Duality for Convex Monoids

Every C*-algebra gives rise to an effect module and a convex space of states, which are connected via Kadison duality. We explore this duality in several examples, where the C*-algebra is equipped with the structure of a finite-dimensional Hopf algebra. When the Hopf algebra is the function algebra or group algebra of a finite group, the resulting state spaces form convex monoids. We will prove that both these convex monoids can be obtained from the other one by taking a coproduct of density matrices on the irreducible representations. We will also show that the same holds for a tensor product of a group and a function algebra.


Introduction
States and observables of a physical system are connected via dualities between certain categories.There are several dualities that can be used for this connection.Known examples include the Gelfand duality theorem and the Kadison duality theorem.For a system in classical physics, the state space is modeled by a topological space, and the observables are given by functions on this space.In this way, the algebra of observables forms a commutative C * -algebra.The celebrated Gelfand theorem states that the category of locally compact Hausdorff spaces is dually equivalent to the category of commutative C * -algebras, and thus it provides an intimate connection between states and observables.A useful special case occurs when the C * -algebras under consideration have a unit.Gelfand duality in this setting states that the category of unital C * -algebras is dually equivalent to the category of compact Hausdorff spaces.
Gelfand duality does not apply to quantum mechanical systems, since their algebra of observables is in general a non-commutative C * -algebra.There is no good non-commutative analogue of Gelfand duality, but there is a duality theorem due to Kadison that can be useful to describe quantum systems.Kadison duality is not based on C * -algebras, but on the unit interval within a unital C * -algebra.This unit interval forms a structure called an effect module, and there is a dual equivalence between a certain category of effect modules and a certain category of convex spaces.The state space of a quantum system forms a convex space and the corresponding effect module contains its observables; hence Kadison duality connects states and observables of quantum systems.It does not directly generalize Gelfand duality, since the unit interval of a C * -algebra contains less information than the C * -algebra itself.
When studying physical systems, one often wants to take the symmetry group of the system into account.In the C * -algebraic picture, this leads to quantum groups.For ordinary Gelfand duality, we use locally compact Hausdorff spaces as state spaces.If we take the symmetry of a system into account, the state space becomes a locally compact group.On the dual side, this gives a coalgebra structure on the C * -algebra, making it into a structure called a quantum group.There is an analogue of the Gelfand duality theorem that takes the symmetry into account.This theorem states that the category of compact (Hausdorff) groups is dually equivalent to the category of commutative compact quantum groups.
Summarizing, there are two dualities involving topological spaces and C * -algebras: one for systems without symmetry, and one for systems with symmetry.Furthermore, Kadison duality relates convex spaces and effect modules for systems without symmetry.In this article we shall will describe a variant of Kadison duality for systems with symmetry.This will lead to a notion of a quantum group whose underlying algebra is an effect module instead of a C * -algebra.Schematically, we wish to complete the following diagram: The categories and functors occuring in this diagram will be explained in more detail in the next section.We will restrict our attention to finite groups.In the theory of C * -algebraic quantum groups, there is only one way to assign a commutative quantum group or Hopf-algebra to any finite group.We show that there are two ways to assign an effect module (and a dual convex space) to a finite group, arising from two different Hopf algebras associated to the group.Both ways to form "effect quantum groups" are related via a version of Pontryagin duality.The outline of this paper is as follows.Section 2 contains preliminary material about convex spaces, effect modules, and quantum groups.In particular we will describe the various dualities that connect these objects.In Section 3 we will determine the effect modules and convex spaces associated to the group algebra and the function algebra of a finite group.The two convex spaces obtained in this way are both convex monoids, that is, monoids in the category of convex spaces.The connection between these two monoids will be established in Section 4. We will prove that both convex monoids determine each other via essentially the same construction: if V 1 , . . ., V k are the irreducible linear representations of either of these monoids, then the coproduct DM( ) is a convex monoid isomorphic to the other one.Finally, in Section 5, we will prove a related result for the tensor product of a group and a function algebra.

Preliminaries
We will present the dualities alluded to in the Introduction in more detail here.The most basic duality that we will use is Gelfand duality.Throughout this paper, we will assume that all C * -algebras we encounter have a unit.Write C * for the category of C * -algebras with *-homomorphisms as maps.The full subcategory of commutative C * -algebras is denoted cC * .Furthermore write KHaus for the category of compact Hausdorff spaces with continuous maps.If X is a compact Hausdorff space, then the collection C(X) of complex-valued functions on X is a commutative C * -algebra with pointwise operations.This construction gives a contravariant functor C from KHaus to cC * by letting it act on morphisms via precomposition.The Gelfand spectrum provides a functor in the other direction: if A is a commutative C * -algebra, then its spectrum Spec A = Hom cC * (A, C) is a compact Hausdorff space.The spectrum construction forms a contravariant functor from cC * to KHaus, again using precomposition.
Theorem 1 (Gelfand).The compositions C •Spec and Spec •C are naturally equivalent to the identity functor.Hence the categories KHaus and cC * are dually equivalent.
There is a more general version of Gelfand duality involving non-unital C * -algebras and locally compact spaces, but we will only be concerned with compact spaces in the remainder of this article.
The Gelfand Theorem justifies viewing C * -algebras as a non-commutative generalization of spaces.Similarly it is useful to have a non-commutative generalization of topological groups.This gives the notion of a quantum group.There are several definitions of quantum groups; here we will use the compact quantum groups from Woronowicz [9].For a general overview of the theory of quantum groups see [8].
If G is a compact Hausdorff group, then its function algebra C(G) is a commutative C * -algebra.It can be made into a compact quantum group by defining ∆ : This construction provides a group-theoretic analogue of Gelfand duality.Instead of compact spaces, we use compact groups.They constitute a category KGrp with continuous homomorphisms as maps.Morphisms between compact quantum groups are unital * -homomorphisms preserving the comultiplication.They make compact quantum groups into a category KQGrp.As in Gelfand duality, we want to consider the full subcategory CKQGrp of commutative compact quantum groups.
Theorem 3. The functor C : KGrp op → CKQGrp is a dual equivalence between the category of compact Hausdorff groups and commutative compact quantum groups.
If A is a commutative compact quantum group, then the underlying space of its dual group is the spectrum of A, considered as C * -algebra.The multiplication on G arises from the comultiplication on A.
There is another way to assign a compact quantum group to a finite group G, namely the group algebra C[G].The elements are again functions from G to C, but now the multiplication is given by convolution: The standard basis of C[G] consists of Dirac functions λ g for g ∈ G, defined by λ g (g) = 1 and λ g (h) = 0 for h = g.The convolution product assumes a particularly easy form on these basis vectors, namely λ g * λ h = λ gh .The comultiplication is defined on basis vectors by ∆(λ g ) = λ g ⊗ λ g .

Effect algebras and modules.
Another duality that we will use involves the effects in a C * -algebra.Effects represent probabilistic measurements that can be performed on a physical system.Let A be any C * -algebra.An element a in A is said to be positive if it can be written as a = b * b for some b ∈ A. Positivity can be used to define an order on the self-adjoint part of A, called the Löwner order.Let a, b be self-adjoint elements in A, then we say that a ≤ b if and only if b − a is positive.An effect in A is a self-adjoint a ∈ A for which 0 ≤ a ≤ 1.
Effects in a C * -algebra can be organized into an algebraic structure called an effect module.Effect modules were introduced in [4], based on earlier work on effect algebras, which started in [1].For an overview of the theory about effect algebras, see [2].
Roughly speaking, an effect module looks like a vector space, but the addition is only a partial operation (since the sum of two effects may lie above 1), and we can only multiply by scalars in the unit interval [0, 1].Instead of complements with respect to 0, we have complements with respect to 1.This means that for every effect a there exists an effect b for which a + b = 1.The precise definition is as follows.
Definition 4.An effect module consists of a set A equipped with a partial binary operation ⊞ called addition, a unary operation (−) ⊥ called orthocomplement, a scalar multiplication • : [0, 1] × A → A and constants 0, 1 ∈ A, subject to the following axioms: • The operation ⊞ is commutative, which means that whenever a ⊞ b is defined, then also b ⊞ a is defined, and a ⊞ b = b ⊞ a. • The operation ⊞ is associative, which means that if a ⊞ b and (a ⊞ b) ⊞ c are defined, then also b⊞c and a⊞(b⊞c) are defined, and (a⊞b)⊞c = a⊞(b⊞c).
Effect modules form a category EMod, in which the morphisms are functions preserving addition, orthocomplement, scalar multiplication, and the constants 0 and 1.
The easiest example of an effect module is the unit interval [0, 1].The partial operation is addition, where a ⊞ b is defined if and only if a + b ≤ 1.The orthocomplement is given by a ⊥ = 1 − a, and the scalar multiplication is simply the multiplication on [0, 1].Another example are the effects in a C * -algebra, with the same operations.If A is a C * -algebra, then its collection of effects is denoted Ef (A).Any Hilbert space H gives rise to a C * -algebra B(H), hence to an effect module Ef (B(H)).We will often abbreviate this to Ef (H).
More generally, every partially ordered vector space V over R gives rise to an effect module.Pick an element u ∈ V for which u > 0, then the interval [0, u] = {v ∈ V | 0 ≤ v ≤ u} is an effect module.Addition serves as the partial binary operation, and the orthocomplement is v ⊥ = u − v.The scalar multiplication is obtained by restricting the scalar multiplication from R to [0, 1].In fact, every effect module is an interval in some partially ordered R-vector space, as shown in [4, Theorem 3.1].
To work with infinite-dimensional vector spaces, it is often necessary to require that they are complete in a certain metric.The same holds for effect modules.If A is an effect module, then a state on A is a morphism σ : A → [0, 1].The collection of all states is written as St(A).Define a metric on A via We call the effect module A a Banach effect module if it is complete in its associated metric.Banach effect modules give a full subcategory of EMod written as BEMod.
2.2.Convex spaces.The state space of an effect module is always a compact convex space.We will make this observation more precise by defining a suitable category of compact convex spaces, following [10].A topological vector space is said to be locally convex if its topology has a base of convex open sets.Let KConv be the category whose objects are compact convex subspaces of a locally convex vector space.A subspace X ⊂ V is called convex if, for all x, y ∈ X and λ ∈ [0, 1], we have that λx The state space of an effect module A is contained in the vector space {ϕ : Examples 6.We give some examples of convex spaces and their dual effect modules. ( This can be visualized as the standard simplex whose vertices are points in X.An element f ∈ D(X) is usually written as a formal convex combination x∈X a x x, where the coefficients are the function values a x = f (x).They are subject to the condition x a x = 1.This construction gives a functor D : FinSets → KConv, where on a morphism ϕ : X → Y we define D(ϕ)( x a x x) = x a x ϕ(x).The dual effect module of D(X) is Hom(D(X), [0, 1]), which is isomorphic to [0, 1] X .
(2) In the above example, D(X) can be thought of as the set of discrete probability measures or distributions on X.There is a continuous analogue of this construction.Let X now be a compact Hausdorff space, and let Σ X be its Borel σ-algebra.Denote the space of Radon measures on X by R(X).
A Radon measure is a probability measure µ : Σ X → [0, 1] that satisfies In [3] it is shown that R forms a monad on the category of compact Hausdorff spaces.Its category of Eilenberg-Moore algebras is equivalent to KConv, so convex spaces of the form R(X) can be thought of as the free convex spaces over a compact Hausdorff space.The dual effect module of R(X) is the collection of continuous functions from X into [0, 1].This fact is a categorical reformulation of the Riesz-Markov theorem.To see this, observe that there is a map R(X) → Hom(C(X, [0, 1]), [0, 1]) given by integration, i.e. µ → (−) dµ.The Riesz-Markov theorem states that this map is an isomorphism, so R(X) is the dual of C(X, [0, 1]).This shows that the following diagram, connecting Gelfand and Kadison duality, commutes: (3) Let H be a Hilbert space.A density matrix on H is a positive trace-class operator ρ : H → H with trace 1.The collection of all density matrices forms a convex space denoted DM(H).The importance of this example lies in its connection to the effects on H: there is an isomorphism Ef (H) → Hom(DM(H), [0, 1]), that maps an effect a to the function ρ → tr(ρa).Because this map is an isomorphism, Ef (H) is the dual effect module of DM(H).
There are several ways to construct new convex spaces from old ones.In the remainder of this paper we will sometimes use coproducts and tensor products of convex spaces, so we will describe these briefly here.
The category KConv has all coproducts.The coproduct of two convex spaces can be described geometrically, using the embedding in a locally convex vector space.The following description is a slight modification of the construction in [10].Suppose that X ⊆ V and Y ⊆ W are compact convex subsets of locally convex vector spaces.Then the coproduct X + Y can be embedded in the vector space V ⊕ W ⊕ R. To construct this coproduct, embed X in this larger vector space via the inclusion x → (x, 0, 1), and embed Y via the inclusion y → (0, y, 0).The convex hull of the disjoint union of X and Y is the coproduct of X and Y .This is made precise in the following.Proposition 7. If X ⊆ V and Y ⊆ W are objects in the category KConv, then their coproduct is Proof.Define embeddings i X : X → X+Y and i Y : Y → X+Y via i X (x) = (x, 0, 1) and i Y (y) = (0, y, 0).Given affine maps f : X → Z and g : Y → Z, define h : X + Y → Z by h(rx, (1 − r)y, r) = rf (x) + (1 − r)g(y).
Then h • i X = f and h • i Y = g, so it remains to be shown that h is the unique map with this property.Suppose that h ′ : X + Y → Z is an affine map for which which proves uniqueness.
Example 8. Denote the one-point convex space by 1.The coproduct 1 + • • • + 1 of n copies of this space is the convex hull of n points, embedded in R n−1 in such a way that they are all affinely independent.Therefore this coproduct is the standard simplex D(n).
We continue with a discussion of the tensor product of compact convex spaces.If X, Y , and Z are compact convex spaces, then a map X × Y → Z is called bi-affine is it is affine in both variables separately.A tensor product of X and Y is a compact convex space X ⊗ Y equipped with a bi-affine map ⊗ : X × Y → X ⊗ Y such that for every compact convex space Z and every bi-affine f : X × Y → Z there exists a unique affine map g : X ⊗ Y → Z such that g • ⊗ = f .Semadeni proves in [10] that any two compact convex spaces admit a tensor product, and that it is unique up to isomorphism.
The above tensor product enjoys many good properties.The one-point convex space 1 acts as a unit for the tensor.Furthermore, the tensor product distributes over coproducts.From these two facts, together with the isomorphism D(n) ∼ = 1 + • • • + 1, it can be deduced that the tensor product of standard simplices is D(n) ⊗ D(m) ∼ = D(nm).

Kadison duality for group and function algebras
Let G be a finite group.This gives rise to two Hopf-algebras, or compact quantum groups, namely the function algebra C(G) and the group algebra C[G].Of these two Hopf-algebras, the function algebra is commutative but in general not cocommutative, while for the group algebra, it is the other way round.Therefore the duality from Theorem 3 only applies to the function algebra C(G).However, Kadison duality also applies to unit intervals of non-commutative C * -algebras, so we can use this for both the group algebra and the function algebra.Definition 9. A convex monoid is an object X of the category KConv, together with a continuous multiplication map • : X × X → X and a constant 1 ∈ X, such that • The operation • is affine in both variables separately, that is, (λx • z and similarly for convex combinations on the right.
Equivalently, a convex monoid is a convex space X equipped with a map X⊗X → X that is associative and has a unit.
A variant of quantum groups in the framework of Kadison duality should give a duality between effect modules with a comultiplication and convex monoids.In this section we describe these objects for the Hopf-algebras C(G) and C[G].We will start with the function algebra C(G).In fact, this algebra can be defined for any compact group G, so we will now determine the effect module and convex space associated to C(G) for an arbitrary compact group G.The multiplication on R(G) can also be decribed directly in terms of the multiplication on G. Applying the functor R to the multiplication map , this provides a convex monoid structure on R(G), which is the dual of Ef (C(G)).This convex monoid has been studied categorically in [6].
The multiplication on the group algebra C[G] is more complicated than the one on the function algebra.Therefore the Löwner order on C[G] and the effect module are also more difficult to compute explicitly.The algebra C[G] is simultaneously a C*-algebra and a Hilbert space, and the algebra structure is compatible with the inner product, so C[G] forms a Hilbert algebra.We shall use some general facts about Hilbert algebras to compute the effect module and the state space of C[G].
Proposition 14.Let V be a unitary representation of G. Write the decomposition of V into irreducible representations as Proof.An intertwining effect ε : V → V can be written as a matrix of maps ε ij : n i V i → n j V j .By Schur's Lemma, each ε ij = 0 for i = j.The effects ε ii can in turn be decomposed into an n i × n i matrix of maps V i → V i , and these are all scalar multiples of the identity by Schur's Lemma.Therefore each ε ii corresponds to an effect on C ni .
Proof.By Proposition 13, the effect module . This is a Banach effect module, so we can use the duality between convex compact spaces and Banach effect modules to determine the dual space.Dualizing turns products into coproducts, so the dual space is Dualizing ∆ gives µ on the state space St(C[G]).The convex monoid structure on the state space of the group algebra satisfies (σ • τ )(λ g ) = σ(λ g )τ (λ g ) on basis vectors.

Convex Pontryagin duality for group and function algebras
The group algebra and the function algebra associated to a finite group are both finite-dimensional Hopf algebras.These are related via a non-commutative generalization of Pontryagin duality, see e.g.[8] for details.In the previous section, we found two convex monoids that can be obtained from a finite group: the state space D(G) of the function algebra, and the state space of the group algebra, where the V i are the irreducible representations of G.This section will present a construction to convert these two convex monoids into each other.This construction can be viewed as a convex counterpart of Pontryagin duality.
Definition 16.A linear representation of a convex monoid X consists of a vector space V and a monoid homomorphism ρ : X → Aut(V ) that preserves convex combinations.
As usual, a representation can also be written as an action of X on V , that is, a map X × V → V .A linear representation of a convex monoid is then required to be affine in the first variable and linear in the second variable.We will look at the linear representations of the convex monoid D(G).
Lemma 17.There is a one-to-one correspondence between representations of the finite group G and linear representations of D(G).
Proof.Representations of G are monoid homomorphisms G → Aut(V ), since all monoid homomorphisms between groups are automatically group homomorphisms.Linear representations of D(G) are monoid homomorphisms D(G) → Aut(V ) that are also morphisms of convex spaces.Since D(G) is the free convex space generated by G, it follows that there is a one-to-one correspondence between maps of sets G → Aut(V ) and maps of convex spaces D(G) → Aut(V ).It is easy to check that this equivalence restricts to monoid homomorphisms.This result produces an easy way to construct the state space of C[G] out of the state space of C(G), in the following steps: (1) Let V 1 , . .., V k be the irreducible linear representations of St(C(G)).
(2) Form the convex sets of density matrices DM(V i ) for each i.
We will now check that all 1-dimensional linear representations are of the form ρ g for some g ∈ G.
The map ρ is a representation, so these two expressions must be equal for all states σ and τ .Comparing coefficients shows that at most one a i is equal to 1, and all others are 0. The element a cannot be identically 0, since ρ preserves 1. Hence a is equal to λ g for some g ∈ G, which proves that the maps ρ g are indeed the only 1-dimensional representations.
There is only one density matrix on any 1-dimensional space.Therefore the space DM( We have shown that if we start with the convex monoid St(C(G)) ∼ = D(G) and apply the above construction twice, then we get back a convex space that is isomorphic to the underlying space of the original convex monoid.Now we wish to show that the multiplication is also preserved in this construction, so that we obtain an isomorphism of convex monoids, rather than just convex spaces.For this we have to endow the coproduct of density matrices with a multiplication.It is useful to have an explicit isomorphism between DM( Taking states of a C * -algebra provides a contravariant functor St : C * → KConv.Therefore, applying the state functor to Ψ gives a map St(End( given by α(ρ)(A) = tr(ρA), and hence St(Ψ) = Φ.Since Ψ is an isomorphism and St is a functor, Φ is also an isomorphism.
Using this isomorphism, the multiplication on the coproduct of density matrices can be described explicitly.Since we are working in a coproduct, it suffices to describe T • S, where T ∈ DM(V i ) and S ∈ DM(V j ).Applying the isomorphism Φ from the lemma above gives states λ g → tr(T ρ i (g)) and λ g → tr(Sρ j (g)) on C[G].Multiplying these states pointwise and using properties of the trace gives the map λ g → tr((T ⊗ S)(ρ i ⊗ ρ j )(g)).Since Φ is an isomorphism, there is a unique We define T • S to be this convex combination i λ i T i .With the proposition and lemma above, we have now proven the following result.
Theorem 20.Let G be a finite group, and let V 1 , . . ., V k be the irreducible linear representations of the convex monoid St(C(G)).Then the convex monoid DM(V 1 )+ • • •+DM(V k ) with multiplication described above is isomorphic to the convex monoid St(C[G]) with pointwise multiplication.

Convex Pontryagin duality for a tensor product
The category of finite-dimensional Hopf algebras is self-dual.The dual of a finite-dimensional Hopf algebra A is Â = {f : A → C | f linear}.Its multiplication is derived from the comultiplication on A, and vice versa.The Hopf algebras C(G) and C[G] coming from a finite group G are duals of each other via this construction.
Let A be either C(G) or C[G].The main result from the previous section states that if V 1 , . . ., V k are the irreducible representations of the convex monoid St(A), then DM(V 1 ) + • • • + DM(V k ) is isomorphic to St( Â).This raises the question if this holds for all Hopf algebras.We do not yet know if this is the case in general, but we will now discuss another example of a Hopf algebra for which it holds, so this may be promising for the general case.
Let G be a finite group.Consider the Hopf algebra A = C(G) ⊗ C[G], i.e. the tensor product of the function algebra and the group algebra.This Hopf algebra is neither commutative nor cocommutative.Since dualizing preserves tensor products, the dual of A is isomorphic to A itself.Thus the statement that connects the state space of A to its dual amounts to the following.which is what we wanted to show.
which is locally convex.Therefore St(A) is an object in the category KConv, and St is a contravariant functor from BEMod to KConv.The functor Hom KConv (−, [0, 1]) is a contravariant functor in the other direction.The following result is taken from[5, Theorem 6], but see also[10, Section 4].Theorem 5.The functors St and Hom KConv (−, [0, 1]) are inverses of each other.Hence the categories KConv and BEMod are dually equivalent.

( 3 )
The coproduct (in the category KConv) of all DM(V i ) is the state space of C[G].Since irreducible representations of G are the same as irreducible linear representations of D(G), this construction yields exactly the state space of C[G].A surprising fact is that it works in two directions: if we apply exactly the same construction to the state space of C[G], we end up with the state space of C(G).Proposition 18.Let V 1 , . . ., V k be the irreducible linear representations of the convex monoid St(C[G]).Then the convex space DM Let ρ : St(C[G]) → C be an arbitrary representation.Then the map ρ extends to a function Hom(C[G], C) → C in the double dual of C[G], hence there exists a ∈ C[G] such that σ(a) = ρ(σ) for all states σ on C[G].We will show that a is actually an element in G ⊆ C[G].Express a as a = a 1 λ g1 + • • • + a n λ gn .Then, for any two states σ and τ , algebras, on basis vectors determined by λ g → (ρ 1 (g), . . ., ρ k (g)).We claim that this map is injective.Suppose that a, b ∈ C[G] are such that ρ i (a) = ρ i (b) for all i.Then a and b act in the same way in all irreducible representations of G. Since any representation of G can be decomposed into irreducibles, a and b act in the same way in all representations of G.In particular, they have the same action on the regular representation C[G].Thus a = a • e = b • e = b.Since Ψ is injective and its domain has the same dimension as its codomain, it is an isomorphism.