CONTRIBUTIONS TO GENERAL ALGEBRA 12

Proceedings of the Vienna Conference, June 3–6, 1999

Verlag Johannes Heyn, Klagenfurt 2000

MULTIPLIER ALGEBRAS OF INVOLUTIVE QUANTALES

JAN PASEKA

DEPARTMENT OF MATHEMATICS, MASARYK UNIVERSITY JANÁČKOVO NÁM. 2A 66295 BRNO, CZECH REPUBLIC

                         E-MAIL: PASEKA@MATH.MUNI.CZ

Abstract. The aim of this note is to establish a description of the (semi)multiplier algebra of an involutive quantale.

Keywords and phrases: involutive quantale, (Hilbert) Q-module, Q-bimodule, nondegenerate morphism, (semi)multiplier algebra.

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

Investigations in the theory of C^{∗}-algebras make often use of Hilbert C^{∗}- modules as a tool for proving. Research fields benefiting from it include operator K-theory, index theory for operator-valued conditional expectations, group representation theory and others. Beside this, the theory of Hilbert C^{∗}-modules is very interesting on its own right. Our main purpose here is to expose some more connections between the theory of Hilbert C^{∗}-modules and the lattice analog of them, Hilbert Q-modules. Of course the main motivation for the machinery developed here, is that certain problems concerning (involutive) quantales should be solvable by transferring the methods of Hilbert C^{∗}-modules (see [6]) into the Hilbert Q-modules framework. Similarly, as for C^{∗}-algebras (see [7]) we shall introduce the notion of a multiplier of an involutive quantale and we show that any involutive quantale with a maximal essential extension over sup-lattices with a duality has a unique maximal essential extension to its multiplier.

The results and definitions follow a natural logical sequence, so we begin without further delay. We refer to [4] for additional information and complementary results, and to [1], [5] and [8] for background information on Hilbert C^{∗}-modules and quantales.

In what follows, a complete lattice will be called sup-lattice. Sup-lattice homomorphisms are maps between sup-lattices preserving arbitrary joins. We shall denote, for S, T sup-lattices, SUP (S, T) the sup-lattice of all sup-lattice homomorphisms from S to T, with the supremum given by the pointwise ordering of mappings. Recall that a quantale is a sup-lattice Q 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

for all x, x_{i} ∈ Q, i ∈ I (I is a set). 1 denotes the greatest element of Q, 0 is the smallest element of Q. A quantale Q is said to be unital if there is an element e ∈ Q such that e · a = a = a · e for all a ∈ Q. A subquantale Q′ of a quantale Q is a subset of Q closed under ⋁ and ·. Since the operators a · − and − · b : Q → Q, a, b ∈ Q preserve arbitrary joins, they have right adjoints. Explicitly, they are given by

a →_{r} c = ⋁{s ∈ Q|a · s ≤ c} and b →_{l} d = ⋁{t ∈ Q|t · b ≤ d}

respectively.

An involution on a sup-lattice S is a unary operation such that

\begin{aligned}a^{∗∗} & = & a, \\ (⋁ a_{i})^{∗} & = & ⋁ a_{i}^{∗} \end{aligned}

for all a, a_{i} ∈ S. An involution on a quantale Q is an involution on the suplattice Q such that

(a · b)^{∗} = b^{∗} · a^{∗},

for all a, b ∈ Q. A sup-lattice (quantale) with the involution is said to be involutive.

By a morphism of (involutive) quantales will be meant a ⋁- (^{∗}-) and · - preserving mapping f : Q → Q′. If a morphism preserves the unital element we say that it is unital.

Let A be a subset of a quantale Q. We shall denote by comm(A) the commutant of A in Q (see [3]) defined as

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

Similarly, we write bicomm(A) for the bicommutant of A in Q defined as

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

By the quantale Q(M) = SUP (M, M) of endomorphisms of the sup-lattice M (see [2]) will be meant the unital quantale of sup-preserving mappings from M to itself, with the multiplication corresponding to the composition of mappings, and with the unit given by the identity mapping. Note that Q(M) ≅ (M ⊗ M^{op})^{op}; here ⊗ is a tensor product of sup-lattices.

In [3], there was introduced the idea of the Cayley representation C(Q) = Q(Q[e] × Q[e]^{op}) of an arbitrary involutive quantale Q; here the involutive quantale embedding α_{Q} : Q → C(Q) is defined by α_{Q}(a) = (a · −, a^{∗} →_{r} −), for all a ∈ Q, and Q[e] is the unital involutive quantale obtained by adding the unit e to Q, putting e = 0 ∨ e and freely generating it by the set Q ∪ {e} if Q is non-unital (see [3]), otherwise we define Q[e] = Q.

Let Q be a quantale. A left module over Q (shortly a left Q-module) is a sup-lattice M, together with a module action • : Q × M → M satisfying

\begin{aligned}(a · b) • m & = & a • (b • m) \\ (⋁ S) • m & = & ⋁\{s • m : s ∈ S\} \\ a • ⋁ X & = & ⋁\{a • x : x ∈ X\} \end{aligned}

for all a, b ∈ Q, m ∈ M, S ⊆ Q, X ⊆ M. Let M and N be modules over Q and let f : M → N be a sup-lattice homomorphism. f is a module homomorphism if f(a • m) = a • f(m) for all a ∈ Q, m ∈ M.

Note first that if M is a sup-lattice then M is a left Q(M)-module such that f • m = f(m) for all f ∈ Q(M) and all m ∈ M. Secondly, we may dually define the notion of a right Q-module. Moreover, all propositions stated for left Q-modules are valid in a dualized form for right Q-modules.

The theory of Hilbert Q-modules (we refer the reader to [4] for details and examples) is a generalization of the theory of complete semilattices with a duality and it is the natural framework for the study of modules over an involutive quantale Q endowed with Q-valued inner products.

Let Q be an involutive quantale, M a right (left) Q-module with a right (left) module action ⋄ ( •). We say that M is a right (left) Hilbert Q-module (right (left) strict Hilbert Q-module) if M is equipped with a map ⟨−, −⟩ : M × M → Q, called the inner product, such that for all a ∈ Q, m, n ∈ M and m_{i} ∈ M, where i ∈ I, the conditions (1)-(5) ((1)-(6)) are satisfied

⟨m, n⟩ · a = ⟨m, n ⋄ a⟩ (a · ⟨m, n⟩ = ⟨a • m, n⟩); (1)

⋁_{i∈I}⟨m_{i}, n⟩ = ⟨⋁_{i∈I} m_{i}, n⟩; (2)

⋁_{i∈I}⟨m, m_{i}⟩ = ⟨m, ⋁_{i∈I} m_{i}⟩; (3)

⟨m, n⟩^{∗} = ⟨n, m⟩; (4) ⟨−, m⟩ = ⟨−, n⟩ (⟨m, −⟩ = ⟨n, −⟩) implies m = n; (5)

⟨m, m⟩ = 0 implies m = 0. (6) 3

Let f : M → N a map between right (left) Hilbert Q-modules. We say that a map f^{∗} : N → M is a ^{∗}-adjoint to f and f is adjointable if

⟨f(m), n⟩ = ⟨m, f^{∗}(n)⟩

for all m ∈ M, n ∈ N. Automatically, f is then a Q-module homomorphism.

The ⋁-semilattice of all adjointable maps from N to M is denoted by A_{Q}(N, M). For m ∈ M and n ∈ N, define a fr-operator Θ_{m,n} : N → M by Θ_{m,n}(p) = m ⋄ ⟨n, p⟩ = ⟨p, n⟩ • m for all p ∈ N. We shall denote by K_{Q}(N, M) the sub-⋁-semilattice of A_{Q}(N, M) generated by the set {Θ_{m,n} : m ∈ M, n ∈ N} and the elements of K_{Q}(N, M) we will call compact operators.

Note that A_{Q}(M), the ⋁-semilattice of all adjointable maps from M to itself, is an involutive quantale. Moreover, if Q = 2, where 2 is the 2-element Boolean algebra then we have Q(M) = A_{2}(M). Namely, ⋁-semilattices with a duality (complete ortholattices) are exactly the Hilbert (strict) 2-modules.

1. Multiplier algebras

Definition 1.1. Let Q be an involutive quantale, M an involutive ⋁-semilattice with an involution ^{#}, M both a left Q-module with the action • and a right Q-module with the action ⋄ such that

\begin{aligned}(a • m) ⋄ b & = & a • (m ⋄ b), \\ (a • m)^{#} & = & m^{#} ⋄ a^{∗} \end{aligned}

for all a, b ∈ Q, m ∈ M. Then we shall say that M is an involutive bimodule over Q. An involutive bimodule homomorphism is both a left- and rightmodule homomorphism preserving involution. Moreover, if Q is an involutive ⋁-subsemilattice of M we shall say that Q is a semiideal of M and M is a semiextension of Q, if

a • b = a · b = a ⋄ b and a • m, m ⋄ b ∈ Q

for all a, b ∈ Q and all m ∈ M. If moreover M is an involutive quantale, Q an involutive subquantale and a semiideal of M we call Q an ideal of M and M an extension of Q. We call a semiideal (ideal) Q essential if for all m, n ∈ M,

a • m = a • n and m ⋄ b = n ⋄ b for all a, b ∈ Q ⟹ m = n.

We shall say that a semiextension (extension) M of Q is maximal if any other such semiextension (extension) M′ of Q can be embedded in it.

Recall that Q is essential in M iff for all m, n ∈ M, a • m = a • n for all a ∈ Q ⟹ m = n iff for all m, n ∈ M, m ⋄ b = n ⋄ b for all b ∈ Q ⟹ m = n. Moreover, Q is essential in Q iff Q is a Hilbert Q-module.

Let Q be an involutive quantale, M a (left) Hilbert Q-module and M^{⊙ ⋆} the ⋁-semilattice of all Q-module homomorphisms from M to Q. Then we know that M^{⊙ ⋆} is a left Q-module with the module multiplication (c • f)(m) = f(m) · c^{∗}, for all c ∈ Q, m ∈ M, f ∈ M^{⊙ ⋆}. We have a canonical embedding ι_{M} : M → M^{⊙ ⋆} defined by m ↦ ⟨−, m⟩, m ∈ M. Namely, let a ∈ Q, m, n ∈ M. Then ι_{M}(a • m)(n) = ⟨n, a • m⟩ = ⟨a • m, n⟩^{∗} = (a · ⟨m, n⟩)^{∗} = ⟨n, m⟩ · a^{∗} = (a • ⟨−, m⟩)(n) = (a • ι_{M}(m))(n). Note that for T ∈ Hom_{Q}(M, M) we have a Q-module homomorphism T^{⊙ ⋆} : M^{⊙ ⋆} → M^{⊙ ⋆} defined by the prescription (T^{⊙ ⋆}(f))(m) = f(T (m)) for all f ∈ M^{⊙ ⋆}, m ∈ M. Namely, we have (a • T^{⊙ ⋆}(f))(m) = T^{⊙ ⋆}(f)(m) · a^{∗} = f(T (m)) · a^{∗} = (a • f)(T (m)) = (T^{⊙ ⋆}(a • f))(m). Evidently, (T_{1} ∘ T_{2})^{⊙ ⋆} = T2^{⊙ ⋆} ∘ T1^{⊙ ⋆}. Similarly, for any T ∈ Hom_{Q}(M, M^{⊙ ⋆}) we have a ⋁-preserving map T^{#} : M → M^{⊙ ⋆} defined by the (T^{#}(m))(n) = (T (n)(m))^{∗} for all m, n ∈ M. Evidently, the map ^{#} is an involution on the ⋁-semilattice Hom_{Q}(M, M^{⊙ ⋆}) since ^{#} preserves arbitrary joins and any T^{#} is a module homomorphism by (a • T^{#}(m))(n) = T^{#}(m)(n)·a^{∗} = (T (n)(m))^{∗} · a^{∗} = (a · T (n)(m))^{∗} = (T (n)(a • m))^{∗} = T^{#}(a • m)(n).

Lemma 1.2. Let Q and Q′ be involutive quantales, M a Hilbert Q-module. Then an involutive quantale homomorphism ρ : Q′ → A_{Q}(M) determines a structure of a ρ(Q′) bimodule on Hom_{Q}(M, M^{⊙ ⋆}) by the following prescription:

ρ(a) • T = ρ(a^{∗})^{⊙ ⋆} ∘ T, T ⋄ ρ(a) = T ∘ ρ(a)

for all a ∈ Q′, T ∈ Hom_{Q}(M, M^{⊙ ⋆}).

Proof. Let us check the left module axioms. Let a, b ∈ Q′, T ∈ Hom_{Q}(M, M^{⊙ ⋆}). Then

(ρ(a) ∘ ρ(b)) • T = ρ(a · b) • T = ρ((a · b)^{∗})^{⊙ ⋆} ∘ T = ρ(b^{∗} · a^{∗})^{⊙ ⋆} ∘ T
= ((ρ(a^{∗})^{⊙ ⋆} ∘ ρ(b^{∗})^{⊙ ⋆}) ∘ T = ρ(a^{∗})^{⊙ ⋆} ∘ (ρ(b^{∗})^{⊙ ⋆} ∘ T)
= ρ(a^{∗})^{⊙ ⋆} ∘ (ρ(b) • T) = ρ(a) • (ρ(b) • T).

Similarly, we have

\begin{aligned}(⋁_{i∈I} ρ(a_{i})) • T & = & ρ(⋁_{i∈I} a_{i}) • T = ρ(⋁_{i∈I} a_{i}^{∗})^{⊙ ⋆} ∘ T \\ & = & ⋁_{i∈I}(ρ(a_{i}^{∗})^{⊙ ⋆} ∘ T) = ⋁_{i∈I}(ρ(a_{i}) • T) \end{aligned}

and

ρ(a) • (⋁_{i∈I} T_{i}) = ρ(a^{∗})^{⊙ ⋆} ∘ (⋁_{i∈I} T_{i}) = ⋁_{i∈I} ρ(a^{∗})^{⊙ ⋆} ∘ T_{i} = ⋁_{i∈I} ρ(a) • T_{i}.

By the same arguments we can prove that Hom_{Q}(M, M^{⊙ ⋆}) is a right module. Let us check that it is an involutive bimodule. We have, for all a, b ∈ Q′ and all T ∈ Hom_{Q}(M, M^{⊙ ⋆}),

\begin{aligned}(ρ(a) • T) ⋄ ρ(b) & = & (ρ(a^{∗})^{⊙ ⋆} ∘ T) ∘ ρ(b) \\ & = & ρ(a^{∗})^{⊙ ⋆} ∘ (T ∘ ρ(b)) = ρ(a) • (T ⋄ ρ(b)) \end{aligned}

and

(ρ(a) • T)^{#}(m)(n) = (ρ(a^{∗})^{⊙ ⋆} ∘ T)^{#}(m)(n) = [((ρ(a^{∗})^{⊙ ⋆}(T (n))) (m)]^{∗}
= [T (n)(ρ(a^{∗})(m))]^{∗} = T^{#}(ρ(a)^{∗}(m)(n)
= (T^{#} ∘ ρ(a)^{∗})(m)(n) = (T^{#} ⋄ ρ(a)^{∗})(m)(n).

Note that we have a ⋁- and involution-preserving embedding j_{M} : A_{Q}(M) → Hom_{Q}(M, M^{⊙ ⋆}) defined by T ↦ i_{M} ∘ T i.e. j_{M}(T)(m) = ⟨−, T (m)⟩ for all m ∈ M. Namely, we have j_{M}(T)^{#} = j_{M}(T^{∗}), j_{M}(⋁_{i∈I} T_{i}) = ⋁_{i∈I} j_{M}(T_{i}) and T^{⊙ ⋆} ∘ i_{M} = T^{∗}.

Definition 1.3. Let Q and Q′ be involutive quantales, M a Hilbert Q-module. We say that an involutive quantale homomorphism α : Q′ → A_{Q}(M) is nondegenerate if M = {⋁_{i∈I} α(a_{i})(m_{i}) : a_{i} ∈ Q′, m_{i} ∈ M}. We shall say that Q′ is non-degenerate if it is a Hilbert Q′-module and the involutive quantale homomorphism α_{Q′} : Q′ → Q(Q′ × Q′^{op}) defined by α_{Q′}(a) = (a · −, a^{∗} →_{r} −), for all a ∈ Q′, is non-degenerate.

It may be remarked that Q′ is non-degenerate iff it is essential in itself, Q′ · Q′ = Q′ and Q′ →_{r} Q′ = Q′. Evidently, any unital involutive quantale is non-degenerate.

Lemma 1.4. Let Q and Q′ be involutive quantales, M a Hilbert Q-module, Q′ a semiideal of an involutive Q′ bimodule N, α : Q′ → A_{Q}(M) a nondegenerate involutive quantale homomorphism. Then α extends uniquely to an involutive bimodule homomorphism \overline{α} : N → Hom_{Q}(M, M^{⊙ ⋆}), i.e.

\overline{α}(a) = ι_{M}(α(a)), \overline{α}(a • n) = α(a) • \overline{α}(n), \overline{α}(n ⋄ b) = \overline{α}(n) ⋄ α(b),

for all a, b ∈ Q′, n ∈ N. If α is injective and Q′ essential in N then \overline{α} is also injective.

Proof. Let us define \overline{α}(n)(⋁_{i∈I} α(a_{i})(m_{i})) = ⋁_{i∈I}(ι_{M} ∘ (α(n ⋄ a_{i}))(m_{i}) for all a_{i} ∈ Q′, m_{i} ∈ M. First, let us show the correctness of our definition. Let ⋁_{i∈I} α(a_{i})(m_{i}) = ⋁_{j∈J} α(b_{j})(r_{j}) and let y = ⋁_{k∈K} α(c_{k})(q_{k}) ∈ M. Then

\overline{α}(n)(⋁_{i∈I} α(a_{i})(m_{i}))(y) = ⋁_{i∈I}(ι_{M} ∘ (α(n ⋄ a_{i}))(m_{i})(y)
= ⋁_{i∈I} ⟨⋁_{k∈K} α(c_{k})(q_{k}), α(n ⋄ a_{i})(m_{i})⟩
= ⋁_{i∈I,k∈K} ⟨(α(n ⋄ a_{i})^{∗} ∘ α(c_{k}))(q_{k}), m_{i}⟩
= ⋁_{i∈I,k∈K} ⟨(α(a_{i}^{∗} • n^{∗} ⋄ c_{k})(q_{k}), m_{i}⟩
= ⋁_{i∈I,k∈K} ⟨(α(n^{∗} ⋄ c_{k})(q_{k}), α(a_{i})(m_{i})⟩
= ⋁_{k∈K} ⟨(α(n^{∗} ⋄ c_{k})(q_{k}), ⋁_{i∈I} α(a_{i})(m_{i})⟩
= ⋁_{k∈K} ⟨(α(n^{∗} ⋄ c_{k})(q_{k}), ⋁_{j∈J} α(b_{j})(r_{j})⟩
= \overline{α}(n)(⋁_{j∈J} α(b_{j})(r_{j}))(y). 6

Now, let us check that \overline{α} preserves arbitrary joins and involution. Assume n_{k} ∈ N, k ∈ K. Then we have

\begin{aligned}\overline{α}(⋁_{k∈K} n_{k})(⋁_{i∈I} α(a_{i})(m_{i})) & = & ⋁_{i∈I}(ι_{M} ∘ α((⋁_{k∈K} n_{k}) ⋄ a_{i}))(m_{i}) \\ ⋁_{i∈I,k∈K}(ι_{M} ∘ α(n_{k} ⋄ a_{i}))(m_{i}) & = & ⋁_{k∈K} \overline{α}(n_{k})(⋁_{i∈I} α(a_{i})(m_{i})) \end{aligned}

and, for y = ⋁_{j∈J} α(b_{j})(r_{j}),

\begin{aligned}\overline{α}(n)^{#}(⋁_{i∈I} α(a_{i})(m_{i}))(y) & = & \overline{α}(n)(y)(⋁_{i∈I} α(a_{i})(m_{i}))^{∗} = \\ ⟨⋁_{i∈I} α(a_{i})(m_{i}), ⋁_{j∈J} α(n ⋄ b_{j})(r_{j})⟩^{∗} & = & ⟨⋁_{j∈J} α(n ⋄ b_{j})(r_{j}), ⋁_{i∈I} α(a_{i})(m_{i})⟩ \\ = ⋁_{i∈I,j∈J} ⟨α(n ⋄ b_{j})(r_{j}), α(a_{i})(m_{i})⟩ & = & ⋁_{i∈I,j∈J} ⟨α(a_{i}^{∗} • n ⋄ b_{j})(r_{j}), m_{i}⟩ \\ = ⋁_{i∈I,j∈J} ⟨α(b_{j})(r_{j}), α(n^{∗} ⋄ a_{i})(m_{i})⟩ & = & \overline{α}(n^{∗})(⋁_{i∈I} α(a_{i})(m_{i}))(y). \end{aligned}

Let us show that \overline{α}(a • n) = α(a) • \overline{α}(n). We have

\begin{aligned}\overline{α}(a • n)(⋁_{i∈I} α(a_{i})(m_{i}))(y) & = & ⋁_{i∈I}(ι_{M} ∘ α(a • n ⋄ a_{i}))(m_{i})(y) = \\ ⋁_{i∈I} ⟨y, α(a • n ⋄ a_{i})(m_{i}))⟩ & = & ⋁_{i∈I} ⟨α(a)^{∗}(y), α(n ⋄ a_{i})(m_{i}))⟩ = \\ ⟨α(a^{∗})(y), ⋁_{i∈I} α(n ⋄ a_{i})(m_{i}))⟩ & = & ι_{M}(⋁_{i∈I} α(n ⋄ a_{i})(m_{i}))(α(a^{∗})(y)) = \\ \overline{α}(n)(⋁_{i∈I} α(a_{i})(m_{i}))(α(a^{∗})(y)) & = & α(a^{∗})^{⊙ ⋆}(\overline{α}(n)(⋁_{i∈I} α(a_{i})(m_{i})))(y) = \\ (α(a^{∗})^{⊙ ⋆} ∘ \overline{α}(n))(⋁_{i∈I} α(a_{i})(m_{i}))(y) & = & (α(a) • \overline{α}(n))(⋁_{i∈I} α(a_{i})(m_{i}))(y). \end{aligned}

Similarly, we have

\overline{α}(n ⋄ b)(⋁_{i∈I} α(a_{i})(m_{i})) = ⋁_{i∈I} α(n ⋄ b ⋄ a_{i})(m_{i}) =
\overline{α}(n)(⋁_{i∈I} α(b ⋄ a_{i})(m_{i})) = \overline{α}(n)(α(b)(⋁_{i∈I} α(a_{i})(m_{i})))
= (\overline{α}(n) ⋄ α(b))(⋁_{i∈I} α(a_{i})(m_{i})).

Now, let β : N → Hom_{Q}(M, M^{⊙ ⋆}) be an involutive bimodule homomorphism such that β(a) = ι_{M}(α(a)), β(a • n) = α(a) • β(n), β(n ⋄ b) = β(n) ⋄ α(b). Then

β(n)(⋁_{i∈I} α(a_{i})(m_{i})) = ⋁_{i∈I}(β(n) ∘ α(a_{i}))(m_{i})
= ⋁_{i∈I}(β(n) ∘ α(a_{i}))(m_{i})
= ⋁_{i∈I} β(n ⋄ a_{i})(m_{i})
= ⋁_{i∈I}(ι_{M} ∘ α(n ⋄ a_{i}))(m_{i})
= \overline{α}(n)(⋁_{i∈I} α(a_{i})(m_{i})).

Finally, if α is injective, we have, for all n_{1}, n_{2} ∈ N and all a ∈ Q′ such that \overline{α}(n_{1}) = \overline{α}(n_{2}), the following:

ι_{M} ∘ α(n_{1} ⋄ a) = \overline{α}(n_{1} ⋄ a) = (\overline{α}(n_{1}) ∘ α)(a) = (\overline{α}(n_{2}) ∘ α)(a) = ι_{M} ∘ α(n_{2} ⋄ a).

This gives us that α(n_{1} ⋄ a) = α(n_{2} ⋄ a) for all a ∈ Q′ i.e. n_{1} ⋄ a = n_{2} ⋄ a i.e. n_{1} = n_{2}.

Corollary 1.5. Let Q and Q′ be involutive quantales, M a Hilbert Q-module, Q′ an involutive subquantale and an ideal of an involutive quantale Q′′, α : Q′ → A_{Q}(M) a non-degenerate involutive quantale homomorphism. Then α extends uniquely to an involutive quantale homomorphism \overline{\overline{α}} : Q′′ → A_{Q}(M). If α is injective and Q′ essential in Q′′ then \overline{\overline{α}} is also injective.

Proof. Let us define

\overline{\overline{α}}(n)(⋁_{i∈I} α(a_{i})(m_{i})) = ⋁_{i∈I} α(n · a_{i})(m_{i})

for all n ∈ Q′. Then \overline{α} = ι_{M} ∘ \overline{\overline{α}} i.e. by the injectivity of ι_{M} is \overline{\overline{α}} correctly defined. It is easy to verify that \overline{α}(n_{1} ·n_{2}) = ι_{M} ∘\overline{\overline{α}}(n_{1})∘\overline{\overline{α}}(n_{2}) i.e. \overline{\overline{α}}(n_{1} ·n_{2}) = \overline{\overline{α}}(n_{1}) ∘ \overline{\overline{α}}(n_{2}). Evidently, \overline{\overline{α}} preserves arbitrary joins. Note that

\begin{aligned}⟨y, \overline{\overline{α}}(n^{∗})(⋁_{i∈I} α(a_{i})(m_{i}))⟩ & = & ⟨y, ⋁_{i∈I} α(n^{∗} · a_{i})(m_{i})⟩ = \\ ⟨α(n)(y), ⋁_{i∈I} α(a_{i})(m_{i})⟩ & = & ⟨\overline{\overline{α}}(n)(y), ⋁_{i∈I} α(a_{i})(m_{i})⟩ \end{aligned}

i.e. \overline{\overline{α}} preserves involution. If α is injective we have by the injectivity of \overline{α} that \overline{\overline{α}} is injective.

Definition 1.6. Let Q, Q′ be involutive quantales and let M be a Hilbert Qmodule. Suppose that α : Q′ → A_{Q}(M) is a non-degenerate injective involutive quantale homomorphism. We say that the set

SM_{M}(Q′) = \{T ∈ Hom_{Q}(M, M^{⊙ ⋆}) : T ⋄ α(Q′) ⊆ \overline{α}(Q′) and α(Q′) • T ⊆ \overline{α}(Q′)\}

is a (M, α)-semimultiplier of the involutive quantale Q′. Similarly, we say that the set

M_{M}(Q′) = {T ∈ A_{Q}(M) : T ∘ α(Q′) ⊆ α(Q′) and α(Q′) ∘ T ⊆ α(Q′)}

is a (M, α)-multiplier of the involutive quantale Q′.

Evidently, from the proof of lemma 1.4 (corollary 1.5) we easily see that SM_{M}(Q′) is a semiextension of α(Q′) (M_{M}(Q′) is an extension of α(Q′)). Applying the non-degeneracy we have that, for all T_{1}, T_{2}, T_{1} ∘ α(a) = T_{2} ∘ α(a) for all a ∈ Q′ implies T_{1} = T_{2} i.e. our semiextension (extension) is essential. Moreover, for any non-degenerate involutive quantale Q′, we have a (Q′ × Q′^{op}, α_{Q′})-semimultiplier (multiplier) over 2. We then have the following.

Lemma 1.7. Let Q, Q′ be involutive quantales and let M be a Hilbert Qmodule. Suppose that α : Q′ → A_{Q}(M) is a non-degenerate injective involutive quantale homomorphism. Then SM_{M}(Q′) (M_{M}(Q′)) is a maximal essential semiextension (extension) of α(Q′).

Proof. Let X be any essential semiextension of α(Q′). Then we have an injective embedding α̃ : X → Hom_{Q}(M, M^{⊙ ⋆}) such that α̃(a) = ι_{M}(α(a)), α̃(a • n) = α(a) • α̃(n) ∈ \overline{α}(Q′), α̃(n ⋄ b) = α̃(n) ⋄ α(b) ∈ \overline{α}(Q′) for all a, b ∈ Q′ and all n ∈ X i.e. α̃(n) ∈ SM_{M}(Q′). The extension part of the proof follows the same ideas.

Proposition 1.8. Let Q′ be an involutive quantale with a maximal essential semiextension (extension) over 2. Then Q′ has a unique (up to isomorphism) maximal essential semiextension (extension), identical on Q′. In particular, all (M, α)-semimultipliers ( (M, α)-multipliers) are isomorphic.

Proof. Let α_{0} : Q′ → Q(\overline{M}) be a maximal semiextension (extension) over 2 and let X be a maximal essential extension of Q′, α : Q′ → X. We shall show that X is isomorphic to the (\overline{M}, α_{0})-semiextension SM\overline{M}(Q′). By maximality of our semiextension there are involutive bimodule monomorphisms β : X → SM\overline{M}(Q′) and γ : SM\overline{M}(Q′) → X such that β_{/α(Q′)} ∘ γ_{/α_{0}(Q′)} = id_{α_{0}(Q′)} and γ_{/α_{0}(Q′)} ∘ β_{/α(Q′)} = id_{α(Q′)}. Then β ∘ γ : SM\overline{M}(Q′) → SM\overline{M}(Q′) ⊆ Hom_{2}(\overline{M}, \overline{M}^{⊙ ⋆}) ≃ A_{2}(\overline{M}, \overline{M}^{op}) ≃ Q(\overline{M}) and β ∘γ_{/α_{0}(Q′)} = id_{α_{0}(Q′)}. By lemma 1.4 id_{α_{0}(Q′)} extends both to β ∘ γ and to id_{SM\overline{M} (Q′)} i.e. from the uniqueness we have that β ∘ γ = id_{SM\overline{M} (Q′)}. The multiplier part is evident.

Theorem 1.9. Let Q′ be an involutive quantale with a maximal essential semiextension (extension) α_{0} : Q′ → Q(\overline{M}) over 2. Then M\overline{M}(Q′) ≃ SM\overline{M}(Q′) ≃ {T ∈ bicomm(α_{0}(Q′)) : T ∘ α_{0}(Q′) ⊆ α_{0}(Q′) and α_{0}(Q′) ∘ T ⊆ α_{0}(Q′)} i.e. any maximal essential semimultiplier is isomorphic to any maximal essential multiplier and both are isomorphic to the involutive quantale M(Q′) = {T ∈ bicomm(α_{0}(Q′)) : T ∘ α_{0}(Q′) ⊆ α_{0}(Q′) and α_{0}(Q′) ∘ T ⊆ α_{0}(Q′)}.

Proof. Note that we have a one-to-one correspondence between α_{0}(a)^{⊙ ⋆} and α_{0}(a^{∗}), for all a ∈ Q′, since α_{0}(a)^{⊙ ⋆} = α_{0}(b)^{⊙ ⋆} implies α_{0}(a) = α_{0}(b) i.e. a = b. Any element of S ∈ Hom_{2}(\overline{M}, \overline{M}^{⊙ ⋆}) is of the form S = ι\overline{M} ∘ T, T ∈ Q(\overline{M}) and the same holds for the module actions. So we have that SM\overline{M}(Q′) ≃ M\overline{M}(Q′). Evidently, M\overline{M}(Q′) ⊇ \{T ∈ bicomm(α_{0}(Q′)) : T ∘ α_{0}(Q′) ⊆ α_{0}(Q′) and α_{0}(Q′) ∘ T ⊆ α_{0}(Q′)\}. Let us show that M\overline{M}(Q′) ⊆ bicomm(α_{0}(Q′)). Assume T ∈ M\overline{M}(Q′), R ∈ comm(α_{0}(Q′)), a, b ∈ Q′. Then

\begin{aligned}T ∘ R ∘ α_{0}(b) & = & T ∘ α_{0}(b) ∘ R & = & R ∘ T ∘ α_{0}(b), \\ α_{0}(a) ∘ T ∘ R & = & R ∘ α_{0}(a) ∘ T & = & α_{0}(a) ∘ R ∘ T \end{aligned}

i.e. T ∘ R = R ∘ T i.e. T ∈ bicomm(α_{0}(Q′)).

Theorem 1.10. Let Q be an involutive quantale, M a Hilbert Q-module such that M = {⋁_{i∈I} ⟨m_{i}, p_{i}⟩ • r_{i} : m_{i}, p_{i}, r_{i} ∈ M}. Then A_{Q}(M) ≃ M(K_{Q}(M)) and Hom_{Q}(M, M^{⊙ ⋆}) ≃ SM(K_{Q}(M)).

Proof. We have an embedding (inclusion) ρ_{M} : K_{Q}(M) → A_{Q}(M) and evidently this embedding is non-degenerate. So we have

M(K_{Q}(M)) ≃ {T ∈ Hom_{Q}(M, M^{⊙ ⋆}) : T ∘ ι_{M} ∘ K_{Q}(M) ⊆ ι_{M} ∘ K_{Q}(M) and

(K_{Q}(M))^{⊙ ⋆} ∘ T ⊆ ι_{M} ∘ K_{Q}(M)} = Hom_{Q}(M, M^{⊙ ⋆}) 9

and

\begin{matrix}M(K_{Q}(M)) ≃ \{ & T ∈ A_{Q}(M) : T ∘ K_{Q}(M) ⊆ K_{Q}(M) and \\ (K_{Q}(M)) ∘ T ⊆ K_{Q}(M)\} = A_{Q}(M). \end{matrix}

Corollary 1.11. Let Q be a non-degenerate involutive quantale. Then A_{Q}(Q) ≃ M(K_{Q}(Q)).

Proof. We have that Q is a Hilbert Q-module with the standard multiplication satisfying the assumptions of Theorem 1.10. The rest is evident.

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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