thm. 4.14,
Thm. 7.12], see also [18, Thm. 3.3]. The implications (iv) $\implies$ (v) and
(v) $\implies$ (i) are easy.

∎

######  Lemma A.2.

Assume that $G$ has general comparability and let $g\in G$. Let
$n\in\mathbb{N}$, $\epsilon>0$ and $m_{0},\dots,m_{N}\in\mathbb{Z}$ be such
that $m_{i}\leq m_{i+1}$ and $m_{0}\leq-\|g\|$, $m_{N}\geq\|g\|$. Then there
are elements $u_{1},\dots,u_{N}\in P(g)$, $\sum_{i}u_{i}=u$ such that

$\|ng-\sum_{i}m_{i}u_{i}\|\leq\max_{i}(m_{i+1}-m_{i}).$

###### Proof.

This proof is very similar to the proof of [30, Thm. 3.22], we give the proof
here for completeness. Similarly as in [30, Lemma 3.21], we may construct a
nonincreasing sequence $q_{i}$, $i=0,\dots N$ such that $q_{i}\in P_{\pm}(ng-
m_{i}u)$. Put $q_{0}=u$, $q_{N}=0$ and for $i=0,\dots,N-2$ put

$q_{i+1}:=r_{i+1}\wedge q_{i}$

where $r_{i+1}$ is any element in $P_{\pm}(ng-m_{i+1}u)$. We will check that
$q_{i+1}\in P_{\pm}(ng-m_{i+1}u)$ (cf [15, Thm. 3.7]). Indeed, we have

$J_{q_{i+1}}(ng-m_{i+1}u)=J_{q_{i+1}}J_{r_{i+1}}(ng-m_{i+1}u)\geq 0$

and since $q_{i+1}^{\prime}=r_{i+1}^{\prime}+r_{i+1}\wedge q_{i}^{\prime}$ and
all these elements are in $P(g)$, we obtain

$J_{q_{i+1}^{\prime}}(ng-
m_{i+1}u)=(J_{r_{i+1}^{\prime}}+J_{r_{i+1}}J_{q_{i}^{\prime}})(ng-
m_{i+1}u)\leq J_{r_{i+1}^{\prime}}(ng-
m_{i+1}u)+J_{r_{i+1}}J_{q_{i}^{\prime}}(ng-m_{i}u)\leq 0.$

Next, for $i=1,\dots,N$, put

$u_{i}:=q_{i-1}-q_{i}=q_{i-1}\wedge q_{i}^{\prime},$

then $u_{i}\in P(g)$ and $\sum_{i=1}^{N}u_{i}=u$. We also have

$\displaystyle J_{u_{i}}(ng-m_{i}u)=J_{q_{i-1}}J_{q_{i}^{\prime}}(ng-
m_{i}u)\leq 0,\quad J_{u_{i}}(ng-m_{i-1}u)=J_{q_{i}^{\prime}}J_{q_{i-1}}(ng-
m_{i-1}u)\geq 0,$

so that $m_{i-1}u_{i}\leq J_{u_{i}}(ng)\leq m_{i}u_{i}$ and hence

$-(m_{i}-m_{i-1})u_{i}\leq J_{u_{i}}(ng)-m_{i}u_{i}\leq 0.$

Summing over $i$ now gives the result.

∎

Rickart property. We say that $G$ has the _Rickart property_ (or that $G$ is
_Rickart_) if there is a mapping $*:G\to P$, called the _Rickart mapping_ ,
such that for all $g\in G$ and $p\in P$, $p\leq g^{*}\,\Leftrightarrow\,g\in
C(p),J_{p}(g)=0$. In this case, the unit interval $E$ has the projection cover
property, with the projection cover obtained as

$a^{\text{\rm o}}=a^{**}=(a^{*})^{\prime},\qquad\mbox{for }a\in E.$ (A.1)

In particular, $P$ is an OML. If $G$ has general comparability, then the
projection cover property of $E$ is equivalent to the Rickart property of $G$,
see [15, Thm. 6.5].

Some important properties of the Rickart mapping are collected in the
following lemma.

######  Lemma A.3.

[15, Lemma 6.2] For all $g,h\in G$ and all $p\in P$ we have: (i) $g^{*}\in
P(g)$ and $J_{g^{*}}(g)=0$. (ii) $p^{*}=p^{\prime}$. (iii)
$g^{**}:=(g^{*})^{*}=(g^{*})^{\prime}$. (iv) $0\leq g\leq h\implies g^{**}\leq
h^{**}$. (v) $J_{p}(g)=g\,\Leftrightarrow\,g^{**}\leq p$.

Orthogonal decompositions. An _orthogonal decomposition_ of an element $g\in
G$ is defined as

$g=g_{+}-g_{-},\qquad g_{+},g_{-}\in G^{+}$

such that there is a projection $p\in P$ satisfying $J_{p}(g)=g_{+}$ and
$J_{p^{\prime}}(g)=-g_{-}$. A projection $p\in P$ defines an orthogonal
decomposition of $g$ if and only if $g\in C_{G}(p)$ and $J_{p^{\prime}}(g)\leq
0\leq J_{p}(g)$.

######  Lemma A.4.

Assume that $G$ has general comparability. Then

  1. (i)

[15, Lemma 4.2] For all $g\in G$, there is a unique orthogonal decomposition.
This decomposition is defined by any element in $P_{\pm}(g)$.

  2. (ii)

[18, Thm. 3.1] If $G$ has also the Rickart property, then $g_{+}^{**}\in
P_{\pm}(g)$. Moreover, $g_{+}^{**}$ is the smallest projection defining the
orthogonal decomposition of $g$.

  3. (iii)

Let $q$ be any projection such that $g\in C_{G}(q)$. Then
$(J_{q}(q))_{\pm}=J_{q}(g_{\pm})$.

###### Proof.

The last part of statement (ii) was proved in [18, Thm. 3.1] in a slightly
weaker form, namely that $g_{+}^{**}$ is the smallest element in $P_{\pm}(g)$.
The difference is that not all projections defining the orthogonal
decomposition are in the bicommutant $P(g)$. However, the proof remains the
same: it follows immediately from $J_{p}(g_{+})=g_{+}$, which implies
$g_{+}^{**}\leq p$, see Lemma A.3 (v). For (iii), let $p\in P_{\pm}(g)$, then
$p$ commutes with $q$ and $J_{p}(g)=g_{+}$, $J_{p^{\prime}}(g)=-g_{-}$. We
then have

$\displaystyle J_{p}(J_{q}(g))=J_{q}(J_{p}(g))=J_{q}(g_{+})\geq 0,\qquad
J_{p^{\prime}}(J_{q}(g))=J_{q}(J_{p^{\prime}}(g))=-J_{q}(g_{-})\leq 0.$

It follows that $p$ defines the orthogonal decomposition of $J_{q}(g)$ and

$\displaystyle J_{q}(g)_{+}=J_{p}J_{q}(g)=J_{q}(g_{+}),\qquad
J_{q}(g)_{-}=-J_{p^{\prime}}(J_{q}(g))=J_{q}(-J_{p^{\prime}}(g))=J_{q}(g_{-}).$

∎

Spectrality. Assume that $G$ has both the general comparability and the
Rickart property. In that case, we will say that $G$ is _spectral_. For $g\in
G$ and $\lambda\in\mathbb{Q}$, let

$p_{g,\lambda}:=((ng-mu)_{+})^{*},\qquad\lambda=\frac{m}{n},\ n>0.$ (A.2)

The element $p_{g,\lambda}$ is well defined, in the sense that it does not
depend on the expression $\lambda=\frac{m}{n}$. The family of projections
$(p_{g,\lambda})_{\lambda\in\mathbb{Q}}$, introduced in [18, Def. 4.1], is
called the _rational spectral resolution_ of $g$. Put

$\displaystyle l_{g}$ $\displaystyle:=\sup\\{m/n:\ m,n\in\mathbb{Z},0<n,mu\leq
ng\\}=\sup\\{\lambda\in\mathbb{Q}:\ p_{g,\lambda}=0\\}$ $\displaystyle u_{g}$
$\displaystyle:=\inf\\{m/n:\ m,n\in\mathbb{Z},0<n,mu\geq
ng\\}=\inf\\{\lambda\in\mathbb{Q}:\ p_{g,\lambda}=u\\}.$

The rational spectral resolution has a number of properties resembling the
spectral resolution for self-adjoint operators, see [18, Thms. 4.1, 4.4]. If
$G$ is archimedean, we have the following characterization.

######  Theorem A.5.

Let $G$ be archimedean and spectral. For $g\in G$, the rational spectral
resolution is the unique family of projections
$\\{p_{\lambda}\\}_{\lambda\in\mathbb{Q}}$ such that

  1. (i)

for $\lambda<l_{g}$, $p_{\lambda}=0$ and for $\lambda\geq u_{g}$,
$p_{g,\lambda}=u$,

  2. (ii)

for $\lambda<\mu$ we have $p_{g,\lambda}\leq p_{g,\mu}$.

  3. (iii)

$\bigwedge_{\lambda<\mu}p_{g,\mu}=p_{g,\lambda}$, for all
$\lambda\in\mathbb{Q}$,

  4. (iv)

for $\lambda=\frac{m}{n}$, $n\in\mathbb{N}$, $m\in\mathbb{Z}$, we have
$nJ_{p_{g,\lambda}}(g)\leq mp_{g,\lambda}$, $mp_{g,\lambda}^{\prime}\leq
nJ_{p_{g,\lambda}^{\prime}}(g)$.

###### Proof.

Assume that $\\{p_{\lambda}\\}$ satisfies the properties (i)-(iv). Let
$\lambda=\frac{m}{n}$, $m\in\mathbb{Z}$, then we have from (iv) that
$J_{p_{\lambda}}(ng-mu)\leq 0$ and $J_{p_{\lambda}^{\prime}}(ng-pu)\geq 0$, so
that $p_{\lambda}^{\prime}$ defines the orthogonal decomposition of $ng-mu$,
$J_{p_{\lambda}^{\prime}}(ng-pu)=(ng-pu)_{+}$. Let $\mu>\lambda$,
$\mu\in\mathbb{Q}$, then there are some $k\in\mathbb{N}$ and
$l_{1},l_{2}\in\mathbb{Z}$, $l_{1}<l_{2}$ such that $\lambda=\frac{l_{1}}{k}$
and $\mu=\frac{l_{2}}{k}$. By (ii), $p_{\mu}^{\prime}\leq
p_{\lambda}^{\prime}$ and therefore

$\displaystyle J_{p_{\mu}^{\prime}}((kg-l_{1}u)_{+})$
$\displaystyle=J_{p_{\mu}^{\prime}}J_{p_{\lambda}^{\prime}}(kg-
l_{1}u)=J_{p_{\mu}^{\prime}}(kg-l_{2}u+(l_{2}-l_{1})u)=(kg-
l_{2}u)_{+}+(l_{2}-l_{1})p_{\mu}^{\prime}$
$\displaystyle\geq(l_{2}-l_{1})p_{\mu}^{\prime}.$

Let $r=(kg-l_{1}u)_{+}^{\text{\rm o}}=(kg-l_{1}u)_{+}^{**}\in P_{\pm}(kg-
l_{1}u)$, then since $r\in P(g)$, $r$ commutes with $p_{\mu}$ and we obtain
$(l_{2}-l_{1})p_{\mu}^{\prime}\leq J_{p_{\mu}^{\prime}}((kg-l_{1}u)_{+})\leq
r$. Since $r$ is principal, this entails $p_{\mu}^{\prime}\leq r$ for all
$\mu>\lambda$. It follows that
$r^{\prime}\leq\bigwedge_{\mu>\lambda}p_{\mu}=p_{\lambda}$. On the other hand,
by Lemma A.4 (ii), we obtain $r\leq P_{\lambda}^{\prime}$, so that
$p_{\lambda}=r^{\prime}=(kg-l_{1}u)_{+}^{*}$ is the spectral projection of
$g$. The converse is clear from the properties of the spectral resolution,
[18, Thm. 4.1, 4.4].

∎

######  Theorem A.6.

[18, Thm. 4.4] Let $G$ be archimedean and spectral, $g\in G$. Then

  1. (i)

Each element $g\in G$ is uniquely determined by its rational spectral
resolution.

  2. (ii)

For $q\in P$, we have $g\in C(q)$ if and only if $p_{g,\lambda}\in C(q)$ for
all $\lambda\in\mathbb{Q}$.

## Acknowledgements

The authors are indebted to Gejza Jenča for discussions and valuable comments,
and to Ryszard Kostecki for pointing out the references [4] and [5]. We are
grateful to the anonymous referee for a number of inspiring remarks that
helped us to improve the paper. The research was supported by the grants VEGA
2/0142/20 and the Slovak Research and Development Agency grant APVV-20-0069.

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