Embeddings of quantales into simple quantales

JAN PASEKA^{∗} DAVID KRUML

Department of Mathematics, Masaryk University Janáčkovo nám. 2a 66295 Brno, Czech Republic e-mail: {paseka, kruml}@math.muni.cz

Abstract Any (involutive) quantale is embeddable into a simple (I-simple involutive) quantale that is, into a (involutive) quantale with no non-trivial quotients.

Keywords and phrases: quantale, involutive quantale, simple quantale, bicommutant, C^{∗}-algebra.

AMS Subject Classification: 06B10, 06B99, 06F05, 06-99, 54A05, 54-99

In this note we shall show that any (involutive) quantale is embeddable into a simple (I-simple involutive) quantale that is, into a (involutive) quantale with no non-trivial quotients. Moreover, motivated by the von Neuman’s bicommutant theorem for C^{∗}-algebras we can show that the bicommutant of the image of a (involutive) unital quantale under the above embedding is exactly this image.

Our motivating source was the paper [6] where the investigation of I-simple involutive quantales is developed. Such involutive quantales could play a similar role as do points in topological spaces or irreducible representations in C^{∗}-algebras (see [2]). This note is closely related to the papers [3] and [4] where the interested reader can find unexplained terms and notation concerning the subject. For facts concerning quantales in general we refer to [7].

A quantale is a complete lattice K with an associative binary multiplication satisfying

x · ⋁_{i∈I} x_{i} = ⋁_{i∈I} x · x_{i} and (⋁_{i∈I} x_{i}) · x = ⋁_{i∈I} x_{i} · x

∗Financial Support of the Grant Agency of the Czech Republic under the grant No. 201/96/0119 is gratefully acknowledged.

for all x, x_{i} ∈ K, i ∈ I (I is a set). 1 denotes the greatest element of K, 0 is the smallest element of K. A quantale K is said to be unital if there is an element e ∈ K such that e · a = a = a · e for all a ∈ K. A subquantale K′ of a quantale K is a subset of K closed under ⋁ and ·.

By an involutive quantale will be meant a quantale K together with an involution ^{∗} satisfying

a^{∗∗} = a, (a · b)^{∗} = b^{∗} · a^{∗} and (⋁ a_{i})^{∗} = ⋁ a_{i}^{∗}

for all a, b, a_{i} ∈ K, i ∈ I. In the event that K is also unital, then necessarily e is selfadjoint i.e.

e = e^{∗}.

By a morphism of quantales will be meant a ⋁- and · -preserving mapping

f : K → K′.

Similarly, a morphism of involutive quantales is a morphism of quantales that is ∗ -preserving.

A non-trivial (involutive) quantale K is said to be simple (I-simple) if any surjective homomorphism K → K′ (with involutive K′) is either an isomorphism or a constant morphism.

By the quantale Q(L) of endomorphisms of the sup-lattice L will be meant the simple unital quantale of sup-preserving mappings from L to itself, with the supremum given by the pointwise ordering of mappings, with the multiplication corresponding to the composition of mappings, and with the unit given by the identity mapping.

Moreover, for any sup-lattice L with duality, the quantale Q(L) ([1]) is itself an I-simple quantale ([6]).

Lemma 1 Let K be a unital quantale with the unit e. Then we have a quantale embedding i_{K} : K → Q(K) such that i_{K}(e) = id_{K}.

Proof. Let us define i_{K} as follows:

i_{K}(a)(b) = a · b

for all a, b ∈ K. Then i_{K}(a) preserves all suprema in K i.e. it belongs to Q(K). Let us show that i_{K} is a quantale embedding. Let a′, a′′ ∈ K, i_{K}(a′) = i_{K}(a′′). Then a′ = i_{K}(a′)(e) = i_{K}(a′′)(e) = a′′. Evidently, i_{K}(e) = id_{K}. Now, let a, b ∈ K. Then, for all c ∈ K, (i_{K}(a) ∘ i_{K}(b))(c) = i_{K}(a)(i_{K}(b)(c)) = i_{K}(a)(b·c) = a·(b·c) = (a · b) · c = i_{K}(a · b)(c) i.e. i_{K}(a) ∘ i_{K}(b) = i_{K}(a · b). Now, assume T ⊆ K. Then, for all c ∈ K, i_{K}(⋁ T)(c) = (⋁ T) · c = ⋁_{t∈T} t · c = ⋁_{t∈T} i_{K}(t)(c) = (⋁_{t∈T} i_{K}(t))(c) i.e. i_{K}(⋁ T) = ⋁_{t∈T} i_{K}(t).

Lemma 2 Let K be a non-unital quantale, K[e] = {a ∨ ε : a ∈ K, ε ∈ {0, e}},e arbitrary such that e ∉ K. Then K[e] is an unital quantale and we have a quantale embedding i_{e} : K → K[e]. Moreover, if K is an involutive quantale we have that K[e] is involutive as well.

Proof. Note that we shall identify the formal join a ∨ 0 with a for any a ∈ K, 0 ∨ e with e. We may then define the supremum

⋁_{i}(a_{i} ∨ ε_{i}) {= (⋁_{i} a_{i}) ∨ e if ∃i ε_{i} = e ⋁_{i} a_{i} otherwise.

Note that K[e] is isomorphic as a complete lattice with the cartesian product of K and a 2-element lattice.

Similarly, we may define the multiplication on K[e] as follows:

(a ∨ ε′) · (b ∨ ε′′) = \{ \begin{matrix}a ·_{K} b & if ε′ = ε′′ = 0 \\ a ·_{K} b ∨_{K} b & if ε′ = e, ε′′ = 0 \\ a ·_{K} b ∨_{K} a & if ε′ = 0, ε′′ = e \\ (a ·_{K} b ∨_{K} a ∨_{K} b) ∨ e & if ε′ = ε′′ = e. \end{matrix}

It is an easy task to check that K[e] is a unital quantale with the unit e. The embedding i_{e} : K → K[e] is defined as follows:

i_{e}(a) = a ∨ 0 = a

for all a ∈ K. Now assume that K is involutive. Then we shall define the involution ^{∗} on K[e] as follows:

(a ∨ ε)^{∗} := a^{∗^{K}} ∨ ε

for all a ∈ K and ε ∈ {0, e}. Again, it is evident that ^{∗} satisfies x^{∗∗} = x, x^{∗} · y^{∗} = (y · x)^{∗} for all x, y ∈ K[e] and preserves arbitrary suprema. Moreover, the embedding i_{e} : K → K[e] preserves involution.

Note that the above construction corresponds to the extension of a non-unital C^{∗}-algebra to a unital C^{∗}-algebra (see [5]).

Theorem 3 Let K be a quantale. Then we have a quantale embedding e_{K} : K → Q(S) for a sup-semilattice S.

Proof. For simplicity, if K is unital, we put K[e] = K and i_{e} = id_{K}. Now, let K be a quantale. We put S = K[e], e_{K} = i_{K[e]}∘ i_{e}. Then e_{K} is a quantale embedding.

Corollary 4 Any quantale is embeddable into a simple quantale.

Lemma 5 Let K be an involutive unital quantale with the unit e. Then we have an involutive quantale embedding I_{K} : K → Q(S) such that S = K^{op} × K is a complete lattice with duality and I_{K}(e) = (id_{K}, id_{K}) = e_{Q(S)}.

Proof. First, note that S is a complete lattice having duality i.e. a unary operation′ such that for all s ∈ S, T ⊆ S we have that s = s′′ and (⋁ T)′ = ⋀_{t∈T} t′. Namely, we may define, for all a, b ∈ K, (a, b)′ = (b, a). In fact, it is easy to show that S is an involutive unital Girard quantale (see [7]).

Let us define now the map I_{K} : K → Q(S) as follows:

I_{K}(a) = (r_{K}(a^{∗}), i_{K}(a))

for all a ∈ K; here r_{K}(a) is the right adjoint to the sup-preserving map i_{K}(a). Then i_{K}(a) resp. r_{K}(a^{∗}) preserve all suprema in K resp. in K^{op} i.e. the pair (r_{K}(a^{∗}), i_{K}(a)) belongs to Q(S). By Lemma 1 we know that I_{K} is an embedding. Let us show that it is a morphism of involutive quantales.

Now, let a, b ∈ K. Again by Lemma 1 we have that i_{K}(a) ∘ i_{K}(b) = i_{K}(a · b). Let us show that r_{K}(a^{∗}) ∘ r_{K}(b^{∗}) = r_{K}((a · b)^{∗}). Evidently, r_{K}(a^{∗}) ∘ r_{K}(b^{∗}) is a right adjoint to i_{K}(b^{∗}) ∘ i_{K}(a^{∗}), r_{K}((a · b)^{∗}) is a right adjoint to i_{K}((a · b)^{∗}) and the left (right) adjoints coincide. Now, assume T ⊆ K. Then by Lemma 1 i_{K}(⋁ T) = ⋁_{t∈T} i_{K}(t) and both r_{K}((⋁ T)^{∗}), ⋀_{t∈T} r_{K}(t^{∗}) are right adjoints to ⋁(i_{K}(T^{∗})) i.e. they coincide.

Now, we shall show that I_{K} preserves involution. First, let us denote the right adjoint to I_{K}(a) by J_{K}(a). Evidently, J_{K}(a) = (i_{K}(a^{∗}), r_{K}(a)). Note that we have I_{K}(a)^{∗} = ′ ∘ J_{K}(a) ∘ ′ (see [4]). Then, for all c, d ∈ K,

\begin{aligned}I_{K}(a)^{∗}(c, d) & = & (J_{K}(a)((c, d)′))′ = (J_{K}(a)((d, c)))′ \\ & = & (i_{K}(a^{∗})(d), r_{K}(a)(c))′ = (r_{K}(a)(c), i_{K}(a^{∗})(d)) = I_{K}(a^{∗})(c, d). \end{aligned}

Evidently, I_{K}(e) = e_{Q(S)}.

Theorem 6 Let K be an involutive quantale. Then we have an involutive quantale embedding E_{K} : K → Q(S) for a complete lattice S with duality.

Proof. The proof copies the proof of Theorem 3.

Corollary 7 Any involutive quantale is embeddable into an I-simple quantale.

Note that we shall denote, for each quantale K, K^{d} the quantale (K, ⋁, ·_{d}, 0) with the same ⋁-semilattice as K, the multiplication ·_{d} is defined as follows: a ·_{d} b = b · a for all a, b ∈ K.

Lemma 8 Let A be a subset of a quantale K. We shall denote by comm(A) the commutant of A in K defined as

comm(A) = {b ∈ K : (∀a ∈ A)(a · b = b · a)}.

Then comm(A) is a subquantale of K. Moreover, if K is unital with the unit e then e ∈ comm(A).

Proof. Let b_{1}, b_{2} ∈ comm(A). Then, for all a ∈ A, we have

a · (b_{1} · b_{2}) = (a · b_{1}) · b_{2} = (b_{1} · a) · b_{2} = b_{1} · (a · b_{2}) = b_{1} · (b_{2} · a) = (b_{1} · b_{2}) · a,

i.e. b_{1} · b_{2} ∈ comm(A). Similarly, let b_{i} ∈ comm(A), i ∈ I. Then, for all a ∈ A, we have

a · (⋁_{i} b_{i}) = ⋁_{i}(a · b_{i}) = ⋁_{i}(b_{i} · a) = (⋁_{i} b_{i}) · a,

i.e. ⋁_{i} b_{i} ∈ comm(A). Evidently, 0 ∈ comm(A). Moreover, if e is the unit of K we have, for all a ∈ A, that e · a = a · e i.e. e ∈ comm(A).

Corollary 9 Let A be a subset of an involutive quantale K such that A = A^{∗}. Then comm(A) is an involutive subquantale of K.

Proof. Let b ∈ comm(A). Then, for all a ∈ A, we have

a · b^{∗} = (a · b^{∗})^{∗∗} = (b · a^{∗})^{∗} = (a^{∗} · b)^{∗} = b^{∗} · a,

i.e. b^{∗} ∈ comm(A). The rest follows from Lemma 8.

Definition 10 Let A be a subset of a quantale K. We shall denote by bicomm(A) the bicommutant of A in K defined as

bicomm(A) = comm(comm(A)).

Corollary 11 Let A, B be subsets of a quantale K. Then A ⊆ bicomm(A), comm(A) = comm(bicomm(A)), A ⊆ B implies comm(B) ⊆ comm(A).

Definition 12 Let A, B be subsets of a quantale K. We shall say that A is commutative if A ⊆ comm(A). We shall say that A and B are commuting subsets if B ⊆ comm(A).

Note that by Corollary 11 A and B are commuting subsets if A ⊆ comm(B).

Proposition 13 Let K be a unital quantale, i_{K} : K → Q(K) the quantale embedding from Theorem 1. Then K^{d} ≅ comm(i_{K}(K)) and K ≅ bicomm(i_{K}(K)).

Proof. Let f ∈ comm(i_{K}(K)). Then, for all a, b ∈ K, we have f(i_{K}(a)(b)) = i_{K}(a)(f(b)) i.e. f(a·b) = a·f(b). Let us put b = e. Then f(a) = f(a·e) = a·f(e). Evidently, any such f is of the form j_{K}(c); here j_{K}(c)(b) = b·c. It is easy to check that the subquantale j_{K}(K) is isomorphic to K^{d} and any of its element belongs to comm(i_{(}K)). Since Q(K) = Q(K^{d}) we have that K = K^{dd} ≅ bicomm(i_{K}(K)) = comm(i_{K^{d}}(K^{d})).

Corollary 14 Let K be a quantale, e_{K} : K → Q(S) the quantale embedding from Theorem 3. Then K[e]^{d} ≅ comm(e_{K}(K)) and K[e] ≅ bicomm(e_{K}(K)).

Proof. It follows immediately from Lemma 2 and Proposition 13.

Proposition 15 Let K be a unital involutive quantale, I_{K} : K → Q(S) the quantale embedding from Theorem 5. Then K ≅ bicomm(I_{K}(K)).

Proof. Let f ∈ comm(I_{K}(K)). Then, for all a, b, c ∈ K, we have f(I_{K}(a)(b, c)) = I_{K}(a)(f(b, c)) i.e. f(r_{K}(a^{∗})(b), i_{K}(a)(c)) = (r_{K}(a^{∗})(π_{1}(f(b, c))), i_{K}(a)(π_{2}(f(b, c)))). Let us put b = 1. Then

f(r_{K}(a^{∗})(1), i_{K}(a)(c)) = f(1, i_{K}(a)(c)) = (r_{K}(a^{∗})(π_{1}(f(1, c))), i_{K}(a)(π_{2}(f(1, c)))).

We have π_{1}(f(1, i_{K}(a)(c))) = r_{K}(a^{∗})(π_{1}(f(1, c))). Let us put f_{1} = π_{1} ∘ f(1, −). Then f_{1} : K → K^{op}, f_{1} preserves arbitrary joins. Then, for all a, c ∈ K, we have f_{1}(a · c) = r_{K}(a^{∗})(f_{1}(c)). Assume c = e. Then f_{1}(a) = f_{1}(a · e) = r_{K}(a^{∗})(f_{1}(e)) i.e. f_{1}(e) = r_{K}(e^{∗})(f_{1}(e)). Since, for all d ∈ K, we have d = r_{K}(e^{∗})(d) the map f_{1} is of the form f_{1}(h) = r_{K}(h^{∗})(u) for some suitable element u ∈ K. Similarly, let us put f_{2} = π_{2} ∘ f(1, −). Then f_{2} : K → K, f_{2} preserves arbitrary joins. Then, for all a, c ∈ K, we have f_{2}(a · c) = i_{K}(a)(f(c)). Then by Proposition 13 f_{2} = j_{K}(v) for some suitable element v ∈ K.

Now, let a, b be arbitrary, c = 0. Then

f(r_{K}(a^{∗})(b), 0) = f(r_{K}(a^{∗})(b), i_{K}(a)(0)) = (r_{K}(a^{∗})(π_{1}(f(b, 0))), i_{K}(a)(π_{2}(f(b, 0)))).

We put f_{3} = π_{1} ∘ f(−, 0) : K^{op} → K^{op}. Then f_{3} preserves arbitrary joins in K^{op} and, for all a, b ∈ K, f_{3}(r_{K}(a^{∗})(b)) = r_{K}(a^{∗})(f_{3}(b)). By an elementary computation, f_{3}^{l}(i_{K}(a^{∗})(d)) = i_{K}(a^{∗})(f_{3}^{l}(d)) for all a, d ∈ K, f_{3}^{l} being the left adjoint to f_{3}. Again by Proposition 13, f_{3}^{l} = j_{K}(w) for some suitable element w ∈ K. Then f_{3} = l_{K}(w); here l_{K}(w) is the right adjoint to the sup-preserving map j_{K}(w).

To the end, let us put f_{4} = π_{2} ∘ f(−, 0) : K^{op} → K. Again, f_{4} is a suppreserving map and, for all a, b ∈ K, f_{4}(r_{K}(a^{∗})(b)) = i_{K}(a)(f_{4}(b)). We shall denote by K(4) the set of all such mappings f_{4} : K^{op} → K. Then f(b, c) = (f_{1}, f_{2})(1, c) ∨ (f_{3}, f_{4})(b, 0) i.e.

f(b, c) = (r_{K}(c^{∗})(u) ∧_{K} l_{K}(w)(b), j_{K}(v)(c) ∨_{K} g(b))

for suitable elements u, v, w ∈ K and g ∈ K(4).

Now, let us check that any element h ∈ bicomm(I_{K}(K)) is of the form I_{K}(x) for some suitable element x ∈ K. Let us denote by g_{0} the constant function such that for each element of y ∈ K^{op}, g_{0}(y) = 0. Then g_{0} ∈ K(4). Similarly, the function r_{1}(c) = r_{K}(c^{∗})(1) = 1 for all c ∈ K, r_{1} : K → K^{op} is join-preserving. We shall define also functions l_{e}(b) = l_{K}(e)(b) = b, l_{0}(b) = l_{K}(0)(b) = 1 for all b ∈ K^{op}, l_{0}, l_{e} : K^{op} → K are join-preserving. Dually, we shall define functions j_{e}(c) = j_{K}(e)(c) = c, j_{0}(c) = j_{K}(0)(c) = 0 for all c ∈ K, j_{0}, j_{e} : K → K are join-preserving.

Now, let us introduce functions

\begin{aligned}F^{1}(u)(b, c) & = & (r_{K}(c^{∗})(u), c) = (r_{K}(c^{∗})(u) ∧_{K} l_{0}(b), j_{e}(c) ∨_{K} g_{0}(b)), \\ F^{2}(v)(b, c) & = & (b, c · v) = (r_{1}(c) ∧_{K} l_{e}(b), j_{K}(v)(c) ∨_{K} g_{0}(b)), \\ F^{3}(w)(b, c) & = & (l_{K}(w)(b), c) = (r_{1}(c) ∧_{K} l_{K}(w)(b), j_{e}(c) ∨_{K} g_{0}(b)), \\ F^{4}(g)(b, c) & = & (b, g(b)) = (r_{1}(c) ∧_{K} l_{e}(b), j_{0}(c) ∨_{K} g(b)) \end{aligned}

for all elements u, v, w ∈ K and g ∈ K(4). Then F^{1}(u), F^{2}(v), F^{3}(w), F^{4}(g) ∈ comm(I_{K}(K)) i.e.

F^{1}(u) ∘ h= h ∘ F^{1}(u) for all u ∈ K,
F^{2}(v) ∘ h = h ∘ F^{2}(v)  for all v ∈ K,
F^{3}(w) ∘ h= h ∘ F^{3}(w) for all w ∈ K,
F^{4}(g) ∘ h = h ∘ F^{4}(g)  for all g ∈ K(4).

Note that, h(b, c) = h(b, 0) ∨ h(1, c). First, let us compute the function H_{1} : K^{op} → K^{op} × K such that H_{1}(b) = h(b, 0) for all b ∈ K^{op}. We shall define H_{11} : K^{op} → K^{op}, H_{12} : K^{op} → K such that H_{11} = π_{1} ∘ H_{1}, H_{12} = π_{2} ∘ H_{1}. Then

\begin{aligned}F^{3}(w)(h(b, 0)) & = & (l_{K}(w)(π_{1}(h(b, 0))), π_{2}(h(b, 0))) \\ h(F^{3}(w)(b, 0)) & = & h(l_{K}(w)(b, 0)) \end{aligned}

i.e.

l_{K}(w)(H_{11}(b)) = H_{11}(l_{K}(w)(b)), H_{12}(b) = H_{12}(l_{K}(w)(b))

for all b ∈ K^{op}, w ∈ K. Let us put w = 0. Then H_{12}(b) = H_{12}(l_{K}(0)(b)) = H_{12}(1) for all b ∈ K^{op}. We have then by the same argument as for f_{3} that H_{11}(b) = r_{K}(y^{∗})(b) for an element y ∈ K. Altogether, H_{1}(b) = (r_{K}(y^{∗})(b), 0).

Similarly, we shall compute the function H_{2} : K → K^{op} × K such that H_{2}(c) = h(1, c) for all c ∈ K. We shall define functions H_{21} : K → K^{op}, H_{22} : K → K such that H_{21} = π_{1} ∘ H_{2}, H_{22} = π_{2} ∘ H_{2}. Then

\begin{aligned}F^{2}(v)(h(1, c)) & = & (π_{1}(h(1, c)), π_{2}(h(1, c)) · v) \\ h(F^{2}(v)(1, c)) & = & h(1, c · v) \end{aligned}

i.e.

H_{21}(c) = H_{21}(c · v), H_{22}(c) · v = H_{22}(c · v)

for all c, v ∈ K. Let us put c = e. Then 1 = H_{21}(e) = H_{21}(v), H_{22}(v) = H_{22}(e) · v for all v ∈ K. We have that H_{2}(c) = (1, i_{K}(x)(c)) for a suitable element x ∈ K. So we obtain that h(b, c) = H_{1}(b) ∨ H_{2}(c) = (r_{K}(y^{∗})(b), i_{K}(x)(c)). Let us prove that x = y. We have, for all u ∈ K, (b, c) ∈ K^{op} × K,

\begin{aligned}F^{1}(u)(h(b, c)) & = & (r_{K}((x · c)^{∗})(u), x · c) \\ h(F^{1}(u)(b, c)) & = & (r_{K}(y^{∗})(r_{K}(c^{∗})(u)), x · c). \end{aligned}

This gives us that

r_{K}(c^{∗} · x^{∗})(u) = r_{K}((x · c)^{∗})(u) = r_{K}(y^{∗})(r_{K}(c^{∗})(u)) = r_{K}(c^{∗} · y^{∗})(u).

Let us put c = e. Then r_{K}(x^{∗})(u) = r_{K}(y^{∗})(u)) for all u ∈ K i.e. x^{∗} = y^{∗} i.e. x = y. So we have proved that any element of bicomm(I_{K}(K)) lies in I_{K}(K). The other implication is evident.

Corollary 16 Let K be an involutive quantale, E_{K} : K → Q(S) the involutive quantale embedding from Theorem 6. Then K[e] ≅ bicomm(E_{K}(K)).

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