CONTRIBUTIONS TO GENERAL ALGEBRA 12

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

Verlag Johannes Heyn, Klagenfurt 2000

HERMITIAN KERNELS, HILBERT Q-MODULES AND ANDO DILATION

JAN PASEKA

Abstract. A theory of hermitian kernels in the context of Hilbert Qmodules is presented. Applications are given, including an extension of the generalized Naimark dilation theorem and the KSGNS construction for Hilbert Q-modules and dilations of isometric representations.

In his influential paper [3], P. Halmos proved that every contraction on a Hilbert space has a unitary dilation. This is later generalized by Sz.-Nagy to his celebrated power dilation theorem: for every contraction T on a Hilbert space H, there is a unitary operator U on a Hilbert space H′ and an isometry V from H to H′ such that every power T^{n} of T dilates to the unitary U^{n} i.e. the Sz.-Nagy dilation applies not just to T, but to the discrete semigroup generated by T. W. Stinespring [16] has proved that any completely positive map P of a space of bounded linear operators B(H) on a Hilbert space H into itself can be represented in the form P (x) = V^{∗}∘π(x)∘V for all x ∈ B(H); here V : H → H′ is a bounded linear operator and π is a representation of B(H) on some Hilbert space H′. The method used to produce a representation from a completely positive map is called the Gelfand-Naimark-Segal construction, or GNS construction, and provides one of the basic tools of C^{∗}-algebra theory (see [4]). The concept of a quantale was introduced in [7] as ”a quantization” of the term locale.

In this paper we present analogues of the above theorems in the fully algebraic context of Hilbert Q-modules.

The organization of the paper is as follows. The first part of the paper recalls the notion of a Hilbert Q-module and of an adjointable map. Further, we introduce the notion of a hermitian kernel playing in the context of Hilbert Q-modules similar role as a positive kernel does for Hilbert C^{∗}-modules. We prove a lattice-theoretical version of the Kolmogorov decomposition theorem for Hilbert Q-modules. Using this result, we show first that a hermitian kernel has a minimal Naimark dilation if and only if it is a Toeplitz kernel and secondly we obtain a concrete representation of Hilbert Q-modules.

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

In the second part of the paper we show that for each isometric adjointable representation on a Hilbert Q-module M of an Ore semigroup we can find a unitary representation on a Hilbert Q-module M′ of the corresponding group and containing M as a Hilbert Q-submodule. Applying this result we obtain an analogue of the celebrated theorem of Ando [1]. Our version of the theorem is also in a sense more general. Moreover, we obtain the KSGNS construction for Hilbert Q-modules and involutive quantales and the GNS construction for ⋁-semilattices and arbitrary quantales. Subsequently, in the last part of the paper we make some concluding remarks.

The paper is closely related to the papers [9], [11] and [13] where the interested reader can find unexplained terms, notation and motivation concerning the subject. For facts concerning quantales in general we refer to [15]. We refer the reader to [4] and [6] for missing backround of C^{∗}-algebras and Hilbert C^{∗}-modules. The monograph [18] will be our reference for the terminologies and results of the Sz.-Nagy-Foias contraction theory.

1. Hermitian kernels and Hilbert Q-modules

As we shall be studying hermitian kernels in the context of Hilbert Qmodule theory, we recall now some basic terminology, notation and results of that theory. The theory of Hilbert Q-modules (we refer the reader to [13] for details and examples) is a generalization of the theory of complete semilattices with a duality and it studies modules over an involutive quantale Q endowed with Q-valued inner products. Throughout this section, Q will be an involutive quantale (almost always unital).

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)

HERMITIAN KERNELS, HILBERT Q-MODULES AND ANDO DILATION 319

⟨m, m⟩ = 0 implies m = 0. (6)

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

⟨f(m), n⟩ = ⟨m, f^{∗}(n)⟩

for all m ∈ M, n ∈ N. Automatically, f is then a Q-module homomorphism i.e. f ∈ H_{Q}(M, N).

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

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).

Let f : M → N be a map between pre-Hilbert Q-modules M, N. 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})⟩. Any inner-product preserving map between Hilbert Q-modules is injective. We say that f is isometric if f is an inner-product preserving module homomorphism. Note that if f is a isometric map between Hilbert Q-modules then f may not be adjointable.

Let f : M → N be an adjointable map. Then f is said to be unitary if f^{∗} ∘ f = id_{M} and f ∘ f^{∗} = id_{N}. In this case f is an isometric surjective Q-module homomorphism. Conversely, if f has these properties, it is unitary. If a unitary map from M to N exists, then M and N are said to be unitarily equivalent. A subset P ⊆ M is said to be a Hilbert Q-submodule of M if P is a Q-submodule of M and a Hilbert Q-module w.r.t. the induced inner-product on P.

Definition 1.1. Let T be an involutive semigroup and let Q, Q′ be involutive quantales, M, M′, M_{v} Hilbert Q-modules, S an arbitrary non-empty set. A map k : S × S → Q′ is said to be a hermitian kernel if k(s, t)^{∗} = k(t, s), for all s, t ∈ S i.e. k is a hermitian matrix over Q′.

Let f : T → Q′ be a map. We shall say that f is hermitian if the map k_{f} : T × T → Q′ defined by k_{f}(a, b) = f(a^{∗} · b), for all a, b ∈ T is a hermitian kernel.

We shall say that f : Q → Q′ is completely hermitian if it is hermitian ⋁-preserving map.

Let φ ∈ A_{Q}(M), ψ ∈ A_{Q}(M′). The operator ψ is said to dilate the operator φ if there is an isometry w : M → M′ such that φ = w^{∗} ∘ ψ ∘ w. We say that ψ is a dilation of φ.

A map v : S → A_{Q}(M, M_{v}) is said to be a Kolmogorov decomposition for a kernel k, k : S × S → A_{Q}(M) if k(s, t) = v(s)^{∗}v(t), for all s, t ∈ S. Moreover, if M_{v} is a strict Hilbert Q-module, we shall say that v is a strict Kolmogorov decomposition for a kernel k. If any p ∈ M_{v} is of the form p = ⋁{v(s_{i})(m_{i}) : s_{i} ∈ S, m_{i} ∈ M, i ∈ I}, then v will be said to be minimal.

Note that, for a unital involutive quantale Q, a map f : Q → Q′ is completely hermitian if and only if it is a ^{∗}- and ⋁-preserving map.

Every hermitian kernel with values in A_{Q}(M) has an essentially unique minimal Kolmogorov decomposition. This is the content of the following result. The proof follows similar ideas as the proof for Hilbert C^{∗}-modules in [9].

Theorem 1.2. Let Q be an involutive quantale, M a Hilbert Q-module, S an arbitrary non-empty set and k a hermitian kernel from S × S to A_{Q}(M). Then there exists a minimal Kolmogorov decomposition for k. Moreover, if v : S → A_{Q}(M, M_{v}) and w : S → A_{Q}(M, M_{w}) are any two such minimal Kolmogorov decompositions, then there exists a unique unitary U : M_{v} → M_{w} such that U ∘ v(s) = w(s) for all a ∈ S.

Proof. First, let us prove that M^{S} is a pre-Hilbert Q-module. Evidently, it is a Q-module. We shall endow M^{S} with a pre-inner product ⟨−, −⟩_{S} : M^{S} × M^{S} → Q by setting

⟨f, g⟩_{S} = ⋁_{s,t∈S} ⟨k(s, t)(f(t)), g(s)⟩ . (7)

Now, let us prove the property (1). We have, for all a ∈ Q and all f, g : S → M,

a · ⟨f, g⟩_{S}=a · ⋁_{s,t∈S} ⟨k(s, t)(f(t)), g(s)⟩ = ⋁_{s,t∈S} a · ⟨k(s, t)(f(t)), g(s)⟩
=⋁_{s,t∈S} ⟨k(s, t)(a • f(t)), g(s)⟩ = ⋁_{s,t∈S} ⟨k(s, t)((a • f)(t)), g(s)⟩
=⟨a • f, g⟩_{S} .

Let us check the property (4) from the definition of pre-inner product. For all f, g : S → M, we have

⟨f, g⟩_{S} = ⋁_{s,t∈S} ⟨k(s, t)(f(t)), g(s)⟩ = ⋁_{s,t∈S} ⟨f(t), k(s, t)^{∗}(g(s))⟩
= ⋁_{s,t∈S} ⟨f(t), k(t, s)(g(s))⟩ = ⋁_{s,t∈S} ⟨k(t, s)(g(s)), f(t)⟩^{∗}
= (⋁_{s,t∈S} ⟨k(t, s)(g(s)), f(t)⟩)^{∗} = ⟨g, f⟩_{S}^{∗} .

We shall prove the property (3) (note that the property (2) follows from the properties (3) and (4)). We have, for all f, g_{i} : S → M, i ∈ I,

⋁_{i∈I} ⟨f, g_{i}⟩_{S} = ⋁_{i∈I} ⋁_{s,t∈S} ⟨k(s, t)(f(t)), g_{i}(s)⟩
= ⋁_{s,t∈S} ⟨k(s, t)(f(t)), ⋁_{i∈I} g_{i}(s)⟩
= ⋁_{s,t∈S} ⟨k(s, t)(f(t)), (⋁_{i∈I} g_{i})(s)⟩
= ⟨f, ⋁_{i∈I} g_{i}⟩_{S} .

Hence, M^{S} is a pre-Hilbert Q-module. For any map f : S → M, we shall define a map K(f) : S → M by setting K(f)(s) = ⋁_{t∈S} k(s, t)(f(t)) and we shall denote by M_{v} the set of all these maps K(f). Evidently, M_{v} is a Q-submodule of the Q−module M^{S}. Namely, if K(f_{i}) ∈ M_{v}, i ∈ I, then (⋁_{i∈I} K(f_{i}))(s) = ⋁_{i∈I,t∈S} k(s, t)(f_{i}(t)) = ⋁_{t∈S} k(s, t)((⋁_{i∈I} f_{i})(t)) = K(⋁_{i∈I} f_{i})(s), for all s ∈ S. Similarly, for all a ∈ Q, K(f) ∈ M_{v}, s ∈ S, we have

\begin{aligned}(a • K(f))(s) & = & ⋁_{t∈S} a • k(s, t)(f(t)) = ⋁_{t∈S} k(s, t)(a • f(t)) \\ & = & ⋁_{t∈S} k(s, t)((a • f)(t)) = K(a • f)(s). \end{aligned}

Moreover, we may endow M_{v} with an inner-product by setting

⟨K(f), K(g)⟩_{v} = ⟨f, g⟩_{S} . (8)

To prove the correctness of the definition (8) we shall check that, for all f_{1}, f_{2}, g_{1}, g_{2} : S → M such that K(f_{1}) = K(f_{2}) and K(g_{1}) = K(g_{2}), we have

⟨K(f_{1}), K(g_{1})⟩_{v} = ⋁_{s,t∈S} ⟨k(s, t)(f_{1}(t)), g_{1}(s)⟩
= ⋁_{s∈S} ⟨⋁_{t∈T} k(s, t)(f_{1}(t)), g_{1}(s)⟩
= ⋁_{s∈S} ⟨K(f_{1})(s), g_{1}(s)⟩ = ⋁_{s∈S} ⟨K(f_{2})(s), g_{1}(s)⟩ =
= ⟨K(f_{2}), K(g_{1})⟩_{v} = ⟨K(g_{1}), K(f_{2})⟩_{v}^{∗}
= ⟨K(g_{2}), K(f_{2})⟩_{v}^{∗} = ⟨K(f_{2}), K(g_{2})⟩_{v} .

Moreover, we have that M_{v} is a pre-Hilbert Q-module. Let us check the last property (5). First of all, we shall define, for all m ∈ M and all t ∈ S, a map e_{m}^{t} : S → M as follows:

e_{m}^{t}(s) = \{ \begin{matrix}m & if t = s, \\ 0 & otherwise \end{matrix}

for all s ∈ S. Let us compute K(e_{m}^{t}). We have K(e_{m}^{t})(s) = ⋁_{t′∈S} k(s, t′)e_{m}^{t}(t′) = k(s, t)(m). Then K(e_{m}^{t}) = k(−, t)(m).

Let g_{1}, g_{2} : S → M be arbitrary, fixed mappings and let us assume that, for all maps f : S → M, we have ⟨K(f), K(g_{1})⟩_{v} = ⟨K(f), K(g_{2})⟩_{v}. Then, for all t ∈ S and all m ∈ M, we have

⟨K(e_{m}^{t}), K(g_{1})⟩_{v} = ⋁_{s∈S} ⟨K(e_{m}^{t})(s), g_{1}(s)⟩ = ⋁_{s∈S} ⟨k(s, t)(m), g_{1}(s)⟩
= ⋁_{s∈S} ⟨m, k(s, t)^{∗}(g_{1}(s))⟩ = ⋁_{s∈S} ⟨m, k(t, s)(g_{1}(s))⟩
= ⟨m, ⋁_{s∈S} k(t, s)(g_{1}(s))⟩ = ⟨m, K(g_{1})(t)⟩ .

Similarly, for all t ∈ S and all m ∈ M, we have

⟨K(e_{m}^{t}), K(g_{2})⟩_{v} = ⟨m, K(g_{2})(t)⟩ .

Then, we have that ⟨m, K(g_{1})(t)⟩ = ⟨m, K(g_{2})(t)⟩ for all m ∈ M i.e. K(g_{1})(t) = K(g_{2})(t) i.e. K(g_{1}) = K(g_{2}). Thus, M_{v} is a Hilbert Q-module.

Now, let us find a minimal Kolmogorov decomposition v for k. For all s ∈ S, we shall define maps v(s) : M → M_{v} and u(s) : M_{v} → M as follows:

v(s)(m) = K(e_{m}^{s}) = k(−, s)(m) and u(s)(K(f)) = K(f)(s),

for each m ∈ M and all maps f : S → M. Then we have, for each m ∈ M and all maps f : S → M,

⟨m, u(s)(K(f))⟩ = ⟨m, K(f)(s)⟩ = ⟨K(e_{m}^{s}), K(f)⟩_{v} = ⟨v(s)(m), K(f)⟩_{v}

i.e. v(s) and u(s) are adjointable. Moreover, if s, t ∈ S and m, n ∈ M, then

\begin{aligned}⟨(v(s)^{∗} ∘ v(t))(m), n⟩ & = & ⟨v(t)(m), v(s)(n)⟩_{v} = ⟨K(e_{m}^{t}), K(e_{n}^{s})⟩_{v} \\ & = & ⋁_{s′∈S} ⟨k(s′, t)(m), K(e_{n}^{s})(s′)⟩ = ⟨k(s, t)(m), n⟩ \end{aligned}

i.e. k(s, t)(m) = (v(s)^{∗} ∘ v(t))(m) i.e. k(s, t) = v(s)^{∗} ∘ v(t). To prove the minimality of v note that any map f : S → M can be written as f = ⋁{e_{m}^{t} : t ∈ S, m = f(t)}. Since K(e^{t}(m)) = v(t)(m) we have that K(f) = K(⋁{e_{m}^{t} : t ∈ S, m = f(t)}) = ⋁{K(e_{m}^{t}) : t ∈ S, m = f(t)} = ⋁{v(t)(m) : t ∈ S, m = f(t)}.

Suppose now that v : S → A_{Q}(M, M_{v}) and w : S → A_{Q}(M, M_{w}) are any two minimal Kolmogorov decompositions for k. We shall define a map U : M_{v} → M_{w} as follows: U(⋁_{i∈I} v(s_{i})(m_{i})) = ⋁_{i∈I} w(s_{i})(m_{i}) for all s ∈ S and all m ∈ M. First, let us check the correctness of U. Assume that ⋁_{i∈I} v(s_{i})(m_{i}) = ⋁_{j∈J} v(t_{j})(n_{j}), s_{i}, t_{j} ∈ S, m_{i}, n_{j} ∈ M, i ∈ I, j ∈ J. Then we have

⟨⋁_{i∈I} w(s_{i})(m_{i}), w(s)(m)⟩_{w} = ⟨w(s)^{∗}(⋁_{i∈I} w(s_{i})(m_{i})), m⟩ =
⟨⋁_{i∈I} w(s)^{∗}(w(s_{i})(m_{i})), m⟩ = ⟨⋁_{i∈I} k(s, s_{i})(m_{i}), m⟩ =
⟨⋁_{i∈I} v(s)^{∗}(v(s_{i})(m_{i})), m⟩ = ⟨⋁_{i∈I} v(s_{i})(m_{i}), v(s)(m)⟩_{v} =
⟨⋁_{j∈J} v(t_{j})(n_{j}), v(s)(m)⟩_{v} = ⟨⋁_{j∈J} v(s)^{∗}(v(t_{j})(n_{j})), m⟩ =
⟨⋁_{j∈J} k(s, t_{j})(n_{j}), m⟩ = ⟨⋁_{j∈J} w(s)^{∗}(w(t_{j})(n_{j})), m⟩ =
⟨⋁_{j∈J} w(t_{j})(n_{j}), w(s)(m)⟩_{w} ,

for all s ∈ S and all m ∈ M. This implies ⋁_{i∈I} w(s_{i})(m_{i}) = ⋁_{j∈J} w(t_{j})(n_{j}). Similarly, we have a map U′ : M_{w} → M_{v} defined as follows: U′(⋁_{i∈I} w(s_{i})(m_{i})) = ⋁_{i∈I} v(s_{i})(m_{i}) for all s ∈ S and all m ∈ M. Clearly, U and U′ are inverse to each other. Now, we have

⟨U(⋁_{i∈I} v(s_{i})(m_{i})), ⋁_{j∈J} w(t_{j})(n_{j})⟩_{w} = ⟨⋁_{i∈I} w(s_{i})(m_{i}), ⋁_{j∈J} w(t_{j})(n_{j})⟩_{w} = ⋁_{i∈I,j∈J} ⟨w(s_{i})(m_{i}), w(t_{j})(n_{j})⟩_{w} = ⋁_{i∈I,j∈J} ⟨w(t_{j})^{∗}(w(s_{i})(m_{i})), n_{j}⟩ = ⋁_{i∈I,j∈J} ⟨k(t_{j}, s_{i})(m_{i}), n_{j}⟩ = ⋁_{i∈I,j∈J} ⟨m_{i}, k(s_{i}, t_{j})(n_{j})⟩ = ⋁_{i∈I,j∈J} ⟨m_{i}, v(s_{i})^{∗}(v(t_{j})(n_{j}))⟩ = ⋁_{i∈I,j∈J} ⟨v(s_{i})(m_{i}), v(t_{j})(n_{j})⟩_{v} =

⟨⋁_{i∈I} v(s_{i})(m_{i}), ⋁_{j∈J} v(t_{j})(n_{j})⟩_{v} = ⟨⋁_{i∈I} v(s_{i})(m_{i}), U′(⋁_{j∈J} w(t_{j})(n_{j})⟩_{v} ,

for all s_{i}, t_{j} ∈ S, m_{i}, n_{j} ∈ M, i ∈ I, j ∈ J. Hence, U is an adjointable map with U^{∗} = U′ = U^{−1}. Thus, U is a unitary map. Also, U(v(s)) = w(s), for all s ∈ S.

Theorem 1.3. Let Q be an involutive quantale, M a strict Hilbert Q-module, S an arbitrary non-empty set and k a hermitian kernel from S ×S to A_{Q}(M). Then there exists a minimal strict Kolmogorov decomposition for k. Moreover, if v : S → A_{Q}(M, M_{v}) and w : S → A_{Q}(M, M_{w}) are any two such minimal strict Kolmogorov decompositions, then there exists a unique unitary U : M_{v} → M_{w} such that U ∘ v(s) = w(s) for all s ∈ S.

Proof. It is enough to check that the inner-product ⟨−, −⟩_{v} defined in the proof of theorem 1.2 satisfies the condition 6. Let f : S → M be any map. Then ⟨K(f), K(f)⟩ = 0 implies

⟨K(f), K(f)⟩_{v}=⋁_{s,t∈S} ⟨k(s, t)(f(t)), f(s)⟩ = ⋁_{s,t∈S} ⟨v(s)^{∗}(v(t)(f(t))), f(s)⟩ =

⋁_{s,t∈S} ⟨v(t)(f(t)), v(s)(f(s))⟩ = ⟨⋁_{t∈S} v(t)(f(t)), ⋁_{s∈S} v(s)(f(s))⟩

=⟨⋁_{t∈S} v(t)(f(t)), ⋁_{t∈S} v(t)(f(t))⟩ = 0.

Since M is strict we have that ⋁_{t∈S} v(t)(f(t)) = 0 i.e. v(s)^{∗}(⋁_{t∈S} v(t)(f(t))) = 0 for all s ∈ S i.e. K(f)(s) = ⋁_{t∈S} k(s, t)(f(t)) = ⋁_{t∈S} v^{∗}(s)(v(t)(f(t))) = 0 for all s ∈ S i.e. K(f) = 0.

The following definition was motivated by the paper [14].

Definition 1.4. Let S be a unital semigroup with unit e, Q an involutive quantale, M a Hilbert Q-module, k : S × S → A_{Q}(M, M) a hermitian kernel such that

k(s, t) = k(cs, ct)

for all c, s, t ∈ S. Then k is said to be a Toeplitz kernel. We say that a Toeplitz kernel is unital if k(e, e) = id_{M}.

We say that a hermitian kernel k : S × S → A_{Q}(M, M) admits a (unital) Naimark dilation if there is a Hilbert Q-module M_{v}, an adjointable map (an isometry) w ∈ A_{Q}(M, M_{v}) and a semigroup {U_{s}}_{s∈S} of isometries on M_{v} that dilates k, i.e.

• U_{s} ∘ U_{t} = U_{st},

• U_{e} = id_{M_{v}},

• k(s, t) = (U_{s} ∘ w)^{∗} ∘ U_{t} ∘ w and U_{s} ∈ A_{Q}(M_{v}, U_{s}(M_{v})) for all s, t ∈ S.

The Naimark dilation is called strict if M_{v} is a strict Hilbert Q-module, minimal if M_{v} = {⋁_{s∈T,m∈N} U_{s}(w(m)) : T ⊆ S, N ⊆ M}, adjointable if U_{s} ∈ A_{Q}(M_{v}, M_{v}) for any s ∈ S and unitary if U_{s} is unitary for each s ∈ S.

Theorem 1.5. Let S be a unital semigroup with unit e, Q an involutive quantale, M a Hilbert Q-module, k : S ×S → A_{Q}(M, M) a hermitian kernel. Then k is a (unital) Toeplitz kernel if and only if it admits a minimal (unital) Naimark dilation. In this case the minimal Naimark dilation is unique up to an isomorphism. Moreover, if zS = S for any z ∈ S then the Naimark dilation is adjointable and unitary.

Proof. Let v be a minimal Kolmogorov decomposition for k, z ∈ S arbitrary, fixed. Let x, y ∈ S. Then

v(zx)^{∗} ∘ v(zy) = k(zx, zy) = k(x, y) = v(x)^{∗} ∘ v(y).

Let us put v_{z}(x) = v(zx). Then v_{z} : S → A_{Q}(M, M_{v}) is another Kolmogorov decomposition for k.

Let us define a map V_{z} : M^{S} → M^{S} by the prescription

V_{z}(f)(x) = ⋁{f(s) : zs = x}

for all x ∈ S. Note that V_{z}(M^{S}) = {f ∈ M^{S} : f(s) = 0 for any s ∉ zS}. We shall show that V_{z} is a module homomorphism. Let F ⊆ M^{S}. Then

V_{z}(⋁ F)(x) = ⋁{ ⋁_{f∈F} f(s) : zs = x} = ⋁_{f∈F} ⋁{f(s) : zs = x} = ⋁_{f∈F} V_{z}(f)(x).

Similarly, let a ∈ Q, f ∈ M^{S}. Then

(a • V_{z}(f))(x) = a • ⋁{f(s) : zs = x} = ⋁{a • f(s) : zs = x} = V_{z}(a • f)(x).

We shall prove that V_{z} is an isometry. Evidently,

⟨V_{z}(f), V_{z}(g)⟩_{S} = ⋁_{x,y∈S} ⟨k(x, y)(V_{z}(f)(y)), V_{z}(g)(x)⟩
= ⋁_{x,y∈S} ⟨k(x, y)(V_{z}(f)(y)), V_{z}(g)(x)⟩
= ⋁_{x,y∈S} ⟨k(x, y)(⋁{f(t) : zt = y}), ⋁{g(s) : zs = x}⟩
= ⋁_{x,y∈S} ⟨⋁{k(x, y)(f(t)) : zt = y}, ⋁{g(s) : zs = x}⟩
= ⋁_{x,y∈S} ⟨⋁{k(zs, zt)(f(t)) : zt = y}, ⋁{g(s) : zs = x}⟩
= ⋁_{s,t∈S} ⟨k(s, t)(f(t)), g(s)⟩ = ⟨f, g⟩_{S} .

Evidently, M_{v_{z}} = {K(V_{z}(f)) : f ∈ M^{S}} ⊆ M_{v} is a pre-Hilbert Q-submodule of M_{v}. The validity of the last property (5) follows by the same arguments as in theorem 1.2 applied on functions e_{m}^{zt} ∈ V_{z}(M^{S}), t ∈ S. So we have that M_{v_{z}} is a Hilbert Q-module.

Let us define a map V′_{z} : M^{S} → M^{S} by

V′_{z}(g)(v) = g(zv)

for all v ∈ S. Then, for all f ∈ M^{S} and g ∈ V_{z}(M^{S}),

⟨V_{z}(f), g⟩_{S} = ⋁_{s,t∈S} ⟨k(s, t)(V_{z}(f)(t)), g(s)⟩
= ⋁_{s,t∈zS} ⟨k(s, t)(⋁{f(v) : zv = t}), g(s)⟩
= ⋁_{s∈zS,v∈S} ⟨k(s, zv)(f(v)), g(s)⟩
= ⋁_{u∈S,v∈S} ⟨k(zu, zv)(f(v)), g(zu)⟩
= ⋁_{u∈S,v∈S} ⟨k(u, v)(f(v)), V′_{z}(g)(u)⟩ = ⟨f, V′_{z}(g)⟩_{S} .

This gives us, for all f ∈ M^{S} and all g_{1}, g_{2} ∈ V_{z}(M^{S}) such that K(g_{1}) = K(g_{2}),

\begin{aligned}⟨K(f), K(V′_{z}(g_{1}))⟩_{S} & = & ⟨f, V′_{z}(g)_{1}⟩_{S} = ⟨V_{z}(f), g_{1}⟩_{S} \\ & = & ⟨V_{z}(f), g_{2}⟩_{S} = ⟨f, V′_{z}(g_{2})⟩_{S} = ⟨K(f), K(V′_{z}(g_{2}))⟩_{S} \end{aligned}

i.e. K(V′_{z}(g_{1})) = K(V′_{z}(g_{2})). Recall that M_{v} ⊆ V′_{z}(V_{z}(M^{S})). Namely, since K(f)(s) = ⋁_{t∈S} k(zs, zt)(f(t)) we have

V′_{z}(V_{z}(K(f)))(s) = V_{z}(K(f))(zs) = ⋁{K(f)(s′) : zs′ = zs} = K(f)(s).

Let us define a map U_{z} : M_{v} → M_{v} by U_{z}(K(f)) = K(V_{z}(f)). Let us check that U_{z} is correctly defined. Assume that K(f_{1}) = K(f_{2}), g ∈ V_{z}(M^{S}). We have

\begin{aligned}⟨K(V_{z}(f_{1})), K(g)⟩_{S} & = & ⟨V_{z}(f_{1}), g⟩_{S} = ⟨f_{1}, V′_{z}(g)⟩_{S} \\ & = & ⟨f_{2}, V′_{z}(g)⟩_{S} = ⟨K(V_{z}(f_{2})), K(g)⟩_{S} \end{aligned}

i.e. K(V_{z}(f_{1})) = K(V_{z}(f_{2})). U_{z} is a module homomorphism such that U_{z}(v(s)m) = K(V_{z}(e_{m}^{s})) = K(e_{m}^{zs}) = v(zs)(m). Similarly, for all f ∈ M^{S} and all g ∈ V_{z}(M^{S}),

⟨U_{z}(K(f)), K(g)⟩_{S} = ⟨V_{z}(f), g⟩_{S} = ⟨f, V′_{z}(g)⟩_{S} = ⟨K(f), K(V′_{z}(g))⟩_{S}

i.e. the restriction U_{z} : M_{v} → M_{v_{z}} is adjointable with an adjoint U_{z}^{∗}. We shall prove that U_{z} is an isometry such that U_{z}^{∗} ∘ U_{z} = id_{M_{v}}. Clearly,

\begin{matrix}⟨(U_{z}^{∗} ∘ U_{z})(K(f)), K(g)⟩_{S} & = & ⟨U_{z}(K(f)), U_{z}(K(g))⟩_{S} = ⟨K(V_{z}(f)), K(V_{z}(g))⟩_{S} \\ = & ⟨V_{z}(f), V_{z}(g)⟩_{S} = ⟨f, g⟩_{S} = ⟨K(f), K(g)⟩_{S} . \end{matrix}

Moreover, we have

(U_{s} ∘ U_{t})(K(f)) = U_{s}(K(V_{t}(f))) = K(V_{s}(V_{t}(f))) = K(V_{st}(f)) = U_{st}(K(f))

i.e. U_{s} ∘ U_{t} = U_{st}. Similarly,

U_{e}(K(f)) = K(V_{e}(f)) = K(f) = id_{M_{v}}(K(f))

i.e. U_{e} = id_{M_{v}}. Thus, the map, U : S → H_{Q}(M_{v}) is an isometric representation of S. Let us put w = v(e). Then

(U_{s}∘w)^{∗}∘U_{t}∘w = (U_{s}∘v(e))^{∗})∘(U_{t}∘v(e)) = v(se)^{∗}∘v(te) = v(s)^{∗}∘v(t) = k(s, t).

Moreover, if k is a unital Toeplitz kernel then id_{M} = k(e, e) = v(e)^{∗}v(e) = w^{∗}w i.e. w is an isometry.

Now, we shall prove that our Naimark dilation is minimal. We have

M_{v} = {⋁_{s∈T,m∈N} v(s)(m) : T ⊆ S, N ⊆ M}
= {⋁_{s∈T,m∈N} U(s)(v(e)(m)) : T ⊆ S, N ⊆ M}
= {⋁_{s∈T,m∈N} U(s)(w(m)) : T ⊆ S, N ⊆ M}.

To prove the uniqueness, let there is a Hilbert Q-module M_{v′}, an isometry w′ ∈ A_{Q}(M, M_{v′}) and a semigroup {U′_{s}}_{s∈S} of isometries on M_{v′} that dilates k. We shall define an operator W : M_{v} → M_{v′} by W (⋁_{i∈I} v(s_{i})(m_{i})) = ⋁_{i∈I} v′(s_{i})(m_{i})) i.e. W (⋁_{i∈I} U_{s_{i}}(v(e)(m_{i})) = ⋁_{i∈I} U′_{s_{i}}(v′(e)(m_{i})).

Since, for any m, m′ ∈ M and s, t ∈ S,

⟨U_{t}(v(e)(m)), U_{s}(v(e)(m′))⟩ = ⟨k(s, t)(m), m′⟩ = ⟨U′_{t}(v′(e)(m)), U′_{s}(v′(e)(m′))⟩

one can easily see that the operator W is correctly defined, isometric, and in view of minimality it is onto.

Assume that zS = S for any z ∈ S. Then M_{v} = M_{v_{z}} and U_{z} : M_{v} → M_{v_{z}} is adjointable. This gives us that U_{z} ∈ A(M_{v}). We have U_{z}^{∗} ∘ U_{z} = id_{M_{v}}. Let us show that U_{z} ∘ U_{z}^{∗} = id_{M_{v}}. Let g ∈ M_{v} = M_{v_{z}}. Then g = U_{z}(f) for some f ∈ M_{v}. We have (U_{z} ∘ U_{z}^{∗})(g) = U_{z}(U_{z}^{∗}(U_{z}(f))) = U_{z}(f) = g = id_{M_{v}}(g).

Let us prove the converse of this theorem. Assume that {U_{s}}_{s∈S} is an isometric representation of S on a Hilbert Q-module M_{v}, w : M → M_{v} an adjointable map. Since k : S × S → A_{Q}(M) is defined by k(s, t) = (U_{s} ∘ w)^{∗} ∘ U_{t} ∘ w, we have k(cs, ct) = (U_{cs} ∘ w)^{∗} ∘ U_{ct} ∘ w = (U_{c} ∘ U_{s} ∘ w)^{∗} ∘ U_{c} ∘ U_{t} ∘ w = (U_{s} ∘ w)^{∗} ∘ U_{c}^{∗} ∘ U_{c} ∘ U_{t} ∘ w = k(s, t). Moreover, if w is an isometry we have that k(e, e) = w^{∗} ∘ U_{e}^{∗} ∘ U_{e} ∘ w = w^{∗} ∘ w = id_{M}. So we infer that k is a (unital) Toeplitz kernel.

Corollary 1.6. Let S be a unital semigroup with unit e, Q an involutive quantale, M a strict Hilbert Q-module, k : S × S → A_{Q}(M, M) a hermitian kernel. Then k is a (unital) Toeplitz kernel if and only if it admits a minimal strict (unital) Naimark dilation. In this case the minimal Naimark dilation is unique up to an isomorphism.

As an easy corollary, we obtain the following analogue of the well known Naimark dilation theorem for groups (see [10]).

Theorem 1.7. Let Q be an involutive quantale, M a (strict) Hilbert Qmodule, G a group, ρ : G → A_{Q}(M) a map such that k(x, y) = ρ(x^{−1}y) is a hermitian kernel from G × G to A_{Q}(M). Then there exists a minimal (strict) Kolmogorov decomposition v : G → A_{Q}(M, M_{v}) for k. Moreover, there exists an adjointable map w : M → M_{v} and a unitary representantion u of G on M_{v} such that ρ(x) = w^{∗} ∘ u(x) ∘ w, for all x ∈ G and M_{v} = {⋁_{i∈I}(u(x_{i}) ∘ w)(m_{i}) : x_{i} ∈ G, m_{i} ∈ M, i ∈ I}. Moreover, if u : G → A_{Q}(M_{v}, M_{v}), u′ : G → A_{Q}(M_{v′}, M_{v′}) are any two such minimal unitary representations, then there exists a unique unitary W : M_{v} → M_{v′} such that W ∘ v(g) = v(g)′ and u(g) = W^{∗} ∘ u′(g) ∘ W for all g ∈ G.

Proof. Let v be a minimal Kolmogorov decomposition for k. Let x, y, z ∈ G. Then

v(zx)^{∗} ∘ v(zy) = ρ(x^{−1}z^{−1}zy) = ρ(x^{−1}y) = v(x)^{∗} ∘ v(y)

i.e. k is a Toeplitz kernel. The rest is evident.

The unitary representation u : G → A_{Q}(M_{v}) is called a dilation of ρ. Moreover, if ρ(e) = id_{M} then the map w is an isometry.

The following proposition is the Hilbert Q-module version of the far reaching Sz.-Nagy Dilation Theorem.

Proposition 1.8. Let Q be an involutive quantale, M a (strict) Hilbert Qmodule, φ ∈ A_{Q}(M). Then there is a (strict) unitary dilation ψ ∈ A_{Q}(M_{v}) of φ such that every power φ^{n} dilates to the unitary ψ^{n}.

Proof. Let us define a map ρ_{Z} : Z → A_{Q}(M) by

ρ_{Z}(m) = \{ \begin{matrix}φ^{m}; & if m > 0 \\ (φ^{∗})^{m}; & if − m > 0 \\ id_{M}; & if m = 0 > 0. \end{matrix}

and a map k_{Z} : Z × Z → A_{Q}(M) by

k_{Z}(m, n) = ρ_{Z}(n − m).

Evidently, k_{Z} is a Toeplitz kernel satisfying the assumptions of the theorem 1.7 i.e. there is a unitary representation u of Z on M_{v} such that ρ_{Z}(m) = w^{∗} ∘ u(m) ∘ w, u(m) = u(1)^{m}, ρ_{Z}(m) = φ^{m} for m > 0, ψ = u(1) i.e. φ^{m} = w^{∗} ∘ ψ^{m} ∘ w.

Definition 1.9. Let S and T be ⋁-lattices with a duality and let Q ⊆ Q(S) be an involutive subquantale of Q(S). Let M be a ⋁-sublattice of A_{2}(S, T) and suppose that the following conditions are satisfied:

1. If f ∈ M, a ∈ Q, then f ∘ a ∈ M;

2. If f, g ∈ M then f^{∗} ∘ g ∈ Q;

3. M is a right Hilbert Q-module.

We call M a concrete Q-module.

The following result states that all Hilbert Q-modules can be represented as concrete Hilbert ones.

Theorem 1.10. Let Q be an involutive quantale and let M be a Hilbert Qmodule. Then there is a faithful representation π : Q → Q(S), S being a ⋁- lattice with a duality, and an isometric adjointable isomorphism μ : M → F, F being a concrete Hilbert π(Q)-submodule of A_{2}(S, T), T being a ⋁-lattice with a duality such that

⟨μ(m), μ(n)⟩ = π(⟨m, n⟩) and μ(m ⋄ a) = μ(m) ∘ π(a) (9)

for all m, n ∈ M, a ∈ Q.

Proof. Let π : Q → Q(S) be any faithful representation of Q (see [12]). Then we have a kernel k : M × M → Q(S) defined by k(m, n) = π(⟨m, n⟩). Namely, we have k(m, n)^{∗} = π(⟨m, n⟩)^{∗} = π(⟨m, n⟩^{∗}) = π(⟨n, m⟩) = k(n, m) for all m, n ∈ M. Then, by the theorem 1.2, k admits a classical minimal Kolmogorov decomposition v : M → A_{2}(S, T), where T is a ⋁-lattice with a duality such that k(m, n) = v(m)^{∗} ∘ v(n) and t ∈ T iff t = ⋁_{i∈I} v(m_{i})(s_{i}) for suitable m_{i} ∈ M and s_{i} ∈ S, i ∈ I. Let us put F = v(M). Then F is a ⋁-sublattice of A_{2}(S, T) for which F ∘ π(Q) ⊆ F and F^{∗} ∘ F ⊆ π(Q). Namely, we have, for all a ∈ Q, m, n ∈ M, s ∈ S, (π(a)^{∗} ∘ v(m)^{∗} ∘ v(n))(s) = (π(a)^{∗} ∘ π(⟨m, n⟩))(s) = (π(a^{∗}·⟨m, n⟩)(s) = (π(⟨a^{∗} • m, n⟩)(s) = k(a^{∗} • m, n)(s) = (v(a^{∗} • m)^{∗}∘v(n))(s) i.e. π(a)^{∗} ∘v(m)^{∗} = v(a^{∗} • m)^{∗} i.e. v(m)∘π(a) = v(a^{∗} • m) = v(m ⋄ a), since T is a Hilbert 2-module ⋁-generated by the elements of the form v(n)(s), n ∈ M, s ∈ S. Let us put μ : M → F by μ(m) = v(m) for all m ∈ M. We shall prove that μ is an injective map. Let v(m_{1}) = v(m_{2}). Then, for all n ∈ M, a ∈ S, π(⟨m_{1}, n⟩)(s) = (v(m_{1})^{∗} ∘ v(n))(s) = (v(m_{2})^{∗} ∘ v(n))(s) = π(⟨m_{2}, n⟩)(s) i.e. π(⟨m_{1}, n⟩) = π(⟨m_{2}, n⟩) for all n ∈ M i.e. ⟨m_{1}, n⟩ = ⟨m_{2}, n⟩ for all n ∈ M i.e. m_{1} = m_{2}.

Similarly, for all m, n ∈ M, we have v(m)^{∗} ∘ v(n) = k(m, n) = π(⟨m, n⟩) ∈ π(Q). So we may define a scalar product on F by ⟨v(m), v(n)⟩ = k(m, n) = π(⟨m, n⟩). It is an easy task to check that the π(Q)-module F is a Hilbert π(Q)-module.

2. Dilations of isometries and the KSGNS construction

Definition 2.1. An Ore semigroup S (see [5]) is a cancellative semigroup such that S · s ∩ S · t ≠ ∅ for every pair s, t ∈ S.

Let us recall that abelian cancellative semigroups and normal generating semigroups (see [8]) are Ore semigroups. Moreover, any Ore semigroup S can be preordered by a relation x ⪯_{r} y iff y ∈ Sx or x = y. This preorder is directed i.e. for all x, y ∈ S there is an element z ∈ S such that x ⪯_{r} z and y ⪯_{r} z.

Recall the following well known theorem (see [2], [5]).

Theorem 2.2. A semigroup S can be embedded in a group G with S^{−1}S = G if and only if it is an Ore semigroup. In this case, the group is determined up to canonical isomorphism and every semigroup homomorphism φ : S → H from S into a group H extends to a group homomorphism φ : G → H.

The following theorem is an easy generalization of the Theorem 2.3. in [8]. Let us recall that a map φ : S → S′ is an isometric representation of a semigroup S into an involutive monoid S′ with an involution ^{∗} and the unit e if φ(s)^{∗}φ(s) = e for all s ∈ S.

Theorem 2.3. Let S be an Ore semigroup, G = S^{−1}S a group and let φ : S → S′ be an isometric representation of S into an involutive monoid S′.

Then there exists a unique extension φ′ : G → S′ of φ having the following properties:

1. φ′(xs) = φ′(x)φ(s) for all x ∈ G and s ∈ S;

2. φ′(x)^{∗} = φ′(x^{−1}) for all x ∈ G.

Proof. Since S is an Ore semigroup, every element x of G can be written in the form x = s^{−1}t, for some elements s and t of S. So, the uniqueness of φ′ is trivial. To get existence, we set φ′(x) = φ(s)^{∗}φ(t), if x = s^{−1}t, where s, t ∈ S. We shall show that φ′ is well defined. Let us suppose that we can also write x = u^{−1}v, where u, v ∈ S. This gives us that t^{−1} = v^{−1}us^{−1}. Then there are elements m, n ∈ S such that mt = nv i.e. m = nvt^{−1} = nvv^{−1}us^{−1} = nus^{−1} i.e. ms = nu. Applying φ, we have φ(m)φ(s) = φ(n)φ(u) i.e. φ(n)^{∗}φ(m)φ(s) = φ(u). We then have φ(s)^{∗}φ(t) = φ(s)^{∗}φ(m)^{∗}φ(m)φ(t) = φ(s)^{∗}φ(m)^{∗}φ(n)φ(v) = (φ(n)^{∗}φ(m)φ(s))^{∗}φ(v) = φ(u)^{∗}φ(v). Evidently, the conditions (1) and (2) are satisfied.

Proposition 2.4. Let Q be an involutive quantale, M a Hilbert Q-module, S an Ore semigroup, G = S^{−1}S a group and let ρ : S → A_{Q}(M) be an isometric representation. Then there exists a unique extension ρ′ : G → A_{Q}(M) having the following properties:

1. ρ′(xs) = ρ′(x) ∘ ρ(s) for all x ∈ G and s ∈ S;

2. ρ′(x)^{∗} = ρ′(x^{−1}) for all x ∈ G.

Moreover, ρ′ is hermitian.

Proof. Since A_{Q}(M) is an involutive monoid the first part of the proposition follows from the theorem 2.3. It remains to show that ρ′ is hermitian. Let us put k′(x, y) = ρ′(x^{−1}y) for all x, y ∈ G. Then we have k′(x, y)^{∗} = ρ′(x^{−1}y)^{∗} = ρ′(y^{−1}x) = k′(y, x) i.e. k′ is a kernel.

Theorem 2.5. Let Q be an involutive quantale, M a (strict) Hilbert Qmodule, S an Ore semigroup, G = S^{−1}S a group and let ρ : S → A_{Q}(M) be an isometric representation. Then there exists a (strict) Hilbert Q-module M′, an isometry w : M → M′ and a unitary representation u : G → A_{Q}(M′) such that ρ(s) = w^{∗} ∘ u(s) ∘ w, for all s ∈ S and M′ = {⋁_{i∈I}(u(x_{i}) ∘ w)(m_{i}) : x_{i} ∈ G, m_{i} ∈ M, i ∈ I}.

Proof. By the proposition 2.4 we have a hermitian map ρ′ : G → A_{Q}(M). Applying to it the theorem 1.7 we have the required isometry and the unitary representation.

Suppose that H is a Hilbert space, while T_{1}, T_{2} are two commutative contractions on H. A celebrated theorem of Ando [1], [17] states the existence of a larger Hilbert space H′ and of a pair of commutative unitaries U_{1}, U_{2} on H′ and an isometry W : H → H′ such that for all integers n_{1}, n_{2} ≥ 0 we have

T_{1}^{n_{1}} ∘ T_{2}^{n_{2}} = W^{∗} ∘ U_{1}^{n_{1}} ∘ U_{2}^{n_{2}} ∘ W.

We shall now prove an analogue of this theorem for Hilbert Q-modules.

Definition 2.6. Let S be a non-empty subset, Q an involutive quantale, M a Hilbert Q-module, {T_{s}}_{s∈S} ⊆ A_{Q}(M) and T_{s} ∘ T_{t} = T_{t} ∘ T_{s} for all s, t ∈ S. We say that a system {V_{s}}_{s∈S} ⊆ A_{Q}(M′) in a Hilbert Q-module M′ is said to be an Ando dilation of the system {T_{s}}_{s∈S} if

• there is an isometry W ∈ A_{Q}(M, M′),

• V_{s} ∘ V_{t} = V_{t} ∘ V_{s} for all s, t ∈ S,

• T_{s_{1}}^{n_{1}} ∘⋯∘T_{s_{l}}^{n_{l}} = W^{∗}∘V_{s_{1}}^{n_{1}} ∘⋯∘U_{s_{l}}^{n_{l}} ∘W for all n_{1}, …, n_{l} ∈ N, s_{1}, …, s_{l} ∈ S, l ∈ N.

The Ando dilation is called strict if M′ is a strict Hilbert Q-module and minimal if

M′ = {⋁_{i} V_{s_{1}}^{n_{1}}∘⋯∘V_{s_{m}}^{n_{m}}∘W (x_{i}) : s_{1}, …, s_{l} ∈ S, n_{1}, …, n_{l} ∈ N, l ∈ N, x_{i} ∈ M}.

The Ando dilation {V_{s}}_{s∈S} is said to be isometric (unitary) if any of its elements is an isometry (unitary).

Theorem 2.7. Let Q be an involutive quantale, M a (strict) Hilbert Qmodule and {T_{s}}_{s∈S} ⊆ A_{Q}(M) a set of commuting isometries. Then there exists a (strict) Hilbert Q-module M′, an isometry W : M → M′ and unitary commuting operators {T_{s}}_{s∈S} ⊆ A_{Q}(M′) such that T_{s_{1}}^{n_{1}} ∘ ⋯ ∘ T_{s_{l}}^{n_{l}} = W^{∗} ∘ U_{s_{1}}^{n_{1}} ∘ ⋯ ∘ U_{s_{l}}^{n_{l}} ∘ W for all n_{1}, …, n_{l} ∈ N, s_{1}, …, s_{l} ∈ S, l ∈ N and M′ is minimal. Moreover, any two minimal unitary Ando dilations of {T_{s}}_{s∈S} are unitarily equivalent.

Proof. Let us take the free commutative semigroup F_{c}(S) over the set S. Then we may assume that F_{c}(S) is a subsemigroup of N^{S} generated by the el-

ements e_{s} = (e_{st})_{t∈S}, e_{st} = \{ \begin{matrix}1 & if s = t \\ 0 & otherwise. \end{matrix} Then F_{c}(S) is an Ore semigroup

since any commutative cancellative semigroup is Ore. Evidently, the free commutative group G = G_{c}(S) over set S satisfies G = (−F_{c}(S)) + F_{c}(S) ⊆ Z^{S}. Let us define a map ρ : F_{c}(S) → A_{Q}(M) by ρ(0) = id_{M} and ρ(e_{s}) = T_{s}. Clearly, ρ is an isometric representation of F_{c}(S). By the theorem 2.5 there exists a Hilbert Q-module M′, an isometry W : M → M′ and a unitary representantion u : G → A_{Q}(M′) such that ρ(s) = W^{∗} ∘ u(s) ∘ W, for all s ∈ S. Let us put U_{s} = u(e_{s}). Then {U_{s}}_{s∈S} are commuting unitaries and u(n_{1} · e_{s_{1}} + ⋯ + n_{l} · e_{s_{l}}) = U_{s_{1}}^{n_{1}} ∘ ⋯ ∘ U_{s_{m}}^{n_{m}}. We have M′ = ⋁ u(G)(W (M)) = {⋁_{i} V_{s_{1}}^{n_{1}} ∘ ⋯ ∘ V_{s_{m}}^{n_{m}} ∘ W (x_{i}) : s_{1}, …, s_{l} ∈ S, n_{1}, …, n_{l} ∈ N, l ∈ N, x_{i} ∈ M}. The unitary equivalence of two minimal unitary Ando dilations follows from theorem 1.2.

Corollary 2.8. Let M be a complete ortholattice and {T_{s}}_{s∈S} ⊆ Q(M) a set of commuting ⋁-preserving injective maps on M. Then there exists a complete ortholattice M′, an injective ⋁-preserving map W : M → M′ and commuting ⋁-preserving automorphisms {T_{s}}_{s∈S} ⊆ Q(M′) such that T_{s_{1}}^{n_{1}} ∘ ⋯ ∘ T_{s_{m}}^{n_{m}} = W^{∗} ∘U_{s_{1}}^{n_{1}} ∘⋯∘U_{s_{m}}^{n_{m}} ∘W for all (n_{1}, …, n_{m}) ∈ N^{m}, (s_{1}, …, s_{m}) ∈ S^{m}, m ∈ N and M′ is minimal. Moreover, any two minimal unitary Ando dilations of {T_{s}}_{s∈S} are ⋁-isomorphic.

The next result gives us a lattice-theoretical version of the Stinespring dilation theorem for involutive quantales.

Theorem 2.9. Let Q and Q′ be involutive quantales, Q′ unital, let M be a (strict) Hilbert Q-module and let ρ : Q′ → A_{Q}(M) be a completely hermitian map. Then there is a (strict) Hilbert Q-module M_{ρ}, a homomorphism of involutive quantales π_{ρ} : Q′ → A_{Q}(M_{ρ}) and an element w_{ρ} ∈ A_{Q}(M, M_{ρ}) such that ρ(b) = w_{ρ}^{∗} ∘π_{ρ}(b)∘w_{ρ}, for all b ∈ Q′. Moreover, M_{ρ} = {⋁_{i∈I}(π_{ρ}(b_{i})∘w_{ρ})(m_{i}) : b_{i} ∈ Q′, m_{i} ∈ M, i ∈ I}. In this case the minimal Stinespring dilation is unique up to an isomorphism.

Proof. We know, by the theorem 1.2, that the kernel k : Q′ × Q′ → A_{Q}(M), k(a, b) = ρ(a^{∗} · b), for all a, b ∈ Q′, has a minimal Kolmogorov decomposition v : Q′ → A_{Q}(M, M_{v}). Moreover, v is ⋁-preserving. Namely, we have, for all m ∈ M and all c ∈ Q′, that

(v(⋁_{i∈I} b_{i})^{∗} ∘ v(c))(m) = ρ((⋁_{i∈I} b_{i})^{∗} · c)(m) = ρ(⋁_{i∈I} b_{i}^{∗} · c)(m)
= ⋁_{i∈I} ρ(b_{i}^{∗} · c)(m) = ⋁_{i∈I}(v(b_{i})^{∗} ∘ v(c))(m)
= ((⋁_{i∈I} v(b_{i})^{∗}) ∘ v(c))(m).

Hence, since any element of M_{v} is a join of the elements v(c)(m), c ∈ Q′, m ∈ M, we have that v(⋁_{i∈I} b_{i})^{∗} = ⋁_{i∈I} v(b_{i})^{∗} i.e. v(⋁_{i∈I} b_{i}) = ⋁_{i∈I} v(b_{i}), for all b_{i} ∈ Q′, i ∈ I.

Now, let us put M_{ρ} = M_{v} and π_{ρ}(b)(⋁_{i∈I} v(c_{i})(m_{i})) = ⋁_{i∈I} v(b · c_{i})(m_{i}). First of all, let us prove the correctness of our definition. Assume that ⋁_{i∈I} v(c_{i})(m_{i}) = ⋁_{j∈J} v(d_{j})(n_{j}), c_{i}, d_{j} ∈ Q′, m_{i}, n_{j} ∈ M, i ∈ I, j ∈ J. Then we have

⟨⋁_{i∈I} v(b · c_{i})(m_{i}), v(d)(m)⟩_{v} = ⟨v(d)^{∗}(⋁_{i∈I} v(b · c_{i})(m_{i})), m⟩ =

⟨⋁_{i∈I} v(d)^{∗}(v(b · c_{i})(m_{i})), m⟩ = ⟨⋁_{i∈I} k(d, b · c_{i})(m_{i}), m⟩ =
⟨⋁_{i∈I} ρ(d^{∗} · (b · c_{i}))(m_{i}), m⟩ = ⟨⋁_{i∈I} ρ((d^{∗} · b) · c_{i})(m_{i}), m⟩ =
⟨⋁_{i∈I} k(b^{∗} · d, c_{i})(m_{i}), m⟩ = ⟨⋁_{i∈I} v(b^{∗} · d)^{∗}(v(c_{i})(m_{i})), m⟩ =
⟨⋁_{i∈I} v(c_{i})(m_{i}), v(b^{∗} · d)(m)⟩_{v} = ⟨⋁_{j∈J} v(d_{j})(n_{j}), v(b^{∗} · d)(m)⟩_{v} =
⟨⋁_{j∈J} v(b · d_{j})(n_{j}), v(d)(m)⟩_{v} ,

for all s ∈ S and all m ∈ M. This implies that ⋁_{i∈I} v(b · c_{i})(m_{i}) = ⋁_{j∈J} v(b · d_{j})(n_{j}).

Let us show that π_{ρ} is a homomorphism of involutive quantales. Evidently, it preserves arbitrary suprema. Let a, b ∈ Q′. Then, for all c_{i} ∈ Q′, m_{i} ∈ M, i ∈ I, we have π_{ρ}(a · b)(⋁_{i∈I} v(c_{i})(m_{i})) = ⋁_{i∈I} v((a · b) · c_{i})(m_{i}) = ⋁_{i∈I} v(a · (b · c_{i}))(m_{i}) = π_{ρ}(a)(⋁_{i∈I} v(b · c_{i})(m_{i})) = π_{ρ}(a)(π_{ρ}(b)(⋁_{i∈I} v(c_{i})(m_{i}))) = (π_{ρ}(a) ∘ π_{ρ}(b))(⋁_{i∈I} v(c_{i})(m_{i})) i.e. π_{ρ}(a · b) = π_{ρ}(a) ∘ π_{ρ}(b). Now, let a ∈ Q′. We have to show that π_{ρ}(a)^{∗} = π_{ρ}(a^{∗}). Assume that c_{i}, d_{j} ∈ Q′, m_{i}, n_{j} ∈ M, i ∈ I, j ∈ J. We have, for all d ∈ Q′ and all m ∈ M

⟨π_{ρ}(a)(v(c)(m)), v(d)(n)⟩_{v} = ⟨v(a · c)(m), v(d)(n)⟩_{v} =

⟨v(d)^{∗}(v(a · c)(m)), n⟩ = ⟨k(d, a · c)(m), n⟩ =
⟨ρ(d^{∗} · (a · c))(m), n⟩ = ⟨ρ((d^{∗} · a) · c))(m), n⟩ =
⟨k(a^{∗} · d, c)(m), n⟩ = ⟨v(a^{∗} · d)^{∗}(v(c)(m)), n⟩ =
⟨v(c)(m), v(a^{∗} · d)(n)⟩_{v} = ⟨v(c)(m), π_{ρ}(a^{∗})(v(d)(n))⟩_{v} ,

i.e. π_{ρ}(a) is adjointable to π_{ρ}(a^{∗}). We shall put w_{ρ} = v(e). Let us show that ρ(b) = w_{ρ}^{∗} ∘ π_{ρ}(b) ∘ w_{ρ}, for all b ∈ Q′. We have, for all m, n ∈ M,

\begin{aligned}⟨(w_{ρ}^{∗} ∘ π_{ρ}(b) ∘ w_{ρ})(m), n⟩ & = & ⟨(v(e)^{∗}(π_{ρ}(b)(v(e)(m))), n⟩ \\ & = & ⟨(v(e)^{∗} ∘ v(b))(m), n⟩ = ⟨ρ(b)(m), n⟩ . \end{aligned}

Note that we have m′ ∈ M_{ρ} iff m′ = ⋁_{a∈A} v(a)(m_{a}) = ⋁_{a∈A}(π_{ρ}(a) ∘ v(e))(m_{a}) = ⋁_{a∈A}(π_{ρ}(a) ∘ w_{ρ})(m_{a}) for suitably chosen m_{a} ∈ M, a ∈ A ⊆ Q′.

To prove the uniqueness, let there is a Hilbert Q-module N, an isometry w ∈ A_{Q}(M, N) and an involutive quantale homomorphism π : Q′ → A_{Q}(N) such that ρ(b) = w^{∗} ∘ π(b) ∘ w, for all b ∈ Q′ and N = {⋁_{i∈I}(π(b_{i}) ∘ w)(m_{i}) : b_{i} ∈ Q′, m_{i} ∈ M, i ∈ I}. We shall define an operator W : M_{ρ} → N by

W (⋁_{i∈I}(π_{ρ}(b_{i}) ∘ w_{ρ})(m_{i})) = ⋁_{i∈I}(π(b_{i}) ∘ w)(m_{i})).

Since

\begin{aligned}⟨π_{ρ}(b) ∘ w_{ρ})(m), π_{ρ}(b′) ∘ w_{ρ})(m′)⟩ & = & ⟨k(b, b′)(m), m′⟩ \\ & = & ⟨π(b) ∘ w)(m), π(b′) ∘ w)(m′)⟩ \end{aligned}

for any m, m′ ∈ M and b, b′ ∈ Q′, we can easily obtain by the same considerations as in theorem 1.2 that the operator W is unitary. Moreover, w = W ∘ w_{ρ} and π(a) = W ∘ π_{ρ}(a) ∘ W^{−1}, for all a ∈ Q′.

The unique (up to unitary equivalence) triple (M_{ρ}, π_{ρ}, w_{ρ}) obtained from ρ as in theorem 2.9 will be called the KSGNS construction (see [6]) associated with ρ (for Kasparov, Stinespring, Gel’fand, Naimark, Segal).

The last construction we consider is the case of ⋁-semilattices (without a duality). Let us recall the following lemma.

Lemma 2.10. Let L be ⋁-semilattice, f : L → L a ⋁ −preserving map. Then the map \overline{f} = f × f^{⊣} : L × L^{op} → L × L^{op} is ⋁-preserving and L × L^{op} is a ⋁- semilattice with a duality ′; here f^{⊣} is the right adjoint to f and (a, b)′ = (b, a). Moreover, there is an injective ⋁-preserving map ι_{f} : L → L × L^{op} and a surjective ⋁-preserving map ν_{f} : L × L^{op} → L such that f = ν_{f} ∘ (f × f^{⊣}) ∘ ι_{f}.

Proof. Let us define ι_{f}(l) = (l, 1) and ν_{f}(l, k) = l for all l, k ∈ L. Then both maps preserve arbitrary joins, ι_{f} is an injective map and ν_{f} is an surjective map. Since f^{⊣} preserves arbitrary meets in L it preserves arbitrary joins in L^{op}.

Then f × f^{⊣} is ⋁-preserving and (ν_{f} ∘ (f × f^{⊣}) ∘ ι_{f})(l) = (ν_{f} ∘ (f × f^{⊣}))(l, 1) = ν_{f}(f(l), f^{⊣}(1)) = f(l).

Proposition 2.11. L be ⋁-semilattice, f : L → L a ⋁ −preserving map. Then there is a ⋁-semilattice K with a duality, an injective ⋁-preserving map i_{L} : L → K, a surjective ⋁-preserving map n_{L} : K → L and ⋁-automorphism g : K → K such that every power f^{n} dilates to the ⋁-automorphism g^{n} i.e. f^{n} = n_{L} ∘ g^{n} ∘ i_{L}.

Proof. By the theorem 1.8 there is a unitary dilation g : K → K of the map f ×f^{op} such that (f ×f^{op})^{n} = w^{∗} ∘g^{n} ∘w. Let us put i_{L} = w ∘ι_{f}, n_{L} = ν_{f} ∘w^{∗}. Since w is an injective ⋁-preserving map and w^{∗} a surjective ⋁-preserving map we have that i_{L} : L → K is injective ⋁-preserving map and n_{L} : L → K is surjective ⋁-preserving map. We have, for all l ∈ L, (n_{L} ∘ g^{n} ∘ i_{L})(l) = ν_{f}((w^{∗} ∘ g^{n} ∘ w)(ι_{f}(l))) = ν_{f}((f × f^{op})^{n})(l, 1)) = ν_{f}(f^{n}(l), (f^{op})^{n}(1)) = f^{n}(l).

Similarily, we have the following

Lemma 2.12. Let Q be a quantale, Q^{d} a quantale with the same join operation ⋁ and a new operation of multiplication ·_{c} defined by a ·_{c} b = b · a, L a ⋁- semilattice with a duality, f : Q → Q(L) a ⋁-preserving map. Then we have an involutive ⋁-preserving map F : Q × Q^{d} → Q(L × L^{op}), an injective morphism of quantales i_{Q} : Q → Q × Q_{d} such that νid ∘ F (i_{Q}(a)) ∘ ιid = f(a) for all a ∈ Q.

Proof. First, we put i_{Q}(a) = (a, 0) for all a ∈ Q. Clearly, i_{Q} is an injective morphism of quantales. Note that evidently Q × Q^{d} is an involutive quantale with involution (a, b)^{∗} = (b, a).

Let us define now the map F : Q × Q^{d} → Q(L × L^{op}) as follows:

F (a, b) = f(a) × f(b)^{⊣} : L × L^{op} → L × L^{op}

for all a, b ∈ Q. Then f(a) resp. f(b)^{⊣} preserve all suprema in L resp. in L^{op} i.e. the pair f(a) × f(b)^{⊣} belongs to Q(L × L^{op}). Let us show that F preserves arbitrary joins and the involution.

Now, assume X ⊆ Q × Q^{d}. Then F (⋁ X) = F (⋁{(a, b) : (a, b) ∈ X}) = F (⋁{a : (a, b) ∈ X}, ⋁{b : (a, b) ∈ X}) = f(⋁{a : (a, b) ∈ X}) × f^{⊣}(⋁{b : (a, b) ∈ X}) = ⋁{f(a) × f(b)^{⊣} : (a, b) ∈ X} = ⋁{F (a, b) : (a, b) ∈ X} = ⋁ F (X).

Now, we shall show that F preserves involution. First, let us denote the right adjoint to F (a, b) by G(a, b). Evidently, G(a, b) = (f(a)^{⊣}, f(b)). Note that we have F (a, b)^{∗} = ′ ∘ G(a, b) ∘ ′ (see [11], [12]). Then, for all l, k ∈ L,

\begin{aligned}F (a, b)^{∗}(l, k) & = & (G(a, b)((l, k)′))′ = (G(a, b)((k, l)))′ = (f(a)^{⊣}(k), f(b)(l))′ \\ & = & (f(b)(l), f(a)^{⊣}(k)) = F (b, a)(l, k) = F ((a, b)^{∗})(l, k). \end{aligned}

We have, for all l ∈ L,

(νid ∘ F (i_{Q}(a)) ∘ ιid)(l) = (νid ∘ F (a, 0))(l, 1) = (νid(f(a), f(0)^{⊣}))(l, 1)
= νid(f(a)(l), f(0)^{⊣}(1)) = f(a)(l).

Recall that (unital) quantales play for involutive quantales the same role as operator algebras (norm closed subalgebras of the space B(H) of bounded linear operators on a Hilbert space H) to C^{∗}-algebras. Similarly as for operator algebras we shall establish the Gelfand-Naimark-Segal construction for arbitrary quantales.

Theorem 2.13. Let Q be a unital quantale, let L be a ⋁-semilattice and let ρ : Q → Q(L) be a ⋁-preserving map. Then there is a ⋁-semilattice L_{ρ} with a duality, a quantale homomorphism π_{ρ} : Q → Q(L_{ρ}), an injective ⋁-preserving map w_{ρ} : L → L_{ρ} and a surjective ⋁-preserving map v_{ρ} : L_{ρ} → L such that ρ(b) = v_{ρ} ∘ π_{ρ}(b) ∘ w_{ρ}, for all b ∈ Q.

Proof. Let us apply the theorem 2.9 to the induced map F : Q × Q^{d} → Q(L × L^{op}) from the preceding lemma. Then we have a ⋁-semilattice L_{F} with a duality, a homomorphism of unital quantales π_{F} : Q × Q^{d} → Q(L_{F}), and an injective ⋁-preserving map w_{F} : L × L^{op} → L_{F} such that F (b, c) = w_{F}^{∗} ∘ π_{F} (b, c) ∘ w_{F}, for all (b, c) ∈ Q × Q^{d}.

Let us put L_{ρ} = L_{F}, π_{ρ} = π_{F} ∘ i_{Q}, w_{ρ} = w_{F} ∘ ι_{id_{L}}, v_{ρ} = ν_{id_{L}} ∘ w_{F}^{∗}.

We have, for all b ∈ Q,

(v_{ρ} ∘ π_{ρ}(b) ∘ w_{ρ})(l)=(v_{ρ} ∘ π_{F} (i_{Q}(b)) ∘ w_{F})(l, 1)
=ν_{id_{L}} (w_{F}^{∗} ∘ π_{F} (i_{Q}(b)) ∘ w_{F})(l, 1))
=ν_{id_{L}}(F (b, 0)(l, 1)) = ν_{id_{L}}((ρ(b) × ρ(0)^{⊣})(l, 1)) = ρ(b)(l).

3. Conclusion and future work

We have introduced hermitian kernels for Hilbert Q-modules and established basic dilation theorems in the Hilbert Q-module context. We have no doubt that other applications of this technology will follow in the course of time. In particular, we used purely elementary algebraic methods that suggest that there should be a common denominator between the category of Hilbert spaces and bounded mappings (Hilbert C^{∗}-modules and adjointable mappings) and the category of ⋁-semilattices with a duality and ⋁-preserving mappings (Hilbert Q-modules and adjointable mappings).

There are several topics in our approach we would like to study in more detail. We hope that after some effort using the KSGNS construction we may obtain more information what the right concept of a point in a quantale should be. Another possibility is to study tensor products of involutive quantales using their representations via ⋁-semilattices with a duality.

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.

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

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