Derivations of quasi *-algebras

The spatiality of derivations of quasi *-algebras is investigated by means of representation theory. Moreover, in view of physical applications, the spatiality of the limit of a family of spatial derivations is considered.


Introduction
In the so-called algebraic approach to quantum systems, one of the basic problems to solve consists in the rigorous definition of the algebraic dynamics, i.e. the time evolution of observables and/or states. For instance, in quantum statistical mechanics or in quantum field theory one tries to recover the dynamics by performing a certain limit of the strictly local dynamics. However, this can be successfully done only for few models and under quite strong topological assumptions (see, for instance, [1] and references therein). In many physical models the use of local observables corresponds, roughly speaking, to the introduction of some cut-off (and to its successive removal) and this is in a sense a general and frequently used procedure, see [2,3,4,5] for conservative and [6,7] for dissipative systems. Introducing a cut-off means that in the description of some physical system, we know a regularized hamiltonian H L , where L is a certain parameter closely depending on the nature of the system under consideration. The role of the commutator [H L , A], A being an observable of the physical system (in a sense that will be made clearer in the following), is crucial in the analysis of the dynamics of the system. We have discussed several properties of this map in a recent paper, [8], focusing our attention mainly on the existence of the algebraic dynamics α t given a family of operators H L as above. Here, in a certain sense, we reverse the point of view. We start with a (generalized) derivation δ and we first consider the following problem: under which conditions is the map δ spatial (i.e., is implemented by a certain operator)? The spatiality of derivations is a very classical problem when formulated in *-algebras and it as been extensively studied in the literature in a large variety of situations, mostly depending on the topological structure of the *-algebras under consideration (C*-algebras, von Neumann algebras, O*-algebras, etc. See [1,9,10,11]). In this paper we consider a more general set-up, turning our attention to derivations taking their values in a quasi *-algebra. This choice is motivated by possible applications to the physical situations described above. Indeed, if A 0 denotes the *-algebra of local observables of the system, in order to perform the so-called thermodynamical limits of certain local observables, one endows A 0 with a locally convex topology τ , conveniently chosen for this aim (the so called physical topology). The completion A of A 0 [τ ], where thermodynamical limits mostly live, may fail to be an algebra but it is in general a quasi *-algebra [5,12,11]. For these reasons we start with considering, given a quasi *-algebra (A, A 0 ), a derivation δ defined in A 0 taking its values in A, and investigate its spatiality. In particular, we consider the case where δ is the limit of a net {δ L } of spatial derivations of A 0 and give conditions for its spatiality and for the implementing operator to be the limit, in some sense, of the operators H L implementing the {δ L }'s.
The paper is organized as follows: In the next section we give the essential definitions of the algebraic structures needed in the sequel.
In Section 3, the possibility of extending δ beyond A 0 , through a notion of τ −closability is investigated. Section 4 is devoted to the analysis of the spatiality of *-derivations which are induced by *-representations, and of the spatiality of the limit of a net of spatial *-derivations. We also extend our results to the situation where the *-representation, instead of living in Hilbert space, takes its values in a quasi *-algebra of operators in rigged Hilbert space (qu*-representation).

The mathematical framework
Let A be a linear space and A 0 a * -algebra contained in A. We say that A is a quasi * -algebra with distinguished * -algebra A 0 (or, simply, over A 0 ) if (i) the left multiplication ax and the right multiplication xa of an element a of A and an element x of A 0 which extend the multiplication of A 0 are always defined and bilinear; (ii) x 1 (x 2 a) = (x 1 x 2 )a and x 1 (ax 2 ) = (x 1 a)x 2 , for each x 1 , x 2 ∈ A 0 and a ∈ A; (iii) an involution * which extends the involution of A 0 is defined in A with the property (ax) * = x * a * and (xa) * = a * x * for each x ∈ A 0 and a ∈ A.
A quasi * -algebra (A, A 0 ) is said to have a unit I if there exists an element I ∈ A 0 such that aI = Ia = a, ∀a ∈ A. In this paper we will always assume that the quasi * -algebra under consideration have an identity. Let A 0 [τ ] be a locally convex * -algebra. Then the completion A 0 [τ ] of A 0 [τ ] is a quasi *algebra over A 0 equipped with the following left and right multiplications: for any x ∈ A 0 and a ∈ A, where {x α } is a net in A 0 which converges to a w.r.t. the topology τ . Furthermore, the left and right multiplications are separately continuous. A * -invariant subspace A of A 0 [τ ] containing A 0 is said to be a (quasi-) * -subalgebra of A 0 [τ ] if ax and xa in A for any x ∈ A 0 and a ∈ A.
Then we have and similarly, for each x 1 , x 2 ∈ A 0 and a ∈ A, which implies that A is a quasi * -algebra over A 0 , and furthermore, A[τ ] is a locally convex space containing A 0 as dense subspace and the right and left multiplications are separately continuous. Hence, A is said to be a locally convex quasi * -algebra over A 0 .
In a series of papers ( [13]- [16]) we have considered a special class of quasi *-algebras, called CQ*-algebras, which arise as completions of C*-algebras. They can be introduced in the following way: Let A be a right Banach module over the C*-algebra A ♭ with involution ♭ and C*-norm . ♭ , and further with isometric involution * and such that Since * is isometric, the space A ♯ is itself, as it is easily seen, a C*-algebra with respect to the involution x ♯ := (x * ) ♭ * and the norm When also ♭ = ♯, we indicate a proper CQ*-algebra with the notation (A, * , A 0 ), since * is the only relevant involution and A 0 = A ♯ = A ♭ .
An example of CQ*-algebra is provided by certain subspaces of B(H +1 , H −1 ), B(H +1 ), B(H −1 ), the spaces of operators acting on a triplet (scale) of Hilbert spaces generated in canonical way by an unbounded operator S ≥ 1 1. For details, see [13,14,11]. From a purely algebraic point of view, each CQ*-algebra can be considered as an example of partial *-algebra, [17,11], by which we mean a vector space A with involution a → a * [i.e. (a + λb) * = a * + λb * ; a = a * * ] and a subset Γ ⊂ A × A such that (i) (a, b) ∈ Γ implies (b * , a * ) ∈ Γ ; (ii) (a, b) and (a, c) ∈ Γ imply (a, b + λc) ∈ Γ ; and (iii) if (a, b) ∈ Γ, then there exists an element ab ∈ A and for this multiplication (which is not supposed to be associative) the following properties hold: if (a, b) ∈ Γ and (a, c) ∈ Γ then ab + ac = a(b + c) and (ab) In the following we also need the concept of *-representation.
Let D be a dense domain in Hilbert space H. As usual we denote with L † (D) the space of all closable operators A with domain D, such that D(A * ) ⊃ D and both A and A * leave D invariant. As is known, D is a *-algebra with the usual operations A + B, λA, AB and the involution A † = A * | D . Let now A be a locally convex quasi *-algebra over A 0 and π o be a *-representation of A 0 , that is, a *-homomorphism from A 0 into the *-algebra L † (D), for some dense domain D. In general, extending π o beyond A 0 will force us to abandon the invariance of the domain D. That is, for A ∈ A\A 0 , the extended representative π(A) will only belong to L † (D, H), which is defined as the set of all closable operators X in H such that D(X) = D and D(X * ) ⊃ D and it is a partial *-algebra (called partial O*-algebra on D) with the usual operations X +Y , λX, the involution X † = X * |D and the weak product X2Y ≡ X † * Y whenever Y D ⊂ D(X † * ) and X † D ⊂ D(Y * ).
It is also known that, defining on D a suitable (graph) topology, one can build up the rigged Hilbert space D ⊂ H ⊂ D ′ , where D ′ is the dual of D, [18], and one has Let (A, A 0 ) be a quasi *-algebra, D π a dense domain in a certain Hilbert space H π , and π a linear map from A into L † (D π , H π ) such that: (i) π(a * ) = π(a) † , ∀a ∈ A; (ii) if a ∈ A, x ∈ A 0 , then π(a)2π(x) is well defined and π(ax) = π(a)2π(x).
For an overview on partial *-algebras and related topics we refer to [11].
As we see, the *-derivation is originally defined only on A 0 . Nevertheless, it is clear that this is not the unique possibility at hand: δ could also be defined on the whole A, or in a subset of A containing A 0 , under some continuity or closability assumption. Since continuity of δ is a rather strong requirement, we consider here a weaker condition: If δ is a τ -closable *-derivation then we define Now, for any a ∈ D(δ), we put and the following lemma holds: Proof -First we observe that D(δ) is a complex vector space. In particular, it is closed under involution. In fact, from the definition itself, if a ∈ D(δ) then there exists a net {x α } τ -converging to a. But, since the involution is τ -continuous, the net {x * α } is τ -converging to a * ∈ A. We conclude that whenever a belongs to D(δ), a * ∈ D(δ).
Next we show that the multiplication between an element a ∈ D(δ) and x ∈ A 0 is welldefined. We consider here the product ax. The proof of the existence of xa is similar. Since Recalling now that the multiplication is separately continuous and since, by assumptions, , which shows that ax belongs to D(δ) and that δ(ax) = τ − lim α δ(x α x).
This Lemma shows that, under some assumptions, it is possible to extend δ to a set larger than A 0 which, also if it is different from A, is a quasi *-algebra over A 0 itself. This result suggests the following rather general definition: Remark:-Because of the previous results, if δ 0 is τ -closable then its closure δ 0 is a *derivation defined on D(δ 0 ). Now we look for conditions for a *-derivation δ to be closable, making use of some duality result. For that we first recall that if (A[τ ], A 0 ) is a locally convex quasi *-algebra and δ is a *-derivation of A 0 , we can define the adjoint derivation δ ′ acting on a subspace D(δ ′ ) of the dual space A ′ of A. The derivation δ ′ is first defined, for ω ∈ A ′ and x ∈ A 0 , by (δ ′ ω)(x) = ω(δ(x)) and then extended to the domain D(δ ′ ) = {ω ∈ A ′ : δ ′ ω has a continuous extension to A}.
A classical result, [19], states that δ is τ -closable if, and only if, We now prove the following result.
Proof -First we notice that condition (1) above implies that ω ∈ D(δ ′ ). Secondly, let x, y, z ∈ A 0 . Since ω(xδ(y)z) = ω(δ(xyz)) − ω(δ(x)yz) − ω(xyδ(z)), we have, as a consequence of the continuity of ω • δ and of ω itself: where we have also used the continuity of the multiplication. C x,z is a suitable positive constant depending on both x and z. Let us further define a new linear functional ω x,z (y) = ω(xyz). Of course we have |ω(xyz)| ≤ D x,z p α (y), for some seminorm p α and a positive constant D x,z . It follows that ω x,z has a continuous extension to A, which we still denote with the same symbol. Moreover, since (δ ′ ω x,z )(y) = ω x,z (δ(y)) = ω(xδ(y)z), we have |(δ ′ ω x,z )(y)| ≤ C x,z p σ (y), for every y ∈ A 0 . This implies that ω x,z belongs to D(δ ′ ) or, in other words, that ω x,z has a continuous extension to A. For this reason we have D(δ ′ ) ⊃ linear span{ω x,z : x, z ∈ A 0 }, and this set is dense in A ′ . Were it not so, then there would exists a non zero element y ∈ A 0 such that ω x,z (y) = 0 for all x, z ∈ A 0 . But, this is in contrast with the faithfulness of the GNS-representation π ω since we would also have ω(xyz) =< π ω (y)λ ω (z), λ ω (x * ) >= 0 for all x, z ∈ A 0 , which, in turn, would imply that π ω (y) = 0.

Spatiality of *-derivations induced by *-representations
Let (A, A 0 ) be a quasi *-algebra and δ be a *-derivation of A 0 as defined in the previous section. Let π be a *-representation of (A, A 0 ). We will always assume that whenever x ∈ A 0 is such that π(x) = 0, π(δ(x)) = 0 as well. Under this assumption, the linear map is well-defined on π(A 0 ) with values in π(A) and it is a *-derivation of π(A 0 ). We call δ π the *-derivation induced by π.
Given such a representation π and its dense domain D π , we consider the usual graph topology t † generated by the seminorms Calling D ′ π the conjugate dual of D π we get the usual rigged Hilbert space , and with L † (D π ) the *-algebra of all operators A in H π such that both A and its adjoint A * map D π into itself. In this case, can be extended to all of D ′ π in the following way: Therefore the multiplication of X ∈ L(D π , D ′ π ) and A ∈ L † (D π ) can always be defined: With these definitions it is known that (L(D π , D ′ π ), L † (D π )) is a quasi *-algebra. We can now prove the following Theorem 4.1 Let (A, A 0 ) be a locally convex quasi *-algebra with identity and δ be a *derivation of A 0 .
there exists H = H † ∈ L(D π , D ′ π ) such that Hξ 0 ∈ H π and (ii) There exists a positive linear functional f on A 0 such that: for some continuous seminorm p of τ and, denoting withf the continuous extension of f to A, the following inequality holds: for some positive constant C.
(iii) There exists a positive sesquilinear form ϕ on A × A such that: for some continuous seminorm p of τ ; and ϕ satisfies the following inequality: for some positive constant C.
Proof -First we show that (i) implies (ii). We recall that the ultra-cyclicity of the vector ξ 0 means that D π = π(A 0 )ξ 0 . Therefore, the map defined as is a positive linear functional on A 0 . Moreover, since f (x * x) = π(x)ξ 0 2 , equation (6) follows because of the (τ − τ s )-continuity of π. As for equation (7), it is clear first of all that f has a unique extension to A defined asf due the (τ − τ s )-continuity of π. Therefore we have, using (5), so that inequality (7) follows with C = Hξ 0 .
Let us now prove that (ii) implies (iii). For that we define a sesquilinear form ϕ in the following way: let a, b be in A and let {x α }, {y β } be two nets in A 0 , τ -converging respectively to a and b. We put It is readily checked that ϕ is well-defined. The proofs of (8), (9) and (10) are easy consequences of definition (13) together with the properties of f .
To conclude the proof, we still have to check that (iii) implies (i). Given ϕ as in (iii) above, we consider the GNS-construction generated by ϕ.
Let H ϕ be the conjugate space of H ϕ , with inner product ¿From now on we will indicate with the same symbol < ., . > all the inner products, whenever no possibility of confusion arises. Let M ϕ be the subspace of H ϕ ⊕ H ϕ spanned by the vectors Inequality (10), together with the equality {λ ϕ (x), λ ϕ (x * )} 2 = ϕ(x, x) + ϕ(x * , x * ), shows that f ϕ is indeed continuous, so that by Riesz's Lemma, there exists a vector {ξ 1 , ξ 2 } ∈ H ϕ ⊕ H ϕ such that Using the invariance of ϕ we also deduce that which, together with the previous result, gives where we have introduced the vector η as Now we define the operator H in the following way: whereπ ϕ indicates the extension of π ϕ , defined in the usual way, which we need to introduce since η belongs to H ϕ and not to D πϕ , in general.
First of all, we notice that from (18) Hλ ϕ (1 1) = η ∈ H ϕ , as stated in (i). Moreover, H is also well-defined and symmetric since for all x, y ∈ A 0 This last equality follows from equation (16). We finally have to prove that H implements the derivation δ πϕ . For this, let x, y, z ∈ A 0 . Then we have Again, we made use of equation (16).
Remark:-If we add to a spatial *-derivation δ 0 a perturbation δ p such that δ = δ 0 + δ p is again a *-derivation, it is reasonable to consider the question as to whether δ is still spatial. The answer is positive under very general (and natural) assumptions: since δ 0 is spatial, the above Proposition states that there exists a positive linear functional f on A 0 whose extensionf satisfies, among the others, inequality (7): |f (δ 0 (x))| ≤ C( f (x * x) + f (xx * )), for all x ∈ A 0 . If we require that δ p satisfies the inequality |f (δ p (x))| ≤ |f (δ 0 (x))|, for all x ∈ A 0 , which is exactly what we expect since δ p is small compared to δ 0 , we first deduce that δ p is spatial and, since, for all x ∈ A 0 , |f (δ(x))| ≤ 2C( f (x * x) + f (xx * )), using the same Proposition we deduce that δ is spatial too. If H, H 0 and H p denote the operators that implement, respectively, δ, δ 0 and δ p , we also get the equality i[H, A]ψ = i[H 0 + H p , A]ψ, for all A ∈ L † (D π ) and ψ ∈ D π .
The problem of the spatiality of a derivation is particularly interesting when dealing with quantum systems with infinite degrees of freedom. The reason is that for these systems we need to introduce a regularizing cut-off in their descriptions and remove this cut-off only at the very end. Specifically, something like this can happen: the physical system S is associated to, say, the whole space R 3 . In order to describe the dynamics of S the canonical approach (see [9] and references therein) consists in considering a subspace V ⊂ R 3 , the physical system S V which naturally lives in this region, and to write down the so-called local hamiltonian H V for S V . This hamiltonian is a self-adjoint bounded operator which implements the infinitesimal dynamics δ V of S V . To obtain information about the dynamics for S we need to compute a limit (in V ) to remove the cutoff. This can be a problem already at this infinitesimal level (see also [8] and references therein) and becomes harder and harder, in general, when the interest is moved to the finite form of the algebraic dynamics, that is, when we try to integrate the derivation. Among the other things, for instance, it may happen that the net H V or the related net δ V (or both), does not converge in any reasonable topology, or that δ V is not spatial. Another possibility that may occur is the following: H V converges (in some topology) to a certain operator H, δ V converges (in some other topology) to a certain *-derivation δ, but δ is not spatial or, even if it is, H is not the operator which implements δ.
However, under some reasonable conditions, all these possibilities can be controlled. The situation is governed by the next Proposition, which is based on the assumption that there exists a (τ − τ s )-continuous *-representation π in the Hilbert space H π , which is ultra-cyclic with ultra-cyclic vector ξ 0 , and a family of *-derivations (in the sense of Definition 3.1) {δ n : n ∈ N} of the *-algebra A 0 with identity. We define a related family of *-derivations δ (n) π induced by π defined on π(A 0 ) and with values in π(A): Proposition 4.2 Suppose that: π is spatial, that is, there exists an operator H n such that H n ξ 0 ∈ H π and δ (n) π (π(x)) = i{H n • π(x) − π(x) • H n }, ∀x ∈ A 0 ; (iii) sup n H n ξ 0 =: L < ∞. Then: (a) ∃ δ(x) = τ − lim δ n (x), for all x ∈ A 0 , which is a *-derivation of A 0 ; (b) δ π , the *-derivation induced by π, is well-defined and spatial; (c) if H is the self-adjoint operator which implements δ π , if < (H n − H)ξ 0 , ξ >→ 0 for all ξ ∈ D π then H n converges weakly to H.
Proof -(a) This first statement is trivial.

Example 1: A radiation model
In this example the representation π is just the identity map. Let us consider a model of n free bosons, [20], whose dynamics is given by the hamiltonian, H = n i=1 a † i a i . Here a i and a † i are respectively the annihilation and creation operators for the i-th mode. They satisfy the following CCR Let Q L be the projection operator on the subspace of H with at most L bosons. This operator can be written considering the spectral decomposition of l . Let us now define a bounded operator H L in H by H L = Q L HQ L . It is easy to check that, for any vector Φ M with M bosons (i.e., an eigenstate of the number operator In particular, for instance, sup L H L Φ 0 = 0. It may be worth remarking that all the vectors Φ M are cyclic. Denoting with δ L the derivation implemented by H L and δ the one implemented by H, it is clear that all the assumptions of the previous Proposition are satisfied, so that, in particular, the weak convergence of H L to H follows. This is not surprising since it is known that H L converges to H strongly on a dense domain, [20]. .

Example 2: A mean-field spin model
The situation described here is quite different from the one in the previous example. First of all, [3,4], there exists no hamiltonian for the whole physical system but only for a finite volume subsystem: where i and j are the indices of the lattice site, σ i 3 is the third component of the Pauli matrices, V is the volume cut-off and |V | is the number of the lattice sites in V . It is convenient to introduce the mean magnetization operator σ V 3 = 1 |V | i∈V σ i 3 . Let us indicate with ↑ i and ↓ i the eigenstates of σ i 3 with eigenvalues +1 and −1, respectively. We define Φ ↑ = ⊗ i∈V ↑ i . It is clear that σ V 3 Φ ↑ = Φ ↑ , which implies that H V Φ ↑ = |V |Φ ↑ , which in turns implies that sup V H V Φ ↑ = ∞. This means that the cyclic vector Φ ↑ does not satisfy the main assumption of Proposition 4.2, and for this reason nothing can be said about the convergence of H V . However, it is possible to consider a different cyclic vector which is again an eigenstate of σ V 3 . Its eigenvalue depends on the volume V . However, it is where ǫ V can take only the values 0, 1. Analogously we have This means that this vector satisfies the assumptions of Proposition 4.2, so that the derivation δ V (.) = i[H V , .] converges to a derivation δ which is spatial and implemented by H, and that H V is weakly convergent to H.
As we see, contrary to the previous example, the choice of the cyclic vector which we take as our starting point, is very important in order to be able to prove the existence of δ, its spatiality and convergence of H V to a limit operator. It is also worth remarking that the same conclusions could also be found replacing Φ 0 with any vector which can be obtained as a local perturbation of Φ 0 itself.
Remark:-All the results we have proved above can be specialized to a CQ*-algebras, which can be considered as particular example of locally convex quasi *-algebras. The main difference in this case concerns statement (c) of Proposition 4.2: the weak convergence of H n to H, in this case, is replaced by a strong convergence. More in details, referring to the Example of Section 2 and calling Ω ∈ H +1 a cyclic vector, we can prove that, if (H n − H)Ω −1 → 0, then (H n − H)AΩ −1 → 0 for all A ∈ B(H +1 ).
The following result gives an interplay between the results of this and of the previous sections. In particular, we consider now the possibility of extending the domain of definition of the derivation δ (as we did in Section 3) defined as a limit of a net of derivations δ n (as we have done in this section). For this we first need the following definition: We can now prove the following Proposition 4.4 Let δ be the τ -limit of a uniformly τ -continuous sequence {δ n } of *-derivations such that the set is τ -dense in A 0 . Then, δ is a *-derivation and, denoting withδ n the continuous extension of δ n to A, we have: {x ∈ A : ∃τ − lim nδn (x)} = A.
Proof -The proof that δ is a *-derivation is trivial. Let a be a generic element in A. Since, by assumption, D(δ) is τ -dense in A 0 , and therefore in A, there exists a net {x α } ⊂ D(δ) τ -converging to a. This means that for any continuous seminorms p and for any ǫ > 0 there exists α p,ǫ such that p(a − x α ) < ǫ for all α > α p,ǫ .
All the results obtained in this section rely on the fact that there exists one underlying Hilbert space related to the representation, in the case of locally convex quasi *-algebras, or to triplets of Hilbert spaces for CQ*-algebras. However, it is known that in some physically relevant situation like in quantum field theory, the relevant operators are the quantum fields and these operators belong to L(D, D ′ ) for suitable D, instead of being in some L † (D, H). This motivates our interest for the next result, which extends in a non trivial way Proposition 4.2. Before stating the Proposition, we need to introduce some definitions.
Let (A, A 0 ) be a quasi *-algebra and π 0 a *-representation of A 0 on the domain D π 0 ⊂ H π 0 . This means that π 0 maps A 0 into L † (D π 0 ) and that π 0 is a *-homomorphism of *-algebras. As usual, we endow D π 0 with the topology t † , the graph topology generated by L † (D π 0 ): in this way we get the rigged Hilbert space π 0 we consider the strong dual topology t ′ † defined by the seminorms In L(D π 0 , D ′ π 0 ) we consider the quasi-strong topology τ qs defined by the seminorms and the uniform topology τ D , defined by the seminorms Definition 4.5 Let (A, A 0 ) and π 0 be as above. A linear map π : A → L(D π , D ′ π ) is called a qu*-representation of A associated with π 0 if π extends π 0 and π(a * ) = π(a) † ∀a ∈ A; π(ax) = π(a)π 0 (x) ∀a ∈ A, x ∈ A 0 . Theorem 4.6 Let (A, A 0 ) be a locally convex quasi *-algebra with identity and with topology τ and δ be a *-derivation of A 0 . Then the following statements are equivalent: (i) There exists a (τ − τ qs )-continuous, ultra-cyclic qu*-representation π of (A, A 0 ), with ultra-cyclic vector ξ 0 such that the *-derivation δ π induced by π is spatial, i.e. there exists (b) Letf be the continuous extension of f to A, then the following inequalities hold: for some continuous seminorm p; for some positive constant γ a ; for some positive constant C.
We now claim that A/N f ⊂ D ′ π 0 , the dual space of D π 0 [t † ]. This follows from the joint continuity of ϕ π 0 , which gives the following estimate |Ω(x, y)| ≤ γ A ′ π 0 (x)λ f (1 1) A ′ π 0 (y)λ f (1 1) which holds for all x, y ∈ A 0 , for suitable γ > 0 and A ′ ∈ L † (D π 0 ). Using the extension of (27) to A 0 × A and (35) we find which implies that λ f (a) ∈ D ′ π 0 . We can now extend π 0 to A in a natural way: for a ∈ A we put π(a)λ f (x) = λ f (ax), for all x ∈ A 0 . For each a ∈ A, π(a) is well-defined and maps D π 0 [t † ] into D ′ π 0 [t ′ † ] continuously. Moreover π is (τ −τ qs )-continuous. The induced derivation δ π is well-defined, as is easily checked, and its spatiality can be proven by repeating essentially the same steps as in Proposition 4.1.
Remark:-In the so-called Wightman formulation of quantum field theory see, e.g. [21]), the point-like A(x), x ∈ R 4 , can be a very singular mathematical object such as a sesquilinear form depending on x and defined on D × D, where D is a dense domain in Hilbert space H. The smeared field is an operator-valued distribution f ∈ S(R 4 ) → L † (D), S(R 4 ) being the space of Schwartz test functions. If f has support contained in a bounded region O of R 4 , then A(f ) is affiliated with the local von Neumann algebra A(O) of all observables in O.
A reasonable approach [22,23] consists in considering the point-like field A(x), for each x ∈ R 4 , as an element of L(D, D ′ ), once a locally convex topology on D has been defined. A crucial physical prescription is that the field must be covariant under the action of a unitary representation U (g) of some transformation group (such as the Poincaré or Lorentz group) and, as is known, the infinitesimal generator H of time translations gives the energy operator of the system which defines in natural way a spatial *-derivation of the quasi *-algebra (A, A 0 ) of observables.
There could be however a different approach. This occurs when a field x → A(x) is defined on the basis of some heuristic considerations. In order that A(x) represent a reasonable physical solution of the problem under consideration, covariance under some Lie algebra of infinitesimal transformation must be imposed. For the infinitesimal time translations this amounts to find some *-derivation δ of the quasi *-algebra obtained by taking the weak completion of the *algebra A 0 generated by the local von Neumann algebras A(O), with O a bounded region of R 4 . But, of course, a number of problems arise.
The first one consists in finding an appropriate domain D for the family of operators {A(f ); f ∈ S(R 4 )} and an appropriate topology on D, in such a way that A(x) ∈ L(D, D ′ ) for every x ∈ R 4 . Once this is done, if the identical representation has the properties required in Theorem 4.6, then a symmetric operator H implementing δ can be found and one expects H to be the energy operator of the system. But, as is well-known, the problem of integrating δ is far to be solved even in much more regular situations than those considered here. We hope to discuss these problems in a future paper.