The interplay between representable functionals and derivations on Banach quasi *-algebras
aa r X i v : . [ m a t h . F A ] S e p THE INTERPLAY BETWEEN REPRESENTABLEFUNCTIONALS AND DERIVATIONS ON BANACHQUASI *-ALGEBRAS
MARIA STELLA ADAMO
Abstract.
This note aims to highlight the link between repre-sentable functionals and derivations on a Banach quasi *-algebra,i.e. a mathematical structure that can be seen as the completion ofa normed *-algebra in the case the multiplication is only separatelycontinuous. Representable functionals and derivations have beeninvestigated in previous papers for their importance concerning thestudy of the structure properties of a Banach quasi *-algebra andapplications to quantum models. Introduction and preliminaries
Representable functionals constitute an important tool to investigate(locally convex) quasi *-algebras for being those linear functionals thatallow a GNS-like construction (see [1, 3, 7, 12, 14]). In particular, weare interested in the case of a Banach quasi *-algebra, i.e. a locallyconvex quasi *-algebra whose topology is generated by a single normin the case of separately continuous multiplication (see [3, 5, 6, 9]).As a consequence, the multiplication is defined only for certain couplesof elements. On the other hand, in the last decades derivations andtheir properties have been extensively studied for their use in describingphysical phenomena (see, for instance, [2, 4, 8, 10, 11]).In the classical context, a derivation δ is a linear mapping definedon a dense *-subalgebra D ( δ ) of a normed *-algebra A [ k · k ] for whichthe Leibnitz rule δ ( xy ) = δ ( x ) y + xδ ( y ) , ∀ x, y ∈ D ( δ ) Mathematics Subject Classification.
Primary 46L08, 46L57; Secondary46L89, 47L60.
Key words and phrases.
Representable functionals, weak *-derivations, infinites-imal generators of weak Banach quasi *-algebras. holds. In the framework of a Banach quasi *-algebra, the main issueconcerns the definition of a weaker form of the Leibnitz rule, suitablefor the new situation. This question has been addressed to the paper[2], where weak *-derivations appear in the *-semisimple case. In thelatter paper, conditions for a weak *-derivation to be the generator ofa *-automorphisms group are given, in particular a uniformly boundednorm continuous group has a closed generator. In the case δ is a specialkind of weak *-derivation, we are able to give a condition of closabilitydepending on a certain representable functional.Going in details, we remind some definitions about Banach quasi*-algebras and representations before focusing our attention on theproperties of representable functionals, in particular we are interestedin notions like fully representability and *-semisimplicity, importantfor the ongoing work (Section 2). Sufficiently many sesquilinear formsare needed to define weak *-derivations and study them when they aregiven as the infinitesimal generator of a weak *-automorphisms group.For a general weak *-derivation, representable functionals still play animportant role in their study (Section 3).We start giving some preliminary notions. For further details werefer to [3]. Definition 1. A quasi *–algebra ( A , A ) is a pair consisting of a vectorspace A and a *–algebra A contained in A as a subspace and such 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 ).We now introduce a suitable class of operators in order to representabstract quasi *-algebras. These operators are defined on a commondense domain D with values on a Hilbert space H and admit an adjointwith the same property. Hence, these operators are closable .Let H be a Hilbert space with inner product h·|·i and let D be a denselinear subspace of H . We denote by L † ( D , H ) the set of all closableoperators X in H such that the domain of X is D and the domain of EPRESENTABLE FUNCTIONALS AND DERIVATIONS 3 the adjoint X ∗ contains D , i.e. L † ( D , H ) = { X : D → H : D ( X ) = D , D ( X ∗ ) ⊃ D} . L † ( D , H ) is a C − vector space with the usual sum X + Y and scalarmultiplication λX for every X, Y ∈ L † ( D , H ) and λ ∈ C . If we definethe following involution † and partial multiplication (cid:3) X X † ≡ X ∗ ↾ D and X (cid:3) Y = X †∗ Y defined whenever X is a left multiplier of Y (or Y a right multiplier of X ), i.e. Y D ⊂ D ( X †∗ ) and X † D ⊂ D ( Y ∗ ), then L † ( D , H ) is a partial*-algebra defined in [3].In L † ( D , H ) several topologies can be introduced (see, [3]). In par-ticular, the weak topology t w defined by the family of seminorms p ξ,η ( X ) = h Xξ | η i , ξ, η ∈ D is related to a characterization of continuity of representable functionalsgiven in a recent work [1].Due to the structure properties of quasi *-algebras, we define a *-representation similarly to the classical case, except for the requirementon the multiplications, in the following way. Definition 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 subspace ofthe 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 • faithful if a = 0 implies π ( a ) = 0; • cyclic if π ( A ) ξ is dense in H π for some ξ ∈ D π .If ( A , A ) has a unit , then we suppose that π ( ) = I D , the identityoperator of D .The closure e π of a *-representation π of the quasi *-algebra ( A , A )in L † ( D π , H π ) is defined as e π : A → L † ( e D π , H π ) , a π ( a ) ↾ e D π where e D π is the completion of D π with respect to the graph topology,i.e. the topology defined by the seminorms η ∈ D π
7→ k π ( a ) η k for every a ∈ A . A *-representation π is said to be closed if π = e π .A quasi *-algebra ( A , A ) is called a normed quasi *-algebra if a norm k · k is defined on A with the properties MARIA STELLA ADAMO (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 3. If ( A , k · k ) is a Banach space, we say that ( A , A ) is aBanach quasi *-algebra. The norm topology of A will be denoted by τ n .2. Representability of Banach quasi *-algebras
In this section we examine some properties related to representabilityof Banach quasi *-algebras. Among them, an important role is playedby fully representability and *-semisimplicity . For details, see [1, 3, 9,12, 14].
Theorem 1. [14, Theorem 3.5]
Let ( A , A ) be a quasi *-algebra withunit and let ω be a linear functional on ( A , A ) that satisfies thefollowing conditions: (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 > such that | ω ( a ∗ x ) | ≤ γ a ω ( x ∗ x ) / , ∀ x ∈ A . Then, there exists a triple ( π ω , λ ω , H ω ) such that: • 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.
Definition 4.
A linear functional ω : A → C satisfying (L.1)-(L.3) inTheorem 1 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 ).As in [14, Definition 2.1], given a quasi *-algebra ( A , A ), we denoteby Q A ( A ) the set of all sesquilinear forms on A × A such that(i) ϕ ( a, a ) ≥ a ∈ A . EPRESENTABLE FUNCTIONALS AND DERIVATIONS 5 (ii) ϕ ( ax, y ) = ϕ ( x, a ∗ y ) for every a ∈ A and x, y ∈ A Proposition 1. [1, Proposition 2.9]
Let ( A , A ) be a quasi *-algebrawith unit and ω a linear functional on A satisfying (L.1) and (L.2).The following statements are equivalent. (i) ω is representable. (ii) There exist a *-representation π defined on a dense domain D π of a Hilbert space H π and a vector ζ ∈ D π such that ω ( a ) = h π ( a ) ζ | ζ i , ∀ a ∈ A . (iii) There exists a sesquilinear form Ω ω ∈ Q A ( A ) such that ω ( a ) = Ω ω ( a, ) , ∀ a ∈ A . To every ω ∈ R ( A , A ) we can associate two sesquilinear forms. Thefirst one Ω ω , already introduced in (iii) of Proposition 1, can be definedthrough the GNS representation π ω , with cyclic vector ξ ω . In fact, weput(2.1) Ω ω ( a, b ) = h π ω ( a ) ξ ω | π ω ( b ) ξ ω i , a, b ∈ A . As we have seen Ω ω ∈ Q A ( A ) and ω ( a ) = Ω ω ( a, ) , for every a ∈ A . The second sesquilinear form, which we denote by ϕ ω , is defined onlyon A × A by(2.2) ϕ ω ( x, y ) = ω ( y ∗ x ) , x, y ∈ A . It is clear that Ω ω extends ϕ ω . 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[12], if ω ∈ R c ( A , A ), then the sesquilinear form ϕ ω defined in (2.2),is closable ; that is, ϕ ω ( x n , x n ) →
0, for every sequence { x n } ⊂ A suchthat 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 . Thus, a natural question arises: under which conditionsone gets the equality D ( ϕ ω ) = A ? An answer to this question will begiven in Proposition 2.Consider now the set A R := \ ω ∈R c ( A , A ) D ( ϕ ω ) . MARIA STELLA ADAMO If R c ( A , A ) = { } , we put A R = A . Note that, if for every ω ∈R c ( A , A ), ϕ ω is jointly continuous with respect to the topology τ n defined by the norm k · k , we get A R = A .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 [12], we call positive elements of A the membersof A + τ n . We set A + := A + τ n . Definition 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 6. [12, Definition 4.1]
A normed quasi ∗ -algebra ( A [ τ n ] , A ) is called fully representable if R c ( A , A ) is sufficient and A R = A . If ( A , A ) has a unit , the condition of sufficiency required in Defi-nition 6 joined with the following condition of positivity(P) a ∈ A and ω x ( a ) ≥ ω ∈ R c ( A , A ) and x ∈ A ⇒ a ≥ ω ( a ) = 0 for every ω ∈ R c ( A , A ) implies a = 0.We denote by S A ( A ) the subset of Q A ( A ) consisting of all contin-uous sesquilinear forms Ω : A × A → C such that | Ω( a, b ) | ≤ k a kk b k , ∀ a, b ∈ A . By defining k Ω k = sup k a k = k b k =1 | Ω( a, b ) | , one obviously has k Ω k ≤
1, for every Ω ∈ S A ( A ). Definition 7.
A normed quasi *-algebra ( A [ τ n ] , A ) is called *-semi-simple if, for every = a ∈ A , there exists Ω ∈ S A ( A ) such that Ω( a, a ) > . As we mentioned before, if ω ∈ R c ( A , A ), then the form ϕ ω definedin (2.2) is closable. For Banach quasi *-algebras this result can beimproved. Proposition 2. [1, Proposition 3.6]
Let ( A , A ) be a Banach quasi *-algebra with unit , ω ∈ R c ( A , A ) and ϕ ω the associated sesquilinearform on A × A defined as in (2.2) . Then D ( ϕ ω ) = A ; hence ϕ ω iseverywhere defined and bounded. EPRESENTABLE FUNCTIONALS AND DERIVATIONS 7
The previous proposition can be used to show the following result.
Theorem 2. [1, Theorem 3.9]
Let ( A , A ) be a Banach quasi *-algebrawith unit . The following statements are equivalent. (i) R c ( A , A ) is sufficient. (ii) ( A , A ) is fully representable.If the condition of positivity ( P ) holds, (i) and (ii) are equivalent tothe following (iii) ( A , A ) is *-semisimple. Theorem 2 shows the deep relationship between fully representabilityand *-semisimplicity for a Banach quasi *-algebra. Under the conditionof positivity (P), the families of sesquilienar forms involved can beidentified. 3.
Derivations and their closability
In this section, we present an appropriate definition of derivation inthe case of a *-semisimple Banach quasi *-algebra. For detailed discus-sion, see [2, 4, 13]. In the case it is a generator of a *-automorphismsgroup, the derivation is closed and it owns a certain spectrum. Employ-ing sesquilinear forms introduced in the previous section, it is possibleto show a condition of closability for general derivations.Through the sesquilinear forms ϕ ∈ S A ( A ), we can define a newmultiplication in A as in [15].Let ( A , A ) be a *-semisimple Banach quasi *-algebra. Let a, b ∈ A .We say that the weak multiplication a (cid:3) b is well-defined if there existsa (necessarily unique) c ∈ A such that: ϕ ( bx, a ∗ y ) = ϕ ( cx, y ) , ∀ x, y ∈ A , ∀ ϕ ∈ S A ( A ) . In this case, we put a (cid:3) b := c . Definition 8.
Let ( A , A ) be a *-semisimple Banach quasi *-algebra.An element a ∈ A is called bounded if the following equivalent condi-tions hold (i) the multiplication operators L a and R a are k · k− continuous; (ii) R w ( a ) = L w ( a ) = A , where R w ( a ) (resp. L w ( a ) ) is the space ofuniversal right (resp. left) weak multipliers of a . Let ( A , A ) be a *-semisimple Banach quasi *-algebra and θ : A → A a linear bijection. We say that θ is a weak *-automorphism of ( A , A )if (i) θ ( a ∗ ) = θ ( a ) ∗ , for every a ∈ A ; MARIA STELLA ADAMO (ii) θ ( a ) (cid:3) θ ( b ) is well defined if, and only if, a (cid:3) b is well defined and,in this case, θ ( a (cid:3) b ) = θ ( a ) (cid:3) θ ( b ) . By the previous definition, if θ is a weak *-automorphism, then θ − is a weak *-automorphism too.Let ( A , A ) be a *-semisimple Banach quasi *-algebra. Suppose thatfor every fixed t ∈ R , β t is a weak *-automorphism of A . If(i) β ( a ) = a, ∀ a ∈ A (ii) β t + s ( a ) = β t ( β s ( a )), ∀ a ∈ A then we say that β t is a one-parameter group of weak *-automorphismsof ( A , A ). If τ is a topology on A and the map t β t ( a ) is τ -continuous, for every a ∈ A , we say that β t is a τ -continuous weak*-automorphism group.The definition of the infinitesimal generator of β t is now quite natu-ral. If β t is τ -continuous, we set D ( δ τ ) = (cid:26) a ∈ A : lim t → β t ( a ) − at exists in A [ τ ] (cid:27) and δ τ ( a ) = τ − lim t → β t ( a ) − at , a ∈ D ( δ τ ) . If the involution a a ∗ is τ -continuous, then a ∈ D ( δ τ ) implies a ∗ ∈ D ( δ τ ) and δ ( a ∗ ) = δ ( a ) ∗ .We are now giving an appropriate definition of *-derivation, weak-ening the Leibnitz rule thanks to sesquilinear forms. Definition 9. [2, Definition 4.5]
Let ( A , A ) be a *-semisimple Banachquasi *-algebra and δ a linear map of D ( δ ) into A , where D ( δ ) is apartial *-algebra with respect to the weak multiplication (cid:3) . We say that δ is a weak *-derivation of ( A , A ) if (i) A ⊂ D ( δ )(ii) δ ( x ∗ ) = δ ( x ) ∗ , ∀ x ∈ A (iii) if a, b ∈ D ( δ ) and a (cid:3) b is well defined, then a (cid:3) b ∈ D ( δ ) and ϕ ( δ ( a (cid:3) b ) x, y ) = ϕ ( bx, δ ( a ) ∗ y ) + ϕ ( δ ( b ) x, a ∗ y ) , for all ϕ ∈ S A ( A ) , for every x, y ∈ A . Parallel to the case of C*-algebras, to a uniformly bounded normcontinuous weak *-automorphisms group there corresponds a closedweak *-derivation that generates the group (see [11]).
EPRESENTABLE FUNCTIONALS AND DERIVATIONS 9
Theorem 3. [2, Theorem 5.1]
Let δ : D ( δ ) → A [ k · k ] be a weak*-derivation on a *-semisimple Banach quasi *-algebra ( A , A ) . Sup-pose that δ is the infinitesimal generator of a uniformly bounded, τ n -continuous group of weak *-automorphisms of ( A , A ) . Then δ is closed;its resolvent set ρ ( δ ) contains R \ { } and (3.1) k δ ( a ) − λa k ≥ | λ | k a k , a ∈ D ( δ ) , λ ∈ R . The converse of Theorem 3 can be proven assuming further condi-tions, for instance that the domain of the weak *-derivation is made ofbounded elements, which turn out to be satisfied in some interestingsituations such as the weak derivative in L p − spaces. Theorem 4. [2, Theorem 5.3]
Let δ : D ( δ ) ⊂ A b → A [ k · k ] be a closedweak *-derivation on a *-semisimple Banach quasi *-algebra ( A , A ) .Suppose that δ verifies the same conditions on its spectrum of Theorem3 and A is a core for every multiplication operator ˆ L a for a ∈ A , i.e. ˆ L a = L a . Then δ is the infinitesimal generator of a uniformly bounded, τ n -continuous group of weak *-automorphisms of ( A , A ) . In the sequel, we focus our attention on a special case of weak *-derivations, those for which D ( δ ) = A . These weak *-derivations arecalled qu*-derivations .Let ( A , A ) be a quasi *-algebra, δ be a qu*-derivation of ( A , A ) and π be a *-representation of ( A , A ). Assume that(3.2) whenever x ∈ A is such that π ( x ) = 0 , then π ( δ ( x )) = 0 . Under this assumption, the linear map δ π ( π ( x )) := π ( δ ( x )) , x ∈ A is well defined on π ( A ) with values in π ( A ) and it is easily checkedthat δ π is a qu*-derivation of A named induced by π . Definition 10.
Let ( A , A ) be a quasi *-algebra, δ be a qu*-derivationof ( A , A ) . Furthermore, let π be a cyclic *-representation of ( A , A ) with cyclic vector ξ satisfying the assumption (3.2) . The induced qu*-derivation δ π is spatial if there exists H = H † ∈ L ( D π , H π ) such that δ π ( π ( x )) = i [ H, π ( x )] , x ∈ A . Proposition 3.
Let ( A , A ) be a Banach quasi *-algebra with unit and let δ be a qu*-derivation of ( A , A ) . Suppose that there exists arepresentable and continuous functional ω with ω ( δ ( x )) = 0 for x ∈ A and let ( H ω , π ω , λ ω ) the GNS-construction associated to ω . Suppose that π ω is a faithful *-representation of ( A , A ) . Then there exists anelement H = H † of L † ( λ ω ( A )) such that π ω ( δ ( x )) = − i [ H, π ω ( x )] , ∀ x ∈ A and δ is closable.Proof. Define H on λ ω ( A ) by Hλ ω ( x ) = iπ ω ( δ ( x )) ξ ω , x ∈ A where ξ ω = λ ω ( ). We first prove that H is well defined. We have h π ω ( δ ( x )) ξ ω | π ω ( y ) ξ ω i = h π ω ( y ∗ ) π ω ( δ ( x )) ξ ω | ξ ω i = h π ω ( y ∗ δ ( x )) ξ ω | ξ ω i = h π ω ( δ ( y ∗ x ) − δ ( y ∗ ) x ) ξ ω | ξ ω i = −h π ω ( δ ( y ∗ ) x ) ξ ω | ξ ω i = −h π ω ( x ) ξ ω | π ω ( δ ( y )) ξ ω i . Hence if λ ω ( x ) = π ω ( x ) ξ ω = 0, it follows that h π ω ( δ ( x ) ξ ω | π ω ( y ) ξ ω i = 0,for every y ∈ A . This in turn implies that π ω ( δ ( x )) ξ ω = 0.The above computation shows also that H is symmetric. Indeed, h Hλ ω ( x ) | λ ω ( y ) i = i h π ω ( δ ( x )) ξ ω | π ω ( y ) ξ ω i = − i h π ω ( x ) ξ ω | π ω ( δ ( y )) ξ ω i = h λ ω ( x ) | Hλ ω ( y ) i . Finally, if x ∈ A , π ω ( δ ( x )) λ ω ( y ) = π ω ( δ ( x )) (cid:3) π ω ( y ) ξ ω = π ω ( δ ( xy )) ξ ω − π ω ( x ) (cid:3) π ω ( δ ( y )) ξ ω = − iHπ ω ( x ) λ ω ( y ) + iπ ω ( x ) Hλ ω ( y )= − i [ H, π ω ( x )] λ ω ( y ) , ∀ y ∈ A . (cid:3) Consider now a sequence x n ∈ A such that k x n k → w ∈ A for which k δ ( x n ) − w k → n → ∞ . Then, for every y, z ∈ A , EPRESENTABLE FUNCTIONALS AND DERIVATIONS 11 we have ω ( z ∗ w ∗ x ) = h π ω ( x ) ξ ω | π ω ( w ) (cid:3) π ω ( z ) ξ ω i = lim n →∞ h π ω ( x ) ξ ω | π ω ( δ ( x n )) (cid:3) π ω ( z ) ξ ω i = i lim n →∞ h π ω ( x ) ξ ω | Hπ ω ( x n ) (cid:3) π ω ( z ) ξ ω i− i lim n →∞ h π ω ( x ) ξ ω | π ω ( x n ) (cid:3) Hπ ω ( z ) ξ ω i = − i lim n →∞ Ω ω ( x, x n δ ( z )) → ξ ω and the densityof A , we obtain π ω ( w ) = 0. We conclude by the faithfulness of π ω . (cid:3) It would be of interest to study the closure of the qu*-derivation inProposition 3. Indeed, the existence of a representable and continuousfunctional makes that the Banach quasi *-algebra ( A , A ) automatically*-semisimple. If a ∈ A is such that Ω( a, a ) = 0 for every Ω ∈ S A ( A ),then in particular S A ( A ) ∋ Ω xω ( a, a ) := Ω ω ( ax, ax ) = 0 , ∀ x ∈ A , Ω ∈ S A ( A ) . This translates into π ω ( a ) λ ω ( x ) = 0 for every x ∈ A and so a = 0again by faithfulness. Acknowledgement:
The author is grateful to the Organizers ofthe International Conference on Topological Algebras and Applications2018 for taking care of this beautiful conference and the University ofTallinn (Estonia) for its hospitality.
References [1]
M. S. Adamo, C. Trapani , Representable and continuous functionals on aBanach quasi ∗ − algebra , Mediterr. J. Math. (2017) : 157.[2] M. S. Adamo, C. Trapani , Unbounded derivations and *-automorphismsgroups of Banach quasi *-algebras , arXiv:1807.11525.[3]
J.-P. Antoine, A. Inoue, C. Trapani , Partial *-Algebras and their Oper-ator Realizations , Math. Appl., , Kluwer Academic, Dordrecht, 2003.[4]
J.-P. Antoine, A. Inoue, C. Trapani , O*-dynamical systems and *deriva-tions of unbounded operator algebras , Math. Nachr., : 5-28 (1999).[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, A. Inoue and
C. Trapani , Representable linear functionalson partial *-algebras , Mediterranean J. Math., (2012) 153-163 [8] F. Bagarello, A. Inoue, C. Trapani
Representations and derivations ofquasi *-algebras induced by local modifications of states , J. Math. Anal. Appl., (2009), 615-623.[9]
F. Bagarello, C. Trapani , CQ ∗ − algebras: Structure properties , Publ.RIMS, Kyoto Univ. (1996), 85-116.[10] F. Bagarello, C. Trapani , The Heisenberg dynamics of spin systems: Aquasi *-algebra approach , J. Math. Phys. (1996), 4219-4234.[11] O. Bratteli, D. W. Robinson , Unbounded derivations of C*-algebras ,Comm. Math. Phys. (1975) n.3, 253–268.[12] M. Fragoulopoulou, C. Trapani, S. Triolo , Locally convex quasi *-algebras with sufficiently many *-representations , J. Math. Anal. Appl. (2012) 1180 - 1193.[13]
E. Hille, R. Phillips , Functional analysis and semi-groups , vol. 31, part 1,American Mathematical Society, 1996.[14]
C. Trapani , *-Representations, seminorms and structure properties ofnormed quasi *-algebras , Studia Math. (2008) 47-75.[15] C. Trapani , Bounded elements and spectrum in Banach quasi *-algebras ,Studia Mathematica (2006), 249-273.
Dipartimento di Matematica e Informatica, Universit`a di Palermo,I-90123 Palermo, Italy
E-mail address ::