Ordered group-valued probability, positive operators, and integral representations
T. Kroupa
math.FA math.LO 06D35 (Primary) 97H50, 47H07 (Secondary)


Ordered group-valued probability]Ordered group-valued probability, positive operators, and integral representations


T. Kroupa]Tomáš Kroupa
The Czech Academy of Sciences
 Institute of Information Theory and Automation
 Prague, Czech Republic
http://staff.utia.cas.cz/kroupa/
 kroupa@utia.cas.cz   

The work on this paper has been supported from the GAČR grant project GA17-04630S

06D35, 97H50, 47H07



Probability maps are additive and normalised maps taking values in the  unit interval of a  lattice ordered Abelian group. They appear in  theory of affine representations and they are also a semantic counterpart of Hájek's probability logic. In this paper we obtain a correspondence between probability maps and positive operators of certain Riesz spaces, which extends the well-known representation theorem of real-valued MV-algebraic states by positive linear functionals. When the codomain algebra contains all continuous functions, the set of all probability maps is convex, and we prove that its extreme points coincide with homomorphisms. We also show that probability maps can be viewed as a collection of states indexed by maximal ideals of a codomain algebra, and we characterise this collection in special cases.


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    December 30, 2023
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§ INTRODUCTION

States are real-valued functions on MV-algebras (or, equivalently, unital lattice ordered Abelian groups), which are counterparts of real-valued probability measures of Boolean algebras. See <cit.> for a detailed treatment of states and their relation to ordered algebras, mathematical logic, and probability theory. In this paper we make an attempt at a systematic study of probability maps, that is, states taking values in more general ordered groups than , with regard to their representation by positive operators of certain Riesz spaces. The achieved results generalise important theorems about real-valued states, such as their correspondence to positive linear functionals and  the integral representation theorem. When compared with <cit.>, we allow for a more general codomain of probability maps than Riesz MV-algebras. The assumption of Dedekind completeness is frequently adopted when working with the codomain of states or positive operators. However, in this paper it is used 
 only in Theorem <ref>. 


The research on states is related to a number of interesting topics in algebraic logic, ordered groups, and probability on MV-algebras. Let us mention just a few of them as a motivation for this paper and as a way of recalling previous results obtained in this area. Our list is by no means exhaustive.


  Generalised states In an effort to develop an algebraic semantics (in the  varietal sense) for Hájek's probability logic <cit.>, the author and Marra introduced the notion of generalised state and investigated the associated two-sorted variety of algebras <cit.>. This approach opens space for an equational treatment of probability and makes it possible to apply universal constructions in this field.

  Affine representations The classical theory of affine representations is used to classify certain types of ordered Abelian groups; see <cit.>. The natural evaluation map from Example <ref>, which sends the universe of algebra into the set of affine continuous functions on the state space, is a prototypical example of a probability map since it is additive and positive, but it does not preserve lattice operations. Affine representations of MV-algebras are investigated in <cit.>.

  Non-archimedean probability Probability measures with co­do­mains more general than [0,1]  can be considered naturally; see <cit.>. It is well-known that the set of all positive integers  admits no uniform probability distribution in the classical sense. On the other hand, there is a uniform probability map on the finite-cofinite Boolean algebra ℬ over . Assume the codomain of a probability map is Chang's algebra C{0,ε,2ε,…,1-2ε,1-ε,1}, where the symbol ε represents the smallest infinitesimal greater than 0.  Then  pℬ→ C defined as p(A)Aε, when A is finite, and p(A) 1-A̅ε, if A is cofinite, is a probability map. It can be thought of as a “uniform distribution” over  since p({n})=ε for each n∈.


The mathematical tools used in this paper are not only algebraic, but they also involve  convexity and infinite-dimensional geometry, which is dictated by the rich structure we are dealing with (spaces of positive operators). This wide arsenal of techniques makes it possible to draw a parallel between certain ordered algebras and their positive maps, which is witnessed by some of the main results of this paper such as Theorem <ref>.


It is well-known that the class of MV-algebras is equational, whereas unital Abelian ℓ-groups do not form a variety of algebras. Interestingly enough, the positivity and the additivity conditions of group homomorphisms take form of two simple equational laws in case of probability maps between MV-algebras; cf. Definition <ref> and Lemma <ref>.

The article is structured as follows. We fix our notation and terminology in Section <ref>. In particular, we repeat basic facts about MV-algebras and positive operators between Riesz spaces, which form a necessary background for dealing with real-valued states of MV-algebras and their generalisations. Throughout the paper we use the apparatus of compact convex sets and affine continuous maps; the reader is referred to Appendix <ref> for summary. Our main results are formulated in Section <ref>.


  * Theorem <ref> says that there is a one-to-one correspondence between probability maps and positive operators.

  * Corollary <ref> identifies extreme points of the convex set of all probability maps when the codomain is the algebra of all continuous functions.

  * Theorem <ref> provides sufficient conditions for an integral representation of a probability map by a vector measure.

In Section <ref> we formulate a framework for studying dual probability maps, which can be fully described in certain simple cases. The paper concludes with an outlook towards further research (Section <ref>).



§ PRELIMINARIES

First, we recall basic definitions and facts about MV-algebras <cit.>. Then we discuss positive operators between Riesz spaces of continuous functions, which is instrumental in parts of theory of MV-algebraic states <cit.>.


 §.§ MV-algebras

We follow the notation and terminology of <cit.>, to which we refer the reader for background. Given an MV-algebra (M,⊕,,0), we define as usual the binary
operation ⊙ and the constant 1 together with the lattice
supremum ∨ and infimum ∧ on M. The lattice order of M is
denoted by ≤.
[Standard MV-algebra]
Let [0,1] be the real unit interval equipped with the operations a⊕ b min(a+b,1), a  1-a, and the constant 0. This structure is an MV-algebra and its order ≤ coincides with the usual total order of the real unit interval [0,1].


[MV-algebra of continuous functions]
Let X be a compact Hausdorff topological space and C(X) be the collection of all continuous functions X→ [0,1]. Since ⊕ and  are continuous functions, C(X) becomes an MV-algebra equipped with the operations of the product MV-algebra [0,1]^X. The MV-algebra C(X) is separating: For every x,y∈ X with x≠ y, there is an element a∈ C(X) such that a(x)≠ a(y).


By McNaughton theorem <cit.> the free MV-algebra over n generators coincides with the algebra of certain continuous functions defined on the hypercube [0,1]^n.
[The free MV-algebra]
Let n∈. We call a [0,1]^n→ [0,1] a McNaughton function if a is continuous and piecewise-linear, where each linear piece has integer coefficients only. Let F_MV(n) be the MV-algebra of all n-variable McNaughton functions, where F_MV(n) is considered as an MV-subalgebra of C([0,1]^n). Then F_MV(n) is precisely the free n-generated MV-algebra. Loosely speaking,  McNaughton functions [0,1]^n→ [0,1] are many-valued generalisations of Boolean functions {0,1}^n→{0,1}.


An ideal of an MV-algebra M is a subset 𝔦 of M containing 0, closed with respect to ⊕, and if a∈𝔦, b∈ M and b≤ a, then b ∈𝔦.
The notions of MV-homomorphism and quotient are defined as usual. By M we denote the nonempty compact Hausdorff topological space of maximal ideals  of M. For any MV-algebra M, let M be the radical of M,

    M ⋂ M.

Always 0∈ M. Nonzero elements of M are called infinitesimals. An MV-algebra M is said to be semisimple if it has no infinitesimals, that is, M={0}.


Let M be an MV-algebra. 


  * For any ∈ M there exist a unique MV-subalgebra I_ of [0,1] together with a unique MV-isomorphism 
    h_ A/ → I_.


  * Let the map a∈ M ↦ a^*∈ [0,1]^ M be defined by 

    a^*()  h_(a/ ),   ∈ M.

Then the map ^* is an MV-homomorphism of M onto a separating MV-subalgebra M^* of C( M). Moreover, the map ^* is an MV-isomorphism onto M^* if, and only if, M is semisimple.


The first part of Theorem <ref> is an MV-algebraic variant of Hölder's theorem.
For any MV-algebra M, the quotient algebra M/  M is semisimple, and the maximal ideal spaces M and (M/  M) are homeomorphic; see <cit.>.

Let (G,u) be a unital lattice ordered Abelian group <cit.> (unital ℓ-group, for short), where u is an order unit. Put Γ(G)  [0,u], where 0 is the neutral element of G and [0,u] denotes the corresponding order interval in G.  For all a,b∈Γ(G) we define

    a⊕ b  (a+b) ∧ u,      a  u-a,

where +,∧, and - are the operations of G. Then (Γ(G),⊕,,0) becomes an MV-algebra. 
Mundici's theory of Γ functor <cit.> says that Γ(G) is  the most general example of an MV-algebra. Specifically, for every MV-algebra M there is a unique unital ℓ-group Ξ(M) such that M is isomorphic to Γ(Ξ(M)).

We define a⊖ b a⊙ b, for all a,b∈ M. In Section <ref> we make an ample use of several MV-algebraic identities:


  *  b∧ a=b⊖ (a⊙ b)

  * 	a⊕ b=a⊕ (b∧ a)

  * 	a⊙ (b∧ a)=0
 
In particular,  <ref> and <ref>  say that any element in M of the form a⊕ b can be additively cut into two disjoint parts, a and b∧ a. 
In order to show that <ref> is true, recall that, by the definition of ∧ in M, we have 

    b∧ a =b⊙ ( b ⊕ a)=b⊙ (a⊙ b)=b⊖ (a⊙ b).

 By <cit.> the  operation ⊕ distributes over ∧, 

    a⊕ (b∧ a) = (a⊕ b) ∧ (a⊕ a)=a⊕ b,

which proves <ref>. Finally, <ref> holds,

    a⊙ (b∧ a)=a⊙ (b⊖ (a⊙ b))=(a⊙ b) ⊙ (a⊙ b)=0.




 §.§ Positive operators

The standard reference for Banach lattices of continuous functions and their positive operators is <cit.>. Positive operators between Riesz spaces are thoroughly studied also in <cit.>. 

Let X and Y be compact Hausdorff spaces. By C_(X) we denote the Banach lattice of all continuous functions a X→, endowed with the  supremum norm ., where 
    asup {a(x)| x∈ X},      a∈ C_(X).
 A unital positive linear operator is a linear mapping L C_(X)→ C_(Y) that is positive (L(a)≥ 0 for all a≥ 0) and unital (L(1)=1). Any such L is necessarily a monotone mapping (a≤ b implies L(a)≤ L(b)) and a continuous linear operator (there is some C>0 such that L(a)≤ Ca for all a∈ C_(X)). Let 
^+(C_(X),C_(Y))
be the set of all unital positive linear operators C_(X)→ C_(Y). Then ^+(C_(X),C_(Y))
is clearly a convex subset of the linear space of all maps C_(X)→ C_(Y). Ellis-Phelps theorem (see <cit.>) then describes the extreme boundary of ^+(C_(X),C_(Y)). We use a formulation of this result from <cit.>.


Let L∈^+(C_(X),C_(Y)). Then the following are equivalent:


  * L  is an extreme point of ^+(C_(X),C_(Y)).

  * L is a lattice homomorphism.

  * There is a unique continuous map f Y→ X such that L(a)=a∘ f, for all a∈ C_(X).



If Y={y}, then C_(Y) and  are identified, and any F ∈^+(C_(X),) is called a unital positive linear functional. The  shorter notation ^+(C_(X)) will be used in place of ^+(C_(X),). The space ^+(C_(X)) is endowed with the subspace weak^*-topology that is inherited from the dual space C^*_(X) of C_(X). This makes ^+(C_(X)) into a compact convex subset of the locally convex Hausdorff space C^*_(X). Let Δ(X) be the compact convex set of all regular Borel probability measures over X (see Example <ref>). Given a probability measure μ∈Δ(X), define

    F_μ(a) ∫_X a dμ,      a∈ C_(X),

and observe that F_μ∈^+(C_(X)). Then Riesz theorem  can be formulated as follows. 

The map Δ(X)→^+(C_(X)) given by 

    μ↦ F_μ

is an affine homeomorphism.






 §.§ States

Let M be an MV-algebra. A state of M is a real function s M→ [0,1] satisfying s(1)=1 and the following condition for all a,b∈ M:

    If a⊙ b=0, then s(a⊕ b)=s(a)+s(b).

Every MV-algebra carries at least one state as a consequence of Theorem <ref> and the fact that M≠∅. Namely take an arbitrary 𝔪∈ M, put 

    s_𝔪(a) a^*(𝔪),    a∈ M,

and observe that the MV-homomorphism s_𝔪 M→ [0,1] is a state of M. In fact every MV-homomorphism h M→ [0,1] is necessarily of the form h=s_𝔪, for some 𝔪∈ M.

States of M are in one-one correspondence with states of the enveloping ℓ-group of M. Specifically, a state of G, where (G,u) is unital ℓ-group, is a unital positive real homomorphism of G, that is, a group homomorphism s G→ such that s[G^+]⊆^+ and s(u)=1. See <cit.> for a thorough exposition of ℓ-group states. 

Let M be an MV-algebra and Ξ(M) be its enveloping ℓ-group. Then the restriction of a state s of Ξ(M) to a function s↾ M→ [0,1] determines a bijection between the states of Ξ(M) and the states of M.


 See <cit.>. 

By ^M we denote the locally convex Hausdorff space of real functions over M equipped with the product topology. The state space of M is a nonempty set

    M {s |s M→ [0,1] is a state}.

We will use the convergence of nets in topological spaces <cit.>.
A net of states (s_γ) converges to a state s in M if it converges pointwise,

    s_γ(a) → s(a)  for every a∈ M.

A straightforward verification proves that  M is a compact convex subset of the Tychonoff cube [0,1]^M⊆^M. By Krein-Milman theorem we know that M coincides with the closed convex hull of the extreme boundary M, which is precisely the set of all MV-homomorphisms into [0,1],

    M = {s_𝔪|𝔪∈ M}.

Since the map 𝔪∈ M ↦ s_𝔪∈ M is bijective and continuous, the extreme boundary M is compact.

From the perspective of states, infinitesimal elements are irrelevant. This fact is in sharp contrast with behavior of states on pseudo MV-algebras <cit.>.

For any MV-algebra M, the state spaces M and (M/ M) are affinely homeomorphic.

In the rest of this section we will always assume that an MV-algebra M is  semisimple. If it is not the case, we can simply replace M with M/  M in the  arguments involving the corresponding state spaces.

 The main characterisation of states is based on integral representation of certain linear functionals. The reader is invited to consult Appendix <ref> for all the concepts related to compact convex sets. Let Δ( M) be the Bauer simplex of all regular Borel probability measures on M and ^+(C_( M)) be the Bauer simplex of all unital positive linear functionals on C_( M). For any μ∈Δ( M), observe that the function s_μ M→ [0,1] defined by

    s_μ(a)∫_ M a^* dμ,      a∈ M,

is a state of M. Note that the restriction of any  L ∈^+(C_( M)) to M^* is also a state.

Let M be a semisimple MV-algebra. Then:


  * The restriction map

    L∈^+(C_( M)) ↦ L  ↾ M^*∈ M^*

is an affine homeomorphism of ^+(C_( M)) onto M^*.

  * The map 

    μ∈Δ( M) ↦ s_μ∈ M

is an affine homeomorphism of M onto Δ( M).


The difficult part of the theorem says that states of a (semisimple) MV-algebra are uniquely extendible to positive linear functionals of the ambient Banach lattice of continuous functions. Then it is a consequence of Riesz theorem that any state s∈ M is of the form s=s_μ, for a unique measure μ∈Δ( M).

 Summing up, the state space M of any MV-algebra M is a Bauer simplex that is affinely homeomorphic to any of the Bauer simplices appearing on the list below:


  * The state space of the maximal semisimple quotient M/ M.

  * The state space of the enveloping ℓ-group Ξ(M).

  * Unital positive linear functionals ^+(C_( M)).

  * Regular Borel probability measures Δ( M).





§ PROBABILITY MAPS
 
 A probability map (Definition <ref>) is an additive normalised map of an MV-algebra M into a more general codomain MV-algebra N than the standard algebra [0,1]. At the same time, Definition <ref> extends the concept of “generalised state" introduced by the author and Marra in <cit.>, where the domain MV-algebra M is taken to be a Boolean algebra. Unlike other presentations of probability-like maps of MV-algebras or similar structures, the one in Definition <ref> is purely equational. Nevertheless, Proposition <ref> below shows that  probability maps admit also a more traditional definition that is analogous to the defining condition (<ref>) of [0,1]-valued states.

Throughout this section, M and N denote MV-algebras.
 Same symbols are used to denote the operations and the constants in both M and N; their meaning will always be clear from the context.

A probability map is a function p M→ N satisfying the following identities for all a,b∈ M.


  *  p(a⊕ b)=p(a)⊕ p(b∧ a)

  *  p( a)= p(a)

  *  p(1)=1


It is not difficult to show that the axioms <ref>–<ref> are independent.
Note that p is also normalised at 0 since, by <ref> and <ref>, p(0)= p(1 )=0.
Every MV-homomorphism is a probability map as <ref> follows from <ref> in this case. The axiomatisation in Definition <ref> is inspired by that of internal states of MV-algebras studied by Flaminio and Montagna <cit.>. An internal state of an MV-algebra M is a mapping σ M→ M satisfying the conditions (1)–(4) for all a,b∈ M:


  * σ(0)=0

  * σ( a)=σ(a)

  * σ(a⊕ b)=σ(a) ⊕σ(b⊖ (a⊙ b))

  * σ(σ(a)⊕σ(b))=σ(a)⊕σ(b)

By <ref> we can rewrite (P1) as
		
			
  *  p(a⊕ b)=p(a) ⊕ p(b⊖ (a⊙ b))

which is among the defining conditions of internal states. Thus, we can equivalently define a probability map to be a function p M→ N satisfying <ref>, (P2), and (P3). This also shows that every internal state of M is a probability map M→ M. However, the converse is typically not true. Whereas any internal state is an idempotent map,  there are already MV-endomorphisms that fail to be idempotent.



We will collect useful elementary properties of probability maps into this lemma.


	Let p M→ N be a probability map. For all a,b∈ M:
	
		
  * If a≤ b, then p(a)≤ p(b).
		
  * p(a⊕ b)≤ p(a)⊕ p(b), and if a⊙ b=0, then p(a⊕ b)=p(a) ⊕ p(b).
		
  * p(a⊖ b)≥ p(a)⊖ p(b), and if b≤ a, then p(a⊖ b)= p(a)⊖ p(b).
		
  * p(a⊙ b)≥ p(a)⊙ p(b), and if a⊙ b=0, then p(a)⊙ p(b)=0.
		


	(1)  Let a≤ b. Then b=a⊕ (b⊖ a) and using <ref> we get
	
    p(b)=p(a⊕ (b⊖ a))=p(a)⊕ p((b⊖ a)∧ a) ≥ p(a).

	
	(2) It follows from <ref> and (1) that 
	
    p(a⊕ b)=p(a) ⊕ p(b⊖ (a⊙ b)) ≤ p(a) ⊕ p(b).

	Let a⊙ b=0. Then 
	p(a⊕ b)=p(a) ⊕ p(b⊖ 0)=p(a)⊕ p(b).


	(3) By the definition of ⊖, <ref>, and (2),
	
    p(a⊖ b)    =p( (  a⊕ b))= p( a⊕ b)
       ≥ ( p(a)⊕ p(b))=p(a)⊙ p(b)=p(a)⊖ p(b).

	Let b≤ a. Then a⊙ b=0 and the equality in the  formula above follows from (2).

	(4) We get
	
    p(a⊙ b)=p(a⊖ b)≥ p(a)⊖ p( b)=p(a)⊙ p( b)=p(a)⊙ p(b).

	Let a⊙ b=0. Then 
	p(a)⊙ p(b) ≤ p(a⊙ b)=p(0)=0.

Probability maps can be viewed as genuinely additive maps between MV-algebras.
	

		Let p M→ N satisfy p(0)=0 and p(1)=1. The following are equivalent.
		
			
  * p is a probability map.
			
  * p(a⊕ b)=p(a)+p(b)-p(a⊙ b), for every a,b ∈ M, where + and - are operations in the enveloping unital ℓ-group Ξ(N) of N.
			
  * If a⊙ b=0, then p(a⊕ b)=p(a) ⊕ p(b) and p(a)⊙ p(b)=0, for all a,b∈ M.
			
  * If a⊙ b=0, then p(a⊕ b)=p(a) + p(b), for all a,b∈ M. 			
		
	
	
 (1) ⇒ (2)  By Lemma <ref>, <ref>, and the fact that 
 
    c+ d=(c⊕ d)+(c⊙ d)

  for all c,d∈ M (see <cit.>), we obtain
		
    p(a⊕ b) =p(a) ⊕ (p(b)⊖ p(a⊙ b))=p(a) + (p(b) - p(a⊙ b)).
 
		
 (2) ⇒ (3) Let a⊙ b=0. Then p(a)+p(b) =p(a⊕ b)≤ 1, which means that p(a⊕ b)=p(a)⊕ p(b) and p(a)⊙ p(b)=0.
 
 
 (3) ⇒ (1) The identity a⊙ a=0 and assumption yield p(a)⊙ p( a)=0. Further,
		
    1=p(1)=p(a⊕ a)=p(a)⊕ p(  a),   for all a ∈ M.

		Therefore p( a)= p(a) by <cit.> and thus <ref> in Definition <ref> holds. Finally, we  prove <ref>. Using <ref>–<ref>, we obtain
		
    p(a⊕ b)=p(a⊕ (b∧ a ))=p(a) ⊕ p(b∧ a ),    a,b ∈ M.

		Hence, p is a probability map. 
		
The conditions (3) and (4) are equivalent since c⊙ d=0 holds in N if, and only if, c+d≤ 1 in the enveloping ℓ-group Ξ(N) of N.		
	





Put

    (M,N) {p |p M→ N is a probability map}.

 Using
 the defining condition (<ref>) of real-valued states and Proposition <ref>(4), one can immediately see that states of MV-algebras are exactly the probability maps whose codomain N is the standard MV-algebra [0,1]. This means that (M,[0,1]) is the same as the classical state space (<ref>) of M,

    (M,[0,1]) =  M.

The state space M is always nonempty for any M, whereas it can easily happen (M,N)=∅. For example, let M [0,1] and N be the two-element Boolean algebra {0,1}. Any p∈([0,1],{0,1}) must  necessarily be a state p [0,1]→ [0,1] whose range is {0,1}. But there is no such s since the only existing state of [0,1] is the identity map. Hence, ([0,1],{0,1})=∅.

In general, even when the set (M,N) is nonempty, it fails to be convex.

In this example we put M=N=F_MV(1), where F_MV(1) is as in Example <ref>. Then (M,M)≠∅ since any MV-endomorphism of M is a probability map. In addition, there are probability maps p M→ M that are not MV-endomorphisms. For example, consider the probability map defined as

    p(a)(x)   x· a(0) + (1-x)· a(1),

for all a∈ M and every  x∈ [0,1]. Then p∈(M,M) is not an MV-endomorphism since it fails to preserve suprema. Indeed, take a𝕀, where 𝕀 is the identity function on [0,1]: 

    p(𝕀∨𝕀)=1 ≠𝕀∨𝕀 = p(𝕀) ∨ p(𝕀).

With (M,M) being a subset of the real linear space C_([0,1])^M of all maps M→ C_([0,1]), it is sensible to ask if (M,M)  is a convex set. This is clearly not the case. It is enough to consider MV-endomorphisms p_1 and p_2 of M defined by

    p_1(a) a,    p_2(a)(x)  a(1-x),    a∈ M, x∈ [0,1].

Observe that the range of the map q 12 (p_1+p_2) is not even a subset of M and so q∉(M,M).

The previous example suggests that convexity of (M,N) might be attained   when the universe of algebra N is a convex set in some linear space. This occurs, for example, when the codomain N is a Riesz MV-algebra <cit.>. However, such a level of generality is not needed in the paper, and we will always consider a convex codomain of a very special kind, the MV-algebra C(X) introduced in Example <ref>. Then the next result is immediate.

Let M be an MV-algebra and Y be a compact Hausdorff space. Then (M,C(Y)) is a nonempty convex subset of the real linear space C_(Y)^M.


By Theorem <ref> an MV-algebra M has at least one MV-homomorphism  h M→ [0,1]. Define p_h M→ C(Y) as

    p_h(a)(y)  h(a),      a∈ M,   y∈ Y.

Then each p_h(a) is a constant function. It is easy to see that p_h is an MV-homomorphism. Thus, (M,C(Y))≠∅.

Consider p,q∈(M,C(Y)) and α∈ [0,1].
 We need to check that the map r M→ C_(Y) defined by
rα p + (1-α)q
belongs to (M,C(Y)). Since α a + (1-α)b ∈ C(Y), for all a,b∈ C(Y), the map r ranges in C(Y). Clearly, r satisfies r(1)=1, and it takes only a routine verification to show that r  is additive in the sense of  Proposition <ref>(4). 


The following example appears in many different contexts with mild variations, ranging from ℓ-groups <cit.>  over MV-algebras <cit.> to integral representation theory for compact convex sets <cit.>.

[Affine representation]
Natural affine representations of partially ordered groups are based on affine continuous functions on their state spaces <cit.>. We recall  that (A) denotes the linear space of all real-valued affine continuous functions on a convex set A in some linear space; see Appendix <ref>. The affine representation of an MV-algebra can be constructed as follows. Let M be an MV-algebra, M be its state space, and consider the linear space ( M) that is partially ordered by the pointwise order of real functions on M. An order unit in ( M) is taken to be the constant function 1. Since M is a Bauer simplex, it is a direct consequence of <cit.> that ( M) is in fact lattice ordered, hence a Riesz space. An affine representation of M is the (Riesz) MV-algebra

    M_AffΓ(( M)).

For any a∈ M consider a function

    â(s)  s(a),    s∈ M,

which is clearly affine and continuous, â∈ M_Aff. Then an injective mapping ϕ M→ M_Aff defined by
    ϕ(a)â

is a probability map that is not an MV-homomorphism. Indeed, the equation ϕ(a∨ b)=ϕ(a)∨ϕ(b) cannot be satisfied for all a,b∈ M, as there are states s∈ M such that s(a∨ b)≠ s(a∨ b). Observe that such states are not extreme points of M.






Let M be an MV-algebra and N be a Riesz MV-algebra <cit.>. Boccuto et al. <cit.> adopt the following definition. A generalised state is a map s M→ N satisfying s(1)=1 and the condition

    s(a⊕ b) = s(a)⊕ s(b),     for all a,b∈ M with a⊙ b=0.

On the one hand, the condition above is weaker than the one in Proposition <ref>(4). On the other, however,  a codomain of  a probability map is not necessarily required to be a Riesz MV-algebra. Summing up, probability maps and generalised states of <cit.> cannot be directly compared.







 §.§ Extension to positive operators

First, we will prove a preparatory lemma, which generalises Theorem <ref> about the extension of states from M onto the states of the enveloping ℓ-group Ξ(M).


Let M and N be MV-algebras. For every p∈(M,N) there is precisely one unital positive group homomorphism f_pΞ(M)→Ξ(N) such that f_p(a) = p(a), for all a∈ M.

 
 Let p M→ N be a probability map and let A_M and A_N be the lattice ordered monoids of good sequences <cit.> of M and N, respectively. For any good sequence 𝐚=(a_m)_m∈∈ M^ there is some n∈ such that a_m=0 for all m > n. For any a∈ M, let a_0 (a,0,0,…) be the good sequence in A_M associated with a.
 
   Put 

    p'(a) p(a)_0,    a∈ M.

Observe that p' is a probability map from M to an isomorphic copy of N, the MV-algebra {b_0 | b ∈ N}.

 We define a map p̂ A_M→ A_N as 
  
    p̂(𝐚)∑_i=1^n p'(a_i),   𝐚=(a_1,…,a_n,0,…) ∈ A_M,

where  ∑ on the right-hand side is the sum of good sequences in A_N. 
Our goal is to prove that p̂ is a unital monoid homomorphism. 
We get 
    p̂(0_0)=p'(0)=p(0)_0=0_0
 and, analogously, p̂(1_0)=1_0. In the next step we will show that the identity

    p̂(𝐚+b_0) = p̂(𝐚) + p̂(b_0)

holds true for every 𝐚=(a_1,…,a_n,0,…) ∈ A_M and  every b∈ M. Put

    a'_1     a_1⊕ b,
    
    	a'_2     a_2⊕ (a_1 ⊙ b),
    
    	a'_3     a_3⊕ (a_2 ⊙ b),
       ⋮
    
    	a'_n     a_n⊕ (a_n-1⊙ b),
    
    	a'_n+1    a_n⊙ b.
	
It is tedious but straightforward to show that +b_0=(a'_1,…,a'_n+1,0,…). Using the definition of p̂(+b_0), we can write

    p̂(+b_0)= ∑_i=1^n+1p'(a'_i)
    =p'(a_1⊕ b) + p'( a_2⊕ (a_1 ⊙ b)) +… + p'(a_n⊙ b).

The elements of a good sequence 𝐚 satisfy the identity

    a_i⊙ a_i+1=a_i+1,    i=1,…,n-1.

Since p' is a probability map we can apply Proposition <ref>(2) to each summand in (<ref>) to get
    p'(a_1)+p'(b)-p'(a_1⊙ b) 
        + p'(a_2)+p'(a_1⊙ b)-p'(a_2⊙ a_1_a_2⊙ b) 
        + p'(a_3)+p'(a_2⊙ b)-p'(a_3⊙ a_2_a_3⊙ b) 
        + …
        + p'(a_n)+p'(a_n-1⊙ b)-p'(a_n⊙ a_n-1_a_n⊙ b) 
        + p'(a_n⊙ b)
       = p'(a_1)+… + p'(a_n) + p'(b)
       =p̂()+p̂(b_0).

Since (<ref>) is satisfied for every ∈ A_M and b∈ M, it follows that 

    p̂( + ) = p̂() + p̂()

  holds for all  ,∈ A_M. Hence, p̂ A_M→ A_N is a unital monoid homomorphism. 
 
 Let Ξ(M) and Ξ(N) be the enveloping ℓ-groups associated with MV-algebras M and N, respectively. By <cit.>, Ξ(M)^+ and A_M are isomorphic as unital lattice ordered Abelian monoids. This means that we can identify the map p̂ with a unital monoid homomorphism Ξ(M)^+→Ξ(N)^+.
 
 
Now, every element x∈Ξ(M) can be written as y-z, for some y,z∈Ξ(M)^+ (see <cit.>). Put f_p(x)p̂(y)-p̂(z), for all x∈Ξ(M)  and some y,z∈Ξ(M)^+ satisfying x=y-z. We can routinely show that f_p(x) does not depend on the choice of y and z and that the map f_pΞ(M)→Ξ(N) is a unital group homomorphism. Since f_p(x)∈Ξ(N)^+ for all x∈Ξ(M)^+, the map f_p is positive and, by construction, it is the unique map satisfying f_p(a) = p(a), for all a∈ M.
  

 
Recall that every semisimple MV-algebra M is isomorphic to the MV-subalgebra M^* of C( M), where ^* is the isomorphism from Theorem  <ref>. 
 
 Let M and N be semisimple MV-algebras and p M→ N be a probability map. There exists a unique unital positive linear operator 
    L_p C_( M)→ C_( N)

 such that p(a)^*=L_p(a^*), for every a∈ M.
 

We will generalise Panti's proof of Proposition 1.1 in <cit.>.
Given p∈(M,N) we can define a probability map p^* from M^*⊆ C_( M) into N^*⊆ C_( N) by putting p^*(a^*) p(a)^*, for all a∈ M. We will show how to lift p^* M^*→ N^* uniquely to a unital positive linear operator L_p C_( M)→ C_( N). The extension consists of three steps.

(1) By Lemma <ref> we extend uniquely p^* to a unital positive group homomorphism f G→ H, where GΞ(M^*) and HΞ(N^*) are the enveloping unital ℓ-groups of M^* and N^*, respectively.

(2) Since G a torsion-free group, it embeds as an ℓ-group into its divisible hull G' <cit.>. Since M is semisimple its enveloping ℓ-group is necessarily Archimedean. Thus, the elements of G are  functions a M→, so that G' can be identified with the following Riesz space over ,

    G'={q· a | a∈ G, q∈}.

We define H' in a completely analogous way to G'. Further, one can routinely check that it is correct to define a map f' G'→ H' by putting f'(q· a) q· f(a), for all q∈ and a∈ G. Clearly, f' is the unique unital positive –linear operator G'→ H'  extending f. Equipped with the sup norm ., the Riesz spaces G' and H' become normed Riesz spaces over . We claim that f' is continuous.

To prove the claim, let a∈ G' and choose any strictly positive q∈ such that q≥a. Then a/q=1/qa≤ 1 yields f'( aq)≤ 1 by monotonicity of f', which implies the inequality

    f'(a)=qf'( aq)≤ q.

Hence, f'(a)≤inf{q∈| q> a}=a. This means that f' is continuous as this inequality holds for any a,b∈ G':

    f'(a)-f'(b)=f'(a-b)≤a-b.


(3) The lattice version of Stone-Weierstrass theorem says that G' is a norm-dense subspace of C_( M). For any a∈ C_( M) we can find a  sequence (a_n) in G' such that a_n-a→ 0 when n→∞. Thus, we can unambiguously define 
    L(a)lim_n→∞ f'(a_n),    a∈ C_( M).

By density the mapping L is the unique continuous linear operator extending f' to C_( M). Since for any a≥ 0 we can find a sequence of positive elements in G' converging to a, the operator L is positive. The proof is finished by putting L_p L and observing that it is the desired extension of p. 


The conclusion of Theorem <ref> remains unchanged when M is not semisimple. This is based on a certain generalisation of Proposition <ref> whose proof we omit, and which says that (M,N) and (M/ M,N) are in bijection. Hence, one obtains a 1-1 correspondence between 


  * probability maps p M→ N and

  * unital positive linear operators  C_( (M/ M))→ C_( N).


 

In order to obtain a bijection between ^+(C_( M), C_( N)) and (M,N), we will assume that the codomain MV-algebra N is of the form C(Y) for some compact Hausdorff space Y. Let L∈^+(C_( M), C_(Y)). Since L is monotone and C(Y)⊆ C_(Y), by restricting the operator L to M^* we obtain a probability map M^*→ C(Y). The following result then generalises the first part of Theorem <ref>.


 Let M be a semisimple MV-algebra and Y be a compact Hausdorff space. Let 
 
    h^+(C_( M), C_(Y))→(M,C(Y))

 be defined by
  
    h(L)(a)  L(a^*),

 for all L∈^+(C_( M), C_(Y)) and every a∈ M. Then h is an affine isomorphism.


The sets (M,C(Y)) and (M^*,C(Y))  are convex (Proposition <ref>), the map h is clearly affine, and it is bijective as a consequence of Theorem <ref>. Thus, h is an affine isomorphism. 


We will take up a natural question of describing the extreme points of convex set 𝒫(M,C(Y)). This task will be presented as a direct application of Lemma <ref> based on the known characterisation of extreme points for the convex set of operators ^+(C_( M), C_(Y)); see Theorem <ref>. In fact, since the map h from Lemma <ref> is an affine isomorphism,  it holds that L∈^+(C_( M), C_(Y)) if, and only if, h(L)∈(M,C(Y)).



 Let M be a semisimple MV-algebra and Y be a compact Hausdorff space. For any p∈𝒫(M,C(Y)), these assertions are equivalent:


  * p  is an extreme point of  (M,C(Y)).

  * p is an MV-homomorphism.




For any p∈(M,C(Y)), define

    L_p h^-1(p),

where h is the affine isomorphism from Lemma <ref>. We will show  that the implication (1) ⇒ (2) holds. Assume p∈(M,C(Y)). Hence, L_p∈^+(C_( M), C_(Y)), which implies that L_p is a lattice homomorphism (Theorem <ref>), and this means that p M→ C(Y) is a 
lattice homomorphism as well. Therefore p satisfies the identity p(a⊕ b)=p(a)⊕ p(b) and it is an MV-homomorphism.
 
In order to prove the converse (2) ⇒ (1), let p∈(M,C(Y)) be an MV-homomorphism. By Theorem <ref> it suffices to check that the operator L_p is a lattice homomorphism. Let GΞ(M^*) and HΞ(C(Y)) be the enveloping ℓ-groups of M^* and C(Y), respectively, and put p^*(a^*) p(a), for all a∈ M. Then the unique extension of p^* to a unital positive group homomorphism f G→ H according to Lemma <ref> is in fact an ℓ-group homomorphism; see <cit.>. Following the proof of Theorem <ref> we need to verify that the extended linear operator f' G'→ H', where G' and H' are divisible hulls of G and H, respectively, is a lattice homomorphism. Since f'(q· a)=q· f(a) for all q∈ and a∈ G, we can routinely check that 

    f'( am ∨ bn )=f' ( am ) ∨ f'( bn)

for all a,b∈ G and positive integers m,n. The analogous identity holds for ∧. Hence, f' G'→ H' is a lattice homomorphism and so is its extension L_p, by continuity of the lattice operations. Since  

    L_p∈^+(C_( M), C_(Y)),

 it results that p is an extreme point of (M,C(Y)) by Lemma <ref>. 
 
 
 For every semisimple MV-algebra M and a compact Hausdorff space Y, the convex set (M,C(Y)) has an extreme point.
 
 
  The map p_h M→ C(Y) defined in (<ref>) is an MV-homomorphism and Corollary <ref> says that p_h is an extreme point of (M,C(Y)).
 
Extreme boundary of (M,C(Y)) can be described  in special cases.
 
 [Stochastic matrices]
 Recall that a stochastic matrix is a real square matrix with nonnegative entries satisfying the condition that the sum of every row is equal to 1.
 Let n ∈ and consider MV-algebras M= N=[0,1]^n. Clearly N is of the form C(Y) for Y{1,…,n}, so Corollary <ref> applies. It is even possible to fully describe the convex set ([0,1]^n,[0,1]^n) as follows. For any p∈([0,1]^n,[0,1]^n), there is a unique unital positive linear mapping L_p^n→^n such that L_p(a)=p(a), for all a (a_1,…,a_n)∈ [0,1]^n.  This means that there exists a unique stochastic matrix 𝐒 of order n such that
    
    p(a)=(𝐒a^⊺)^⊺,     a∈ [0,1]^n.

  Thus, ([0,1]^n,[0,1]^n) is affinely isomorphic to the n(n-1)-dimensional convex polytope in ^n^2 of all stochastic matrices of order n. It is known that the extreme points of this polytope are precisely those stochastic matrices whose entries are {0,1}-valued only; see <cit.>. Since there are n^n of such extreme points and n^n>n(n-1) whenever n>1, this means that ([0,1]^n,[0,1]^n) cannot be an n(n-1)-dimensional (Bauer) simplex.  
  
 
 The previous example shows that the set of all probability maps (M,N) fails to be a simplex even when M and N possess finitely-many maximal ideals. In general, almost none of the characteristic features of the Bauer simplex M is preserved, when passing to a more complicated collection of maps (M,N).
 
 

 §.§ Integral representation

 Every probability map from M into the standard MV-algebra [0,1] is Lebesgue integral (<ref>) with respect to a unique regular Borel measure over M. This is the content of representation theorem whose proof relies on the extension of states onto positive linear functionals and the subsequent application of Riesz theorem for Banach lattice C_( M); see <cit.>. Theorem <ref> makes it possible to pursue this path of reasoning a little further, since the theorem says that a probability map between semisimple MV-algebras has a unique extension to a positive operator between the associated Banach lattices of continuous functions. Therefore it is not unreasonable to ask a question like this: Does a probability map allow for a certain integral representation? Theory of vector measures <cit.> provides the right framework for stating this problem precisely. In that follows only vector measures with values in spaces of the form C_(X) are considered.
 
 Let X and Y be compact Hausdorff spaces. A vector measure is a mapping μ⃗ from Borel sets of X into C_(Y) satisfying the following condition: If A_1,A_2,… is a sequence of pairwise disjoint Borel measurable subsets of X, then
 
    μ⃗ ( ⋃_n=1^∞ A_n ) = ∑_n=1^∞μ⃗(A_n),

where the sum on the right-hand side converges in the norm topology of Banach space C_(Y). We say that  a vector measure μ⃗ is regular when for each Borel set A and every ϵ >0 there exists a compact set K and an open set  O such that K⊆ A ⊆ O and μ⃗(O∖ K)<ϵ, where μ⃗ is the so-called semivariation of μ⃗; see <cit.>. The integral of a∈ C_(X) with respect to a C_(Y)-valued regular vector measure on X is an almost direct generalisation of Lebesgue integral, since the construction starts from an elementary integral of simple functions as in the classical case. Thus, for any a∈ C_(X), the integral

    ∫_X a  dμ⃗

 has a tangible meaning and the assignment
 
    L_μ⃗ (a) ∫_X a  dμ⃗,      a∈ C_(X),

determines a bounded linear operator L_μ⃗ C_(X)→ C_(Y). Now, the question from a previous paragraph amounts to asking for Riesz theorem for operators. Specifically: Given a bounded linear operator L C_(X) → C_(Y), is there a regular vector measure μ⃗ on the Borel sets of X with values in C_(Y) such that L= L_μ⃗? It turns out that this question has a positive answer if and only if the operator L is weakly compact, that is, L maps bounded sets in C_(X) to relatively weakly compact subsets of C_(Y). This follows from Bartle-Dunford-Schwartz theorem; see <cit.>. Using this result it is very easy to find a positive operator L for which no representing regular vector measure exists. Indeed, consider the identity operator C_([0,1])→ C_([0,1]) and note that (the weak closure of) the unit ball in C_([0,1]) fails to be weakly compact. Hence the identity is not a weakly compact operator. 

The above discussion implies that there are probability maps having no integral representation in the sense introduced above. However, we can still look for conditions on C_(X) and C_(Y) guaranteeing that any bounded linear operator between the two spaces is weakly compact. Such sufficient conditions are provided by Grothendieck's theorem, which is the main ingredient of this theorem.

Let M be a complete MV-algebra and N be a semisimple MV-algebra with metrisable N. For every probability map p M→ N there is a C_( N)-valued regular vector measure μ⃗ on M such that

    p(a)^* = ∫_ M a^*dμ⃗,      a∈ M.


 Every σ-complete MV-algebra is semisimple <cit.>. A fortiori, M is semisimple. It follows from <cit.> that the enveloping ℓ-group Ξ(M) is Dedekind complete. Since the space of extreme states of M is homeomorphic to the space of extreme states of Ξ(M), where the latter is extremally disconnected by <cit.>, the space M is extremally disconnected, too. Since N is metrisable, the Banach space C_( N) is separable. Then Grothendieck's theorem <cit.> says that any bounded linear operator C_( M)→ C_( N) is weakly compact. In particular, the positive operator L C_( M)→ C_( N) extending p (Theorem <ref>)  is weakly compact. Thus there exists a regular vector measure μ⃗ (<cit.>) such that 

    L(a) = ∫_ M a dμ⃗,      a∈ C_( M).

Since p(a)^*=L(a^*) for all a∈ M, this completes the proof. 





§ DUAL MAPS

An alternative representation of probability maps is studied in this section. A possible motivation can be Example <ref>, where a unique stochastic matrix 𝐒 is associated with a probability map p∈([0,1]^n,[0,1]^n), such that the identity (<ref>) holds. The matrix is of the form 
    𝐒[ 𝐬_1;   ⋮; 𝐬_n ]

with each 𝐬_i (s_i1,…,s_in)∈ [0,1]^n satisfying ∑_j=1^ns_ij=1. From here n states s_1,…,s_n∈ [0,1]^n can be defined:

    s_i(a) ∑_j=1^n s_ij· a_j,      a (a_1,…,a_j)∈ [0,1]^n,   i=1,…,n.

 By (<ref>) the relation between p and (s_1,…,s_n) is

    p(a)_i = s_i(a),     for all a∈ [0,1]^n and i=1,…,n.

Since [0,1]^n can be identified with {1,…,n}, one can think of (s_1,…,s_n) as an n-tuple of states indexed by maximal ideals of [0,1]^n. The above construction can be carried out for any probability map. We spell out the details in the rest of this sectiom.

Let M and N be arbitrary MV-algebras and p∈(M,N). For any 𝔫∈ N, define a function p_𝔫 M→ [0,1] as

    p_𝔫(a)  p(a)^*(𝔫),      a∈ M.

Since p is a probability map and ^* N→ C( N) is an MV-homomorphism (Theorem <ref>), the function  p_𝔫 is a state of M.
Let p∈(M,N). The dual of p is the mapping 
    p' N → M
 
such that p'() p_, where p_ is as in (<ref>). Put

    '(M,N) {p'|p' is the dual of some p∈(M,N)}.



In this lemma basic properties of dual maps are collected. 

Let M and N be MV-algebras. The following hold true.


  * p' is continuous for every  p∈(M,N).

  * If N is semisimple, then p↦ p' is a bijection.




(1) Let (_γ) be a net in N converging to some ∈ N. We need to show that the limit of a net of states (p'(𝔫_γ)) in M is p'(). Let a∈ M. Then, by (<ref>), (<ref>), and continuity of p(a)^*∈ N^*, 

    p'(𝔫_γ)(a)=p_𝔫_γ(a)=p(a)^*(𝔫_γ) →  p(a)^*(𝔫) = p_𝔫(a)=p'()(a).


(2) The assertion is trivially true when the cardinality of (M,N) is strictly smaller than two. Let p,q∈(M,N) with p≠ q. Then there is a∈ M having the property p(a)≠ q(a). Since N is semisimple, the map ^* from N onto N^* is injective (Theorem <ref>), which implies that there is some ∈ N satisfying p(a)^*() ≠ q(a)^*(). By (<ref>) this inequality reads as
p'_𝔫(a)≠ q'_𝔫(a). Hence, p'≠ q'. 


The set of duals of probability maps can  easily be described in some special cases, which depend on the structure of N.


 §.§ Codomain N=C(Y)

Let M be an arbitrary semisimple MV-algebra and N C(Y) for a compact Hausdorff space Y. Since C(Y) is homeomorphic to Y (see <cit.>), we will tacitly identify the elements of Y with maximal ideals in  C(Y). Let f Y→ M be any continuous map. Then one can routinely check that the map p M→ C(Y) defined by p(a)(y) f(y)(a), for all a∈ M and all y∈ Y, is a probability map and its dual is p'=f.   Thus, by Lemma <ref>, the dual of (M,C(Y)) is

    '(M,C(Y)) = {f |f Y → M is continuous}.

The set '(M,C(Y)) is clearly a nonempty convex subset of the real linear space ( M)^Y. In particular, '(M,C(Y)) is affinely isomorphic to the convex set (M,C(Y)) under the bijection p↦ p', which is a direct consequence of Lemma <ref> and the definition of p'.



Let M be a semisimple MV-algebra and Y be a compact Hausdorff space. Then (M,C(Y))≠∅ and the following assertions are equivalent for every p∈(M,C(Y)).


  * p∈(M,C(Y)).

  * p'∈'(M,C(Y)).

  * p is an MV-homomorphism.

  * p'[Y]⊆ M.

  * There is a unique continuous map f Y → M satisfying the equality p(a)=a^*∘ f, for all a∈ M.



By Corollary <ref>, (M,C(Y))≠∅.
 Since the map p↦ p' is an affine isomorphism of (M,C(Y)) onto '(M,C(Y)), claims (1) and (2) are equivalent. The equivalence of (1) and (3) follows from Corollary <ref>.  We prove that (3) implies (4). Let p M→ C(Y) be an MV-homomorphism. By (<ref>) we need only show that p'(y) is an MV-homomorphism M→ [0,1], for all y∈ Y. But this is a straightforward consequence of the assumption and the definition of p'. Conversely, let (4) be true and we want to conclude that p is an MV-homomorphism. For all a,b∈ M and every y ∈ Y, the definition of p' yields

    p(a⊕ b)(y)=p'_y(a⊕ b) =p'_y(a) ⊕ p'_y(b) = p(a)(y) ⊕ p(b)(y).

Since p is a probability map and satisfies p(a⊕ b)=p(a)⊕ p(b), it is an MV-homomorphism. The equivalence of (4) and (5) follows from (<ref>) and the fact that M is homeomorphic to M. 




 §.§ Codomain N= Affine representation of M

Assume that M is a semisimple MV-algebra and N M_Aff, where M_Aff is as in Example <ref>. Since M_Aff is a convex subset of ^ M, the set (M,M_Aff) is convex as well. As M_Aff is a separating subalgebra of C( M), the space M_Aff is homeomorphic to M, so we may consider dual maps as mappings M → M. For any p∈(M,M_Aff), it is easy to see that the dual map p' M → M is  continuous (Lemma <ref>)  and affine. Conversely, let a map f M → M be affine continuous. Then the map p_f M→ M_Aff defined by p_f(a)=â∘ f, for all a∈ M, is a probability map satisfying p_f'=f. We can conclude

    '(M,M_Aff) = {f |f M → M is affine continuous},

and the convex sets (M,M_Aff) and '(M,M_Aff) are affinely isomorphic.


The reader familiar with the construction of state space functor in <cit.> might point out an alternative notion of dual map. Namely consider a probability map p M→ N and a map p^⋆ N→ M defined by precomposing with p. Specifically,

    p^⋆(s)  s∘ p,   for all s∈ N.

Since N and N are homeomorphic under ↦ s_, where s_ is as in (<ref>), we get

    p^⋆(s_)(a) = s_(p(a)) =p(a)^*()=p'_(a),

for all a∈ M. Hence, p^⋆(s_)=p'(), for any ∈ N. Since p' is a continuous function that coincides with p^⋆ over N and p^⋆ is clearly an affine continuous functions, then Lemma <ref> says that p^⋆ is the unique extension of p' over the entire state space N. Therefore the map p', which is defined on a less complicated domain than p^⋆, contains the same amount of information about p as p^⋆. 







§ FURTHER RESEARCH

It is clear from Section <ref> that one cannot obtain a general integral representation theorem for any probability map M→ N, even though both M and  N are relatively well-behaved. However, it makes sense to ask whether the conclusion of Theorem <ref> can be extended beyond the assumptions on M and N stated in that theorem. Specifically, find the conditions on a probability map p M→ N which guarantee that p extends to a weakly compact positive operator.


The material presented in Section <ref> leads to the following question. Given MV-algebras M and N, can we characterise duals of probability maps M→ N as continuous functions N→ M satisfying special properties? It is known that this problem has a solution only in special cases, such as when M, N are finitely presented algebras and probability maps are homomorphisms M→ N. Indeed, it follows from   <cit.> and Theorem <ref> that the duals of such homomorphisms are precisely continuous piecewise linear maps N→ M with integer coefficients. Therefore, one might at least ask if  the duals of probability maps M→ N between two finitely presented MV-algebras can be characterised as a special class of continuous functions N→ M.










§ COMPACT CONVEX SETS

The book <cit.> is a classical reference concerning compact convex sets in infinite-dimensional spaces and integral representation theorems. Since the importance of this theory for classifying state spaces of partially ordered Abelian groups cannot be overestimated, the interested reader might also want to check <cit.>. See <cit.> for all the unexplained terminology and results related to functional analysis and basic convexity.

In this section we assume that K and L are real locally convex Hausdorff spaces. Let A⊆ K and B⊆ L be convex sets. A map f A→ B is affine when f(α x + (1-α)y)=α f(x) + (1-α)f(y), for every x,y ∈ A and all α∈ [0,1]. If f is affine and bijective, then f^-1 is affine as well, and we say that f is an affine isomorphism. Assume that the convex sets A and B are in addition compact. Then f A→ B is an affine homeomorphism if f is an affine isomorphism and a homeomorphism.

If A is a nonempty compact subset of K, then it has at least one extreme point. The set A of all extreme points of A is called an extreme boundary. Every compact convex set is generated by its extreme boundary in the following sense.


Let A⊆ K be a nonempty compact convex set. Then A is the closed convex hull of its extreme boundary,

    A =  A.



[Probability measures]
Let X be a compact Hausdorff topological space. By Δ(X) we denote the convex set of all regular Borel probability measures on X.  The set Δ(X) becomes a compact subset of the locally convex space of all regular Borel probability measures on X, which is endowed with the weak^*-topology. In this topology, a net (μ_γ) in Δ(X) converges to μ∈Δ(X)  if and only if lim_γ∫_X f dμ_γ=∫_X f dμ, for every continuous function f X→. Given x∈ X, let δ_x be the corresponding Dirac measure. Then

    Δ(X) = {δ_x| x∈ X}

and the map x∈ X ↦δ_x∈Δ(X) is in fact a homeomorphism. By Krein-Milman theorem the set of all convex combinations of Dirac measures is dense in Δ(X).


There are several caveats that concern usability of Krein-Milman theorem. In practical situations one is mostly interested in a workable description of the extreme boundary A and its measure-topological properties (compactness or measurability). Moreover, easy examples from a finite-dimensional space show that even if A is a convex polytope (so necessarily A is finite and A= A by Minkowski theorem), there can be many convex combinations representing given x∈ A. The compact convex sets, whose properties are on the contrary very convenient, are particular infinite-dimensional simplices called Bauer simplices. We will spell out the definition of Bauer simplex using one of their equivalent characterisations from <cit.>. To this end, we define (A) to be the linear space of all real-valued affine continuous functions on a convex set A⊆ K.

Let ∅≠ A⊆ K be compact convex. We say that x∈ A is the barycenter of μ∈Δ(A) if this condition holds true:

    f(x) = ∫_A f dμ,     for all f∈(A).

Clearly, every point x∈ A is the barycenter of at least one measure μ∈Δ(A) since it suffices to take μδ_x. Conversely, it is not difficult to show that, for every μ∈Δ(A), there is precisely one point x∈ A that is the barycenter of μ. The barycenter mapping is  
    r_AΔ(A) → A
 assigning to every μ∈Δ(A) its unique barycenter r_A(μ)∈ A. The barycenter mapping is surjective (since x=r_A(δ_x) for any x∈ A), affine, and continuous.



Let A be a compact convex subset of K. Then A is called a Bauer simplex if, for every x∈ A, there is a unique μ∈Δ(A) supported by A, and such that x is the barycenter of μ, that is, r_A(μ)=x.

The extreme boundary of a Bauer simplex is necessarily closed, A = A. Any finite-dimensional simplex is a Bauer simplex and so is the set of probability measures Δ(X) from Example <ref>. In the paper we make use of the  following fact, an abstract variant of the Dirichlet problem of the extreme boundary. Although the statement is implicit in many textbooks, we provide its standalone proof.



Let K and L be real locally convex Hausdorff spaces, A⊆ K be a Bauer simplex, and B⊆ L be compact convex. If f A → B is continuous, then there is a unique affine continuous function f̅ A → B such that f̅(x)=f(x), for all x∈ A.


By <cit.>, every Bauer simplex A is affinely homeomorphic to Δ( A) via   the barycenter mapping r_AΔ( A) → A. Let μ_x r^-1_A(x) be the unique representing probability measure for x∈ A. Let f A → B be a continuous map. Define f_♯μ_xμ_x∘ f^-1 and notice that f_♯μ_x∈Δ(B). Hence, it is sensible to define

    f̅(x)  r_B(f_♯μ_x),      x∈ A,

where r_BΔ(B)→ B is the barycenter mapping on B. We will check that f̅ is the desired extension of f.

If x∈ A, then μ_x=δ_x∈Δ( A), and

    f̅(x)=r_B(f_♯δ_x)=r_B(δ_f(x))=f(x).


The mapping f̅ is affine. To show this, let x,y∈ A, α∈ [0,1], and put zα x + (1-α)y. Then, by affinity of barycenter maps, we get

    f̅(z)    =r_B(f_♯μ_z)=r_B(f_♯(αμ_x + (1-α)μ_y) ) 
       =α r_B(f_♯μ_x) + (1-α)r_B(f_♯μ_y)=αf̅(x) + (1-α)f̅(y).


To prove continuity of f̅, consider a net (x_γ) converging to some x in A. Since r_A is a homeomorphism, this gives μ_x_γ→μ_x in Δ( A). Hence, f_♯μ_x_γ→ f_♯μ_x in Δ(B). By continuity of r_B this implies 

    f̅(x_γ)=r_B(f_♯μ_x_γ) → r_B( f_♯μ_x)=f̅(x).


Let g A→ B be an affine continuous map such that g(x)=f(x), for all x∈ A. We want to conclude that g(x)=f̅(x), for every x∈ A. Proceeding by cases, first assume x∈ A. By the assumption, g(x)=f(x)=f̅(x). Now, let x∑_i=1^nα_i x_i, for some positive integer n, where x_i∈ A and α_i≥ 0 with ∑_i=1^nα_i=1. By affinity of f and g and by the assumption,

    f̅(x)=∑_i=1^nα_if̅(x_i)=∑_i=1^nα_i g(x_i)=g(x).

 Krein-Milman theorem says that the only remaining case is when x∈ A is the limit of a net (x_γ) in A, where each x_γ is a finite convex combination of extreme points. Then, by continuity of f̅ and g together with what we have proved above,

    f̅(x)=lim_γf̅(x_γ)=lim_γ g(x_γ)=g(x).

This finishes the proof. 







 §.§ Acknowledgment


 The author is grateful to Prof. Vincenzo Marra (University of Milan) for many suggestions and inspiring comments.




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