Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

MORITA EQUIVALENCE IN THE CONTEXT OF HILBERT MODULES

JAN PASEKA

Abstract. The Morita equivalence of m-regular involutive quantales in the context of the theory of Hilbert A-modules is presented. The corresponding fundamental representation theorems are shown. We also prove that two commutative m-regular involutive quantales are Morita equivalent if and only if they are isomorphic.

In the paper [5] F. Borceux and E.M. Vitale made a first step in extending the theory of Morita equivalence to quantales. They considered unital quantales and the category of all left modules over these unital quantales which are unital in a natural sense. They proved that two such module categories over unital quantales A and B, say, are equivalent if and only if there exist a unital A−B bimodule M and a unital B −A bimodule N such that M ⊗_{B} N ≃ A and N ⊗_{A} M ≃ B.

The aim of this paper is to extend this theory in the following way: to cover also the case of m-regular (generally non-unital) involutive quantales and Hilbert modules over them. Our motivation to work in this setting comes from the theory of operator algebras, where there is a theory of Morita equivalence for C^{∗}-algebras for the non-unital case (see [4], [8] and [12]). Our presentation is a combination of those in [1], [4] and [5].

This paper is closely related to the papers [9] and [11] where the interested reader can find unexplained terms and notation concerning the subject. For facts concerning quantales and quantale modules in general we refer to [13]. The algebraic background may be found in any account of Morita theory for rings, such as [1] or [3].

The paper is organized as follows. First, we recall the notion of a right Hilbert A-module and related notions. In Section 1 the necessary basic properties of right Hilbert modules are established. Moreover, a categorical characterization of surjective module maps in the category of m-regular right Hilbert modules is given. In section 2 the key result is the Eilenberg-Watts theorem which states that colimit preserving ∗-functors between categories of Hilbert modules correspond to right Hilbert bimodules. Our second result is the fundamental Morita theorem for Hilbert modules. As a consequence we get that two m-regular involutive quantales are strongly Morita equivalent if and only if they are Morita equivalent. Moreover, two commutative m-regular involutive quantales are Morita equivalent if and only if they are isomorphic.

2000 Mathematics Subject Classification. 46M15, 46L05, 18D20, 06F07.

Key words and phrases. Morita equivalence, quantale.

Financial Support by the NATO Research Fellowship Program and by the Ministry of Education under the project MSM 143100009 are gratefully acknowledged.

Jan Paseka, Morita equivalence in the context of Hilbert modules, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 223–251, Topology Atlas, Toronto, 2002; arXiv:math.OA/0204112.

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.

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 suplattice homomorphisms from S to T, with the supremum given by the pointwise ordering of mappings. If S = T we put Q(S) = SUP (S, S). Recall that a quantale is a sup-lattice A 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} ∈ A, i ∈ I (I is a set). 1 denotes the greatest element of A, 0 is the smallest element of A. A quantale A is said to be unital if there is an element e ∈ A such that e · a = a = a · e for all a ∈ A. An opposite quantale A^{d} to A is a sup-lattice with the same join operation as A and with the multiplication a ∘ b = b · a. A subquantale A′ of a quantale A is a subset of A closed under ⋁ and ·. Since the operators a · − and − · b : A → A, a, b ∈ A preserve arbitrary joins, they have right adjoints. Explicitly, they are given by

a →_{r} c = ⋁{s ∈ A|a · s ≤ c} and b →_{l} d = ⋁{t ∈ A|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 A is an involution on the sup-lattice A such that

(a · b)^{∗} = b^{∗} · a^{∗}

for all a, b ∈ A. 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 : A → A′. If a morphism preserves the unital element we say that it is unital.

Let A be a quantale. A right module over A (shortly a right A-module) is a sup-lattice M, together with a module action

−⋄− : M × A → M

satisfying (M1) m ⋄ (a · b) = (m ⋄ a) ⋄ b (M2) (⋁ X) ⋄ a = ⋁{x ⋄ a : x ∈ X} (M3) m ⋄ ⋁ S = ⋁{m ⋄ s : s ∈ S} for all a, b ∈ A, m ∈ M, S ⊆ A, X ⊆ M. So we have two maps

− →_{L} − : M × A → M, − →_{R} − : M × M → A

such that, for all a ∈ A, m, n ∈ M,

m ⋄ a ≤ n iff a ≤ m →_{R} n iff m ≤ a →_{L} n.

M is called a unital A-module if A is a unital quantale with the unit e and m ⋄ e = m for all m ∈ M.

Let M and N be modules over A and let f : M → N be a sup-lattice homomorphism. f is a module homomorphism if f(m ⋄ a) = f(m) ⋄ a for all a ∈ A, m ∈ M. We shall denote by MOD_{A} the category of right A−modules and module homomorphisms.

For 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 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 and that M is (right) separated by A if m ⋄ (−) = n ⋄ (−) implies m = n. We say that M is m-regular if it is both separated by A and essential. An involutive quantale A is called m-regular if it is m-regular as an A-module. Then, evidently 1 · 1 = 1 in A and a · (−) = b · (−) implies a = b.

Note that we may dually define the notion of a (unital) left A-module with a left multiplication •. We then have two maps

− →_{l} − : M × M → A, − →_{r} − : A × M → M

such that, for all a ∈ A, m, n ∈ M,

a • m ≤ n iff a ≤ m →_{l} n iff m ≤ a →_{r} n.

The theory of Hilbert A-modules (we refer the reader to [9] 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.

Let A be an involutive quantale, M a right (left) A-module. We say that M is a right (left) Hilbert A-module (right (left) strict Hilbert A-module), right (left) pre-Hilbert A-module if M is equipped with a map

⟨−, −⟩ : M × M → A

called the inner product, such that for all a ∈ A, m, n ∈ M and m_{i} ∈ M, where i ∈ I, the conditions (0.1)–(0.5) ((0.1)–(0.6), (0.1)–(0.4)) are satisfied.

⟨m, n⟩ · a = ⟨m, n ⋄ a⟩ (a · ⟨m, n⟩ = ⟨a • m, n⟩); (0.1)

⋁_{i∈I}⟨m_{i}, n⟩ = ⟨⋁_{i∈I} m_{i}, n⟩; (0.2)

⋁_{i∈I}⟨m, m_{i}⟩ = ⟨m, ⋁_{i∈I} m_{i}⟩; (0.3)

⟨m, n⟩^{∗} = ⟨n, m⟩; (0.4) ⟨−, m⟩ = ⟨−, n⟩ (⟨m, −⟩ = ⟨n, −⟩) implies m = n; (0.5) ⟨m, m⟩ = 0 implies m = 0. (0.6)

If A is an involutive quantale, let HMOD_{A} be the category of right Hilbert A-modules with morphisms the usual A-module maps. The full subcategory of m-regular right Hilbert A-modules is denoted mreg − HMOD_{A}.

Let A be an involutive quantale, f : M → N a map between right (left) pre-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. Evidently, any adjointable map is a module homomorphism. If f is adjointable we put

f^{∗} = ⋁{g : N → M; g is a ^{∗}-adjoint to f}.

Note that f ≤ f^{∗∗} and f^{∗} = f^{∗∗∗}. If M and N are Hilbert A-modules the ∗-adjoint to f is uniquely determined by property (0.5) i.e. f = f^{∗∗}. The set of all adjointable maps from M to N is denoted by A_{A}(M, N). We shall denote by mreg − Hilb_{A} the category of m-regular right Hilbert A-modules and adjointable mappings. 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 [9] 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⟩.

Similarly, let A and B be involutive quantales, and let M and X be right Hilbert B-modules. We say that M is a right Hilbert A − B bimodule if it is a left A-module satisfying

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

for all a ∈ A, x, y ∈ M, and b ∈ B. We say that F is an m-regular right Hilbert A − B bimodule if it is both an m-regular left A-module and an m-regular right B-module. An isomorphism of right Hilbert bimodules is a bijective ⋁-preserving map Φ: M → N such that

(i) Φ(a • x) = a • Φ(x), (ii) Φ(x ⋄ b) = Φ(x) ⋄ b, and (iii) ⟨Φ(x), Φ(y)⟩_{B} = ⟨x, y⟩_{B}

i.e. Φ is an isometric surjective bimodule homomorphism. In particular, Φ is a unitary B-module map.

We say that a (full m-regular) right Hilbert A − B bimodule X is an (imprimitivity) A − B bimodule if X is also a (full m-regular) 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^{∗}⟩ .

Suppose that M is a right Hilbert A-module, N is a right Hilbert A − B bimodule. Then the sup-semilattice tensor product M ⊗ N is a pre-Hilbert B-module under the pre-inner product given on simple tensors by

⟨x_{1} ⊗ y_{1}, x_{2} ⊗ y_{2}⟩_{M⊗N} = ⟨y_{1}, ⟨x_{1}, x_{2}⟩ • y_{2}⟩

for all x_{1}, x_{2} ∈ M and y_{1}, y_{2} ∈ N. We then denote by M ˙⊗_{A}N the corresponding Hilbert A-module (M ⊗N)_{R_{H}} and we say that M ˙⊗_{A}N is a interior tensor product of M and N over A. It is shown in [11] that our interior tensor product has similar properties as its Hilbert C^{∗}-module counterpart. Note only that a right Hilbert A-module is m-regular if and only if M ˙⊗_{A}A ≃ M via the standard isomorphism m ˙⊗_{A}a ↦ m ⋄ a.

We shall sometimes use C_{A} for some of the following categories: mreg − HMOD_{A}, mreg − Hilb_{A}. Similarly, we have the categories of left modules AMOD, _{A}HMOD, mreg −_{A}HMOD, mreg −_{A}Hilb).

In this paper we are concerned with functors between categories of modules. Such functors, e.g. F : mreg−Hilb_{A} → mreg−Hilb_{B} are assumed to be join-preserving (SUP-functors) on ⋁-semilattices of morphisms. Thus the map T ↦ F (T) from mreg − Hilb_{A}(X, W) to mreg − Hilb_{B}(F (X), F (W)) is join-preserving, for all pairs of objects X, W ∈ C_{A}.

In what follows let us assume that A and B are m-regular involutive quantales. We say that a SUP-functor

F : mreg − Hilb_{A} → mreg − Hilb_{B}

is a ∗-functor if F (T^{∗}) = F (T)^{∗} for all adjointable A-module maps T. In particular, from [11] we know that the function

(−) ˙⊗_{A}N : mreg − Hilb_{A} → mreg − Hilb_{B}

which assigns to each right Hilbert A-module M the inner tensor product M ˙⊗_{A}N 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 preserving biproducts.

We say two ∗-functors

F_{1}, F_{2} : mreg − Hilb_{A} → mreg − Hilb_{B}

are (naturally) unitarilly isomorphic, if they are naturally isomorphic via a natural transformation τ in the sense of category theory [1], with the natural transformations being unitaries i.e. τ_{F_{1}(M)}^{∗} ∘ τ_{F_{1}(M)} = id_{F (M_{1})} and τ_{F (M_{1})}∘τ_{F (M_{1})}^{∗} = id_{F (M_{2})}. In this case we write F_{1} ≅ F_{2} unitarilly. Similarly, we say that a ∗-functor F : mreg − Hilb_{A} → mreg − Hilb_{B} is a unitary equivalence functor if there is a ∗-functor G : mreg − Hilb_{B} → mreg − Hilb_{A} such that we have natural unitary isomorphisms η : GF → Id_{mreg−Hilb_{A}} and ζ : F G → Id_{mreg−Hilb_{B}}.

1. Hilbert modules

In this section we develop the basic properties of Hilbert modules not mentioned in [9]. Let us begin with a small observation about Hilbert modules.

Lemma 1.1. For any involutive quantale A ∈ HMOD_{A} and any right Hilbert A-module M, A is right separating for M.

Proof. Assume that m ⋄ (−) = n ⋄ (−) i.e. m ⋄ a = n ⋄ a for all a ∈ A. Hence, for all p ∈ M and all a ∈ A we have that ⟨p, m ⋄ a⟩ = ⟨p, n ⋄ a⟩ i.e. ⟨p, m⟩ · a = ⟨p, n⟩ · a i.e. ⟨p, m⟩ = ⟨p, n⟩. So we get that m = n. □

The preceding observation then yields

Corollary 1.2. Let A be an involutive quantale, A ∈ HMOD_{A}. Then a morphism f in C_{A} ∈ {mreg−HMOD_{A}, mreg−Hilb_{A}} is mono iff the map f is one-to-one.

Proof. Evidently, any injective morphism in C_{A} is a monomorphism. Conversely, let f : M → N be a monomorphism in C_{A}, f(m_{1}) = f(m_{2}). Then f ∘ (m_{1} ⋄ (−)) = f ∘ (m_{2} ⋄ (−)) i.e. m_{1} ⋄ (−) = m_{2} ⋄ (−). Hence m_{1} = m_{2} and f is injective. □

Note that in the category Hilb_{A} over A ∈ HMOD_{A} an adjointable map is mono iff f^{∗} is epi. This immediately yields that for any surjective adjointable map its adjoint is one-to-one.

Lemma 1.3. The category C_{A} ∈ {mreg − HMOD_{A}, mreg − Hilb_{A}} has biproducts.

Proof. We know from [9] that Hilb_{A} has biproducts (with injections i_{j} and projections π_{j}) and these are exactly cartesian products. The lemma follows then from the fact that evidently any cartesian product of m-regular modules is m-regular. □

As usual, for any A ∈ HMOD_{A}, A^{J} will be viewed as a Hilbert A-module equipped with the inner-product

⟨(a_{j})_{j∈J}, (b_{j})_{j∈J}⟩ = ⋁_{j∈J} a_{j}^{∗}b_{j}.

Let us observe that A^{J} will often stand for the set of column matrices over A of the type S × 1. In that way, the above inner-product can be expressed, for v = (a_{j})_{j∈J} and w = (b_{j})_{j∈J}, as ⟨v, w⟩ = v^{∗}w. Note that v^{∗} refers to the conjugate-transpose matrix. We shall denote, for any a ∈ A and all j ∈ J, by a_{j} an element of A^{J} defined as follows:

π_{k}(a_{j}) = \{ \begin{matrix}a & if k = j \\ 0 & otherwise. \end{matrix}

For each J-tuple μ = (m_{j})_{j∈J} in the Hilbert A-module M^{J}, we denote by Ω_{μ} the operator in A_{A}(A^{J}, M) defined by

Ω_{μ}((a_{j})_{j∈J}) = ⋁_{j∈J} m_{j} ⋄ a_{j}, (a_{j})_{j∈J} ∈ A^{J}.

It is easy to see that Ω_{μ}^{∗} is given by

Ω_{μ}^{∗}(x) = (⟨m_{j}, x⟩)_{j}, x ∈ M.

If ν = (n_{j})_{j∈J} is an J-tuple of elements of N, then the operator T = Ω_{ν} ∘Ω_{μ}^{∗} is in A_{A}(M, N). In particular, for all m ∈ M and n ∈ N, Ω_{(n)} = n^{∼}, Ω_{(m)}^{∗} = m^{⋆} and defining

Θ_{n,m}(x) = (Ω_{(n)} ∘ Ω_{(m)}^{∗})(x) = n ⋄ ⟨m, x⟨

we have

T (x) = ⋁_{j∈J} Θ_{n_{j},m_{j}}(x), x ∈ M.

Maps such as T are exactly compact operators in the sense of [9] 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 a composition of a compact operator 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 operators on Hilbert A-modules may be found in [9].

Lemma 1.4. Let A be an m-regular involutive quantale and let M ∈ mreg− HMOD_{A}, M is full. Then K_{A}(M) is an m-regular involutive quantale and M is an m-regular full left Hilbert K_{A}(M)-module.

Proof. Evidently, K_{A}(M) is an involutive quantale (see [9]). Let f, g ∈ K_{A}(M). Assume that f ∘ Θ_{y,x} = g ∘ Θ_{y,x} for all y, x ∈ X. This gives us Θ_{f(y),x} = Θ_{g(y),x} and since M is full then f(y) ⋄ a = g(y) ⋄ a for all a ∈ A i.e. f = g. Similarly,

Θ_{y,x} = Θ_{⋁_{i∈I} y_{i}⋄a_{i},x}
= ⋁_{i∈I} Θ_{y_{i}⋄a_{i},x}
= ⋁_{i∈I} Θ_{y_{i}⋄⋁_{j∈J} ⟨v_{ij},u_{ij}⟩,x}
= ⋁_{i∈I,j∈J} Θ_{y_{i}⋄⟨v_{ij},u_{ij}⟩,x}
= ⋁_{i∈I,j∈J} Θ_{y_{i},v_{ij}} ∘ Θ_{u_{ij},x}

for suitable elements y_{i}, u_{ij}, v_{ij}, i ∈ I, j ∈ J. This gives us that K_{A}(M) is m-regular.

We shall define the module action on • : K_{A}(M) × M → M by

f • m = f(m)

and the inner product _{K_{A}(M)}⟨−, −⟩ : M × M → K_{A}(M) by

K_{A}(M)⟨y, x⟩ = Θ_{y,x}.

Then evidently the pre-Hilbert module conditions are satisfied,

M = K_{A}(M) • M, _{K_{A}(M)}⟨M, M⟩ = K_{A}(M).

Assume that _{K_{A}(M)}⟨y_{1}, −⟩ = _{K_{A}(M)}⟨y_{2}, −⟩ i.e. y_{1} ⋄ a = y_{2} ⋄ a for all a ∈ A. Hence y_{1} = y_{2}. □

Proposition 1.5. Let A be an m-regular involutive quantale and let M ∈ mreg − HMOD_{A}. For each μ = (μ_{j})_{j∈J} in M^{J} one has that Ω_{μ} is in K_{A}(A^{J}, M) and hence also that Ω_{μ}^{∗} is in K_{A}(M, A^{J}). Moreover, K_{A}(A^{J}, M) ≃ M^{J} as sup-lattices.

Proof. It is obviously enough to consider the case |J| = 1. Let μ = (m) and ⋁_{λ∈Λ} m_{λ} ⋄ a_{λ} = m. Therefore we have for all a in A

Ω_{μ}(a) = m ⋄ a
= (⋁_{λ∈Λ} m_{λ} ⋄ a_{λ}) ⋄ a
= Ω_{(m_{λ})_{λ∈Λ}}Ω_{(a_{λ}^{∗})_{λ∈Λ}}^{∗}(a).

We have that

Ω_{(m)} ∘ Ω_{(a)}^{∗}(b) = Ω_{(m⋄a^{∗})}(b)

i.e. any generator of K_{A}(A^{J}, M) has the form Ω_{(m)}. Since Ω_{(m)} is compact for all m we have that

K_{A}(A^{J}, M) ≃ {Ω_{(m)} : m ∈ M} ≃ M.
□

Lemma 1.6. Let A be a unital involutive quantale and let M ∈ mreg − HMOD_{A}. Then HMOD_{A}(A^{J}, M) = K_{A}(A^{J}, M) ≃ M.

Proof. Let f : A^{J} → M be any module map, x = (x_{j})_{j∈J}. Then

f(x) = ⋁_{j∈J} f(e_{j}) ⋄ x_{j} = ⋁_{j∈J} (Ω_{f(e_{j})} ∘ i_{j} ∘ Ω_{e_{j}}^{∗}) (x).

Hence,

f = ⋁_{j∈J} Ω_{f(e_{j})} ∘ i_{j} ∘ Ω_{e_{j}}^{∗} ∈ K_{A}(A^{J}, M).

□

Corollary 1.7. Let A be a unital involutive quantale. Then id_{A^{J}} is in K_{A}(A^{J}).

Definition 1.8. Let A be an involutive quantale. A Hilbert A-module M will be said to be a nuclear module if the identity operator id_{M} is in K_{A}(M).

Note that any m-regular involutive quantale A ≃ K_{A}(A) is nuclear if and only if it is unital. We shall now give the complete characterization of nuclear Hilbert modules over unital involutive quantales.

Proposition 1.9. Let A be a unital involutive quantale and let M ∈ ess − HMOD_{A}. The following conditions are equivalent:

(1) M is nuclear.

(2) M is a retract of A^{J} in Hilb_{A}.

Proof. (1) ⟹ (2). Assume M to be nuclear. Then id_{M} = Ω_{ν}Ω_{μ}^{∗} where μ and ν are in M^{J}.

(2) ⟹ (1). Let M be a retract of A^{J} in Hilb_{A}, r : A^{J} → M the retraction and i : M → A^{J} the embedding such that r ∘ i = id_{M}. Since id_{A^{J}} ∈ K_{A}(A^{J}) we have that id_{M} = r ∘ id_{A^{J}} ∘ i ∈ K_{A}(M). □

Sometimes we shall need the following lemma, a part of which is contained in [9].

Lemma 1.10. Let A be an involutive quantale and let M be a left (right) pre-Hilbert A-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 A-module. Moreover, j_{H} : M → M_{R_{H}} is an adjointable map and if f : N → M is a module (adjointable) map then there is a unique module (adjointable) map \overline{f} : N → M_{R_{H}} such that \overline{f} = j_{H} ∘ f; here j_{H}(m) = ⋁{n ∈ M : (m, n) ∈ R_{H}}. Similarly, if g : M → P is an adjointable map then there is a unique adjointable map ĝ : M_{R_{H}} → P such that ĝ = g ∘ j_{H}^{∗}.

Proof. The main part of this lemma was proved in [9] and the adjointability of j_{H} follows from its definition. So let f : N → M be a module (adjointable) map. We shall put \overline{f}(n) = j_{H}(f(n)). Evidently, \overline{f} is a module map. Let us check that \overline{f} is adjointable. Assume n ∈ N, j_{H}(m) ∈ M_{R_{H}}. Then

⟨\overline{f}(n), j_{H}(m)⟩ = ⟨f(n), j_{H}(m)⟩ = ⟨n, f^{∗}(j_{H}(m))⟩ .

Clearly, such \overline{f} is uniquely determined.

Similarly, let g : M → P be an adjointable map. We define

ĝ(m) = g(j_{H}^{∗}(m)) = g(m).

Since ĝ is a composition of adjointable maps it is adjointable. The uniqueness is evident. □

Theorem 1.11. Let A be an involutive quantale. Then the category of pre-Hilbert A-modules has limits of arbitrary diagrams.

Proof. The proof follows general category theoretic principles. We describe the limit of the diagram

⟨(M_{i})_{i∈O}, (f_{j}: M_{d(j)} → M_{c(j)})_{j∈J}⟩

as a set of particular elements of the product of all M_{i}’s, the so-called commuting tuples.

M = {(x_{i})_{i∈O} ∈ ∏_{i∈O} M_{i} : ∀j ∈ J x_{c(j)} = f_{j}(x_{d(j)})}.

Evidently, M is a A-submodule of the product, that is, M is a pre-Hilbert A-module because the coordinatewise supremum of commuting tuples is commuting as all f_{j} are module maps. This also proves that the projections π_{j}: ∏_{i∈O} M_{i} → M_{j} restricted to M are module maps. They give us the maps needed to complement M to a cone.

Given any other cone ⟨E, (g_{i}: E → M_{i})_{i∈O}⟩, we define the mediating morphism h: E → M by h(x) = (g_{i}(x))_{i∈O}. Again, it is obvious that this is well-defined and a module map, and that it is the only possible choice. □

We also have the dual:

Theorem 1.12. Let A be an involutive quantale. Then the category of pre-Hilbert A-modules has colimits of arbitrary diagrams.

Definition 1.13. Let C be a subcategory of MOD_{A} and let M be a module in C.

(1) M is called faithful if m⋄a = m⋄b for all m ∈ M implies that a = b.

(2) M generates X ∈ C if there is a direct sum ∐_{γ∈Γ} M of copies of M and an epimorphism φ : ∐_{γ∈Γ} M → X in C. M is a generator for C, if M generates all modules X ∈ C.

(3) M cogenerates X ∈ C if there is a direct product ∏_{γ∈Γ} M of copies of M and a monomorphism ψ : X → ∏_{γ∈Γ} M in C. M is a cogenerator for C, if M cogenerates all modules X ∈ C.

Lemma 1.14. Let A be a right separating involutive quantale, M be a full right Hilbert A-module. Then M is faithful.

Proof. Let a, b ∈ A such that m⋄a = m⋄b for all m ∈ M. Then ⟨n, m ⋄ a⟩ = ⟨n, m ⋄ b⟩ for all m, n ∈ M i.e. ⟨n, m⟩ · a = ⟨n, m⟩ · b for all m, n ∈ M i.e. a = b. □

Lemma 1.15. Let A be an m-regular involutive quantale. Then A is both a generator and a cogenerator for C_{A} ∈ {mreg − HMOD_{A}, mreg − Hilb_{A}}. More exactly, for every M in C_{A}, M is both a quotient module and a submodule of the sum ⨆_{m∈M} A ≅ ∏_{m∈M} A = A^{M}.

Proof. We shall use the compact maps m^{∼} : A → M and m^{⋆} : M → A, m ∈ M.

[figure: A^{M}; p_{M}; M; i_{M}; A^{M}]

[figure: i_{m}; π_{m}]

A [figure: m^{∼}; m^{⋆}]

They give us compact maps p_{M} : A^{M} → M, p_{M} = Ω_{(m)_{m∈M}} and i_{M} : M → A^{M}, i_{M} = Ω_{(m)_{m∈M}}^{∗}; here π_{m} ∘ i_{M} = m^{⋆} and p_{M} ∘ ι_{m} = m^{∼}. Since M is a Hilbert A-module i_{M} is injective and since M is essential p_{M} is surjective. Namely, for any n ∈ M, there is an element u_{n} ∈ A^{M} defined by

p_{m}(u_{n}) = ⋁{a ∈ A : m ⋄ a ≤ n}

such that n = ⋁_{m∈M} m ⋄ p_{m}(u_{n}) = p_{M}(u_{n}). □

Note that the surjective module map p_{M} : A^{M} → M may be defined for any essential A-module M. Similarly as in [6] we have the following proposition.

Proposition 1.16. Let A be an m-regular involutive quantale. Then U ∈ mreg−HMOD_{A} (U ∈ mreg−Hilb_{A}) is faithful if and only if it cogenerates a generator.

Proof. Suppose that U is faithful. Then the module homomorphism (adjointable map) f : A → U^{U} defined by f(a)(u) = u ⋄ a is monic and U cogenerates the generator A. Conversely, let M be a generator for mreg − HMOD_{A} and U a cogenerator of M. Let u ⋄ a = u ⋄ b for all u ∈ U. Then m ⋄ a = m ⋄ b since M is embeddable into the product of copies of U. Since we have an epimorphism g : M^{J} → A such that a^{∼} ∘ g = b^{∼} ∘ g we obtain that c · a = c · b for all c ∈ A i.e. a = b. □

Proposition 1.17. Let A be an m-regular involutive quantale. Then, for every surjective module homomorphism p : P → M in mreg − HMOD_{A}, p is the coker of a pair (u, v) of compact arrows from the following diagram.

(1.1)  [figure: A^{J}; u; v; P; p; M.]

Proof. Note that we know e.g. from [5] or [7] that in the category MOD_{A} of modules over A the diagram

[figure: D; u′; v′; P; p; M.]

exists; here D = {(x, y) ∈ P × P : p(x) = p(y)} is a right A-module that is right separated, u′, v′ are the respective projections. Then p is the coker of the pair (u′, v′) in this category. Let us form the following diagram.

[figure: ess(D); in_{D}; D; u′; v′; P; p; M]

[figure: f]

Z\begin{matrix}g \\  \end{matrix}

Here Z is in mreg − HMOD_{A} and f is a module map from P to Z and g is defined by g(m) = f(x) for m = p(x). Let us show that our definition is correct. Let m ∈ M, m = p(x) = p(y). Then f(x) = f(y) iff f(x) ⋄ a = f(y) ⋄ a for all a ∈ A i.e. f(x ⋄ a) = f(y ⋄ a) for all a ∈ A. But the last condition evidently holds since (x, y) ∈ D gives us (x ⋄ a, y ⋄ a) ∈ ess(D) i.e. f(x ⋄ a) = f(y ⋄ a). Since ess(D) is m-regular it is a surjective image of A^{ess(D)} by the module map p_{ess(D)}. Then we have the diagram

[figure: A^{ess(D)}; p_{ess(D)}; ess(D); in_{D}; D; u′; v′; P; p; M.]

Evidently, p is the coker of the pair

(u, v) = (u′ ∘ in_{D} ∘ p_{ess(D)}, v′ ∘ in_{D} ∘ p_{ess(D)}

in the category of m-regular Hilbert A-modules. Recall that

u = ⋁_{(d_{1},d_{2})∈ess(D)} u′ ∘ in_{D} ∘ ((d_{1}, d_{2}) ⋄ (−)) = Ω_{(d_{1})_{(d_{1},d_{2})∈ess(D)}}

and similarly

v = Ω_{(d_{2})_{(d_{1},d_{2})∈ess(D)}}

i.e. by Lemma 1.5 u and v are compact. □

Lemma 1.18. Let A be an unital involutive quantale. Then mreg−HMOD_{A} and mreg − Hilb_{A} have free objects.

Proof. Let X be an arbitrary set. We put F_{A}(X) = A^{X}. Evidently, F_{A}(X) is an m-regular right Hilbert A-module. Let us show that it is free over X. Evidently, the set {e_{x} : x ∈ X} generates F_{A}(X) as a submodule and we have an inclusion ι_{X} : X → F_{A}(X) defined by x ↦ e_{x}. By standard considerations we can check that, for any map f : X → M, M being an m-regular right Hilbert A-module, there is a unique A-module map g : F_{A}(X) → M such that f = g ∘ i_{X}. Moreover, let us define a map h : M → F_{A}(X) by h(n) = (⟨f(x), n⟩)_{x∈X}. Then, for all (a_{x})_{x∈X} ∈ A^{X} and for all n ∈ M, we have

⟨g(a_{x})_{x∈X}, n⟩ = ⟨⋁_{x∈X} f(x) ⋄ a_{x}, n⟩
= ⋁_{x∈X} ⟨f(x) ⋄ a_{x}, n⟩
= ⋁_{x∈X} a_{x}^{∗} ⟨f(x), n⟩
= ⟨(a_{x})_{x∈X}, h(n)⟩ .

Then h = g^{∗} i.e. mreg−Hilb_{A} has free objects. Note that g = Ω_{(f(x))_{x∈X}}. □

We also have the following.

Lemma 1.19. Let A be an m-regular involutive quantale. Assume that M, N are m-regular right Hilbert A-modules and that f : M → N is an epimorphism in C_{A} ∈ {mreg − HMOD_{A}, mreg − Hilb_{A}}. Then f(M) is a right Hilbert A-module, f(M) ∈ C_{A} and the induced surjective map \overline{f} : M → f(M) is adjointable whenever f is. Moreover, we have an innerproduct preserving module embedding f̂ from f(M) to N such that f(M) separates elements of N.

Proof. Evidently, f(M) is a pre-Hilbert A-module and, whenever M is essential then also f(M) is essential.

Let us show that f(M) separates N. Let n_{1}, n_{2} ∈ N, n_{1} ≠ n_{2}. Then ⟨n_{1}, −⟩ ≠ ⟨n_{2}, −⟩ i.e. ⟨n_{1}, −⟩ ∘ f ≠ ⟨n_{2}, −⟩ ∘ f i.e. there is an element p ∈ M such that ⟨n_{1}, f(p⟩) ≠ ⟨n_{2}, f(p)⟩. In particular,

⟨f(m_{1}), −⟩_{f(M)} = ⟨f(m_{2}), −⟩_{f(M)} implies f(m_{1}) = f(m_{2})

i.e. f(M) is an m-regular right Hilbert A-module. Clearly, for an adjointable f, \overline{f} is adjointable since

⟨\overline{f}(m_{1}), f(m_{2})⟩ = ⟨f(m_{1}), f(m_{2})⟩ = ⟨m_{1}, f^{∗}(f(m_{2}))⟩ .

Note that the inclusion f̂ : f(M) → N, f(m) ↦ f(m) is evidently a module map preserving inner-product. □

Corollary 1.20. Let A be an m-regular involutive quantale. Assume that M, N are Hilbert A-modules and f : M → N is an epimorphism in C_{A} ∈ {mreg − HMOD_{A}, mreg − Hilb_{A}}. Then f is a surjective map iff f̂ is adjointable.

Proof. Evidently, if f is surjective then f̂ = id_{N}. Conversely, let f̂ be adjointable and assume that n ∉ f(M). Then, for any m ∈ M there is an element u_{m} ∈ M such that

⟨f(m), f(u_{m})⟩ ≠ ⟨n, f(u_{m})⟩ = ⟨n, f̂(f(u_{m}))⟩ = ⟨f̂^{∗}(n), f(u_{m})⟩ .

But f̂^{∗}(n) = f(m) for some m ∈ M, a contradiction. □

Corollary 1.21. Let A be an m-regular involutive quantale. Assume that M, N are m-regular right Hilbert A-modules and f is an epimorphism in C_{A} ∈ {mreg − HMOD_{A}, mreg − Hilb_{A}}. Then f is a surjective map iff f is a coker in C_{A}.

Proof. Evidently, if f is a surjective map so it is a coker of maps u and v by Proposition 1.17. Conversely, assume that f is a coker of u and v. Then we have the commutative diagram.

[figure: P; u; v; M; f; N]

[figure: \overline{f}; s; f̂]

f(M)

Here s : N → f(M) is the unique module (adjointable) map such that \overline{f} = s ∘ f. Then

f̂∘ (s ∘ f) = f̂∘ \overline{f} = f

i.e. f̂∘ s = id_{N}. Hence f̂ is onto i.e. f(M) = N. □

Definition 1.22. Let A be an m-regular involutive quantale, M, N ∈ C_{A}, C_{A} ∈ {mreg−HMOD_{A}, mreg−Hilb_{A}}. Assume that f : M → N is a morphism in C_{A}. We shall denote by _{f}C_{A} the category with objects all triples ⟨e, X, m⟩, e : M → X an epimorphism in C_{A}, m : N → X an injective

module map such that e = m ∘ f, and as arrows ⟨e, X, m⟩ \overline{\begin{matrix}g \\ \end{matrix}}⟨e′, X′, m′⟩ all module maps g : X → X′ such that e′ = g ∘ e and m′ = g ∘ m as maps. In pictures,

objects ⟨e, X, m⟩ : [figure: M; f; N; e; m; X];

arrows ⟨e, X, m⟩ \overline{\begin{matrix}g \\ \end{matrix}}⟨e′, X′, m′⟩ : [figure: M; f; N; e; e′; m; m′; X; X′; g],

with the diagram commutative.

Proposition 1.23. Let A be an m-regular involutive quantale, Y, Z ∈ C_{A}, C_{A} ∈ {mreg − HMOD_{A}, mreg − Hilb_{A}}. Assume that q : Y → Z is an epimorphism in C_{A}, \overline{q} : Y → q(Y) is the induced surjective module (adjointable) map and q̂: q(Y) → Z is the induced injective module map. Then q is surjective if and only if the triple ⟨q, Z, q̂⟩ is couniversal in \overline{q}C_{A}.

Proof. Let q be surjective and let ⟨e, X, m⟩ be in \overline{q}C_{A}. Then from the first diagram we have that ⟨q, Z, q̂⟩ is couniversal since q = \overline{q} and q̂ = id_{Z}. Conversely, suppose that ⟨q, Z, q̂⟩ is couniversal. From the second diagram we have that ⟨\overline{q}, q(Y), id_{q(Y)}⟩ is couniversal as well. Since there is an isomorphism h : Z → q(Y) such that \overline{q} = h ∘ q we have that q = h^{−1} ∘ \overline{q} is surjective.

[figure: Y; \overline{q}; q(Y) = Z; e; q; m; q̂ = id_{Z}; X; Z; m; Y; \overline{q}; q(Y); e; \overline{q}; m; id_{q(Y)}; X; q(Y); m]

□

Definition 1.24. Let A be an m-regular involutive quantale, M ∈ C_{A}, C_{A} ∈ {mreg−HMOD_{A}, mreg−Hilb_{A}}. Then M is called weakly projective in C_{A} if to every diagram in C_{A}

\begin{aligned}& & M \\ Y & \overline{q} & Z & \begin{matrix}φ \\ \end{matrix} \end{aligned}

such that q is a surjective morphism there is a morphism ψ : M → Y making the diagram commutative.

Proposition 1.25. Let A be an unital involutive quantale. Then A is weakly projective in C_{A}, C_{A} ∈ {mreg − HMOD_{A}, mreg − Hilb_{A}}.

Proof. Assume that we have the diagram from 1.24. Then there is an element y ∈ Y such that q(y) = φ(e). Define ψ : A → Y by ψ(a) = y ⋄ a. Then evidently ψ is an adjointable A-module map such that q ∘ ψ. □

Corollary 1.26. Let A be an unital involutive quantale. Then A^{J} is weakly projective in C_{A}, C_{A} ∈ {mreg − HMOD_{A}, mreg − Hilb_{A}} for any index set J.

Proposition 1.27. Let A be an unital involutive quantale, M ∈ C_{A}, C_{A} ∈ {mreg−HMOD_{A}, mreg−Hilb_{A}} Then M is weakly projective in C_{A} if and only if M is a retract of a free Hilbert A-module.

Proof. Assume that M is weakly projective. Since M is a surjective image of A^{J} by some q : A^{J} → M we have a morphism ψ : M → A^{J} (taking φ = id_{M}) such that id_{M} = q ∘ ψ. Conversely, let M be a retract of a free Hilbert Amodule A^{J}, r : A^{J} → M the retraction and i : M → A^{J} the embedding such that r ∘ i = id_{M}. Let Y, Z be Hilbert A-modules, q : Y → Z a surjective morphism, φ : M → Z a morphism. Since A^{J} is weakly projective there is a morphism ψ′ : A^{J} → Y such that q ∘ ψ′ = φ ∘ r. Let us put ψ = ψ′ ∘ i. Then

q ∘ ψ = q ∘ ψ′ ∘ i = φ ∘ r ∘ i = φ.
□

2. The Eilenberg-Watts and fundamental Morita theorems for Hilbert modules

Definition 2.1. We say that two m-regular involutive quantales A and B are Morita equivalent if there exist ∗-functors F : mreg−Hilb_{A} → mreg−Hilb_{B} and G : mreg − Hilb_{B} → mreg − Hilb_{A}, such that F G ≅ Id and GF ≅ Id unitarilly. Such F and G will be called equivalence functors.

Theorem 2.2 (Eilenberg-Watts theorem for m-regular involutive quantales). Let A and B be m-regular involutive quantales and let F be a colimitpreserving *-functor from mreg − Hilb_{A} to mreg − Hilb_{B}. Then there is a right Hilbert A − B bimodule Z such that F (−) is naturally unitarily isomorphic to the interior tensor product (−) ˙⊗_{A}Z. That is, there is a natural isomorphism between these functors, which implements a unitary isomorphism F (Y) ≅ Y ˙⊗_{A}Z for all Y ∈ mreg − Hilb_{A}.

Proof. Define Z = F (A). Then Z is an m-regular right Hilbert B-module. We make Z into a left A-module by defining a • z = F (L(a))(z), for a ∈ A, z ∈ Z. Here L(a) : A → A is the adjointable map b ↦ ab. Let us check that Z is a right Hilbert A − B bimodule. We have

⟨a • x, y⟩ = ⟨F (L(a))(x), y⟩
= ⟨x, F (L(a))^{∗}(y)⟩
= ⟨x, F (L(a)^{∗})(y)⟩
= ⟨x, F (L(a^{∗}))(y)⟩
= ⟨x, a^{∗} • y⟩ .

Now, we shall check that Z is essential with respect to •. Note that we know that the map p : A^{A} → A, p = ⋁_{a∈A} L(a) ∘ π_{a} is surjective i.e. it is a coker of adjointable maps in rm mreg−Hilb_{A}. Then F (p) : F (A)^{A} → F (A) is a coker in mreg − Hilb_{B} i.e.

F (p) = ⋁_{a∈A} F (L(a)) ∘ F (π_{a}) = ⋁_{a∈A} a • F (π_{a})

is surjective by Corollary 1.21.

It follows that, for all Y ∈ mreg − Hilb_{A}, the interior tensor product Y ˙⊗_{A}Z is well defined. We define an adjointable map τ_{Y} : Y ˙⊗_{A}Z → F (Y) by τ_{Y} (y ˙⊗_{A}z) = F (L(y))(z), where L(y) : A → Y is the map L(y)(a) = y ⋄a. Let us prove that τ_{Y} is a unitary natural transformation.

First note that for Y = A^{J} this is easy, in fact τ_{A^{J}} is the canonical unitary isomorphism from

A^{J} ˙⊗_{A}F (A) ≅ (A ˙⊗_{A}F (A))^{J} ≅ F (A)^{J} ≅ F (A^{J}).

This is because if y = (a_{j})_{j∈J} ∈ A^{J}, z ∈ F (A) then

(a_{j})_{j∈J} ˙⊗_{A}z ↦ (a_{j} ˙⊗_{A}z)_{j∈J} ↦ (a_{j} • z)_{j∈J}
= (F (L(a_{j}))(z))_{j∈J} ↦ ⋁_{j∈J} F (i_{j})(F (L(a_{j}))(z))
= ⋁_{j∈J} F (i_{j} ∘ L(a_{j}))(z)
= ⋁_{j∈J} F (L(i_{j}(a_{j})))(z)
= F (⋁_{j∈J} L(i_{j}(a_{j})))(z)
= F (L(⋁_{j∈J} i_{j}(a_{j})))(z)
= F (L(y))(z)
= τ_{Y} (y ˙⊗_{A}z).

Now fix Y ∈ mreg − Hilb_{A}. We have then the following commutative diagram for a suitable index set J such that p_{Y} is the coker of adjointable maps u, v.

(2.1)  [figure: A^{J}; u; v; A^{Y}; p_{Y}; Y]

Applying F, (−) ˙⊗_{A}Z and again F we get a diagram.

(2.2)

[figure: F (A^{J}); F (u); F (v); F (A^{Y}); F (p_{Y}); F (Y)]

[figure: A^{J} ˙⊗_{A}Z; τ_{A^{J}}^{∗}; u ˙⊗_{A}id_{Z}; v ˙⊗_{A}id_{Z}; A^{Y} ˙⊗_{A}Z; τ_{A^{Y}}^{∗}; p_{Y} ˙⊗_{A}id_{Z}; Y ˙⊗_{A}Z; τ_{Y}^{∗}]

[figure: F (A^{J}); τ_{A^{J}}; F (u); F (v); F (A^{Y}); τ_{A^{Y}}; F (p_{Y}); F (Y); τ_{Y}]

We shall first check that the diagram is commutative.

Note that

τ_{Y} (p_{Y} (w) ˙⊗_{A}z) = F (L(p_{Y} (w)))(z)
= F (p_{Y})F (L(w))(z)
= F (p_{Y})τ_{A^{Y}} (w ˙⊗_{A}z)

for w ∈ A^{Y}, z ∈ Z. Similarly, let y ∈ Y, w = (a_{x})_{x∈Y} ∈ A^{Y}, z, z′ ∈ Z. Then

⟨p_{Y} ˙⊗_{A}id_{Z}((a_{x})_{x∈Y} ˙⊗_{A}z′), y ˙⊗_{A}z⟩ = ⟨z′, ⟨p_{Y} (a_{x})_{x∈Y} , y⟩ • z⟩
= ⟨⟨y, p_{Y} (a_{x})_{x∈Y} ⟩ • z′, z⟩
= ⋁_{x∈Y} ⟨⟨y, x ⋄ a_{x}⟩ • z′, z⟩
= ⋁_{x∈Y} ⟨F (L(⟨y, x ⋄ a_{x}⟩))(z′), z⟩

and

⟨(τ_{Y}^{∗} ∘ F (p_{Y}) ∘ τ_{A^{Y}})((a_{x})_{x∈Y} ˙⊗_{A}z′), y ˙⊗_{A}z⟩
= ⟨(F (p_{Y}) ∘ τ_{A^{Y}})((a_{x})_{x∈Y} ˙⊗_{A}z′), τ_{Y} (y ˙⊗_{A}z)⟩
= ⟨(F (L(y)^{∗}) ∘ F (p_{Y}) ∘ τ_{A^{Y}})((a_{x})_{x∈Y} ˙⊗_{A}z′), z⟩
= ⟨⋁_{x∈Y} (F (L(y)^{∗} ∘ p_{Y} ∘ i_{x} ∘ L(a_{x}))(z′), z⟩
= ⋁_{x∈Y} ⟨(F ((⟨y, −⟩) ∘ (x ⋄ (−)) ∘ (a_{x} · (−)))(z′), z⟩
= ⋁_{x∈Y} ⟨(F (L(⟨y, x ⋄ a_{x}⟩))(z′), z⟩ .

Then, since the upper and lower horizontal lines of the diagram (2.2) produce the respective cokers F (p_{Y}), the composite right square of this diagram is a pushout in mreg − Hilb_{B}.

(2.3)  [figure: F (A^{Y}); F (p_{Y}); F (Y); F (A^{Y}); id_{F (A^{Y} )}; F (p_{Y}); F (Y); τ_{Y} ∘ τ_{Y}^{∗}]

i.e. τ_{Y} ∘ τ_{Y}^{∗} = id_{F (Y)}. Similarly, we have this commutative diagram.

(2.4)

A^{Y} ˙⊗_{A}Z \overline{p_{Y} ˙⊗_{A}id_{Z}} Y ˙⊗_{A}Z

[figure: A^{Y} ˙⊗_{A}Z; id_{A^{Y} ˙⊗_{A}Z}; p_{Y} ˙⊗_{A}id_{Z}; Y ˙⊗_{A}Z; id_{Y ˙⊗_{A}Z}; τ_{Y}^{∗} ∘ τ_{Y}]

Since p_{Y} ˙⊗_{A}id_{Z} is a surjective adjointable map it is an epimorphism and we have τ_{Y}^{∗} ∘ τ_{Y} = id_{Y ˙⊗_{A}Z}.

Let f : X → Y be a morphism in mreg − Hilb_{A}, x ∈ X and z ∈ Z. Then

(τ_{Y} ∘ (f ˙⊗_{A}id_{Z}))(x ˙⊗_{A}z) = F (L(f(x)))(z)
= F (f ∘ L(x))(z)
= F (f)(F (L(x))(z))
= (F (f) ∘ τ_{X})(x ˙⊗_{A}z)

and since τ_{X}, τ_{Y} are unitary also

τ_{Y}^{∗} ∘ F (f) = (f ˙⊗_{A}id_{Z}))τ_{X}^{∗}

i.e. τ_{Y} , τ_{Y}^{∗} are natural transformations. □

Lemma 2.3. Let A and B be m-regular involutive quantales and let X_{1} and X_{2} be essential right Hilbert A − B bimodules such that F_{1} = (−) ˙⊗_{A}X_{1} and F_{2} = (−) ˙⊗_{A}X_{2} are functors from mreg − Hilb_{A} to mreg − Hilb_{B}. Then F_{1} and F_{2} are equivalent if and only if X_{1} ≅ X_{2} unitarily and as bimodules.

Proof. Assume that α : F_{1} → F_{2} is a natural isomorphism such that α_{M} : F_{1}(M) → F_{2}(M) is a unitary map. Then we have the following diagram.

[figure: A ˙⊗_{A}X_{1}; α_{A}; A ˙⊗_{A}X_{1}; X_{1}; κ_{X_{1}}; h; X_{2}; χ_{X_{2}}]

Here χ_{X_{2}}(a ˙⊗_{A}x) = a • x, κ_{X_{1}} = χ_{X_{1}}^{∗} and h = χ_{X_{2}} ∘ α_{A} ∘ κ_{X_{1}}. Left multiplication in A are adjointable maps, so a natural transformation α_{A} has to preserve them. From [11] we know that κ_{X_{1}} and χ_{X_{2}} are unitary maps and bimodule homomorphisms. Hence h is unitary and bimodule homomorphism. The other direction is evident. □

The following theorem is an involutive quantale version of Morita’s fundamental theorem.

Theorem 2.4. Let A and B be m-regular involutive quantales. Then A and B are Morita-equivalent if and only if there are an essential right Hilbert A − B bimodule X and an essential right Hilbert B − A bimodule Y such that X ˙⊗_{B}Y ≅ A and Y ˙⊗_{A}X ≅ B as bimodules.

Proof. We have F (−) ≅ − ˙⊗_{A}X and G(−) ≅ − ˙⊗_{B}Y. Composing these two we obtain

A ≅ GF (A) ≅ A ˙⊗_{A}X ˙⊗_{B}Y ≅ X ˙⊗_{B}Y.

Similarly, Y ˙⊗_{A}X ≅ B, and this identification and the last are unitarily, and as bimodules, the latter exactly as in pure algebra [3].

Conversely, let λ : X ˙⊗_{B}Y → A be the unitary isomorphism, F (−) ≅ − ˙⊗_{A}X and G(−) ≅ − ˙⊗_{B}Y. Then

(GF)(M) ≅ M ˙⊗_{A}X ˙⊗_{B}Y.

Let us define

τ_{M} = π_{M} ∘ (id_{M} ˙⊗_{A}λ),

where π_{M} : M ˙ ⊗_{A}A → M is the canonical unitary isomorphism defined by m ˙ ⊗_{A}a ↦ m ⋄ a. We get a natural isomorphism τ : GF → Id_{mreg−Hilb_{A}}. By symmetry, A is Morita equivalent to B. □

Corollary 2.5. Let A and B be unital involutive quantales. Then A and B are Morita-equivalent if and only if their categories of weakly projective m-regular right Hilbert modules are equivalent.

Proof. It follows from the fact that weak projectivity is a categorical property. Hence weakly projective objects are mapped on weakly projective objects. Conversely, any equivalence between categories of weakly projective m-regular right Hilbert modules can be easily extended to an equivalence of m-regular right Hilbert modules. □

We recall that if A and B are m-regular involutive quantales then an imprimitivity Hilbert A − B bimodule is an A − B bimodule X, which is a full m-regular right Hilbert B-module, and also a full m-regular left Hilbert A-module, such that _{A}⟨x, y⟩z = x⟨y, z⟩_{B} whenever x, y, z ∈ X. The existence of such an X is the definition of A and B being strongly Morita equivalent. Our theorem gives a functorial characterization of such an X, and of strong Morita equivalence of A and B.

Theorem 2.6. Let A and B be m-regular involutive quantales. Suppose that F: mreg − Hilb_{A} → mreg − Hilb_{B}, G: mreg − Hilb_{B} → mreg − Hilb_{A} are equivalence *-functors, with F G ≅ Id and GF ≅ Id via unitary natural isomorphisms. Then A and B are strongly Morita equivalent. Moreover, the A − B bimodule X = F (A) from the Eilenberg-Watts theorem is an imprimitivity bimodule implementing the strong Morita equivalence. As in that theorem, F (−) ≅ − ˙⊗_{A}X naturally and unitarily. Conversely, any A−B imprimitivity bimodule X implements such a functorial isomorphism.

Proof. It follows from the fact that X is an imprimitivity A − B bimodule if and only if there is an essential right Hilbert B − A bimodule Y such that X ˙⊗_{B}Y ≅ A and Y ˙⊗_{A}X ≅ B as right-Hilbert bimodules (see [11]) and Theorem 2.4. □

Proposition 2.7. Let A, B and C be m-regular involutive quantales. Then the unitary isomorphism classes of equivalence functors mreg − Hilb_{A} → mreg−Hilb_{B} are in a 1-1 correspondence with the unitary equivalence classes of imprimitivity A−B bimodules. Composition of such functors corresponds to the interior tensor product of the bimodules.

Proof. Every imprimitivity A − B bimodule X gives rise to an equivalence (−) ˙⊗_{A}X : mreg − Hilb_{A} → mreg − Hilb_{B}, the isomorphism type depends only on the isomorphism type of X (as we saw in Lemma 2.3). Conversely, if F : mreg−Hilb_{A} → mreg−Hilb_{B} is an equivalence, F (A) is an imprimitivity A−B bimodule and its isomorphism type depends only on that of F. If Y is an imprimitivity B −C bimodule of an equivalence functor, the composition of the equivalences mreg − Hilb_{A} → mreg − Hilb_{B} → mreg − Hilb_{C} is given by (−) ˙⊗_{A}(X ˙⊗_{B}Y). In particular, X ˙⊗_{B}Y is an imprimitivity A − C bimodule. □

Corollary 2.8. Let A be an m-regular involutive quantale. Then the unitary isomorphism classes of self-equivalence functors mreg − Hilb_{A} → mreg − Hilb_{A} under a composition form a group isomorphic to the group of the unitary equivalence classes of imprimitivity A − A bimodules.

Corollary 2.9. Let A and B be m-regular involutive quantales. Then A and B are Morita-equivalent if and only if A^{d} and B^{d} are Morita-equivalent.

Proof. Let A and B are Morita-equivalent. Then we have an imprimitivity A − B bimodule X. Then Y with the same order and inner products as X and equipped with operations a •_{d} x = a^{∗} • x and x ⋄_{d} b = x ⋄ x^{∗} is evidently an imprimitivity A^{d} − B^{d} bimodule. □

Proposition 2.10. Let A be an m-regular involutive quantale. Then A is Morita equivalent to the matrix quantale M^{J}(A).

Proof. Note that M^{J}(A) ≅ K_{A}(A^{J}, A^{J}). □

Proposition 2.11. Let S be a sup-semilatticce with a duality. Then the involutive subquantale Q_{0}(S) of Q(S) that is generated by right-sided elements of Q(S) is Morita equivalent to the 2-element Boolean algebra 2. In particular, M^{J}(2) is Morita equivalent to 2.

Proof. Recall that Q_{0}(S) ≅ K_{2}(S, S). □

Lemma 2.12. Let E be an imprimitivity Hilbert B-A bimodule with A = K_{2}(S_{A}), S_{A} being a sup-lattice with a duality. Then there is a sup-lattice with a duality S_{B} such that B ≅ K_{2}(S_{B}) and E ≅ K_{2}(S_{A}, S_{B}).

Proof. We write ι for the identity representation of K_{2}(S_{A}) on S_{A} and put S_{B} = E ˙⊗_{A}S_{A}. Left multiplication in the first component of S_{B} then defines a faithful representation of B ≃ K_{A}(E) onto S_{B} i.e. B ≃ K_{2}(S_{B}) (see [11, Corollary 1.13]).

Now we shall define a mapping

Θ: E → A_{2}(S_{A}, S_{B}), x ↦ (s ↦ x ˙⊗_{A}s).

Namely,

⟨Θ(x)(s), y ˙⊗_{A}t⟩ = ⟨x ˙⊗_{A}s, y ˙⊗_{A}t⟩ = ⟨s, ⟨x, y⟩_{A} • t⟩

i.e.

Θ(x)^{∗}(y ˙⊗_{A}t) = ⟨x, y⟩_{A} • t.

We also observe that

Θ(x)^{∗}Θ(y) = ⟨x, y⟩_{A} .

If a ∈ A, then

Θ(x ⋄ a)(s) = (x ⋄ a) ˙⊗_{A}s = x ˙⊗_{A}a • s,

so that

Θ(E) = Θ(E ⋄ A) ⊆ Θ(E) ∘ K_{2}(S_{A}) ⊆ K_{2}(S_{A}, S_{B}).

Similarly,

Θ_{y ˙⊗_{A}t,s} = y ˙⊗_{A}t ⋄ ⟨s, −⟩ = Θ(y) ∘ Θ_{t,s}

i.e.

K_{2}(S_{A}, S_{B}) ⊆ Θ(E) ∘ K_{2}(S_{A}) ⊆ Θ(E).

So we conclude that

Θ(E) = K_{2}(S_{A}, S_{B}).

It is a straightforward computation that Θ is an isomorphism of Hilbert bimodules. □

Lemma 2.13. For any m-regular involutive quantale A,

A_{A}(A) ∩_{A}A(A) = HMOD_{A}(A) ∩_{A}HMOD(A).

Proof. It is enough to show that any bimodule endomorphism f : A → A is adjointable. Note that

⟨m, f(n)⟩ = m^{∗} · f(n)
= f(m^{∗} · n)
= f(m^{∗}) · n
= (f(m^{∗})^{∗})^{∗} · n
= ⟨f(m^{∗})^{∗}, n⟩ μ for all m, n ∈ A

i.e. f has an adjoint. □

Definition 2.14. Let A be an m-regular involutive quantale. The set of adjointable natural transformations from the identity functor Id_{mreg−Hilb_{A}} to itself is called ANat(A). It is a unital commutative involutive quantale with the composition of natural transformations as multiplication, with the adjoint of a natural transformation as involution

[figure: M; η_{M}; M; N; f; η_{N}; N; f] ⟼ [figure: M; η_{M}^{∗}; M; N; f^{∗}; η_{N}^{∗}; N; f^{∗}]

and with the join given by

[figure: M; η_{i}_{M}; M; N; f; η_{i}_{N}; N; f], i ∈ I ⟼ [figure: M; ⋁_{i} η_{i}_{M}; M; N; f; ⋁_{i} η_{i}_{N}; N; f].

The centre of A is the unital commutative involutive quantale Cen(A) = A_{A}(A) ∩_{A}A(A).

The elements of Cen(A) are by Lemma 2.13 A-bimodule endomorphisms on A. Note that evidently Cen(A) is commutative since, for all a ∈ A, a = ⋁_{j} a_{j} · b_{j}, f, g ∈ Cen(A), we have

(f ∘ g)(a) = (f ∘ g)(⋁_{j} a_{j} · b_{j})
= ⋁_{j} f(g(a_{j} · b_{j}))
= ⋁_{j} f(g(a_{j}) · b_{j})
= ⋁_{j} g(a_{j}) · f(b_{j})
= ⋁_{j} g(f(a_{j} · b_{j}))
= (g ∘ f)(⋁_{j} a_{j} · b_{j})
= (g ∘ f)(a).

Whenever A is commutative we can see A as a complete ^{∗}-ideal of Cen(A) by identifying an element a ∈ A with the bimodule endomorphism on A induced by multiplication by a.

The following theorem is based on the theorem 4.2 from [2] for involutive rings.

Theorem 2.15. Let A, B be Morita equivalent m-regular involutive quantales. Then

(1) Cen(A) and Cen(B) are isomorphic as involutive quantales.

(2) If A and B are commutative then A and B are isomorphic as involutive quantales.

Proof. (1). By Theorem 2.6 there is an A−B imprimitivity bimodule X implementing the Morita equivalence. We shall now define a map γ : Cen(A) →

246 JAN PASEKA

Cen(B) by the prescription

γ(f)(⋁_{i} ⟨x_{i}, r_{i} • y_{i}⟩_{B}) = ⋁_{i} ⟨x_{i}, f(r_{i}) • y_{i}⟩_{B}

for all x_{i}, y_{i} ∈ X and r_{i} ∈ A. Let us check that γ is well defined. Assume that

⋁_{i} ⟨x_{i}, r_{i} • y_{i}⟩_{B} = ⋁_{j} ⟨u_{j}, p_{j} • v_{j}⟩_{B} .

Let z, t ∈ X and r′ ∈ A. Then

⟨z, r′ • t⟩_{B} · (⋁_{i} ⟨x_{i}, f(r_{i}) • y_{i}⟩_{B}) = ⋁_{i} ⟨z, (r′ · ⟨t, x_{i}⟩_{A} · f(r_{i})) • y_{i}⟩_{B}
= ⋁_{i} ⟨z, (f(r′) · ⟨t, x_{i}⟩_{A} · r_{i}) • y_{i}⟩_{B}
= ⟨z, f(r′) • t⟩_{B} · (⋁_{i} ⟨x_{i}, r_{i} • y_{i}⟩_{B})
= ⟨z, f(r′) • t⟩_{B} · (⋁_{j} ⟨u_{j}, p_{j} • v_{j}⟩_{B})
= ⟨z, r′ • t⟩_{B} · (⋁_{j} ⟨u_{j}, f(p_{j}) • v_{j}⟩_{B}).

We show that γ(f) ∈ Cen(B). Let b_{j} ∈ B,

b_{j} = ⋁_{i∈I_{j}} ⟨x_{i}^{j}, r_{i}^{j} • y_{i}^{j}⟩_{B} .

Then

γ(f)(⋁_{j} b_{j}) = γ(f)(⋁_{j} ⋁_{i∈I_{j}} ⟨x_{i}^{j}, r_{i}^{j} • y_{i}^{j}⟩_{B})
= ⋁_{j} ⋁_{i∈I_{j}} ⟨x_{i}^{j}, f(r_{i}^{j}) • y_{i}^{j}⟩_{B}
= ⋁_{j} γ(f)(⋁_{i∈I_{j}} ⟨x_{i}^{j}, j_{i} • y_{i}^{j}⟩_{B})
= ⋁_{j} γ(f)(b_{j})

i.e. γ(f) is a sup-lattice homomorphism. Now, let b, c ∈ B,

b = ⋁_{i} ⟨x_{i}, r_{i} • y_{i}⟩_{B} .

Then

γ(f)(b · c) = γ(f)(⋁_{i} ⟨x_{i}, r_{i} • y_{i} ⋄ c⟩_{B})
= ⋁_{i} ⟨x_{i}, f(r_{i}) • y_{i} ⋄ c⟩_{B}
= ⋁_{i} ⟨x_{i}, f(r_{i}) • y_{i}⟩_{B} · c
= γ(f)(b) · c

and

γ(f)(c · b) = γ(f)(⋁_{i} ⟨x_{i} ⋄ c^{∗}, r_{i} • y_{i}⟩_{B})
= ⋁_{i} ⟨x_{i} ⋄ c^{∗}, f(r_{i}) • y_{i}⟩_{B}
= c · ⋁_{i} ⟨x_{i}, f(r_{i}) • y_{i}⟩_{B}
= c · γ(f)(b)

i.e. γ(f) is an bimodule endomorphism.

Let us check that γ is an involutive quantale homomorphism from Cen(A) to Cen(B). Let f, g, f_{j} ∈ Cen(A), j ∈ J. Then

γ(⋁_{j} f_{j})(⋁_{i} ⟨x_{i}, r_{i} • y_{i}⟩_{B}) = ⋁_{i} ⟨x_{i}, ⋁_{j} f_{j}(r_{i}) • y_{i}⟩_{B}
= ⋁_{j} ⋁_{i} ⟨x_{i}, f_{j}(r_{i}) • y_{i}⟩_{B}
= ⋁_{j} γ(f_{j})(⋁_{i} ⟨x_{i}, r_{i} • y_{i}⟩_{B}),
γ(f^{∗})(⋁_{i} ⟨x_{i}, r_{i} • y_{i}⟩_{B}) = ⋁_{i} ⟨x_{i}, f^{∗}(r_{i}) • y_{i}⟩_{B}
= ⋁_{i} ⟨x_{i}, f(r_{i}^{∗})^{∗} • y_{i}⟩_{B}
= ⋁_{i} ⟨f(r_{i}^{∗}) • x_{i}, y_{i}⟩_{B}
= (⋁_{i} ⟨y_{i}, f(r_{i}^{∗}) • x_{i}⟩_{B})^{∗}
= (γ(f)(⋁_{i} ⟨y_{i}, r_{i}^{∗} • x_{i}⟩_{B}))^{∗}
= γ(f)^{∗}(⋁_{i} ⟨y_{i}, r_{i}^{∗} • x_{i}⟩_{B}^{∗})
= γ(f)^{∗}(⋁_{i} ⟨r_{i}^{∗} • x_{i}, y_{i}⟩_{B})
= γ(f)^{∗}(⋁_{i} ⟨x_{i}, r_{i} • y_{i}⟩_{B})

and

γ(f ∘ g)(⋁_{i} ⟨x_{i}, r_{i} • y_{i}⟩_{B}) = ⋁_{i} ⟨x_{i}, (f ∘ g)(r_{i}) • y_{i}⟩_{B}
= ⋁_{i} ⟨x_{i}, f(g(r_{i}) • y_{i}⟩_{B}
= γ(f)(⋁_{i} ⟨x_{i}, g(r_{i}) • y_{i}⟩_{B})
= (γ(f) ∘ γ(g))(⋁_{i} ⟨x_{i}, ⋁_{j} r_{i} • y_{i}⟩_{B}).

Similarly, we have an involutive quantale homomorphism δ from Cen(B) to Cen(A) defined by

δ(g)(⋁_{i}_{A} ⟨x_{i} ⋄ s_{i}, y_{i}⟩) = ⋁_{i}_{A} ⟨x_{i} ⋄ g(s_{i}), y_{i}⟩

for g ∈ Cen(B) and x_{i}, y_{i} ∈ X, s_{i} ∈ B. One can easily check that δ is the inverse of γ i.e. γ provides an involutive quantale isomorphism from Cen(A) to Cen(B).

(2). Let us assume that A and B are commutative i.e. each of these involutive quantales can be viewed as a complete ^{∗}-ideal of its centre. It is enough to show that γ(A) = B and δ(B) = A. Assume that r ∈ A and x, y ∈ X. Then γ(r)(⟨x, y⟩_{B}) = ⟨x, r • y⟩_{B}. Since r = ⋁_{j} _{A}⟨z_{j} ⋄ s_{j}, t_{j}⟩ for suitable z_{j}, t_{j} ∈ X and s_{j} ∈ B and B is commutative we have

γ(r)(⟨x, y⟩_{B}) = ⋁_{j} ⟨x,_{A}⟨z_{j} ⋄ s_{j}, t_{j}⟩ • y⟩_{B}
= ⋁_{j} ⟨x, (z_{j} ⋄ s_{j}) ⟨t_{j}, y⟩_{B}⟩_{B}
= ⋁_{j} ⟨x, z_{j} ⋄ (s_{j} · ⟨t_{j}, y⟩_{B})⟩_{B}
= ⋁_{j} ⟨x, z_{j} ⋄ (⟨t_{j}, y⟩_{B} · s_{j})⟩_{B}
= ⋁_{j} ⟨x,_{A}⟨z_{j}, t_{j}⟩ • y⟩_{B} · s_{j}
= (⋁_{j} s_{j} · γ(_{A}⟨z_{j}, t_{j}⟩)) • ⟨x, y⟩_{B} .

Since B is an ideal in Cen(B) we have that

q = ⋁_{j} s_{j} · γ(_{A}⟨z_{j}, t_{j}⟩) ∈ B.

Hence γ(r) = q i.e. γ(A) ⊆ B and therefore also A ⊆ δ(B). Similarly δ(B) ⊆ A and B ⊆ γ(A). □

Lemma 2.16. Let A be an m-regular involutive quantale. Then Cen(A) and ANat(A) are isomorphic as involutive quantales.

Proof. Let ψ ∈ Cen(A) and let M ∈ m-reg−Hilb_{A}. Then, since A ≅ K_{A}(A) as involutive quantales M is a right Hilbert K_{A}(A)-module. In particular, we have a right action ⋄_{K_{A}(A)} : M × K_{A}(A) → M that gives rise to a nondegenerate involutive quantale homomorphism f : K_{A}(A) → A_{A}(M). Then we can find its unique extension g : A_{A}(A) → A_{A}(M) by Corollary 1.5 in [10] i.e. M ∈ MOD_{A_{A}(A)} with the right action ⋄_{A_{A}(A)} : M × A_{A}(A) → M. Moreover the map σ_{M} : M → M given by m ↦ m⋄_{A_{A}(A)}ψ is adjointable. Namely,

b · ⟨n, σ_{M}(m)⟩ · a = b · ⟨n, m ⋄ ψ⟩ · a
= b · ⟨n, (m ⋄ ψ) • a⟩
= b · ⟨n, m ⋄ ψ(a)⟩
= b · ⟨n, m⟩ · ψ(a)
= b · ψ(⟨n, m⟩ · a)
= ψ(b · ⟨n, m⟩) · a
= ψ(b) · ⟨n, m⟩ · a
= ⟨n ⋄ ψ(b)^{∗}, m⟩ · a
= ⟨n ⋄ ψ^{∗}(b^{∗}), m⟩ · a
= ⟨n ⋄ ψ^{∗} ∘ b^{∗}, m⟩ · a
= b · ⟨n ⋄ ψ^{∗}, m⟩ · a.

Therefore ⟨n, σ_{M}(m)⟩ = ⟨n ⋄ ψ^{∗}, m⟩ i.e. σ_{M} is adjointable. Hence the map Φ : Cen(A) → ANat(A) given by ψ ⟼ (σ_{M} : M → M, m ↦ m⋄_{A_{A}(A)}ψ) defines an adjointable natural transformation since any adjointable map preserves the right action. This is easily checked since

(h ∘ σ_{M})(m) ⋄ a = h(σ_{M}(m) ⋄ a)
= h(m ⋄ ψ ⋄ a)
= h(m ⋄ ψ(a))
= h(m) ⋄ ψ(a)
= h(m) ⋄ ψ ⋄ a
= σ_{M}(h(m)) ⋄ a
= (σ_{M} ∘ h)(m) ⋄ a

i.e. h ∘ σ_{M} = σ_{M} ∘ h. Conversely, if σ : Id_{mreg−Hilb_{A}} → Id_{mreg−Hilb_{A}} is an adjointable natural transformation then σ_{A} ∈ Cen(A). □

The following theorem provides a fully categorical proof of the first part of the Theorem 2.15.

Theorem 2.17. Let A, B be Morita equivalent m-regular involutive quantales. Then ANat(A) and ANat(B) are isomorphic as involutive quantales.

Proof. Let F : mreg − Hilb_{A} → mreg − Hilb_{B} be the unitary equivalence functor. Then there is a functor G : mreg−Hilb_{B} → mreg−Hilb_{A} such that we have natural unitary isomorphisms η : GF → Id_{mreg−Hilb_{A}} and ζ : F G → Id_{mreg−Hilb_{B}}. Similarly as in [6] let us define maps T : ANat(A) → ANat(B) and S : ANat(B) → ANat(A) by the prescription

σ ↦ (ζ_{N} ∘ F (σ_{G(N)}) ∘ ζ_{N}^{−1})_{N} and ρ ↦ (η_{M} ∘ G(ρ_{F (M)}) ∘ η_{M}^{−1})_{M}.

Note that since η_{M} is a unitary isomorphism we have that

σ_{GF (M)} = GF (σ_{M}) and σ_{GF G(N)} = GF (σ_{G(N)}).

So we have the following pair of commuting diagrams, the second diagram is an application of the functor F to the first diagram.

GF GF (M) \overline{\begin{matrix}GF (σ_{GF (M)}) \\  \end{matrix}}GF GF (M)

[figure: GF (M); G(ζ_{F (M)}); \begin{matrix}GF (σ_{M}) \\ σ_{GF (M)} \end{matrix}; GF (M); G(ζ_{F (M)})]

[figure: M; η_{M}; σ_{M}; M; η_{F M}]
F GF GF (M) \overline{\begin{matrix}F GF (σ_{F GF (M)}) \\  \end{matrix}}F GF GF (M)

[figure: F GF (M); F G(ζ_{F (M)}); \begin{matrix}F GF (σ_{M}) \\ F (σ_{GF (M)}) \end{matrix}; F GF (M); F G(ζ_{F (M)})]

[figure: F (M); F (η_{M}); F (σ_{M}); F (M); F (η_{F M})]

Hence, from the second diagram, we have

F (σ_{M}) ∘ F (η_{M}) ∘ F G(ζ_{F (M)}) = F (η_{M}) ∘ F G(ζ_{F (M)}) ∘ F GF (σ_{F GF (M)}).

In particular,

F (σ_{M}) = F (η_{M}) ∘ F G(ζ_{F (M)}) ∘ F GF (σ_{F GF (M)}) ∘ F G(ζ_{F (M)}^{−1}) ∘ F (η_{M}^{−1})

i.e.

F (σ_{M}) = F (η_{M}) ∘ F G(T (σ)_{F (M)}) ∘ F (η_{M}^{−1}).

Hence σ_{M} = ST (σ)_{M}. Similarly, ρ_{N} = T S(ρ)_{N} i.e. ST = id_{ANat(A)} and T S = id_{ANat(B)}. Note that we have the following commuting diagram

[figure: GF G(N); \begin{matrix}GF (σ_{G(N)}) \\ σ_{GF G(N)} \end{matrix}; GF G(N); G(N); G(ζ_{N}); \begin{matrix}σ_{G(N)} \\ G(ζ_{N}) ∘ GF (σ_{G(N)}) ∘ G(ζ_{N}^{−1}) \end{matrix}; G(N); G(ζ_{N})]

This gives us that

σ_{G(N)} = G(ζ_{N}) ∘ GF (σ_{G(N)}) ∘ G(ζ_{N}^{−1})

i.e. the diagram

[figure: G(N_{1}); \begin{matrix}σ_{G(N_{1})} \\ G(ζ_{N_{1}}) ∘ GF (σ_{G(N_{1})}) ∘ G(ζ_{N_{1}}^{−1}) \end{matrix}; G(N_{1}); G(N_{2}); G(f); \begin{matrix}σ_{G(N_{2})} \\ G(ζ_{N_{2}}) ∘ GF (σ_{G(N_{2})}) ∘ G(ζ_{N_{2}}^{−1}) \end{matrix}; G(N_{2}); G(f)]

commutes. Hence also the following diagram commutes

[figure: N_{1}; ζ_{N_{1}} ∘ F (σ_{G(N_{1})}) ∘ ζ_{N_{1}}^{−1}; N_{1}; N_{2}; f; ζ_{N_{2}} ∘ F (σ_{G(N_{2})}) ∘ ζ_{N_{2}}^{−1}; N_{2}; f]

i.e. T (σ) is a natural transformation and it is evidently adjointable. Similarly for S(ρ). Evidently, T and S are involutive quantale isomorphisms. □

Corollary 2.18. A unital involutive quantale is Morita equivalent to a commutative m-regular involutive quantale C if and only if it is equivalent to its own centre.

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[9] Jan Paseka, Hilbert Q-modules and nuclear ideals in the category of ⋁-semilattices with a duality, CTCS ’99: Conference on Category Theory and Computer Science (Edinburgh), Elsevier, Amsterdam, 1999, pp. Paper No. 29019, 19 pp. (electronic). MR 2001k:03143

[10] , Multiplier algebras of involutive quantales, Accepted in Proceedings of the Summer School on General Algebra and Ordered Sets.

ftp://www.math.muni.cz/pub/math/people/Paseka/papers, 1999.

[11] , Interior tensor product of Hilbert modules, Contributions to general algebra, 13 (Velké Karlovice, 1999/Dresden, 2000), Heyn, Klagenfurt, 2001, pp. 253–263. MR 1 854 589

[12] Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C^{∗}- algebras, American Mathematical Society, Providence, RI, 1998. MR 2000c:46108

[13] Kimmo I. Rosenthal, Quantales and their applications, Pitman Research Notes in Mathematics Series, no. 234, Longman Scientific & Technical, Harlow, 1990. MR 92e:06028

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

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