The three types of normal sequential eﬀect algebras
Abraham Westerbaan
1
, Bas Westerbaan
1,2
, and John van de Wetering
1
1
2
University College London
A sequential eﬀect algebra (SEA) is an eﬀect algebra equipped with a sequential
product operation modeled after the uders product
(
a, b
)
7→
ab
a on C
-algebras.
A SEA is called normal when it has all suprema of directed sets, and the sequential
product interacts suitably with these suprema. The eﬀects on a Hilbert space and the
unit interval of a von Neumann or JBW algebra are examples of normal SEAs that are
in addition convex, i.e. possess a suitable action of the real unit interval on the algebra.
Complete Boolean algebras form normal SEAs too, which are convex only when
0 = 1
.
We show that any normal SEA E splits as a direct sum E
=
E
b
E
c
E
ac
of a
complete Boolean algebra E
b
, a convex normal SEA E
c
, and a newly identiﬁed type of
normal SEA E
ac
we dub purely almost-convex.
Along the way we show, among other things, that a SEA which contains only idem-
potents must be a Boolean algebra; and we establish a spectral theorem using which
we settle for the class of normal SEAs a problem of Gudder regarding the uniqueness of
square roots. After establishing our main result, we propose a simple extra axiom for
normal SEAs that excludes the seemingly pathological a-convex SEAs. We conclude
the paper by a study of SEAs with an associative sequential product. We ﬁnd that
associativity forces normal SEAs satisfying our new axiom to be commutative, shed-
ding light on the question of why the sequential product in quantum theory should be
non-associative.
1 Introduction
Understanding the properties and foundations of quantum theory requires contrasting it with
hypothetical alternative physical theories and mathematical abstractions. By studying these alter-
natives it becomes clearer which parts of quantum theory are special to it, and which are present
in any reasonable physical theory.
A framework that has been used extensively to study such alternatives is that of generalised
probabilistic theories (GPTs) [6]. GPTs have built into their deﬁnition the classical concepts of
probability theory, and so the state and eﬀect spaces of hypothetical physical systems are modelled
by convex sets. While convexity is a useful and powerful property, it precludes the study of physical
theories that have a more exotic notion of probability. For instance, deterministic or possibilistic
‘probabilities’ studied in contextuality [1, 48] or systems where probabilities can vary spatially [12].
To study systems with this broader notion of probability, a more general structure than convex
sets is needed. This paper is about eﬀect algebras, which were introduced in 1994 by Foulis and
Bennett [13].
Eﬀect algebras generalise and abstract the unit interval of eﬀects in a C
-algebra. The study
of eﬀect algebras has become a ﬂourishing ﬁeld on its own [7, 10, 14, 24, 27, 35, 45] and covers a
variety of topics [34, 36, 47, 51]. An eﬀect algebra that has an action of the real unit interval on it
is called a convex eﬀect algebra [25]. Such eﬀect algebras essentially recover the standard notion
of an eﬀect space in GPTs.
Abraham Westerbaan: bram@westerbaan.name
Bas Westerbaan: bas@westerbaan.name
John van de Wetering: john@vandewetering.name
Accepted in Quantum 2020-12-22, click title to verify. Published under CC-BY 4.0. 1
arXiv:2004.12749v2 [quant-ph] 22 Dec 2020
Eﬀect algebras, or even just convex eﬀect algebras are very general structures. So, as in
GPTs where additional assumptions are often needed on the physical theory to make interesting
statements, we will impose some additional structure on eﬀect algebras.
The eﬀects of a quantum system, i.e. the positive subunital operators on some Hilbert space or,
more generally, positive subunital elements of a C
-algebra or Jordan algebra, have some interesting
algebraic structure (aside from the eﬀect algebra operations). The one we focus on in this paper
is the sequential product, also known as the L¨uders product
(
a, b
)
7→
ab
a. This product models
the act of ﬁrst measuring the eﬀect a and then measuring the eﬀect b.
In order to study the sequential product in the abstract, Gudder and Greechie introduced in
2002 [20] the notion of a sequential eﬀect algebra (SEA), that models the sequential product on
an eﬀect algebra as a binary operation satisfying certain properties. SEAs have been studied by
several authors, see e.g. [1618, 21, 22, 28, 38, 39, 49, 52, 53, 55, 58, 61].
Adopting the language of GPTs (and speciﬁcally that of reconstructions of quantum theory) we
can view the assumed existence of a sequential product on our eﬀect algebras as a physical postulate
(indeed, such a sequential product has been used for a reconstruction of quantum theory before [55]).
But as is the case for GPTs, we will also need some structural ‘background’ assumption. Indeed,
in most work dealing with GPTs it is assumed that the sets of states and eﬀects are closed in a
suitable topology. This models the operational assumption that when we can arbitrarily closely
approximate an eﬀect, that this eﬀect is indeed a physical eﬀect itself. Such assumptions are
routinely used to, for instance, remove the possibility of inﬁnitesimal eﬀects.
In a (sequential) eﬀect algebra we however have no natural notion of topology and hence no
direct way to require such a closure property. But as an eﬀect algebra is a partially ordered set, we
can require that this order is directed complete, meaning that any upwards directed subset has a
supremum. Indeed, this condition is routinely used in theoretical computer science to give meaning
to algorithms with loops or recursion in a branch called domain theory [2]. In the ﬁeld of operator
algebras, one possible characterisation of von Neumann algebra’s is that they are C
-algebras that
are bounded directed-complete and have a separating set of normal states [40, 46, 60].
Motivated by the structure of von Neumann algebras, we call a sequential eﬀect algebra normal
when every directed set has a supremum, the sequential product preserves these suprema in the
second argument, and an eﬀect commutes with such a suprema provided it commutes with all
elements in the directed set. The set of eﬀects on a Hilbert space is a convex normal SEA. More
generally, the unit interval of any JBW-algebra (and so in particular that of any von Neumann
algebra) is also a convex normal SEA [56]. As a rather diﬀerent example, any complete Boolean
algebra is a normal SEA, which is not convex.
A priori, a (normal) SEA is a rather abstract object and there could potentially be rather exotic
examples of them. The main result of this paper is to show that for normal SEAs this is, in a
sense, not the case.
We will show that any normal SEA is isomorphic to a direct sum E
b
E
c
E
ac
, where E
b
is
a complete Boolean algebra, E
c
is a convex normal SEA and E
ac
is a normal SEA that is purely
a-convex, a new type of eﬀect algebra we will deﬁne later on. We will show there is no overlap: for
instance, there is no normal SEA that is both purely a-convex and Boolean.
In much the same way as a C
-algebra is the union of its commutative subspaces, we show
that a purely a-convex SEA is a union of convex SEAs. Hence, normal SEAs come essentially in
two main types: Boolean algebras that model classical deterministic logic, and convex SEAs that
ﬁt into the standard GPT framework. Hence, rather than having convex structure as a starting
assumption, we show that it can be derived as a consequence of our other assumptions.
Along the way we will establish several smaller results that might be of independent interest.
We show any SEA where all elements are sharp (i.e. idempotent) is a Boolean algebra, and that
consequently any SEA with a ﬁnite number of elements is a Boolean algebra. We introduce a
new axiom for SEAs that exclude the seemingly pathological purely a-convex SEAs. Combining
this with our main result shows that normal SEAs satisfying this additional axiom neatly split up
into a Boolean algebra and a convex normal SEA. Finally, we study SEAs where the sequential
product is associative. We ﬁnd that associative normal SEAs satisfying our additional axiom must
be commutative, and we are able to completely classify the associative normal purely a-convex
factors, i.e. SEAs with trivial center.
This work relies on recent advances made in the representation theory of directed-complete
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eﬀect monoids [59]. An eﬀect monoid is an eﬀect algebra with an additional associative (not
necessarily commutative) multiplication operation that is additive in both arguments. Crucially,
any commutative normal SEA is a directed-complete eﬀect monoid.
Section 2 contains the basic deﬁnitions and recalls the necessary results from [59]. Then in
Section 3 we show that a (normal) SEA where every element is idempotent must be a (complete)
Boolean algebra. In Section 4 we prove our main technical results that show that any normal
SEA splits up into a Boolean algebra and an a-convex normal SEA. Then in Section 5 we improve
this result by showing that an a-convex normal SEA splits up into a convex part and a purely
a-convex part, and we introduce a new axiom that excludes the seemingly pathological purely a-
convex normal SEAs. In Section 6 we study the consequences of our representation theorem for the
existence of non-commutative associative sequential products. Finally, in Section 7 we speculate
on possible future avenues and consequences of our results.
2 Preliminaries
Deﬁnition 1.
An
eﬀect algebra
(EA) [
13
] is a set
E
with distinguished element 0
E
, partial
binary operation
>
(called
sum
) and (total) unary operation
a 7→ a
(called
complement
),
satisfying the following axioms, writing a b whenever a > b is deﬁned and deﬁning 1 := 0
.
Commutativity: if a b, then b a and a > b = b > a.
Zero: a 0 and a > 0 = a.
Associativity: if
a b
and (
a > b
)
c
, then
b c
,
a
(
b > c
), and (
a > b
)
> c
=
a >
(
b > c
).
The complement a
is the unique element with a > a
= 1.
If a 1, then a = 0.
For
a, b E
we write
a b
whenever there is a
c E
with
a > c
=
b
. This turns
E
into a poset
with minimum 0 and maximum 1. The map
a 7→ a
is an order anti-isomorphism. Furthermore,
a b
if and only if
a b
. If
a b
, then the element
c
with
a > c
=
b
is unique and is denoted
by b a.
Remark 2.
We pronounce
a b
as
a
is summable with
b
. In the literature this is more often
referred to as orthogonality, but we avoid this terminology, as it clashes with the orthogonality for
sequential eﬀect algebras we will see in Deﬁnition 15. We use the symbol
>
to denote the eﬀect
algebra sum following for instance [8, 33]. The symbol is also commonly used in the literature.
Example 3.
Let
B
be a Boolean algebra (or more generally, an orthomodular poset [
31
,
43
,
44
].)
Then
B
is an eﬀect algebra with the partial addition deﬁned by
x y x y
= 0 and in that
case
x> y
=
xy
. The complement, ( )
, is given by the complement, ( )
(or the orthocomplement
for an orthomodular lattice.) The lattice order coincides with the eﬀect algebra order (deﬁned
above). See e.g. [61, Prop. 27] or [13, §5].
Example 4.
Let
G
be an ordered abelian group (such as an ordered vector space, for example a
C
-algebra,) and let
u G
be any positive element. Then any interval [0
, u
]
G
=
{a G
; 0
a u}
forms an eﬀect algebra, where the eﬀect algebra sum
a > b
of
a
and
b
from the interval is deﬁned
when
a
+
b u
and in that case coincides with the group addition:
a > b
=
a
+
b
. The complement
is given by a
= u a. The eﬀect algebra order on [0, u]
G
coincides with the regular order on G.
In particular, the set of eﬀects [0
,
1]
C
of a unital C
-algebra
C
forms an eﬀect algebra with
a b a + b 1, and a
= 1 a.
Remark 5.
In GPTs, the set of eﬀects of a system is often modelled by the unit interval of an
ordered vector space. Hence, the above example demonstrates how these eﬀect spaces ﬁt into
the notion of an eﬀect algebra. The corresponding addition of eﬀects in a GPT is often called
coarse-graining.
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Example 6.
If
E
and
F
are eﬀect algebras, then its Cartesian product
E F
with component-wise
operations is again an eﬀect algebra.
1
We will often implicitly use the following basic facts about eﬀect algebras.
Proposition 7. In any eﬀect algebra we have (see e.g. [11] or [62, §175V])
1. (involution) a
⊥⊥
= a;
2. (positivity) if a > b = 0, then a = 0 and b = 0;
3. (cancellation) if a > b = a > c, then b = c;
4. a b iﬀ b
a
and
5. a b iﬀ a b
.
Some eﬀect algebras are more closely related to ordered vector spaces (like the example of a
C
-algebra above or the eﬀect spaces coming from a GPT). We call these eﬀect algebras convex:
Deﬁnition 8.
A
convex action
on an eﬀect algebra
E
is a map
·
: [0
,
1]
×E E
, where [0
,
1] is
the regular unit interval, obeying the following axioms for all a, b E and λ, µ [0, 1]:
λ · (µ · a) = (λµ) ·a.
If λ + µ 1, then λ · a µ · a and λ · a > µ · a = (λ + µ) · a.
1 · a = a.
λ · (a > b) = λ · a > λ · b.
A
convex eﬀect algebra
[
25
] is an eﬀect algebra endowed with such a convex action
2
. We will
say that an eﬀect algebra E is convex when there is at least one convex action on E.
Example 9.
Let
V
be an ordered real vector space (such as the space of self-adjoint elements of
a C
-algebra). Then any interval [0
, u
]
V
where
u
0 is a convex eﬀect algebra with the obvious
action of the real unit interval. Conversely, for any convex eﬀect algebra
E
, we can ﬁnd an ordered
real vector space
V
and
u V
such that
E
is isomorphic as a convex eﬀect algebra to [0
, u
]
V
[
26
].
Remark 10.
Convex eﬀect algebras have been well-studied, see e.g. [
19
,
32
,
36
,
53
,
54
]. In
the literature on eﬀectus theory, convex eﬀect algebras are also called eﬀect modules [
8
,
9
]. In
Deﬁnition 45 we introduce a-convex (almost convex) eﬀect algebras, by dropping the last axiom.
Deﬁnition 11.
Let
E
be a partially ordered set (such as an eﬀect algebra). A subset
S
of
E
is
called
directed
when it is non-empty, and for any
a, b S
there exists a
c S
such that
a, b c
.
We say that E is directed complete when every directed subset S of E has a supremum,
W
S.
Example 12. Any complete Boolean algebra is a directed-complete eﬀect algebra.
Example 13.
Let
A
be a unital C
-algebra. Then [0
,
1]
A
is a directed-complete eﬀect algebra
if and only if
A
itself is bounded-directed complete, that is: if every bounded set of self-adjoint
elements has a least upper bound. Such C
-algebras are called monotone complete or monotone
closed, [
41
,
§
2], and include all von Neumann algebras. A commutative unital C
-algebra, being
of the form
C
(
X
) for some compact Hausdorﬀ space
X
, is bounded-directed complete if and only
if X is extremally disconnected [15, Sections 1H & 3N.6].
Remark 14.
Because the complement acts as an order anti-isomorphism, a directed-complete
eﬀect algebra also has inﬁma for all
ﬁltered
, i.e. downwards directed, sets. In particular, if we
have a decreasing sequence
a
1
a
2
a
3
···
in a directed-complete eﬀect algebra, then this has
an inﬁmum
V
n
a
n
.
1
This is in fact a categorical product with the obvious projectors and as morphisms maps
f
with:
f
(1) = 1
and a b implies f (a) f(b) and f(a) > f(b) = f(a > b).
2
Usually, a convex eﬀect algebra is deﬁned simply as an eﬀect algebra
E
where for every
a E
and
λ
[0
,
1]
there is an element
λa E
satisfying the axioms above. However, in this paper we face the possibility of an eﬀect
algebra having many inequivalent convex actions. This is why we are explicit in deﬁning a convex eﬀect algebra as
an eﬀect algebra equipped with a convex action.
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2.1 Sequential eﬀect algebras
We now come to the deﬁnition of the central object of study of this paper, the sequential eﬀect
algebra. Before we give the formal deﬁnition, let us give some motivation for the assumptions we
will require. The sequential product a b of two eﬀects a and b represents the sequential measure-
ment of ﬁrst a and then b. An important diﬀerence between classical and quantum systems is that
in a classical system we can measure without disturbance, and hence the order of measurement is
not important: a b
=
b a for all eﬀects a and b. In a quantum system this is generally not the
case, and the order of measurement is important. However, what is interesting in quantum theory
is that some measurements are compatible, meaning that the order of measurement for those mea-
surements is not important.
3
We will use the symbol a | b to denote that a and b are compatible
eﬀects. For such compatible eﬀects a, b and c we can expect that doing ‘classical operations’ or
‘classical post-processing’ on these eﬀects retains compatibility. For instance, if a | b, and we negate
the outcome of a to get a
, we still expect a
| b, as negation can be done classically without
interacting with the quantum system. Similarly, we would expect a | b > c, as addition of eﬀects
corresponds to classically coarse-graining the measurement outcomes.
Let us now give the formal deﬁnition:
Deﬁnition 15.
A
sequential eﬀect algebra
(SEA) [
20
]
E
is an eﬀect algebra with an additional
(total) binary operation
, called the
sequential product
, satisfying the axioms listed below,
where a, b, c E. Elements a and b are said to commute, written a | b, whenever a b = b a.
S1. a (b > c) = a b > a c whenever b c.
S2. 1 a = a.
S3. a b = 0 = b a = 0.
S4. If a | b, then a | b
and a (b c) = (a b) c for all c.
S5. If c | a and c | b then also c | a b and if furthermore a b, then c | a > b.
A SEA E is called normal when E is directed complete, and
S6. Given directed S E we have a
W
S =
W
sS
a s, and a |
W
S when a | s for all s S.
We say
E
is
commutative
whenever
a | b
for all
a, b E
. An element
a E
is
central
if it
commutes with every element in
E
. The
center Z
(
E
) of
E
is the set of all central elements. We
call
p E
an
idempotent
whenever
p
2
=
p p
=
p
(or equivalently
p p
= 0). We call
E Boolean
if every element is an idempotent
4
. We say a, b E are orthogonal, provided that a b = 0.
The discussion above Deﬁnition 15 should make clear why we have axioms S2, S4 and S5. Axiom
S3 is less operationally motivated, but is a useful property as it makes the notion of orthogonality
well-behaved. Axiom S6 is our way of stating continuity of the product without having access to
an actual topology. Indeed, if E is the unit interval of a von Neumann algebra, then Axiom S6 says
that the product is ultraweakly continuous in the second argument. Axiom S1 can be understood
operationally as follows. Consider an ensemble of systems all prepared in the same state and that
we measure them all with the eﬀect a. We then measure some of the states with b and others with
c. By adding up the probabilities of success of these measurement outcomes we model the eﬀect
a
(
b > c
)
as we have a and then b or c’. Alternatively, we could measure one half of the states
with a b and the other with a c and add up the outcomes to get
(
a b
)
>
(
a c
)
. But looking at
the interactions we have with the system, these two protocols are obviously equivalent, and hence
we should have a
(
b > c
) = (
a b
)
>
(
a c
)
. Note that we do not in general expect this property
to hold in the other argument:
(
a > b
)
c 6
= (
a c
)
>
(
b c
)
. Indeed, the addition is a classical
operation, and there is no way in general to perform the measurement
(
a > b
)
c as it requires the
measurement a or b to be performed on a single system.
The motivating example of a (normal) sequential eﬀect algebra is the unit interval of a C
-algebra:
3
The word ‘compatible’ is used for this purpose when it comes to sequential eﬀect algebras [
20
]. However, in
other parts of the literature ‘compatible’ is used for a diﬀerent notion, and the term ‘non-disturbing’ might be used
4
We will see that a Boolean SEA is a Boolean algebra, see Proposition 44.
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Example 16.
If
A
is a C
-algebra, then the EA [0
,
1]
A
is a SEA with the sequential product
deﬁned by
a b
:=
ab
a
, see [
20
]. If
A
is furthermore bounded-directed complete (for instance if
it is a von Neumann algebra), then [0
,
1]
A
is a normal SEA with the same sequential product. It is
also possible to deﬁne a sequential product using only the Jordan algebra structure. In particular,
any JB-algebra is a convex SEA, while any JBW-algebra is a convex normal SEA. For the details
we refer to [56].
We will use the following properties without further reference in the remainder of the paper.
Proposition 17 (Cf. §3 of [20]). Let E be a SEA with a, b, c, p E with p idempotent.
1. a 0 = 0 a = 0 and a 1 = 1 a = a.
2. a b a.
3. If a b, then c a c b.
4. p a iﬀ p a = p iﬀ a p = p iﬀ a
p = 0 iﬀ p a
= 0.
5. a p iﬀ p a = a iﬀ a p = a iﬀ a p
= 0 iﬀ p
a = 0.
6. p
is idempotent.
7. If p a, then a is idempotent if and only if p > a is idempotent.
Proof.
Concerning 1: Since 0 = 0
>
0 by S1, we have
a
0 =
a
0
> a
0, and so
a
0 = 0, by
cancellativity of the addition in an eﬀect algebra. Then 0
a
= 0 too, by S3. In particular,
a |
0,
and so a | 0
= 1, by S4. With S2, we get a 1 = 1 a = a.
Point 3: when
a b
, we have
b
=
a >
(
b a
), and so
c b
=
c a > c
(
b a
)
c a
, by S1.
Taking b = 1, we get c a c 1 = c, by 1, and so we have 2.
To show pt. 5, we will prove that we have the implications
a p p
a
= 0
a p
=
0
a p
=
a p a
= 0
a p
and thus they are all equivalent. So ﬁrst suppose that
a p
.
Then
p
a p
p
= 0, by 3, so
p
a
= 0. Now suppose instead that
p
a
= 0, which is
equivalent to
a p
= 0 by S3, which in turn is equivalent to
a
=
a
1 =
a
(
p > p
) =
a p
by S2.
Since then
a | p
, we get
a | p
, by S4, and so
p a
=
a p
=
a
. Since
p a
=
a
on its own entails
that a p, by 2, we are back where we started, and therefore done.
For 4, note that
p a
iﬀ
a
p
iﬀ
a
p
= 0 iﬀ
p a
= 0 by 5. Moreover, since
p a
=
p
is equivalent to
p a
= 0 by S1, and
a p
=
p
entails
p a
by 2, the only thing left to show is
that
p a
implies
a p
=
p
. So suppose that
p a
a
p
= 0 =
p a
= 0
and
p a
=
p
. Since thus
a | p
, and so
a | p
by S4, we get
a p
=
p a
=
p
. We continue with
pt. 6. Note that
a
is idempotent iﬀ
a a
= 0. Thus
p p
= 0 and so
p
p
= 0 by S3. Hence
p
is indeed idempotent.
Finally, we move to point 7. Assume
a
is idempotent. Clearly
p p > a
and so (
p > a
)
p
=
p
by 4. Similarly (
p > a
)
a
=
a
. Thus (
p > a
)
(
p > a
) = ((
p > a
)
p
)
>
((
p > a
)
a
) =
p a
, as desired.
Conversely, assume
p > a
is an idempotent. Note
p
and (
p > a
)
are summable idempotents and so
by the previous point
a
=
p
>
(
p > a
)
is idempotent as well. Thus
a
is indeed idempotent.
The next ﬁve lemmas were originally proven in [59] for eﬀect monoids, and we will need them
for our results in the context of SEAs.
Lemma 18.
Let
E
be a SEA with
p, a, b E
and
p
idempotent. If
a, b p
and
a > b
exists,
then a > b p.
Proof.
Because
a p
, we have
p
a
= 0. Since similarly,
p
b
= 0, we have
p
(
a > b
) = 0, and
so a > b p.
Lemma 19. In a SEA, a a
is summable with itself for any element a.
Proof.
Note that since
a | a
, we have
a | a
by S4, or in other words
a a
=
a
a
. Since
1 =
a> a
=
a
(
a> a
)
> a
(
a> a
) =
a a > a a
> a
a > a
a
=
a
2
>
2(
a a
)
>
(
a
)
2
,
we see that a a
is indeed summable with itself.
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Lemma 20.
Let
E
be a directed-complete eﬀect algebra, and let
S E
be some directed non-empty
subset. Let a E be such that a s for all s S. Then a
W
S and a >
W
S =
W
sS
a > s.
Proof.
The map
b 7→ a > b
, giving an order isomorphism from [0
, a
]
E
to [
a,
1]
E
(with inverse
b 7→ b a
,) preserves (and reﬂects) suprema. In particular,
W
S
, the supremum of
S
in
E
, which is
the supremum of
S
in [0
, a
]
E
too, is mapped to
W
sS
a > s
, the supremum of the
a > s
in
E
, and
in [a, 1]
E
too.
Lemma 21.
The only element
a
of a directed-complete eﬀect algebra
E
for which the
n
-fold
sum na exists for all n is zero.
Proof. We have a >
W
n
na =
W
n
a > na =
W
n
(n + 1)a =
W
n
na, and so a = 0.
Lemma 22. Let E be a normal SEA and suppose a E satisﬁes a
2
= 0. Then a = 0.
Proof.
Since
a
2
= 0 we have
a
=
a
1 =
a
(
a > a
) =
a a
, and hence (see Lemma 19)
a
is
summable with itself. But furthermore (a > a)
2
= 4a
2
= 0, and so (a > a)
2
= 0.
Continuing in this fashion, we see that 2
n
a
exists for every
n N
and (2
n
a
)
2
= 0. Hence, for
any m N the sum ma exists so that by Lemma 21, we have a = 0.
Proposition 23.
Assume
E
is a SEA with idempotent
p E
. Write
p E
:=
{p a
;
a E}
for
the
left corner
by
p
. The sequential product of
E
restricts to
p E
and in fact, with partial sum
and zero of
E
and complement
a 7→ p a
, the set
p E
is an SEA. If
E
is normal, then
p E
is
normal as well.
Proof.
By Prop. 17 we have
p E
=
{a
;
a E a p}
and so
p E
is an eﬀect algebra. For
a, b
p E
, we have
a b a p
and so
a b p E
. As the sequential product, zero and addition
of
p E
and
E
coincide almost all axioms for an SEA hold trivially. Only the ﬁrst part of S4
(which involves the orthocomplement) remains. So assume
a, b p E
with
a | b
. We have to show
that
a | pb
. Note
a p
= 0 and so
a | p
. Thus by S5, we have
a | b> p
and as
b> p
= (
pb
)
we have by S4 (for E), a | p b. Thus p E is indeed an SEA.
Now assume
E
is normal. As a principal downset of
E
, the suprema computed within
p E
are the same as computed in
E
and so
p E
is directed complete. As additionally the sequential
product of p E is the restriction of that of E, the axiom S6 holds trivially.
Proposition 24. Let p be a central idempotent in a SEA E. Then E
=
p E p
E.
Proof.
Note that the map
a 7→
(
p a, p
a
) is additive and unital. It is order reﬂecting, because if
p a p b
and
p
a p
b
then
a
=
a p > a p
=
p a > p
a p b > p
b
=
b
, using
the centrality of
p
(and hence
p
by S4). It is obviously surjective because for
a p E
we have
p a = a, and similarly for b p
E, and hence a > b 7→ (a, b).
To show that it preserves the sequential product we note that
p (a b) = (p a) b = (p (p a)) b = ((p a) p) b = (p a) (p b)
and similarly for p
.
Deﬁnition 25.
We call an idempotent
p E Boolean
when
p E
is Boolean, i.e. when all
a p
in E are idempotent.
The following deﬁnition and result will be crucial for our arguments, as it relates sequential
eﬀect algebras to commutative eﬀect monoids (see next section).
Deﬁnition 26.
Let
S E
be a subset of elements of a SEA
E
. The
commutant of S
is deﬁned
as S
0
:= {a E; a | s for all s S}. The bicommutant of S is deﬁned simply as S
00
:= (S
0
)
0
.
Remark 27.
Let
H
be some Hilbert space, and let
E
= [0
,
1]
B(H )
. For any
a E
, the
bicommutant
{a}
00
is the set of eﬀects of the least commutative von Neumann algebra of
B
(
H
)
containing
a
by the bicommutant theorem. In general this is false, consider for instance
E
= [0
,
1]
2
(with component-wise standard product) and
a
= (0
,
1) then
{a}
00
=
E
, while
{
0
} ×
[0
,
1] is a
smaller commutative subalgebra containing a.
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Proposition 28
(Cf. [
53
], Proposition III.12)
.
Let
S E
be a set of mutually commuting elements
in a normal SEA E, then S
00
is a commutative normal SEA.
Proof.
To start oﬀ, assume
S E
an arbitrary subset of
E
. Clearly 0
,
1
S
0
. If
a, b S
0
, then for
any
s S
, we have
a | s
and
b | s
so that
a b | s
and
a
| s
. If
a b
, then
a > b | s
. Thus
S
0
is
closed under partial sum, complement and sequential product. As a direct axiom of a normal SEA,
the set S
0
is also closed under directed suprema. Hence S
0
is a sub-normal-SEA of E.
As
S
was arbitrary, we see that (
S
0
)
0
=
S
00
is a sub-normal-SEA of
E
as well. Assume the
elements of
S
are pairwise commuting. Then
S S
0
so that
S
00
S
0
. By deﬁnition
S
00
commutes
with all elements from
S
0
and so in particular with all elements from
S
00
itself. Hence
S
00
is
commutative.
2.2 Eﬀect monoids
Before we can proceed with the theory of (normal) SEAs we must discuss eﬀect monoids. Roughly
speaking, eﬀect monoids are the classical counterparts to SEAs, being endowed with an associative
and biadditive multiplication ·. Note that we do not require an eﬀect monoid to be commutative,
although commutativity often follows automatically, for example, in the (for us relevant) case that
the eﬀect monoid is directed complete.
Deﬁnition 29.
An
eﬀect monoid
(EM) is an eﬀect algebra (
M, > ,
0
,
( )
, ·
(total) binary operation ·, such that the following conditions hold for all x, y, z M .
Unit: x · 1 = x = 1 · x.
Distributivity: If
y z
, then
x · y x · z
,
y · x z · x
with
x ·
(
y > z
) = (
x · y
)
>
(
x · z
)
and (y > z) · x = (y · x) > (z ·x). In other words: · is bi-additive.
Associativity: x · (y · z) = (x · y) · z.
We call an eﬀect monoid
M commutative
if
x · y
=
y · x
for all
x, y M
; an element
p
of
M
idempotent
whenever
p
2
=
p · p
=
p
; elements
x
,
y
of
M orthogonal
when
x · y
=
y · x
= 0. An
eﬀect monoid is Boolean if all its elements are idempotents.
Example 30.
Any Boolean algebra (
B,
0
,
1
, , ,
( )
) is an eﬀect algebra by Example 3, and,
moreover, a Boolean commutative eﬀect monoid with multiplication deﬁned by
x · y
=
x y
.
Conversely, any Boolean eﬀect monoid is a Boolean algebra [59, Proposition 47].
Example 31.
The unit interval [0
,
1]
R
of any (partially) ordered unital ring
R
(in which the sum
a + b and product a · b of positive elements a and b are again positive) is an eﬀect monoid.
For example, let
X
be a compact Hausdorﬀ space. The space of continuous complex-valued
functions
C
(
X
) is a commutative unital C
-algebra (and conversely by the Gelfand theorem, any
commutative C
-algebra with unit is of this form) and hence its unit interval [0
,
1]
C(X)
=
{f
:
X
[0, 1]} is a commutative eﬀect monoid.
In [
8
, Ex. 4.3.9] and [
61
, Cor. 51] two diﬀerent non-commutative eﬀect monoids are constructed.
The latter also provides an example of an eﬀect monoid that is not a SEA (since
e
3
· e
2
= 0, while
e
2
· e
3
= e
5
).
Note that an eﬀect monoid that additionally satisﬁes the implication a · b
= 0 =
b · a
= 0
is
a SEA.
Example 32.
A commutative SEA is a commutative eﬀect monoid and vice versa. Moreover, a
normal commutative SEA is a directed-complete commutative eﬀect monoid and vice versa
5
.
Remark 33.
In Ref. [
28
], “distributive” sequential eﬀect algebras were introduced. These are the
same thing as eﬀect monoids satisfying the condition a · b = 0 b · a = 0.
Example 34.
Let
M
be an eﬀect monoid and let
p M
be some idempotent. The subset
pM
:=
{p·e
;
e M}
is called the
left corner
by
p
and is an eﬀect monoid with (
p·e
)
:=
p·e
and all other
operations inherited from
M
. The map
e 7→
(
p · e, p
· e
) is an isomorphism
M
=
pM p
M
[
59
,
Corollary 21]. Analogous facts hold for the right corner M p := {e · p; e M}.
5
While the ﬁrst claim is obvious, the astute reader might notice that the second claim requires showing that the
product in a directed-complete commutative eﬀect monoid is always normal. This is done in [59, Theorem 43].
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2.2.1 Representation theorem for directed-complete eﬀect monoids
We will need the following representation theorem for directed-complete eﬀect monoids from [59].
(In fact, a representation theory for the more general class of ω-directed complete eﬀect monoids
is established there, which we do not need here).
Theorem 35.
[
59
] In every directed-complete eﬀect monoid
M
there is an idempotent
p
such
that
pM
is a convex eﬀect algebra and
p
M
is Boolean. Furthermore, there is an extremally-
disconnected compact Hausdorﬀ space X such that pM
=
[0, 1]
C(X)
.
Corollary 36
(Spectral Theorem for normal SEA)
.
Let
E
be a normal SEA and let
a E
. Then
there is a extremally-disconnected compact Hausdorﬀ space
X
and complete Boolean algebra
B
such that {a}
00
=
[0, 1]
C(X)
B.
Proof. Combine Example 32, Proposition 28 and Theorem 35.
The previous allows us to answer Problem 20 of [18] for the special case of normal SEAs.
Corollary 37. Any element a of a normal SEA E has a unique square root.
Proof.
Note that
a {a}
00
. There is an idempotent
p {a}
00
such that
p ◦{a}
00
contains only
idempotents and
p
◦{a}
00
=
[0
,
1]
C(X)
for some extremally-disconnected compact Hausdorﬀ space
X
.
Write
a
i
:=
p a
and
a
c
:=
p
a
. Then there is a unique
b p
◦{a}
00
with
b
2
=
a
c
(because this
is true in
C
(
X
)). Deﬁne
a
:=
a
i
> b
. As
a
i
is idempotent, it is easy to see that
a
2
=
a
and that
in fact
a is the unique such element within {a}
00
.
To prove uniqueness, assume
c
2
=
a
for some
c E
. As
{c}
00
is a sub-algebra of mutually
commuting elements we have
a
=
c
2
{c}
00
and hence
a | c
. As
a {a}
00
, we must then
also have
c |
a
. Consider
B
:=
{
a, a, c}
00
. Reasoning as before,
a
has a unique root in
B
,
hence c =
a.
Note that if a SEA is not normal, it does not need to have unique square roots [50].
2.3 An interesting sequential eﬀect algebra
Before we continue, let us construct a concrete example of a sequential eﬀect algebra that is not
directed complete, in order to serve as a foil of some of the other properties we will prove later.
First, the following is a construction for eﬀect monoids.
Example 38.
Let
V
be an ordered vector space. Let
R
be the space of linear functions
f
:
V V
.
For
f, g R
we set
f g
when
f
(
v
)
g
(
v
) for all
v
0 in
V
. This makes
R
into an ordered vector
space. The set
M
:= [0
, id
]
R
:=
{f R
; 0
f id}
f > g
:=
f
+
g
that is deﬁned when
f id g
is a convex eﬀect algebra [
25
]. Furthermore, deﬁning a product via
composition,
f · g
:=
f g
, makes it an eﬀect monoid. Indeed, this product obviously distributes
over the addition and has id as the identity. That 0 f · g id follows because for all v V
+
we
have 0 f(g(v)) f(v) v = id(v), since 0 g(v), f(v) v = id(v) by assumption.
We can use this construction to construct a speciﬁc eﬀect monoid that also happens to be a
non-commutative sequential eﬀect algebra.
Example 39.
Consider
V
:=
R
2
with the positive cone deﬁned by (
a, b
)
>
0 iﬀ
a
+
b >
0. Let
R
and
M
be deﬁned as in Example 38. Of course
R
is just the space of 2
×
2 real matrices. With
some straightforward calculation it can then be veriﬁed that
A =
a b
c d
M A = 0 or A = id or 1 > a + c = b + d > 0.
Deﬁne a map
τ
:
M
[0
,
1] by
τ
(
a b
c d
) =
a
+
c
=
b
+
d
. Then it is straightforward to check
that
τ
is monotone (
A B
=
τ
(
A
)
τ
(
B
)), multiplicative (
τ
(
A · B
) =
τ
(
A
)
τ
(
B
)), and
A
= 0
iﬀ
τ
(
A
) = 0. As a result
A · B
= 0 iﬀ
A
= 0 or
B
= 0. Hence
M
satisﬁes
A · B
= 0 iﬀ
B · A
= 0,
making M a SEA.
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This example is interesting because of how close its structure is to that of a SEA coming from
a C
-algebra while still being diﬀerent in crucial ways. First of all, it is convex and it contains no
non-zero inﬁnitesimal elements: if nA M for all n, then A
= 0
. As far as we are aware, this
is the ﬁrst example of a non-commutative convex SEA without inﬁnitesimal elements that is not
somehow related to examples from quantum theory. It might hence serve as a counter-example to
reasonable sounding hypotheses or characterisations of (convex) SEAs.
Remark 40.
An eﬀect algebra that does not contain inﬁnitesimal elements is often called
Archimedean. For a convex eﬀect algebra
E
=
[0
,
1]
V
there is however also a stronger notion
that is referred to as being Archimedean. Namely, that for any element of the corresponding vector
space
a V
, if
na
1 for all
n N
then
a
0. We will refer to a convex eﬀect algebra that
has this property as strongly Archimedean. Having no non-zero inﬁnitesimals is equivalent to the
order unit semi-norm
kak
:=
inf{λ R
;
λ
1
a λ
1
}
being a norm, while being strongly
Archimedean also requires the positive cone of V to be closed in this norm.
Coming back to Example 39, let W denote the space spanned by M in R. Then W is a (non-
Archimedean) ordered vector space where its order-unit semi-norm is in fact a norm. This norm
satisﬁes kAk
:=
inf{λ R
>0
;
λid A λid}
=
|τ
(
A
)
| and as a consequence we see that
the sequential product is continuous in this norm. This is signiﬁcant because in Ref. [55] it was
shown that a convex ﬁnite-dimensional strongly Archimedean SEA where the sequential product
is continuous in the norm must be order-isomorphic to the unit interval of a Euclidean Jordan
algebra, a type of structure closely related to C
-algebras, and hence quantum theory. It was
already noted in [55] that this result does not hold if the SEA contains inﬁnitesimal elements, but
it was left open whether the stronger notion of being Archimedean was necessary, or if having no
non-zero inﬁnitesimals was suﬃcient. As Example 39 is not order-isomorphic to the unit interval
of a Euclidean Jordan algebra we indeed see that the mere absence of inﬁnitesimal elements does
not suﬃce for the result.
Example 39 has some further noteworthy properties. Its product is non-commutative, but
unlike the standard sequential product on for instance a C
-algebra, is also associative. Finally, it
is ‘close’ to being directed complete, in the sense that any non-empty directed set S has a minimal
upper bound, but, unless
W
τ
(
S
) = 1
, such upper bounds are not unique, and hence S does not
have a supremum.
3 Boolean sequential eﬀect algebras
Naming sequential eﬀect algebras where every element is idempotent ‘Boolean’ of course suggests
that such an eﬀect algebra must actually be a Boolean algebra. This is indeed the case, and as far
as we know has not been observed before. Let us therefore prove this before we continue on to our
main results. Note that these results are a strict generalization of those in Ref. [52].
Lemma 41.
Let
a
and
b
be elements of a SEA such that
a b
is idempotent. Then
a b b
.
Moreover, if a b
is idempotent too, then a b = b a.
Proof.
Since
a b a
, and
a b
is an idempotent, we have
a b | a
and
a b
= (
a b
)
a
, by point 4
of Proposition 17. Then, using S4, we see that
a b
= (
a b
)
(
a b
) = ((
a b
)
a
)
b
= (
a b
)
b
,
and so a b b, by point 4 of Proposition 17.
Now suppose that
a b
is an idempotent too. Since then
a b
b
, we have
b
(
a b
) = 0,
and so
b
(
a b
) =
b a
. On the other hand,
a b
=
b
(
a b
), because
a b b
, so altogether we
get a b = b (a b) = b a.
Corollary 42. A Boolean idempotent of a SEA is central.
Proof.
Let
E
be a SEA and let
a E
be a Boolean idempotent
a
. Then
a
commutes with
every
b E
by Lemma 41, because
a b
and
a b
are idempotents, on account of being below the
Boolean idempotent a.
Proposition 43.
Let
p
and
q
be idempotents of a SEA
E
. Then
p q
is an idempotent if and only
if p and q commute. Moreover, in that case p q is the inﬁmum of p and q in E.
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Proof. If p and q commute, then it is straightforward to show that (p q)
2
= p q.
Now for the other direction, suppose
p q
is idempotent. Then
p q
is idempotent too, and
so p and q commute, by Lemma 41.
Finally, we must show that
p q
is the inﬁmum of
p
and
q
in
E
. By Lemma 41 we know that
p q
is a lower bound of
p
and
q
. To show that
p q
is the greatest lower bound, let
r p, q
be given.
Since
p
and
q
are idempotents, we have
r p
=
r
=
r q
and
r | p
, so
r
(
p q
) = (
r p
)
q
=
r q
=
r
.
It follows that r p q, and so we conclude that indeed p q = p q.
Recall that a SEA is Boolean when every element is idempotent.
Proposition 44.
Let
E
be a Boolean SEA. Then
E
is a Boolean algebra. Furthermore, for all
a, b E, a b = a b. If E is normal, then E is complete as a Boolean algebra.
Proof.
Let
a, b E
. As
a
,
b
and
a b
are idempotent by assumption, by the previous proposition
they commute and
a b
=
a b
. We conclude that
E
is commutative, and hence it is a (directed-
complete) Boolean eﬀect monoid. Ref. [
59
, Proposition 47] then shows that it is a (complete)
Boolean algebra.
Using our later results it will turn out that any SEA containing only a ﬁnite number of elements
must be a Boolean algebra (see Corollary 60).
4 Almost-convex sequential eﬀect algebras
In this section we study what we call almost-convex SEAs (abbreviated to a-convex). We do this
because they naturally arise in a structure theorem for normal SEAs that we prove at the end of
this section.
The content of this section is rather technical, with items 4955 serving as preparation for the
main results: Theorem 57 that characterises convex normal SEAs among a-convex normal SEAs
in several diﬀerent ways, and Theorems 58 and 59 that together show that any normal SEA splits
up into a direct sum of a Boolean algebra and an a-convex eﬀect algebra.
In the next section we will show that an a-convex normal SEA factors into a convex normal
SEA and a ‘purely a-convex’ normal SEA (Proposition 62).
Deﬁnition 45.
An
a-convex action
(almost-convex action) on an eﬀect algebra
E
is a map
·
: [0
,
1]
× E E
, where [0
,
1] is the standard real unit interval, satisfying the following axioms for
all a, b E and λ, µ [0, 1]:
λ · (µ · a) = (λµ) · a.
If λ + µ 1, then λ · a µ · a and λ · a > µ · a = (λ + µ) · a.
1 · a = a.
An
a-convex eﬀect algebra
is an eﬀect algebra
E
endowed with such an a-convex action. When
we say that an eﬀect algebra
E
is a-convex we mean that there’s at least one a-convex action on
E
(but note that in general, there might be more then one a-convex action on a given eﬀect algebra.)
Remark 46.
Recall from Deﬁnition 8 that a convex action on an eﬀect algebra is an a-convex
action that in addition satisﬁes the requirement that for all summable a and b, and λ [0, 1]
λ · (a > b) = λ · a > λ · b.
The deﬁnition of a-convexity is strictly weaker then that of convexity. Before we demonstrate
this with an explicit example, we ﬁrst recall the following method for constructing new (sequential)
eﬀect algebras by taking the ‘disjoint union’ of eﬀect algebras:
Deﬁnition 47.
Let
I
be some indexing set, and let
E
α
be an eﬀect algebra for each
α I
. The
horizontal sum
[
13
] of the
E
α
is then deﬁned as
HS
(
E
α
, I
) :=

αI
E
α
/
, the disjoint union
modulo the identiﬁcation of all the zeros and and all the ones. Explicitly, denoting an element of

αI
E
α
by (
a, α
) where
α I
and
a E
α
we set (
a, α
)
(
b, β
) iﬀ
a
=
b
= 1 or
a
=
b
= 0 or
a
=
b
Accepted in Quantum 2020-12-22, click title to verify. Published under CC-BY 4.0. 11
and
α
=
β
. We deﬁne the complement as (
a, α
)
= (
a
, α
). The summability relation is deﬁned
as follows. For elements (
a, α
) and (
b, β
) we set (
a, α
)
(
b, β
) iﬀ
α
=
β
and
a b
, or if
a
= 0 or
b
= 0. The sum (
a, α
)
>
(
b, β
) for summable elements (
a, α
) and (
b, β
) is deﬁned by case distinction
as follows. If
a
= 0, then (
a, α
)
>
(
b, β
) = (
b, β
). If
b
= 0, then (
a, α
)
>
(
b, β
) = (
a, α
). If
a 6
= 0 and
b 6
= 0 then necessarily
α
=
β
for summable elements and we set (
a, α
)
>
(
b, α
) = (
a > b, α
). This
complement, summability relation, and addition makes the horizontal sum an eﬀect algebra.
This deﬁnition might look a bit arbitrary, but note that the horizontal sum of two eﬀect
algebras is in fact the coproduct in the category of eﬀect algebras with unital morphisms [33].
The horizontal sum of two sequential eﬀect algebras does not have to again be a sequential eﬀect
algebra. Necessary and suﬃcient conditions were found in Ref. [20, Theorem 8.2] for a horizontal
sum of sequential eﬀect algebras to allow a sequential product. Let us give a simple example of
such a SEA coming from a horizontal sum.
Example 48.
Let
H
be the horizontal sum of the unit interval with itself. I.e.
H
is the disjoint
union of the unit interval with itself, that we will call the left [0
,
1]
L
and the right [0
,
1]
R
interval,
where 0
L
= 0
R
and 1
L
= 1
R
are identiﬁed. This is an eﬀect algebra where addition is only
deﬁned when elements are both from the left respectively the right interval, and the complement is
λ
L
= (1
λ
)
L
(same for right). It is easy to see that this eﬀect algebra is directed complete (as
each of the unit intervals is). It is also a normal SEA with the product
λ
A
µ
B
= (
λµ
)
A
where
A, B {L, R}
. We can give
H
two diﬀerent a-convex structures, determined by either setting
λ ·
1 =
λ
L
or
λ ·
1 =
λ
R
. For every other element there is a unique choice given by
λ ·µ
A
= (
λµ
)
A
for
A {L, R}
. It is straightforward to check that either choice for
λ ·
1 gives an a-convex action on
H
.
This however does not make
H
a convex eﬀect algebra, because (supposing without loss of generality
that λ · 1 = λ
L
) we get λ · (
1
2
R
>
1
2
R
) = λ · 1 = λ
L
, while λ ·
1
2
R
> λ ·
1
2
R
= (λ
1
2
)
R
> (λ
1
2
)
R
= λ
R
.
On a normal SEA E an a-convex action yields an additive map ϕ :
[0
,
1]
E given by ϕ
(
λ
) =
λ ·
1
.
We will prove its converse, but that will require some preparation.
Recall that a normal SEA also has inﬁma of decreasing sequences (Remark 14).
Deﬁnition 49. Let E be a normal SEA. For a E we deﬁne its ﬂoor to be bac :=
V
n
a
n
.
Lemma 50.
Let
E
be a normal SEA. For
a E
, we have
bac a
. Furthermore,
bac
is the largest
idempotent below a.
Proof.
To start, since
a
commutes with each
a
n
, we have
a | bac
=
V
n
a
n
, and
baca
=
a ◦bac
=
V
n
a a
n
=
V
n
a
n+1
=
V
n
a
n
=
bac
. Then
baca
n
=
bac
for all
n
, and so
bac
2
=
V
n
baca
n
=
bac
.
Hence bac is an idempotent.
Let
p
be an idempotent below
a
; we must show that
p bac
. Since
p a
, we have
p a
=
a p
=
p
. Note that
p a
2
=
p
(
a a
) = (
p a
)
a
=
p a
=
p
, using here that
a | p
. By a similar
reasoning we get p a
n
= p for all n. Thus p ◦bac =
V
n
p a
n
= p, and so p bac.
Lemma 51.
Given an element
a
of a normal SEA
E
with
bac
= 0 and
n N
>0
, there is a
unique a
0
E with a = na
0
. Moreover, a
0
{a}
00
.
Proof.
Concerning uniqueness, suppose for now that there is an
a
0
{a}
00
with
na
0
=
a
, and
let
b E
with
a
=
nb
be given. Since
b | nb
=
a
, we have
b {a}
0
, and thus
b | a
0
, using here
that
a
0
{a}
00
. It follows that
b
and
a
0
are both part of the commutative normal SEA
{b, a
0
}
00
.
Now, the representation theory for directed-complete eﬀect monoids (cf. Section 2.2.1) gives us that
nb
=
na
0
implies
b
=
a
0
(indeed, this implication holds for commutative
C
-algebras, and trivially
for Boolean algebras). Hence, a
0
is unique.
For the existence of
a
0
{a}
00
we can assume without loss of generality that
E
=
{a}
00
, and,
moreover, that
E
=
B C
for some complete Boolean algebra
B
, and some convex normal eﬀect
monoid
C
, again by the representation theory for directed-complete eﬀect monoids. Then
a
= (
b, c
)
for some
b B
and
c C
. We claim that
b
= 0. For this, note that
b
must be an idempotent
since
B
is Boolean, and so
bbc
=
b
. Since 0 =
bac
=
b
(
b, c
)
c
= (
bbc, bcc
) = (
b, bcc
), we have
b
= 0.
Finally, deﬁne a
0
:= (0,
1
n
· c), using here that C is convex, and observe that na
0
= a.
Note that the condition bac
= 0
is necessary in the previous Lemma. Indeed, in Example 48
taking a = 1, we get 2(
1
2
L
) = a = 2(
1
2
R
) while
1
2
L
6=
1
2
R
.
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Proposition 52. Let ϕ: [0, 1] E be an additive map into a normal SEA E.
1. ϕ is normal: we have ϕ(
W
D) =
W
λD
ϕ(λ) for every directed subset D of [0, 1].
2. ϕ(λ) | ϕ(µ) for all λ, µ [0, 1].
3. bϕ(λ)c = 0 for any λ [0, 1).
4. If ϕ(λ) = ψ(λ) for some λ (0, 1) and additive ψ : [0, 1] E, then ϕ = ψ.
Proof.
1. Since
W
λD
ϕ
(
λ
)
ϕ
(
W
D
) the diﬀerence
ϕ
(
W
D
)
W
λD
ϕ
(
λ
) exists; we must show
that it is zero. Let a natural number n > 0 be given, and pick µ D with
W
D µ
1
n
. Since
ϕ(
W
D)
W
λD
ϕ(λ) ϕ(
W
D) ϕ(µ) = ϕ(
W
D µ) ϕ(
1
n
),
using that
ϕ
(
µ
)
W
λD
ϕ
(
λ
) in the ﬁrst inequality, we see then that
ϕ
(
W
D
)
W
λD
ϕ
(
λ
) has an
n-fold sum for all n, and hence must therefore be zero by Lemma 21.
2. Let
n, m >
0 be natural numbers. Since
ϕ
(
1
nm
) commutes with itself, it commutes with
(
1
nm
) =
ϕ
(
1
m
) by S5. But then
(
1
nm
) =
ϕ
(
1
n
) commutes with
ϕ
(
1
m
) too, again by S5. Going
on like this we see that
ϕ
(
k
n
)
| ϕ
(

m
) for all natural numbers
k n
and
m
. Whence
ϕ
(
q
)
| ϕ
(
r
)
for all rational
q, r
[0
,
1]. Now let
x, y
[0
,
1] be arbitrary, and pick directed sets
C, D
[0
,
1]
of rational numbers with
W
C
=
x
and
W
D
=
y
. Then
ϕ
(
c
)
| ϕ
(
d
) for all
c C
and
d D
,
and so
ϕ
(
x
) =
W
cC
ϕ
(
c
)
| ϕ
(
d
) for all
d D
, by S6, and the fact that
ϕ
is normal. But
then ϕ(x) |
W
dD
ϕ(d) = ϕ(y) too.
3. Pick a natural number
n >
0 with
λ
1
1
n
. Then
bϕ
(
λ
)
c bϕ
(1
1
n
)
c
and so it suﬃces to
show that
bϕ
(1
1
n
)
c
= 0. To this end let
p
be an idempotent with
p ϕ
(1
1
n
). We have to show
that
p
= 0. Since
p ϕ
(1
1
n
) =
ϕ
(1)
ϕ
(
1
n
)
1
ϕ
(
1
n
), and so
ϕ
(
1
n
)
p
, we have
ϕ
(1) =
(
1
n
)
p
by Lemma 18, and whence
p ϕ
(1)
. On the other hand
p ϕ
(1
1
n
)
ϕ
(1), so
p
, being below both
ϕ
(1) and
ϕ
(1)
, is summable with itself. Since
p
is an idempotent, we get
p > p p by Lemma 18, and so p = 0, as desired.
4. Since
ϕ
(
λ
) =
ψ
(
λ
), we have
ϕ
(
λ
n
) =
ψ
(
λ
n
) for all natural numbers
n >
0, by Lemma 51, using
here that
bϕ
(
λ
)
c
=
bψ
(
λ
)
c
= 0 by point 3. Since numbers of the form
n
lie dense in [0
,
1], point 1
entails that ϕ = ψ.
Deﬁnition 53. Let E be an eﬀect algebra. We call an element h E a half when h > h = 1.
Proposition 54. Let E be a normal SEA. A half is central iﬀ it is unique.
Proof. Given elements h and g of E with h > h = 1 = g > g, and h g = g h, we have
h = h 1 = h (g > g) = h g > h g = g h > g h = g (h > h) = g,
so commuting halves are equal. In particular, a half is unique when it is central.
For the converse, suppose that
h
is the only half in
E
, and let
a
be an element of
E
. We must
show that
a
and
h
commute. To this end, note that
a h > a
h
is a half, and so
h
=
a h > a
h
,
by uniqueness of
h
. Since
a h
commutes with
a h > a h
=
a
, and, similarly,
a
h
commutes
with
a
, and thus with
a
too, we see that
a
commutes with
a h > a
h
=
h
. Whence
h
is
central.
Proposition 55.
Let
E
be a normal SEA. Any unital, additive map
ϕ:
[0
,
1]
E
gives an a-convex
action
·
ϕ
on
E
via
λ ·
ϕ
a
=
a ϕ
(
λ
). Moreover, for an a-convex action
·
, an a-convex action
·
is of
this form iﬀ λ · a = a (λ · 1) for all a E and λ [0, 1].
Proof.
(
·
ϕ
is an a-convex action) Clearly 1
·
ϕ
a
=
a
and (
λ
+
µ
)
·
ϕ
a
=
λ·
ϕ
a> µ·
ϕ
a
for
λ
+
µ
1 and
a
E
. The only diﬃculty here is in establishing the last remaining condition, that
µ·
ϕ
(
λ·
ϕ
a
) = (
µλ
)
·
ϕ
a
given
µ, λ
[0
,
1] and
a E
. Since for ﬁxed
λ
and
a
both
µ 7→ µ ·
ϕ
(
λ ·
ϕ
a
) and
µ 7→
(
µλ
)
·
ϕ
a
give
,
1]
E
, it suﬃces by point 4 of Proposition 52 to show that
1
2
·
ϕ
(
λ·
ϕ
a
) = (
1
2
λ
)
·
ϕ
a
.
For
λ
= 1, this is obvious enough, and for
λ <
1 this follows immediately from the fact that
λ ·
ϕ
a
has a unique half by Lemma 51, because
bλ ·
ϕ
ac
= 0 by point 3 of Proposition 52. Note that we
could not prove
1
2
·
ϕ
(
λ ·
ϕ
a
) = (
1
2
λ
)
·
ϕ
a
here by simply applying point 4 of Proposition 52 again as
it’s a priori not clear that λ 7→
1
2
·
ϕ
(λ ·
ϕ
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(The condition
λ · a
=
a
(
λ ·
1)) Note that
a
(
λ ·
ϕ
1) =
a
(1
ϕ
(
λ
)) =
a ϕ
(
λ
) =
λ ·
ϕ
a
for
ϕ
. Conversely, for an a-convex action
·
with
λ · a
=
a
(
λ ·
1) for all
λ
[0
,
1]
and a E, we immediately get · = ·
ϕ
where ϕ(λ) = λ · 1.
One might think that the process of constructing an a-convex action from a unital additive map,
and retrieving a unital additive map from an a-convex action are each others inverse, i.e. that we
always have the condition λ · a
=
a
(
λ ·
1)
stated in the proposition above. Example 56 below
shows that this is not the case. For convex actions, however, the equation λ·a
=
a
(
λ·
1)
does hold
(because there is only one a-convex action in the presence of a convex action, see Theorem 57).
Example 56.
Let
H
=
HS
([0
,
1]
,
[0
,
1]) be the horizontal sum of [0
,
1] with itself from Example 48,
equipped with the a-convex action determined by
λ ·
1 =
λ
L
. Now let
E
=
HS
(
H H,
[0
,
1]),
the horizontal sum of
H H
and [0
,
1]. We deﬁne a sequential product by case distinction: Let
(
a, b
)
,
(
a
0
, b
0
)
H H
and take
µ, µ
0
[0
,
1]. We set
µ µ
0
=
µµ
0
and
µ
(
a, b
) =
1
2
(
µa
+
µb
), where
we interpret
a
and
b
as elements of [0
,
1]. We set (
a, b
)
(
a
0
, b
0
) = (
a a
0
, b b
0
) and (
a, b
)
µ
=
(a µ
R
, b µ
R
) where we interpret µ as µ
R
in H. So in particular (1, 0) µ = (µ
R
, 0).
Now we equip
E
with an a-convex action. For (
a, b
)
H H
with (
a, b
)
6
= (1
,
1) we set
λ ·
(
a, b
) = (
λ · a, λ · b
), and for
µ
[0
,
1] with
µ 6
= 1 we set
λ · µ
=
λµ
. Finally, for 1
E
we
set
λ ·
1 =
λ
R
, i.e. an element of [0
,
1]. We now see that
λ ·
(1
,
0) = (
λ ·
1
,
0) = (
λ
L
,
0), while
(1, 0) (λ · 1) = (1, 0) λ = (1 λ
R
, 0) = (λ
R
, 0).
Theorem 57. For a normal SEA E the following are equivalent.
1. There is a convex action on E.
2. There is precisely one a-convex action on E.
3. There is a central h E with h > h = 1.
4. There is precisely one h E with h > h = 1.
5. For each a E there is precisely one b E with b > b = a.
6.
There is an a-convex action on
E
, and any such a-convex action
·
can be restricted to
Z
(
E
)
in the sense that λ · a is central for all λ [0, 1] and a Z(E).
7. Z(E) is convex.
Moreover, in that case
λ · a
=
a
(
λ ·
1) for the unique a-convex action
·
on
E
, and all
a E
and λ [0, 1].
Proof.
We begin by proving that 15 are equivalent by producing the loop of implications
143521. Along the way, we establish 24 as it is needed to prove 21.
(1
4) Let
·
be a convex action on
E
. Then 1 has a unique half: if
h > h
= 1 for some
h E
,
then h = (
1
2
>
1
2
) · h =
1
2
· h >
1
2
· h =
1
2
· (h > h) =
1
2
· 1.
(43) Follows immediately from Proposition 54.
(3
5) Let
a E
be given; we must show that
a
has a unique half. Let
h
be a central element
such that
h > h
= 1. Since
b
=
b
(
h > h
) = (
b h
)
>
(
b h
) = (
h b
)
>
(
h b
) =
h
(
b > b
) =
h a
for any b E with a = b > b, we see that a has a unique half.
(5
2) Assume each
a E
has a unique half. For uniqueness, let
·
1
and
·
2
be a-convex actions
on
E
. To prove that
·
1
=
·
2
, we must show that (
·
)
·
1
a
= (
·
)
·
2
a
for given
a E
, and for this it
suﬃces to show that
1
2
·
1
a
=
1
2
·
2
a
, by Proposition 52.4. But since both
1
2
·
1
a
and
1
2
·
2
a
are halves
of a, this follows by assumption.
It remains to be shown that there is at least one a-convex action on
E
. To this end, let
h
be a
half of 1, and note that
{h}
00
being a directed-complete eﬀect monoid with a half is isomorphic
to [0
,
1]
C(X)
for some extremally-disconnected compact Hausdorﬀ space
X
, via some isomorphism
Φ
:
[0
,
1]
C(X)
{h}
00
. The assignment
λ 7→
Φ(
λ
), where is the function on
X
that is constant one,
ϕ:
[0
,
1]
E
, and so
E
has a a-convex action given by
λ · a
=
a ϕ
(
λ
)
by Proposition 55.
(2
4) Suppose that
E
has a unique a-convex action
·
. Then clearly 1 has a half given by
1
2
·
1.
Concerning uniqueness, let
h E
with
h > h
= 1 be given. Considering the directed-complete
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eﬀect monoid
{h}
00
we can ﬁnd a unital, additive map
ϕ:
[0
,
1]
E
with
ϕ
(
1
2
) =
h
, which yields
an a-convex action
·
ϕ
on
E
given by
λ ·
ϕ
a
=
a ϕ
(
λ
) by Proposition 55. Since there is only one
a-convex action on E, we get · = ·
ϕ
, and thus
1
2
· 1 =
1
2
·
ϕ
1 = ϕ(
1
2
) = h.
(2
1) Let
·
be the unique a-convex action on
E
. Given summable
a, b E
we must show that
λ · a > λ · b
=
λ ·
(
a > b
) for all
λ
[0
,
1]. Note that by Proposition 52.4 it suﬃces to show that
1
2
·a >
1
2
·b
=
1
2
·
(
a > b
). Since both sides of this equation are clearly halves of
a > b
, we are done if
halves are unique—which indeed they are, since we have already established that 2435.
Whence 1, 2, 3, 4, and 5 are equivalent. We continue by showing that 6 is equivalent to 15.
(6
3) Let
·
be an a-convex action on
E
. Then, clearly,
h
:=
1
2
·
1 is a central element that
obeys h > h = 1.
(15
6) Let
·
be the unique a-convex action on
E
from 2. We must show that
·
can be restricted
to
Z
(
E
). Since
λ ·
0
a
=
a
(
λ · a
) deﬁnes an a-convex action
·
0
on
E
, and
·
is unique, we have
·
=
·
0
,
and so
λ