CONTRIBUTIONS TO GENERAL ALGEBRA 13

Proceedings of the Dresden Conference 2000 and the Summer School 1999

Verlag Johannes Heyn, Klagenfurt 2000

INTERIOR TENSOR PRODUCT OF HILBERT MODULES

JAN PASEKA

Abstract. The interior tensor product in the context of the theory of Hilbert modules over involutive quantales is studied. Associativity, functoriality and representability via compact maps of the interior tensor product are established.

It is the purpose of this article to give an analog of the interior tensor product for Hilbert C^{∗}-modules in the context of involutive quantales.

This paper is closely related to the papers [2] and [3] (the same volume of Contributions to General Algebra as this paper) where the interested reader can find unexplained terms, notation and general properties concerning the Hilbert modules over involutive quantales. For facts concerning quantales and quantale modules in general we refer to [7]. We shall also assume some familiarity with the theory of tensor products of sup-semilattices. For motivation on Hilbert C^{∗}-modules and transfered notations to Hilbert modules over involutive quantales we recommend the first three chapters of [1].

The paper is organized as follows. First, we recall the notions not introduced in [2] and [3]. Then we recast the theory of the interior tensor product for Hilbert C^{∗}-modules (see [1]) in the quantale language of Hilbert A-modules over an involutive quantale A and deduce corresponding results here. We establish basic properties of the interior tensor product – associativity, functoriality and representability via compact maps.

In order to allow applications to involutive quantales, the interior tensor product plays a crucial role in proving the Eilenberg-Watts theorem and the fundamental Morita theorem for Hilbert modules (see [4]). Moreover, we were able to introduce in [5], using the interior tensor product, the Rieffel induction for involutive quantales (for motivation see [6]), i.e. we obtained a construction of a representation π_{A} of A from a representation π_{B} of B.

Financial Support of the NATO Research Fellowship Program and of the Grant Agency of the Czech Republic under the grant No. 201/99/0310 are gratefully acknowledged.

The present work was largely developed during the author’s visit to the Mathematics Department at the Université Catholique de Louvain, whose members he would like to thank for their warm hospitality.

Preliminaries

Let us begin by establishing the common symbols and notations in this paper.

Let A be a quantale. We shall denote by MOD_{A} (_{A}MOD) the category of right (left) A−modules with the right (left) action ⋄ ( •) and right (left) module homomorphisms. For a right A-module X in MOD_{A} the submodule ess(X) = X ⋄ A generated by the elements x ⋄ a is called the essential part of X. If ess(X) = X we say that X is (right) essential. The full subcategory of essential A-modules is denoted ess − MOD_{A}. We shall say that A is right separating for the A-module M if m ⋄ (−) = n ⋄ (−) implies m = n. M is called right faithful if (−) ⋄ a = (−) ⋄ b implies that a = b.

Note that we may dually define the notion of a left (essential, separating, faithful) A-module with a left multiplication •.

The theory of right (left) Hilbert A-modules (we refer the reader to [2] 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 A endowed with A-valued inner products. Note only that for each right Hilbert A-module M there is a conjugate left Hilbert A-module M^{∗}: as an sup-semilattice M^{∗} is just M with a left module multiplication a • m = m ⋄ a^{∗} and a left A-valued inner product _{A}⟨m, n⟩ = ⟨m, n⟩_{A} (see Lemma 1.5 in [2]). Recall that a Hilbert A-module M is said to be full if ⟨M, M⟩ = {⋁_{j∈J} ⟨x_{j}, y_{j}⟩ : x_{j}, y_{j} ∈ M, j ∈ J} = A. We shall denote by HMOD_{A} the category of right Hilbert A−modules with morphisms the usual A-module maps. The full subcategory of essential right Hilbert A-modules is denoted ess − HMOD_{A}. If we write A ∈ ess − HMOD_{A} we mean that A is an essential right Hilbert A-module with respect to the usual quantale multiplication. Then, evidently 1 · 1 = 1 in A and a · (−) = b · (−) implies a = b.

Let f : M → N be a map between right (left) Hilbert A-modules. We say that a map g : N → M is a ^{∗}-adjoint to f and f is adjointable if ⟨f(m), n⟩ = ⟨m, g(n)⟩ for all m ∈ M, n ∈ N. Note that the ^{∗}-adjoint g to f is uniquely determined. Namely, if ⟨f(m), n⟩ = ⟨m, g(n)⟩ = ⟨m, h(n)⟩ then g(n) = h(n) for all n i.e. g = h. We then put f^{∗} = g. Evidently, any adjointable map is a right (left) module homomorphism (see [2]). The set of all adjointable maps from M to N is denoted by A_{A}(M, N), A_{A}(M) = A_{A}(M, M). We shall denote by Hilb_{A} (ess−Hilb_{A}, ess_{f} −Hilb_{A}) the category of (essential, essential full) right Hilbert A-modules and adjointable maps.

Let B be an involutive quantale such that φ: B → A_{A}(M) is an involutive quantale homomorphism. Then φ is said to be nondegenerate if M = φ(B)(M), i.e. for any m ∈ M we have that m = ⋁_{j∈J} φ(b_{j})(m_{j}) for suitable elements m_{j} ∈ M and b_{j} ∈ B, j ∈ J.

We say that an adjointable map f : M → N is an isometry if, for all m_{1}, m_{2} ∈ M, ⟨m_{1}, m_{2}⟩ = ⟨f(m_{1}), f(m_{2})⟩. This is equivalent to f^{∗} ∘ f = id_{M}. Similarly, an adjointable map f : M → N is unitary if f^{∗} ∘ f = id_{M} and f ∘ f^{∗} = id_{N}. Note that any surjective isometry is necessarily unitary.

Recall that from [2] we know that, for any right Hilbert A-module M and for all m ∈ M, the map m^{∼} : A → M defined by a ↦ m ⋄ a has a ^{∗}-adjoint m^{⋆} : M → A defined by n ↦ ⟨m, n⟩. The maps of the form

T = ⋁_{j∈J} Θ_{n_{j},m_{j}},

here Θ_{n,m} = n ⋄ ⟨m, −⟩ = n^{∼} ∘m^{⋆}, m ∈ M, n ∈ N are said to be compact maps and the set of all compact maps will be denoted K_{A}(M, N) or just K_{A}(M) in case M = N. Note that any compact map is an adjointable map and that a composition of a compact map with an adjointable one is again compact, i.e. K_{A}(N, P) ∘ A_{A}(M, N) ⊆ K_{A}(M, P) and A_{A}(N, P) ∘ K_{A}(M, N) ⊆ K_{A}(M, P). An expository treatment of compact maps on Hilbert A-modules may be found in [2].

1. Interior tensor product of Hilbert modules

Definition 1.1. Let A and B be involutive quantales, and let F and X be right Hilbert B-modules. We say that F is a right Hilbert A–B bimodule if it is a left A-module satisfying

a • (x ⋄ b) = (a • x) ⋄ b and ⟨a • x, y⟩_{B} = ⟨x, a^{∗} • y⟩_{B}

for all a ∈ A, x, y ∈ F, and b ∈ B.

We say that a (full essential) right Hilbert A − B bimodule X is an (imprimitivity) A−B bimodule if X is also a (full essential) left Hilbert A-module in such a way that

A⟨x, y⟩ • z = x ⋄ ⟨y, z⟩_{B} and _{A}⟨x ⋄ b, y⟩ = _{A}⟨x, y ⋄ b^{∗}⟩ .

Recall that the property a • (x ⋄ b) = (a • x) ⋄ b is redundant — it follows from the property ⟨a • x, y⟩_{B} = ⟨x, a^{∗} • y⟩_{B} (see Lemma 1.2) — but we include it to emphasize that F is a balanced bimodule.

Lemma 1.2. Let A and B be involutive quantales, and let F be a right Hilbert B-module. Then F is a right Hilbert A–B bimodule if and only if we have an involutive quantale homomorphism φ: A → A_{B}(F) such that a • x = φ(a)(x).

Proof. Let F be a right Hilbert A–B bimodule. We put φ(a)(x) = a • x. Let us show that φ(a) ∈ A_{B}(F). We have

⟨φ(a)(x), y⟩=⟨a • x, y⟩ = ⟨x, a^{∗} • x⟩ = ⟨x, φ(a^{∗})(y)⟩ ,

i.e. the adjointable map to φ(a) is φ(a^{∗}). Hence φ(a)^{∗} = φ(a^{∗}). Evidently, φ preserves arbitrary joins and multiplication since F is a left A-module.

Conversely, let φ: A → A_{B}(F) be an involutive quantale homomorphism and let us put a • x = φ(a)(x). Then F is a left A-module. Let a ∈ A, b ∈ B, x, y ∈ F. Then a • (x ⋄ b) = φ(a)(x ⋄ b) = φ(a)(x) ⋄ b = (a • x) ⋄ b and ⟨a • x, y⟩ = ⟨φ(a)(x), y⟩ = ⟨x, φ(a^{∗})(y)⟩ = ⟨x, a^{∗} • x⟩. □

Example 1.3. 1. A sup-semilattice S with a duality is an imprimitivity K_{2}(S) − 2 bimodule with _{K_{2}(S)}⟨s, t⟩ = Θ_{s,t} (see [2], Lemma 2.12).

2. If B is an involutive quantale, B ∈ HMOD_{B} then B becomes a right-Hilbert B–B bimodule by putting

a • b = a · b, b ⋄ c = b · c and ⟨a, b⟩_{B} = a^{∗}b

for a, b, c ∈ B. More generally, if φ: A → A_{B}(B) is an involutive quantale homomorphism, then B becomes a right Hilbert A–B bimodule with a left action given by

a • b = φ(a)(b).

3. Let X be a right Hilbert B-module. Then we have an involutive quantale embedding ι_{X} : K_{B}(X) → A_{B}(X) and the identity map id_{A_{B}(X)} : A_{B}(X) → A_{B}(X). Hence X is both a right Hilbert K_{B}(X) − B bimodule and a right Hilbert A_{B}(X) − B bimodule.

Moreover, if X is essential and full as a right Hilbert B-module then X is essential as a left K_{B}(X)(A_{B}(X))-module, i.e. ι_{X} (id_{A_{B}(X)}) are nondegenerate involutive quantale homomorphisms. In fact, X is then an imprimitivity K_{B}(X) − B bimodule; here _{K_{B}(X)}⟨x, y⟩ = Θ_{x,y}.

4. Let X, Y be full essential right Hilbert B-modules. Then K_{B}(Y, X) may be regarded as an imprimitivity K_{B}(X) − K_{B}(Y) bimodule.

Lemma 1.4. Let A and B be involutive quantales, A essential, and let X be a right essential full Hilbert A–B bimodule. Then X is an imprimitivity A–B bimodule if and only if we have an involutive quantale isomorphism φ: A → K_{B}(X) such that a • x = φ(a)(x).

Proof. Let X be an imprimitivity A–B bimodule, x, y ∈ X. Then φ(_{A}⟨x, y⟩) = A⟨x, y⟩ • (−) = x ⋄ ⟨y, −⟩_{B} = Θ_{x,y}. Since A is essential and any element from K_{B}(X) is a join of elements of the form Θ_{x,y}, φ is a surjective involutive quantale homomorphism. Let φ(a_{1}) = φ(a_{2}), a_{1} = ⋁_{j∈J} _{A}⟨x_{j}^{1}, y_{j}^{1}⟩,

a_{2} = ⋁_{k∈K} _{A}⟨x_{k}^{2}, y_{k}^{2}⟩. Then, for all z, w ∈ X, _{A}⟨⋁_{j∈J} ⟨x_{j}^{1}, y_{j}^{1}⟩ • z, w⟩ =

A⟨⋁_{k∈K} ⟨x_{k}^{2}, y_{k}^{2}⟩ • z, w⟩. Hence ⋁_{j∈J} _{A}⟨x_{j}^{1}, y_{j}^{1}⟩ · _{A}⟨z, w⟩ = ⋁_{k∈K} _{A}⟨x_{k}^{2}, y_{k}^{2}⟩ · 4

A⟨z, w⟩, i.e. a_{1} = a_{2}. Thus φ is an isomorphism. The converse follows from the preceding example, part 3. □

Proposition 1.5. Let A, B be involutive quantales. Suppose that E is a left Hilbert A-module and F is a right Hilbert A − B bimodule. Then the sup-semilattice of left A-compact maps _{A}K(E, F) generated by the elements LΘ_{y,x} = _{A}⟨−, x⟩ • y, x ∈ E, y ∈ F and equipped with the natural right Bmodule action (f ⋄ b)(m) = f(m) ⋄ b, for all m ∈ E and all b ∈ B, and with the inner product given by

⟨Θ_{y_{1},x_{1}}, Θ_{y_{2},x_{2}}⟩_{K_{A}(E,F)} = ⟨y_{1}, ⟨x_{1}, x_{2}⟩ • y_{2}⟩

for all x_{1}, x_{2} ∈ E and y_{1}, y_{2} ∈ F is a right Hilbert B-module.

Proof. Since E is a left Hilbert A module we shall work with the maps LΘ_{y,x} = ⟨−, x⟩ • y and _{A}K(E, F) is a sup-subsemilattice of the sup-lattice AMOD(E, F) of homomorphisms between left modules E and F generated by elements of the form Θ_{y,x}. Evidently, _{A}K(E, F) is a B-module. The rest proceeds similarly as in the Proposition 1.20 in [2]. □

Proposition 1.6. Let A, B be involutive quantales. Suppose that E is a right Hilbert A-module, F is a right Hilbert A−B bimodule. Then the sup-semilattice tensor product E ⊗ F is a pre-Hilbert B-module under the pre-inner product given on simple tensors by

⟨x_{1} ⊗ y_{1}, x_{2} ⊗ y_{2}⟩_{E⊗F} = ⟨y_{1}, ⟨x_{1}, x_{2}⟩ • y_{2}⟩

for all x_{1}, x_{2} ∈ E and y_{1}, y_{2} ∈ F.

Proof. Let us recall that from the definition of ⟨−, −⟩_{E⊗F} on generators we get a map from (E⊗F)×(E⊗F) to B that preserves joins in each variable. The action of B is given by (x⊗y) ⋄ b = x⊗(y ⋄ b), for all b ∈ B, x ∈ E and y ∈ F. Let u ∈ E and v ∈ F. Then, we have ⟨u ⊗ v, (x ⊗ y) ⋄ b⟩ = ⟨u ⊗ v, x ⊗ (y ⋄ b)⟩ = ⟨v, ⟨u, x⟩ • (y ⋄ b)⟩ = ⟨v, (⟨u, x⟩ • y) ⋄ b⟩ = ⟨v, ⟨u, x⟩ • y⟩ · b = ⟨u ⊗ v, x ⊗ y⟩ · b and similarly, ⟨u ⊗ v, x ⊗ y⟩^{∗} =⟨v, ⟨u, x⟩ • y⟩^{∗} =⟨⟨u, x⟩ • y, v⟩=⟨y, ⟨u, x⟩^{∗} • v⟩= ⟨y, ⟨x, u⟩ • v⟩ = ⟨x ⊗ y, u ⊗ v⟩. □

(E ⊗ F)_{R_{H}} is by definition the quotient of E ⊗ F, the elements of which are the top elements of equivalence classes of the relation R_{H} = {(U, V) ∈ (E ⊗ F) × (E ⊗ F) : ⟨U, W ⟩ = ⟨V, W ⟩ for all W ∈ E ⊗ F }. We shall denote by j_{R_{H}} : E ⊗ F → (E ⊗ F)_{R_{H}} the corresponding module nucleus; j_{R_{H}} (U) = ⋁{V ∈ E ⊗ F : (U, V) ∈ R_{H}}.

Definition 1.7. Let A, B be involutive quantales. Suppose that E is a right Hilbert A-module, F is a right Hilbert A − B bimodule. Then the Hilbert Bmodule (E ⊗ F)_{R_{H}} will be denoted by E ˙⊗_{A}F and we shall speak about the interior module tensor product over A.

We shall sometimes write x ˙⊗_{A}y for the element j_{R_{H}} (x⊗y). Recall that, for all a ∈ A and all x ∈ E, y ∈ F, we have that (x ⋄ a) ˙⊗_{A}y = x ˙⊗_{A}a • y. Namely, we have that, for all u ∈ E and v ∈ F,

\begin{matrix}⟨u ⊗ v, (x ⋄ a) ⊗ y⟩ & = & ⟨v, ⟨u, x ⋄ a⟩ • y⟩ = ⟨v, (⟨u, x⟩ · a) • y⟩ \\ = & ⟨v, ⟨u, x⟩ • (a • y)⟩ = ⟨u ⊗ v, x ⊗ a • y⟩ , \end{matrix}

i.e. ((x ⋄ a)⊗y, x⊗a • y) ∈ R_{H}.

Similarly, for all b ∈ B and all x ∈ E, y ∈ F, we have that x ˙⊗_{A}(y ⋄ b) = (x ˙⊗_{A}y) ⋄ b. This follows from the fact that we have a B-module action ⋄_{E⊗F} : (E ⊗ F) × B → E ⊗ F defined by (x ⊗ y) ⋄_{E⊗F} b = x⊗(y ⋄ b) and that j_{R_{H}} is a B-module map.

Let us point out that, for F ∈ ess − HMOD_{B}, we have that E ˙⊗_{A}F ∈ ess − HMOD_{B}. Similarly, E ˙⊗_{A}F is full whenever E and F are full and F is left essential.

Moreover, for the interior tensor product construction we have

Proposition 1.8. Let A, B be involutive quantales. Suppose further that E_{1}, E_{2} are right Hilbert A-modules, F_{1}, F_{2} are right Hilbert A − B bimodules, that f : E_{1} → E_{2} is an adjointable right A-module map, g : F_{1} → F_{2} is an adjointable right B-module map and a left A-module map. Then f ˙⊗_{A}g : E_{1} ˙⊗_{A}F_{1} → E_{2} ˙⊗_{A}F_{2} is an adjointable right B-module map.

Proof. Let us assume that f and g are both adjointable. We define f ˙⊗_{A}g = j_{R_{H}}^{2} ∘ (f⊗g) ∘ j_{R_{H}}^{1}^{⊢}; here j_{R_{H}}^{1}^{⊢} is the right adjoint to j_{R_{H}}^{1} in posets, i.e. (f ˙⊗_{A}g)(x) = j_{R_{H}}^{2} (f⊗g)(x), for all x ∈ E_{1} ˙⊗_{A}F_{1} ⊆ E_{1} ⊗ F_{1}. We shall prove that (f ˙⊗_{A}g) is adjointable. We have

⟨(f⊗g)(p ⊗ q), u⊗v⟩ = ⟨f(p)⊗g(q)), u⊗v⟩ = ⟨g(q), ⟨f(p), u⟩ • v⟩

= ⟨⟨f(p), u⟩^{∗} • g(q), v⟩ = ⟨⟨u, f(p)⟩ • g(q), v⟩ = ⟨g(⟨u, f(p)⟩ • q), v⟩ = ⟨⟨f^{∗}(u), p⟩ • q, g^{∗}(v)⟩ = ⟨q, ⟨p, f^{∗}(u)⟩ • g^{∗}(v)⟩ = ⟨p ⊗ q, (f^{∗}⊗g^{∗})(u⊗v)⟩ .

Applying the above computation we obtain

\begin{matrix}⟨ & (f ˙⊗_{A}g)(s ˙⊗_{A}t), m ˙⊗_{A}n⟩ = ⋁_{p⊗q≤s ˙⊗_{A}t,u⊗v≤m ˙⊗_{A}n} ⟨f⊗g)(p ⊗ q), u⊗v⟩ \\ = ⋁_{p⊗q≤s ˙⊗_{A}t,u⊗v≤m ˙⊗_{A}n} ⟨p ⊗ q, (f^{∗}⊗g^{∗})(u⊗v)⟩ = ⟨s ˙⊗_{A}t, (f^{∗} ˙⊗_{A}g^{∗})(m ˙⊗_{A}n)⟩ , \end{matrix}

i.e. the ^{∗}-adjoint to f ˙⊗_{A}g is the map f^{∗} ˙⊗_{A}g^{∗}. In fact, from the above computation we obtain that for adjointable f and g we have (f ˙⊗_{A}g)(s ˙⊗_{A}t) = f(s) ˙⊗_{A}g(t). □

Definition 1.9. Let A, B be involutive quantales. We say that Ψ : Hilb_{A} → Hilb_{B} is a SUP-functor if Ψ(⋁_{i∈I} T_{i}) = ⋁_{i∈I} Ψ(T_{i}) for all {T_{i} : i ∈ I} ⊆ A_{A}(M, N), M, N ∈ Hilb_{A}. A SUP-functor Ψ is a *-functor if Ψ(T^{∗}) = Ψ(T)^{∗} for all adjointable maps T.

Theorem 1.10. Let A, B be involutive quantales. Suppose that F is a right Hilbert A − B bimodule. Then the function (−) ˙⊗_{A}F : Hilb_{A} → Hilb_{B} which assigns to each right Hilbert A-module E the inner tensor product E ˙⊗_{A}F and to each adjointable map f between right Hilbert A-modules the adjointable map f ˙⊗_{A}id_{F} between right Hilbert B-modules is a *-functor. If moreover F is left faithful, i.e. the induced map φ : A → A_{B}(F) is an injective homomorphism of involutive quantales then (−) ˙⊗_{A}F is faithful on adjointable mappings.

Proof. Note that evidently (g ∘ f) ˙⊗_{A}id_{F} = (g ˙⊗_{A}id_{F}) ∘ (f ˙⊗_{A}id_{F}). Since we have (id_{E} ˙⊗_{A}id_{F})(s ˙⊗_{A}t) = j_{R_{H}} (s ˙⊗_{A}t) = s ˙⊗_{A}t, (−) ˙⊗_{A}F is a functor.

Evidently, since (−) ˙⊗_{A}F preserves arbitrary joins it is a SUP-functor. Since interior tensor product preserves adjointability it is a *-functor. Suppose that F is left faithful. Let t_{1}, t_{2} ∈ A_{A}(E_{1}, E_{2}) such that t_{1} ˙⊗_{A}id_{F} = t_{2} ˙⊗_{A}id_{F}. Then, for all x ∈ E_{1}, u ∈ E_{2} and all y, v ∈ F, we have that ⟨y, ⟨t_{1}(x), u⟩ • v⟩ = ⟨t_{1}(x) ˙⊗_{A}y, u ˙⊗_{A}v⟩=⟨(t_{1} ˙⊗_{A}id_{F})(x ˙⊗_{A}y), u ˙⊗_{A}v⟩=⟨(t_{2} ˙⊗_{A}id_{F})(x ˙⊗_{A}y), u ˙⊗_{A}v⟩= ⟨t_{2}(x) ˙⊗_{A}y, u ˙⊗_{A}v⟩ = ⟨y, ⟨t_{2}(x), u⟩ • v⟩ , i.e. ⟨t_{1}(x), u⟩ • v = ⟨t_{2}(x), u⟩ • v. Then ⟨t_{1}(x), u⟩ = ⟨t_{2}(x), u⟩, i.e. t_{1}(x) = t_{2}(x). Hence t_{1} = t_{2}. □

Corollary 1.11. Let A, B be involutive quantales. Suppose that E is a right Hilbert A-module, F is a right Hilbert A − B bimodule with the induced homomorphism φ : A → A_{B}(F) of involutive quantales. Then, for all t ∈ A_{A}(E), we have a map t ˙⊗_{A}id_{F} : E ˙⊗_{A}F → E ˙⊗_{A}F defined on generators by (t ˙⊗_{A}id_{F})(x ˙⊗_{A}y) = t(x) ˙⊗_{A}y. This gives us a unital involutive quantale homomorphism φ_{∗} : A_{A}(E) → A_{B}(E ˙⊗_{A}F), t ↦ t ˙⊗_{A}id_{F}. If φ is injective then so is φ_{∗}.

Proposition 1.12. Let A, B be involutive quantales. Suppose that E_{1} is a right Hilbert A-module, E_{2} is a right essential Hilbert A-module, F is a right Hilbert A − B bimodule with the induced involutive quantale homomorphism φ : A → K_{B}(F). Then we have the restriction (−)^{K} ˙⊗_{A}id_{F} : K_{A}(E_{1}, E_{2}) → K_{B}(E_{1} ˙⊗_{A}F, E_{2} ˙⊗_{A}F). If φ is surjective (injective) then so is (−)^{K} ˙⊗_{A}id_{F}.

Proof. Let us recall that φ is also a map from A to A_{B}(F). Applying the same procedure as for C^{∗}-modules in [1], Lemma 4.6 and Proposition 4.7, we may define, for any element x ∈ E_{j}, 1 ≤ j ≤ 2 an adjointable map Θ_{x} : F → E_{j} ˙⊗_{A}F, f ↦ x ˙⊗_{A}f. Its adjoint is then the map Θ_{x}^{∗} : E_{j} ˙⊗_{A}F → F, u ˙⊗_{A}v ↦ φ(⟨x, u⟩)(v). One can easily check that (Θ_{x ⋄ a,y}) ˙⊗_{A}id_{F} = Θ_{x} ∘ φ(a) ∘ Θ_{y}^{∗} and that this element is in K_{B}(E_{1} ˙⊗_{A}F, E_{2} ˙⊗_{A}F) since φ(a) ∈ K_{B}(F). For completeness, we sketch the arguments of [1]. Let a ∈ A, y, u ∈ E_{1}, x ∈ E_{2} and v ∈ F. Then

\begin{aligned}(Θ_{x} ∘ φ(a)∘ & Θ_{y}^{∗})(u ˙⊗_{A}v) = Θ_{x}(φ(a)(φ(⟨y, u⟩)(v))) = x ˙⊗_{A}(φ(a)(φ(⟨y, u⟩)(v))) \\ & = x ⋄ (a · ⟨y, u⟩) ˙⊗_{A}v = Θ_{x ⋄ a,y}(u) ˙⊗_{A}v = (Θ_{x ⋄ a,y} ˙⊗_{A}id_{F})(u ˙⊗v). \end{aligned}

By E_{2} ⋄ A = E_{2}, we have that any Θ_{x,y} ˙⊗_{A}id_{F} ∈ K_{B}(E_{1} ˙⊗_{A}F, E_{2} ˙⊗_{A}F).

Suppose that φ is surjective. Let y ∈ E_{1}, x ∈ E_{2} and z, w ∈ F. Then we can find an element a ∈ A such that φ(a) = Θ_{z,w}. Let u ∈ E_{1} and v ∈ F.

Then

Θ_{x ˙⊗_{A}z,y ˙⊗_{A}w}(u ˙⊗_{A}v) = (x ˙⊗_{A}z) ⋄ ⟨y ˙⊗_{A}w, u ˙⊗_{A}v⟩ = (x ˙⊗_{A}z) ⋄ ⟨w, φ(⟨y, u⟩)(v)⟩
= x ˙⊗_{A}(z ⋄ ⟨w, φ(⟨y, u⟩)(v)⟩) = x ˙⊗_{A}(Θ_{z,w}(φ(⟨y, u⟩)(v))
= x ˙⊗_{A}(φ(a)(φ(⟨y, u⟩)(v)) = x ˙⊗_{A}(φ(a · ⟨y, u⟩)(v))
= (Θ_{x ⋄ a,y} ˙⊗_{A}id_{F})(u ˙⊗_{A}v).

Since K_{B}(E_{1} ˙⊗_{A}F, E_{2} ˙⊗_{A}F) is generated by the elements of the form Θ_{x ˙⊗_{A}z,y ˙⊗_{A}w} it follows that (−)^{K} ˙⊗_{A}id_{F} is surjective. The injectivity follows from the Proposition 1.10. □

Corollary 1.13. Let A, B be involutive quantales. Suppose that E is a right essential full Hilbert A-module, F is a right Hilbert A − B bimodule and φ : A → K_{B}(F) is the induced nondegenerate involutive quantale homomorphism. Then the restriction φ_{∗}^{K} : K_{A}(E) → K_{B}(E ˙⊗_{A}F) of φ_{∗} is a unital nondegenerate involutive quantale homomorphism. If φ is surjective (injective) then so is φ_{∗}^{K}.

Proof. Evidently, φ_{∗}^{K} is a unital involutive quantale homomorphism. Let us show that it is nondegenerate. Let x, y, u ∈ E, v ∈ F. Then we have φ_{∗}(Θ_{x,y})(u ˙⊗_{A}v) = Θ_{x,y}(u) ˙⊗_{A}v = x ⋄ ⟨y, u⟩ ˙⊗_{A}v = x ˙⊗_{A}φ(⟨y, u⟩)(v). Since E is full and φ is nondegenerate then any element of E ˙⊗_{A}F is a join of elements of this form. □

Recall that the cartesian product ∏_{j∈J} E_{j} of Hilbert modules coincides as usual for quantale modules with their sum ∐_{j∈J} E_{j}. The respective projections π_{j} and injections i_{j} are defined by π_{j}((x_{k})_{k∈J}) = x_{j}, π_{k}(i_{j}(x_{j})) = x_{j} if j = k and π_{k}(i_{j}(x_{j})) = 0 if j ≠ k.

Lemma 1.14. Let A, B be involutive quantales. Suppose that E_{j}, j ∈ J are Hilbert A-modules, F is a Hilbert A − B bimodule. Then we have a unitary isomorphism ∐_{j∈J}(E_{j} ˙⊗_{A}F) ≅ (∐_{j∈J} E_{j}) ˙⊗_{A}F.

Proof. Note that we have unitary maps h : (∐_{j∈J} E_{j}) ˙⊗_{A}F → ∐_{j∈J}(E_{j} ˙⊗_{A}F)

defined by (m_{j})_{j∈J} ˙⊗_{A}n ↦ (m_{j} ˙⊗_{A}n)_{j∈J}, \overline{h} : ∐_{j∈J}(E_{j} ˙⊗_{A}F) → (∐_{j∈J} E_{j}) ˙⊗_{A}F

defined by (m_{j} ˙⊗_{A}n_{j})_{j∈J} ↦ ⋁_{j∈J} i_{j}(m_{j}) ˙⊗_{A}n_{j}. Evidently, h and \overline{h} are inverse of each other. So \overline{h} = h^{∗}. □

Lemma 1.15. Let A, B, C be involutive quantales. Suppose that F is a right Hilbert A–B bimodule and G is a right Hilbert B–C bimodule. Then the interior tensor product F ⊗_{B} G is a right Hilbert A–C bimodule under the left action given on simple tensors by a • (n ˙⊗_{B}o) = (a • n) ˙⊗_{B}o for all m ∈ F and

n ∈ G. 8

Proof. First, let us check that our definition is correct. Let ⋁_{j∈J}(n_{j} ˙⊗_{B}o_{j}) = ⋁_{k∈I}(p_{k} ˙⊗_{B}r_{k}). Then

⟨⋁_{j∈J}(a • n_{j} ˙⊗_{B}o_{j}),u ˙⊗_{B}v⟩ = ⋁_{j∈J} ⟨o_{j}, ⟨a • n_{j}, u⟩ • v⟩

= ⋁_{j∈J} ⟨o_{j}, ⟨n_{j}, a^{∗} • u⟩ • v⟩ = ⟨⋁_{j∈J} n_{j} ˙⊗_{B}o_{j}, (a^{∗} • u) ˙⊗_{B}v⟩ = ⟨⋁_{k∈I} p_{k} ˙⊗_{B}r_{k}, (a^{∗} • u) ˙⊗_{B}v⟩ = ⋁_{k∈I} ⟨r_{k}, ⟨p_{k}, a^{∗} • u⟩ • v⟩ = ⋁_{k∈I} ⟨r_{k}, ⟨a • p_{k}, u⟩ • v⟩ = ⟨⋁_{k∈I}(a • p_{k} ˙⊗_{B}r_{k}), u ˙⊗_{B}v⟩ .

From the definition of the left action and of the interior tensor product it is evident that F ⊗_{B} G is a left A-module and a right Hilbert C-module. Let us check the following:

\begin{matrix}⟨a • (n ˙⊗_{B}o), u ˙⊗_{B}v⟩ & = & ⟨(a • n) ˙⊗_{B}o, u ˙⊗_{B}v⟩ = ⟨n ˙⊗_{B}o, (a^{∗} • u) ˙⊗_{B}v⟩ \\ = & ⟨n ˙⊗_{B}o, a^{∗} • (u ˙⊗_{B}v)⟩ . \end{matrix}

Then F ⊗_{B} G is a right Hilbert A–C bimodule. □

Lemma 1.16. Let A, B be involutive quantales, A ∈ HMOD_{A}, F a right Hilbert A − B bimodule such that F ∈ ess −_{A}MOD. Then the canonical map χ_{F} : A ˙⊗_{A}F → F given by a ˙⊗_{A}n ↦ a • n is a unitary isomorphism of Hilbert B-modules and a bimodule isomorphism of right Hilbert A − B bimodules.

Proof. Let us show that the definition of χ_{F} is correct. Assume ⋁_{j∈J} a_{j} ˙⊗_{A}y_{j} = ⋁_{k∈I} b_{k} ˙⊗_{A}z_{k}. Since F ∈ ess −_{A}MOD any element of F is a join of elements of the form c • u. We have

⟨⋁_{j∈J} a_{j} • y_{j}, c • u⟩=⋁_{j∈J} ⟨a_{j} • y_{j}, c • u⟩ = ⋁_{j∈J} ⟨y_{j}, (a_{j}^{∗} · c) • u⟩
=⋁_{j∈J} ⟨y_{j}, ⟨a_{j}, c⟩ • u⟩ = ⟨⋁_{j∈J} a_{j} ˙⊗_{A}y_{j}, c ˙⊗_{A}u⟩
=⟨⋁_{k∈I} b_{k} ˙⊗_{A}z_{k}, c ˙⊗_{A}u⟩ = ⟨⋁_{k∈I} b_{k} • z_{k}, c • u⟩ .

We claim that χ_{F} is an adjointable map. Let us define a map κ_{F} : F → A ˙⊗F by κ_{F} (y) = ⋁{a ˙⊗_{A}z : a • z ≤ y}. Then we have

⟨χ_{F} (⋁_{j∈J} a_{j} ˙⊗_{A}y_{j}), y⟩ = ⟨⋁_{j∈J} a_{j} • y_{j}, ⋁{a • z : a • z ≤ y}⟩

= ⋁{⟨a_{j} • y_{j}, a • z⟩ : j ∈ J, a • z ≤ y} = ⋁{⟨a_{j} ˙⊗_{A}y_{j}, a ˙⊗_{A}z⟩ : j ∈ J, a ⋄ z ≤ y} = ⟨⋁_{j∈J} a_{j} ˙⊗_{A}y_{j}, κ_{F} (y)⟩ .

To show that χ_{F} is a unitary isomorphism it is enough to note that it is a surjective inner-product preserving adjointable map. We have a • χ_{F} (c ˙⊗_{A}n) = a • (c • n) = (a · c) • n) = χ_{F} (a · c ˙⊗_{A}n). Hence χ_{F} and also κ_{F} are bimodule isomorphisms. □

We have also the following two important lemmas.

Lemma 1.17. Let A be an involutive quantale, A, E ∈ ess−HMOD_{A}. Then the canonical map π_{E} : E ˙⊗_{A}A → E given by m ˙⊗_{A}a ↦ m ⋄ a is a unitary isomorphism of Hilbert A-modules.

Proof. First, let us show that the definition of π_{E} is correct. Assume ⋁_{j∈J} y_{j} ˙⊗_{A}a_{j} = ⋁_{k∈I} z_{k} ˙⊗_{A}b_{k}. Since E ∈ ess − HMOD_{A} any element of E is a join of elements of the form u ⋄ c. We have

⟨⋁_{j∈J} y_{j} ⋄ a_{j}, u ⋄ c⟩=⋁_{j∈J} ⟨y_{j} ⋄ a_{j}, u ⋄ c⟩ = ⋁_{j∈J} a_{j}^{∗} · ⟨y_{j}, u⟩ · c
=⋁_{j∈J} ⟨a_{j}, ⟨y_{j}, u⟩ • c⟩ = ⟨⋁_{j∈J} y_{j} ˙⊗_{A}a_{j}, u ˙⊗_{A}c⟩
=⟨⋁_{k∈I} z_{k} ˙⊗_{A}b_{k}, u ˙⊗_{A}c⟩ = ⟨⋁_{k∈I} z_{k} ⋄ b_{k}, u ⋄ c⟩ .

Now, we shall check that π_{E} is an adjointable map. Let us define a map ι_{E} : E → E ˙⊗A by ι_{E}(y) = ⋁{z ˙⊗_{A}a : z ⋄ a ≤ y}. Then we have

⟨π_{E}(⋁_{j∈J} y_{j} ˙⊗_{A}a_{j}), y⟩ = ⟨⋁_{j∈J} y_{j} ⋄ a_{j}, ⋁{z ⋄ a : z ⋄ a ≤ y}⟩

= ⋁{⟨y_{j} ⋄ a_{j}, z ⋄ a⟩ : j ∈ J, z ⋄ a ≤ y} = ⋁{⟨y_{j} ˙⊗_{A}a_{j}, z ˙⊗_{A}a⟩ : j ∈ J, z ⋄ a ≤ y} = ⟨⋁_{j∈J} y_{j} ˙⊗_{A}a_{j}, ι_{E}(y)⟩ .

Since π_{E} is a surjective inner-product preserving adjointable map it is a unitary isomorphism. □

Lemma 1.18. Let A, B be involutive quantales. Suppose that E is a right Hilbert A-module and F is a right Hilbert A−B bimodule. Then K_{A}(E^{∗}, F) ≅ E ˙⊗_{A}F as right Hilbert B-modules.

Proof. We shall prove that the natural maps δ_{E,F} : K_{A}(E^{∗}, F) → E ˙⊗_{A}F given by ^{L}Θ_{f,e} ↦ e ˙⊗_{A}f and γ_{E,F} : E ˙⊗_{A}F → K_{A}(E^{∗}, F) defined as e ˙⊗_{A}f ↦ LΘ_{f,e} are unitary isomorphisms. First, let us check that they are correctly defined. Assume that ⋁_{i} Θ_{f_{i},e_{i}} = ⋁_{j} Θ_{v_{j},u_{j}}. We have

\begin{matrix}⟨ & δ_{E,F} (⋁_{i} Θ_{f_{i},e_{i}}), e ˙⊗_{A}f⟩ = ⟨⋁_{i} e_{i} ˙⊗_{A}f_{i}, e ˙⊗_{A}f⟩ = ⋁_{i} ⟨e_{i} ˙⊗_{A}f_{i}, e ˙⊗_{A}f⟩ \\ = ⟨⋁_{i} ⟨e, e_{i}⟩ • f_{i}, f⟩ = ⟨⋁_{j} ⟨e, u_{j}⟩ • v_{j}, f⟩ = ⟨δ_{E,F} (⋁_{j} Θ_{v_{j},u_{j}}), e ˙⊗_{A}f⟩ . \end{matrix}

Evidently, δ_{E,F} is inner-product preserving and surjective, i.e. it is a unitary map. The proof for γ_{E,F} goes the same way. □

Proposition 1.19. Let A, B, C be involutive quantales. Suppose that E is a right Hilbert A-module, F is a right Hilbert A–B bimodule and G is a right Hilbert B–C bimodule. Then (E ˙⊗_{A}F) ˙⊗_{B}G ≅ E ˙⊗_{A}(F ˙⊗_{B}G), i.e. the interior tensor product is associative.

Proof. Let us define a map h : (E ˙⊗_{A}F) ˙⊗_{B}G → E ˙⊗_{A}(F ˙⊗_{B}G) on simple tensors by (m ˙⊗_{A}n) ˙⊗_{B}o ↦ m ˙⊗_{A}(n ˙⊗_{B}o). First, let us check the correctness. Assume that ⋁_{j∈J}(m_{j} ˙⊗_{A}n_{j}) ˙⊗_{B}o_{j} = ⋁_{k∈I}(r_{k} ˙⊗_{A}s_{k}) ˙⊗_{B}t_{k}. We have then

⟨m_{j} ˙⊗_{A}(n_{j} ˙⊗_{B}o_{j}), u ⊗_{A} (v ˙⊗_{B}w)⟩ = ⟨n_{j} ˙⊗_{B}o_{j}, ⟨m_{j}, u⟩ • (v ˙⊗_{B}w)⟩ = ⟨n_{j} ˙⊗_{B}o_{j}, (⟨m_{j}, u⟩ • v) ˙⊗_{B}w⟩ = ⟨o_{j}, ⟨n_{j}, ⟨m_{j}, u⟩ • v⟩ • w⟩

= ⟨o_{j}, ⟨m_{j} ˙⊗_{A}n_{j}, u ˙⊗_{A}v⟩ • w⟩ = ⟨(m_{j} ˙⊗_{A}n_{j}) ˙⊗_{B}o_{j}, (u ⊗_{A} v) ˙⊗_{B}w⟩ 10

and similarly

⟨r_{k} ˙⊗_{A}(s_{k} ˙⊗_{B}t_{k}), u ⊗_{A} (v ˙⊗_{B}w)⟩ = ⟨(r_{k} ˙⊗_{A}s_{k}) ˙⊗_{B}t_{k}, (u ⊗_{A} v) ˙⊗_{B}w⟩ .

This gives us that h(⋁_{j∈J}(m_{j} ˙⊗_{A}n_{j}) ˙⊗_{B}o_{j}) = h(⋁_{k∈I}(r_{k} ˙⊗_{A}s_{k}) ˙⊗_{B}t_{k}). Similarly, we may correctly define a map g : E ˙⊗_{A}(F ˙⊗_{B}G) → (E ˙⊗_{A}F) ˙⊗_{B}G on simple tensors by m ˙⊗_{A}(n ˙⊗_{B}o) ↦ (m ˙⊗_{A}n) ˙⊗_{B}o and the maps are inverse of each other. They are also adjointable since we have

\begin{matrix}⟨h((m ˙⊗_{A}n) ˙⊗_{B}o), u ⊗_{A} (v ˙⊗_{B}w)⟩ = ⟨m ˙⊗_{A}(n ˙⊗_{B}o), u ⊗_{A} (v ˙⊗_{B}w)⟩ \\ = ⟨(m ˙⊗_{A}n) ˙⊗_{B}o, (u ⊗_{A} v) ˙⊗_{B}w⟩ = ⟨(m ˙⊗_{A}n) ˙⊗_{B}o, g(u ⊗_{A} (v ˙⊗_{B}w))⟩ . \end{matrix}

So we have a unitary isomorphism between (E ˙⊗_{A}F) ˙⊗_{B}G and E ˙⊗_{A}(F ˙⊗_{B}G). □

Acknowledgements

We want to thank here the anonymous referee for many valuable and perceptive comments that allowed us to highly improve the final version of this paper.

References

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[2] J. Paseka, Hilbert Q-Modules and Nuclear Ideals, Proceedings of the Eight Conference on Category Theory and Computer Science (CTCS ’99), Electronic Notes in Computer Science 24 (1999) 319-338 (http://www.elsevier.nl/locate/entcs/volume29.html).

[3] J. Paseka, Multiplier algebras of involutive quantales, (preprint), (1999), accepted in Proceedings of the Summer School on General Algebra and Ordered Sets.

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[5] J. Paseka, Rieffel induction in the context of Hilbert modules (in preparation), (2000).

[6] I. Raeburn, D.P. Williams, Morita equivalence and Continuous-Trace C^{∗}-algebras, (Mathematical Surveys and Monographs 60, American Mathematical Society, 1998).

[7] K. I. Rosenthal, Quantales and their applications, (Pitman Research Notes in Mathematics Series 234, Longman Scientific & Technical, 1990).

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

  E-mail address: paseka@math.muni.cz