Lobachevskii Journal of Mathematics

http://ljm.ksu.ru

Vol. 18, 2005, 107–125 \begin{matrix}c \\ ◯ \end{matrix} Per K. Jakobsen, V.V. Lychagin

Per K. Jakobsen and V.V. Lychagin

UNIVERSAL SEMIGROUPS

Abstract. In this paper we introduce the notion of a universal semigroup and its dual, the universal cosemigroup. We show that the class of universal semigroups include the class of monoids and is included in the class of semigroups with a product that is an epimorphism. Both inclusions are proper. Semigroups in the category of Banach spaces are Banach algebras and we show that all Banach algebras with an approximate unit are universal and construct a finite dimensional Banach algebra that has no unit but is universal. The property of being universal is thus a generalized unit property.

Contents

1. Introduction 107 2. Universal semigroups 109 2.1. Examples from Sets and general monoidal categories 110 2.2. Examples from the category of Banach spaces 118 3. Universal cosemigroups 122 3.1. Examples from Sets and general monoidal categories 123 References 125

1. Introduction

The notions of monoids and semigroups are well known and appear naturally in many contexts. In this paper we introduce a new algebraic structure, the universal semigroup. Its definition is inspired by the categorical generalization of monoids and semigroups [1]. In a monoidal category a semigroup is a pair ⟨A, μ⟩ where A is an object in the category and μ : A ⊗ A → A is a morphism such that the following diagram commute

[figure: A ⊗ (A ⊗ A); 1_{A} ⊗ μ_{A}; A ⊗ A; μ_{A}; A]
[figure: μ_{A} ⊗ 1_{A}]

(A ⊗ A) ⊗ A ^{α_{A,A,A}}

where α is the associativity constraint in the category. This definition does not only include the usual algebraic notion of semigroups and associative algebras, but also many other algebraic structures. What algebraic structure it represents depends on the choice of monoidal category. Let S_{A} be the diagram we get by removing the right part of the previous diagram. Thus S_{A} is the diagram

A ⊗ (A ⊗ A) \overline{^{1_{A} ⊗ μ_{A}}} A ⊗ A
[figure: μ_{A} ⊗ 1_{A}]

(A ⊗ A) ⊗ A ^{α_{A,A,A}}

Observe that from a categorical point of view ⟨A, μ⟩ is a semigroup if and only if it is a cocone[1] on the diagram S_{A}. In general ⟨A, μ⟩ is not a universal cocone on S_{A}. Recall that ⟨B, g⟩ is a universal cocone on D_{A} if it is a cocone and if for any cocone ⟨C, f⟩, there exists a unique map φ : B ⟶ C such that φ ∘ g = f

[figure: A ⊗ (A ⊗ A); 1_{A} ⊗ μ_{A}; A ⊗ A; g; B]
[figure: μ_{A} ⊗ 1_{A}][figure: ∀f]

[figure: (A ⊗ A) ⊗ A; α_{A,A,A}; C; ∃!ϕ]

The universal cocone could be though of as the ”smallest” cocone on the diagram, the one that gives the best ”commutative fit” to the diagram. In this paper we define ⟨A, μ⟩ to be a universal semigroup if it is a universal cocone on the diagram S_{A} and show that the class of universal semigroups includes the class of monoids, but that there are universal semigroups that are not monoids. For any monoid the product is an epimorphism. We show that this also holds for any universal semigroup but that there are semigroups that are not universal and where the product is an epimorphism. Thus the class of universal semigroups includes all monoids and is included in the class of semigroups with products that are epimorphisms.

The category of Banach spaces is a monoidal category where the monoidal structure is determined by the projective tensor product of Banach spaces. In this category semigroups are Banach algebras. We show that all Banach algebras with an approximate unit are universal and that there are universal Banach algebras that does not have an approximate unit. All C^{∗}−algebras admit an approximate unit[2] and are thus universal as are L_{1} group algebras of locally compact topological groups since they all admit an approximate unit[3].

These results leads to the idea that the universal cocone property for semigroups is a generalized unit condition. Universal semigroups could be expected to share properties with monoids that general semigroups does not.

Note that the property of being a universal semigroup is a typical categorical property in that it depends on which category the semigroup is placed in. One can thus not state that a semigroup is universal without specifying which category we refer to. This may appear a little strange if one is not familiar with the categorical point of view. It is however a fact that even such well known properties as associativity and commutativity of algebraic structures actually depends on the categorical context. It has for instance been shown that both quaternions and Cayley numbers are commutative, associative algebras[6] if placed in the right categories.

2. Universal semigroups

In order to give a categorical definition of universal semigroups or even of semigroups we need the notion of a category with an associative product, or a semimonoidal category.

Definition 1. A semimonoidal category is a triple ⟨ C, ⊗, α⟩, where ⊗ : C × C ⟶ C is a bifunctor on C and α is a natural isomorphism

α : ⊗ ∘ (1_{C} × ⊗) ⟶ ⊗ ∘ (⊗ × 1_{C}),

such that the following MacLane coherence conditions

α_{A⊗B,C,D} ∘ α_{A,B,C⊗D} = (α_{A,B,C} ⊗ 1_{D}) ∘ α_{A,B⊗C,D} ∘ (1_{A} ⊗ α_{B,C,D}),

are satisfied for all objects A, B, C, D

Recall that naturality means that the following identities

((f ⊗ g) ⊗ h) ∘ α_{A,B,C} = α_{A′,B′,C′} ∘ (f ⊗ (g ⊗ h)),

holds for all choices of arrows f : A ⟶ A′, g : B ⟶ B′ , h : C ⟶ C′.

The product ⊗ is thus not strictly associative but associative up to isomorphism. A semimonoidal category contains the structures we need to define semigroups[1].

Definition 2. A semigroup ⟨A, μ_{A}⟩ in a semimonoidal category is an object A and a morphism μ_{A} : A ⊗ A ⟶ A such that the following diagram commute

[figure: A ⊗ (A ⊗ A); 1_{A} ⊗ μ_{A}; A ⊗ A; μ_{A}; A]
[figure: μ_{A} ⊗ 1_{A}]

(A ⊗ A) ⊗ A ^{α_{A,A,A}} .

Let S_{A} be the previous diagram with the right-hand node removed

A ⊗ (A ⊗ A) \overline{^{1_{A} ⊗ μ_{A}}} A ⊗ A
[figure: μ_{A} ⊗ 1_{A}]

(A ⊗ A) ⊗ A ^{α_{A,A,A}} .

Definition 3. A universal semigroup in a semimonoidal category ⟨ C, ⊗, α⟩ is a universal cocone ⟨A, μ_{A}⟩ on the diagram S_{A}.

2.1. Examples from Sets and general monoidal categories. Let us consider a few examples from the category Sets with the usual monoidal structure. A monoid ⟨A, μ, e⟩ in Sets is a semigroup ⟨A, μ⟩ with a neutral element e such that μ(e, x) = μ(x, e) = x for all x in A. Let ⟨C, f⟩ be any cocone on S_{A}. This means that f ∘ (1 × μ) = f ∘ (μ × 1) or f(x, yz) = f(xy, z) for all x, y, z in A. Here we write xy = μ(x, y).The condition for ⟨A, μ⟩ to be a universal cocone on S_{A} is therefore that the equation

φ ∘ μ = f,

has a unique solution for all f such that f(x, yz) = f(xy, z). Let φ and ψ be two solutions. Then we have φ(x) = φ(xe) = f(x, e) = ψ(xe) = ψ(x) for all x in A. So there can be at most one solution. Define a map φ : A ⟶ C by φ(x) = f(x, e). Then we have

(φ ∘ μ)(x, y) = φ(xy) = f(xy, e) = f(x, ye) = f(x, y)

The map φ is therefore a solution and we have proved the following proposition

Proposition 4. Let ⟨A, μ, e⟩ be a monoid in Sets. Then ⟨A, μ⟩ is a universal semigroup.

The class of Universal semigroups in Sets therefore includes all monoids. This is a proper inclusion, there are universal semigroups that are not monoids as the following two examples show.

Let A = {a, b} be a semigroup with two elements and table of multiplication given by

μ a b
a a b
b a b

The conditions on f are for this semigroup given by f(a, a) = f(b, a), f(a, b) = f(b, b). Let φ_{1} and φ_{2} be two solutions of φ ∘ μ = f. Then we must have φ_{1}(a) = φ_{1}(μ(a, a)) = f(a, a) = φ_{2}(μ(a, a)) = φ_{2}(a) and φ_{1}(b) = φ_{1}(μ(b, b)) = f(b, b) = φ_{2}(μ(b, b)) = φ_{2}(b). So φ_{1} = φ_{2} and there is at most one solution. Define a map φ : A ⟶ B by φ(a) = f(a, a), φ(b) = f(b, b). Then we have

φ(μ(a, a)) = φ(a) = f(a, a)
φ(μ(a, b)) = φ(b) = f(b, b) = f(a, b)
φ(μ(b, a)) = φ(a) = f(a, a) = f(b, a)
φ(μ(b, b)) = φ(b) = f(b, b)

This proves that φ is the unique solution to the equation φ ∘ μ = f and as a consequence ⟨A, μ⟩ is a universal semigroup. On the other hand ⟨A, μ⟩ does not have a neutral element and is therefore not a monoid.

As a second example of a universal semigroup in Sets that is not a monoid let us consider a set A and the projection on the first factor μ : A × A → A. Thus μ(x, y) = x. Then we evidently have

[μ ∘ (μ × 1)](x, y, z) = μ(x, z) = x
[μ ∘ (1 × μ)](x, y, z) = μ(x, y) = x

so ⟨A, μ⟩ is a semigroup.The multiplication clearly does not have a unit, so ⟨A, μ⟩ is not a monoid. Let now ⟨C, f⟩ be a cocone on S_{A}. Then f : A × A → C and we have the condition f ∘ (1 × μ) = f ∘ (μ × 1). This means that f(x, y) = f(x, z) for all x, y, z. If φ is any solution of φ ∘ μ = f we must have φ(x) = φ(μ(x, y)) = f(x, y), so there can be at most one solution. Let x_{0} be any element in A. Define the map φ : A ⟶ C by φ(x) = f(x, x_{0}). Then we have

φ(μ(x, y)) = φ(x) = f(x, x_{0}) = f(x, y)

so φ is a solution. Therefore ⟨A, μ⟩ is a universal semigroup that is not a monoid. Projection on the second factor would in a similar way produce a universal semigroup.

We will now prove a few simple results that holds for universal semigroups in any semimonoidal category.

Proposition 5. Let ⟨A, μ_{A}⟩ be a universal semigroup. Then the arrow μ_{A} is epi.

Proof. Let B be any object in the category and let f, g : A ⟶ B be two arrows and assume that f ∘ μ_{A} = g ∘ μ_{A}. Let h = f ∘ μ_{A}. Then the equation

φ ∘ μ_{A} = h

has both f and g as solutions. Since ⟨A, μ_{A}⟩ is a universal semigroup we must have f = g and this proves that μ_{A} is epi. □

Any semigroup in Sets with a nonsurjective product is therefore not a universal semigroup in Sets.The semigroup A = {a, b, c} with table of multiplication given by

μ a b c
a b c c
b c c c
c c c c

does not have a surjective multiplication and is therefore not universal.

We will next show that monoids are universal semigroups in any semimonoidal category. In order to define the notion of a monoid in a semimonoidal category we need a neutral object for the product ⊗. This leads to the well known notion of a monoidal category.

Definition 6. Let C be a category and let e be a object in C. Define two functors L_{e}, R_{e} : C ⟶ C on objects and morphisms by

L_{e}(A) = e ⊗ A,
L_{e}(f) = 1_{e} ⊗ f,
R_{e}(A) = A ⊗ e,
R_{e}(f) = f ⊗ 1_{e}.

A monoidal category is a 6 tuple ⟨ C, ⊗, e, α, β, γ⟩ where ⟨ C, ⊗, α⟩ is a semimonoidal category and β : L_{e} ⟶ 1_{C} and γ : R_{e} ⟶ 1_{C} are natural isomorphisms such that the following identities

(γ_{A} ⊗ 1_{B}) ∘ α_{A,e,B} = (1_{A} ⊗ β_{A}),
β_{e} = γ_{e},

holds for all objects A and B.

Naturality of β, γ means that the following identities

f ∘ β_{a} = β_{B} ∘ (1_{e} ⊗ f),
f ∘ γ_{A} = γ_{B} ∘ (f ⊗ 1_{e}),

holds for all morphism f : A ⟶ B.

These are the MacLane coherence conditions for a monoidal category. They ensure that all diagrams generated using the functors ⊗, L_{e}, R_{e} and the natural isomorphisms α, β, γ will commute.

Definition 7. A monoid in a monoidal category is a triple ⟨A, μ_{A}, u_{A}⟩, where ⟨A, μ_{A}⟩ is a semigroup in the category and where u_{A} : e ⟶ A is a morphism such that the following unit condition holds

μ_{A} ∘ (1_{A} ⊗ u_{A}) = γ_{A}
μ_{A} ∘ (u_{A} ⊗ 1_{A}) = β_{A}

We can now prove that monoids are universal semigroups in any monoidal category.

Proposition 8. Let ⟨A, μ_{A}, u_{A}⟩ be a monoid in a monoidal category C. Then ⟨A, μ_{A}⟩ is a universal semigroup.

Proof. We want to show that ⟨A, μ⟩ is a universal cocone on the diagram S_{A}.

A ⊗ (A ⊗ A) \overline{^{1_{A} ⊗ μ_{A}}} A ⊗ A
[figure: μ_{A} ⊗ 1_{A}]

(A ⊗ A) ⊗ A ^{α_{A,A,A}}

This means that the equation

φ ∘ μ_{A} = f

should have a unique solution for all f : A ⊗ A ⟶ A such that f ∘ (1_{A} ⊗ μ_{A}) = f ∘ (μ_{A} ⊗ 1_{A}) ∘ α_{A,A,A}. Define a map δ_{A} : A → A ⊗ A by δ_{A} = (u_{A} ⊗ 1_{A}) ∘ β_{A}^{−1} . Then we have

μ_{A} ∘ δ_{A} = μ_{A} ∘ (u_{A} ⊗ 1_{A}) ∘ β_{A}^{−1} = β_{A} ∘ β_{A}^{−1} = 1_{A}

This identity show that the equation φ ∘ μ = f can have at most one solution and this solution must be φ = f ∘ δ_{A}. The proof is complete if we can show that this φ really is a solution. From the naturality of α, β and the MacLane coherence conditions we have the following identities

(u_{A} ⊗ (1_{A} ⊗ 1_{A})) ∘ α_{e,A,A}^{−1} = α_{A,A,A}^{−1} ∘ ((u_{A} ⊗ 1_{A}) ⊗ 1_{A}),
β_{A}^{−1} ∘ μ_{A} = (1_{e} ⊗ μ_{A}) ∘ β_{A⊗A}^{−1},
β_{A⊗A}^{−1} = α_{e,A,A}^{−1} ∘ (β_{A}^{−1} ⊗ 1_{A}).

But then we have

φ ∘ μ_{A} = f ∘ δ_{A} ∘ μ_{A}
= f ∘ (u_{A} ⊗ 1_{A}) ∘ β_{A}^{−1} ∘ μ_{A}
= f ∘ (u_{A} ⊗ 1_{A}) ∘ (1_{e} ⊗ μ_{A}) ∘ β_{A⊗A}^{−1}
= f ∘ (u_{A} ⊗ μ_{A}) ∘ β_{A⊗A}^{−1}
= f ∘ (1_{A} ⊗ μ_{A}) ∘ (u_{A} ⊗ (1_{A} ⊗ 1_{A})) ∘ β_{A⊗A}^{−1}
= f ∘ (1_{A} ⊗ μ_{A}) ∘ (u_{A} ⊗ (1_{A} ⊗ 1_{A})) ∘ α_{e,A,A}^{−1} ∘ (β_{A}^{−1} ⊗ 1_{A})
= f ∘ (1_{A} ⊗ μ_{A}) ∘ α_{A,A,A}^{−1} ∘ ((u_{A} ⊗ 1_{A}) ⊗ 1_{A}) ∘ (β_{A}^{−1} ⊗ 1_{A})
= f ∘ (μ_{A} ⊗ 1_{A}) ∘ ((u_{A} ⊗ 1_{A}) ⊗ 1_{A}) ∘ (β_{A}^{−1} ⊗ 1_{A})
= f ∘ (μ_{A} ⊗ 1_{A}) ∘ (δ_{A} ⊗ 1_{A})
= f,

so φ = f ∘ δ_{A} is a solution □

Proposition 5 and 8 show that in any monoidal category the class of universal semigroups includes all monoids and is included in the class of semigroups where the product is an epimorphism. The following proposition show that one can construct simple universal semigroups in most monoidal categories.

Proposition 9. Let A be a object and assume that there is an arrow ϵ_{A} : A ⟶ e from A to the neutral object e in the category. Define an arrow μ_{A} : A ⊗ A ⟶ A by

μ_{A} = β_{A} ∘ (ϵ_{A} ⊗ 1_{A})

Then ⟨A, μ_{A}⟩ is a semigroup. It is a universal semigroup if μ_{A} has a right inverse Proof. It is easy to show, using the monoidal structure and naturality, that the following set of identities holds

(β_{A} ⊗ 1_{A}) ∘ ((1_{e} ⊗ ϵ_{A}) ⊗ 1_{A}) = (ϵ_{A} ⊗ 1_{A}) ∘ (β_{A} ⊗ 1_{A}),
((ϵ_{A} ⊗ ϵ_{A}) ⊗ 1_{A}) ∘ α_{A,A,A} = α_{e,e,A} ∘ (ϵ_{A} ⊗ (ϵ_{A} ⊗ 1_{A})),
(β_{A} ⊗ 1_{A}) ∘ α_{e,e,A} = (1_{e} ⊗ β_{A}).

Using these identities we have

μ_{A}^{l} ∘ (μ_{A}^{l} ⊗ 1_{A}) ∘ α_{A,A,A}
= β_{A} ∘ (ϵ_{A} ⊗ 1_{A}) ∘ (β_{A} ∘ (ϵ_{A} ⊗ 1_{A}) ⊗ 1_{A}) ∘ α_{A,A,A}
= β_{A} ∘ (ϵ_{A} ⊗ 1_{A}) ∘ (β_{A} ⊗ 1_{A}) ∘ ((ϵ_{A} ⊗ 1_{A}) ⊗ 1_{A}) ∘ α_{A,A,A}
= α_{A} ∘ (β_{e} ⊗ 1_{A}) ∘ ((1_{e} ⊗ ϵ_{A}) ⊗ 1_{A}) ∘ ((ϵ_{A} ⊗ 1_{A}) ⊗ 1_{A}) ∘ α_{A,A,A}
= β_{A} ∘ (β_{e} ⊗ 1_{A}) ∘ ((ϵ_{A} ⊗ ϵ_{A}) ⊗ 1_{A}) ∘ α_{A,A,A}
= β_{A} ∘ (β_{e} ⊗ 1_{A}) ∘ α_{e,e,A} ∘ (ϵ_{A} ⊗ (ϵ_{A} ⊗ 1_{A}))
= β_{A} ∘ (1_{e} ⊗ β_{A}) ∘ (ϵ_{A} ⊗ (ϵ_{A} ⊗ 1_{A}))
= β_{A} ∘ (ϵ_{A} ⊗ β_{A} ∘ (ϵ_{A} ⊗ 1_{A}))
= β_{A} ∘ (ϵ_{A} ⊗ 1_{A}) ∘ (1_{A} ⊗ β_{A} ∘ (ϵ_{A} ⊗ 1_{A}))
= μ_{A}^{l} ∘ (1_{A} ⊗ μ_{A}^{l}),

and this proves that ⟨A, μ_{A}⟩ is a semigroup. In order to prove that it is universal when μ_{A} has a right inverse we have to prove that the equation φ ∘ μ_{A} = f has a unique solution φ : A ⟶ B for all f : A ⊗ A ⟶ B such that f ∘ (1_{A} ⊗ μ_{A}) = f ∘ (μ_{A} ⊗ 1_{A}) ∘ α_{A,A,A}. Let the right inverse for μ_{A} be δ_{A}. We thus have the identity μ_{A} ∘ δ_{A} = 1_{A}. But then the only possible solution of the equation φ ∘ μ_{A} = f is φ = f ∘ δ_{A}. We must now show that this in fact is a solution. The following identities follow from unit coherence and naturality

δ_{A} ∘ β_{A} = β_{A⊗A} ∘ (1_{e} ⊗ δ_{A}),
α_{e,A,A} ∘ (ϵ_{A} ⊗ (1_{A} ⊗ 1_{A})) = ((ϵ_{A} ⊗ 1_{A}) ⊗ 1_{A}) ∘ α_{A,A,A},
(β_{A} ⊗ 1_{A}) ∘ α_{e,A,A} = β_{A⊗A}.

But then we have

f ∘ δ_{A}^{l} ∘ μ_{A}^{l}
= f ∘ δ_{A}^{l} ∘ β_{A} ∘ (ϵ_{A} ⊗ 1_{A})
= f ∘ β_{A⊗A} ∘ (1_{e} ⊗ δ_{A}^{l}) ∘ (ϵ_{A} ⊗ 1_{A})
= f ∘ β_{A⊗A} ∘ (ϵ_{A} ⊗ δ_{A}^{l})
= f ∘ β_{A⊗A} ∘ (ϵ_{A} ⊗ (1_{A} ⊗ 1_{A})) ∘ (1_{A} ⊗ δ_{A}^{l})
= f ∘ (β_{A} ⊗ 1_{A}) ∘ α_{e,A,A} ∘ (ϵ_{A} ⊗ (1_{A} ⊗ 1_{A})) ∘ (1_{A} ⊗ δ_{A}^{l})
= f ∘ (β_{A} ⊗ 1_{A}) ∘ ((ϵ_{A} ⊗ 1_{A}) ⊗ 1_{A}) ∘ α_{A,A,A} ∘ (1_{A} ⊗ δ_{A}^{l})
= f ∘ (μ_{A}^{l} ⊗ 1_{A}) ∘ α_{A,A,A} ∘ (1_{A} ⊗ δ_{A}^{l})
= f ∘ (1_{A} ⊗ μ_{A}^{l}) ∘ (1_{A} ⊗ δ_{A}^{l})
= f.
□

A special case of the previous proposition is

Corollary 10. Assume that there exists a map u_{A} : e ⟶ A such that ϵ_{A} ∘ u_{A} = 1_{e}. Then ⟨A, μ_{A}⟩ is a universal semigroup.

Proof. Define an arrows δ_{A} : A ⟶ A ⊗ A by

δ_{A} = (u_{A} ⊗ 1_{A}) ∘ β_{A}^{−1}

Then we have

μ_{A} ∘ δ_{A} = β_{A} ∘ (ϵ_{A} ⊗ 1_{A}) ∘ (u_{A} ⊗ 1_{A}) ∘ β_{A}
= β_{A} ∘ (ϵ_{A} ∘ u_{A} ⊗ 1_{A}) ∘ β_{A}^{−1}
= β_{A} ∘ β_{A}^{−1} = 1_{A}.

But this show that δ_{A} is the right inverse of μ_{A} and the result follows from the previous proposition. □

In a similar way we find that ⟨A, μ_{A}⟩ is a universal semigroup if we define

μ_{A} = β_{A} ∘ (1_{A} ⊗ ϵ_{A})

and assume that μ_{A} has a right inverse. Note that in general the universal semigroups described in these propositions are not monoids.

Let us consider a few examples of the previous construction. Let us first consider the case of Sets with the cartesian product as monoidal structure and neutral object given by the set T = {∗}. Since T is the terminal object in Sets there exists a unique map ϵ_{A} : A ⟶ T from any set A to the terminal T. This map is obviously defined by ϵ_{A}(x) = ∗ for all x ∈ A. In Sets the natural isomorphism β_{A} : T × A ⟶ A is given by β_{A}(∗, x) = x. Let x_{0} be any element in A and define a map u_{A} : T ⟶ A by u_{A}(∗) = x_{0}. Then ϵ_{A} ∘ u_{A} = 1_{e} and we can conclude that ⟨A, μ_{A}⟩ is a universal semigroups. The product is explicitly given by

μ_{A}(x, y) = [β_{A} ∘ (ϵ_{A} × 1_{A})](x, y) = β_{A}(∗, y) = y

so μ_{A} is the projection on the second factor. We have previously shown directly that these maps gives universal semigroups.

Let V ect_{ℝ}be the category of real vector spaces with linear maps as arrows.In this category tensor product of vector spaces ⊗ = ⊗_{ℝ} is a monoidal structure. The neutral object is ℝ and the arrows β_{A} and its inverse β_{A}^{−1}are given by

β_{A}(r ⊗ a) = ra
β_{A}^{−1}(a) = 1 ⊗ a

where r is any real number, a a element in A and 1 is the unit in ℝ. Note that a semigroup in this category is a real associative algebra.

Let A be a vector space with a positive definite inner product I_{A} : A ⊗ A ⟶ ℝ and let a ∈ A. Define a map ϵ_{A} : A ⟶ ℝ by

ϵ_{A}(b) = I_{A}(a ⊗ b)

This is clearly a linear map. Define now the map μ_{A} : A ⊗ A ⟶ A by μ_{A} = β_{A} ∘ (ϵ_{A} ⊗ 1_{A}). Explicitly we have

μ_{A}(x ⊗ y) = β_{A}(I_{A}(a ⊗ x) ⊗ y) = I_{A}(a ⊗ x)y

Then the general theory show that ⟨A, μ_{A}⟩ is a semigroup in V ect_{ℝ}.. Now define a map u_{A} : ℝ ⟶ A by

u_{A}(r) = \frac{ra}{I_{A}(a, a)}

Then u_{A} is a linear map and

(ϵ_{A} ∘ u_{A})(r) = ϵ_{A}(\frac{ra}{I_{A}(a, a)}) = \frac{r}{I_{A}(a, a)}ϵ_{A}(a) = r

so we have ϵ_{A} ∘ u_{A} = 1 and the semigroup ⟨A, μ_{A}⟩ is in fact a universal semigroup. This example be generalized in the following way. We consider a subcategory of topological vector spaces over a field F that is closed with respect to tensor product. Such subcategories certainly exists. Let α be a continuous linear functional α : A ⟶ ℝ. The morphism ϵ_{A} : A ⟶ F is now defined by

ϵ_{A}(a) = α(a)

and the product in the semigroup ⟨A, μ_{A}⟩ is

μ_{A}(a ⊗ a′) = α(a)b

The resulting semigroup is a universal semigroup because the continuous linear map u_{A} : F ⟶ A defined by u_{A}(r) = r(α(b))^{−1}b,where b ∈ A and α(b) ≠ 0, satisfy ϵ_{A} ∘ u_{A} = 1_{A}.

We have seen that in general the class of universal semigroups is strictly larger than the class of monoids. We also know that the class of universal semigroups is included in the class of semigroups with a product rule that is an epimorphisms. In general this last inclusion is also proper as the following example show.

Let A be a real vector space of dimension three with basis {i, j, k}. Define a product, μ on A by the following multiplication table

μ i j k i i 0 k j j 0 0 k 0 0 0 .

By direct calculation one can show that ⟨A, μ⟩ is a semigroup in V ect_{ℝ},or in other words, a real associative algebra. The product is clearly an epimorphism. The chosen basis for A induce in the usual way a basis for A⊗A. Let now V be a one dimensional vector space and let f : A⊗A → V be a linear map that is zero on all basis vectors except on the vector j ⊗ k where it has a nonzero value. It is a straight forward calculation to show that ⟨V, f⟩ is a cocone on the diagram S_{A}. If φ is a solution of the equation φ ∘ μ = f we must have

φ(i) = φ(μ(i ⊗ i)) = f(i ⊗ i) = 0,
φ(j) = φ(μ(j ⊗ i)) = f(j ⊗ i) = 0,
φ(k) = φ(μ(i ⊗ k)) = f(i ⊗ k) = 0,

so φ = 0. But φ = 0 is not a solution of the equation φ ∘ μ = f because f ≠ 0. The equation φ ∘ μ = f therefore have no solution for the cocone ⟨V, f⟩ and then by definition ⟨A, μ⟩ is not a universal algebra.

2.2. Examples from the category of Banach spaces. Let B be the category where objects are Banach spaces and where morphisms are bounded linear maps. The Banach spaces are vector spaces over a field F where F is ℝ or ℂ. We introduce a monoidal structure in this category by defining X ⊗ Y to be the projective tensor product[4] of Banach spaces. The unit object for this product is the Banach space F and the natural isomorphisms α,β and γ are the standard ones. The standard interpretation of the projective tensor product in terms of properties of bilinear maps[5] shows that ⟨A, μ⟩ is a semigroups in B iff it is a Banach algebra. Recall that an approximate unit[3] in a Banach algebra A is a net {u_{λ}}_{λ∈Λ} such that ∥u_{λ}∥ ≤ 1 for all λ ∈ Λ and such that for all elements a ∈ A we have

lim_{λ} u_{λ}a = a,
lim_{λ} au_{λ} = a.

Recall that a morphism f : X → Y in any category is a epimorphism iff for all morphisms g, h : Y → Z with f ∘ g = f ∘ h we have g = h. In the category of Sets epimorphisms are exactly the surjective maps. In the category of Banach spaces it is easy to show that any morphism with a dense image is a epimorphism. Using this observation we first prove the following result

Proposition 11. Let ⟨A, μ⟩ be a Banach algebra with an approximate unit. Then μ : A ⊗ A → A is a epimorphism.

Proof. Let m = μ(A ⊗ A) ⊂ A be the image of μ. We must show that the image is dense in A. Let a ∈ A and let {u_{λ}} be an approximate unit in A. Define a net a_{λ} = u_{λ}a in A. Then we have by the defining property of an approximate unit that

lim_{λ} a_{λ} = a.

But this means that the closure of m is A and thus that m is dense in A. □

Therefore Banach algebras with an approximate unit is included in the class of algebras where the product is an epimorphism. We now use this result to prove our main result in this section.

Theorem 12. Let ⟨A, μ⟩ be a Banach algebra with an approximate unit. Then ⟨A, μ⟩ is a universal semigroup in the category of Banach spaces.

Proof. Let {u_{λ}}_{λ∈Λ} be the approximate unit. For each λ ∈ Λ define a linear map δ_{λ} : A → A ⊗ A by

δ_{λ}(a) = u_{λ} ⊗ a.

Then since the projective norm is a cross norm we have

∥δ_{λ}(a)∥ = ∥u_{λ} ⊗ a∥ = ∥u_{λ}∥∥a∥ ≤ ∥a∥,

and therefore each δ_{λ} is bounded with ∥δ_{λ}∥ ≤ 1. Let now B be any other Banach space and let f : A ⊗ A → B be a bounded linear map that satisfy

f ∘ (μ ⊗ 1_{A}) = f ∘ (1_{A} ⊗ μ).

Note that because of linearity and continuity of f this condition holds iff

f(ab ⊗ c) = f(a ⊗ bc),

holds for all elements a, b and c in A.

We must show that the equation φ ∘ μ = f has one and only one solution. We know from proposition 11 that μ is a epimorphism so there can be at most one solution. For each λ ∈ Λ define φ_{λ} : A → B by

φ_{λ} = f ∘ δ_{λ}.

This map is continuous and

∥φ_{λ}∥ ≤ ∥f∥∥δ_{λ}∥ ≤ ∥f∥,

so the family of maps {φ_{λ}}_{λ∈Λ} is uniformly bounded in λ by ∥f∥. Let

m = {∑_{n} a_{n}b_{n} | a_{n}, b_{n} ∈ A} ⊂ μ(A ⊗ A).

Then for a ∈ m we have

φ_{λ}(a) = φ_{λ}(∑_{n} a_{n}b_{n}) = f(u_{λ} ⊗ ∑_{n} a_{n}b_{n}) = f(∑_{n} u_{λ}a_{n} ⊗ b_{n}),

and therefore by continuity

lim_{λ} φ_{λ}(a) = lim_{λ} f(∑_{n} u_{λ}a_{n}⊗b_{n}) = f(∑_{n} lim_{λ}(u_{λ}a_{n})⊗b_{n}) = f(∑_{n} a_{n}⊗b_{n}).

Thus lim_{λ} φ_{λ}(a) exists for all a ∈ m. We know that

R = {∑_{n} a_{n} ⊗ b_{n} | a_{n}, b_{n} ∈ A},

is dense in A⊗A. Therefore m = μ(R) is dense in A and so m is dense in A. Let now a ∈ A. Then there exists a sequence {a_{n}} in m with a_{n} → a and if λ, β ∈ Λ we have

∥φ_{λ}(a) − φ_{β}(a)∥ = ∥φ_{λ}(a) − φ_{λ}(a_{n}) + φ_{λ}(a_{n}) − φ_{β}(a) − φ_{β}(a_{n}) + φ_{β}(a_{n})∥
≤ ∥φ_{λ}(a − a_{n})∥ + ∥φ_{λ}(a_{n}) − φ_{β}(a_{n})∥ + ∥φ_{β}(a) − φ_{β}(a_{n})∥
≤ (∥φ_{λ}∥ + ∥φ_{β}∥)∥a − a_{n}∥ + ∥φ_{λ}(a_{n}) − φ_{β}(a_{n})∥
≤ 2∥f∥∥a − a_{n}∥ + ∥φ_{λ}(a_{n}) − φ_{β}(a_{n})∥.

Let now n be fixed and so large that 2∥f∥∥a − a_{n}∥ < \frac{1}{2}ϵ. Since a_{n} ∈ m we have

a_{n} = ∑_{j=1}^{p} a_{j}^{n}b_{j}^{n},

and therefore

φ_{λ}(a_{n}) − φ_{β}(a_{n}) = f(u_{λ} ⊗ ∑_{j=1}^{p} a_{j}^{n}b_{j}^{n}) − f(u_{β} ⊗ ∑_{j=1}^{p} a_{j}^{n}b_{j}^{n})

= f(∑_{j=1}^{p}(u_{λ}a_{j}^{n} − u_{β}a_{j}^{n}) ⊗ b_{j}^{n}),

and so

∥φ_{λ}(a_{n}) − φ_{β}(a_{n})∥ ≤ ∥f∥M_{n} ∑_{j=1}^{p} ∥u_{λ}a_{j}^{n} − u_{β}a_{j}^{n}∥,

where we have assumed without loss of generality that ∥b_{j}^{n}∥ ≤ M_{n} for j = 1...p. But u_{λ} is an approximate unit and therefore u_{λ}a_{j}^{n} converges ∀j and is thus Cauchy since A is a Banach space. But then there exists λ_{j} ∈ Λ j = 1..p such that

∥u_{λ}a_{j}^{n} − u_{β}a_{j}^{n}∥ < \frac{1}{2pϵ∥f∥M_{n}} .

Since Λ is a directed set there exists a element λ_{0} such that all these inequalities holds when λ, β > λ_{0}. But then for such λ and β we have using the previously derived inequalities that

∥φ_{λ}(a) − φ_{β}(a)∥ < ϵ,

and this means that the net {φ_{λ}(a)}_{λ∈Λ} is Cauchy and therefore converges. We can now define a map φ : A → B by

φ(a) = lim_{λ} φ_{λ}(a).

This map is linear and bounded because

∥φ(a)∥ = lim_{λ} ∥φ_{λ}(a)∥ ≤ ∥f∥∥a∥.

But for any element ξ = ∑_{j} a_{j} ⊗ b_{j} in the dense set R ⊂ A ⊗ A we have

φ(μ(ξ)) = lim_{λ} φ_{λ}(∑_{j} a_{j}b_{j}) = lim_{λ} f(∑_{j} u_{λ}a_{j} ⊗b_{j}) = f(∑_{j} a_{j} ⊗b_{j}) = f(ξ).

Therefore the bounded maps φ ∘ μ and f agree on a dense set and we can conclude that

φ ∘ μ = f,

and this shows that the algebra A is universal. □

Universal Banach algebras thus include all Banach algebras with an approximate unit. On the other hand it is not hard to construct Banach algebras without unit that are universal, so the inclusion is proper. Let ⟨A, μ⟩ be a three dimensional algebra with basis {i, j, k} and table of multiplication given by

μ i j k i i 0 0 j j 0 0 k 0 0 k .

By direct calculation one can easily show that this algebra is universal but does not have a unit (or approximate unit).

3. Universal cosemigroups

To any categorical concept described in terms of diagrams there is a dual concept that we get by reversing all arrows. For the notion of a universal semigroup this procedure leads to the notion of a universal cosemigroup. Let ⟨C, ⊗, α⟩ be a semimonoidal category. A cosemigroup in C is a pair ⟨A, δ_{A}⟩ where A is a object in C and δ_{A} : A ⟶ A ⊗ A is a arrow in C and where the following diagram commute.

[figure: A ⊗ (A ⊗ A); 1_{A} ⊗ δ_{A}; A ⊗ A; δ_{A}; A]
[figure: δ_{A} ⊗ 1_{A}]

(A ⊗ A) ⊗ A ^{α_{A,A,A}}

Let CS_{A} be the diagram we get by removing the right-hand node.

A ⊗ (A ⊗ A) ^{1_{A} ⊗ δ_{A}} A ⊗ A
[figure: δ_{A} ⊗ 1_{A}]

(A ⊗ A) ⊗ A ^{α_{A,A,A}}

Evidently ⟨A, δ_{A}⟩ is a cosemigroup iff ⟨A, δ_{A}⟩ is a cone on the diagram CS_{A}. In general a cosemigroup does not give a universal cone.

By definition ⟨B, f⟩ is a universal cone on the diagram CS_{A} if it is a cone and if for all cones ⟨C, g⟩ there exists a unique arrow φ : C ⟶ B in C such that

f ∘ φ = g

Definition 13. A universal cosemigroup in a semimonoidal category ⟨ C, ⊗, α⟩ is a cosemigroup ⟨A, δ_{A}⟩ such that ⟨A, δ_{A}⟩ is a universal cocone on the diagram CS_{A}.

3.1. Examples from Sets and general monoidal categories. We have the following result that show that not all cosemigroups are universal.

Proposition 14. Let ⟨A, δ_{A}⟩ be a universal cosemigroup. Then δ_{A} is a monomorphism.

Proof. Let B be any object in the category and let f, g : B ⟶ A be two arrows and assume that δ_{A}∘ f = δ_{A} ∘ g. Let h = δ_{A} ∘ f. Then the equation

δ_{A} ∘ φ = h

has both f and g as solutions. Since ⟨A, δ_{A}⟩ is a universal cosemigroup we must have f = g and this proves that δ_{A} is a monomorphism. □

We will now illustrate this definition with several examples. Let us first work in the category Sets. This is a monoidal category with cartesian product as product bifunctor and with trivial associativity constraint. Let A be any set and define a map δ_{A} : A ⟶ A × A by δ_{A}(x) = (x, x). This is the diagonal map in Sets. We have

[(1_{A} × δ_{A}) ∘ δ_{A}](x) = (1_{A} × δ_{A})(x, x) = (x, x, x)
[(δ_{A} × 1_{A}) ∘ δ_{A}](x) = (δ_{A} × 1_{A})(x, x) = (x, x, x)

so ⟨A, δ_{A}⟩ is a cone. Let ⟨B, f⟩ be any cone and consider the system

δ_{A} ∘ φ = f

Since ⟨B, f⟩ is a cone we have (1_{A} × δ_{A}) ∘ f = (δ_{A} × 1_{A}) ∘ f. We have f : B ⟶ A × A so we can write f(x) = (f_{1}(x), f_{2}(x)). The cone condition then gives

(f_{1}(x), f_{2}(x), f_{2}(x)) = (f_{1}(x), f_{1}(x), f_{2}(x))

so we must have f_{1}(x) = f_{2}(x)for all x in A. Let φ be any solution to δ_{A} ∘ φ = f. Then we have (φ(x), φ(x)) = (f_{1}(x), f_{2}(x)) or φ = f_{1}. So there is at most one solution. Let φ = f_{1} then we find

[δ_{A} ∘ φ](x) = δ_{A}(f_{1}(x)) = (f_{1}(x), f_{1}(x)) = (f_{1}(x), f_{2}(x)) = f(x)

so φ = f_{1} is a solution. We have therefore proved

Proposition 15. Let δ_{A} : A ⟶ A × A be the diagonal map of sets. Then ⟨A, δ_{A}⟩ is a universal cosemigroup in Sets.

As our next example let us consider a pointed set. This is a set A with a chosen point x_{0} ∈ A. Define a map δ_{A} : A ⟶ A×A by δ_{A}(x) = (x_{0}, x). Then we have

[(1_{A} × δ_{A}) ∘ δ_{A}](x) = (1_{A} × δ_{A})(x_{0}, x) = (x_{0}, x_{0}, x)
[(δ_{A} × 1_{A}) ∘ δ_{A}](x) = (δ_{A} × 1_{A})(x_{0}, x) = (x_{0}, x_{0}, x)

so ⟨A, δ_{A}⟩ is a cone. Let ⟨B, f⟩ be any cone. This means that f : B ⟶ A × A and (1_{A} × δ_{A}) ∘ f = (δ_{A} × 1_{A}) ∘ f. As above we write f(x) = (f_{1}(x), f_{2}(x)) and the cone condition becomes

(f_{1}(x), x_{0}, f_{2}(x)) = (x_{0}, f_{1}(x), f_{2}(x))

so we must have f_{1}(x) = x_{0}. But then we have

[δ_{A} ∘ φ](x) = δ_{A}(φ(x)) = (x_{0}, φ(x)) = (f_{1}(x), f_{2}(x)) = f(x)

if we choose φ(x) = f_{2}(x). This is clearly the only solution. We therefore have

Proposition 16. Let ⟨A, x_{0}⟩ be a pointed set. Define δ_{A} : A ⟶ A × A by δ_{A}(x) = (x_{0}, x). Then ⟨A, δ_{A}⟩ is a universal cosemigroup.

The map δ_{A} : A ⟶ A×A defined by δ_{A}(x) = (x, x_{0}) will similarly give a universal cosemigroup. If ⟨A, μ_{A}, e⟩ is a monoid we get two universal cosemigroups by choosing x_{0} = e.

Finite sets offer many examples of universal cosemigroups. Consider the set A = {a, b, c}. Define δ_{A} : A ⟶ A × A by

δ_{A}(a) = (a, a)
δ_{A}(b) = (b, a)
δ_{A}(c) = (c, c)

Then we have

[(δ_{A} × 1_{A}) ∘ δ_{A}](a) = (δ_{A} × 1_{A})(a, a) = (a, a, a)
[(1_{A} × δ_{A}) ∘ δ_{A}](a) = (1_{A} × δ_{A})(a, a) = (a, a, a)
[(δ_{A} × 1_{A}) ∘ δ_{A}](b) = (δ_{A} × 1_{A})(b, a) = (b, a, a)
[(1_{A} × δ_{A}) ∘ δ_{A}](b) = (1_{A} × δ_{A})(b, a) = (b, a, a)
[(δ_{A} × 1_{A}) ∘ δ_{A}](c) = (δ_{A} × 1_{A})(c, c) = (c, c, c)
[(1_{A} × δ_{A}) ∘ δ_{A}](c) = (1_{A} × δ_{A})(c, c) = (c, c, c)

so ⟨A, δ_{A}⟩ is a cone. Next let ⟨B, f⟩ be any cone. This implies that (δ_{A} × 1_{A}) ∘ f = (1_{A} × δ_{A}) ∘ f. Direct computation show that (δ_{A} × 1_{A})(x, y) = (1_{A} × δ_{A})(x, y) iff (x, y) is a element in the range of δ_{A}. We therefore can conclude that ⟨B, f⟩ is a cone if the image of f is included in the range of δ_{A}. Since δ_{A} is injective there is at most one solution to the equation δ_{A} ∘ φ = f. Let b be any element in B. Then there exists a unique element a in A such that δ_{A}(a) = f(b). This is true since the range of f is equal to the range of δ_{A} and δ_{A} is injective. Define φ(b) = a, then φ is well defined and clearly a solution to the equation δ_{A} ∘ φ = f. This proves that ⟨A, δ_{A}⟩ is a universal cosemigroup.

We have seen that a necessary condition for a cosemigroup ⟨A, δ_{A}⟩ to be a universal cosemigroup is that δ_{A} is a monomorphism. This is not sufficient in general. Let A = {a, b, c} and define a map δ_{A} : A ⟶ A×A by

δ_{A}(a) = (a, a),
δ_{A}(b) = (b, a),
δ_{A}(c) = (a, c).

The map δ_{A} is a monomorphism and a direct calculation like the one above show that ⟨A, δ_{A}⟩ is a cosemigroup but that it is not a universal cosemigroup. In general one can prove by dualizing the proof for monoids that comonoid in a monoidal category are universal cosemigroups. Thus universal cosemigroups is a class that contains all comonoids and is included in all cosemigroups with a coproduct that is a monomorphism.

References

[1] S. MacLane, Categories for the Working Mathematician, Springer 1991.

[2] R. V. Kadison, Fundamentals of The Theory of Operator Algebras, Volume 1, Academic Press,1983.

[3] C. E. Richart,Van Nostrand, Banach Algebras, 1960.

[4] G. Köthe, Topological Vector Spaces II, Springer,1979.

[5] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Memoires American Mathematical Society, 16, 1955.

[6] Hilja Lisa Huru, Quantization and associativity constraints of the category of graded modules, Cand. Scient. Thesis in Mathematics,University of Tromso,2002.

University of Tromso,9020 Tromso, Norway E-mail address: perj@math.uit.no

Received May 24, 2005