QUANTALES

JAN PASEKA and JIŘÍ ROSICKÝ

Department of Mathematics, Masaryk University, Janáčkovo nám. 2a, 66295 Brno, Czech Republic

           e-mail: {paseka, rosicky}@math.muni.cz

1 Introduction

The term quantale was suggested by C.J. Mulvey at the Oberwolfach Category Meeting (see [14]) as ”a quantization” of the term locale. Locales form an ordertheoretic counterpart of topological spaces and are therefore able to describe commutative C^{∗}-algebras. More precisely, a commutative C^{∗}-algebra A is completely determined by its lattice I(A) of closed ideals which is order isomorphic to the lattice of open sets of its spectrum. The aim of C.J. Mulvey has been to find a substitute of locales which could play the same role for general C^{∗}-algebras. He was motivated by C.A. Akemann [1], [2] and R. Giles and H. Kummer [7] who had found a structure on the lattice R(A) of all closed right ideals of a C^{∗}-algebra A which makes possible to determine A. The suggestion of C.J. Mulvey was to consider R(A) as a lattice equippped with a multiplication. In the commutative case, this multiplication is the meet operation, i.e., the quantale R(A) is the locale I(A).

The concept of a quantale is, of course, much older. It goes back to 1930’s when M. Ward and R.P. Dilworth have realized that the theory of ideals in rings can be conveniently formulated in terms of a lattice of ideals equipped with a multiplication (see [24], [25], [26] and [5]). Even, as L. Skula has informed us, this idea is contained in W. Krull [12]. An important moment in the development of the theory of quantales was the realization that quantales give a semantics for propositional linear logic in the same way as Boolean algebras give a semantics for classical propositional logic (see [8]). In fact, the appearance of quantales in linear logic was anticipated by the use of multiplicative structures in relevance logic and Lambek calculus.

Nowadays, there is a monograph of K. Rosenthal [20] devoted to quantales. Our aim is to survey the theory of quantales, mainly in their connections with C^{∗}-algebras. We also add some recent developments.

2 Locales and quantales

In what follows, a complete lattice will be called a sup-lattice. Sup-lattice homomorphisms are maps between sup-lattices preserving arbitrary joins. Evidently, any sup-lattice isomorphism is a complete lattice isomorphism (i.e., it preserves all meets). A sup-lattice homomorphism will be called strong if it preserves the top element 1. Any sup-lattice homomorphism f : L_{1} → L_{2} between sup-lattices has a right adjoint f_{∗} : L_{2} → L_{1} defined by

f_{∗}(a) = ⋁{s ∈ L_{1}|f(s) ≤ a}

for all a ∈ L_{2}. Then f_{∗} preserves arbitrary meets.

A frame is a sup-lattice L in which the infinite distributive law

a ∧ ⋁ S = {a ∧ s|s ∈ S}

holds for all a ∈ L, S ⊆ L. In fact, frames are exactly complete Heyting algebras and the Heyting implication is given by

a ⇒ b = ⋁{s ∈ L|s ∧ a ≤ b}.

Frame homomorphisms are defined as sup-lattice homomorphisms between frames preserving finite meets (including the empty one), especially they preserve the distinguished elements 0 (the bottom) and 1 (the top). The study of frames was motivated by the following observation. Let X be a topological space and Ω(X) the complete lattice of open subsets of X. Since the finite meets and arbitrary joins correspond to the set-theoretic operations of intersection and union, Ω(X) is a frame. Moreover, if f : X → Y is a continuous map of spaces, the induced map f^{−1} : Ω(Y) → Ω(X) is a frame homomorphism. The functor X ↦ Ω(X), f ↦ f^{−1} is contravariant and thus the category of ”generalized spaces” should be the opposite of the category of frames. Its objects are then called locales and morphisms locale maps.

Recall that a point of a space X may be identified with a continuous map from the one-point space 1 to X. Therefore, a point of a frame L is, alternatively, a frame homomorphism h : L → 2 into the two-element frame 2 = {0, 1} ≅ Ω(1). It is easy to see that points h : L → 2 correspond to prime elements p ∈ L, i.e., to elements 1 ≠ p ∈ L such that

x ∧ y ≤ p ⟹ x ≤ p or y ≤ p

for any x, y ∈ L. This correspondence is given by

p = ⋁{x ∈ L|h(x) = 0}.

The spectrum P (L) of a frame L is a space whose elements are primes of L and whose open sets are the sets {p ∈ P (L)|x ≰ p} where x ∈ L. L is said to be spatial if the surjective frame homomorphism x ↦ {p ∈ P (L)|x ≰ p} is an isomorphism. It happens if and only if any element x ∈ L is a meet of primes, i.e. if L has ”enough” points to separate distinct elements of L.

One can treat frames as generalized topological spaces. For instance, a frame L is called compact if, for any S ⊆ L with 1 = ⋁ S, there is a finite subset S′ ⊆ S such that 1 = ⋁ S′. Let a, b ∈ L. We say that b is rather below a, and write b ⊲ a, if there is an element c ∈ L such that c ∧ b = 0, c ∨ a = 1. A frame L in which each element is a join of elements rather below it is said to be regular. Then a topological space X is compact (regular) iff the frame Ω(X) is compact (regular). Consult P.T. Johnstone [9] for more information about frames.

Replacing in the definition of a frame the commutative operation ∧ with a general associative binary operation, we obtain a ”non-commutative generalized space”. Namely, by a quantale, we shall mean a sup-lattice Q equipped with an associative multiplication · which distributes over arbitrary joins:

a · ⋁ S = ⋁{a · s|s ∈ S} and ⋁ S · a = ⋁{s · a|s ∈ S}

for all a ∈ Q, S ⊆ Q. A quantale homomorphism is defined as a sup-lattice homomorphism between quantales preserving the multiplication. If a quantale Q is a monoid then the unit will be denoted by e and such a Q is called unital. A quantale homomorphism is said to be unital if it preserves the unit e. Let us mention that, despite the fact that the term ”quantale” is made up from that of a ”locale”, our quantale homomorphisms correspond to frame homomorphisms and not to locale maps.

Let x be an element of a quantale Q. It is called idempotent if x · x = x and right-sided (left-sided) if x·1 ≤ x (1·x ≤ x). It is called two-sided if it is both rightsided and left-sided. The set of all right-sided (left-sided, two-sided, idempotent) elements of a quantale Q is denoted by R(Q) (L(Q), I(Q), E(Q)). We shall say that a quantale Q is right-sided (left-sided, two-sided, idempotent, right-idempotent, leftidempotent) if Q = R(Q) (Q = L(Q), Q = I(Q), Q = E(Q), R(Q) ⊆ E(Q), L(Q) ⊆ E(Q)). A subset T ⊆ Q is called a subquantale of Q if it is closed under all joins and multiplication in Q. It may be remarked that R(Q), L(Q) and I(Q) are subquantales of Q closed under arbitrary meets, Q · R(Q) ⊆ R(Q) and L(Q) · Q ⊆ L(Q). A quantale congruence on Q is an equivalence relation on Q which is a subquantale of Q × Q. A subquantale of Q is said to be strong if it contains the top element of Q. A quantale 1 = {0} is called trivial.

Recall that frames are exactly two-sided idempotent quantales. Moreover, any frame homomorphism is a strong and unital quantale homomorphism. Another important examples are quantales Rid(R) (Lid(R), Id(R)) of all right-sided (leftsided, two-sided) ideals of a ring R, with set inclusion as partial order and the familiar multiplication of right-sided (left-sided, two-sided) ideals. Note that the ring R is not required to have a unit here; if it does have a unit then R is the right unit (left unit, unit) of Rid(R) (Lid(R), Id(R)).

Since a · − : Q → Q and − · a : Q → Q are sup-lattice homomorphisms for any a ∈ Q, they have right adjoints a →_{r} − : Q → Q and a →_{l} − : Q → Q. Explicitly, they are given by

a →_{r} x = ⋁_{a·y≤x} y and a →_{l} x = ⋁_{y·b≤d} y.

For any a ∈ Q, we have a →_{r} 0 ∈ R(Q) and a →_{l} 0 ∈ L(Q). A quantale Q is discrete if

(a →_{l} 0) →_{r} 0 = 0 for any a ∈ Q

and

(b →_{r} 0) →_{l} 0 = 0 for any b ∈ Q.

In any discrete quantale Q we have L(Q) ≃ R(Q)^{op}.

A quantale Q is said to be a factor if I(Q) = {0, 1}. Every sup-lattice L can be seen as a right-sided idempotent factor by providing it with the following multiplication:

x · y = x if y ≠ 0 and x · 0 = 0,

for all x, y ∈ L. Conversely, any right-sided idempotent factor has to have such multiplication. This follows from the fact that in right-sided idempotent quantales the multiplication is given by the formula

x · y = x ∧ ŷ

where ŷ is the smallest two-sided element greater or equal that y. In what follows, whenever we consider a sup-lattice as a quantale we will have in mind the just described one.

Especially, if H is a Hilbert space then the complete orthocomplemented (in fact orthomodular) lattice L(H) of all closed subspaces of H is a right-sided idempotent factor. Recall that a lattice L is called orthocomplemented if there is a unary operation ^{⊥} : L → L such that a^{⊥⊥} = a, a ≤ b implies b^{⊥} ≤ a^{⊥} and a ∨ a^{⊥} = 1 for all a, b ∈ L.

Let L be a lattice. Atoms of L are elements a ∈ L such that x ≤ a implies x = 0 or x = a, for all x ∈ L. Similarly, dual atoms of L are exactly atoms of the dual lattice L^{op}. L is atomic (dually atomic) if each element is a join (meet) of atoms (dual atoms). Let L_{1}, L_{2} be dually atomic sup-lattices and f : L_{1} → L_{2} a sup-lattice homomorphism. We say that f is surjective on dual atoms if for any dual atom x ∈ L_{2} there is a dual atom y ∈ L_{1} such that f(y) = x.

Lemma 2.1 Let f : L_{1} → L_{2} be a strong sup-lattice homomorphism between dually atomic sup-lattices. Then f is surjective on dual atoms iff f_{∗} is injective on dual atoms (i.e., it injectively maps dual atoms of L_{2} into dual atoms of L_{1}).

Proof. Let f be surjective on dual atoms and consider a dual atom x ∈ L_{2}. Since x = f(y) for a dual atom y ∈ L_{1} and f(1) = 1, we have y = f_{∗}(x). Since f(y_{1}) = x_{1} ≠ x_{2} = f(y_{2}) implies y_{1} ≠ y_{2}, f_{∗} is injective on dual atoms.

Conversely, let f_{∗} be injective on dual atoms. Let x be a dual atom in L_{2}. Then f_{∗}(x) is a dual atom in L_{1}. It suffices to show that f(f_{∗}(x)) = x. We have that f(f_{∗}(x)) ≤ x. Assume that f(f_{∗}(x)) < x. Then there is a dual atom f(f_{∗}(x)) < y ∈ L_{2}, x ≠ y. We have

y ≥ f(f_{∗}(x)) ∨ f(f_{∗}(y)) = f(f_{∗}(x) ∨ f_{∗}(y)) = f(1) = 1,

which is a contradiction.

3 Quantales and C^{∗}-algebras

Recall that a C^{∗}-algebra is a complex Banach algebra A with an involution ^{∗} satifying

||x^{∗}x|| = ||x||^{2}

for all x ∈ A. C^{∗}-algebras are precisely closed subalgebras of algebras B(H) of all bounded linear operators on a Hilbert space H. Commutative C^{∗}-algebras can be characterized as algebras C_{0}(X) of complex-valued continuous functions vanishing at infinity on locally compact regular topological spaces X. It is the well-known Gelfand duality. In a special case, it says that commutative unital C^{∗}-algebras correspond to algebras C(X) of complex-valued continuous functions on compact regular topological spaces X. In both cases, we obtain a duality of relevant categories (cf. [6] or [11]).

Let A be a C^{∗}-algebra and denote by R(A) its lattice of closed right ideals. Define the multiplication on R(A) by taking for J_{1} · J_{2} the closure \overline{J_{1}J_{2}} in the norm topology of the algebraic product

J_{1}J_{2} = {xy|x ∈ J_{1}, y ∈ J_{2}}

of J_{1}, J_{2} ∈ R(A). We then have the following basic result.

Lemma 3.1 ([14]), [3] R(A) is an idempotent right-sided quantale for any C^{∗}- algebra A.

Recall that R(A) is dually atomic, dual atoms of R(A) are maximal regular closed right ideals and that they correspond to pure states of A.

Any homomorphism f : A → B of C^{∗}-algebras induces a sup-lattice homomorphism R(f) : R(A) → R(B) given by taking R(f)(J) as the closed right ideal generated by f(J). But, in general, R(f) does not preserve the multiplication. The reason is that R(f)(J_{1} · J_{2}) is generated by f(J_{1}J_{2}) = f(J_{1})f(J_{2}) while R(f)(I_{1}) · R(f)(I_{2}) is generated by f(J_{1})Bf(J_{2}). To see a difference, it suffices to consider the embedding of diagonal matrices into the full matrix algebra M_{n} = B(ℂ^{n}) for n ≥ 2. R(f) preserves multiplication if either f is surjective or both A and B are commutative.

Remarks 3.2 (see [4]): (i) A is commutative iff R(A) is commutative, i.e., iff R(A) is a frame.

(ii) A is elementary, i.e. isomorphic to the algebra K(H) of all compact operators on a Hilbert space H, iff R(A) is an atomic factor. In that case R(A) ≅ L(H).

(iii) A is dual, i.e. a C^{∗}-subalgebra of an elementary one, iff R(A) is atomic. In that case R(A) ≅ ∏_{t∈T} L(H_{t}).

Let A be a C^{∗}-algebra and p : A → B(H) a non-zero irreducible representation. This means that p is a C^{∗}-homomorphism such that p ≠ 0 and the only closed linear subspaces of the Hilbert space H invariant under p are {0} and H. Then p(A) is dense in B(H) in the weak topology and, consequently, for any J ∈ R(A), the weak closure cl(p(J)) is a weakly closed right ideal in B(H). Since the lattice of weakly closed right ideals of B(H) is isomorphic to L(H), we get a mapping

p̃: R(A) → L(H)

see ([4]).

Proposition 3.3 Let p : A → B(H) be a non-zero irreducible representation of a C^{∗}-algebra A. Then p̃: R(A) → L(H) is a strong homomorphism of quantales which is surjective on dual atoms.

Proof. Following [4], p̃ is a strong homomorphism of quantales. Due to Lemma 2.1, it suffices to prove that p̃_{∗} is injective on dual atoms. Dual atoms of L(H) are precisely orthocomplements ⟨x⟩^{⊥} of one-dimensional subspaces of H. The weakly closed right ideal J_{x} of B(H) corresponding to ⟨x⟩^{⊥} is defined as

J_{x} = {f ∈ B(H)|f(x) = 0}.

Hence

p̃_{∗}(⟨x⟩^{⊥}) = {a ∈ A|cl({p(a)}) ⊆ J_{x}} = {a ∈ A|p(a)(x) = 0}.

Following [6] Theorem 2.9.5, p̃_{∗}(⟨x⟩^{⊥}) is a dual atom in R(A).

Assume that ⟨x⟩ and ⟨y⟩ are distinct one-dimensional subspaces of H. Then x and y are linearly independent and, following the transitivity theorem ([11] 10.2.1), for any 0 ≠ z ∈ H there is a ∈ A such that p(a)(x) = z and p(a)(y) = 0. Hence p̃_{∗}(⟨x⟩^{⊥}) ≠ p̃_{∗}(⟨y⟩^{⊥}) because a ∈ p̃_{∗}(⟨y⟩^{⊥}) but a ∉ p̃_{∗}(⟨x⟩^{⊥}).

Remark 3.4 As we will explain in Remark 5.4, surjectivity of p̃ on dual atoms also follows from [16].

The main result of C.A. Akemann, R. Giles and H. Kummer is that a unital C^{∗}- algebra A is determined by the knowledge of homomorphisms p̃: R(A) → L(H), 0 ≠ p an irreducible representation. But, in general, we cannot recognize these homomorphisms p̃ from the quantale R(A) only.

Proposition 3.5 ([22]) Let A be a C^{∗}-algebra. Then the quantale R(A) is determined by the Jordan algebra A_{h} of hermitian elements of A.

If A is commutative then any non-zero irreducible representation p : A → B(H) is one-dimensional, i.e., H ≅ ℂ. Hence L(H) ≅ 2 and p̃ are precisely points of a frame R(A). But a much larger class of C^{∗}-algebras A can be reconstructed from R(A).

Theorem 3.6 ([4]) Any postliminary unital C^{∗}-algebra A is determined by its quantale R(A).

A consequence of [1], [2] and [7] is the following result

Proposition 3.7 ([4]) Let A be a C^{∗}-algebra and J_{1}, J_{2} two distinct elements of R(A). Then there exists a non-zero irreducible representation p : A → B(H) such that p̃(J_{1}) ≠ p̃(J_{2}).

Remark 3.8 ([22]) One can put more structure on R(A). Let J_{1}∘J_{2} be the closure in the norm topology of J_{1}^{∗}J_{2}. The new multiplication ∘ on R(A) is commutative but not associative and J_{1} ∘ J_{2} is always two-sided. The disjointness relation

J_{1}⊥J_{2} iff J_{1} ∘ J_{2} = 0

makes possible to introduce topological properties, like regularity, for R(A). One can prove then that R(A) is compact regular for any unital C^{∗}-algebra A. But, still, ∘ is determined by the Jordan part A_{h} only.

4 Involutive quantales

An involution (see [15]) on a quantale Q is a unary operation ^{∗} such that

\begin{aligned}a^{∗∗} & = & a, \\ (a · b)^{∗} & = & b^{∗} · a^{∗}, \\ (⋁_{i∈I} a_{i})^{∗} & = & ⋁_{i∈I} a_{i}^{∗} \end{aligned}

for all a, a_{i}, b ∈ Q, i ∈ I. A quantale with the involution is said to be involutive. An involutive homomorphism is a quantale homomorphism of involutive quantales which also preserves the involution. An involutive quantale Q is called Gelfand if a · a^{∗} · a = a for all a ∈ R(Q). An element a ∈ Q is said to be hermitian if a = a^{∗}. We put H(Q) = {a ∈ Q|a = a^{∗}}. Since the involution is a complete lattice isomorphism, H(Q) is closed under arbitrary joins and meets. Moreover, the involution establishes a sup-lattice isomorphism between R(Q) and L(Q).

It may be remarked that, for an involutive quantale Q and any a, b ∈ Q,

a^{∗} →_{r} 0 = ⋁{c ∈ Q|c^{∗} · a = 0} and (b^{∗} →_{l} 0) = ⋁{d ∈ Q|b · d^{∗} = 0}.

An involutive quantale Q is said to be an I-factor if I(Q) ∩ H(Q) = {0, 1} i.e. if 0 and 1 are the only hermitian two-sided elements of Q.

Examples 4.1 (see [15]): (i) Any frame L is a unital, involutive, commutative Gelfand quantale with the multiplication given by the meet ∧, with unit given by the top element 1 and ^{∗} = id_{L}.

(ii) The set Rel(X) of relations on a set X, ordered by the inclusion, is a unital involutive quantale with multiplication given by the composition of relations

T_{1} · T_{2} = {(x, z) ∈ X × X|∃y ∈ X(x, y) ∈ T_{1} and (y, z) ∈ T_{2}},

with the equality relation providing the unit and the involution defined as the inverse relation

T^{∗} = {(x, y) ∈ X × X|(y, x) ∈ T }.

The right-sided elements of Rel(X) are those relations R_{A}, for A ⊆ X, given by

R_{A} = (X − A) × X.

Similarly, the left-sided elements of Rel(X) are such relations L_{B}, for B ⊆ X, given by

L_{B} = X × B.

Note that R_{A}^{∗} = L_{X−A}. Moreover, there is a well-known bijective correspondence between the relations on X and sup-preserving maps from the power set 2^{X} to itself. Namely, for all T ∈ Rel(X) and all φ : 2^{X} → 2^{X}, T ↦ φ_{T} , φ ↦ R_{φ} such that

φ_{T} (A) = {x ∈ X|∃y ∈ A (x, y) ∈ T }, R_{φ} = {(x, y) ∈ X × X|x ∈ φ({y})}.

For any sup-lattice S, we will denote by Q(S) the sup-lattice of all sup-lattice endomorphisms f : S → S (with the pointwise ordering f ≤ g iff f(x) ≤ g(x) for all x ∈ S). Then Q(S) is a unital quantale; multiplication is the composition and unit is id_{S}. Moreover, Q(S) is an involutive quantale provided that S is a sup-lattice with a duality, i.e., with a unary operation ′ such that

\begin{aligned}s′′ & = & s \\ (⋁_{i∈I} s_{i})′ & = & ⋀_{i∈I} s′_{i} \end{aligned}

for all s, s_{i} ∈ S, i ∈ I. In fact, the involution ^{∗} on Q(S) is then given by

φ^{∗}(s) = ( ⋁_{φ(t)≤s′} t)′

(see [19]). In a special case when S is an orthocomplemented sup-lattice and s′ = s^{⊥} for s ∈ S (i.e., duality is given by orthocomplements), the calculations were done in C.M. Mulvey and J.W. Pelletier [15] (the general case is completely analogous). Following Example 4.1 (ii), Rel(X) ≅ Q(2^{X}) for any set X.

Right-sided elements of Q(S) are precisely mappings

ρ_{s}(t) = \{ \begin{matrix}s & t ≠ 0 \\ 0 & otherwise \end{matrix}

where s ∈ S. Similarly, left-sided elements are precisely mappings

λ_{s}(t) = \{ \begin{matrix}0 & t ≤ s \\ 1 & otherwise. \end{matrix}

Thus R(Q(S)) ≅ L(Q(S))^{op}. If S is equipped with a duality then R(Q(S)) ≅ L(Q(S)). Moreover, ρ_{s}^{∗} = λ_{s′} and λ_{s}^{∗} = ρ_{s′}. Since

(λ_{u} ∘ ρ_{v})(t) = \{ \begin{matrix}0 & if t = 0 or v ≤ u \\ 1 & otherwise, \end{matrix}

we have that ρ_{s}^{⊥} = ρ_{s}^{∗} →_{r} 0 = λ_{s′} →_{r} 0 = ρ_{s′} i.e. the sup-lattice isomorphism between S and R(Q(S)) preserves the duality.

It may be remarked that ρ_{s} ∘ ρ_{s} = ρ_{s} and λ_{s} ∘ λ_{s} = λ_{s}, i.e., Q(S) is rightidempotent and left-idempotent.

In this way, one obtains

Proposition 4.2 ([15], [19]).Let S be a sup-lattice with a duality. Then Q(S) is a right-idempotent, discrete involutive quantale. Moreover, S is orthocomplemented iff Q(S) is a Gelfand quantale.

Proof. Everything, except that, for any Gelfand quantale Q(S), S ≃ R(Q(S)) is orthocomplemented, is in [15] and [19]. Let a ∈ S. Then ρ_{a} ∈ R(Q(S)) and ρ_{a} ∧ ρ_{a}^{⊥} = 0 by [15] i.e. ρ_{1} = ρ_{a} ∨ ρ_{a}^{⊥} = ρ_{a} ∨ ρ_{a′} = ρ_{a∨a′} which exactly means that a′ ∨ a = 1.

Moreover, the joins of a right-sided and a left-sided element are of the form

(ρ_{u} ∨ λ_{v})(t) = \{ \begin{matrix}0 & t = 0 \\ u & 0 ≠ t ≤ v \\ 1 & otherwise. \end{matrix}

Lemma 4.3 (see [13]) Let S be a sup-lattice. Then any element of Q(S) is a meet of elements of the form ρ_{s} ∨ λ_{t}, s, t ∈ S.

Remark 4.4 The sup-lattice Q(S) can be described by means of a tensor product of sup-lattices

Q(S) ≅ (S ⊗ S^{op})^{op}

(see [10]). This isomorphism is given by

f ↦ ⋁_{f(t)≤s} t ⊗ s.

Elements ρ_{s}∨λ_{t} correspond to t⊗s, which explains Lemma 4.3. If S is a sup-lattice with a duality then the involution in Q(S) is induced by

(t ⊗ s)^{∗} = s′ ⊗ t′.

Remark 4.5 For any quantale Q, we get a map

μ : Q → Q(R(Q))

given by μ(a)(c) = a · c, which clearly preserves multiplication and arbitrary joins. Moreover, if Q is unital then μ is unital as well. If Q is involutive and R(Q) a discrete quantale then μ is an involutive quantale homomorphism (see [16], [19]). Moreover, μ is strong iff Q is a factor.

5 Mulvey-Pelletier spectrum

Let A be a C^{∗}-algebra and denote by M(A) its lattice of closed linear subspaces. Define the multiplication on M(A) by taking for X · Y the closed subspace generated by XY. Then M(A) is an involutive quantale. R(A) is the quantale of right-sided elements of M(A). Moreover, the multiplication ∘ from 3.8 is calculated as

J_{1} ∘ J_{2} = J_{1}^{∗} · J_{2}

in M(A). For a homomorphism f : A → B of C^{∗}-algebras, put

M(f(X)) = \overline{f(X)}

for any X ∈ M(A). It is easy to see that M : C^{∗} → IQ is a functor. Here C^{∗} denotes the category of C^{∗}-algebras and IQ the category of involutive quantales.

Let p : A → B(H) be a non-zero irreducible representation of a C^{∗}-algebra A. For any X ∈ M(A), put

p̂(X)(V) = \overline{\{∑_{i=1}^{n} p(x_{i})(v_{i})|x_{i} ∈ X, v_{i} ∈ V, n ∈ ℕ\}}.

We get a mapping p̂: M(A) → Q(L(H)).

Lemma 5.1 [16] Let A be a C^{∗}-algebra. Then p̂: M(A) → Q(L(H)) is a strong involutive homomorphism of involutive quantales.

Proof. It is easy to see that p̂ is a homomorphism of quantales. The fact that p̂ is strong follows from the transitivity theorem. Indeed, having 0 ≠ X ∈ L(H) and u ∈ H, we can find 0 ≠ x ∈ X and a ∈ A such that p(a)(x) = u. Hence p̂(A) = 1_{Q(L(H))}. It remains to prove that p̂ preserves involution, which we will not recall here.

Remark 5.2 The restriction of p̂ : M(A) → Q(L(H)) on right-sided elements is equal to p̃ : R(A) → L(H) from chapter 3. In fact, since p̂ is strong, p̂(J) ∈ R(Q(L(H))) ≅ L(H) for any J ∈ R(A). Hence p̂(J) = ρ_{X} for some X ∈ L(H). It is equivalent to the fact that p̂(J)(H) = X, which exactly means that cl(p(J)) is generated by the projection on X. Hence p̃(J) = p̂(J).

Theorem 5.3 ([16]). Let A be a C^{∗}-algebra, S a non-empty atomic orthocomplemented sup-lattice and f : R(A) → S a strong homomorphism of quantales which is surjective on dual atoms. Then f ≅ p̃ for a non-zero irreducible representation p : A → B(H) iff f can be extended to an involutive quantale homomorphism M(A) → Q(S).

Remark 5.4 (1) Under f ≅ p̃ we mean that there is a sup-lattice isomorphism h : S → L(H) such that h ∘ f = p̃.

(2) In the just stated Theorem, we have reformulated the main result of [16] in order to stress the role of quantales R(A). Mulvey and Pelletier [16] use pure states instead of dual atoms x in right-sided elements. They are related by

s = x ∨ x^{∗}.

The necessity in our formulation also follows from Proposition 3.3.

A consequence of Theorem 5.3 is that we can reconstruct a C^{∗}-algebra A from its involutive quantale M(A).

6 Simple and spatial quantales

We have seen in chapter 2 that points of a frame are frame homomorphisms L → 2. The frame 2 can be characterized among frames as being simple, it means, to have exactly two quotients 2 and 1. It is the usual algebraic concept of simplicity. Geometrically, in the dual category of locales, it means that 2 has exactly two subspaces, the zero subspace and the space itself, which is the geometric meaning of a point. It is natural to apply the same approach to (involutive) quantales.

A non-trivial (involutive) quantale Q is said to be (I-simple ) simple if any surjective homomorphism of (involutive) quantales is either an isomorphism or a constant homomorphism. Evidently, any simple involutive quantale is I-simple.

A quantale Q will be called faithful if, whenever

x · a = y · a and b · x = b · y

for all a ∈ R(Q) and b ∈ L(Q), then

x = y

for all x, y ∈ Q.

We say that the multiplication of a quantale Q is trivial if x · y = 0 for each x, y ∈ Q; it is equivalent to 1 · 1 = 0. It is evident that the only simple quantale with a trivial multiplication is the quantale 0_{2} = {0, 1}. Analogously, the only I-simple quantale with a trivial multiplication is the quantale 0_{2} with the identity involution.

Theorem 6.1 (i) An (involutive) quantale with a non-trivial multiplication is (I-)simple iff it is a faithful discrete (I-)factor.

(ii) Any (I-)simple (involutive) quantale with a non-trivial multiplication is right-idempotent.

Proof. (1) A quantale with a non-trivial multiplication is simple iff it is a faithful discrete factor (see [17]).

(2) Any faithful discrete I-factor is I-simple and, conversely, any right idempotent I-simple involutive quantale Q is a faithful discrete I-factor (see [19]).

It immediately follows from (1) and (2) that (ii) ⟹ (i).

Assume first that Q is a simple quantale with a non-trivial multiplication and 0 ≠ b ∈ R(Q). Then, following (1) and [17] 3.3 and 2.2, we have that b · 1 = b and 1 · b = 1. Hence

b = b · 1 = b · 1 · b = b · b,

which means that Q is right-idempotent.

Now, let Q be an I-simple involutive quantale with a non-trivial multiplication which is not simple. Then Q is an I-factor (see [19]) and, by [17] 2.2, 1 · x · 1 = 1 for all 0 ≠ x ∈ H(Q). Let a ∈ I(Q) − {0, 1}. Then a must be non-hermitian, a^{∗} ∈ I(Q) and a ∨ a^{∗}, a · a^{∗}, a^{∗} · a, a^{∗} ∧ a ∈ I(Q) ∩ H(Q). Hence, a ∨ a^{∗} = 1 and a · a^{∗} = a^{∗} · a = a^{∗} ∧ a = 0.

Let us define a relation σ_{a} ⊆ Q × Q by

xσ_{a}y iff c · x · d = c · y · d and c · x^{∗} · d = c · y^{∗} · d

for all c ∈ L(Q), c ≤ a and d ∈ R(Q), d ≤ a. Evidently, σ_{a} is an equivalence relation closed under involution and arbitrary joins. Let x_{1}σ_{a}y_{1} and x_{2}σ_{a}y_{2}. Then, for all c ∈ L(Q), c ≤ a and d ∈ R(Q), d ≤ a, c · x_{1} · x_{2} · d = c · x_{1} · (x_{2} · d) = c · y_{1} · (x_{2} · d) = (c · y_{1}) · x_{2} · d = (c · y_{1}) · y_{2} · d = c · y_{1} · y_{2} · d and, similarly, c · x_{2}^{∗} · x_{1}^{∗} · d = c · y_{2}^{∗} · y_{1}^{∗} · d since c · y_{1}, c · x_{2}^{∗} ∈ L(Q) and x_{2} · d, y_{1}^{∗} · d ∈ R(Q). Therefore, σ_{a} is an involutive quantale congruence on Q, i.e., either σ_{a} = Q × Q or σ_{a} = id_{Q}. Since a · a · a ∨ (a · a · a)^{∗} = (a ∨ a^{∗})^{3} = 1, we have a · a · a ≠ 0 and thus (0, a) ∉ σ_{a}. Therefore σ_{a} = id_{Q}.

Let c ∈ L(Q), c ≤ a and d ∈ R(Q), d ≤ a. Then

\begin{aligned}c · a · d & = & c · (a ∨ a^{∗}) · d = c · 1 · d = c · 1 · 1 · d = c · (a ∨ a^{∗}) · 1 · d \\ & = & c · a · 1 · d ∨ c · a^{∗} · 1 · d = c · a · 1 · d \end{aligned}

and

c · a^{∗} · d = 0 = c · (a · 1)^{∗} · d.

Therefore (a, a · 1) ∈ σ_{a}, i.e., a = a · 1. Consequently,

a = a · 1 = a · (a ∨ a^{∗}) = a · a.

It means that I(Q) is an idempotent 2-sided quantale, i.e., a frame. Since a^{∗} is a complement of a for any a ∈ I(Q) − {0, 1}, I(Q) is a Boolean algebra. It follows from the proof of [19] 4.4. that I(Q) is a four-element Boolean algebra. We will write I(Q) = {0, a, a^{∗}, 1}.

Now, let 0 ≠ b ∈ R(Q). Consider c ∈ L(Q), c ≤ a and d ∈ R(Q), d ≤ a. Since c · b ∈ I(Q) and c · b ≤ a, we either have c · b = 0 or c · b = a. In the first case, c · b · d = c · b · b · d. Assume that c · b = a. Since a · b ∈ I(Q) and a = c · b ≤ a · b ≤ a, we have a · b = b. Hence,

c · b · d = a · d = a · b · d = c · b · b · d.

We have proved that c · b · d = c · b · b · d. At the same time, since b^{∗} · d ∈ I(Q), we have

c · b^{∗} · d = c · b^{∗} · d · b^{∗} · d ≤ c · b^{∗} · 1 · b^{∗} · d ≤ c · b^{∗} · b^{∗} · d ≤ c · 1 · b^{∗} · d ≤ c · b^{∗} · d.

Hence c · b^{∗} · d = c · (b · b)^{∗} · d, which yields that b · b = b.

Remark 6.2 (1) The above characterization was at first found for right-idem-potent involutive quantales in [19]. Then it was extended (and strengthened) to quantales in [17].

(2) Any (I-)simple (involutive) quantale with a non-trivial multiplication is also left-idempotent. It is evident for involutive quantales and for quantales it follows from the proof of [17] 2.3.

Corollary 6.3 ([19], [17]) Let S be a sup-lattice (with a duality). Then the (involutive) quantale Q(S) is simple.

Corollary 6.4 Let Q be a simple (involutive) quantale which is not isomorphic to 0_{2}. Then Q is isomorphic to a strong (involutive) subquantale of Q(S) for some sup-lattice S (with a duality).

Proof. Consider μ : Q → Q(R(Q)) from 4.5. Then, since Q is a factor, μ is strong. Moreover, if Q is involutive and discrete we have that R(Q) is equipped with a duality and μ is an involutive quantale homomorphism. The injectivity of μ follows from the faithfulness of Q.

Remark 6.5 (1) A four-element Boolean algebra is an example of an I-simple involutive quantale which is not simple (see [19]). More generally, any involutive quantale Q(S) × Q(S^{op}), where S is a sup-lattice, is I-simple but not simple.

(2) Any I-simple involutive quantale is isomorphic to a strong involutive subquantale of Q(S) × Q(S^{op}) for some sup-lattice S (see [13]).

Theorem 6.6 ([18]) Any (involutive) quantale has an (involutive) embedding into an (I-)simple (involutive) quantale.

Hence, the definition of (I-)spatiality by means of enough (involutive) quantale homomorphisms into (I-)simple quantales is too week. We use strong homomorphisms for defining points, i.e., we work in the category of (involutive) quantales and strong (involutive) homomorphisms (it does not change the concept of (I- )simplicity because any surjective (involutive) quantale homomorphism of (involutive) quantales is strong). It is also supported by Lemma 5.1 – any non-zero irreducible representation of a C^{∗}-algebra should yield a point of an associated quantale. Another concept of a point is introduced in [16] (see Remark 6.17 at the end of our paper).

An (involutive) quantale Q is called (I)-spatial if it has enough points. It means that for any two distinct elements x, y ∈ Q there exists a strong (involutive) quantale homomorphism f : Q → P into an (I)-simple quantale P such that f(x) ≠ f(y).

Proposition 6.7 Any (I-)spatial (involutive) quantale satisfying 1 · 1 = 1 is both right- and left-idempotent.

Proof follows from Theorem 6.1 and Remark 6.2.

Proposition 6.8 ([13]) (i) A quantale satisfying 1 · 1 = 1 is spatial iff it has enough strong quantale homomorphisms from Q into quantales Q(S).

(ii) An involutive quantale satisfying 1 · 1 = 1 is I-spatial iff it has enough strong quantale homomorphisms from Q into quantales Q(S) × Q(S^{op}).

Proof follows from 6.4 and 6.5.

Remark 6.9 More generally, any (I-)spatial (involutive) quantale is isomorphic to an (involutive) strong subquantale of a product of quantales of the form Q(S) (Q(S) × Q(S^{op})) and of the copies of 0_{2}.

Proposition 6.10 ([13]) An I-spatial involutive quantale Q is spatial and, conversely, any spatial involutive quantale satisfying 1 · 1 = 1 is I-spatial.

Proof. Assume first that Q is an I-spatial involutive quantale and x, y ∈ Q, x ≠ y. Then there exists a strong involutive quantale homomorphism f : Q → P into an I-simple quantale P such that f(x) ≠ f(y). Hence, either P = 0_{2} or P is a strong subquantale of Q(S) × Q(S^{op}) by Remark 6.5. In the first case, f : Q → P is a strong quantale homomorphism separating x and y. In the second case, the projections p_{1} : Q(S) × Q(S^{op}) → Q(S) and p_{2} : Q(S) × Q(S^{op}) → Q(S^{op}) are strong quantale homomorphisms and either p_{1} ∘ f or p_{2} ∘ f separates x and y.

Conversely, let Q be a spatial involutive quantale satisfying 1 · 1 = 1. By Proposition 6.8 we have enough strong quantale homomorphisms into quantales Q(S). Any strong quantale homomorphism f : Q → Q(S) induces a strong involutive quantale homomorphism g : Q → Q(S) × Q(S^{op}) defined by g(a) = (f(a), f(a^{∗})_{∗}), for all a ∈ Q. Hence, Q is an I-spatial involutive quantale.

Example 6.11 of a spatial involutive quantale which is not I-spatial.

Consider the quantale Q = 0_{2} × 0_{2} with the involution

(a, b)^{∗} = (b, a).

Then Q is evidently spatial. Assume that there is a strong involutive quantale homomorphism f : Q → P into an I-simple involutive quantale P satisfying f(0, 1) ≠ f(1, 0). Since Q has a trivial multiplication, P = 0_{2}. Let, for instance, f(0, 1) = 0 and f(1, 0) = 1. Then

1 = f(1, 0) = (f(0, 1))^{∗} = 0^{∗},

which is a contradiction.

D. Kruml [13] has achieved to find a characterization of spatial quantales by introducing an appropriate concept of a prime element in quantales.

Definition 6.12 An element p ≠ 1 of a quantale Q is called prime if

r · l ≤ p ⟹ r ≤ p or l ≤ p

for all r ∈ R(Q), l ∈ L(Q).

It is clear that prime elements of a frame coincide with those from chapter 2.

Theorem 6.13 ([13]) A quantale Q satisfying 1 · 1 = 1 is spatial iff each its elements is a meet of primes.

Remark 6.14 (i) If Q is an idempotent right-sided quantale then 1 ≠ p ∈ Q is a prime iff

x · y ≤ p implies x ≤ p or y ≤ p

for all x, y ∈ Q. In fact, whenever p is a prime and x · y ≤ p then

x · (1 · y) = (x · 1) · y = x · y ≤ p

and therefore

x ≤ p or y = y · y ≤ 1 · y ≤ p.

It has been shown in [21] that, in an idempotent right-sided quantale Q, any element is an intersection of primes iff Q has enough strong homomorphisms into idempotent right-sided factor quantales.

(ii) Elements ρ_{s} ∨ λ_{t} from 4.3 are primes in Q(S).

Proposition 6.15 Let A be a C^{∗}-algebra and J_{1}, J_{2} two distinct right-sided elements of M(A). Then there is a strong involutive quantale homomorphism f : M(A) → Q(L(H)), where H is a Hilbert space, such that f(J_{1}) ≠ f(J_{2}).

Proof follows from Proposition 3.7 and Remark 5.2. Hence M(A) is ”spatial on right-sided elements”. But M(A) is not necessarily spatial, as the following example shows.

Example 6.16 (D. Kruml) Let A = B(ℂ^{2}) be the C^{∗}-algebra of complex 2 × 2matrices. As a linear space, A ≅ ℂ^{4}. Let

X = \{( \begin{matrix}z & x \\ y & x + y \end{matrix} ) |x, y, z ∈ ℂ\} .

Then X is a dual atom in M(A). We will show that X is not prime, which implies that it cannot be an intersection of primes. Consider

X_{1} = \{( \begin{matrix}a & b \\ 0 & 0 \end{matrix} ) |a, b ∈ ℂ\} and X_{2} = \{( \begin{matrix}c & 0 \\ d & 0 \end{matrix} ) |c, d ∈ ℂ\} .

Then X_{1} ∈ R(A), X_{2} ∈ L(A), X_{1} ⊈ X, X_{2} ⊈ X and X_{1} · X_{2} ⊆ X because

( \begin{matrix}a & b \\ 0 & 0 \end{matrix} ) ( \begin{matrix}c & 0 \\ d & 0 \end{matrix} ) = ( \begin{matrix}ac + bd & 0 \\ 0 & 0 \end{matrix} ) ∈ X.

Remark 6.17 Another definition of points of a (Gelfand) quantale is introduced in [16]. There are defined as involutive quantale homomorphisms f : Q → Q(S) where S is an atomic orthocomplemented lattice and, for each atom s ∈ S, we have

{f(q)(s)|q ∈ Q} = S.

Evidently, such homomorphisms are strong. If A is a C^{∗}-algebra and p : A → B(H) a non-zero irreducible representation then p̂: M(A) → Q(L(H)) is a point in this sense (see [16]).

Acknowledgements

Financial Support of the Grant Agency of the Czech Republic under the grant No. 201/99/0310 is gratefully acknowledged.

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