From Kleisli Categories to Commutative C*-algebras: Probabilistic Gelfand Duality

C*-algebras form rather general and rich mathematical structures that can be studied with different morphisms (preserving multiplication, or not), and with different properties (commutative, or not). These various options can be used to incorporate various styles of computation (set-theoretic, probabilistic, quantum) inside categories of C*-algebras. At first, this paper concentrates on the commutative case and shows that there are functors from several Kleisli categories, of monads that are relevant to model probabilistic computations, to categories of C*-algebras. This yields a new probabilistic version of Gelfand duality, involving the"Radon"monad on the category of compact Hausdorff spaces. We then show that the state space functor from C*-algebras to Eilenberg-Moore algebras of the Radon monad is full and faithful. This allows us to obtain an appropriately commuting state-and-effect triangle for C*-algebras.


INTRODUCTION
There are several notions of computation.We have the classical notion of computation, probabilistic computation, where a computer may make random choices, and quantum computation, which uses quantum mechanical interference and measurement.Normally we would consider classical computation to be done on sets, probabilistic computation on spaces with a measure, and quantum computation on Hilbert spaces.We can instead use categories with C * -algebras as objects and a choice of either *-homomorphisms (called MIU-map below) or positive unital maps as the morphisms.The general outline is represented in this table.While the quantum case is an important source of motivation, we will be concerned with the classical and probabilistic cases in this article.In particular, we will relate the alternative method of representing probabilistic computation, using monads, to the C * -algebraic approach.
In recent years the methods and tools of category theory have been applied to Hilbert spaces -see e.g.[1] and the references there -and also to C * -algebras, see for instance [29,26].In this paper we show that clearly distinguishing different types of homomorphisms of C * -algebras already brings quite some clarity.Moreover, we demonstrate the relevance of monads (and their Kleisli and Eilenberg-Moore categories) in this field.The aforementioned paper [29] concerns itself with only the *-homomorphisms (i.e. with the MIU-maps in our terminology).
Giry [12, I.4] described how we can consider a stochastic process as being a diagram in the Kleisli category of the Giry monad on measure spaces.By using the Radon monad on compact spaces instead, we can get a different category of stochastic processes on compact spaces as diagrams in the (opposite of the) category of commutative C * -algebras with PU-maps.This allows the quantum generalization to taking diagrams in the category of non-commutative C * -algebras.The relationship to quantum computation is that B(H), the algebra of all bounded operators on a Hilbert space is a C * -algebra, and for every C * -algebra A, there is a Hilbert space H such that A is isomorphic to a norm-closed *-subalgebra of B(H).Unitary maps U : H → H define MIU maps a → U * aU : B(H) → B(H).The category of C * -algebras allows us to represent measurement with maps from a commutative C * -algebra to B(H).We can also represent composite systems that are partly quantum and partly classical.Girard also used certain special C * -algebras, von Neumann algebras, for his Geometry of Interaction [11].

PRELIMINARIES ON C * -ALGEBRAS
We write Vect = Vect C for the category of vector spaces over the complex numbers C.This category has direct product V ⊕ W , forming a biproduct (both a product and a coproduct) and tensors V ⊗ W , which distribute over ⊕.The tensor unit is the space C of complex numbers.The unit for ⊕ is the singleton (null) space 0. We write V for the vector space with the same vectors/elements as V , but with conjugate scalar product: z • V v = z • V v.This makes Vect an involutive category, see [16].
A *-algebra is an involutive monoid A in the category Vect.Thus, A is itself a vector space, carries a multiplication • : A ⊗ A → A, linear in each argument, and has a unit 1 ∈ A. Moreover, there is an involution map (−) * : A → A, preserving 0 and + and satisfying: Here we have written a fat dot • for scalar multiplication, to distinguish it from the algebra's multiplication •.For z = a + bi ∈ C we have the conjugate z = a − bi.Often we omit the multiplication dot • and simply write xy for x • y.Similarly, the scalar multiplication • is often omitted.We then rely on the context to distinguish the two multiplications.
A C * -algebra is a *-algebra A with a norm − : A → R ≥0 in which it is complete, satisfying the conditions x = 0 iff x = 0 and: The last equation x * • x = x 2 , is the C * identity and distinguishes C * -algebras from Banach *-algebras.In the current setting, each C * -algebra is unital, i.e. has a (multiplicative) unit 1.A C * -algebra is called commutative if its multiplication is commutative, and finite-dimensional is it has finite dimension when considered as a vector space.An element x in a C * -algebra A is called positive if it can be written in the form x = y * • y.We write A + ⊆ A for the subset of positive elements in A. This subset is a cone, which is to say it is closed under addition and scalar multiplication with positive real numbers.The multiplication x • y of two positive elements need not be positive in general (think of matrices).The square x 2 = x • x of a self-adjoint element x = x * , however, is obviously positive.In a commutative C * -algebra the positive elements are closed under multiplication.A cone A + in a vector space defines a partial order as follows. x This is defines an order on every C * -algebra.
There are mainly two options when it comes to maps between C * -algebras.The difference between them plays an important role in this paper.Definition 2.1.We define two categories Cstar MIU and Cstar PU with C * -algebras as objects, but with different morphisms.
(1) A morphism f : A → B in Cstar MIU is a linear map preserving multiplication (M), involution (I), and unit (U).Explicitly, this means for all x, y ∈ A, Often such "MIU" maps are called *-homomorphisms.(2) A morphism f : A → B in Cstar PU is a linear map that preserves positive elements and the unit.This means that f restricts to a function A + → B + .Alternatively, for each x ∈ A there is an y ∈ B with f (x * x) = y * y.
For both X = MIU and X = PU there are obvious full subcategories of commutative and/or finite-dimensional C * -algebras, as described in: Clearly, each "MIU" map is also a "PU" map, so that we have inclusions Cstar MIU ֒→ Cstar PU , also for the various subcategories.A map that preserves positive elements is called positive itself; and a unit preserving map is called unital.
For a category B one often writes B(X, Y ) or Hom(X, Y ) for the "homset" of morphisms X → Y in B. For C * -algebras A, B we write Hom MIU (A, B) = Cstar MIU (A, B) and Hom PU (A, B) = Cstar PU (A, B) for the homsets of MIU-and PU-maps.For the special case where B is the algebra C of complex numbers we define sets of "states" and of "multiplicative states" as: There is also the commonly used notion of completely positive maps, which is a stronger condition than positivity but weaker than being MIU.These maps are important when defining the tensor of C * -algebras as a functor, as the tensor of positive maps need not be positive.They are also widely considered to represent the physically realizable transformations.Positive, but non-completely positive maps of C * -algebras also have their uses, as entanglement witnesses for example [14, theorem 2].Since we mainly consider the commutative case, where positive and completely positive coincide, we do not consider the category of C * -algebras with completely positive maps any further in this paper.However, since a completely positive unital map is what is known as a channel in quantum information, then theorem 5.1 shows that every channel in Mislove's sense [27] is a channel in this sense.
Moreover, a PU-map Proof.An element x is called self-adjoint if x * = x.Each self-adjoint x can be written uniquely as a difference x = x p − x n of positive elements x p , x n , with x p x n = x n x p = 0 and x p , x n ≤ x , see [21,Proposition 4.2.3 (iii)]; as a result f (x * ) = f (x) = f (x) * , for a PU-map f .Next, an arbitrary element y can be written uniquely as y = y r + iy i for self-adjoint elements y r = 1 2 (y + y * ), y i = 1 2i (y − y * ), so that y r , y i ≤ y .Then f (y * ) = f (y) * .Preservation of the order is trivial.
For positive x we have x ≤ x • 1, and thus f (x) ≤ x • 1, which gives f (x) ≤ x .An arbitrary element x can be written as linear combination of four positive elements x i , as in In fact, it can be shown that a PU-map satisfies f (x) ≤ x , see [31, corollary 1].But this sharpening is not needed here.
We next recall two famous adjunctions involving compact Hausdorff spaces.The first one is due to Manes [25] and describes compact Hausdorff spaces as monadic over Sets, via the ultrafilter monad.The second one is known as Gelfand duality, relating compact Hausdorff spaces and commutative C * -algebras.Notice that this result involves the "MIU" maps.

MStat
On the left the functor U sends a set X to the ultrafilters on the powerset P(X).And on the right the equivalence of categories is given by sending a compact Hausdorff space X to the commutative C * -algebra C(X) = Cont(X, C) of continuous functions X → C. The "weak-* topology" on states will be discussed below.
The multiplicative states on a commutative C * -algebra can equivalently be described as maximal ideals, or also as so-called pure states (see below).
Corollary 2.4.For each finite-dimensional commutative C * -algebra A there is an n ∈ N with A ∼ = C n in FdCCstar MIU .
Proof.By the previous theorem there is a compact Hausdorff space X such that A is MIU-isomorphic to the algebra of continuous maps X → C.This X must be finite, and since a finite Hausdorff space is discrete, all maps X → C are continuous.Let n ∈ N be the number of elements in X; then we have an isomorphism A ∼ = C n .
As we can already see in the above theorem, it is the opposite of a category of C * -algebras that provides the most natural setting for computations.This is in line with what is often called the Heisenberg picture.In a logical setting it corresponds to computation of weakest preconditions, going backwards.The situation may be compared to the category of complete Heyting algebras, which is most usefully known in opposite form, as the category of locales, see [20].
The set of states Stat(A) = Hom PU (A, C) can be equipped with the weak-* topology, which is the coarsest (smallest) topology in which all evaluation maps ev x = λs.s(x) : Hom PU (A, C) → C, for x ∈ A, are continuous.We introduce the category CCLcvx, which first appeared in [36], in order to extend Stat to a functor.
The category CCLcvx has as its objects compact convex subsets of (Hausdorff) locally convex vector spaces.More accurately, the objects are pairs (V, X) where V is a (Hausdorff) locally convex space, and X is a compact convex subset of V .The maps (V, X) → (W, Y ) are continuous, affine maps X → Y .Note that if (V, X) and (W, Y ) are isomorphic, while X is necessarily homeomorphic to Y , V need not bear any particular relation to W at all.We can see CCLcvx forms a category, as identity maps are affine and continuous and both of these attributes of a map are preserved under composition.We remark at this point that we have a forgetful functor U : CCLcvx → CH, taking the underlying compact Hausdorff space of X.We recall that a function (between convex sets) is called affine if it preserves convex sums.We will see shortly that such affine maps are homomorphisms of Eilenberg-Moore algebras for the distribution monad D.
Proof.For each finite collection h i ∈ Hom PU (A, C) with r i ∈ [0, 1] satisfying i r i = 1, the function h = i r i h i is again a state.Moreover, such convex sums are preserved by precomposition, making the maps (−) • f affine.
The fact that dual space of A, given the weak-* topology is a locally convex space is standard, and only uses that A is a Banach space.This implies that the space of states is Hausdorff.The space of states is closed since any net of states that converges in the weak-* topology converges to a state.The space of states is also bounded as each state has norm 1. Therefore the state space is a closed and bounded and hence compact by the Banach-Alaoglu Theorem.
Precomposition with the identity map gives the same state again, so Stat preserves identity maps.Since composition of PU-maps is associative, Stat preserves composition, and hence is a functor.
2.1.Effect modules.Effect algebras have been introduced in mathematical physics [9], in the investigation of quantum probability, see [8] for an overview.An effect algebra is a partial commutative monoid (M, 0, ) with an orthocomplement (−) ⊥ .One writes x ⊥ y if x y is defined.The formulation of the commutativity and associativity requirements is a bit involved, but essentially straightforward.The orthocomplement satisfies x ⊥⊥ = x and x x ⊥ = 1, where 1 = 0 ⊥ .There is always a partial order, given by x ≤ y iff x z = y, for some z.The main example is the unit interval [0, 1] ⊆ R, where addition + is obviously partial, commutative, associative, and has 0 as unit; moreover, the orthocomplement is r ⊥ = 1 − r.We write EA for the category of effect algebras, with morphism preserving and 1 -and thus all other structure.
For each set X, the set [0, 1] X of fuzzy predicates on X is an effect algebra, via pointwise operations.Each Boolean algebra B is an effect algebra with x ⊥ y iff x∧y = ⊥; then x y = x∨y.In a quantum setting, the main example is the set of effects Ef (H) = {E : H → H | 0 ≤ E ≤ I} on a Hilbert space H, see e.g.[8,13].
An effect module is an "effect" version of a vector space.It involves an effect algebra M with a scalar multiplication s • x ∈ M , where s ∈ [0, 1] and x ∈ M .This scalar multiplication is required to be a suitable homomorphism in each variable separately.The algebras [0, 1] X and Ef (H) are clearly such effect modules.Maps in EMod are EA maps that are additionally required to commute with scalar multiplication.
For a C * -algebra A the subset A + ֒→ A of positive elements carries a partial order ≤ defined on self-adjoint elements in (2.1).We write [0, 1] A ⊆ A + ⊆ A for the subset of positive elements below the unit.The elements in [0, 1] A will be called effects (or sometimes also: predicates).For instance, for the C * -algebra B(H) of bounded operators on a Hilbert space H the unit interval We claim that [0, 1] A is an effect algebra and carries a [0, 1] ⊆ R scalar multiplication, thus making it an effect module.
• Since A with 0, + is a partially ordered Abelian group, [0, 1] A is a so-called interval effect algebra, with x ⊥ y iff x + y ≤ 1, and in that case x y = x + y.The orthocomplement x ⊥ is given by 1 − x. • For r ∈ [0, 1] and x ∈ [0, 1] A the scalar multiplications rx and (1 − r)x are positive, and their sum is This restriction is a map of effect modules.Hence we get a "predicate" functor Cstar PU → EMod.Lemma 2.6.The functor [0, 1] (−) : Cstar PU → EMod is full and faithful.
Proof.Any PU-map f : A → B is completely determined (and defined by) its action on [0, 1] A : for a non-zero positive element x ∈ A we use x ≤ x 1 and thus ).An arbitrary element y ∈ A can be written uniquely as linear sum of four positive elements (see Lemma 2.2), determining f (y).
The (finite, discrete probability) distribution monad D : Sets → Sets sends a set X to the set Such an element ϕ ∈ D(X) may be identified with a finite, formal convex sum i r i x i with x i ∈ X and r i ∈ [0, 1] satisfying i r i = 1.The unit η : X → D(X) and multiplication µ : D 2 (X) → D(X) of this monad are given by singleton/Dirac convex sum and by matrix multiplication: A convex set is an Eilenberg-Moore algebra of this monad: it consists of a carrier set X in which actual sums i r i x i ∈ X exist for all convex combinations.We write Conv = EM(D) for the category of convex sets, with "affine" functions preserving convex sums.Effect modules and convex sets are related via a basic adjunction [19], obtained by "homming into [0, 1]", as in: algebra, with pointwise addition, multiplication and involution, and with the uniform/supremum norm: In fact it is a typical example of a commutative W * -algebra, but we do not require this fact.This yields a functor ℓ ∞ : Sets → (CCstar MIU ) op , where for h : ; it preserves the (pointwise) operations.We have the following result.
By composition and uniqueness of adjoints we get: When we restrict to the full subcategory FinSets ֒→ Sets of finite sets we obtain a functor ℓ ∞ = C (−) : FinSets → (FdCCstar MIU ) op .The next result is then a well-known special case of Gelfand duality (Theorem 2.3).We elaborate the proof in some detail because it is important to see where the preservation of multiplication plays a role.Proposition 3.2.The functor C (−) : FinSets → (FdCCstar MIU ) op is an equivalence of categories.
Proof.It is easy to see that the functor C (−) is faithful.The crucial part is to see that it is full.So assume we have two finite sets, seen as natural numbers n, m, and a MIU-homomorphism h : C m → C n .For j ∈ m, let |j ∈ C m be the standard base vector with 1 at the j-th position and 0 elsewhere.Since this |j is positive, so is h(|j ), and thus we may write it as h(|j ) = (r 1j , . . ., r nj ), with r ij ∈ R ≥0 .Because |j •|j = |j , and h preserves multiplication, we get h(|j and thus r 2 ij = r ij .This means r ij ∈ {0, 1}, so that h is a (binary) Boolean matrix.But h is also unital, and so: For each i ∈ n there is thus precisely one j ∈ m with r ij = 1 -so that h is a "functional" Boolean matrix.This yields the required function f : n → m with C f = h.Corollary 2.4 says that the functor C (−) : FinSets → (FdCCstar MIU ) op is essentially surjective on objects, and thus an equivalence.This proof demonstrates that preservation of multiplication, as required for "MIU" maps, is a rather strong condition.We make this more explicit.

DISCRETE PROBABILISTIC COMPUTATIONS IN C * -ALGEBRAS
We turn to probabilistic computations and will see that we remain in the world of commutative C * -algebras, but with PU-maps (positive unital) instead of MIU-maps.Recall that the set of states Stat(A) of a C * -algebra A contains the PU-maps A → C. Lemma 4.1.Sending a set X to the set of states of the C * -algebra ℓ ∞ (X) yields the (underlying functor of the) expectation monad E from [18]: the mapping X → Stat(ℓ ∞ (X)) is isomorphic to the expectation monad E : Sets → Sets, defined in [18] via effect module homomorphisms: Proof.The predicate/effect functor [0, 1] (−) : Cstar PU → EMod is full and faithful by Lemma 2.6, and so: The isomorphism α : Hom PU (C n , C) −→ D(n) follows because the expectation and distribution monad coincide on finite sets, see [18].Explicitly, it is given by α The unit η and multiplication µ structure on E(X) ∼ = Hom PU (ℓ ∞ (X), C) is very much like for "continuation" or "double dual" monads, see [23,28,15], with: For an arbitrary monad T = (T, η, µ) on a category B we write Kℓ(T ) for the Kleisli category of T .Its objects are the same as those of B, but its maps X → Y are the maps X → T (Y ) in B. The unit η : X → T (X) is the identity map X → X in Kℓ(T ); and composition of f : X → Y and g : Y → Z in Kℓ(T ) is given by g ⊙ f = µ • T (g) • f .Maps in such a Kleisli category are understood as computations with outcomes of type T , see [28].For a monad T : Sets → Sets we write Kℓ N (T ) ֒→ Kℓ(T ) for the full subcategory with numbers n ∈ N as objects, considered as n-element sets.Proposition 4.2.The expectation monad E(X) ∼ = Hom PU (ℓ ∞ (X), C) gives rise to a full and faithful functor: Proof.First we need to see that algebras via the pointwise definitions of the relevant constructions.
We check that C E preserves (Kleisli) identities and composition: Further, C E is obviously faithful, and it is full since for h : We turn to the finite case, like in the previous section.We do so by considering the Kleisli category Kℓ N (E) obtained by restricting to objects n ∈ N. Since the expectation monad E and the distribution monad D coincide on finite sets, we have Kℓ N (E) ∼ = Kℓ N (D).Maps n → m in this category are probabilistic transition matrices n → D(m).The following equivalence is known, see e.g.[24], although possibly not in this categorical form.
It is given by This equivalence (4.2) may be read as: the category FdCCstar PU of finite-dimensional commutative C * -algebras, with positive unital maps, is the Lawvere theory of the distribution monad D.
Proof.Fullness and faithfulness of the functor C D follow from Proposition 4.2, using the isomorphism Hom PU (C n , C) ∼ = D(n) from Lemma 4.1.This functor C D is essentially surjective on objects by Corollary 2.4, using the fact that a MIU-map is a PU-map.

CONTINUOUS PROBABILISTIC COMPUTATIONS
The question arises if the full and faithful functor Kℓ(E) → (CCstar PU ) op from Proposition 4.2 can be turned into an equivalence of categories, but not just for the finite case like in Proposition 4.3.In order to make this work we have to lift the expectation monad E on Sets to the category CH of compact Hausdorff spaces.As lifting we use what we call the Radon monad R, defined on X ∈ CH as: where, as usual, C(X) = {f : X → C | f is continuous}; notice that the functions f ∈ C(X) are automatically bounded, since X is compact.We have implicitly applied the forgetful functor from CCLcvx → CH to make R into an endofunctor of CH.The elements of R(X) are related to measures in the following way.If µ is a probability measure on the Borel sets of X, integration of continuous functions with respect to µ gives X -dµ ∈ R(X).A Radon probability measure, or an inner regular probability measure, is one such that µ(S) = sup K⊆S µ(K) where K ranges over compact sets.The map from measures to elements of R(X) is a bijection [30, Thm.2.14], and accordingly we shall sometimes refer to elements of R(X) as measures.Therefore the Radon monad can be considered as a variant of the Giry monad.It differs in that it uses the topology of a space, and that in the case of a non-Polish space there can be non-Radon measures [10, 434K (d), page 192].This Radon monad R is not new: we shall see later that it occurs in [36,Theorem 3] as the monad of an adjunction ("probability measure" is used to mean "Radon probability measure" in that article).It has been used more recently in [27].However, our duality theorem below is not known in the literature.
From Proposition 2.5 it is immediate that R(X) is again a compact Hausdorff space.The unit η : X → R(X) and multiplication µ : R 2 (X) → R(X) are defined as for the expectation monad, namely as η(x)(v) = v(x) and µ(g)(v) = g λh.h(v) .We check that η is continuous.Recall from the proof of Proposition 2.5 that a basic open in R(X) is of the form ev −1 We are now ready to state our main, new duality result.It may be understood as a probabilistic version of Gelfand duality, for commutative C * -algebras with PU maps instead of the MIU maps originally used (see Theorem 2.3).
Theorem 5.1.The Radon monad (5.1) yields an equivalence of categories: Proof.We define a functor C R : Kℓ(R) → (CCstar PU ) op like in (4.1), namely by: The fact that C R is a full and faithful functor follows as in the proof of Proposition 4.2.This functor is essentially surjective on objects by ordinary Gelfand duality (Theorem 2.3).
We investigate the Radon monad R a bit further, in particular its relation to the distribution monad D on Sets.Lemma 5.2.There is a map of monads (U, τ ) : R → D in: where U is the forgetful functor and τ commutes appropriately with the units and multiplications of the monads D and R. (Such a map is called a "monad functor" in [35, §1].)As a result the forgetful functor lifts to the associated categories of Eilenberg-Moore algebras: Hence the carrier of an R-algebra is a convex compact Hausdorff space, and every algebra map is an affine function.
Proof.For X ∈ CH and ϕ ∈ D(U X), that is for ϕ : U X → [0, 1] with finite support and It is easy to see that τ is a linear map C(X) → C that preserves positive elements and the unit.Moreover, it commutes appropriately with the units and multiplications.For instance: The continuous dual space of C(X) can be ordered using (2.1), by taking the positive cone to be those linear functionals that map positive functions to positive numbers.Definition 5.3.A state φ ∈ R(X) = Hom PU (C(X), C) is a pure state if for for each positive linear functional such that ψ ≤ φ, i.e. such that φ − ψ is positive, there exists an α ∈ [0, 1] such that ψ = αφ.Lemma 5.4.For a compact Hausdorff space X, the subset of unit (or Dirac) measures {η(x) | x ∈ X} ⊆ R(X) is the set of extreme points of the set of Radon measures R(X) -where η(x) = η R (x) = ev x = λh.h(x) is the unit of the monad R.
Proof.We rely on the basic fact, see [7, 2.5.2, page 43], that Dirac measures η(x) ∈ R(X) are "pure" states.We prove the above lemma by showing that the pure states are precisely the extreme points of the convex set R(X). • , where no two elements of {φ, φ 1 , φ 2 } are the same.Then φ ≥ α 1 φ 1 , since for a positive function f ∈ C(X) one has Hence φ is an extreme point.
(1) The maps τ X : D(U X) → U R(X) from (5.2) are injective; as a result, the unit/Dirac maps η : X → R(X) are also injective.(2) The maps τ X : D(U X) U R(X) are dense.
We can conclude that the unit X → R(X) is also injective, since its underlying function can be written as the composite To show that the image of τ X is dense, we proceed as follows.By Lemmas 5.4 and 5.2, the extreme points of R(X) are and are thus in the image of τ : D(U X) U R(X).Since every convex combination of η R (x) comes from a formal convex sum ϕ ∈ D(U X), all convex combinations of extreme points are in the image of τ X .Using Proposition 2.5, R(X) can be considered an object of CCLcvx, i.e. a compact convex subset of a locally convex space.Accordingly, we may apply the Krein-Milman theorem [6, Proposition 7.4, page 142] to conclude the set of convex combinations of extreme points is dense.

is faithful, and an EM(D) map comes from an EM(R) map if and only if it is continuous.
We shall follow the convention of writing A(X, Y ) for the homset of continuous and affine functions X → Y .
Proof.Clearly, each algebra map is both continuous and affine.For the converse, if f : X → Y is continuous, it is a map in the category CH of compact Hausdorff spaces.Since it is affine, both triangles commute in: there is at most one such map.Therefore f is an algebra map.
The category EM(R) of Eilenberg-Moore algebras of the Radon monad may thus be understood as a suitable category of convex compact Hausdorff spaces, with affine continuous maps between them.In the next section, we see how to use a result from [36] to relate this to CCLcvx, which is a category of "concrete" convex sets.Using this theorem, it will be shown that "observability" conditions like in [18] always hold for algebras of R.

Świrszcz's Theorem and Noncommutative
C * -algebras.In this section we show that the Radon monad arises from an adjunction in [36] enabling us to use Świrszcz's theorem 3 from that paper to show that the categories CCLcvx and EM(R) are equivalent, which we can then apply to represent noncommutative C * -algebras.The adjunction in question has U : CCLcvx → CH as the right adjoint, and the details of the construction of the left adjoint are not given.In order to prove that R is the monad arising from this adjunction, we need to know its unit and counit, so our next task is to define the left adjoint explicitly.Of course, any other left adjoint will be naturally isomorphic.
We begin as follows.We define Ś : To show that Ś is the left adjoint to U , we use the unit and counit definition of an adjunction.We already know the unit, η X : X → U ( Ś(X)), as we gave it when defining the unit of R. To define the counit we use the notion of barycentre.
We can understand the intuitive notion of barycentre by thinking of a probability measure µ on the unit square [0, 1] 2 .If we wanted to find the centre of mass of µ, which we shall call b ∈ [0, 1] 2 , we would take for the x and y coordinates.We can see that x and y are continuous affine functions from [0, 1] 2 → R, assigning each point to its x and y coordinate respectively.Therefore we can rewrite the above as This is the idea behind the following standard definition.
Definition 5.8.If X ∈ CCLcvx and φ ∈ Ś(U (X)), then a point x ∈ X is a barycentre for φ if for all continuous affine functions f from X → R we have that φ(f ) = f (x).
The theorem that every φ has a barycentre when X is a compact subset of a locally convex space is standard and is proven in [3, proposition I.2.1 and I.2.2].
We will require the following important lemma, one of sevaral variants of the Hahn-Banach separation lemma, and some of its corollaries, which give an affine analogue of Urysohn's lemma for objects in CCLcvx.Lemma 5.9.If V is a locally convex topological vector space, X a closed convex subset and Y a compact convex subset that is disjoint from X, then there exists a continuous linear functional For proof, see either [6, theorem IV.3.9] or [33, II.4.2 corollary 1].
Corollary 5.10.Let (K, V ) ∈ Obj(CCLcvx).In the following X, Y will be arbitrary closed disjoint subsets of K, x, y arbitrary distinct points of K.
(i) There is a φ ∈ A(K, R) and an Proof.(i) Apply lemma 5.9 to obtain φ ′ : V → R separating X from Y .Since K has the subspace topology, φ = φ ′ | K is continuous, and since φ ′ is linear, φ is affine, hence φ ∈ A(K, R).We also keep the properties that φ(X) ⊆ (α, ∞) and φ(Y ) ⊆ (−∞, α).(ii) Since points are compact and convex, we can restrict the above to that case, and we have φ ∈ A(K, R) such that φ(x) > α and φ(y) < α.Therefore φ(x) = φ(y).If |φ(x)−φ(y)| = 0 then φ(x) = φ(y), so it must be false, and since the absolute value of a number is non-negative, we have that |φ(x) − φ(y)| > 0. (iii) We use (i) and obtain φ ′ ∈ A(K, R) and α ′ ∈ R. Since the image of a compact space is compact, and a compact subset of R is closed and bounded, the numbers exist, and φ ′ can be considered as an affine continuous map , otherwise we define it without dividing by anything, though this can only happen if one of X or Y is empty.The iamge of φ is contained in [0, 1], and φ is affine and continuous, being the composition of affine and continuous maps.We define again not doing the division if it is zero.We have that φ(X) ⊆ (α, ∞), and since the image of φ is contained in [0, 1], this implies φ(X) ⊆ (α, 1].The proof that φ(Y ) ⊆ [0, α) is similar.(iv) This is proven using (iii), again using the fact that points are closed, convex sets.The argument for |φ(x) − φ(y)| > 0 is the same as for (ii).
Using the properties proven above, we can start to define the counit of the adjunction.
(i) For every φ ∈ Ś(U (X)) the barycentre is unique.The function ε X : Ś(U (X)) → X mapping φ to its barycentre is well defined.(ii) This ε X is an affine map.

Proof.
(i) We show the barycentre is unique as follows.Let (V, X) be an object of CCLcvx, V being the locally convex space and X the compact convex subset.Let x, x ′ ∈ X be barycentres of φ ∈ Ś(U ).Suppose for a contradiction that x = x ′ .By corollary 5.10 (ii), there is an f ∈ A(X, R) such that f (x) = f (x ′ ).Since x and x ′ are both barycentres of φ, a contradiction.So we have x = x ′ .Therefore ε X is well-defined, at least as a function between sets.(ii) To show that ε X is affine, consider two Radon measures φ, ψ ∈ Ś(U (X)), such that ε X (φ) = x and ε X (ψ) = y, i.e. these are the barycentres.To show that ε X (αφ Given an continuous affine function f : X → R, we have Lemma 5.12.The barycentre map ε X is continuous, hence a map in CCLcvx. Proof.We now show that ε X is continuous.We use the filter-theoretic definition of continuity.Given φ ∈ Ś(U (X)), with barycentre x, we want to show that for every neighbourhood V of x, there is a neighbourhood U of φ such that ε X (U ) ⊆ V .It suffices to prove this for a chosen set of basic neighbourhoods, so we choose open neighbourhoods for X and for Ś(U (X)) we choose finite intersections of elements of the following subbasis of closed neighbourhoods: We find the neighbourhood of φ using a compactness argument.Consider the following subset of X.
,ǫ for all values of f and ǫ, we have that x is in this intersection.We will show that As we already know x is an element of the left hand side, we will show that if x ′ ∈ X and x ′ = x, then x ′ is not an element of the left hand side.So since x = x ′ , by Corollary 5.10(ii) there is an We show that x ′ ∈ ε X (U f,f (x),ǫ ) and therefore is not in (5.3) by showing there is an open set containing x ′ that is disjoint from ε X (U f,f (x),ǫ ).The open set we choose is which is open because f is continuous.Assume for a contradiction that there is some and there is some ψ ∈ U f,f (x),ǫ of which x ′′ is the barycentre, i.e. for all g ∈ A(X, R) ψ(g) = g(x ′′ ).Therefore it must be the case that ψ(f ) = f (x ′′ ), and so the inequality deriving from If we combine this with (5.5) and use the triangle inequality, we get |f (5.4).Therefore the assumption that x ′′ could exist is wrong, so x ′ is in an open set outside ε X (U f,f (x),ǫ ), and hence x ′ ∈ ε X (U f,f (x),ǫ ).This establishes that (5.3) is the case.Now consider X \ V , which is a closed set that does not contain x, since V is an open neighbourhood of x.We therefore have The right hand side is a family of closed subsets of a compact space with empty intersection.Therefore there is a finite subfamily also having empty intersection.We use the numbers i ∈ {1, . . ., n} as an index set, and take {ǫ i }, {f i } such that we have an arbitrary open neighbourhood of ε X (φ), we have that ε X is continuous at φ. Since the choice of φ was arbitrary, ε X is continuous.Proof.We must show that Ś(U (X)) Suppose that φ ∈ Ś(U (X)) and ε X (φ) = x, i.e. x is the barycentre of φ.It suffices to show that f (x) is the barycentre of Ś(U (f )(φ).Let h ∈ C(Y ), and we have by definition that We want to show that if h is affine, then Ś(U (f ))(φ)(h) = h(f (x)), as this would show f (x) is the barycentre.Since h • f is the composite of continuous, affine functions, it is also continuous and affine, and so, using that x is the barycentre of φ, we have that φ(h which is what we were required to prove.
Taken together, the preceding three lemmas define the counit.We can now move on to showing that this is actually an adjunction.
Theorem 5.14.The functor Ś : CH → CCLcvx is the left adjoint to U : CCLcvx → CH Proof.We show that the unit-counit diagrams commute.
First we must show that the following commutes: In other words, we must show that for all y ∈ U Y , y is the barycentre of η U Y (y).Using the definition of η, we have that for any affine continuous function f : because that is already true for all continuous functions f ∈ C(X).Therefore x is the barycentre of η U Y (x), and so the diagram commutes.
The second diagram we must consider is the following: This time, we need to show that φ ∈ Ś(X) is the barycentre of the measure Ś(η X )(φ).So consider an affine continuous function k : Ś(X) → R. We want to show that Ś(η X )(φ)(k) = k(φ) for all φ ∈ Ś(X).To do this, we use Lemma 5.5.We show the diagram commutes on the convex combinations of extreme points, and since this is a dense subset, the diagram commutes by continuity.So let {x 1 , . . .x n } be a finite subset of X, and a finite convex combination of extreme points of Ś(X).Now with the last step holding because k is an affine function.
As explained before, this shows Ś(η X )(φ)(k) = k(φ) for all φ ∈ Ś(X), and hence the diagram commutes.Thus we have that Ś is the left adjoint to U .Now that we have defined the adjunction Ś ⊣ U , we can move on to proving that R is not only the same functor as the monad derived from Ś ⊣ U but also the same as a monad.In order to do this, we require a few lemmas concerning the definition of µ we gave at the start of Section 5.The map µ was defined using λh.h(v).Since we need to prove certain properties about it, we give this map a name, and generalize it somewhat for later use.If A is a (possibly noncommutative) C * -algebra, we define In the special case we had earlier, we were using ζ C(X) for a compact Hausdorff space X, since C(X) sa = C R (X), the real-valued functions.We can see that Lemma 5.15.The map ζ A is a bijection between A sa and A(Stat(A), R). ζ C(X) is a bijection between C R (X) and A( Ś(X), R).In fact, the bijection is an isomorphism of ordered R-vector spaces with unit, taking these to be defined pointwise on A(Stat(A), R).
The proof can be found in Proof.We have by definition that R = U Ś and η = η.Therefore we only need to show that µ = U ε Ś.What we need to show then, is that if X is a compact Hausdorff space and φ ∈ Ś(U ( Ś(X))), then µ(φ) is the barycentre of φ.That is to say, for all f ∈ A( Ś(X), R), φ(f ) = f (µ X (φ)).Using Lemma 5.15, we reduce to showing that for all f ∈ C R (X), we have φ Theorem 5.17 ( Świrszcz's theorem).The forgetful functor U : CCLcvx → CH is monadic, i.e.

CCLcvx ≃ EM(U • Ś). By Theorem 5.16, CCLcvx ≃ EM(R).
This comes from [36,Theorem 3].A proof not using any monadicity theorems can be found in [34, Proposition 7.3].5.1.1.Non-commutative C * -algebras and EM(R).In the following section we shall show that the category Cstar PU embeds fully and faithfully in EM(R).To do this, we use the fact that EM(R) ≃ CCLcvx, and also the functor Stat : Cstar PU → CCLcvx.
We begin with a standard separation result from the theory of C * -algebras.for all φ ∈ Stat(A) implies a = b.In other words, A is separated by its states, or A has "sufficiently many states".
Proof.In [21, theorem 4.3.4(i)] we have that if φ(a) = 0 for all φ ∈ Stat(A), then a = 0. We simply apply this to a − b.
On the set A(X, C), for X ∈ Obj(CCLcvx), we can define a C-vector space structure, a positive cone, and a distinguished unit, simply by using the fact that C has these things and defining them pointwise.The positive cone is [0, ∞) ⊆ C and the unit is 1.Given these definitions, we can prove the complexification of Lemma 5.15.

Lemma 5.19. For each C * -algebra A, the map ξ
is an isomorphism of complex vector spaces preserving the positive cone and unit in both directions.
Proof.First we show that the map ξ A is C-linear and preserves * .For C-linearity, let z ∈ C, φ ∈ Stat(A) and a ∈ A. Then To show that it preserves * , where for f ∈ A(Stat(A), C), f * is calculated pointwise, we use the fact that every positive linear functional on A, and hence every state, is self-adjoint, as described in Lemma 2.2, i.e. φ(a * ) = φ(a).
Thus we have and so ξ A (a * ) = ξ A (a) * .From Lemma 5.15 we have that ξ restricts to an isomorphism ζ : A sa ∼ = A(Stat(A), R) as an ordered vector space with unit.We extend this to complex numbers as follows.Given a ∈ A, we can define its real and imaginary parts as and we see that ℜ(a) + iℑ(a) = a.Similarly, using pointwise complex conjugation as * , we can define real and imaginary parts of an affine continuous map from Stat(A) → C, and the self-adjoint elements are maps Stat(A) → C. Since we know that η X has an inverse for self-adjoint elements, we can define the inverse as We show this is the inverse of ξ A .For one way For the other way, with a, b ∈ A sa , The positive elements of A(Stat(A), C) are given by functions whose image is contained in the positive reals, [0, ∞) ⊆ C. We need to show that if a ∈ A(Stat(A), [0, ∞)), then so is A(g, C)(a).This is easily accomplished as before.If φ ∈ Stat(B), then A(g, C)(a)(φ) = (a • g)(φ) = a(g(φ)).8.6].We do not give the characterization here as it involves many further definitions.Since there are PU-maps that are not completely positive, Stat is not a full functor when restricted to Cstar cPU .In fact, whether or not a map is completely positive or not depends on the orientation (in the sense of [4]) and cannot be defined purely from the EM(R) structure of the state space.This can be seen by the fact that the transpose map, the archetypal positive but not completely positive map, is selfinverse, and hence an isomorphism as a PU map, and so by the above result defines an isomorphism in EM(R) on the state space.

STATES AND EFFECTS
We start with a simple observation.Lemma 6.1.The unit interval [0, 1] is a compact convex subset of the locally convex space R, and therefore carries a R-algebra structure by Theorem 5.17.The algebra map R([0, 1]) → [0, 1] maps each measure to its mean value.
For an arbitrary R-algebra X, the homset of algebra maps: is an effect module, with pointwise operations.Recall from Proposition 5.7 that this homset is the affine and continuous functions X → [0, 1].Taken all together, we have defined a functor A(−, [0, 1]) : EM(R) → EMod op .
In [18] it is shown that for an effect module M , the homset EMod(M, [0, 1]) is a convex compact Hausdorff space.In fact, it carries an R-algebra structure: V V q q q q q q q q q q q (6.1)Such diagrams appear in [15] as a categorical representation of the duality between states and effects, with the Schrödinger picture on the right vertex of the triangle, and the Heisenberg picture on the left vertex of the triangle (see also [17]).In these diagrams: • The map Kℓ(R) → EMod op on the left is the "predicate" functor, sending a space X to the predicates on X, given by the effect module Cont(X, [0, 1]) of continuous functions X → [0, 1], or for C * algebras mapping A to the effects [0, 1] A .For C * -algebras this was shown to be full and faithful in Lemma 2.6, and for Kℓ(R) we combine Lemma 2. • The "state" functor Kℓ(R) → EM(R) is the standard full and faithful "comparison" functor from a Kleisli category to a category of Eilenberg-Moore algebras.In the C * -algebra case it is the functor Stat, combined with the equivalence from Theorem 5.17 We summarise what we have just shown.Theorem 6.2.The diagrams (6.1) are commuting "state-and-effect" triangles.

Proposition 2 . 5 .
For each C * -algebra A, the set of states Stat(A) = Hom PU (A, C) is convex, and is a compact Hausdorff subspace of the dual space of A given the weak-* topology.Each PU-map f : A → B yields an affine continuous function Stat(f ) = (−) • f : Stat(B) → Stat(A).This defines a functor Stat : (Cstar PU ) op → CCLcvx.

Proposition 4 . 3 .
The functor C E from (4.1) restricts in the finite case to an equivalence of categories:

Lemma 5 . 6 .Proposition 5 . 7 .
Let X, Y be compact Hausdorff spaces.Each Eilenberg-Moore algebra α : R(X) → X is an affine function.For each continuous map f :X → Y , the function R(f ) : R(X) → R(Y ) is affine.Proof.This follows from naturality of τ : DU ⇒ U R. Let α : R(X) → X and β : R(Y ) → Y be two Eilenberg-Moore algebras of the Radon monad R. A function f : X → Y is an algebra homomorphism if and only if f is both continuous and affine.As a result, the functor EM(R) → EM(D) = Conv from Lemma 5.2