Some remarks on the GNS representations of topological ∗ -algebras
aa r X i v : . [ m a t h - ph ] F e b Some remarks on the GNS representations of topological ∗ -algebras S. M. Iguri ∗ and M. A. Castagnino † Instituto de Astronom´ıa y F´ısica del Espacio (CONICET-UBA).C. C. 67 - Suc. 28, 1428 Buenos Aires, Argentina.andDpto. de F´ısica, FCEyN, Universidad de Buenos Aires.Ciudad Universitaria Pab. I, 1428 Buenos Aires, Argentina.
Abstract
After an appropriate restatement of the GNS construction for topological ∗ -algebras weprove that there exists an isomorphism among the set Cycl( A ) of weakly continuous stronglycyclic ∗ -representations of a barreled dual-separable ∗ -algebra with unit A , the space Hilb A ( A ∗ )of the Hilbert spaces that are continuously embedded in A ∗ and are ∗ -invariant under the dualleft regular action of A and the set of the corresponding reproducing kernels. We show thatthese isomorphisms are cone morphisms and we prove many interesting results that follow fromthis fact. We discuss how these results can be used to describe cyclic representations on moregeneral inner product spaces.2000 Mathematics Subject Classification : 16G99, 46H15, 47L90, 81P99, 81R15.Keywords : topological ∗ -algebra, cyclic representation, GNS construction. ∗ e-mail: [email protected] † e-mail: [email protected] Introduction
Quantum statistical mechanics and quantum field theories are believed to be fully described inpurely algebraic terms, the so-called C ∗ -algebraic approach (see [1, 2, 3] for textbooks and [4, 5] forrecent reviews on the subject) being the most appealing one. Despite of the successful aspects ofthe C ∗ -algebraic approach, in order to find abstract counterparts for all observable magnitudes inan algebraic approach it is mandatory to consider ∗ -algebras with less restrictive topologies thanthe ones derived from C ∗ -norms [6, 7, 8, 9]. Moreover, if quantum gauge theories are also assumedto be described in algebraic terms, the appropriate representation spaces would be more generalinner product spaces than Hilbert spaces [10] and in that case there is no compelling reason tobelieve that the ∗ -algebra describing the observable content of the theory should be a normableone.One of the fundamentals of the C ∗ -algebraic approach is the Gelfand-Naimark-Segal (GNS)theorem. The so-called GNS construction is an important tool from both the physical and thestrictly mathematical points of view. It characterizes the building blocks of the representationtheory of C ∗ -algebras, i.e., their cyclic representations and defines in this way the bridge betweenthe formalism and the physical reality.During the 70’s the systematic study of the representations of algebras of unbounded operatorsbegun with Powers [11, 12, 13]. In the seminal paper of Powers there is a version of the GNStheorem but unfortunately it makes no mention on the topological properties of the represented ∗ -algebra. The lack of information on this topology gives to the construction generality but, on theother hand, it restricts its scope. There were other statements of the GNS theorem during the lastyears [14, 15, 16, 17] assuming more or less restrictive conditions on the topological nature of the ∗ -algebra and there are even versions of the GNS construction on non-necessarily definite positiveinner product spaces [18, 19, 20]. The aim of this paper is to complement all these treatments.2e will restate the GNS theorem for a wide class of topological ∗ -algebras this restatementallowing us to prove that there exists a continuous bijection between the space of GNS represen-tations and a set of Hilbert spaces continuously embedded in the dual space of the ∗ -algebra inhands, an idea already suggested in [21, 22]. More explicitly, if A is a barreled dual-separable ∗ -algebra with unit we will prove that the set Cycl( A ) of weakly continuous strongly cyclic ∗ -representations of A is isomorphic to the set Hilb A ( A ∗ ) of the Hilbert subspaces of A ∗ that are ∗ -invariant under the left dual regular action of A on A ∗ . In turn this bijection can be extendedto a multiple isomorphism among these spaces, the space of continuous positive functionals over A and the corresponding space of invariant positive operators.This characterization of the space of GNS representations will also allow us to transport thecone structure already defined on Hilb A ( A ∗ ) to Cycl( A ) and to prove some remarkable conse-quences of this fact. Let us mention that this strictly convex cone structure on Cycl( A ) can beused for describing GNS representations over spaces with non-necessarily positive definite innerproduct [23] and it could be useful for studying deformation theory of GNS representations [24].The paper is organized as follows. In section 2 we review the main results of Schwartz’stheory of Hilbert subspaces [25] and their associated reproducing kernels, the most importantone being the natural bijection between the set of Hilbert subspaces of a given topological spaceand the set of positive operators mapping its dual on it. In section 3 we present those aspectsof the representation theory of topological algebras needed in the sequel. We have essentiallyfollowed [26] but some concepts were slightly modified. In section 4 we show that for a barreleddual-separable ∗ -agebra there is a one-to-one correspondence between its GNS representations andthose Hilbert subspaces of its dual that are invariant under the dual left regular action. We alsoshow that this map is a cone morphism for the cone structure already defined on this last spacein [25]. In section 5 we derive many consequences of the previous sections. Finally, in section 6we present our conclusions. 3 Hilbert subspaces and reproducing kernels
In this section we will review some definitions and we will introduce a few items of notation con-cerning the theory of the Hilbert spaces that can be continuously embedded in a quasi-complete locally convex Hausdorff separable vector space over the field of complex numbers C [25]. We willdenote any space fulfilling these requirements by E .Let us first recall the definition of a Hilbert subspace of E . A linear subspace H ⊆ E is calleda Hilbert subspace of E whenever H is equipped with a definite positive inner product ( · , · ) turningit into a Hilbert space and the inclusion of H into E is a continuous map, the norm k · k = ( · , · ) / defining a topology on H finer than the one induced by E .When dealing with the Hilbert subspaces of E it is convenient to consider the topological anti-dual space of E instead of its dual E ′ , the reason being that every Hilbert space can be canonicallyidentified through the Riesz isomorphism with its anti-dual. The anti-dual space E ∗ of E is theconjugate of E ′ , i.e., it is defined as a topological vector space over C with an anti-isomorphismmapping it onto E ′ . Under this map the canonical bilinear form on E ′ × E becomes a sesquilinearmap on E ∗ × E which we will denote as h x | φ i for all x ∈ E ∗ and all φ ∈ E . Notice that thisbracket is anti-linear in its first argument while it is linear in the second one. We will refer tothe elements of E ∗ as functionals over E even when it should be remembered that they are notelements of the dual. If E is a Hilbert space and, as we have already said, we identify the elementsof E with those of E ∗ , the duality bracket reduces to the inner product on E .As in the case for a strictly dual system, given a continuous map T : E → F we introduce its adjoint as the linear map T ∗ : F ∗ → E ∗ defined by the identity h T ∗ x | φ i = h x | T φ i for all x ∈ F ∗ and all φ ∈ E . It is a continuous map provided that E ∗ and F ∗ are equipped with their weak or,alternatively, their strong dual topologies. If both spaces are Hilbert spaces, the adjoint of a map A topological vector space E is said to be quasi-complete if every bounded closed subset of E is complete [27]. Notice that, up to isomorphisms, E ∗ is unique. T ∗ = T † . In the case that T is an anti-linear operator,the expression defining its adjoint must be replaced by h T ∗ x | φ i = h x | T φ i for all x ∈ F ∗ and all φ ∈ E .Let H be a Hilbert subspace of E and let J be the inclusion map of H into E . By theRiesz representation theorem, given x ∈ E ∗ , there exists a unique element J ∗ x ∈ H such that( J ∗ x, ξ ) = h x | J ξ i for all ξ ∈ H . Let us denote by Hx = J J ∗ x the same element regarded as anelement of E . The operator H mapping E ∗ into E is called the reproducing operator of H . Itis a continuous operator when E and E ∗ are equipped with their weak topologies σ ( E ∗ , E ) and σ ( E, E ∗ ), respectively.The reproducing operators of Hilbert subspaces have many remarkable properties. For in-stance, they are all hermitian , where by a hermitian operator we mean a linear map T : E ∗ → E satisfying h x | T y i = h y | T x i , for any pair x, y ∈ E ∗ . In fact, for all x, y ∈ E ∗ we have h x | Hy i =( J ∗ x, J ∗ y ) = ( J ∗ y, J ∗ x ) = h y | Hx i . Moreover, setting x = y in the last expression it follows that H is a positive operator , i.e., h x | Hx i = ( J ∗ x, J ∗ x ) ≥ x ∈ E ∗ . In addition, it is possible toprove a Cauchy-Schwartz like identity, i.e., |h x | Hy i| ≤ h x | Hx i h y | Hy i for all x, y ∈ E ∗ .Let us denote by L ( E ) the set of all continuous operators mapping E ∗ into E , these spacesbeing endowed with their weak topologies, and let L + ( E ) be the proper strictly convex cone ofpositive elements of L ( E ). Reproducing operators belong to L + ( E ).The map that assigns to each element T ∈ L ( E ) the form given by h x | T y i with x, y ∈ E ∗ is analgebraic isomorphism mapping L ( E ) onto the space of separately weakly continuous sesquilinearforms on E ∗ , i.e., the kernels on E ∗ . As it should be clear, when this map is restricted to L + ( E )it gives an isomorphism onto the positive kernels on E ∗ . In this context, if the operator in L + ( E )is the reproducing operator of a Hilbert subspace H of E , the corresponding sesquilinear form on E ∗ is called the reproducing kernel of H in E .Let Hilb( E ) be the set of all Hilbert subspaces of E . As it was already proved by Schwartz525], it is possible to endow Hilb( E ) with a proper strictly convex cone structure. Let us brieflyoutline the corresponding definitions. Sum of Hilbert subspaces:
Let H and H be two Hilbert subspaces of E , J and J beingthe respective inclusion maps. Let H × H be the Hilbert space product of H and H . Finally,let Φ : H × H → E be the continuous map given by Φ( ξ , ξ ) = J ξ + J ξ and consider thequotient space ( H × H ) / ker(Φ) equipped with its canonical Hilbert space structure. The sumof H and H is defined as the image space Φ( H × H ) ⊆ E endowed with the unique normthat makes the canonical linear bijection between ( H × H ) / ker(Φ) and Φ( H × H ) an isometricisomorphism. We will denote this space by H + H . The norm on H + H is explicitly givenby k ξ k = inf {k ξ k + k ξ k } where k · k (resp., k · k ) is the norm on H (resp., H ) and theinfimum is taken over those pairs ( ξ , ξ ) ∈ H × H such that ξ = Φ( ξ , ξ ). If ker(Φ) = 0, then H ∩ H = { } and H + H is simply the Hilbert space direct sum of both spaces and in thatcase we will write, as it is usual, H ⊕ H .The definition of the sum of two Hilbert subspaces does agree with a more general constructionconcerning the Hilbert subspaces of spaces that are images under continuous mappings. Let E and F be two quasi-complete locally convex Hausdorff separable vector spaces over C and let T : E → F be a continuous linear map. Consider a Hilbert subspace H of E and let us denoteby J the inclusion map of H into E . Since T J : H → F is also a continuous map, its kernel is aclosed linear subspace of H . The image space of H under T J , endowed with the Hilbert structuremaking the restriction of
T J to ker(
T J ) ⊥ a linear isometry, is a Hilbert subspace of F . We willsimply denote this space by T ( H ). Multiplication by non-negative real numbers in Hilb ( E ) : The multiplication law onHilb( E ) by non-negative real numbers is defined as follows. Let H be a Hilbert subspace of E and let λ be a positive real number. The space λ H is the Hilbert subspace of E with underlyinglinear space equal to H and the norm on λ H being defined as (1 / √ λ ) times the original norm on6 . The action of R > on Hilb( E ) is extended to R ≥ setting λ H equal to { } when λ = 0. Order in Hilb ( E ) : Finally, a partial order, compatible with the structures given above, isdefined on Hilb( E ) in the following way. If H and H are two Hilbert subspaces of E , we willwrite H ≤ H if H ⊆ H and the inclusion of H into H is an operator of norm at most 1, i.e., H belongs to Hilb( H ).If H and H are two Hilbert subspaces of E it is easy to check that H ∩H = { } if and only if H and H are mutually excluding for the order relation in Hilb( E ), i.e., if for any Hilbert subspace K of E such that K ≤ H and K ≤ H it follows that K = { } . We will say that a Hilbert subspace H is indecomposable if it does not admit a non-trivial decomposition as a direct sum of Hilbertsubspaces, i.e., if for any decomposition H = H + H , H and H being two mutually excludingHilbert subspaces of E , it is possible to prove that H = { } or H = { } . We will denote by [0 , H ]the interval in Hilb( E ) between { } and H , i.e., [0 , H ] = {H ′ ∈ Hilb( E ) : 0 ≤ H ′ ≤ H} . We will saythat H is an extremal element of Hilb( E ) if the interval [0 , H ] equals { λ H ∈
Hilb( E ) : 0 ≤ λ ≤ } .Notice that every extremal element of Hilb( E ) is an indecomposable Hilbert subspace of E butthe converse of this statement is not generally true.We are now in a position to recall the most important result of Schwartz Hilbert subspacestheory. We will only sketch the proof and we refer to [25] for more details. Theorem 2.1.
The map that assigns to each Hilbert subspace of E its reproducing operator is abijection from Hilb ( E ) onto L + ( E ) .Proof. Let H be a Hilbert subspace of E and let us denote by J the inclusion of H in E . We willprove that H is determined by its reproducing operator H . First, notice that J being an inclusion,it follows that J ∗ is a dense range projection, i.e., J ∗ E ∗ is a dense linear subspace of H . On theother hand, there is no element in H orthogonal to J ∗ E ∗ but the origin. Since the unit ball B of H is weakly compact, J B is weakly closed in E , and this set being convex, it is closed for the7riginal topology on E . It follows that J B is the closure in E of the set { Hx ∈ E : h x | Hx i / ≤ } and this proves that H is fully determined by H . Moreover, given φ ∈ E , it belongs to H if andonly if sup {|h x | φ i| / h x | Hx i / } < + ∞ , where the supremum is taken over the elements x ∈ E such that h x | Hx i >
0. The value of this expression equals k φ k .Now, let us prove that given H ∈ L + ( E ), we can define a Hilbert subspace H of E in sucha way that its reproducing operator equals it. Let the quotient space E ∗ / ker( H ) be equippedwith the Hilbert space structure derived from the sesquilinear form induced by H . The canonicalinjection of E ∗ / ker( H ) into E has a one-to-one extension to the completion. It is in order toprove this statement that it is essential to assume that E is a quasi-complete separable vectorspace [25]. The image space under this mapping, endowed with the unique Hilbert space structureturning it into an isometry, is a Hilbert subspace of E , its reproducing operator being H . Proposition 2.2.
The bijection between L + ( E ) and Hilb ( E ) is a cone morphism for the usualcone structure on L + ( E ) and the cone structure we have already introduced for Hilb ( E ) . Moreexplicitly, let H , H and H be three Hilbert subspaces of E , let H , H and H be their respectivereproducing operators. Let λ be a non-negative real number. Then:1. H = H + H if and only if H = H + H ,2. H = λ H if and only if H = λH , and3. H ≤ H if and only if H ≤ H .Proof. See [25], Prop. 11-13, p. 158-160. ∗ -algebras The purpose of this section is to restate the classical GNS theorem for topological ∗ -algebras. Inorder to do that we will first recall some basic facts on representations and ∗ -representations of8 -algebras on Hilbert spaces by (non necessarily bounded) linear operators. We will mainly follow[26] except for some minor changes in some definitions that will be justified in the rest of thepaper. We will always denote by A an associative algebra over C . When assuming A to be analgebra with unit, we will denote the unit in A by e .A representation π of A on a Hilbert space H is a map from A into a set of linear operators,all of them defined on a common domain D , such that the following conditions are fulfilled:1. D is a dense subspace of H ,2. D is invariant under the action of A , i.e., π ( x ) D ⊆ D for all x ∈ A , and3. A acts linearly and multiplicatively on D , i.e., for all x, y ∈ A and all λ ∈ C we have: π ( x + y ) = π ( x ) + π ( y ), π ( λx ) = λπ ( x ) and π ( xy ) = π ( x ) π ( y ).If the algebra has a unit, it is also assumed that4. π ( e ) equals the identity operator on D , i.e., π ( e ) = Id D .Let π and π be two representations of A on the Hilbert spaces H and H and let D and D be their respective domains. We will say that π is an algebraic extension of π or that π is an algebraic subrepresentation of π , and we will write π ⊆ π , if D ⊆ D , H is a linear subspaceof H and π ( x ) ↾ D equals π ( x ) for every x ∈ A . If, in addition, the scalar product on H isthe restriction to H of the scalar product on H , i.e., π ( x ) ⊆ π ( x ) for all x ∈ A , we will saythat π is an extension of π or that π is a subrepresentation of π . In this last case we will write π ≤ π . Remark 3.1.
In the next section we will find that the conditions imposed on H and H for π to be a subrepresentation of π can be conveniently modified. We will consider a less restrictivenotion of extension of a representation asking H to be a Hilbert subspace of H , i.e., we will ssume that H is continuously embedded in H , the corresponding inclusion being an operator ofnorm at most . Let us notice that all the contents of the present section will remain being valid. In order to define a concept analogous to the one of a closed operator but for a representation,we will proceed as usual endowing the domain of a given representation with a topology inducedby the action of the algebra on it. Let π be a representation of A on a Hilbert space H and let D be its domain. The graph topology on D is the locally convex topology generated by the family ofseminorms { p x = k π ( x ) ·k} where k · k is the norm on H and x runs over A . The graph topologycan be characterized as the weakest locally convex topology on D which makes the embedding of D into its completion relative to the topology determined by the norm k · k + k π ( x ) · k a continuousmapping for every x ∈ A . In this context, the graph topology can be viewed as a projectivetopology in the sense of the theory of locally convex spaces [27]. When A has a unit, the graphtopology is always finer than the one induced by H on D . Clearly, the graph topology is generatedby a single norm, the one on H , if and only if the image of each element of A through π can beextended to a bounded operator on H .If D is complete when equipped with the graph topology we will say that π is a closed rep-resentation . A representation π will be called a closable representation of A if π ( x ) is a closableoperator on D for all x ∈ A .Given a closable representation of A on a Hilbert space H , let π ( x ) be the closure of theoperator π ( x ), the domain of π ( x ) being the common domain D for all x ∈ A . Let us denote by D x the domain of π ( x ) for every x ∈ A . Finally, let D be the completion of D in T x ∈ A D x relativeto the graph topology. D is the domain of a closed representation π of A in H defined as π ( x ) = π ( x ) ↾ D (1)for all x ∈ A . This representation is called the closure of π , and it is the minimal closed extensionof it. Of course [11] π is closed if and only if π is closable and it equals π .10et us now assume that A is a ⋆ -algebra. Like in the case of a single operator acting on aHilbert space, we will define an adjoint of a given representation.Suppose that π is a representation of A on a Hilbert space H and let D be its domain. For all x ∈ A , let π ( x ) ∗ be the adjoint of π ( x ) and let D ∗ x be its domain. Further, let D ∗ = T x ∈ A D ∗ x andlet us denote by H ∗ the completion of D ∗ in H . The adjoint representation of π is defined as therepresentation π ∗ of A on H ∗ with domain D ∗ given by π ∗ ( x ) = π ( x ∗ ) ∗ ↾ D ∗ (2)for all x ∈ A .The adjoint of a given representation is always a closed representation and it is the largest oneamong those representations ˜ π of A on H ∗ with domain ˜ D that satisfies ( ξ, ˜ π ( x ∗ ) χ ) = ( π ( x ) ξ, χ )for all x ∈ A , χ ∈ ˜ D and ξ ∈ D .We will say that a representation π is adjointable (resp. biclosed ) if H = H ∗ (resp. if it equalsits biadjoint representation, i.e., if π ⋆⋆ = ( π ∗ ) ∗ ).All concepts above suggest the following definition originally introduced by Powers in [11]. Let π be a representation of a ∗ -algebra A on a Hilbert space H and let D be its dense domain. Wewill say that π is a hermitian representation , or simply a ∗ -representation of A , if π ≤ π ∗ , in otherwords, if for all χ, ξ ∈ D and all x ∈ A , π satisfies( ξ, π ( x ∗ ) χ ) = ( π ( x ) ξ, χ ) (3)Notice that every ∗ -representation is necessarily adjointable.If A is a Banach ∗ -algebra then a ∗ -representation π of A on a Hilbert space H is closed if andonly if D equals H . On the other hand, if π is a ∗ -representation of a ∗ -algebra A on a Hilbert space H and D = H , it follows from the closed graph theorem that π is a bounded representation, i.e., π maps A into bounded operators on H . These facts are clear evidences that the previous definitionis a consistent generalization of the usual concept of ∗ -representation by bounded operators.11he adjoint representation of a given ∗ -representation π may fail to be a ∗ -representation as itis the case of the adjoint of a single hermitian operator acting on a Hilbert space. But as we havealready said it is actually a closed representation extending π . Moreover, every ∗ -representationextending π is necessarily a restriction of π ∗ .We will say that a ∗ -representation π of a ∗ -algebra A is a maximal (resp. self-adjoint , resp. essentially self-adjoint ) ∗ -representation if every ∗ -representation extending π equals it (resp. if π = π ∗ , resp. if its closure is a self-adjoint representation.)Some general properties of ∗ -representations are collected in the following proposition. Theproof can be found in [26]. Proposition 3.2.
Let π be a ∗ -representation of A in a Hilbert space H with domain D .1. π and π ⋆⋆ are both ∗ -representations of A , and π ≤ π ≤ π ⋆⋆ ≤ π ∗ . Moreover, one has that D = T x ∈ A D x .2. π is self-adjoint if and only if D ∗ ⊆ D .3. π ∗ is self-adjoint if and only if it is a ∗ -representation.4. If π is self-adjoint then any ∗ -representation extending π in the same Hilbert space equals it. Among those representations that usually appear in quantum statistical mechanics and quan-tum field theories, cyclic ones play a particularly relevant role. When dealing with algebras ofnon-necessarily bounded operators two definitions of cyclicity are available.Let π be a representation of an algebra A on a Hilbert space H and let D be its domain. Avector ξ ∈ D is said to be a cyclic vector if the set π ( A ) ξ = { π ( x ) ξ : x ∈ A } is dense in H .In that case we will say that the representation is a cyclic representation . If π ( A ) ξ is dense in D when endowed with the graph topology, then ξ is said to be a strongly cyclic vector of π . Arepresentation having a strongly cyclic vector will be called a strongly cyclic representation .12et π be a ∗ -representation of a ∗ -algebra A on a Hilbert space H and let ξ ∈ D . Let ˆ π be therestriction of π to π ( A ) ξ . It follows that ξ is strongly cyclic if and only if the closure of ˆ π is anextension of π .From now on, we will consider the case in which A is a topological ∗ -algebra.Let ρ be a functional on A . If for all x ∈ A such a functional satisfies ρ ( x ∗ x ) ≥ positive functional . Continuous positive functionals over A conform a proper strictlyconvex weakly closed cone in A ∗ that we will denote A ∗ + . While the sum and the multiplicationlaw by non-negative real numbers are the ones induced by restricting the ordinary sum and scalarmultiplication on A ∗ , the order on A ∗ + is defined as follows: given ρ , ρ ∈ A ∗ + , one writes ρ ≤ ρ if and only if ρ ( x ∗ x ) ≤ ρ ( x ∗ x ) for all x ∈ A .Let us recall that A being a unital ∗ -algebra and ρ being a positive functional on A , for all x, y ∈ A , it follows [15] that ρ ( x ∗ y ) = ρ ( y ∗ x ) ∗ . In particular, ρ is hermitian , i.e., ρ ( x ∗ ) = ρ ( x ) ∗ for all x ∈ A . Moreover, for all x, y ∈ A it follows that | ρ ( x ∗ y ) | ≤ ρ ( x ∗ x ) ρ ( y ∗ y ) and ρ is Hilbertbounded , i.e., there exists a constant B satisfying, for all x ∈ A , | ρ ( x ) | ≤ Bρ ( x ∗ x ). The Hilbertbound k ρ k ≡ sup {| ρ ( x ) | : x ∈ A, ρ ( x ∗ x ) ≤ } equals ρ ( e ), where e is the unit in A .We will say that a continuous positive functional on A is an extremal element of A ∗ + ifit is indecomposable as a sum of continuous positive functionals that are not multiples of ρ .Equivalently, a continuous positive functional on A is extremal in A ∗ + if and only if the interval[0 , ρ ] = { ρ ′ ∈ A ∗ + : 0 ≤ ρ ′ ≤ ρ } equals { λρ : 0 ≤ λ ≤ } .Finally, we can state the GNS theorem for topological ∗ -algebras. Its proof mainly followsthe steps of the original proof for C ∗ -algebras (see, f.e., [1]). More details for the case of general ∗ -algebras can be found in [11], [15] and [26]. Notice that they all discuss pre- ∗ -representationsand only consider ∗ -representations for the normable case. This fact distinguishes our version ofthe theorem from theirs. Recall that we are calling functionals those elements in A ∗ . heorem 3.3. For each continuous positive functional ρ on a topological ∗ -algebra A with unitthere is a closed weakly continuous strongly cyclic ∗ -representation π of A on a Hilbert space H with domain D such that ρ ( x ) = ( π ( x ) ξ, ξ ) (4) for all x ∈ A , ξ ∈ D being a strongly cyclic vector of π . The representation π is determined by ρ up to unitary equivalence. Furthermore, if ρ is an extremal functional then the correspondingrepresentation is topologically irreducible.Proof. Let ρ be a continuous positive functional on A and consider the quotient space A ρ = A/N ρ ,where N ρ = { x ∈ A : ρ ( y ∗ x ) = 0 for all y ∈ A } is the so-called Gelfand ideal of ρ . Let us denoteby φ ρ the canonical projection of A onto A ρ and let ( · , · ) : A ρ × A ρ → C be the form on A ρ definedby ( φ ρ x, φ ρ y ) = ρ ( y ∗ x ) for all x, y ∈ A . It is straightforward to check that this form is a positivenon-degenerate sesquilinear form on A ρ endowing it with a pre-Hilbert structure. We will denoteby H the completion of A ρ with respect to the norm k · k = ( · , · ) / .Since N ρ is a left ideal of A , the map π assigning to every element x ∈ A the (non-necessarilybounded) operator π ( x ) on H with domain A ρ given by π ( x ) φ ρ y = φ ρ ( xy ) for all y ∈ A defines arepresentation of A . It is actually a ∗ -representation of A (see [21] for the details). The closure π of π is the closed ∗ -representation of A whose existence is claimed in the theorem. Recall that thedomain D of π is the completion of A ρ with respect to the graph topology. The weak continuityof π is a direct consequence of the continuity of ρ . On the other hand, setting ξ = φ ρ e , it followsthat π ( x ) ξ = π ( x ) φ ρ e = φ ρ x and, therefore, π ( A ) ξ equals A ρ , which is a dense subspace of D ,showing that π is a strongly cyclic representation of A and that ξ is a strongly cyclic vector of π .Finally, for all x ∈ A we have that ( π ( x ) ξ, ξ ) = ( π ( x ) φ ρ e, φ ρ e ) = ( φ ρ x, φ ρ e ) = ρ ( x ), and the firststatement of the theorem is proved.In order to prove that π is determined by ρ up to unitary equivalence it is necessary to14how that there exists a unitary operator intertwining any two ∗ -representations satisfying (4).Explicitly, let π and π be two closed weakly continuous strongly cyclic ∗ -representations of A , let H (resp., H ) be the Hilbert space on which π (resp., π ) acts, let D (resp., D ) itsdomain, and let ξ ∈ D (resp., ξ ∈ D ) be a strongly cyclic vector of π (resp., π ) such that( π ( x ) ξ , ξ ) = ( π ( x ) ξ , ξ ) , where ( · , · ) (resp., ( · , · ) ) is the inner product on H (resp., H ).We need to show that there exists a unitary operator U mapping H onto H such that, for all x ∈ A , U π ( x ) = π ( x ) U . In [11] it was proved that such an operator is obtained by extendingthe operator U : D → D given by U π ( x ) ξ = π ( x ) ξ for all x ∈ A , to an operator from H into H .The proof of the last statement of the theorem concerning extremal functionals on A andtopologically irreducible representations of A can be found in [26].Given a continuous positive functional ρ on A , the representation built as in the theorem iscalled the GNS representation of A associated with ρ . The vector ξ in (4) is sometimes referredas a normalizing vector of π .GNS representations are usually constructed from states instead of positive functionals. Statesover A are defined as those continuous functionals that are positive and, in addition, satisfy ρ ( e ) =1. The set of states is a weakly closed convex section of A ∗ + . Pure states are defined in analogy withextremal positive functionals as those that are indecomposable as convex combinations of otherstates. In that context it is possible to prove a stronger statement than the last one in Theorem3.4. In fact, if GNS representations are built upon states, the cyclic vector ξ is necessarily anormal vector, i.e., ( ξ, ξ ) = 1, and the space of pure states is in one-to-one correspondence withthe collection of unitary equivalence classes of topologically irreducible GNS representations of A .At this point it will be convenient to introduce the space Cycl( A ) of those pairs of the form( π, ξ ) where π is a closed weakly continuous strongly cyclic ∗ -representation of A and ξ is a15articular strongly cyclic vector of π . We will endow this space with an equivalence relation asfollows: given ( π , ξ ) and ( π , ξ ) in Cycl( A ) we will say that they are unitarily equivalent , andwe will denote it by ( π , ξ ) ∼ ( π , ξ ), if there exists a unitary operator U intertwining π and π , i.e., π ( x ) U = U π ( x ) for all x ∈ A , and, in addition, we have that ξ = U ξ . Notice that thisnotion of unitary equivalence for cyclic ∗ -representations is stronger than the usual one where norequirements on the corresponding cyclic vectors are imposed.The motivations for introducing the space Cycl( A ) and their unitary equivalence classes arisefrom both physical and mathematical interests. From a strictly physical point of view, thereare cases (f.e., when a symmetry is spontaneously broken) where the usual notion of unitaryequivalence of ∗ -representations is not sufficient to ensure a complete identification of two physicalsituations. In a more mathematical context, the space Cycl( A ) and their unitary equivalenceclasses are relevant in our discussion since the quotient Cycl( A ) / ∼ is precisely the space on whichthe GNS mapping becomes a bijection as it is proved in the following proposition. Proposition 3.4.
The map assigning to each continuous positive functional ρ on a topological ∗ -algebra with unit A the GNS representation of A associated with ρ defines, up to unitary equiv-alence, a one-to-one mapping from A ∗ + onto Cycl ( A ) .Proof. Theorem 3.4 asserts that the GNS mapping actually maps, up to unitary equivalence, A ∗ + into Cycl( A ). Therefore, we only need to check that this map is, in fact, a bijection. Let ρ and ρ be two continuous positive functionals on A and let us assume that the correspondingGNS representations π and π are unitarily equivalent as elements of Cycl( A ). Let us denoteby H and H the Hilbert spaces on which π and π act, and let ξ and ξ be their respectivestrongly cyclic vectors. It follows that there exists a unitary operator U mapping H onto H satisfying U π ( x ) = π ( x ) U for all x ∈ A and U ξ = ξ . But then, for all x ∈ A , ( π ( x ) ξ , ξ ) =( U π ( x ) ξ , U ξ ) = ( π ( x ) U ξ , U ξ ) = ( π ( x ) ξ , ξ ) , i.e., ρ = ρ . On the other hand, given16n arbitrary element π in Cycl( A ) with strongly cyclic vector ξ , the GNS representation of A associated with the positive functional ρ = ( π ( · ) ξ, ξ ) is unitarily equivalent to π , the proof beingidentical to the one of the unicity statement in Theorem 2.The previous proposition shows that the space of unitary equivalence classes of Cycl( A ) can besuitably endowed with a proper strictly convex cone structure, the one inherited from A ∗ + throughthe GNS map. In the next section we will explicitly define this structure after relating the GNSrepresentations of a ∗ -algebra with some of the Hilbert subspaces embedded in its anti-dual space.Throughout the rest of the paper we will identify the elements in Cycl( A ) with their respec-tive canonical GNS representatives and we will omit any explicit reference to the cyclic vectorassociated with each representation in Cycl( A ). Therefore, instead of saying that ( π, ξ ) belongsto the unitary equivalence class in Cycl( A ) corresponding to the GNS representation associatedwith the functional ρ = ( π ( · ) ξ, ξ ) we will simply say that π is an element of Cycl( A ). In the previous sections we have recalled the theory of the Hilbert subspaces of a quasi-completelocally convex Hausdorff separable vector space E and we have discussed the GNS constructionfor a topological unital ∗ -algebra A . In this section, and following an idea already suggested in[21] and [22], we will establish a connection between both approaches showing that there existsa bijection between Cycl( A ) and a particular subcone of Hilb( A ∗ ), this statement being valid fora wide class of topological ∗ -algebras. This characterization of Cycl( A ) will further allow us toexplicitly endow it with a cone structure in such a way that this bijection actually becomes a conemorphism.Let us begin with some general remarks on the continuous representations of a topological ∗ -algebra over a vector space and their restrictions to its Hilbert subspaces. Let π be a strongly17ontinuous representation of A on E , i.e., a separately continuous map ( x, φ ) → π ( x ) φ from A × E into E such that π ( xy ) φ = π ( x ) π ( y ) φ for all x, y ∈ A and all φ ∈ E . Let us denote by π ∗ the dual representation of π , i.e., the representation of A on E ∗ defined by h π ∗ ( x ) y | φ i = h y | π ( x ∗ ) φ i for all x ∈ A , all y ∈ E ∗ and all φ ∈ E . The representation π ∗ should not be confused with theHilbert adjoint representation we have defined in Section 3. As it is the case for π , π ∗ is also astrongly continuous representation of A whenever E ∗ is equipped with the weak topology, as wewill assume from now on.Given a Hilbert subspace H of E with reproducing operator H , we will say that H is invariantunder π or π -invariant if π ( x ) HE ∗ ⊆ HE ∗ for all x ∈ A . If π ( x ) H = Hπ ∗ ( x ) for all x ∈ A we willsay that H is ∗ -invariant under π or π - ∗ -invariant . Obviously, any π - ∗ -invariant Hilbert subspaceof E is invariant under π .The motivation for introducing invariant and ∗ -invariant Hilbert subspaces is the following.If H is a π -invariant Hilbert subspace of E , the restriction of π to HE ∗ defines a representationof A on H in the sense of Powers [11]. This fact should be clear since H can be thought as thecompletion of HE ∗ with respect to the norm given by k Hx k = h x | Hx i / for all x ∈ E ∗ , i.e., HE ∗ is an invariant dense subspace of H .When H is a π - ∗ -invariant Hilbert subspace of E , π defines by restriction to HE ∗ a ∗ -representation on H . In fact, for all x ∈ A one has that π ( x ∗ ) equals π ( x ) ∗ on HE ∗ (see [21] for thedetails). In this case, not only the restriction of π to HE ∗ defines a ∗ -representation but also itsclosure, which exists since every ∗ -representation is closable. The domain D of this representation,that we will denote also by π as for the representation acting on E , is the completion of HE ∗ inthe graph topology. 18et us assume that A is a barreled dual-separable ∗ -algebra with unit . Since the weak dualof any barreled space is necessarily a quasi-complete space, Schwartz’s theory of Hilbert subspacesapplies to A ∗ and we can set E = A ∗ in the previous discussion. Under this identification E ∗ isisomorphic to A .Further, let us consider the particular case in which π is the dual representation of the leftregular action of A on itself, i.e., the representation of A on A ∗ defined through h y | π ( x ) φ i = h x ∗ y | φ i for all x, y ∈ A and all φ ∈ A ∗ . We will denote by Hilb A ( A ∗ ) the subcone of Hilb( A ) composedby those Hilbert subspaces of A ∗ that are ∗ -invariant under the π . Theorem 4.1.
For every π - ∗ -invariant Hilbert subspace H of A ∗ there exists a closed weaklycontinuous strongly cyclic ∗ -representation of A acting on it, this representation being identifiablewith the GNS representation of A associated with the functional ρ = He where, as before, H isthe reproducing operator of H and e is the identity of A . The correspondence defined in this wayis a bijection from Hilb A ( A ∗ ) into Cycl ( A ) .Proof. Let us first check that in this situation the restriction of π to HA defines a representationwhose closure is a weakly continuous strongly cyclic ∗ -representation of A on H . The weakcontinuity of π follows immediately, via polarizability, from the continuity of H , the strongcontinuity of π on A ∗ and the quasi-completeness of A ∗ . The strong cyclicity of π is a consequenceof the existence of a unit in A . Setting ξ = He it follows that π ( A ) ξ equals HA , and since thisspace is a subspace of D that is dense for the graph topology, it follows that ξ is a strongly cyclic Recall that a barreled algebra is a topological algebra in which every barrel, i.e., every absorbing, convex,balanced and closed subset, is a zero neighborhood [27]. Notice that this conditions are fulfilled in the particular case in which A is separable by itself and { } is theintersection of a numerable set of environments. Recall that we are using the same notation for denoting both the representation acting on A ∗ and the closedone on H defined by restriction π . On the other hand, we have that ρ ( x ) = h x | He i = ( π ( x ) He, He ) for all x ∈ A .By virtue of Theorem 3.4 it follows that π can be identified with the GNS representation of A associated with ρ = He .We need to prove that the mapping we have defined is a one-to-one correspondence betweenHilb A ( A ∗ ) and Cycl( A ). First, consider two different π - ∗ -invariant Hilbert subspaces of A ∗ . Letus denote them by H and H . Let H and H be their respective reproducing operators and let π and π be the corresponding representations of A . Since H does not equal H , it follows that ρ = H e is a functional on A different from ρ = H e . If it were not the case, a contradiction isobtained from the cyclicity of both functionals and the fact that π and π are restrictions of thesame representation over A ∗ . Finally, since π and π are the GNS representations of A associatedwith ρ and ρ , respectively, from the unicity statement in theorem 2 it follows that π and π cannot be simultaneously identified with the same GNS representation.Let us finally check that given a closed weakly continuous strongly cyclic ∗ -representation π of A on a a Hilbert space H with domain D and cyclic vector ξ it is possible to constructa π - ∗ -invariant Hilbert subspace of A ∗ in such a way that the corresponding representation isequivalent to π . Consider the correspondence x → π ( x ) ξ . Let us denote it by T . Since π is weaklycontinuous, T is a continuous operator mapping A into H . On the other hand, π is strongly cyclicand then T is a dense range operator. It follows that T ∗ , the adjoint of T , is an injective continuousmap from H into A ∗ . Let H be the image of H through T ∗ with the transported Hilbert spacestructure. The operator H = T ∗ T from A into A ∗ is a positive operator reproducing H in A ∗ . Itis easy to see that H is actually a π - ∗ -invariant Hilbert subspace of A ∗ . In fact, for all x, y ∈ A we have that h y | π ( x ) He i = h x ∗ y | He i = ( T ( x ∗ y ) , T e ) = ( π ( x ∗ ) T y, T e ). But since by hypothesis π is a ∗ -representation, we have that ( π ( x ∗ ) T y, T e ) = (
T y, π ( x ) T e ) = (
T y, T x ) = h y | Hx i , i.e., In what follows we omit any reference to the inclusion J of H into A ∗ . Therefore, instead of writing ( J ∗ x, J ∗ y )for all x, y ∈ A , we will write ( Hx, Hy ). ( x ) He = Hx , and then, π ( x ) Hy = H ( xy ) for all x, y ∈ A . Finally, ( π ( x ) ξ, ξ ) = ( T x, T e ) = h x | T ∗ T e i = h x | He i and, then, π can be fully identified with π .We have found that, for a barreled dual-separable ∗ -algebra with unit A , there exists a multiplebijection among the following spaces:1. the set Cycl( A ) of unitary equivalence classes of GNS representations of A ,2. the space A ∗ + of positive continuous functionals on A ,3. the cone of Hilb A ( A ∗ ) of Hilbert subspaces of A ∗ that are ∗ -invariant under the dual leftregular action π of A ,4. the subfamily of L + ( A ∗ ) of continuous π - ∗ -invariant positive operators mapping A ∗ into A ,and5. the space of π - ∗ -invariant positive kernels on A .The natural cone structures on the last four listed spaces are compatible with the bijectionconnecting them. It is then customary to transport such a structure on Cycl( A ). We introducethe following definitions. Addition law in Cycl ( A ) : As we will see in the next paragraphs, the multiplication law bynon-negative real numbers and the order can be easily defined on Cycl( A ) without any referenceto the bijection connecting this space with any other of those listed above. It is not the case for thesum. In fact, in order to properly define the sum of GNS representations it is mandatory to embedthe corresponding representation spaces in a common domain. Let π and π be two elements ofCycl( A ) and let H and H be the two Hilbert subspaces of A ∗ on which they act. Recalling thatthe domain of π (resp. π ) is the completion in the graph topology of H A (resp. H A ) where H (resp. H ) is the reproducing operator of H (resp. H ), let us consider the representation on21 + H with domain ( H + H ) A given by ( H + H ) y → π ( x ) H y + π ( x ) H y for all x, y ∈ A .This is actually a well defined representation on H + H since for any other y ′ ∈ A such that H y ′ + H y ′ = H y + H y it follows that h z | π ( x ) H y ′ + π ( x ) H y ′ i = h x ∗ z | H y ′ + H y ′ i = h x ∗ z | H y + H y i = h z | π ( x ) H y + π ( x ) H y i for all z ∈ A and then π ( x ) H y ′ + π ( x ) H y ′ equals π ( x ) H y + π ( x ) H y . Moreover, since π and π are both weakly continuous strongly cyclic ∗ -representations of A , it is also the case for it. We will define the sum of π and π as the closureof this representation and we will denote it by π + π . Of course, π + π can be identified withthe GNS representation of A associated with the positive functional H e + H e . Multiplication by non-negative real numbers in Cycl ( A ) : Given a closed weakly con-tinuous strongly cyclic ∗ -representation π of A acting on the π - ∗ -invariant Hilbert subspace H of A ∗ we will define the representation λπ for every λ > λ H and al-gebraically coincides with π . If ξ is the strongly cyclic vector associated with π , we will set λξ to be the corresponding cyclic vector of λπ . The action of R > on Cycl( A ) is extended to R ≥ by identifying the representation 0 π with the trivial representation of A . It is straightforward tocheck that λπ is a weakly continuous strongly cyclic ∗ -representation identifiable with the GNSrepresentation of A associated with λHe where, as before, H is the reproducing operator of H .Notice that even when we are not identifying π and λπ as GNS representations of A , λπ is unitarily equivalent to π in the usual sense for every λ >
0. Consequently, extremal ele-ments in Cycl( A ), i.e., those GNS representations of A obtained as in Theorem 4.1 from extremal π - ∗ -invariant Hilbert subspaces of A ∗ , are necessarily associated with topologically irreduciblerepresentations. Order in Cycl ( A ) : A partial order on Cycl( A ) compatible with the operations we have justdefined on this space has been already mentioned in the previous section. We will write π ≤ π , π and π being two elements in Cycl( A ), if and only if π extends π in the sense of Remark 3.1.Finally we can state the following proposition. The proof straightforwardly follows from the22revious definitions. Proposition 4.2.
The canonical bijection between Cycl ( A ) and Hilb A ( A ∗ ) is an isomorphism forthe cone structures we have defined. Explicitly, let π , π and π be three weakly continuous stronglycyclic ∗ -representations of A and let H , H and H be three elements of Hilb A ( A ∗ ) . Let λ be anon-negative real number. It follows that1. π = π + π if and only if H = H + H ,2. π equals λπ if and only if H = λ H , and3. π ≤ π if and only if H ≤ H . ( A ) and Hilb A ( A ∗ ) As we have already mentioned, the cone structure defined on Cycl( A ) have many interestingconsequences. In this section we will derive some of them. As before, A will denote a barreleddual-separable ∗ -algebra with unit. Hilbert subspaces of A ∗ that are ∗ -invariant under the leftdual regular action of A on A ∗ will be simply refered as ∗ -invariant Hilbert subspaces of A ∗ . Proposition 5.1.
Let π and π be two elements in Cycl ( A ) . π belongs to [0 , π ] if and only ifthere exists a representation π in Cycl ( A ) such that π = π + π . In that case, π is unique. Wewill denote it by π = π − π .Proof. Let H and H be the ∗ -invariant Hilbert subspaces of A ∗ associated with π and π , re-spectively. If there exists a representation π ∈ Cycl( A ) such that π = π + π it follows that H = H + H , H being the ∗ -invariant Hilbert subspace of A ∗ associated with π and then, H ≤ H , i.e., π ≤ π . Conversely, if π ∈ [0 , π ], H ≤ H and this inequality extends to the23eproducing operators, i.e., H ≤ H , where H (resp., H ) is the reproducing operator of H (resp., H ). It follows that H − H is a positive operator reproducing a ∗ -invariant Hilbert subspace of A ∗ , say H , whose associated representation π in Cycl( A ) satisfies π = π + π . The uniquenessof π follows from the uniqueness of the operator H − H . Proposition 5.2.
Let π , π ∈ Cycl ( A ) . Then, π is an algebraic subrepresentation of π if andonly if there exists λ > such that π ∈ [0 , λπ ] .Proof. If there exists λ > λπ extends π it straightforwardly follows that λπ alge-braically extends π and the same is true for π since it is unitarily equivalent, in the ordinarysense, to λπ . Let us assume that π is an algebraic subrepresentation of π . Let H and H the ∗ -invariant Hilbert subspaces of A ∗ associated with π and π , respectively. Since the inclusion of H into A ∗ is continuous, its graph is closed in H × A ∗ and it is also the case for its graph in H × H . It follows from the closed graph theorem that the inclusion of H into H is continuous,and √ λ denoting its norm, we finally obtain that H ≤ λ H . Therefore, π is in [0 , λπ ] as wewanted to prove. Proposition 5.3.
Let π and π be two representations in Cycl ( A ) . It follows that π and π aremutually excluding, i.e., π + π is unitarily equivalent to π ⊕ π if and only if π ≤ π , π ≤ π with π ∈ Cycl ( A ) implies π = 0 .Proof. Let H and H be the ∗ -invariant Hilbert subspaces of A ∗ associated with π and π ,respectively. If π + π is unitarily equivalent to π ⊕ π it follows that H ∩ H = { } andthen, if there exists a ∗ -invariant Hilbert subspace H of A ∗ such that H ≤ H and H ≤ H , wehave H = { } , i.e., for any π ∈ Cycl( A ) such that π ≤ π and π ≤ π it follows that π is thetrivial representation. Conversely, assume that for any π ∈ Cycl( A ) satisfying π ≤ π and π ≤ π one has π = 0 and let us define on H ∩ H the form ( ξ, χ ) = ( ξ, χ ) H + ( ξ, χ ) H . This form24s a positive definite inner product making H ∩ H into a Hilbert subspace of A ∗ [25]. Since itclearly is a ∗ -invariant form, he have that H ∩ H belongs to Hilb A ( A ∗ ). But H ∩ H ≤ H and H ∩ H ≤ H , and then H ∩ H = { } . Consequently, π + π is unitarily equivalent to π ⊕ π ,as we wanted to prove. Proposition 5.4.
Let π and π be two elements of Cycl ( A ) . Then, π is a subrepresentation of π in the ordinary sense if and only if π − π ∈ Cycl ( A ) and π and π − π are mutually excluding. Inthat case, π − π is also a subrepresentation of π in the ordinary sense and π is unitarily equivalentto π ⊕ ( π − π ) .Proof. Let us first suppose that π is unitarily equivalent to a subrepresentation of π in theordinary sense. It follows that the ∗ -invariant Hilbert subspace H of A ∗ associated with π is asubspace of the one associated with π with the induced Hilbert space structure. Let us considerthe space H ⊥ orthogonal to H in H . It is also a ∗ -invariant Hilbert subspace of A ∗ and it equals H − H . Accordingly, we have that π − π belongs to Cycl( A ) and that it is unitarily equivalent toa subrepresentation of π in the ordinary sense, since the Hilbert space structure of H ⊥ is the oneinduced by H . On the other hand, since H ∩ H ⊥ = { } , it follows from the previous propositionthat π is unitarily equivalent to π ⊕ ( π − π ).Reciprocally, let us assume that π − π belongs to Cycl( A ) and that π and π − π are mutuallyexcluding. If we denote by H the ∗ -invariant Hilbert subspace of A ∗ associated with π − π , then H ∩ H = { } , H = H + H and H ⊂ H with the induced Hilbert space structure, i.e., π is asubrepresentation of π in the ordinary sense as we wanted to prove. Proposition 5.5.
Let ( π i ) i ∈ I be a decreasing filtering system of representations in Cycl ( A ) . Itfollows that π = inf { π i : i ∈ I } exists in Cycl ( A ) .Proof. I is a right filtering set of indices such that, for i, j ∈ I , i ≤ j , we have that π i is asubrepresentation of π j . Let ( H i ) i ∈ I be the ∗ -invariant Hilbert subspaces of A ∗ associated with25 π i ) i ∈ I , respectively, and let ( H i ) i ∈ I be their corresponding reproducing operators. The space H = inf {H i : i ∈ I } exists in Hilb( A ∗ ) and its reproducing operator is H = inf { H i : i ∈ I } whichequals lim i H i in L ( A ∗ ) when this space is endowed with the weak uniform convergence topology[25]. Since for every i ∈ I , H i is ∗ -invariant under the dual left regular action of A , it follows that H ∈
Hilb A ( A ∗ ). The corresponding GNS representation is the one whose existence is claimed inthe proposition. Further, notice that the Hilbert space on which π acts is the subspace of ∩ i ∈ I H i composed by those φ ∈ A ∗ such that k φ k := sup {k φ k i : i ∈ I } = lim i k φ k i < + ∞ . Proposition 5.6.
Let ( π i ) i ∈ I be an increasing filtering system of elements of Cycl ( A ) and let ( ξ i ) i ∈ I be their corresponding normalizing vectors. Then, ( π i ) i ∈ I is majorized in Cycl ( A ) if andonly if sup {k π i ( x ) ξ i k i : i ∈ I } < + ∞ (5) for all x ∈ A .Proof. Here, the set I is as in the previous proposition but now for i, j ∈ I , i ≤ j , we have that π i is an extension of π j . Let ( H i ) i ∈ I be the ∗ -invariant Hilbert subspaces of A ∗ associated with( π i ) i ∈ I , respectively, ( H i ) i ∈ I being their reproducing operators. Notice that k π i ( x ) ξ i k i = h x, H i x i for all i ∈ I and all x ∈ A . It follows that the condition (5) is necessary and sufficient for ( H i ) i ∈ I tobe majorized in Hilb A ( A ∗ ). In this case, H = sup {H i : i ∈ I } has H = sup { H i : i ∈ I } = lim i H i as a reproducing operator where, as before, the limit is taken in L ( A ∗ ) with this space endowedwith the weak uniform convergence topology. Since H is obviously ∗ -invariant under the dual leftregular action of A , the proposition is proved. Notice, in addition, that H can be characterized asthe q -completion in A ∗ of the space ∪ i ∈ I H i equiped with the pre-Hilbert structure derived fromthe norm k φ k := inf {k φ k i : i ∈ I } = lim i k φ k i . Proposition 5.7.
Let ( π i ) i ∈ I be a collection of elements of Cycl ( A ) . For every i ∈ I , let usdenote by H i the ∗ -invariant Hilbert subspace of A ∗ associated with π i , respectively. Consider he (abstract) Hilbert sum ˆ ⊕ i ∈ I H i , the elements of this space being those sequences ( φ i ) i ∈ I with φ i ∈ H i such that k ( φ i ) i ∈ I k = X i ∈ I k φ i k i < + ∞ (6) where k · k i denotes the norm in H i . Finally, let ⊕ i ∈ I H i be the dense linear subspace of ˆ ⊕ i ∈ I H i composed by those sequences ( φ i ) i ∈ I in which all the φ i are nul but a finite number of them andthe pre-Hilbert structure inherited from ˆ ⊕ i ∈ I H i . It follows that the sums P i ∈ I ′ π i , I ′ denoting thefinite subsets of I , are majorized in Cycl ( A ) if and only if the application Φ mapping ⊕ i ∈ I H i into A ∗ defined by Φ ( ⊕ i ∈ I φ i ) = X i ∈ I φ i (7) is continuous.Proof. Let us assume that the finite sums P i ∈ I ′ π i , I ′ being any finite subsets of I , are majorizedin Cycl( A ). From Proposition 5.6 we have that P i ∈ I h x | H i x i < + ∞ for all x ∈ A , where weare denoting by H i the reproducing operator of H i for all i ∈ I , respectively. In order to seethat the mapping given by Eq. (7) is continuous we must prove that the image of the unitball B in ˆ ⊕ i ∈ I H i under Φ is weakly bounded. Now, let x be an element of A and let φ bein Φ( B ). φ equals P i ∈ I ′ φ i for a finite subset I ′ in I and P i ∈ I ′ k φ i k i < + ∞ . It follows that |h x | φ i| ≤ P i ∈ I ′ |h x | φ i i| ≤ (cid:0)P i ∈ I ′ k φ i k i (cid:1) (cid:0)P i ∈ I ′ h x | H i x i (cid:1) ≤ P i ∈ I ′ h x | H i x i , what proves that Φ( B )is weakly bounded. Reciprocally, let us assume that Φ is a continuous mapping. Let us considerthe extension of it to a continuous operator mapping ˆ ⊕ i ∈ I H i into A ∗ . If we denote by ˆΦ suchextension and ( φ i ) i ∈ I is an element of ˆ ⊕ i ∈ I H i , it follows that it equals the limit following thefiltering system of finite subsets I ′ of I of those sequences ( φ ′ i ) i ∈ I whose components satisfy φ ′ i = φ i for all i ∈ I ′ and φ ′ i = 0 otherwise. Consequently, ˆΦ (( φ i ) i ∈ I ) is the limit taken in A ∗ of P i ∈ I ′ φ i . Further, we have the following factorization for ˆΦ: ˆ ⊕ i ∈ I H i → (cid:0) ˆ ⊕ i ∈ I H i (cid:1) / ker ( ˆΦ) → A ∗ .The first mapping is the canonical projection of ˆ ⊕ i ∈ I H i onto (cid:0) ˆ ⊕ i ∈ I H i (cid:1) / ker ( ˆΦ), while the second27ne, that we will denote by ˜Φ, is an isomorphism from (cid:0) ˆ ⊕ i ∈ I H i (cid:1) / ker ( ˆΦ) onto the image ofˆ ⊕ i ∈ I H i under ˆΦ. Assuming (cid:0) ˆ ⊕ i ∈ I H i (cid:1) / ker ( ˆΦ) endowed with the Hilbert structure derived fromthe quotient norm, let us consider on Φ (cid:0) ˆ ⊕ i ∈ I H i (cid:1) this structure transported by ˜Φ, i.e., the metricstructure making ˜Φ an isometric isomorphism. Explicitly, the norm on Φ (cid:0) ˆ ⊕ i ∈ I H i (cid:1) is given by k φ k I = inf { P i ∈ I k φ i k i : P i ∈ I φ i = φ } . If I ′ is a finite subset of I it follows that any element φ in P i ∈ I ′ H i pick ups the form P i ∈ I ′ φ i with φ i ∈ H i for all i ∈ I ′ , and then k φ k I ′ = inf { P i ∈ I ′ k φ i k i : P i ∈ I ′ φ i = φ } . This shows that P i ∈ I ′ H i is a Hilbert subspace of Φ (cid:0) ˆ ⊕ i ∈ I H i (cid:1) . Further, sinceΦ (cid:0) ˆ ⊕ i ∈ I H i (cid:1) is clearly a ∗ -invariant Hilbert subspace of A ∗ , it follows that the finite sums of theform P i ∈ I ′ π i are majorized by the element in Cycl( A ∗ ) associated with Φ (cid:0) ˆ ⊕ i ∈ I H i (cid:1) .If a given sequence ( π i ) i ∈ I of elements of Cycl( A ) satisfies the conditions of the previousproposition, it is called a summable sequence in Cycl( A ). The element in Cycl( A ∗ ) associatedwith Φ (cid:0) ˆ ⊕ i ∈ I H i (cid:1) with the Hilbert structure transported by ˜Φ is the sum of ( π i ) i ∈ I and we willdenote it by P i ∈ I π i . Corollary 5.8.
Let ( π i ) i ∈ I be a collection of elements of Cycl ( A ) . Then, there exists a represen-tation π ∈ Cycl ( A ) such that π = ⊕ i ∈ I π i if and only if1. P i ∈ I π i is well defined, and2. if φ i ∈ H i such that P i ∈ I k φ i k i < + ∞ then P i ∈ I π i ( x ) φ i = 0 for all x ∈ A implies φ i = 0 for all i ∈ I .Proof. For the existence of such a representation it is necessary and sufficient that ( π i ) i ∈ I is asummable sequence in Cycl( A ), i.e., P i ∈ I π i is well defined, and that the map ˆΦ in the previousproposition is an isomorphism, i.e., the second condition.The definition of infinite sums of representations in Cycl( A ) has a natural generalization tointegrals when A by itself is weakly separable. We will briefly comment on this issue.28et Γ be a locally compact measure space and let us denote by µ its measure. We will say the amapping γ → π γ from Γ into Cycl( A ) is integrable if, for every x ∈ A , the function γ → k π γ ( x ) ξ γ k γ is integrable, where we are denoting by ξ γ the normalizing vector of π γ , for every γ ∈ Γ. Themapping γ → H γ , where H γ is the ∗ -invariant Hilbert subspace of A ∗ associated with π γ for each γ ∈ Γ, respectively, will also be referred as an integral mapping from Γ into Hilb A ( A ∗ ).In [25] it was proved that given an integral map from Γ into Hilb A ( A ∗ ) there exists a continuousmapping ˆΦ from R ⊕ Γ H γ dµ ( γ ) into A ∗ defined byˆΦ (( φ γ ) γ ∈ Γ ) = Z Γ φ γ dµ ( γ ) (8)the second term in this equation being the weak integral of a scalarly integrable function. Thespace R ⊕ Γ H γ dµ ( γ ) is the space of measurable vector fields γ → φ γ ∈ H γ such that R Γ k φ γ k γ dµ ( γ ) < + ∞ , where k · k γ is the norm on H γ for every γ ∈ Γ, endowed with the Hilbert space structurederived from the norm k ( φ γ ) γ ∈ Γ k = R Γ k φ γ k γ dµ ( γ ).As it is the case for infinite sums of Hilbert subspaces, the operator ˆΦ can be decomposed as R ⊕ Γ H γ dµ ( γ ) → (cid:16)R ⊕ Γ H γ dµ ( γ ) (cid:17) / ker ( ˆΦ) → A ∗ . The first map appearing in this factorization isthe canonical projection of R ⊕ Γ H γ dµ ( γ ) onto (cid:16)R ⊕ Γ H γ dµ ( γ ) (cid:17) / ker ( ˆΦ) while the second one is theisomorphism from (cid:16)R ⊕ Γ H γ dµ ( γ ) (cid:17) / ker ( ˆΦ) onto ˆΦ (cid:16)R ⊕ Γ H γ dµ ( γ ) (cid:17) . This last space, if equippedwith the transported Hilbert space structure of (cid:16)R ⊕ Γ H γ dµ ( γ ) (cid:17) / ker ( ˆΦ), is a Hilbert subspaceof A ∗ that we will denote by R Γ H γ dµ ( γ ). When all the vector fields γ → φ γ take values in ∗ -invariant Hilbert subspaces of A ∗ , i.e., when the integral mapping under consideration maps Γinto Hilb A ( A ∗ ), the space R Γ H γ dµ ( γ ) turn to be also a ∗ -invariant Hilbert subspace of A ∗ , itsassociated representation in Cycl( A ) being called the integral of the map γ → π γ . Of course, wewill denote it by R Γ π γ dµ ( γ ).Finally, we will discuss which is the effect of a continuous algebra ∗ -morphism in this context.Let A and A be two barreled dual-separable ∗ -algebras with unit and let α be a strongly contin-29ous ∗ -homomorphism from A into A . We will denote by π (resp., π ) the dual represntationof the left regular action of A (resp., A ) on its antidual space. We can prove the followingproposition. Proposition 5.9.
Let H be a π - ∗ -invariant Hilbert subspace of A ∗ and let H be the Hilbertsubspace of A ∗ that is image of H under the transpose mapping of α . It follows that H is ∗ -invariant under the dual left regular action of A .Proof. Let H and H be the reproducing operators of H and H , respectively. Recall that H = α ∗ H α . Let x , y be an arbitrary pair of elements of A . Then h y | α ∗ π ( αx ) φ i = h αy | π ( αx ) φ i = h ( αx ) ∗ αy | φ i = h α ( x ∗ y ) | φ i = h x ∗ y | α ∗ φ i = h y | π ( x ) α ∗ φ i , for all φ ∈ A ∗ , i.e., we have that α ∗ π ( αx ) equals π ( x ) α ∗ on A ∗ . It follows that α ∗ π ( αx ) H α = π ( x ) α ∗ H α = π ( x ) H . Now, let us assume that H is ∗ -invariant under π . Under thisassumption we have α ∗ π ( αx ) H αy = α ∗ H [( αx ) αy ] = α ∗ H α ( x y ) = H ( x y ). But then,for all x , y ∈ A we have that π ( x ) H y = H ( x y ), i.e., H = α ∗ ( H ) is ∗ -invariant under π , as we wanted to prove.The previous proposition shows that the mapping assigning to every Hilbert subspace H of A ∗ the Hilbert subspace of A ∗ given by H = α ∗ ( H ), when restricted to those Hilbert subspacesthat are ∗ -invariant under the dual left regular action of A , defines a mapping from Hilb A ( A ∗ )into Hilb A ( A ∗ ). Consequently, we have a well defined mapping from Cycl( A ) into Cycl( A ) thatwe will also denote by α ∗ . It is straightforward to prove that this application is a cone structurepreserving mapping. Let us conclude this paper summarizing the main results we have obtained.30fter recalling in section 2 the main aspects of Schwartz’s theory on Hilbert subspaces oftopological vector spaces, in section 3 we have discussed some basics of the representation theoryof algebras of unbounded operators and we have restated the GNS construction theorem for general ∗ -algebras. In section 4 we have proved that for a wide class of topological ∗ -algebras, i.e., barreleddual-separable unital ∗ -algebras, their weakly continuous strongly cyclic ∗ -representations are inone-to-one correspondence with the Hilbert spaces continuously embedded in its dual that are ∗ -invariant under the dual left regular action of the algebra in hands. After explicitly endowingthe first of these spaces with a cone structure we have proved that this correspondence actuallyis a cone isomorphism. Finally, in section 5 we have proved many consequences of the existenceof such an isomorphism: we described the connection between the order of GNS representationsand the usual concept of subrepresentation, we defined the difference of GNS representations, weproved a couple of propositions concerning the existence of extremal representations of filteringsystems and we discussed the effect of ∗ -algebra morphisms.As we have already mention these results could be useful for studying continuity aspects ofthe deformation of GNS representations. In a forthcoming paper we will prove that the similarityclasses of GNS representations of a given barreled dual-separable ∗ -algebra with unit A on innerproduct spaces are in one-to-one correspondence with the elements of the canonical real expansionof L + ( A ∗ ) showing that Kolmogorov functionals exhaustively define the GNS representations onKrein spaces. References [1] O. Bratelli and D. W. Robinson, “Operator algebras and quantum statistical mechanics. VolI: C ⋆ and W ⋆ -algebras, symmetry groups, decomposition of states.” Springer-Verlag, NewYork (1979) (Texts and monographs in physics).312] O. Bratelli and D. W. Robinson, “Operator algebras and quantum statistical mechanics. VolII: Equilibrium states. Models in quantum statistical mechanics.” Springer-Verlag, New York(1981) (Texts and monographs in physics).[3] R. Haag, “Local quantum physics. Fields, particles, algebras.” Springer, Berlin (1996).[4] D. Buchholz and R. Haag, “The quest for understanding in relativistic quantum physics.” J.Math. Phys. , 3674 (2000) [arXiv:hep-th/9910243].[5] H. J. Borchers, “On revolutionizing quantum field theory with Tomita’s modular theory.” J.Math. Phys. , 3604 (2000).[6] D. G. Arb´o, M. A. Castagnino, F. H. Gaioli and S. M. Iguri, “Minimal irre-versible quantum mechanics. The decay of unstable states.” Physica A , 469 (2000)[arXiv:quant-ph/0005041].[7] F. Bagarello, “Applications of topological*-algebras of unbounded operators.” J. Math. Phys. , 6091 (1998) [arXiv:math/9803133].[8] A. Inoue and K. Takesue, “Spatial theory for algebras of unbounded operators II.” Proc. Am.Math. Soc. , 295 (1983).[9] A. V. Voronin, V. N. Sushko and S. S. Khoruzhii, “Algebras of unbounded operators andvacuum superselection rules in quantum field theory I. Some properties of Op ∗ -algebras andvector states on them.” Theor. Math. Phys. , 335 (1984).[10] S. Albeverio, H. Gottschalk and J. Wu, “Models of local relativistic quantum fieldswith indefinite metric (in all dimensions).” Commun. Math. Phys. , 509 (1997)[arXiv:math-ph/0409057]. 3211] R. T. Powers, “Self-adjoint algebras of unbounded operators.” Comm. Math. Phys. , 85(1971).[12] R. T. Powers, “Self-adjoint algebras of unbounded operators II.” Trans. Amer. Math. Soc. , 261 (1974).[13] S. Gudder and W. Scruggs, “Unbounded representations of ∗ -algebras.” Pac. J. Math. ,369 (1977).[14] T. W. Palmer, “Banach algebras and the general theory of ⋆ -algebras. Vol. I: Algebras andBanach algebras.” Cambridge University Press, Cambridge (1994) (Encyclopedia of mathe-matics and its applications; Vol. 49).[15] T. W. Palmer, “Banach algebras and the general theory of ⋆ -algebras. Vol. II: ⋆ -algebras.”Cambridge University Press, Cambridge (2001) (Encyclopedia of mathematics and its appli-cations; Vol. 79).[16] S. M. Iguri and M. A. Castagnino, “The formulation of quantum mechanics in terms ofnuclear algebras.” Int. J. Theor. Phys. , 143 (1999).[17] J. Stochel and S. Todorov, “Characterizations of cyclic *-representations with equal weakand strong commutants.” J. Math. Phys. , 4190 (1992).[18] J. P. Antoine and S. Ota, “Unbounded GNS representations of a ∗ -algebra in a Krein space.”Lett. Math. , 267 (1989).[19] G. Hofmann, “On GNS representations on inner product spaces I. The structure of therepresentation space.” Comm. Math. Phys. , 299 (1998).3320] M. Mnatsakanova, G. Morchio, F. Strocchi and Y. Vernov, “Irreducible representa-tions of the Heisenberg algebra in Krein spaces.” J. Math. Phys. , 2969 (1998)[arXiv:math-ph/0211025].[21] A. Belanger and E. G. F. Thomas, “Positive forms on nuclear ∗ -algebras and their integralrepresentations.” Pac. J. Math. , 410 (1990).[22] T. Constantinescu and A. Gheondea, “Representations of Hermitian Kernels byMeans of Krein Spaces II. Invariant Kernels.” Comm. Math. Phys. , 409 (2001)[arXiv:math/0007183].[23] S. M. Iguri and M. A. Castagnino, “GNS representations on Krein spaces.” In preparation.[24] S. Waldmann, “Remarks on the deformation of GNS representations of *-algebras.” Rept.Math. Phys. , 389 (2001) [arXiv:math/0012071].[25] L. Schwartz, “Sous-espaces hilbertiens d’espaces vercoriels topologiques et noyaux associ´es.”Journal d’Anal. Math. , (1964) 115.[26] K. Schm¨udgen, “Unbounded operator algebras and representation theory.” Basel-Boston-Berlin, Birkh¨auser Verlag (1990) (Operator Theory: Advences and Applications; Vol. 37).[27] F. Treves, “Topological vector spaces, distributions and kernels.” Academic Press, New York-Londres (1967).[28] L. Turowska, “On the complexity of the description of ⋆ -algebra representations by unboundedoperators.” Proc. Am. Math. Soc.130