About tensor products of Hilbert quasi *-algebras and their representability properties
aa r X i v : . [ m a t h . F A ] F e b ABOUT TENSOR PRODUCTS OF HILBERT QUASI*-ALGEBRAS AND THEIR REPRESENTABILITYPROPERTIES
MARIA STELLA ADAMO
Abstract.
This note aims to investigate the tensor product oftwo given Hilbert quasi *-algebras and its properties. The con-struction proposed in this note turns out to be again a Hilbertquasi *-algebra, thus interesting representability properties studedin [2] are maintained. Furthermore, if two functionals are rep-resentable and continuous respectively on the two Hilbert quasi*-algebras, then so is their tensor product. Introduction and basic notions
The study of (locally convex) quasi *-algebras was initiated by G.Lassner in 1988 (see [17, 18]) to give a rigorous solution to some prob-lems coming from Quantum Statistical Mechanics. Since then, a widerange of literature appeared on this topic, both regarding applicationsand purely mathematical aspects of quasi *-algebras (see [4, 5, 6, 8, 14]).A special interest has been shown on the theory of representations ona suitable family of unbounded operators. In this context, new notionslike full representability and *-semisimplicity have been introduced (e.g.[7, 14]). A central role is played by representable functionals , i.e., thosefunctionals that allow a GNS-like construction, in investigating struc-ture properties of locally convex quasi *-algebras (see [2, 14, 23]).The purpose of this note is to exhibit a tensor product constructionfor two given Hilbert quasi *-algebras and examine its properties. Theinterest in tensor products comes from physical phenomena, as they areemployed to investigate two different physical systems as a joint one.In addition, very little is known about tensor products of unboundedoperator algebras (e.g., [3, 12, 13, 15]).Hilbert quasi *-algebras are a particular subclass of locally convexquasi *-algebras arising as completions of Hilbert algebras with respect
Mathematics Subject Classification.
Primary 46A32; Secondary 47L60,46L08.
Key words and phrases.
Tensor products of quasi *-algebras, Hilbert quasi *-algebras, Representable functionals. to the norm defined by their inner product. The main target is to builda new Hilbert quasi *-algebra that encodes the properties of the factors,by looking at them as Hilbert spaces. This would lead to defining thetensor product as a complex linear space that turns out to verify allthe requirements for a quasi *-algebra on the tensor product of the*-algebras of the factors.To obtain a tensor product Hilbert quasi *-algebra, it suffices to equipthe tensor product quasi *-algebra with the inner product obtained asa combination of the two given in the factors and take its completion.In this way, we get a Hilbert quasi *-algebra, owning all the propertiesstudied in [2], in particular, it is fully representable and *-semisimple.Furthermore, given two representable and continuous functionals onfactors, it is showed that their tensor product is still a representable andcontinuous functional. Hence, this highlights that the representabilityof functionals passes from factors to the tensor product.The note is structured as follows. Firstly, preliminary notions aboutquasi *-algebras and representable functionals are recalled. In partic-ular, for normed quasi *-algebras, in Section 2, we remind the notionsof full representability, *-semisimplicity and the results obtained in theBanach case. Among them, for Hilbert quasi *-algebras, the character-ization of representable and continuous functionals as elements of theHilbert space will be employed to show the passing of representabilityfor functionals to their tensor product. The latter fact has been provedin Section 3, in which we discuss the mentioned construction.For the reader’s convenience, we recall some preliminary notions forfuture use. Further details can be found in [4].
Definition 1.1. A quasi *–algebra ( A , A ) is a pair consisting of avector space A and a *-algebra A contained in A as a subspace andsuch that(i) A carries an involution a a ∗ extending the involution of A ;(ii) A is a bimodule over A and the module multiplications extendthe multiplication of A . In particular, the following associativelaws hold:( xa ) y = x ( ay ); a ( xy ) = ( ax ) y, ∀ a ∈ A , x, y ∈ A ;(iii) ( ax ) ∗ = x ∗ a ∗ , for every a ∈ A and x ∈ A .A quasi *-algebra ( A , A ) is unital if there is an element ∈ A , suchthat a = a = a , for all a ∈ A ; is unique and called the unit of( A , A ).A quasi *-algebra ( A , A ) is called a normed quasi *-algebra if anorm k · k is defined on A with the properties (i) k a ∗ k = k a k , ∀ a ∈ A ;(ii) A is dense in A ;(iii) for every x ∈ A , the map R x : a ∈ A → ax ∈ A is continuousin A .The continuity of the involution implies that(iii’) for every x ∈ A , the map L x : a ∈ A → xa ∈ A is continuousin A . Definition 1.2.
If ( A , k · k ) is a Banach space, we say that ( A , A ) isa Banach quasi *-algebra .The norm topology of A will be denoted by τ n .A special class of Banach quasi *-algebras is given by Hilbert quasi*-algebras . Their interest comes from their rich structure, in which thenorm arises from an inner product with certain properties.
Definition 1.3.
Let A be a *-algebra which is also a pre-Hilbert spacewith respect to the inner product h·|·i such that:(1) the map y xy is continuous with respect to the norm definedby the inner product;(2) h xy | z i = h y | x ∗ z i for all x, y, z ∈ A ;(3) h x | y i = h y ∗ | x ∗ i for all x, y ∈ A ;(4) A is total in A .If H denotes the Hilbert space completion of A with respect to theinner product h·|·i , then ( H , A ) is called Hilbert quasi *-algebra .2.
Representable functionals
A convenient tool to study structure properties of quasi *-algebrasis given by representable functionals . Loosely speaking, those mapsconstitute a valid replacement for positive functionals in *-algebras foradmitting a GNS-like triple.In this section, we recall some properties related to representabilityof Banach quasi *-algebras. Among them, important notions are fullrepresentability and *-semisimplicity . For details, see [2, 4, 7, 14, 23].
Definition 2.1.
Let ( A , A ) be a quasi *-algebra. A linear functional ω : A → C satisfying(L.1) ω ( x ∗ x ) ≥ , ∀ x ∈ A ;(L.2) ω ( y ∗ a ∗ x ) = ω ( x ∗ ay ) , ∀ x, y ∈ A , ∀ a ∈ A ;(L.3) ∀ a ∈ A , there exists γ a > | ω ( a ∗ x ) | ≤ γ a ω ( x ∗ x ) / , ∀ x ∈ A . MARIA STELLA ADAMO is called representable on the quasi *-algebra ( A , A ).The family of representable functionals on the quasi *-algebra ( A , A )is denoted by R ( A , A ).The term representable is justified by the existence of a GNS-liketriple of a *-representation π ω , a Hilbert space H ω and a map λ ω . Inthis context, we need a proper definition of *-representation of the quasi*-algebra ( A , A ) and a family of operators on which we represent it.Let H be a Hilbert space and let D be a dense linear subspace of H .We denote by L † ( D , H ) the following family of closable operators L † ( D , H ) = { X : D → H : D ( X ) = D , D ( X ∗ ) ⊃ D} . L † ( D , H ) is a C − vector space with the usual sum and scalar multi-plication. If we define the involution † and partial multiplication (cid:3) as X X † ≡ X ∗ ↾ D and X (cid:3) Y = X †∗ Y, then L † ( D , H ) is a partial *-algebra defined in [4]. Definition 2.2. A *-representation of a quasi *-algebra ( A , A ) is a*-homomorphism π : A → L † ( D π , H π ), where D π is a dense subspaceof the Hilbert space H π , with the following properties:(i) π ( a ∗ ) = π ( a ) † for all a ∈ A ;(ii) if a ∈ A and x ∈ A , then π ( a ) is a left multiplier of π ( x ) and π ( a ) (cid:3) π ( x ) = π ( ax ).A *-representation π is • cyclic if π ( A ) ξ is dense in H π for some ξ ∈ D π ; • closed if π coincides with its closure e π defined in [23, Section 2].If ( A , A ) has a unit , then we suppose that π ( ) = I D , the identityoperator of D . Theorem 2.3. [23] Let ( A , A ) be a quasi *-algebra with unit and let ω be a linear functional on ( A , A ) that satisfies the conditions (L.1)-(L.3) of Definition 2.1. Then, there exists a triple ( π ω , λ ω , H ω ) suchthat: • π ω is a closed cyclic *-representation π ω of ( A , A ), with cyclicvector ξ ω ; • λ ω is a linear map of A into λ ω ( A ) = D π ω , ξ ω = λ ω ( ) and π ω ( a ) λ ω ( x ) = λ ω ( ax ), for every a ∈ A and x ∈ A ; • ω ( a ) = h π ω ( a ) ξ ω | ξ ω i , for every a ∈ A . This representation is unique up to unitary equivalence.
To every ω ∈ R ( A , A ), we can associate the sesquilinear form ϕ ω defined on A × A as(2.1) ϕ ω ( x, y ) := ω ( y ∗ x ) , x, y ∈ A . It is easy to see that(i) ϕ ω ( x, x ) ≥ , for every x ∈ A .(ii) ϕ ω ( xy, z ) = ϕ ω ( y, x ∗ z ) for every x, y, z ∈ A .If ( A , A ) is a normed quasi *-algebra, we denote by R c ( A , A ) thesubset of R ( A , A ) consisting of continuous functionals.As shown in [14] for locally convex quasi *-algebras, if ω ∈ R c ( A , A ),then the sesquilinear form ϕ ω defined in (2.1) is closable ; that is, ϕ ω ( x n , x n ) →
0, for every sequence { x n } ⊂ A such that k x n k → ϕ ω ( x n − x m , x n − x m ) → . In this case, ϕ ω has a closed extension ϕ ω to a dense domain D ( ϕ ω )containing A . For Banach quasi *-algebras this result can be improved. Proposition 2.4. [2] Let ( A , A ) be a Banach quasi *-algebra with unit , ω ∈ R c ( A , A ) and ϕ ω the associated sesquilinear form on A × A defined as in (2.1). Then D ( ϕ ω ) = A ; hence ϕ ω is everywhere definedand bounded.Consider now the set A R := \ ω ∈R c ( A , A ) D ( ϕ ω ) . If R c ( A , A ) = { } , we put A R = A . Now set A + := ( n X k =1 x ∗ k x k , x k ∈ A , n ∈ N ) . Then A + is a wedge in A and we call the elements of A + positiveelements of A . As in [14], we call positive elements of A the membersof A + τ n . We set A + := A + τ n .A linear functional on A is positive if ω ( a ) ≥ a ∈ A + . Definition 2.5.
A family of positive linear functionals F on ( A [ τ n ] , A )is called sufficient if for every a ∈ A + , a = 0, there exists ω ∈ F suchthat ω ( a ) > Definition 2.6.
A normed quasi ∗ -algebra ( A [ τ n ] , A ) is called fullyrepresentable if R c ( A , A ) is sufficient and A R = A .We denote by S A ( A ) the family of all continuous sesquilinear formsΩ : A × A → C such that MARIA STELLA ADAMO (i) Ω( a, a ) ≥ a ∈ A ;(ii) Ω( ax, y ) = Ω( x, a ∗ y ) for every a ∈ A , x, y ∈ A ;(iii) | Ω( a, b ) | ≤ k a kk b k , for all a, b ∈ A . Definition 2.7.
A normed quasi *-algebra ( A [ τ n ] , A ) is called *-semi-simple if, for every 0 = a ∈ A , there exists Ω ∈ S A ( A ) such thatΩ( a, a ) > Theorem 2.8. [2] Let ( A , A ) be a Banach quasi *-algebra with unit . The following statements are equivalent.(i) R c ( A , A ) is sufficient.(ii) ( A , A ) is fully representable.If the following condition of positivity ( P ) a ∈ A and ω ( x ∗ ax ) ≥ ∀ ω ∈ R c ( A , A ) and x ∈ A ⇒ a ∈ A + holds, (i) and (ii) are equivalent to the following(iii) ( A , A ) is *-semisimple. Remark 2.9.
The condition ( P ) is not needed to show the implication( iii ) ⇒ ( ii ) of Theorem 2.8.For a Hilbert quasi *-algebra ( H , A ), representable and continuousfunciontals are in 1-1 correspondence with a certain family of elementsin H . Definition 2.10.
Let ( H , A ) be a Hilbert quasi *-algebra. An element ξ ∈ H is called(i) weakly positive if the operator L ξ : A → H defined as L ξ ( x ) = ξx is positive.(ii) bounded if the operator L ξ : A → H is bounded.The set of all weakly positive (resp. bounded) elements will be denotedas H + w (resp. H b ), see [2, 24].The name weakly positive is justified by the existence of a strongernotion of positivity introduced before (e.g. [14]). The comparisonbetween them has been investigated in [2]. Remark 2.11.
A Hilbert quasi *-algebra ( H , A ) is always fully clos-able , i.e., for every ξ ∈ H the densely defined operator L ξ : A → H isclosable. If ξ ∈ H b , then L ξ is everywhere defined and bounded. Theorem 2.12. [2] Let ( H , A ) be a Hilbert quasi *-algebra. Then ω ∈ R c ( H , A ) if, and only if, there exists a unique weakly positivebounded element η ∈ H such that ω ( ξ ) = h ξ | η i , ∀ ξ ∈ H . Construction and properties of a tensor productHilbert quasi *-algebra
In this section, we will present a way to construct a tensor prod-uct quasi *-algebra of two given Hilbert quasi *-algebras. Our aimis to preserve structural properties of the factors in the tensor prod-uct, especially those related to representability. For further readingon algebraic tensor products see for instance [9], for topological tensorproducts refer to [10, 11, 19].Let ( H , A ) and ( H , B ) be two given Hilbert quasi *-algebras.The first step in this construction consists of building up the algebraictensor product. For convenience, we can assume that both the Hilbertquasi *-algebras are unital.Observe that both H and H are bimodules, respectively over A and B , thus we expect the tensor product to be a bimodule over asuitable *-algebra given by the tensor product *-algebra of A and B . Definition 3.1. [16] Let
H, K, L be Hilbert spaces over the complexfield and R : H × K → L a bilinear map. R is called a Schimdtbilinear map if for each total system { e µ ; µ ∈ Λ } in H and { e ν ; ν ∈ Λ } in K and each z ∈ L we have X {| hR ( e µ , e ν ) | z i | : µ ∈ Λ , ν ∈ Λ } < ∞ . Note that the sum computed above depends only on the choice of z and not on the chosen systems. Definition 3.2. [16, Ch. 2] Let
H, K, L be Hilbert spaces over thecomplex field and R : H × K → L a Schmidt bilinear map. Thenumber sup k z k L ≤ X {| hR ( e µ , e ν ) | z i | : µ ∈ Λ , ν ∈ Λ } is said to be the Schimdt norm of the bilinear map R . Definition 3.3. [16, Ch. 2] Let
H, K,
Θ be Hilbert spaces over thecomplex field and Ψ : H × K → Θ a Schmidt bilinear map. The pair(Θ , Ψ) is a
Hilbert tensor product of H and K if it has the universalproperty with respect to all Schmidt bilinear maps Ψ ′ : H × K → Θ ′ ,that are Schimdt norm contractions, where Θ ′ is some complex Hilbertspace.The pair (Θ , Ψ) exists and it is unique, in the sense that if ( E ′ , Ψ ′ ) isanother tensor product of the Hilbert spaces H, K as in Definition 3.1,then there is an algebraic isomorphism i : Θ → Θ ′ such that i ◦ Ψ = Ψ ′ .Hence, the unique Hilbert tensor product will be denoted by H b ⊗ K . MARIA STELLA ADAMO
Theorem 3.4. [16, Ch. 2] Let S : K → K and T : L → L bebounded operators between Hilbert spaces. Then there exists a uniquebounded operator S b ⊗ T : K b ⊗ L → K b ⊗ L such that ( S b ⊗ T )( x ⊗ y ) = S ( x ) ⊗ T ( y ) for all x ∈ K , y ∈ L . Moreover k S b ⊗ T k = k S k k T k .Having at our hands these notions, we construct the tensor productexplicitly and show that it is a quasi *-algebra. H ⊗H is built as the vector space tensor product of H and H over C . A ⊗ B can be naturally regarded as a subspace of H ⊗ H , since A and B are subspaces of H and H respectively. A and B are also*-algebras, thus their vector space tensor product A ⊗ B becomes a*-algebra if we define the product of two elements z = P ni =1 x i ⊗ y i , z ′ = P mj =1 x ′ j ⊗ y ′ j in A ⊗ B as follows zz ′ := n X i =1 m X j =1 x i x ′ j ⊗ y i y ′ j and an involution ∗ as z ∗ := n X i =1 x ∗ i ⊗ y ∗ i . Note that the product and the involution are well-defined (see [21, pp.188,189]).A general element ζ ∈ H ⊗ H is of the form ζ = n X i =1 ξ i ⊗ η i , ξ i ∈ H , η i ∈ H , ≤ i ≤ n. The C -vector space H ⊗ H becomes a bimodule over the *-algebra A ⊗ B defining the module actions and the involution component-wise, i.e., for every x ⊗ y ∈ A ⊗ B , ξ ⊗ η ∈ H ⊗ H ( x ⊗ y )( ξ ⊗ η ) = xξ ⊗ yη and ( ξ ⊗ η )( x ⊗ y ) = ξx ⊗ ηy ;( ξ ⊗ η ) ∗ = ξ ∗ ⊗ η ∗ . It is easily shown that the preceding operations are well defined, ex-tending those defined in A ⊗ B (see [21, pp. 188-189] and [20, Lemma1.4, p. 361]).All the requirements of Definition 1.1 are easily verified by the prop-erties of the quasi *-algebras ( H , A ) and ( H , B ). Thus, we concludethat ( H ⊗ H , A ⊗ B ) is a quasi *-algebra .The second step consists of providing the quasi *-algebra H ⊗ H with a suitable inner product in the way it becomes a Hilbert quasi*-algebra.
If we denote by h·|·i and h·|·i the inner products of H and H respectively, the most natural choice is given by(3.1) h ζ | ζ ′ i := n X i =1 m X j =1 (cid:10) ξ i | ξ ′ j (cid:11) (cid:10) η i | η ′ j (cid:11) , ∀ ζ , ζ ′ ∈ H ⊗ H , where ζ = P ni =1 ξ i ⊗ η i and ζ ′ = P mj =1 ξ ′ j ⊗ η ′ j . By Lemma 1.1 in [22,Ch. IV], the map defined in (3.1) is a well-defined inner product on H ⊗ H .The norm k·k h induced by the inner product in (3.1) is a cross-norm ,i.e., if ζ = ξ ⊗ η with ξ ∈ H , η ∈ H is an elementary tensor, then k ζ k h = k ξ ⊗ η k h = k ξ k k η k .Taking the completion H b ⊗ h H of H ⊗ h H with respect to thetopology h corresponding to k · k h , we obtain a Hilbert space, whichis the Hilbert tensor product according to Definition 3.1. The tensormap Θ : H × H → H b ⊗ h H is a Schmidt bilinear map.We want to show that H b ⊗ h H can be regarded as a Hilbert quasi*-algebra over A ⊗ B .Conditions (2) - (4) of Definition 1.3 are obviously verified by theproperties of Hilbert quasi *-algebras ( H , A ) and ( H , B ). To show(1), we have to prove the continuity of the left multiplication operators L z : H b ⊗ h H → H b ⊗ h H defined as ζ L z ( ζ ) = ζ z for every z ∈ A ⊗ B . Without loss of generality, we can assume that z = x ⊗ y , for x ⊗ y ∈ A ⊗ B . It is enough to show that the operator L z restricted to H ⊗ H is continuous.The restriction of L x ⊗ y to H ⊗ H coincides with the operator L x ⊗ L y , thus, by Theorem 3.4, it is continuous since the operators L x , L y are continuous.What remains to be shown is that in fact the completion H b ⊗ h H isthe same as A b ⊗ h B . For our aim, it is enough to show that A ⊗ B is h -dense in H ⊗ H . Without loss of generality, we show the statementonly for elementary tensors in H ⊗ H .Let ξ ⊗ η an elementary tensor in H ⊗ H . By the properties ofHilbert quasi *-algebra of H and H , there exist sequences { x n } and { y n } in A and B respectively, such that k x n − ξ k → k y n − η k →
0. By the cross-norm property of k·k h , the sequence { x n ⊗ y n } in A ⊗ B approximates ξ ⊗ η .Since A ⊗ B is dense in H ⊗ H , we can deduce that A b ⊗ h B ∼ = H b ⊗ h H . In conclusion, we have proved the following
Theorem 3.5.
Let ( H , A ) and ( H , B ) be unital Hilbert quasi *-algebras. Then the completion A b ⊗ h B ∼ = H b ⊗ h H of A ⊗ B withrespect to the inner product (3.1) is a unital Hilbert quasi *-algebra. Definition 3.6.
Let ( H , A ), ( H , B ) be unital Hilbert quasi *-alge-bras. Then the Hilbert quasi *-algebra ( H b ⊗ h H , A ⊗ B ) constructedin Theorem 3.5 will be called tensor product Hilbert quasi *-algebra .For the tensor product Hilbert quasi *-algebra ( H b ⊗ h H , A ⊗ B )obtained in Theorem 3.5 apply all the representability results obtainedfor Hilbert quasi *-algebras in [2].If Ω : H b ⊗ h H → C is a representable and continuous functional,then by Proposition 2.4, the domain of the sesquilinear form ϕ Ω is thewhole H b ⊗ h H .Moreover, Hilbert quasi *-algebras are always *-semisimple . Hence,( H b ⊗ h H , A ⊗ B ) is *-semisimple, and, by Theorem 2.8 and Remark2.9, it is also fully representable.We conclude then with the following Corollary 3.7.
Let ( H , A ) and ( H , B ) be unital Hilbert quasi *-algebras. Then the tensor product Hilbert quasi *-algebra ( H b ⊗ h H , A ⊗ B ) is *-semisimple and fully representable.By Remark 2.11, the Hilbert quasi *-algebra ( H b ⊗ h H , A ⊗ B ) isalso fully closable . Note that this property can be now derived directlyby Corollary 3.7.Corollary 3.7 highlights the important fact that the representabil-ity properties are maintained passing from the factors to the tensorproduct. Moreover, given a couple of representable and continuousfunctionals defined on the factors, the tensor product of these func-tionals turns out to be always representable and continuous, as shownin Proposition 4.2 in the next section.The same question about representability has been investigated in amore general setting of Banach quasi *-algebras in [1].4. Representability for tensor product Hilbert quasi*-algebras
In this section, we investigate how representability of functionalspasses to their tensor product. Since for Hilbert quasi *-algebras rep-resentable and continuous functionals have been characterized in terms of weakly positive bounded elements, we first prove a lemma about thetensor product of these elements. Lemma 4.1.
Let ( H b ⊗ h H , A ⊗ B ) be the tensor product Hilbertquasi *-algebra of ( H , A ) and ( H , B ), as in Theorem 3.5. Then, wehave the following inclusions(i) ( H ) + w ⊗ ( H ) + w ⊆ ( H b ⊗ h H ) + w .(ii) ( H ) b ⊗ ( H ) b ⊆ ( H b ⊗ h H ) b . Proof.
Let η ∈ ( H ) + w and η ∈ ( H ) + w . Then, the multiplicationoperators R η : A → H and R η : B → H are densely defined andpositive. Let R ′ η and R ′ η be positive self-adjoint extensions of R η and R η respectively. We have R η ⊗ η = R η ⊗ R η on A ⊗ B .The tensor product η ⊗ η is again a weakly positive element. Indeed,let z = P ni =1 x i ⊗ y i be in A ⊗ B , then h z | R η ⊗ η z i = * n X i =1 x i ⊗ y i | ( R η ⊗ R η ) n X j =1 x j ⊗ y j !+ = n X i,j =1 h x i | R η x j i h y i | R η y j i = n X i,j =1 (cid:10) x i | R ′ η x j (cid:11) (cid:10) y i | R ′ η y j (cid:11) = n X i,j =1 D ( R ′ η ) x i | ( R ′ η ) x j E D ( R ′ η ) y i | ( R ′ η ) y j E = n X i,j =1 (cid:10) x ′ i | x ′ j (cid:11) (cid:10) y ′ i | y ′ j (cid:11) , where x ′ i = ( R ′ η ) x i ∈ H and y ′ i = ( R ′ η ) y i ∈ H , for every i = 1 , . . . , n . Applying the Gram-Schmidt orthogonalization, we canassume that { x ′ i } ni =1 are orthogonal. Hence, h z | R η ⊗ η z i = n X i,j =1 (cid:10) x ′ i | x ′ j (cid:11) (cid:10) y ′ i | y ′ j (cid:11) = n X i =1 h x ′ i | x ′ i i h y ′ i | y ′ i i ≥ . Let χ ∈ ( H ) b and χ ∈ ( H ) b . Then χ ⊗ χ is also bounded.Indeed, the left multiplication operator L χ ⊗ χ : H ⊗H → H b ⊗ h H isequal to L χ ⊗ L χ . Thus, it is bounded by Theorem 3.4 and continuityof L χ and L χ . (cid:3) Proposition 4.2.
Let ( H , A ), ( H , B ) be unital Hilbert quasi *-algebras. Let ( H b ⊗ h H , A ⊗ B ) be the tensor product Hilbert quasi*-algebra. If ω and ω are representable and continuous functionalson H and H respectively, then ω ⊗ ω extends continuously to arepresentable and continuous functional on H b ⊗ h H . Proof.
Let ω and ω be representable and continuous functionals on H and H respectively. By Theorem 2.12, there exist weakly positivebounded elements χ ∈ H and χ ∈ H such that ω ( η ) = h η | χ i , ∀ η ∈ H , (4.1) ω ( η ) = h η | χ i , ∀ η ∈ H . (4.2)Define now ω ⊗ ω on H ⊗ h H as ω ⊗ ω n X i =1 ξ i ⊗ η i ! := n X i =1 ω ( ξ i ) ω ( η i ) , ∀ n X i =1 ξ i ⊗ η i ∈ H ⊗ h H . ω ⊗ ω is a well defined linear map on H ⊗ h H . By (4.1) and (4.2), ω ⊗ ω n X i =1 ξ i ⊗ η i ! = n X i =1 h ξ i | χ i h η i | χ i = * n X i =1 ξ i ⊗ η i | χ ⊗ χ + h for all P ni =1 ξ i ⊗ η i ∈ H ⊗ h H . Hence, ω ⊗ ω is continuous anddenote by Ω its continuous extension to H b ⊗ h H .By Lemma 4.1, χ ⊗ χ is weakly positive and bounded on H b ⊗ h H .Therefore, Ω is representable and continuous. Indeed, if nP k n i =1 x in ⊗ y in o is a sequence of elements in A ⊗ B approximating ψ ∈ H b ⊗ h H , thenΩ is given by Ω( ψ ) := lim n → + ∞ ω ⊗ ω k n X i =1 x in ⊗ y in ! , ∀ ψ ∈ H b ⊗ h H . By the properties of ω and ω , Ω( ψ ) = lim n → + ∞ ω ⊗ ω k n X i =1 x in ⊗ y in ! = lim n → + ∞ * k n X i =1 x in ⊗ y in | χ ⊗ χ + = h ψ | χ ⊗ χ i , for all ψ ∈ H b ⊗ h H . By Theorem 2.12 we get the representability ofΩ on H b ⊗ h H . (cid:3) Final remarks and open problems
A key role in looking at structure properties of locally convex quasi*-algebras is played by representable (and continuous) functionals (see[2, 7, 14, 23]). In this paper we constructed a new Hilbert quasi *-algebra as the tensor product of two given ones and then we studiedits properties, based on the results obtained in [2]. Nevertheless, manyinteresting questions remained open.It would be of interest to understand which is the relationship be-tween bounded elements in the factors ( H , A ) and ( H , B ) and thosein the tensor product ( H b ⊗ h H , A ⊗ B ). A similar question can beposed for weakly positive elements.In (ii) of Lemma 4.1, we have shown that the tensor product of( H ) b and ( H ) b is contained in the C*-algebra ( H b ⊗ h H ) b (see also[2, Proposition 4.9]). It is unknown if it is isomorphic of a certainC*-completion of the tensor product ( H ) b ⊗ ( H ) b .Answering these questions concerns the study of representable andcontinuous functionals on the tensor product Hilbert quasi *-algebra.In particular, if Ω ∈ R c ( H b ⊗ h H , A ⊗ B ), the restrictions ω on H and ω on H of Ω belong to R c ( H , A ) and R c ( H , B ) respectively.By Proposition 4.2, the tensor product ω ⊗ ω extends to a functional e Ω in R c ( H b ⊗ h H , A ⊗ B ). A natural question to ask is under whichconditions this extension corresponds to Ω.The construction given in Theorem 3.5 could be useful to exhibitother examples of Hilbert quasi *-algebras ( H , A ) for which R c ( H , A )is equal to R ( H , A ). By Propositions 5.1 and 5.2 of [2], this is true forthe Hilbert quasi *-algebras ( L ( I, dλ ) , C ( I )) and ( L ( I, dλ ) , L ∞ ( I, dλ )),where I is a compact interval of the real line and λ is the Lebesguemeasure.We leave these questions about tensor product of Hilbert quasi *-algebras for future papers, with the thought that they can contributeto give a better understanding of locally convex quasi *-algebras andtheir tensor products. Acknowledgment:
The author is grateful to the Organizers of the27th International Conference in Operator Theory for this wonderfulconference and the West University of Timisoara for its hospitality. Theauthor also wishes to thank prof. M. Fragoulopoulou for interestingdiscussions and suggestions.
References [1]
M. S. Adamo, M. Fragoulopoulou , Tensor products of normed and Ba-nach quasi *-algebras , in preparation.[2]
M. S. Adamo, C. Trapani , Representable and continuous functionals on aBanach quasi ∗ − algebra , Mediterr. J. Math. (2017) : 157.[3] D. Aerts, I. Daubechies , Physical justification for using the tensor productto describe two quantum systems as one joint system , Helv. Phys. Acta (1978), 661-675.[4] J.-P. Antoine, A. Inoue, C. Trapani , Partial *-Algebras and their Oper-ator Realizations , Math. Appl., , Kluwer Academic, Dordrecht, 2003.[5]
F. Bagarello, M. Fragoulopoulou, A. Inoue, C. Trapani , Structure oflocally convex quasi C ∗ − algebras , J. Math. Soc. Japan, , n.2 (2008), pp.511-549.[6] F. Bagarello, M. Fragoulopoulou, A. Inoue and
C. Trapani , Thecompletion of a C ∗ − algebra with a locally convex topology , J. Operator Theory, (2006), 357-376. .[7] F. Bagarello, C. Trapani , CQ ∗ − algebras: Structure properties , Publ.RIMS, Kyoto Univ. (1996), 85-116.[8] F. Bagarello, C. Trapani , The Heisenberg dynamics of spin systems: Aquasi *-algebra approach , J. Math. Phys. (1996), 4219-4234.[9] L. Chambadal, J.L. Ovaert , Alg`ebre Lin´eaire et Alg`ebre Tensorielle , DunodUniversit´e, Paris, 1968.[10]
A. Defant, K. Floret , Tensor Norms and Operator Ideals , North-Holland,Amsterdam, 1993.[11]
M. Fragoulopoulou , Topological Algebras with Involution , North-Holland,Amsterdam, 2005.[12]
M. Fragoulopoulou, A. Inoue, M. Weigt , Tensor products of generalized B ∗ -algebras , J. Math. Anal. Appl. (2014), 1787-1802.[13] M. Fragoulopoulou, A. Inoue, M. Weigt , Tensor products of unboundedoperator algebras , Rocky Mount. (2014), 895-912.[14] M. Fragoulopoulou, C. Trapani, S. Triolo , Locally convex quasi *-algebras with sufficiently many *-representations , J. Math. Anal. Appl. (2012) 1180 - 1193.[15]
W-D. Heinrichs , Topological tensor products of unbounded operator algebrason Fr`echet domains , Publ. RIMS, Kyoto Univ. (1997), 241-255.[16] A.Ya. Helemskii , Lectures and Exercises on Functional Analysis , Transla-tions of Mathematical Monographs, Vol. 233, Amer. Math. Soc., Providence,Rhode Island, 2006.[17]
G. Lassner , Topological algebras and their applications in quantum statistics,Wiss. Z. KMU-Leipzig, Math. Naturwiss. R. (1981) 572-595.[18] G. Lassner , Algebras of unbounded operators and quantum dynamics , Phys.A (1984) 471-480.[19]
K.B. Laursen , Tensor products of Banach algebras with involution , Trans.Amer. Math. Soc. (1969), 467-487.[20]
A. Mallios , Topological Algebras. Selected topics , North-Holland, Amster-dam, 1986.[21]
G.J. Murphy , C*-Algebras and Operator Theory , Academic Press, 2014. [22] M. Takesaki , Theory of Operator Algebras I , Springer-Verlag, New York,1979.[23]
C. Trapani , *-Representations, seminorms and structure properties ofnormed quasi *-algebras , Studia Math. (2008) 47-75.[24] C. Trapani , Bounded elements and spectrum in Banach quasi *-algebras ,Studia Mathematica (2006), 249-273.
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