Hilbert Q-modules and Nuclear Ideals in the Category of ⋁-semilattices with a Duality

J. Paseka^{1}

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

Abstract

The basics of the theory of Hilbert Q-modules are established. Using this approach to ⋁-semilattices with a duality i.e. Hilbert 2-modules, we describe a nuclear traced ideal and characterize nuclear maps in the respective category.

Quantales, that can be seen as algebras of finite observations, are a generalization of locales studied by Mulvey [12] with the aim of providing a constructive formulation of the foundations of quantum mechanics. They also provide a basis for posetal models of linear logic (see [6,18]). Essentially, quantales are to linear logic what locales (complete Heyting algebras) are to intuitionistic logic and complete Boolean algebras to classical logic. Quantale modules, that can be seen as algebras of finitely observable properties, provide an algebraic framework for operational, denotational and axiomatic semantics (see [1]). Nuclear ideals, a new categorical structure introduced in [4], are slight variants of compact closed categories, and very roughly, they can be viewed as models of multiplicative linear logic. If a monoidal category is traced the Geometry of Interaction programme (see [3,7]) can be used to obtain a compact closed category from it. Moreover, interaction categories proposed by Abramsky (see [2]) as a new paradigm for the semantics of computation can be conveniently treated as quantaloids, i.e., as categories enriched over the category of ⋁- semilattices. Our particular interest here is to develop basic theory of a new mathematical structure closely related to the above important notions and present an interesting example of a traced nuclear ideal which arises from this theory.

Since Girard in his Geometry of Interaction programme (see [6–8]) used the formalism of C^{∗}-algebras and Banach spaces and Abramsky and others (see

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

Preprint submitted to Elsevier Preprint 23 February 2001

[4]), generalizing ordinary binary relations to probabilistic relations with an eye towards certain applications in computer science, studied the category of Hilbert spaces as an example of a tensored ∗-category with a traced nuclear ideal, there arises a natural idea to study a lattice-theoretical counterpart of Hilbert C^{∗}-modules. They provide (see [11]) a conceptual and mathematical framework for modern operator algebra and play a large role in noncommutative topology and geometry, being a noncommutative version of a vector bundle. Intuitively speaking, a Hilbert C^{∗}-module Y is defined exactly like a Hilbert space, except that Y is also a module over a C^{∗}-algebra A, the inner product is A-valued and we have a compatibility condition. Another motivation for their study comes in getting a technical underpinning for the C^{∗}-algebraic approach to quantum groups.

The organization of the paper is as follows. The first part of the paper attempts to begin the investigation of Hilbert Q-modules. The basics of the theory of Hilbert Q-modules are established. Our idea is, similarly as for Hilbert C^{∗}- modules, to generalize a ⋁-semilattice with a duality by allowing the inner product to take values in a (unital) involutive quantale rather than in the two-element Boolean algebra 2. Analogously as for Hilbert C^{∗}-modules the morphisms in the category of Hilbert Q-modules are adjointable maps. A special class of the so called compact operators playing a similar role as finiterank operators is introduced. Moreover, the category of Hilbert Q-modules and adjointable maps between them is an idempotent involutive category with biproducts and an involutive quantaloid. To get insight into the structure of Hilbert Q-modules we explore in the second part of the paper the notion of the tensor product in the category of Hilbert 2-modules. We shall give its explicit description using only the order. We prove that the category of (strict) Hilbert 2-modules is a ∗-tensored category and, motivated by the paper [4], that the class of compact operators forms a traced nuclear ideal for (strict) Hilbert 2-modules. A characterization of nuclear maps and nuclear objects is given. Subsequently, in the last part of the paper we make some concluding remarks.

The paper is closely related to the papers [13], [15], [19] and [4] where the interested reader can find unexplained terms, notation and motivation concerning the subject. For facts concerning quantales in general we refer to [18].

1 Hilbert modules

The theory of Hilbert modules 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. Throughout this section, Q will be an involutive quantale (almost always unital).

Definition 1 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) pre-Hilbert Q-module (right (left) Hilbert Q-module, strict Hilbert Q-module) if M is equipped with a map ⟨−, −⟩ : M ×M → Q, called the pre-inner product (inner product), such that for all a ∈ Q, m, n ∈ M and m_{i} ∈ M, where i ∈ I, the conditions (1)-(4) ((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)

Lemma 2 Let Q be an involutive quantale and let M be a left (right) pre-Hilbert Q-module. Then the factor module M_{R_{H}} defined by the equivalence relation R_{H} = {(m, n) ∈ M × M : ⟨m, p⟩ = ⟨n, p⟩ for all p ∈ M} is a left (right) Hilbert Q-module.

PROOF. We shall define an operator j_{H} : M → M by j_{H}(m) = ⋁{n ∈ M : (m, n) ∈ R_{H}}. Then evidently m ≤ j_{H}(m), (m, j_{H}(m)) ∈ R_{H}, j_{H}(j_{H}(m)) = j_{H}(m), m ≤ n implies n = m ∨ n i.e. (j_{H}(n), j_{H}(m) ∨ j_{H}(n)) ∈ R_{H} i.e. j_{H}(n) ≥ j_{H}(m) ∨ j_{H}(n) ≥ j_{H}(m) for all m, n ∈ M. Moreover, if a ∈ Q and p ∈ M, we have that ⟨m, p⟩ = ⟨j_{H}(m), p⟩ i.e. a · ⟨m, p⟩ = a · ⟨j_{H}(m), p⟩ i.e. ⟨a • m, p⟩ = ⟨a • j_{H}(m), p⟩ i.e. (a • m, a • j_{H}(m)) ∈ R_{H} i.e. a • j_{H}(m) ≤ j_{H}(a • m). So we have proved that j_{H} is a module nucleus on M i.e. M_{R_{H}} is a factor module on M. Note that, for all m, n ∈ M, ⟨m, n⟩ = ⟨j_{H}(m), j_{H}(n)⟩. We shall now define the inner product ⟨−, −⟩_{H} on M_{R_{H}} by ⟨m, n⟩_{H} = ⟨m, n⟩ for all m, n ∈ M_{R_{H}}. Let us prove the conditions (1)-(5). We have a · ⟨m, n⟩_{H} = a · ⟨m, n⟩ = ⟨a • m, n⟩ = ⟨a •_{H}m, n⟩_{H} for all a ∈ Q, m, n ∈ M_{R_{H}}. Similarly, ⋁_{i∈I}⟨m_{i}, n⟩_{H} = ⋁_{i∈I}⟨m_{i}, n⟩ = ⟨⋁_{i∈I} m_{i}, n⟩ = ⟨j_{H}(⋁_{i∈I} m_{i}), n⟩_{H} for all m_{i}, n ∈ M_{R_{H}}, i ∈ I. By the same procedure we have ⋁_{i∈I}⟨n, m_{i}⟩_{H} = ⟨n, j_{H}(⋁_{i∈I} m_{i})⟩_{H}. Evidently, ⟨m, n⟩_{H}^{∗} = ⟨m, n⟩^{∗} = ⟨n, m⟩ = ⟨n, m⟩_{H} for all m, n ∈ M_{R_{H}}. Now, let us assume that ⟨m, −⟩_{H} = ⟨n, −⟩_{H}. Then, for all p ∈ M, we have that ⟨m, j_{H}(p)⟩_{H} = ⟨n, j_{H}(p)⟩_{H} i.e. we have ⟨m, p⟩ = ⟨n, p⟩ i.e. m = j_{H}(m) = j_{H}(n) = n. □

Lemma 3 Let M be a ⋁-semilattice. Then the following conditions are equivalent:

(1) M is a ⋁-semilattice with a duality ′.

(2) M is a left (right) Hilbert 2-module; here 2 is a complete lattice {0, e} equipped with a non-trivial multiplication (· = ∧) such that the involution on it is the identity.

PROOF. It is evident (we put m ⋄ e = m = e • m and ⟨m, n⟩ = 0 iff m ≤ n′). □

Corollary 4 Let M be a ⋁-semilattice. Then the following conditions are equivalent:

(1) M is an ortholattice with an orthocomplement ′.

(2) M is a left (right) strict Hilbert 2-module.

PROOF. (1) ⟹ (2): Let ⟨m, m⟩ = 0. Then m ≤ m′ i.e. m = m ∧ m ≤ m′ ∧ m = 0 i.e. m = 0.

(2) ⟹ (1): We have ⟨m ∧ m′, m ∧ m′⟩ ≤ ⟨m, m′⟩ = 0 i.e. m ∧ m′ = 0. □

A basic example of a right (left) Hilbert module over an involutive unital quantale Q is M = Q with the standard inner product ⟨m, n⟩ = m^{∗} · n and the module action m ⋄ a = m·a (a • m = m·a^{∗}); another example is any complete right (left) ideal of Q with the same inner product.

Note that, if M is both left and right Hilbert module we shall speak about a Hilbert w-bimodule and we have that a • m = m ⋄ a^{∗}. Note that any right (left) Hilbert module gives rise to a Hilbert w-bimodule and conversely.

Lemma 5 Let Q be an involutive quantale, M a ⋁-semilattice. Then the following is equivalent:

(1) M is a right Hilbert Q-module.

(2) M is a left Hilbert Q-module.

(3) M is a Hilbert Q-w-bimodule.

PROOF. (1) ⟹ (2): Let us define, for all a ∈ Q and all m ∈ M, a • m = m ⋄ a^{∗}. Then, for all a ∈ Q and all m, n ∈ M, we have a · ⟨m, n⟩ = (a · ⟨m, n⟩)^{∗∗} = (⟨n, m⟩ · a^{∗})^{∗} = ⟨n, m ⋄ a^{∗}⟩^{∗} = ⟨m ⋄ a^{∗}, n⟩ = ⟨a • m, n⟩ i.e. the property (1) from the definition 1 is satisfied. The properties (2)-(4) from 1 are evident.

(2) ⟹ (1): The proof dualizes the proof of (1) ⟹ (2).

(1) ⟺ (3): It is evident. □

Since the theory of ⋁-semilattices with a duality (complete ortholattices) and ⋁-preserving maps is based on the use of duality (othocomplementation), it is clear that there will be obstacles to develop an analogous theory for Hilbert Q-modules. Nevertheless, it is useful to use ⋁-semilattices with a duality (complete ortholattices) as a guide, adding extra conditions when necessary to obtain a working theory for Hilbert Q-modules. With this in mind and motivated by theory of Hilbert C^{∗}-modules, we now introduce an important class of operators on Hilbert Q-modules.

Definition 6 Let Q be an involutive quantale, 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. Note that the ^{∗}-adjoint to f is uniquely determined by 1, property (5). The set of all adjointable maps from M to N is denoted by A_{Q}(M, N).

Let us point out that, similarly as for Hilbert C^{∗}-modules, the existence of a ∗-adjoint is not automatic, unlike for ⋁-semilattices with a duality (complete ortholattices).

Lemma 7 Let Q be a (unital) quantale, M, N, P right (left) Hilbert Q-modules, f ∈ A_{Q}(M, N), h ∈ A_{Q}(N, P). Then we have

(1) f preserves arbitrary suprema;

(2) f = f^{∗∗};

(3) f is a right (left) module homomorphism;

(4) h ∘ f ∈ A_{Q}(M, P) and (h ∘ f)^{∗} = f^{∗} ∘ h^{∗};

(5) id_{M} = id_{M}^{∗};

(6) A_{Q}(M, N) is a ⋁-subsemilattice of N^{M};

(7) A_{Q}(M, N) ≅ A_{Q}(N, M);

(8) A_{Q}(M, M) is an involutive quantale.

PROOF. 1. Let S ⊆ M. Then, for all n ∈ N, we have ⟨f(⋁ S), n⟩ = ⟨⋁ S, f^{∗}(n)⟩ = ⋁_{s∈S}⟨s, f^{∗}(n)⟩ = ⋁_{s∈S}⟨f(s), n⟩ = ⟨⋁_{s∈S} f(s), n⟩ i.e. ⋁_{s∈S} f(s) = f(⋁ S) from 1 property (5).

2. Let m ∈ M, n ∈ N arbitrary. Then ⟨f(m), n⟩ = ⟨m, f^{∗}(n)⟩ = ⟨f^{∗}(n), m⟩^{∗} = ⟨n, f^{∗∗}(m)⟩^{∗} = ⟨f^{∗∗}(m), n⟩ = i.e. f^{∗∗}(m) = f(m) for all m ∈ M.

3. Let a ∈ Q. Then, for all n ∈ N, ⟨n, f(m ⋄ a)⟩ = ⟨n, f^{∗∗}(m ⋄ a)⟩ = ⟨f^{∗}(n), m ⋄ a⟩ = ⟨f^{∗}(n), m⟩·a = ⟨n, f^{∗∗}(m)⟩·a = ⟨n, f^{∗∗}(m) ⋄ a⟩ = ⟨n, f(m) ⋄ a⟩ i.e. f(m) ⋄ a = f(m ⋄ a).

4. Let m ∈ M, p ∈ P. Then we have ⟨(h ∘ f)(m), p⟩ = ⟨h(f(m)), p⟩ = ⟨f(m), h^{∗}(p)⟩ = ⟨m, f^{∗}(h^{∗}(p))⟩ = ⟨m, (f^{∗} ∘ h^{∗})(p)⟩.

5. Let m, n ∈ M. We have ⟨id_{M}(m), n⟩ = ⟨m, n⟩ = ⟨m, id_{M}(n)⟩ i.e. id_{M} = id_{M}^{∗};

6. Let T ⊆ A_{Q}(M, N). Then ⟨⋁ T (m), n⟩ = ⟨⋁_{g∈T} g(m), n⟩ = ⋁_{g∈T} ⟨g(m), n⟩ = ⋁_{g∈T} ⟨m, g^{∗}(n)⟩ = ⟨m, ⋁_{g∈T} g^{∗}(n)⟩ for all m ∈ M, n ∈ N i.e. the map ⋁_{g∈T} g^{∗} is a ^{∗}-adjoint to ⋁_{g∈T} g i.e. ⋁_{g∈T} g ∈ A_{Q}(M, N).

7. Note that we have bijective mappings ^{∗_{M,N}} : A_{Q}(M, N) → A_{Q}(N, M), ∗_{N,M} : A_{Q}(N, M) → A_{Q}(M, N) that preserve arbitrary suprema such that ∗_{M,N} ∘^{∗_{N,M}} = id_{A_{Q}(N,M)} and ^{∗_{N,M}} ∘^{∗_{M,N}} = id_{A_{Q}(M,N)}.

8. Evidently, A_{Q}(M, M) is a subquantale of the quantale M^{M}. By 2, 4 and 6 we have that it is an involutive quantale. □

Lemma 8 Let M be a ⋁-semilattice with a duality, f : M → M a map. Then the following conditions are equivalent:

(1) f is a ⋁-semilattice morphism.

(2) f is adjointable on the left (right) Hilbert 2-module M,

PROOF. Evidently, (2)⟹(1). Let us show that (1)⟹(2). Assume that f is a ⋁-semilattice morphism i.e. it is an element of the simple involutive quantale Q(M) (see [16]). It is enough to check that the involution in Q(M) is the ^{∗}- adjoint to f. Note that the involution is defined by ′ ∘ f^{⊢}∘′; here f^{⊢} is the right adjoint to f. We have, for all m, n ∈ M, 0 = ⟨f(m), n⟩ iff f(m) ≤ n′ iff m ≤ f^{⊢}(n′) iff m′ ≥ f^{⊢}(n′)′ iff f^{∗}(n) ≤ m′ iff ⟨f^{∗}(n), m⟩ = 0 iff ⟨m, f^{∗}(n)⟩ = 0 i.e. ⟨f(m), n⟩ = ⟨m, f^{∗}(n)⟩ i.e. f ∈ A_{2}(M, M). □

Proposition 9 Let Q be an involutive quantale. Then the category Hilb_{Q} (SHilb_{Q}) of right (strict) Hilbert Q-modules and adjointable maps between them is an involutive category.

PROOF. It follows immediately from the lemma 7. □

Proposition 10 Let Q be an involutive quantale. Then

(1) The null object of Hilb_{Q} (SHilb_{Q}) is the one-element lattice O = {0} with the unique inner product i.e. ⟨0, 0⟩ = 0_{Q}.

(2) Let M_{k}, k ∈ J be right (strict) Hilbert Q-modules. Then the cartesian product ∏_{k∈J} M_{k} is a biproduct in the category Hilb_{Q}(SHilb_{Q}).

PROOF. 1. It is evident that O is initial in the category Hilb_{Q}(SHilb_{Q}). Namely, for any right (strict) Hilbert Q-module M we have a unique ⋁- preserving map 0_{M} : O → M such that ⟨0_{M}(0), m⟩ = 0_{Q} = ⟨0, 0_{M}^{∗}(m)⟩, where 0_{M}^{∗}(m) = 0 for all m ∈ M. Then O is terminal i.e. it is a null object.

2. First, let us show that the cartesian product ∏_{k∈J} M_{k} is a right (strict) Hilbert Q-module. Evidently, it is a right Q-module. Let us define the obvious inner product on ∏_{k∈J} M_{k} as follows:

⟨(m_{k})_{k∈J}, (n_{k})_{k∈J}⟩ = ⋁_{k∈J}⟨m_{k}, n_{k}⟩_{M_{k}}

for all (m_{k})_{k∈J}, (n_{k})_{k∈J} ∈ ∏_{k∈J} M_{k}. Note that the properties (1)–(4) from the definition 1 are evidently satisfied. Let us check the property (5).

Now, let us put p_{j}((m_{k})_{k∈J}) = m_{j} for all (m_{k})_{k∈J} ∈ ∏_{k∈J} M_{k}, j ∈ J and

p_{k}(i_{j}(m_{j})) = \{ \begin{matrix}m_{j} & for & j = k \\ 0_{M_{k}} & for & j ≠ k. \end{matrix}

for all m_{j} ∈ M_{j} and all j, k ∈ J. Then p_{j} ∘ i_{j} = id_{M_{j}} and p_{k} ∘ i_{j} = 0_{M_{j},M_{k}} for j ≠ k.

Assume that ⟨−, (m_{k})_{k∈J}⟩ = ⟨−, (n_{k})_{k∈J}⟩. Then, for all j ∈ J and all q ∈ M_{j}, we have ⟨q, m_{j}⟩ = ⟨i_{j}(q), (m_{k})_{k∈J}⟩ = ⟨i_{j}(q), (n_{k})_{k∈J}⟩ = ⟨q, n_{j}⟩ i.e. n_{j} = m_{j} for all j ∈ J i.e. (m_{k})_{k∈J} = (n_{k})_{k∈J}.

Moreover, for all (m_{k})_{k∈J} ∈ ∏_{k∈J} M_{k} and n_{j} ∈ M_{j}, we have ⟨p_{j}((m_{k})_{k∈J}), n_{j}⟩ = ⟨m_{j}, n_{j}⟩ = ⟨m_{j}, n_{j}⟩∨⋁_{k∈J−{j}}⟨m_{k}, 0_{M_{k}}⟩ = ⟨(m_{k})_{k∈J}, i_{j}(n_{j})⟩. Evidently, ∏_{k∈J} M_{k} is strict whenever M_{k}, k ∈ J are strict. □

Corollary 11 Let Q be a commutative involutive quantale, M, N left (right) Hilbert Q-modules. Then A_{Q}(M, N) is a left (right) Hilbert Q-submodule of N^{M}.

PROOF. It is enough to check that, for all a ∈ Q and all f ∈ A_{Q}(M, N), a • g ∈ A_{Q}(M, N). We have, for all m ∈ M, n ∈ N, ⟨a • f(m), n⟩ = a · ⟨f(m), n⟩ = a · ⟨m, f^{∗}(n)⟩ = (⟨m, f^{∗}(n)⟩^{∗} · a^{∗})^{∗} = (a^{∗} · ⟨m, f^{∗}(n)⟩^{∗})^{∗} = (a^{∗} · ⟨f^{∗}(n), m⟩)^{∗} = ⟨a^{∗} • f^{∗}(n), m⟩^{∗} = ⟨m, a^{∗} • f^{∗}(n)⟩. □

Definition 12 A quantaloid ([18]) is a category K enriched over the category CSem of complete ⋁-semilattices and ⋁-preserving maps. An involutive quantaloid is an involutive category which is a quantaloid and satisfies

( ⋁_{j∈J} f_{j})^{∗} = ⋁_{j∈J} f_{j}^{∗}.

Theorem 13 Let Q be an involutive quantale. The category Hilb_{Q} (SHilb_{Q}) of right (strict) Hilbert Q-modules and adjointable maps between them is an idempotent involutive category with biproducts. Moreover, it is an involutive quantaloid.

PROOF. The existence of biproducts follows from propositions 9 and 10. Let us show that Hilb_{Q} is idempotent. Note that, for a right Hilbert Q-module M, we have △_{M}^{J}(m) = (m)_{j∈J} and ∇_{M}^{J}(m_{k})_{k∈J} = ⋁_{k∈J} m_{k} for all m ∈ M and all (m_{k})_{k∈J} ∈ ∏_{k∈J} M_{k}. Then, for all m ∈ M, (∇_{M}^{J} ∘ △_{M}^{J})(m) = ∇_{M}^{J}(△_{M}^{J}(m)) = ∇_{M}^{J}((m)_{k∈J}) = ⋁_{k∈J} m = m = id_{M}(m). The rest follows from the lemma 7. □

Lemma 14 Let Q be a unital involutive quantale with the standard inner product. Then A_{Q}(Q, Q) = {j_{Q}(a) : a ∈ Q} ≅ Q; here j_{Q}(a)(b) = a · b.

PROOF. Let g ∈ A_{Q}(Q, Q), b, c ∈ Q. Then we have ⟨c, g(b)⟩ = ⟨g^{∗}(c), b⟩. Let us put c = e. Then g(b) = e^{∗} · g(b) = g^{∗}(e)^{∗} · b i.e. g(b) = j_{Q}(a)(b), where a = g^{∗}(e)^{∗}. Conversely, we have to show that, for all a ∈ Q, j_{Q}(a) ∈ A_{Q}(Q, Q). Namely, we have ⟨j_{Q}(a)(b), c⟩ = (a · b)^{∗} · c = b^{∗} · (a^{∗} · c) = ⟨b, j_{Q}(a^{∗})(c)⟩ for all b, c ∈ Q. □

We shall sometimes write a^{⌣} instead of j_{Q}(a). Motivated by the preceding lemma we have

Lemma 15 Let M be a right Hilbert Q-module, Q an involutive quantale with the standard inner product, m ∈ M. Then the map m^{∼} : Q → M defined by a ↦ m ⋄ a has a ^{∗}-adjoint m^{⋆} : M → Q defined by n ↦ ⟨m, n⟩. Moreover, if Q is unital then A_{Q}(M, Q) = {m^{⋆} : m ∈ M} ≅ M and A_{Q}(Q, M) = {m^{∼} : m ∈ M} ≅ M.

PROOF. Let a ∈ Q, n ∈ M. Then ⟨m^{∼}(a), n⟩ = ⟨m ⋄ a, n⟩^{∗∗} = ⟨n, m ⋄ a⟩^{∗} = a^{∗} · ⟨n, m⟩^{∗} = a^{∗} · ⟨m, n⟩ = a^{∗} · m^{⋆}(n) = ⟨a, m^{⋆}(n)⟩. Let g ∈ A_{Q}(Q, M). Then, for all a ∈ Q, n ∈ M, we have ⟨g(a), n⟩ = ⟨a, g^{∗}(n)⟩. Let us put a = e. Then g(e)^{⋆}(n) = ⟨g(e), n⟩ = ⟨e, g^{∗}(n)⟩ = e^{∗} · g^{∗}(n) = g^{∗}(n). So we have that g = g(e)^{∼}. The rest is evident. □

We now introduce a class of operators analogous to the finite-rank operators on a Hilbert space.

Definition 16 Let Q be an involutive quantale, M, N right (left) Hilbert Qmodules. 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 = (m^{∼} ∘ n^{⋆})(p)

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 ⟨Θ_{m,n}(p), q⟩ = ⟨n, p⟩^{∗}⟨m, q⟩.

Lemma 17 Let Q be an involutive quantale, M, N, P right (left) Hilbert Qmodules, m, v ∈ M, n, u ∈ N, g ∈ A_{Q}(M, P), h ∈ A_{Q}(P, N). Then we have

(1) Θ_{m,n}^{∗} = Θ_{n,m};

(2) Θ_{m,n} ∘ Θ_{u,v} = Θ_{m ⋄ ⟨n,u⟩,v} = Θ_{m,v ⋄ ⟨u,n⟩} = Θ_{⟨u,n⟩ • m,v} = Θ_{m,⟨n,u⟩ • v};

(3) g ∘ Θ_{m,n} = Θ_{g(m),n};

(4) Θ_{m,n} ∘ h = Θ_{m,h^{∗}(n)}.

PROOF. 1. Let x ∈ M, y ∈ N. Then we have ⟨Θ_{m,n}(y), x⟩ = ⟨m^{∼}(n^{⋆}(y)), x⟩ = ⟨n^{⋆}(y), m^{⋆}(x)⟩ = ⟨y, n^{∼}(m^{⋆}(x))⟩ = ⟨y, Θ_{n,m}(x)⟩.

2. Let x ∈ M. Then we have (Θ_{m,n} ∘ Θ_{u,v})(x) = Θ_{m,n}(Θ_{u,v}(x)) = (m^{∼} ∘ n^{⋆} ∘ u^{∼} ∘ v^{⋆})(x) = ((m^{∼} ∘ n^{⋆} ∘ u^{∼}) ∘ v^{⋆})(x) = (m^{∼} ∘ (n^{⋆} ∘ u^{∼} ∘ v^{⋆}))(x).

First, let us check that (m^{∼} ∘ n^{⋆} ∘ u^{∼})(a) = Θ_{m,n}(u) ⋄ a. We have m^{∼} ∘ n^{⋆} ∘ u^{∼}(a) = (m^{∼} ∘ n^{⋆})(u^{∼}(a)) = (m^{∼} ∘ n^{⋆})(u ⋄ a) = Θ_{m,n}(u) ⋄ a since Θ_{m,n} is a module homomorphism. This gives us that Θ_{m,n} ∘ Θ_{u,v} = Θ_{m ⋄ ⟨n,u⟩,v}.

Applying the preceding observation on the ^{∗}-adjoint to n^{⋆} ∘ u^{∼} ∘ v^{⋆} we have that v^{∼} ∘ u^{⋆} ∘ n^{∼} = Θ_{v,u}(n)^{⋆} i.e. (n^{⋆} ∘ u^{∼} ∘ v^{⋆})(x) = Θ_{v,u}(n)^{∼}(x). We obtain that Θ_{m,n} ∘ Θ_{u,v} = Θ_{m,v ⋄ ⟨u,n⟩}.

3. First, let us check that g ∘ m^{∼} = g(m)^{∼}. We have, for all a ∈ Q, (g ∘ m^{∼})(a) = g(m^{∼}(a)) = g(m ⋄ a) = g(m) ⋄ a = g(m)^{∼}(a) since g is a module homomorphism.

Now, we have g ∘ Θ_{m,n} = g ∘ (m^{∼} ∘ n^{⋆}) = (g ∘ m^{∼}) ∘ n^{⋆} = g(m)^{∼} ∘ n^{⋆} = Θ_{g(m),n}.

4. Similarly, we have n^{⋆}∘h = (n^{⋆}∘h)^{∗∗} = (h^{∗}∘n^{∼})^{∗} = (h^{∗}(n)^{∼})^{∗} = h^{∗}(n)^{⋆}. Then we compute Θ_{m,n}∘h = (m^{∼}∘n^{⋆})∘h = m^{∼}∘(n^{⋆}∘h) = m^{∼}∘h^{∗}(n)^{⋆} = Θ_{m,h^{∗}(n)^{⋆}}. □

Corollary 18 Let Q be an involutive quantale, M a right Hilbert Q-module. Then K_{Q}(M, M) is an involutive subquantale of A_{Q}(M, M).

Corollary 19 Let Q be an involutive quantale with a standard inner product, M, N, P right Hilbert Q-modules, a, b ∈ Q, m, n ∈ N, g ∈ A_{Q}(M, P), h ∈ A_{Q}(P, N). Then we have

(1) a^{⌣} ∘ b^{⌣} = (a · b)^{⌣};

(2) m^{⋆} ∘ n^{∼} = ⟨m, n⟩^{⌣};

(3) a^{⌣} ∘ m^{⋆} = (a • m)^{⋆}; (4) m^{∼} ∘ a^{⌣} = (m ⋄ a)^{∼};

(5) m^{⋆} ∘ h = h^{∗}(m)^{⋆};

(6) g ∘ m^{∼} = g(m)^{∼}.

PROOF. It follows from the proof of the lemma 17. □

Proposition 20 Let Q be a commutative involutive quantale, N, M (strict) Hilbert Q-modules. Then K_{Q}(N, M) is a (strict) Hilbert Q-module.

PROOF. First, let us define the action • : Q × K_{Q}(N, M) → K_{Q}(N, M) by a • Θ_{m,n} = Θ_{a • m,n} for all a ∈ Q, m ∈ M, n ∈ N. Evidently, K_{Q}(N, M) is a Q-module.

Now, we denote by T_{Q}(N, M) = {Θ_{m,n} : n ∈ N, m ∈ M}. Then we shall define a map σ : T_{Q}(N, M) × T_{Q}(N, M) → Q by σ(Θ_{t,s}, Θ_{v,u}) = ⟨s, u⟩ · ⟨t, v⟩, for all s, u ∈ N, t, v ∈ M. This gives us a map \overline{σ} : K_{Q}(N, M) × K_{Q}(N, M) → Q defined by \overline{σ}(d, f) = ⋁_{i,j} σ(Θ_{t_{j},s_{j}}, Θ_{y_{i},x_{i}}) for all f = ⋁_{i} Θ_{y_{i},x_{i}}, d = ⋁_{j} Θ_{t_{j},s_{j}}. We have, for all f, d ∈ K_{Q}(S, T), f = ⋁_{i} Θ_{y_{i},x_{i}} = ⋁_{k} Θ_{v_{k},u_{k}}, d = ⋁_{j} Θ_{t_{j},s_{j}} = ⋁_{l} Θ_{z_{l},w_{l}}, x_{i}, s_{j}, u_{k}, w_{l} ∈ N, y_{i}, t_{j}, v_{k}, z_{l} ∈ M,

\overline{σ}(⋁_{j} Θ_{t_{j},s_{j}}, ⋁_{i} Θ_{y_{i},x_{i}}) = ⋁_{i,j} σ(Θ_{t_{j},s_{j}}, Θ_{y_{i},x_{i}}) = ⋁_{i,j} ⟨s_{j}, x_{i}⟩ · ⟨t_{j}, y_{i}⟩
= ⋁_{i} ⟨⋁_{j} ⟨s_{j}, x_{i}⟩ • t_{j}, y_{i}⟩ = ⋁_{i} ⟨⋁_{l} ⟨w_{l}, x_{i}⟩ • z_{l}, y_{i}⟩
= ⋁_{i,l} ⟨w_{l}, x_{i}⟩ · ⟨z_{l}, y_{i}⟩ = ⋁_{l} ⟨w_{l}, ⋁_{i} x_{i} ⋄ ⟨z_{l}, y_{i}⟩⟩
= ⋁_{l} ⟨w_{l}, ⋁_{k} u_{k} ⋄ ⟨z_{l}, v_{k}⟩⟩ = ⋁_{k,l} ⟨w_{l}, u_{k}⟩ · ⟨z_{l}, v_{k}⟩
= ⋁_{k,l} σ(Θ_{v_{k},u_{k}}, Θ_{z_{l},w_{l}}) = \overline{σ}(⋁_{l} Θ_{z_{l},w_{l}}, ⋁_{k} Θ_{v_{k},u_{k}})

i.e. our definition of \overline{σ} was correct.

Evidently, for all f, g ∈ K_{Q}(S, T), f = ⋁_{i} Θ_{y_{i},x_{i}}, g = ⋁_{j} Θ_{n_{j},m_{j}}, x_{i}, m_{j} ∈ S, y_{i}, n_{j} ∈ T, \overline{σ}(−, f) = \overline{σ}(−, g) implies f = g. Namely, for all s ∈ S and all t ∈ T, ⟨t, f(s)⟩ = ⟨t, (⋁_{i} Θ_{y_{i},x_{i}}) (s)⟩ = ⟨t, ⋁_{i} Θ_{y_{i},x_{i}}(s)⟩ = ⟨t, ⋁_{i} ⟨s, x_{i}⟩ • y_{i}⟩ = ⋁_{i} ⟨s, x_{i}⟩· ⟨t, y_{i}⟩ = ⋁_{j} ⟨s, m_{j}⟩ · ⟨t, n_{j}⟩ = ⟨t, ⋁_{j} ⟨s, m_{j}⟩ • n_{j}⟩ = ⟨t, (⋁_{j} Θ_{n_{j},m_{j}} ) (s)⟩ = ⟨t, g(s)⟩ i.e. f(s) = g(s) for all s ∈ S i.e. f = g. Moreover, if f = ⋁_{i} Θ_{y_{i},x_{i}}, g = ⋁_{j} Θ_{n_{j},m_{j}}, a ∈ Q then

\overline{σ}(f, g)^{∗} = \overline{σ}(⋁_{i} Θ_{y_{i},x_{i}}, ⋁_{j} Θ_{n_{j},m_{j}})^{∗} = ⋁_{i,j} σ(Θ_{y_{i},x_{i}}, Θ_{n_{j},m_{j}})^{∗}
= ⋁_{i,j}(⟨x_{i}, m_{j}⟩ · ⟨y_{i}, n_{j}⟩)^{∗} = ⋁_{i,j} ⟨y_{i}, n_{j}⟩^{∗} · ⟨x_{i}, m_{j}⟩^{∗}
= ⋁_{i,j} ⟨n_{j}, y_{i}⟩ · ⟨m_{j}, x_{i}⟩ = ⋁_{i,j} ⟨m_{j}, x_{i}⟩ · ⟨n_{j}, y_{i}⟩
= ⋁_{i,j} σ(Θ_{n_{j},m_{j}}, Θ_{y_{i},x_{i}}) = \overline{σ}(⋁_{j} Θ_{n_{j},m_{j}}, ⋁_{i} Θ_{y_{i},x_{i}}) = \overline{σ}(g, f)

and

a · \overline{σ}(f, g) = a · \overline{σ}(⋁_{i} Θ_{y_{i},x_{i}}, ⋁_{j} Θ_{n_{j},m_{j}}) = a · ⋁_{i,j} σ(Θ_{y_{i},x_{i}}, Θ_{n_{j},m_{j}})
= a · ⋁_{i,j} ⟨x_{i}, m_{j}⟩ · ⟨y_{i}, n_{j}⟩ = ⋁_{i,j} ⟨x_{i}, m_{j}⟩ · a · ⟨y_{i}, n_{j}⟩
= ⋁_{i,j} ⟨x_{i}, m_{j}⟩ · ⟨a • y_{i}, n_{j}⟩ = ⋁_{i,j} σ(Θ_{a • y_{i},x_{i}}, Θ_{n_{j},m_{j}})
= \overline{σ}(⋁_{i} Θ_{a • y_{i},x_{i}}, ⋁_{j} Θ_{n_{j},m_{j}}) = \overline{σ}(⋁_{i} a • Θ_{y_{i},x_{i}}, ⋁_{j} Θ_{n_{j},m_{j}})
= \overline{σ}(a • ⋁_{i} Θ_{y_{i},x_{i}}, ⋁_{j} Θ_{n_{j},m_{j}}) = \overline{σ}(a • f, g).

Now, let f_{k} = ⋁_{i_{k}} Θ_{y_{i_{k}} ,x_{i_{k}}}, g = ⋁_{j} Θ_{n_{j},m_{j}}. Then

\begin{aligned}⋁_{k} \overline{σ}(f_{k}, g) & = & ⋁_{k} \overline{σ}(⋁_{i_{k}} Θ_{y_{i_{k}} ,x_{i_{k}}} , ⋁_{j} Θ_{n_{j},m_{j}}) = ⋁_{k,i_{k},j} σ(Θ_{y_{i_{k}} ,x_{i_{k}}} , Θ_{n_{j},m_{j}}) \\ & = & \overline{σ}(⋁_{k} ⋁_{i_{k}} Θ_{y_{i_{k}} ,x_{i_{k}}} , ⋁_{j} Θ_{n_{j},m_{j}}) = \overline{σ}(⋁_{k} f_{k}, g). \end{aligned}

Similarly, we have ⋁_{k} \overline{σ}(g, f_{k}) = \overline{σ}(g, ⋁_{k} f_{k}). Altogether, \overline{σ} defines an innerproduct on K_{Q}(N, M). □

Under a further assumption one can completely characterize A_{Q}(M, M). A Hilbert Q-module is called self-dual when every module morphism φ : M → Q is of the form φ(m) = ⟨n, m⟩ for some n ∈ M i.e. φ = n^{⋆}.

Proposition 21 In a self-dual Hilbert Q-module M, A_{Q}(M, M) coincides with the ⋁-semilattice H_{Q}(M, M) of all module homomorphisms on M.

PROOF. In view of the lemma 7 we have that A_{Q}(M, M) ⊆ H_{Q}(M, M). So we only need to show that a given module homomorphism f ∈ H_{Q}(M, M) is adjointable. Indeed, for fixed n ∈ M define φ_{f,n} : M → Q by φ_{f,n}(m) := ⟨m, f(n)⟩. By self-duality this must equal ⟨g(m), n⟩ for some g(m) ∈ M. So the map g : M → M is by definition f^{∗}. □

Definition 22 Let M, N be Hilbert Q-modules, f : M → N a module homomorphism. We say that f is inner-product preserving if, for all m_{1}, m_{2} ∈ M, ⟨m_{1}, m_{2}⟩ = ⟨f(m_{1}), f(m_{2})⟩.

Note that any inner-product preserving homomorphism is injective. Namely, f(m) = f(n) implies ⟨m, p⟩ = ⟨f(m), f(p)⟩ = ⟨f(n), f(p)⟩ = ⟨n, p⟩ for all p i.e. m = n.

Proposition 23 Let M, N be Hilbert Q-modules, f : M → N a module homomorphism. Then, for the following conditions are equivalent:

(1) f is an adjointable map such that f^{∗} ∘ f = id_{M},

(2) f is an adjointable inner-product preserving map.

PROOF. (1)⟹(2) Let us assume that m, n ∈ M. Then we have ⟨m, n⟩ = ⟨m, (f^{∗} ∘ f)(n)⟩ = ⟨f(m), f(n)⟩.

(2)⟹(1) Let us prove that f^{∗}∘f = id_{M}. Let m, n ∈ M. Then ⟨n, (f^{∗} ∘ f)(m)⟩ = ⟨n, f^{∗}(f(m))⟩ = ⟨f(n), f(m)⟩ = ⟨n, m⟩ = i.e. m = (f^{∗} ∘ f)(m) for all m ∈ M. □

Definition 24 Let M, N be Hilbert Q-modules, f : M → N an adjointable map. Then f is said to be unitary if

f^{∗} ∘ f = id_{M}, f ∘ f^{∗} = id_{N}.  (7)

Lemma 25 Let M, N be Hilbert Q-modules, f : M → N a module homomorphism. Then the following conditions are equivalent:

(1) f is a surjective (bijective) inner-product preserving map,

(2) f is unitary.

PROOF. (1)⟺(2) It follows immediately from 23. □

2 Tensor products and nuclear ideals of Hilbert 2-modules

We study the tensor product S ⊗_{H} T of (strict) Hilbert 2-modules (i.e. objects are ⋁-semilattices with a duality and morphisms are ⋁-preserving maps). We prove the existence of a tensor product in the respective category. Let us recall the following standard definitions.

Definition 26 Let S, T and U are (strict) Hilbert 2-modules. A function f : S × T → U is a bimorphism in the category of (strict) Hilbert 2-modules if the functions g_{s} : T → U defined by g_{s}(t) = f(s, t) and h_{t} : S → U defined by h_{t}(s) = f(s, t) are morphisms of (strict) Hilbert 2-modules for each s ∈ S and t ∈ T.

Definition 27 Let S and T be (strict) Hilbert 2-modules. A (strict) Hilbert 2module S⊗_{H} T is a tensor product of S and T in the category of (strict) Hilbert 2-modules if there exists a canonical bimorphism e_{S,T} : S × T → S ⊗_{H} T such that for any (strict) Hilbert 2-module U and any bimorphism g : S × T → U there is a unique morphism of (strict) Hilbert 2-modules h : S ⊗_{H} T → U satisfying g = h ∘ e_{S,T}.

Note that then f(S × T) generates S ⊗_{H} T.

Lemma 28 Let S be a pre-Hilbert 2-module. If we define an operator ′ : S → S by s′ = ⋁{t ∈ S : ⟨s, t⟩ = 0} for all s ∈ S we have, for all m ∈ S, j_{H}(m) = m′′ and m′ = j_{H}(m′).

PROOF. Note first that ′ is a Galois connection on S. Namely, for all m, n ∈ S, m ≤ n′ iff ⟨m, n⟩ = 0 iff ⟨n, m⟩ = 0 iff n ≤ m′. Evidently, m ≤ n implies n′ ≤ m′, m ≤ m′′. Moreover (m, m′′) ∈ R_{H} i.e. m′′ ≤ j_{H}(m). Conversely, ⟨m, m′⟩ = 0 i.e. ⟨j_{H}(m), m′⟩ = 0 i.e. m′′ ≥ j_{H}(m). We then have m′ = m′′′ = j_{H}(m′). □

Lemma 29 Let S and T be Hilbert 2-modules. Then the tensor product S⊗_{s}T in the category of ⋁-semilattices is a pre-Hilbert 2-module.

PROOF. Note that, since S ⊗_{s} T is a ⋁-semilattice, it is a left 2-module. Its elements are called C-ideals and ⟨X⟩_{s} denotes the least C-ideal containing X.

First, let us define a map [[[ −, −]]] : P(S × T) × P(S × T) → 2 as follows:

[[[ X, Y]]] := ⋁{⟨s_{1}, s_{2}⟩_{S} · ⟨t_{1}, t_{2}⟩ : (s_{1}, t_{1}) ∈ X, (s_{2}, t_{2}) ∈ Y }

for all X, Y ∈ P(S × T). We have, for all X, Y ∈ P(S × T), [[[ X,Y]]] =[[[ ⟨X⟩_{s},Y]]] . Namely, let us put Z = {(s, t) ∈ S × T : [[[ {(s, t)}, Y]]] ≤ [[[ X, Y]]] }. Evidently, X ⊆ Z, {0} × T ⊆ Z, S × {0} ⊆ Z. Let (s_{λ}, t) ∈ Z. Then, for all (u, v) ∈ Y, we have ⟨s_{λ}, u⟩_{S} · ⟨t, v⟩_{T} ≤ [[[ X, Y]]] i.e. ⟨⋁_{λ∈Λ} s_{λ}, u⟩_{S} · ⟨t, v⟩_{T} = ⋁_{λ∈Λ}⟨s_{λ}, u⟩_{S} · ⟨t, v⟩_{T} ≤ [[[ X, Y]]] i.e. (⋁_{λ∈Λ} s_{λ}, t) ∈ Z. Similarly, if (s, t_{λ}) ∈ Z, λ ∈ Λ then (s, ⋁_{λ∈Λ} t_{λ}) ∈ Z.

So we have that Z ∈ S ⊗_{s} T, X ⊆ Z and [[[ X, Y]]] ≤ [[[ ⟨X⟩_{s}, Y]]] ≤ [[[ Z, Y]]] ≤ [[[ X, Y]]] . Note that, since 2 is a commutative quantale, we have [[[ X, Y]]] = [[[ Y, X]]] i.e. [[[ X, Y]]] = [[[ X, ⟨Y ⟩_{s}]]] .

We shall define a pre-inner product ⟨−, −⟩ : (S ⊗_{s} T) × (S ⊗_{s} T) → 2 as follows:

⟨I, J⟩ := [[[ I, J]]]

for all I, J ∈ S ⊗_{s} T. Let us check the conditions (1)-(4). Let a ∈ 2, I, J, J_{λ} ∈ S ⊗_{s} T, λ ∈ Λ. We have a · ⟨I, J⟩ = a · ⋁{⟨s_{1}, s_{2}⟩_{S} · ⟨t_{1}, t_{2}⟩_{T} : (s_{1}, t_{1}) ∈ I, (s_{2}, t_{2}) ∈ J} = ⋁{a · ⟨s_{1}, s_{2}⟩_{T} · ⟨t_{1}, t_{2}⟩_{T} : (s_{1}, t_{1}) ∈ I, (s_{2}, t_{2}) ∈ J} = ⋁{⟨a •_{S}s_{1}, s_{2}⟩_{S} · ⟨t_{1}, t_{2}⟩_{T} : (s_{1}, t_{1}) ∈ I, (s_{2}, t_{2}) ∈ J} = ⋁{⟨p, s_{2}⟩_{T} · ⟨t_{1}, t_{2}⟩_{T} : (p, t_{1}) ∈ a • I, (s_{2}, t_{2}) ∈ J} = ⟨a • I, J⟩ and ⋁_{λ∈Λ} ⟨J_{λ}, J⟩ = ⋁_{λ∈Λ} ⋁{⟨s_{1}, s_{2}⟩_{S} · ⟨t_{1}, t_{2}⟩_{T} : (s_{1}, t_{1}) ∈ J_{λ}, (s_{2}, t_{2}) ∈ J} = ⋁{⟨s_{1}, s_{2}⟩_{T} · ⟨t_{1}, t_{2}⟩_{T} : (s_{1}, t_{1}) ∈ J_{λ}, λ ∈ Λ, (s_{2}, t_{2}) ∈ J} = ⋁{⟨s_{1}, s_{2}⟩_{T} · ⟨t_{1}, t_{2}⟩_{T} : (s_{1}, t_{1}) ∈ ⋁_{λ∈Λ} J_{λ}, (s_{2}, t_{2}) ∈ J} = ⟨⋁_{λ∈Λ} J_{λ}, J⟩. Similarly, we have ⋁_{λ∈Λ} ⟨J, J_{λ}⟩ = ⟨J, ⋁_{λ∈Λ} J_{λ}⟩. Using the fact that ⟨X, Y ⟩ = ⟨Y, X⟩ and the involution on 2 is the identity map, we have ⟨X, Y ⟩^{∗} = ⟨Y, X⟩. □

Theorem 30 Let S and T be (strict) Hilbert 2-modules. Then the factor (strict) Hilbert 2-module S ⊗_{H} T = (S ⊗_{s} T)_{R_{H}} is a tensor product in the category of (strict) Hilbert 2-modules and the tensor product is unique up to isomorphism of Hilbert 2-modules.

PROOF. First, let us show that, for all s ∈ S, (s⊗_{s}1)′ = s′⊗_{s}1 i.e. ⟨(s, 1)⟩_{s}′ = ⟨(s′, 1)⟩_{s}. Let us put Z = {(u, v) ∈ S × T : [[[ (s, 1), (u, v)]]] = 0}. Evidently, Z is an element of S ⊗_{s} T. Then (u, v) ∈ Z iff (⟨s, u⟩_{S} = 0 or ⟨1, v⟩_{T} = 0) iff (u ≤ s′ or v = 0) iff ((u, v) ≤ (s′, 1) or (u, v) ≤ (1, 0)) iff (u, v) ∈ ⟨(s′, 1)⟩_{s}. Then ⟨(s′, 1)⟩_{s} = Z = {(u, v) ∈ S × T : [[[ (s, 1), (u, v)]]] = 0} = {(u, v) ∈ S × T : [[[ ⟨(s, 1)⟩_{s}, (u, v)]]] = 0} = ⟨(s, 1)⟩_{s}′.

Now, we shall prove that s⊗_{s}1 is a j_{H}-fixed element i.e. ⟨(s, 1)⟩_{s} = j_{H}(⟨(s, 1)⟩_{s}) for all s ∈ S. Evidently, ⟨(s, 1)⟩_{s} ⊆ j_{H}(⟨(s, 1)⟩_{s}). Let (u, v) ∈ S × T. Then [[[ (u, v), ⟨(s, 1)⟩_{s}]]] = 0 iff [[[ (u, v), j_{H}(⟨(s, 1)⟩_{s})]]] = 0 i.e. [[[ (s′, 1), j_{H}(⟨(s, 1)⟩_{s})]]] = 0 i.e. j_{H}(⟨(s, 1)⟩_{s}) ⊆ ⟨(s′, 1)⟩_{s}′ = ⟨(s, 1)⟩_{s}.

Similarly, we have, for all t ∈ T, ⟨(1, t)⟩_{s}′ = ⟨(1, t′)⟩_{s} and ⟨(1, t)⟩_{s} = j_{H}(⟨(1, t)⟩_{s}).

Now, let s ∈ S, t ∈ T arbitrary. Then (s ⊗_{s} 1 ∨_{H} 1 ⊗_{s} t)′ = s′ ⊗_{s} t′. Namely, we have (s ⊗_{s} 1 ∨_{H} 1 ⊗_{s} t)′ = (⟨{(s, 1)}⟩_{s} ∨_{H} ⟨{(1, t)}⟩_{s})′ = ⟨{(s, 1)}⟩_{s}′ ∧_{H} ⟨{(1, t)}⟩_{s}′ = ⟨{(s′, 1)}⟩_{s} ∩ ⟨{(1, t′)}⟩_{s} = ⟨{(s′, t′)}⟩_{s} = s′ ⊗_{s} t′.

We shall now define the canonical map f : S × T → S ⊗_{H} T by f(s, t) = s ⊗_{s} t ∈ S ⊗_{H} T. Now, let s ∈ S. Then, for all T′ ⊆ T, we have f(s, ⋁ T′) = s ⊗_{s} ⋁ T′ = ⋁_{S⊗_{s}T} {s ⊗_{s} t : t ∈ T′} = ⋁_{S⊗_{H}T} {s ⊗_{s} t : t ∈ T′} = ⋁_{S⊗_{H}T} {f(s, t) : t ∈ T′} i.e. h_{s} : T → S ⊗_{H} T is ⋁-preserving. Similarly, for all t ∈ T, the map g_{t} : S → S ⊗_{H} T is ⋁-preserving. So we have proved that f is the canonical bimorphism.

Let U be any Hilbert 2-module and let g : S × T → U be any bimorphism. There is a unique morphism of ⋁-semilattices φ : S ⊗_{s} T → U such that g(s, t) = φ(s⊗_{s}t). We define a map φ^{∗} : U → S ⊗_{s} T by φ^{∗}(n) = (φ^{⊣}(n′))′. We then have, for all m ∈ S ⊗_{s} T and all n ∈ U, ⟨φ(m), n⟩_{U} = 0 iff φ(m) ≤ n′ iff m ≤ φ^{⊣}(n′) iff (φ^{⊣}(n′))′ ≤ m′ iff φ^{∗}(n) ≤ m′ iff [[[ m, φ^{∗}(n)]]] = 0. Since φ^{∗}(n) ∈ S ⊗_{H} T by 28 we have a map ψ^{∗} : U → S ⊗_{H} T defined by ψ^{∗}(n) = φ^{∗}(n) for all n ∈ U. Let us assume ψ(m) = φ(m) for all m ∈ S ⊗_{H} T. Then we have ⟨ψ(m), n⟩_{U} = 0 iff ⟨φ(m), n⟩_{U} = 0 iff [[[ m, φ^{∗}(n)]]] = 0 iff ⟨m, ψ^{∗}(n)⟩ = 0 i.e. ψ and ψ^{∗} are adjointable maps in the category of Hilbert 2-modules. Moreover, g(s, t) = φ(s ⊗_{s} t) = ψ(s ⊗_{s} t) = ψ(f(s, t)) for all (s, t) ∈ S × T and ψ is uniquely determined. Finally, since the set {s ⊗_{s} t : (s, t) ∈ S × T } generates the ⋁ −semilattice S ⊗_{s} T it generates also the Hilbert 2-module S ⊗_{H} T. This shows that Hilbert 2-module S ⊗_{H} T and the canonical bimorphism f satisfy the conditions of the definition of a tensor product. The uniqueness of a tensor product is clear from its definition as a solution of a universal problem. Evidently, if S and T are strict Hilbert 2-modules we have that S ⊗_{H} T is also strict. □

Definition 31 Let S, T be Hilbert 2-modules, X ⊆ S × T. We define X^{+} = {(u, v) ∈ S × T : (∀(s, t) ∈ X)(s ≤ u) or (t ≤ v)}, X^{−} = {(u, v) ∈ S × T : (∀(s, t) ∈ X)(s ≥ u) or (t ≥ v)}.

Recall that is well known ([17], [10]) that the maps X ↦ X^{+} and X ↦ X^{−} form a Galois connection from the power set of S × T to itself. Moreover, we have an explicit description of S ⊗_{H} T using only the order on S and T.

Lemma 32 Let S, T be Hilbert 2-modules, X ⊆ S × T. Then X^{+−} = X′′ = j_{H}(X).

PROOF. Note that, for all Y ⊆ S × T we have (c, d) ∈ Y′ iff for all (e, f) ∈ Y we have [[[ {(c, d)},{(e, f)}]]] =0 iff e ≤ c′ or f ≤ d′ for all (e, f) ∈ Y iff (c′, d′) ∈ Y^{+}. Similarly, (a, b) ∈ Y^{−} iff (a, b) ∈ {(e′, f′) : (e, f) ∈ Y }′ iff (a, b) ∈ ⋀_{(e,f)∈Y} (e ⊗_{H} 1 ∨ 1 ⊗_{H} f).

This gives us that, for all (s, t) ∈ S × T, (s, t) ∈ X^{+−} iff (s, t) ∈ {(e′, f′) : (e, f) ∈ X^{+}}′ iff (s, t) ∈ {(c, d) : (c′, d′) ∈ X^{+}}′ iff (s, t) ∈ {(c, d) : (c, d) ∈ X′}′ iff (s, t) ∈ X′′. □

Lemma 33 Let S be a (strict) Hilbert 2-module. Then we have a morphism of Hilbert 2-modules T r : S ⊗_{H} S → 2 such that ⟨s, t⟩ = T r(s ⊗_{H} t) for all s, t ∈ S i.e. ⋁_{i} ⟨s_{i}, t_{i}⟩ = T r(⋁_{i} s_{i} ⊗_{H} t_{i}).

PROOF. It follows immediately from the fact that ⟨−, −⟩ : S × S → 2 is a bimorphism of Hilbert 2-modules. □

Note that, similarly as in [5], we can extend the operation ⊗_{H} to morphisms. We may also easily check that the tensor product in the category of (strict) Hilbert 2-modules is both associative and commutative and we have the unit I = 2. Recall that the notation c_{S,T}: S ⊗_{H} T → T ⊗_{H} S will be used for the symmetry operator.

It is straightforward to see that:

Theorem 34 The category of (strict) Hilbert 2-modules is a symmetric monoidal category.

PROOF. The proof is simply a checking the necessary diagrams. □

Theorem 35 The category of (strict) Hilbert 2-modules is a ∗-tensored category.

PROOF. By the theorem 34 the category is symmetric monoidal and by the theorem 9 it is an involutive category. Assume that f : A → B and g : C → D are ⋁-preserving maps. Let us prove that (f ⊗_{H} g)^{∗} = f^{∗} ⊗_{H} g^{∗}. We have, for all a ∈ A, b ∈ B, c ∈ C, d ∈ D,

⟨a ⊗_{H} b, (f ⊗_{H} g)^{∗}(c ⊗_{H} d)⟩ = ⟨(f ⊗_{H} g)(a ⊗_{H} b), c ⊗_{H} d⟩
= ⟨f(a) ⊗_{H} g(b), c ⊗_{H} d⟩
= ⟨f(a), c⟩ · ⟨g(b), d⟩
= ⟨a, f^{∗}(c)⟩ · ⟨b, g^{∗}(d)⟩
= ⟨a ⊗_{H} b, f^{∗}(c) ⊗_{H} g^{∗}(d)⟩
= ⟨a ⊗_{H} b, (f^{∗} ⊗_{H} g^{∗})(c ⊗_{H} d)⟩ .

This gives us that (f ⊗_{H} g)^{∗}(c ⊗_{H} d) = (f^{∗} ⊗_{H} g^{∗})(c ⊗_{H} d) i.e. (f ⊗_{H} g)^{∗} = f^{∗} ⊗_{H} g^{∗}. As a conjugation, we will take the identity. Evidently, \overline{\overline{A}} = A ≅ A, \overline{A ⊗_{H} B} = A ⊗_{H} B ≅ A ⊗_{H} B = \overline{A} ⊗_{H} \overline{B} and \overline{2} = 2 ≅ 2. Moreover, since for any ⋁ −preserving map f : 2 → 2 we have either f = 0 or f = id_{2} i.e. f^{∗} = 0 or f^{∗} = id_{2} i.e. f = f^{∗}, then

[figure: 2; f^{∗}; 2; \overline{2}; ≅; \overline{f}; \overline{2}; ≅; iff; 2; f; 2; 2; ≅; f; 2.; ≅]

Hence the category of (strict) Hilbert 2-modules is ∗-tensored. □

Lemma 36 For all objects S, T ∈ Hilb_{2} we put N (S, T) = K_{2}(S, T) ⊆ A_{2}(S, T). We will refer to the union of these subsets as N. The class N is closed under composition with arbitrary morphisms of (strict) Hilbert 2modules, closed under ⊗_{H}, closed under ( )^{∗}, and the conjugate functor.

PROOF. It is enough to check that all the above properties are preserved on generators i.e. mappings of the form Θ_{t,s}. By the lemma 17 we have that, for all (strict) Hilbert 2-modules S, T, U, V and for all g ∈ A_{2}(T, V), h ∈ A_{2}(U, S) s ∈ S, t ∈ T, we have that g ∘ Θ_{t,s} = Θ_{g(t),s}, Θ_{t,s} ∘ h = Θ_{t,h^{∗}(s)} and Θ_{t,s}^{∗} = Θ_{s,t} i.e. N is closed under composition with arbitrary morphisms of (strict) Hilbert 2-modules and under under ( )^{∗}.

Now, let S, T, U, V be any (strict) Hilbert 2-modules, s ∈ S, t ∈ T, u ∈ U and v ∈ V. Then, for all x ∈ S, y ∈ U, we have (Θ_{t,s} ⊗_{H} Θ_{v,u})(x ⊗_{H} y) = Θ_{t,s}(x) ⊗_{H} Θ_{v,u}(y) = ⟨x, s⟩ • t ⊗_{H} ⟨y, u⟩ • v = (⟨x, s⟩ · ⟨y, u⟩) • (t ⊗_{H} v) = ⟨x ⊗_{H} y, s ⊗_{H} u⟩ • (t ⊗_{H} v) = Θ_{t⊗_{H}v,s⊗_{H}u}(x ⊗_{H} y) i.e. N is closed under ⊗_{H}.

Since the conjugate functor is the identity the lemma is proved. □

Lemma 37 Let S, T be Hilbert 2-modules. We define a mapping U: S⊗_{H} T → N (S, T) by U(s ⊗_{H} t)(p) = ⟨p, s⟩ • t, where s ⊗_{H} t ∈ S ⊗_{H} T, p ∈ S. Then N (S, T) is a (strict) Hilbert 2-module and U is a unitary isomorphism of S ⊗_{H} T onto N (S, T).

PROOF. Evidently, N (S, T) is a (strict) Hilbert 2-module by the proposition 20 and the map u : S × T → N (S, T) defined by u(s, t) = Θ_{t,s} is a bimorphism of Hilbert 2-modules. This gives us that U is correctly defined. Moreover, since N (S, T) is generated by the elements of the form Θ_{t,s} the morphism U of Hilbert 2-modules is onto. Note that, for all s, u ∈ S, t, v ∈ T, we have ⟨s ⊗_{H} t, u ⊗_{H} v⟩_{S⊗_{H}T} = ⟨s, u⟩_{S} · ⟨t, v⟩_{T} = ⟨Θ_{t,s}, Θ_{v,u}⟩ = ⟨U(s ⊗_{H} t), U(u ⊗_{H} v)⟩ i.e. U is inner-product preserving. By the lemma 25, U is a unitary isomorphism. □

Corollary 38 Let S and T be (strict) Hilbert 2-modules. Then the (strict) Hilbert 2-module N (S, T) is a tensor product in the category of (strict) Hilbert 2-modules. Moreover, we have a morphism of Hilbert 2-modules T r : N (S, S) → 2 such that ⟨s, t⟩ = T r(Θ_{t,s}) for all s, t ∈ S.

Note that T r(Θ_{t,s}) = e iff Θ_{t,s} ∘ Θ_{t,s} = Θ_{t,s} iff s ≰ t′. Let us put z_{S} = ⋁{Θ_{t,s} : s, t ∈ S, s ≤ t′}. Then, for all f ∈ N (S, S), T r(f) = 0 iff f ≤ z_{S}.

Lemma 39 Let S be a Hilbert 2-module. Then f ∈ A_{2}(2, S) iff f = ϵ_{s} for some s ∈ S; here

ϵ_{s}(a) = \{ \begin{matrix}0 & if a = 0, \\ s & if a = e. \end{matrix}

Moreover, S is unitary isomorphic to A_{2}(2, S).

PROOF. It is evident. □

Theorem 40 The class of compact operators forms a traced nuclear ideal for (strict) Hilbert 2-modules.

PROOF. Let S and T be Hilbert 2-modules. By the lemma 39, it is evident that A_{2}(2, \overline{S} ⊗_{H} T) ≅ \overline{S} ⊗_{H} T. So the morphism U, defined in the lemma 37, has an inverse unitary map V such that V gives us the transpose operator by the prescription θ(f) = ϵ_{V (f)}, for all f ∈ N (S, T). It only remains to check the equations. First, let us check the preservation of tensored^{∗}-structure defined in [4]. Let f : S → T, g : M → N be nuclear. Then V (f) ⊗_{H} V (g) = (id_{S} ⊗_{H} c_{T,M} ⊗_{H} id_{N})∘V (f ⊗_{H} g) by a straightforward computation. This gives us the property (a). Let us prove (b). Since, V (Θ_{t,s}^{∗}) = V (Θ_{s,t}) = t ⊗_{H} s = c_{S,T} (s ⊗_{H} t) = (c ∘ V)(Θ_{t,s}) and V is uniquely determined by its action on generators, the property (b) is verified.

Now, let us prove the naturality of θ. We have, for all s ∈ S, t ∈ T, Θ_{t,s} ↦ ϵ_{s⊗_{H}t} ↦ ϵ_{f(s)⊗_{H}g(t)} by the prescription A_{2}(2, f⊗_{H}g)∘θ and Θ_{t,s} ↦ g∘Θ_{t,s}∘f^{∗} = Θ_{g(t),f(s)} ↦ ϵ_{f(s)⊗_{H}g(t)} by the prescription θ∘N (f^{∗}, g) i.e. the required diagram commutes.

To prove compactness of N, it is enough to check the diagram from the definition (see [4]) for f = Θ_{w,s} : S → T and g = Θ_{m,t} : T → M, s ∈ S, t, w ∈ T, m ∈ M. Then, for all p ∈ S, we have g(f(p)) = Θ_{m,⟨w,t⟩ • s}(p) = (⟨p, s⟩ ·

⟨t, w⟩) • m = \{ \begin{matrix}0 & if ⟨s, p⟩ = 0 or ⟨t, w⟩ = 0 \\ m & otherwise \end{matrix} . For the other part of the diagram:

p ↦ e⊗_{H}p ↦ t⊗_{H}m⊗_{H}p ↦ m⊗_{H}t⊗_{H}p ↦ \{ \begin{matrix}0 ↦ 0 & if ⟨s, p⟩ & = 0 or ⟨t, w⟩ = 0 \\ m⊗_{H} & e ↦ m & otherwise \end{matrix} ,

since θ(Θ_{w,s}^{∗})^{∗}(t ⊗_{H} p) ≤ 0 iff (⟨s, p⟩ = 0 or ⟨t, w⟩ = 0).

Now, we shall show that N is traced. Let f : S → T, g : T → S be nuclear maps i.e. f = ⋁_{i} Θ_{t_{i},s_{i}}, g = ⋁_{j} Θ_{u_{j},v_{j}}, s_{i}, u_{j} ∈ S, t_{i}, v_{j} ∈ T. Note that f̂(e) = ⋁_{i} s_{i} ⊗_{H} t_{i}, ĝ^{∗}(e) = ⋁_{j} u_{j} ⊗_{H} v_{j}. Then,

⟨(ĝ ∘ f̂)(e), e⟩ = ⟨f̂(e), ĝ^{∗}(e)⟩ = ⋁_{i,j} ⟨s_{i} ⊗_{H} t_{i}, u_{j} ⊗_{H} v_{j}⟩
= ⋁_{i,j} ⟨s_{i}, u_{j}⟩ · ⟨t_{i}, v_{j}⟩ = ⋁_{i} ⟨⋁_{j} ⟨t_{i}, v_{j}⟩ • u_{j}, s_{i}⟩
= T r(⋁_{i} Θ_{g(t_{i}),s_{i}}) = T r(g ∘ ⋁_{i} Θ_{t_{i},s_{i}}) = T r(g ∘ f).

Similarly, let f′ : S → U, g′ : U → S be nuclear maps such that g′ ∘ f′ = g ∘ f. Then ⟨(ĝ′ ∘ f̂′)(e), e⟩ = T r(g′ ∘ f′) = T r(g ∘ f) = ⟨(ĝ ∘ f̂)(e), e⟩ i.e. ĝ′ ∘ f̂′ = ĝ ∘ f̂. □

Proposition 41 Let S, T be Hilbert 2-modules, f : S → T a morphism of Hilbert 2-modules. Then f is nuclear iff for all u ∈ S, v ∈ T we have (u′, v) ∈ Υ(f)^{−+}, v ≤ f(u) implies v = f(u); here Υ(f) = {(x′, f(x)) : x ∈ S}.

PROOF. Note that Θ_{t,s} ≤ f iff for all x ∈ S (x ≤ s′ or t ≤ f(x)) iff (s, t) ∈ Υ(f)^{−}. Then we have, f is nuclear iff f = ⋁{Θ_{t,s} : (s, t) ∈ Υ(f)^{−}}. Now assume that f is nuclear. Let (u′, v) ∈ Υ(f)^{−+}, v ≤ f(u). Then, for all (s, t) ∈ Υ(f)^{−}, s ≤ u′ or t ≤ v i.e. ⋁{Θ_{t,s}(u) : (s, t) ∈ Υ(f)^{−}} ≤ v i.e. f(u) ≤ v ≤ f(u) i.e. f(u) = v.

Conversely, let for all u ∈ S, v ∈ T we have (u′, v) ∈ Υ(f)^{−+}, v ≤ f(u) implies v = f(u). Let u ∈ S. We put ⋁{Θ_{t,s}(u) : (s, t) ∈ Υ(f)^{−}} = v. Then v ≤ f(u). Moreover, if (s, t) ∈ Υ(f)^{−} and s ≰ u′ then t = Θ_{t,s}(u) ≤ v i.e. (u′, v) ∈ Υ(f)^{−+} i.e. v = f(u) i.e. f = ⋁{Θ_{t,s} : (s, t) ∈ Υ(f)^{−}}. □

Note that a ⋁-preserving map f : S → T, S, T being ⋁-semilattices, is tight (see [20], [21]) iff f = ⋁ E_{b}^{a} for suitable a ∈ S, b ∈ T; here

E_{b}^{a}(x) = \{ \begin{matrix}0 & if x ≤ a \\ b & otherwise. \end{matrix}

Theorem 42 Let S, T be (strict) Hilbert 2-modules, f : S → T a morphism of Hilbert 2-modules. Then f is nuclear iff it is tight.

PROOF. f is nuclear iff f = ⋁{Θ_{t,s} : (s, t) ∈ Υ(f)^{−}} iff f = ⋁{E_{t}^{s′} : (s, t) ∈ Υ(f)^{−}} iff f is tight. □

Theorem 43 Let S be a Hilbert 2-module. Then S is nuclear iff it is completely distributive.

PROOF. We have by [9], S is completely distributive iff id_{S} is tight iff id_{S} is nuclear iff S is nuclear. □

Corollary 44 Let S be a strict Hilbert 2-module. Then S is nuclear iff it is a complete Boolean algebra.

3 Conclusion and future work

We have introduced Hilbert Q-modules, their morphisms and compact operators on them. We have illustrated this notions in the simplest case of Hilbert 2-modules getting a ∗-tensored category with a traced nuclear ideal. It would of course be interesting to have more general examples of Hilbert Q-modules having this nuclear ideal structure.

There are several topics in our approach we would like to study in more detail. We hope that after some effort using the proposition 20 we can show that any category of Hilbert modules over a commutative quantale is a ∗-tensored category with a traced nuclear ideal.

Another possibility is to study tensor products of involutive quantales using the representation of involutive quantales via Hilbert 2-modules (see [14]) similarly as for C^{∗}-algebras. Having a good tensor product of involutive quantales it would be of interest to develop a notion of a Hopf involutive quantale and study related questions.

Acknowledgements

We want to thank here the anonymous referee for many valuable and perceptive comments that allow highly improve the final version of this paper. Paul Taylor’s diagram macros are acknowledged.

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