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DERIVATIONS OF QUASI
∗-ALGEBRAS
F. BAGARELLO, A. INOUE, and C. TRAPANI Received 17 July 2003
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.
2000 Mathematics Subject Classification: 47L60, 47L90.
1. Introduction. In the so-called algebraic approach to quantum systems, one of the basic problems to be solved consists in the rigorous definition of the algebraic dynam- ics, that is, 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 strictlylocaldynamics. However, this can be success- fully done only for few models and under quite strong topological assumptions (see, e.g., [22] and the references therein). In many physical models, the use of local observ- ables corresponds, roughly speaking, to the introduction of somecutoff (and to its successive removal) and this is in a sense a general and frequently used procedure; see [8,10,21,23] for conservative systems and [1,9] for dissipative ones.
Introducing a cutoff means that in the description of some physical system, we know aregularizedHamiltonianHL, whereLis a certain parameter closely depending on the nature of the system under consideration. The role of the commutator[HL,A],Abeing an observable of the physical system (in a sense that will be made clearer in the follow- ing), is crucial in the analysis of the dynamics of the system. We have discussed several properties of this map in a recent paper, [14], focusing our attention mainly on the ex- istence of the algebraic dynamicsαtgiven a family of operatorsHLas above. Here, in a certain sense, we reverse the point of view. We start with a (generalized) derivationδand we first consider the following problem: under which conditions is the mapδspatial (i.e., implemented by a certain operator)? The spatiality of derivations is a very classical problem when formulated in∗-algebras and it has been extensively studied in the liter- ature 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 [2,3,15,22]). In this paper, we consider a more general setup, turning our at- tention to derivations taking their values in a quasi∗-algebra. This choice is motivated by possible applications to the physical situations described above. Indeed, ifᏭ0de- notes the∗-algebra of local observables of the system, in order to perform the so-called thermodynamical limits of certain local observables, one endowsᏭ0with a locally con- vex topologyτ, conveniently chosen for this aim (the so-calledphysical topology). The
completionᏭofᏭ0[τ], where thermodynamical limits mostly live, may fail to be an algebra, but it is in general aquasi∗-algebra[3,21,24]. For these reasons, we start with considering, given a quasi∗-algebra(Ꮽ,Ꮽ0), a derivationδdefined inᏭ0taking its val- ues inᏭ, and investigate its spatiality. In particular, we consider the case whereδ is the limit of a net{δL}of spatial derivations ofᏭ0, and give conditions for its spatiality and for the implementing operator to be the limit, in some sense, of the operatorsHL
implementing the{δL}’s.
The paper is organized as follows. InSection 2, we give the essential definitions of the algebraic structures needed in the sequel. InSection 3, the possibility of extendingδ beyondᏭ0, through a notion ofτ-closability, is investigated.Section 4is 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).
2. The mathematical framework. LetᏭbe a linear space andᏭ0a∗-algebra con- tained in Ꮽ as a subspace. We say that Ꮽ is a quasi ∗-algebra with distinguished
∗-algebraᏭ0(or, simply, overᏭ0) if
(i) the left multiplicationax and the right multiplicationxaof an elementaof Ꮽ and an element x ofᏭ0 which extend the multiplication ofᏭ0are always defined and bilinear;
(ii) x1(x2a)=(x1x2)aandx1(ax2)=(x1a)x2for eachx1,x2∈Ꮽ0anda∈Ꮽ; (iii) an involution ∗which extends the involution of Ꮽ0 is defined inᏭ with the
property(ax)∗=x∗a∗and(xa)∗=a∗x∗for eachx∈Ꮽ0anda∈Ꮽ.
A quasi∗-algebra(Ꮽ,Ꮽ0)is said to have a unitIif there exists an elementI∈Ꮽ0such that aI=Ia=a, for alla∈Ꮽ. In this paper, we will always assume that the quasi
∗-algebra under consideration has an identity.
LetᏭ0[τ]be a locally convex∗-algebra. Then the completionᏭ0[τ]ofᏭ0[τ]is a quasi∗-algebra overᏭ0equipped with the following left and right multiplications: for anyx∈Ꮽ0anda∈Ꮽ,
ax≡lim
α xαx, xa≡lim
α xxα, (2.1)
where{xα}is a net inᏭ0which converges toawith respect to the topologyτ. Fur- thermore, the left and right multiplications are separately continuous. A ∗-invariant subspaceᏭofᏭ0[τ]containingᏭ0is said to be a (quasi-)∗-subalgebra ofᏭ0[τ]ifax andxaare inᏭfor anyx∈Ꮽ0anda∈Ꮽ. Then we have
x1x2a
=lim
α x1x2xα
=lim
α
x1x2xα=x1x2a (2.2)
and similarly,
ax1
x2=a x1x2
, x1
ax2
= x1a
x2, (2.3)
DERIVATIONS OF QUASI -ALGEBRAS 1079 for eachx1,x2∈Ꮽ0anda∈Ꮽ, which implies thatᏭis a quasi∗-algebra overᏭ0, and furthermore,Ꮽ[τ]is a locally convex space containingᏭ0as dense subspace and the right and left multiplications are separately continuous. Hence,Ꮽis said to be alocally convex quasi∗-algebraoverᏭ0.
If(Ꮽ[τ],Ꮽ0)is a locally convex quasi∗-algebra, we indicate with{pα, α∈I}a di- rected set of seminorms which definesτ.
In a series of papers [7, 11, 12, 13], we have considered a special class of quasi∗- algebras, calledCQ∗-algebras, which arises as completions of C∗-algebras. They can be introduced in the following way.
LetᏭbe a right Banach module over theC∗-algebraᏭwith involutionand C∗- norm·, and further with isometric involution∗, such thatᏭ⊂Ꮽ. SetᏭ=(Ꮽ)∗. We say that{Ꮽ,∗,Ꮽ,}is aCQ∗-algebra if
(i) Ꮽis dense inᏭwith respect to its norm·,
(ii) Ꮽ0:=Ꮽ∩Ꮽis dense inᏭwith respect to its norm·, (iii) (ab)∗=b∗a∗, for alla,b∈Ꮽ0,
(iv) y=supa∈Ꮽ,a≤1ay,y∈Ꮽ.
Since∗ is isometric, the space Ꮽ is itself, as it is easily seen, a C∗-algebra with respect to the involutionx:=(x∗)∗and the normx:= x∗.
ACQ∗-algebra is calledproper ifᏭ=Ꮽ. When also=, we indicate a proper CQ∗-algebra with the notation(Ꮽ,∗,Ꮽ0), since ∗is the only relevant involution and Ꮽ0=Ꮽ=Ꮽ.
An example ofCQ∗-algebra is provided by certain subspaces ofᏮ(Ᏼ+1,Ᏼ−1),Ꮾ(Ᏼ+1), andᏮ(Ᏼ−1), the spaces of operators acting on a triplet (scale) of Hilbert spaces gen- erated in a canonical way by an unbounded operatorS≥1. For details, see [3,11,12].
From a purely algebraic point of view, eachCQ∗-algebra can be considered as an ex- ample of partial∗-algebra, [3,4, 5], by which we mean a vector spaceᏭwith involu- tiona→a∗ (i.e.,(a+λb)∗=a∗+λb∗;a=a∗∗) and a subsetΓ ⊂Ꮽ×Ꮽsuch that (i) (a,b)∈Γ implies(b∗,a∗)∈Γ; (ii)(a,b), (a,c)∈Γ imply(a,b+λc)∈Γ; and (iii) if (a,b)∈Γ, then there exists an elementab∈Ꮽand for this multiplication (which is not supposed to be associative) the following properties hold: if(a,b)∈Γ and(a,c)∈Γ, thenab+ac=a(b+c)and(ab)∗=b∗a∗.
In the following, we also need the concept of∗-representation.
LetᏰ be a dense domain in Hilbert spaceᏴ. As usual, we denote withᏸ†(Ᏸ)the space of all closable operatorsAwith domainᏰsuch thatD(A∗)⊃Ᏸand bothAand A∗ leaveᏰinvariant. As it is known,Ᏸis a∗-algebra with the usual operationsA+B, λA,AB, and the involutionA†=A∗|Ᏸ. Now letᏭbe a locally convex quasi∗-algebra over Ꮽ0 and πo a ∗-representation ofᏭ0, that is, a ∗-homomorphism from Ꮽ0 into the∗-algebraᏸ†(Ᏸ), for some dense domainᏰ. In general, extendingπobeyond Ꮽ0
will force us to abandon the invariance of the domainᏰ. That is, forA∈Ꮽ\Ꮽ0, the extended representativeπ(A)will only belong toᏸ†(Ᏸ,Ᏼ)which is defined as the set of all closable operatorsXinᏴsuch thatD(X)=ᏰandD(X∗)⊃Ᏸand it is a partial
∗-algebra (called partial O∗-algebra onᏰ) with the usual operationsX+Y, λX, the involutionX†=X∗|Ᏸ, and the weak productXY≡X†∗Y wheneverYᏰ⊂D(X†∗)and X†Ᏸ⊂D(Y∗).
It is also known that, defining onᏰa suitable (graph) topology, one can build up the rigged Hilbert spaceᏰ⊂Ᏼ⊂Ᏸ, whereᏰis the dual ofᏰ[18] and one has
ᏸ†(Ᏸ)⊂ᏸ(Ᏸ,Ᏸ), (2.4)
whereᏸ(Ᏸ,Ᏸ)denotes the space of all continuous linear maps fromᏰintoᏰ. More- over, under additional topological assumptions, the following inclusions hold:ᏸ†(Ᏸ)⊂ ᏸ†(Ᏸ,Ᏼ)⊂ᏸ(Ᏸ,Ᏸ). A more complete definition will be given inSection 4.
Let(Ꮽ,Ꮽ0)be a quasi∗-algebra,Ᏸπ a dense domain in a certain Hilbert spaceᏴπ, andπ a linear map fromᏭintoᏸ†(Ᏸπ,Ᏼπ)such that
(i) π(a∗)=π(a)†for alla∈Ꮽ;
(ii) ifa∈Ꮽ, x∈Ꮽ0, thenπ(a)π(x)is well defined andπ(ax)=π(a)π(x).
We say that such a mapπis a∗-representation ofᏭ;
(iii) ifπ(Ꮽ0)⊂ᏸ†(Ᏸπ), thenπis a∗-representation of the quasi∗-algebra(Ꮽ,Ꮽ0).
Let π be a∗-representation of Ꮽ. The strong topology τs on π(Ꮽ)is the locally convex topology defined by the following family of seminorms:{pξ(·);ξ∈Ᏸπ}, where pξ(π(a))≡ π(a)ξ, wherea∈Ꮽ,ξ∈Ᏸπ.
For an overview on partial∗-algebras and related topics, we refer to [3].
3. ∗-Derivations and their closability. Let(Ꮽ,Ꮽ0)be a quasi∗-algebra.
Definition3.1. A∗-derivation ofᏭ0is a mapδ:Ꮽ0→Ꮽwith the following prop- erties:
(i) δ(x∗)=δ(x)∗for allx∈Ꮽ0;
(ii) δ(αx+βy)=αδ(x)+βδ(y)for allx,y∈Ꮽ0and for allα,β∈C; (iii) δ(xy)=xδ(y)+δ(x)yfor allx,y∈Ꮽ0.
As we see, the∗-derivation is originally defined only onᏭ0. Nevertheless, it is clear that this is not the unique possibility at hand:δ could also be defined on the whole Ꮽ, or in a subset ofᏭcontainingᏭ0, under some continuity or closability assumption.
Since the continuity ofδ is a rather strong requirement, we consider here a weaker condition.
Definition3.2. A∗-derivationδofᏭ0is said to beτ-closable if for any net{xα} ⊂ Ꮽ0such thatxα τ
→0 andδ(xα)→τ b∈Ꮽ,b=0 results.
Ifδis aτ-closable∗-derivation, then we define D
δ
=
a∈Ꮽ:∃ xα
⊂Ꮽ0s.t.τ−lim
α xα=a, δ xα
converges inᏭ. (3.1)
Now, for anya∈D(δ), we put
δ(a)=τ−lim
α δ xα
, (3.2)
and the following lemma holds.
Lemma3.3. Ifδ(Ꮽ0)⊂Ꮽ0, thenD(δ)is a quasi∗-algebra overᏭ0.
DERIVATIONS OF QUASI -ALGEBRAS 1081 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, ifa∈D(δ), then there exists a net{xα}τ-converging toa. But, since the involution isτ-continuous, the net{x∗α}is τ-converging toa∗∈Ꮽ. We conclude that whenevera∈D(δ),a∗∈D(δ).
Next we show that the multiplication of an element a∈D(δ)andx ∈Ꮽ0 is well defined. We consider here the productax. The proof of the existence ofxais similar.
Sincea∈D(δ), then there exists{xα} ⊂Ꮽ0such thatxα →τ a. Moreover, the net δ(xα) τ-converges to an element b∈Ꮽ:δ(xα) →τ b=δ(a). Recalling now that the multiplication is separately continuous and since, by assumptions,δ(x)∈Ꮽ0, we de- duce thatδ(xαx)=δ(xα)x+xαδ(x)→τ δ(a)x+aδ(x), which shows thataxbelongs toD(δ)and thatδ(ax)=τ−limαδ(xαx).
This lemma shows that, under some assumptions, it is possible to extendδto a set larger thanᏭ0which, also if it is different fromᏭ, is a quasi∗-algebra overᏭ0itself.
This result suggests the following rather general definition.
Definition3.4. Let(Ꮽ,Ꮽ0)be a quasi∗-algebra andᏰa vector subspace ofᏭsuch that(Ᏸ,Ꮽ0)is a quasi∗-algebra. A mapδ:Ᏸ→Ꮽis called a∗-derivation if
(i) δ(Ꮽ0)⊂Ꮽ0andδ0≡δ|Ꮽ0is a∗-derivation ofᏭ0; (ii) δis linear;
(iii) δ(ax)=aδ(x)+δ(a)x=aδ0(x)+δ(a)xfor alla∈Ᏸand for allx∈Ꮽ0. Remark3.5. Because of the previous results, ifδ0isτ-closable, then its closureδ0
is a∗-derivation defined onD(δ0).
Now we look for conditions for a∗-derivationδto be closable, making use of some duality result. For that, we first recall that if(Ꮽ[τ],Ꮽ0)is a locally convex quasi ∗- algebra andδis a∗-derivation ofᏭ0, we can define the adjoint derivationδacting on a subspaceD(δ)of the dual spaceᏭofᏭ. The derivationδis first defined, forω∈Ꮽ andx∈Ꮽ0, by(δω)(x)=ω(δ(x))and then extended to the domain
D(δ)= {ω∈Ꮽ:δωhas a continuous extension toᏭ}. (3.3)
A classical result, [20], states thatδisτ-closable if and only ifD(δ)isσ (Ꮽ,Ꮽ)-dense inᏭ. We now prove the following result.
Proposition3.6. Letδ:Ꮽ0→Ꮽbe a∗-derivation. Assume that there existsω∈Ꮽ such thatω|Ꮽ0 is a positive linear functional onᏭ0and
(1) ω◦δisτ-continuous onᏭ0;
(2) the GNS representationπωofᏭ0is faithful.
Thenδisτ-closable.
Proof. First we notice that condition (3.1) above implies thatω∈D(δ). Secondly, letx,y,z∈Ꮽ0. Sinceω(xδ(y)z)=ω(δ(xyz))−ω(δ(x)yz)−ω(xyδ(z)), we have, as a consequence of the continuity ofω◦δand ofωitself,
ω
xδ(y)z≤pα(xyz)+pβ
δ(x)yz +pγ
xyδ(z)
≤Cx,zpσ(y), (3.4)
where we have also used the continuity of the multiplication.Cx,z is a suitable posi- tive constant depending on bothx and z. We further define a new linear functional ωx,z(y)=ω(xyz). Of course we have|ω(xyz)| ≤Dx,zpα(y)for some seminormpα
and a positive constantDx,z. It follows thatωx,zhas a continuous extension toᏭ, which we still denote with the same symbol. Moreover, since (δωx,z)(y)=ωx,z(δ(y))= ω(xδ(y)z), we have|(δωx,z)(y)| ≤Cx,zpσ(y)for everyy∈Ꮽ0. This implies that ωx,zbelongs toD(δ)or, in other words, thatωx,zhas a continuous extension toᏭ. For this reason, we haveD(δ)⊃linear span{ωx,z:x,z∈Ꮽ0}, and this set is dense inᏭ. Were it not so, then there would exist a nonzero elementy∈Ꮽ0such thatωx,z(y)=0 for allx,z∈Ꮽ0. But this is in contrast with the faithfulness of the GNS-representation πωsince we would also haveω(xyz)= πω(y)λω(z),λω(x∗) =0 for allx,z∈Ꮽ0, which, in turn, would imply thatπω(y)=0.
4. Spatiality of∗-derivations induced by∗-representations. Let(Ꮽ,Ꮽ0)be a quasi
∗-algebra andδa∗-derivation ofᏭ0as defined inSection 3. Letπbe a∗-representation of(Ꮽ,Ꮽ0). We always assume that wheneverx∈Ꮽ0is such thatπ(x)=0,π(δ(x))=0 as well. Under this assumption, the linear map
δππ(x)
=πδ(x), x∈Ꮽ0, (4.1)
is well defined onπ(Ꮽ0)with values inπ(Ꮽ)and it is a∗-derivation ofπ(Ꮽ0). We call δπthe∗-derivationinducedbyπ.
Given such a representationπand its dense domainᏰπ, we consider the usual graph topologyt†generated by the seminorms
ξ∈Ᏸπ → Aξ, A∈ᏸ†Ᏸπ
. (4.2)
CallingᏰπthe conjugate dual ofᏰπ, we get the usual rigged Hilbert spaceᏰπ[t†]⊂ Ᏼπ ⊂Ᏸπ[t†], wheret† denotes the strong dual topology ofᏰπ. As usual, we denote withᏸ(Ᏸπ,Ᏸπ)the space of all continuous linear maps fromᏰπ[t†]intoᏰπ[t†], and withᏸ†(Ᏸπ)the∗-algebra of all operatorsAinᏴπ such that bothAand its adjointA∗ mapᏰπ into itself. In this case,ᏸ†(Ᏸπ)⊂ᏸ(Ᏸπ,Ᏸπ). Each operatorA∈ᏸ†(Ᏸπ)can be extended to all ofᏰπ in the following way:
Aξˆ ,η
= ξ,A†η
∀ξ∈Ᏸπ, η∈Ᏸπ. (4.3) Therefore, the multiplication ofX∈ᏸ(Ᏸπ,Ᏸπ)andA∈ᏸ†(Ᏸπ)can always be defined:
(X◦A)ξ=X(Aξ), (A◦X)ξ=A(Xξ)ˆ ∀ξ∈Ᏸπ. (4.4) With these definitions, it is known that(ᏸ(Ᏸπ,Ᏸπ),ᏸ†(Ᏸπ))is a quasi∗-algebra.
We can now prove the following theorem.
Theorem4.1. Let(Ꮽ,Ꮽ0)be a locally convex quasi∗-algebra with identity andδa
∗-derivation ofᏭ0. Then the following statements are equivalent.
(i)There exists a(τ−τs)-continuous, ultra-cyclic∗-representationπofᏭ, with ultra- cyclic vectorξ0, such that the∗-derivationδπinduced byπis spatial, that is, there exists
DERIVATIONS OF QUASI -ALGEBRAS 1083 H=H†∈ᏸ(Ᏸπ,Ᏸπ)such thatHξ0∈Ᏼπ and
δπ π(x)
=i
H◦π(x)−π(x)◦H
∀x∈Ꮽ0. (4.5)
(ii)There exists a positive linear functionalf onᏭ0such that f
x∗x
≤p(x)2 ∀x∈Ꮽ0, (4.6)
for some continuous seminormpofτand, denoting withf˜the continuous extension of f toᏭ, the following inequality holds:
f˜
δ(x)≤C f x∗x
+ f xx∗
∀x∈Ꮽ0 (4.7)
and for some positive constantC.
(iii)There exists a positive sesquilinear form ϕ onᏭ×Ꮽsuch that ϕis invariant, that is,
ϕ(ax,y)=ϕx,a∗y
∀a∈Ꮽ, x,y∈Ꮽ0; (4.8) ϕisτ-continuous, that is,
ϕ(a,b)≤p(a)p(b) ∀a,b∈Ꮽ (4.9) and for some continuous seminormpofτ; andϕsatisfies the following inequality:
ϕδ(x),1≤C ϕ(x,x)+ ϕx∗,x∗
∀x∈Ꮽ0 (4.10) and for some positive constantC.
Proof. First we show that (i) implies (ii).
We recall that the ultra-cyclicity of the vectorξ0means thatᏰπ=π(Ꮽ0)ξ0. Therefore, the map defined as
f (x)=
π(x)ξ0,ξ0
, x∈Ꮽ0, (4.11)
is a positive linear functional onᏭ0. Moreover, sincef (x∗x)= π(x)ξ02, (4.6) follows because of the(τ−τs)-continuity ofπ. As for (4.7), it is clear first of all thatf has a unique extension toᏭdefined as
f (a)˜ =
π(a)ξ0,ξ0
, a∈Ꮽ, (4.12)
due to the(τ−τs)-continuity ofπ. Therefore, we have, using (4.5), f˜δ(x)=H◦π(x)ξ0,ξ0
−Hξ0,πx∗ξ0
≤Hξ0π(x)ξ0,π(x)ξ01/2
+πx∗ξ0,πx∗ξ01/2 (4.13) so that inequality (4.7) follows withC= Hξ0.
We now prove that (ii) implies (iii). For that, we define a sesquilinear formϕin the following way: leta,b be in Ꮽ and let{xα}, {yβ} be two nets in Ꮽ0, τ-converging, respectively, toaandb. We put
ϕ(a,b)=lim
α,βfyβ∗xα. (4.14)
It is readily checked thatϕis well defined. The proofs of (4.8), (4.9), and (4.10) are easy consequences of definition (4.14) together with the properties off.
To conclude the proof, we still have to check that (iii) implies (i).
Givenϕas in (iii) above, we consider the GNS-construction generated byϕ.
Letᏺϕ = {a∈Ꮽ; ϕ(a,a)=0}; then Ꮽ/ᏺϕ = {λϕ(a)=a+ᏺϕ, a∈Ꮽ}is a pre- Hilbert space with inner productλϕ(a),λϕ(b) =ϕ(a,b),a,b∈λϕ(Ꮽ). We callᏴϕ
the completion ofλϕ(Ꮽ)in the norm · ϕ given by this inner product. It is easy to check thatλϕ(Ꮽ0)is·ϕ-dense inᏴϕ. In fact, due to the definition of locally convex quasi∗-algebra, givena∈Ꮽ, there exists a netxα⊂Ꮽ0such thatxα→τ a. Therefore, we have, using the continuity ofϕ,
λϕ(a)−λϕ
xα2ϕ=λϕ
a−xα2ϕ=ϕ
a−xα,a−xα
≤p
a−xα2
→0. (4.15) We can now define a∗-representationπϕ with ultra-cyclic vectorλϕ(1)as follows:
πϕ(a)λϕ(x)=λϕ(ax), a∈Ꮽ, x∈Ꮽ0. (4.16) In particular, the fact thatλϕ(1)is ultra-cyclic follows from the fact thatπϕ(Ꮽ0)λϕ(1)
=λϕ(Ꮽ0)is dense inᏴϕ. Moreover, the representationπϕis also(τ−τs)-continuous;
in fact, takinga∈Ꮽandx∈Ꮽ0, we have
πϕ(a)λϕ(x)2ϕ=λϕ(ax)2ϕ=ϕ(ax,ax)≤p(ax)2
≤γxp(a)2. (4.17) The last inequality follows from the continuity of the multiplication. This inequality shows that wheneverτ−limαxα=a, thenτs−limαπϕ(xα)=πϕ(a).
This construction produces a∗-representationπϕ with all the properties required forπ in (i). As a consequence, we can define a∗-derivationδπϕ induced byπϕ as in (4.1):δπϕ(πϕ(x))=πϕ(δ(x))forx∈Ꮽ0. The proof of the spatiality ofδπϕgeneralizes the proof of the analogous statement forC∗-algebras (see, e.g., [15]).
LetᏴϕ be the conjugate space ofᏴϕ, with inner product λϕ(x),λϕ(y)
Ᏼϕ=
λϕ(y),λϕ(x)
Ᏼϕ. (4.18)
From now on, we will indicate with the same symbol·,· all the inner products whenever no possibility of confusion arises.
Letᏹϕbe the subspace ofᏴϕ⊕Ᏼϕspanned by the vectors{λϕ(x),λϕ(x∗)},x∈Ꮽ0. We define a linear functionalFϕ onᏹϕby
Fϕ
λϕ(x),λϕ x∗
=iϕ δ(x),1
, x∈Ꮽ0. (4.19)
DERIVATIONS OF QUASI -ALGEBRAS 1085 Inequality (4.10), together with the equality{λϕ(x),λϕ(x∗)}2=ϕ(x,x)+ϕ(x∗,x∗), shows thatfϕ is indeed continuous so that by Riesz’s lemma, there exists a vector {ξ1,ξ2} ∈Ᏼϕ⊕Ᏼϕ such that
Fϕ
λϕ(x),λϕ x∗
=
λϕ(x),λϕ x∗
, ξ1,ξ2
=
λϕ(x),ξ1
+ ξ2,λϕ
x∗
. (4.20)
Using the invariance ofϕ, we also deduce that
Fϕ
λϕ(x),λϕ x∗
=iϕ δ(x),1
= −iϕ δ
x∗ ,1
, (4.21)
which, together with (4.20), gives
1
iϕδ(x),1
=λϕ(x),η
−η,λϕx∗, x∈Ꮽ0, (4.22)
where we have introduced the vectorηas
η=ξ2−ξ1
2 . (4.23)
Now we define the operatorHin the following way:
Hλϕ(x)=1 iλϕ
δ(x)
+πˆϕ(x)η, x∈Ꮽ0, (4.24)
where ˆπϕ indicates the extension ofπϕ, defined in the usual way, which we need to introduce sinceηbelongs toᏴϕand not toᏰπϕin general.
First of all, we notice from (4.24) thatHλϕ(1)=η∈Ᏼϕ, as stated in (i). Moreover,H is also well defined and symmetric since for allx,y∈Ꮽ0,
Hπϕ(x)λϕ(1),πϕ(y)λϕ(1)
−πϕ(x)λϕ(1),Hπϕ(y)λϕ(1)
=
Hλϕ(x),λϕ(y)
−
λϕ(x),Hλϕ(y)
= 1
iλϕδ(x)
+πˆϕ(x)η
,λϕ(y)
−
λϕ(x), 1
iλϕδ(y)
+πˆϕ(y)η
=1 i
ϕ
δ(x),y +ϕ
x,δ(y) +
πˆϕ(x)η,λϕ(y)
−
λϕ(x),πˆϕ(y)η
=1 iϕ
δ y∗x
,1 +
η,λϕ x∗y
− λϕ
y∗x ,η
=0.
(4.25)
This last equality follows from (4.22). We finally have to prove thatHimplements the derivationδπϕ. For this, letx,y,z∈Ꮽ0. Then we have
i
H◦πϕ(x)λϕ(y),λϕ(z)
−
πϕ(x)◦Hλϕ(y),λϕ(y)
=i
Hλϕ(xy),λϕ(z)
−
Hλϕ(y),λϕ x∗y
=i1
iλϕδ(xy)
+πˆϕ(xy)η,λϕ(z)
− 1
iλϕδ(y)
+πˆϕ(y)η,λϕx∗z
=ϕ
δ(x)y,z
= πϕ
δ(x) λϕ
δ(y) ,λϕ
δ(z) .
(4.26)
Again, we made use of (4.22).
Remark4.2. If we add to a spatial∗-derivationδ0aperturbationδpsuch thatδ= δ0+δpis again a∗-derivation, it is reasonable to consider the question as to whetherδis still spatial. The answer is positive under very general (and natural) assumptions: since δ0is spatial, the above proposition states that there exists a positive linear functional f onᏭ0whose extension ˜f satisfies, among the others, inequality (4.7):|f (δ˜ 0(x))| ≤ C( f (x∗x)+ f (xx∗))for all x∈Ꮽ0. If we require thatδpsatisfies the inequality
|f (δ˜ p(x))| ≤ |f (δ˜ 0(x))|for all x∈Ꮽ0, which is exactly what we expect sinceδp is small compared to δ0, we first deduce that δp is spatial and, since for all x ∈Ꮽ0,
|f (δ(x))˜ | ≤2C( f (x∗x)+ f (xx∗)), using the same proposition, we deduce that δ is spatial too. IfH,H0, andHpdenote the operators that implement, respectively,δ, δ0, andδp, we also get the equalityi[H,A]ψ=i[H0+Hp,A]ψfor allA∈ᏸ†(Ᏸπ)and ψ∈Ᏸπ.
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 cutoff in their descriptions and remove this cutoff only at the very end. Specifically, something like this can happen: the physical systemis associated to, say, the whole spaceR3. In order to describe the dynamics of
, the canonical approach (see [15] and the references therein) consists in considering a subspaceV⊂R3, the physical systemV which naturally lives in this region, and writ- ing down the so-called local HamiltonianHV forV. This Hamiltonian is a selfadjoint bounded operator which implements the infinitesimal dynamicsδV ofV. To obtain information about the dynamics for, we need to compute a limit (inV) toremove the cutoff. This can be a problem already at this infinitesimal level (see also [14] and the 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 netHV
or the related netδV (or both) does not converge in any reasonable topology, or that δVis not spatial. Another possibility that may occur is the following:HV converges (in some topology) to a certain operatorHandδV converges (in some other topology) to a certain∗-derivationδ, butδis not spatial or, even if it is,His not the operator which implementsδ.
DERIVATIONS OF QUASI -ALGEBRAS 1087 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 exist a(τ−τs)-continuous∗-representationπin the Hilbert spaceᏴπ, which is ultra-cyclic with ultra-cyclic vectorξ0, and a family of∗-derivations (in the sense of Definition 3.1){δn:n∈N}of the∗-algebraᏭ0with identity. We define a related family of∗-derivationsδ(n)π induced byπ, defined onπ(Ꮽ0), and with values inπ(Ꮽ):
δ(n)π π(x)
=πδn(x), x∈Ꮽ0. (4.27) Proposition4.3. Suppose that
(i) {δn(x)}isτ-Cauchy for allx∈Ꮽ0;
(ii) for eachn∈N,δ(n)π is spatial, that is, there exists an operatorHnsuch that Hn=Hn†∈ᏸᏰπ,Ᏸπ
,
Hnξ0∈Ᏼπ, δ(n)π π(x)
=i
Hn◦π(x)−π(x)◦Hn
∀x∈Ꮽ0;
(4.28)
(iii) supnHnξ0 =:L <∞. Then
(a) there existsδ(x)=τ−limδn(x)for allx∈Ꮽ0, which is a∗-derivation ofᏭ0; (b) δπ, the∗-derivation induced byπ, is well defined and spatial;
(c) ifHis the selfadjoint operator which implementsδπ, and if(Hn−H)ξ0,ξ →0 for allξ∈Dπ, thenHnconverges weakly toH.
Proof. (a) This first statement is trivial.
(b) Fora,b∈Ꮽ, we putϕ(a,b)= π(a)ξ0,π(b)ξ0. Thenϕis an invariant positive sesquilinear form onᏭ×Ꮽsince
ϕ(ax,y)=
π(ax)ξ0,π(y)ξ0
=
π(a)π(x)ξ0,π(y)ξ0
=π(x)ξ0,πa∗π(y)ξ0
=ϕx,a∗y (4.29)
for alla∈Ꮽandx,y∈Ꮽ0.ϕisτ-continuous: ifa,b∈Ꮽ,
ϕ(a,b)=π(a)ξ0,π(b)ξ0≤π(a)ξ0π(b)ξ0≤pα(a)pα(b), (4.30) for some continuous seminormpαonᏭ, because of the(τ−τs)-continuity ofπ.
From this inequality, we deduce that forx∈Ꮽ0, ϕδ(x),1=lim
n ϕδn(x),1
=lim
n Hn◦π(x)ξ0,ξ0
−π(x)◦Hnξ0,ξ0
=lim sup
n
Hn◦π(x)ξ0,ξ0
−
π(x)◦Hnξ0,ξ0
≤lim sup
n
Hnξ0π(x)ξ0+πx∗ξ0
≤L ϕ(x,x)+ ϕx∗,x∗.
(4.31)
This sesquilinear form ϕ satisfies all the conditions required in Theorem 4.1(iii).
Then, following the same steps as in the proof ofTheorem 4.1, (iii)⇒(i), we construct the GNS-representationπϕassociated toϕ. We callᏴϕ,ξϕ, andHϕ, respectively, the Hilbert space, the ultra-cyclic vector, and the symmetric operator implementing the derivation associated toπϕ. Among others, the following equality must be satisfied:
ϕ(a,b)=
π(a)ξ0,π(b)ξ0
=
πϕ(a)ξϕ,πϕ(b)ξϕ
∀a,b∈Ꮽ, (4.32)
which implies that πϕ andπ are unitarily equivalent, that is, there exists a unitary operator U:Ᏼπ →Ᏼϕ such thatUξ0=ξϕ, Uπ(a)U−1=πϕ(a), for all a∈Ꮽ, and U is continuous fromDπ[tπ]intoDϕ[tϕ]. We prove here only this last property. Let x,y∈Ꮽ0; we have
πϕ(y)Uπ(x)ξ0ϕ=Uπ(y)π(x)ξ0ϕ=π(y)π(x)ξ0, (4.33)
which implies thatU∗can be extended to an operatorU†:Ᏸϕ→Ᏸπ. We have now δπϕ
πϕ(x)
=πϕ δ(x)
=Uπ δ(x)
U−1=Uδπ π(x)
U−1, (4.34)
which implies thatδπ(π(x))=U−1δπϕ(πϕ(x))U. Sinceδπϕis well defined, this equal- ity implies that alsoδπ is well defined. Indeed, we have
π(x)=0 ⇒πϕ(x)=0 ⇒δπϕπϕ(x)
=0⇒δππ(x)
=0. (4.35)
Now we defineH=U−1HϕU|Ᏸπ. Then δπ
π(x)
=U−1δπϕ
πϕ(x) U
=iU−1Hϕ◦πϕ(x)−πϕ(x)◦HϕU
=iU−1HϕU◦U−1πϕ(x)U−U−1πϕ(x)U◦U−1HϕU
=i
H◦π(x)−π(x)◦H ,
(4.36)
which allows us to conclude.
(c) Forx,y,z∈Ꮽ0, we have, using the definition ofϕand itsτ-continuity, ϕ
δn(x)y,z
= δ(n)π
π(x)
π(y)ξ0,π(z)ξ0
=i
Hn◦π(x)
π(y)ξ0,π(z)ξ0
−π(x)◦Hnπ(y)ξ0,π(z)ξ0
→ϕ
δ(x)y,z .
(4.37)
DERIVATIONS OF QUASI -ALGEBRAS 1089 Since(π(x)◦Hn)π(y)ξ0,π(z)ξ0 = Hnπ(y)ξ0,π(x∗z)ξ0, we deduce that, tak- ingy=1,(π(x)◦Hn)ξ0,π(z)ξ0 = Hnξ0,π(x∗z)ξ0 → Hξ0,π(x∗z)ξ0because of the assumption onHn. Then, since
ϕδ(x),z
=iH◦π(x)ξ0,π(z)ξ0
−
π(x)◦H
ξ0,π(z)ξ0
, (4.38)
we get, by (4.37), that(Hnπ(x))ξ0,π(z)ξ0 → (Hπ(x))ξ0,π(z)ξ0for allx,z∈Ꮽ0. ThenHnconverges weakly toH.
Example4.4(a radiation model). In this example, the representationπis just the identity map. We consider a model ofnfree bosons, [6], whose dynamics is given by the Hamiltonian,H=n
i=1a†iai. Hereaianda†i are, respectively, the annihilation and creation operators for theith mode. They satisfy the following CCR:
ai,a†j
=1δi,j. (4.39)
LetQLbe the projection operator on the subspace ofᏴwith at mostLbosons. This oper- ator can be written considering the spectral decomposition ofH(i)=a†iai=∞
l=0lEl(i). We have QL =n
i=1
L
l=0El(i). We now define a bounded operatorHL in Ᏼ by HL = QLHQL. It is easy to check that, for any vector ΦM with M bosons (i.e., an eigen- state of the number operator N=H=n
i=1a†iai with eigenvalue M), the condition supLHLΦM<∞is satisfied. In particular, for instance, supLHLΦ0 =0. It may be worth remarking that all the vectors ΦM are cyclic. Denoting withδL the derivation implemented byHLand byδthe one implemented byH, it is clear that all the assump- tions ofProposition 4.3are satisfied so that, in particular, the weak convergence ofHL
toHfollows. This is not surprising since it is known thatHLconverges toHstrongly on a dense domain [6].
Example4.5(a mean-field spin model). The situation described here is quite dif- ferent from the one in the previous example. First of all (see [8,10]) there exists no Hamiltonian for the whole physical system but only for a finite volume subsystem:
HV =(1/|V|)
i,j∈Vσ3iσ3j, whereiand j are the indices of the lattice site,σ3i is the third component of the Pauli matrices,Vis the volume cutoff, and|V|is the number of the lattice sites inV. It is convenient to introduce themean magnetizationoperator σ3V=(1/|V|)
i∈Vσ3i. We indicate with↑iand↓ithe eigenstates ofσ3iwith eigenvalues +1 and−1, respectively. We defineΦ↑= ⊗i∈V ↑i. It is clear thatσ3VΦ↑=Φ↑, which im- plies thatHVΦ↑= |V|Φ↑, which in turn implies that supVHVΦ↑ = ∞. This means that the cyclic vectorΦ↑does not satisfy the main assumption ofProposition 4.3, and for this reason, nothing can be said about the convergence ofHV. However, it is possible to consider a different cyclic vector
Φ0= ···⊗ ↑j−1⊗ ↓j⊗ ↑j+1⊗ ↓j+2⊗··· (4.40)
which is again an eigenstate ofσ3V. Its eigenvalue depends on the volumeV. However, it is clear thatσ3VΦ0 =(1/|V|)Φ0V, whereVcan take only the values 0, 1. Analo- gously, we haveHVΦ0 =(1/|V|)Φ02V→0. This means that this vector satisfies the assumptions ofProposition 4.3 so that the derivationδV(·)=i[HV,·]converges to a derivationδwhich is spatial and implemented byH, and thatHV is weakly convergent toH.
As we see, contrary toExample 4.4, 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 ofδ, its spatiality, and convergence ofHV to a limit operator. It is also worth remarking that the same conclusions could also be found replacingΦ0with any vector which can be obtained as a local perturbation ofΦ0itself.
Remark 4.6. All the results we have proved above can be specialized to CQ∗- algebras, which can be considered as a particular example of locally convex quasi∗- algebras. The main difference in this case concerns statement(c)ofProposition 4.3:
the weak convergence ofHntoH, in this case, is replaced by a strong convergence. In more details, referring to the example ofSection 2and callingΩ∈Ᏼ+1a cyclic vector, we can prove that if(Hn−H)Ω−1→0, then(Hn−H)AΩ−1→0 for allA∈B(Ᏼ+1).
The following result gives an interplay between the results of this section and of the previous sections. In particular, we consider now the possibility of extending the domain of definition of the derivationδ(as we did in Section 3) defined as a limit of a net of derivationsδn (as we have done in this section). For this, we first need the following definition.
Definition4.7. Let(Ꮽ[τ],Ꮽ0)be a locally convex quasi∗-algebra. A sequence{δn} of∗-derivations is called uniformlyτ-continuous if for any continuous seminormpon Ꮽ, there exists a continuous seminormqonᏭsuch that
p δn(x)
≤q(x) ∀x∈Ꮽ0,∀n∈N. (4.41)
We can now prove the following.
Proposition4.8. Letδbe theτ-limit of a uniformlyτ-continuous sequence{δn}of
∗-derivations such that the set
Ᏸ(δ)=
x∈Ꮽ0:∃τ−lim
n δn(x)
(4.42)
isτ-dense inᏭ0. Then,δis a∗-derivation and, denoting withδ˜nthe continuous extension ofδntoᏭ, we have{x∈Ꮽ:∃τ−limnδ˜n(x)} =Ꮽ.
Proof. The proof thatδis a∗-derivation is trivial.
Letabe a generic element inᏭ. Since, by assumption,Ᏸ(δ)isτ-dense inᏭ0, and therefore inᏭ, there exists a net{xα} ⊂D(δ) τ-converging toa. This means that for any continuous seminormspand for any >0, there existsαp,such thatp(a−xα) <
for allα > αp,.
