Tensor products of normed and Banach quasi *-algebras
aa r X i v : . [ m a t h . F A ] F e b TENSOR PRODUCTS OF NORMED AND BANACHQUASI *-ALGEBRAS
MARIA STELLA ADAMO AND MARIA FRAGOULOPOULOU
Abstract.
Quasi *-algebras form an essential class of partial *-algebras, which are algebras of unbounded operators. In this work,we aim to construct tensor products of normed, respectively Ba-nach quasi *-algebras, and study their capacity to preserve someimportant properties of their tensor factors, like for instance, *-semisimplicity and full representability. Introduction
Topological quasi *-algebras appeared in the literature at the begin-ning of the ’80s, last century. They were introduced, in 1981, by G.Lassner [33, 34], to encounter solutions of certain problems in quan-tum statistics and quantum dynamics. But only later (see [44, p. 90]),the initial definition was reformulated in the right way, having thusincluded many more interesting examples.Quasi *-algebras came in light, in 1988; see [47], as well as literaturein [3, 9]. Many results have been published on this topic, which canbe found in the treatise [8], where the reader will also find a corre-sponding rich literature for partial *-algebras, whose a special subclassis given by quasi *-algebras. Note that partial *-algebras are algebrasof unbounded operators (for an extended exhibition of the latter, see[44]). The simplest example of a quasi (resp. partial) *-algebra isthe completion of a locally convex *-algebra with separately contin-uous multiplication. It is clear then that in this case, multiplicationis not everywhere defined. Completions of the previous kind may, forinstance, occur in quantum statistics. Applications of quasi *-algebrascan be found, e.g., in [24, 48].Partial *-algebras were introduced by J-P. Antoine and W. Kar-wowski in [6, 7] and, as we mentioned above, they are algebras of Keywords and phrases: Normed and Banach quasi *-algebra, representable lin-ear functional, sesquilinear form, *-semisimplicity, full representability, tensor prod-uct normed and Banach quasi *-algebra. Mathematics Subject Classification (2010): 46A32, 46K10, 47L60, 47L90. unbounded operators, playing an essential role in quantum field theory(see [8]).In the present paper, an effort is made to investigate topologicaltensor products of normed, respectively Banach quasi *-algebras. Themotivation, apart from the preceding discussion, is assisting from thefact that tensor products are used to describe two quantum systems asone joint system (see, for instance, [4] and [35]), while the physical sig-nificance of tensor products always depends on the applications, whichmay involve wave functions, spin states, oscillators and even more; inthis aspect, see e.g., [15, 26].In the literature, one can find very few articles dealing with tensorproducts of unbounded operator algebras, the oldest one, to our knowl-edge, dating from 1997 (see [27]) and dealing with tensor products ofunbounded operator algebras with Fr´echet domains. Another two ap-peared in 2014 (see [21, 22]) and concern tensor products of generalized B ∗ -algebras, respectively tensor products of generalized W ∗ -algebras.Both kinds of these algebras are unbounded generalizations of standard C ∗ -, respectively W ∗ -algebras, initiated by G.R. Allan (1967, [5]) andA. Inoue (1978, [31]), respectively. The latter author used generalized W ∗ -algebras for developing a Tomita Takesaki theory in algebras ofunbounded operators (1998). For this theory, the reader is referred to[32].The structure of the present paper is as follows: in Section 2, weexhibit the background material needed for our study.The structure of a (normed, resp. Banach) quasi *-algebra ( A , A )(where A is a vector space and A a *-algebra and a subspace of A ,both of them satisfying specific properties) leads to the examinationof the best possible (topological) tensor product of two (normed, resp.Banach) quasi *-algebras.In Section 3, we construct the algebraic tensor product of quasi *-algebras. We were led to our construction mainly from the fact thatthe new object we wanted to have as a quasi *-algebra should be acomplex linear space containing a *-algebra with certain properties.When we are given two quasi *-algebras ( A , A ), ( B , B ), for obtaining A ⊗ B as a quasi *-algebra over A ⊗ B , we consider the latter to bethe algebraic tensor product *-algebra canonically contained in A ⊗ B ,and then we define the left and right multiplications between elementsof A ⊗ B and A ⊗ B .Section 4 gives the construction of a tensor product normed, respec-tively Banach quasi *-algebra, coming from two given normed, respec-tively Banach quasi *-algebras. ENSOR PRODUCTS 3
In Section 5, examples of tensor product Banach quasi *-algebras arepresented.In the final Section 6, we discuss full representability and existence of*-representations on a tensor product normed quasi *-algebra. Since *-semisimplicity is related to both of the preceding concepts, informationis also given for this notion, in the tensor product environment. Moreprecisely, the mentioned concepts are studied in the capacity of passingfrom the considered tensor product to its factors and vice versa (see,e.g., Propositions 6.2, 6.5 and Theorems 6.11, 6.14, 6.18).2.
Notation and background material
All algebras and vector spaces we deal with in this article are over thefield C of complexes. Moreover, all topological spaces are consideredto be Hausdorff. Our basic definitions and notation concerning quasi*-algebras are mainly from [8].In the present section, we exhibit the necessary machinery, terminol-ogy and notation we need throughout this work. PART I:
QUASI *-ALGEBRAS
Definition 2.1. [8, Definition 2.1.9] A quasi *-algebra ( A , A ) is a pairconsisting of a vector space A and a *-algebra A contained in A as asubspace and such that(i) the left multiplication ax and the right multiplication xa of anelement a ∈ A and x ∈ A are always defined and bilinear;(ii) ( xa ) y = x ( ay ) and a ( xy ) = ( ax ) y , for each x, y ∈ A and a ∈ A ;(iii) an involution ∗ is defined in A , which extends the involution of A and has the property ( ax ) ∗ = x ∗ a ∗ and ( xa ) ∗ = a ∗ x ∗ , for all a ∈ A and x ∈ A .For a quasi *-algebra ( A , A ), we shall also use the term quasi *-algebra over A . ◮ Given a quasi *-algebra ( A , A ), the elements of A will always bedenoted by x, y, . . . , and the elements of A by a, b, . . . .We say that a quasi *-algebra ( A , A ) has a unit , if there is a uniqueelement e in A , such that ae = a = ea , for all a ∈ A . Example . Let I = [0 ,
1] be the unit interval and λ the Lebesguemeasure on I . Then, for 1 ≤ p < ∞ , the pair ( L p ( I, λ ) , L ∞ ( I, λ )) is aquasi *-algebra with respect to the usual operations, i.e., the multipli-cation is defined pointwise and the involution is given by the complexconjugate.
MARIA STELLA ADAMO AND MARIA FRAGOULOPOULOU ◮ From now on, writing L p ( I ), p ≥ we shall always mean that I is endowed with the Lesbesque measure , say λ , except if otherwise isspecified. Definition 2.3.
A quasi *-algebra ( A , A ) is called a normed quasi *-algebra if A is a normed space under a norm k·k satisfying the followingconditions:(i) k a ∗ k = k a k , ∀ a ∈ A ;(ii) A is dense in A [ k · k ];(iii) for every x ∈ A , the map R x : a ∈ A [ k · k ] → ax ∈ A [ k · k ] iscontinuous.When A [ k · k ] is a Banach space, we say that ( A [ k · k ] , A ) is a Banachquasi *-algebra .The continuity of the involution implies that(iv) for every x ∈ A , the map L x : a ∈ A [ k · k ] → xa ∈ A [ k · k ] isalso continuous.It is evident from the above that if ( A , A ) has an identity element e , then(a) if ax = 0, respectively xa = 0, for every x ∈ A , then a = 0;(b) if ax = 0, respectively xa = 0, for every a ∈ A , then x = 0.A norm is defined on A as follows: k x k := max {k L x k , k R x k} , x ∈ A , with k L x k , k R x k , the usual operator norms (see [24, beginning of Chap-ter 3]). Then, A [ k · k ] is a normed *-algebra and k ax k ≤ k x k k a k , ∀ a ∈ A , x ∈ A . (2.1)Observe that if ( A [ k · k ] , A ) has an identity e then, without loss ofgenerality, we may suppose that k e k = 1, since taking the equivalentto k · k norm k · k ′ on A defined by k a k ′ := k a k / k e k , a ∈ A , we obviouslyhave k e k ′ = 1. Furthermore, note that the norms k · k , k · k are notcomparable on A , in general. For instance, consider the Banach quasi*-algebra without unit ( L p ( R ) , C c ( R )), where C c ( R ) stands for the *-algebra of continuous functions on R with compact support. Then thenorms k · k p , k · k = k · k ∞ clearly are not comparable on C c ( R ). Butif a normed quasi *-algebra ( A [ k · k ] , A ) has a unit, then (2.1) impliesthat k x k ≤ k x k , for every x ∈ A .Other examples of Banach quasi *-algebras can be found, for in-stance, in [10, 12, 24]. In particular, we have ENSOR PRODUCTS 5
Example . Consider the unit interval I = [0 , L p -space L p ( I )with 1 ≤ p < ∞ and the C*-algebra C ( I ) of all continuous functionson I . Then the pair ( L p ( I ) , C ( I )) is a Banach quasi *-algebra. Example . The pair ( L p ( I ) , L ∞ ( I )) considered in Example 2.2 isanother example of Banach quasi *-algebra.On the other hand, among Banach quasi *-algebras, an essential roleis played by the completion of a Hilbert algebra with respect to thenorm induced by the given inner product. In what follows we firstdefine a Hilbert algebra and then a Hilbert quasi *-algebra. Definition 2.6.
A Hilbert algebra (see [39, Section 11.7]) is a *-algebra A , which is also a pre-Hilbert space with inner product h·|·i , such that(i) for every x ∈ A , the map y xy is continuous, with respectto the norm defined by the inner product;(ii) h xy | z i = h y | x ∗ z i , for all x, y, z ∈ A ;(iii) h x | y i = h y ∗ | x ∗ i , for all x, y ∈ A ;(iv) A is total in A .From (ii) and (iii) it follows that h xy | z i = h x | zy ∗ i , ∀ x, y, z ∈ A . Definition 2.7.
Let A be as in Definition 2.6 and let H denote theHilbert space completion of A , with respect to the norm k · k givenby the inner product. The involution of A extends to the whole of H , since by (iii) it is isometric. The multiplication ξx (or xξ ) of anelement ξ in H with an element x in A is defined by the usual limitprocedure. To avoid trivial instances, we assume that ξ ∈ H , such that ξx = 0 , ∀ x ∈ A , implies ξ = 0 . Under the preceding operations, the pair ( H [ k · k ] , A ) is now a Banachquasi *-algebra, that we call Hilbert quasi *-algebra .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 T in H , such that the domain of T is D and the domain ofits adjoint T ∗ , denoted by D ( T ∗ ), contains D , i.e., L † ( D , H ) = { T : D → H : D ( T ∗ ) ⊇ D} . The set L † ( D , H ) is a C − vector space with the usual sum T + S andscalar multiplication λT , for all T, S ∈ L † ( D , H ) and λ ∈ C . Definethe following involution † and partial multiplication (cid:3) by T T † ≡ T ∗ ↿ D and T (cid:3) S = ( T † ) ∗ S. MARIA STELLA ADAMO AND MARIA FRAGOULOPOULOU
It is clear that the partial multiplication (cid:3) is defined whenever S D ⊆ D (( T † ) ∗ ) and T † D ⊆ D ( S ∗ ) . (2.2)Then L † ( D , H ) becomes a partial *-algebra , in the sense of [8, Definition2.1.1]. In L † ( D , H ) several topologies can be introduced (see, [8]).Here, we will use the weak and strong* topology denoted by τ w , τ s ∗ respectively, which are defined by the families of seminorms p ξ,η ( T ) := |h T ξ | η i| , ξ, η ∈ D , T ∈ L † ( D , H ) ,p ∗ ξ ( T ) := max {k T ξ k , k T † ξ k} , ξ ∈ D , T ∈ L † ( D , H ) . We denote by L † ( D ) the subset of the elements T in L † ( D , H ), suchthat T D ⊆ D and T † D ⊆ D . Then, L † ( D ) is a *-algebra with respectto the involution † and the weak multiplication (cid:3) defined above. It isclear that the inclusions in (2.2) are always valid in L † ( D ). Definition 2.8. A *-representation π , of a quasi *-algebra ( A , A ) in aHilbert space H π , is a linear map π from A in L † ( D π , H π ), where D π isa dense subspace of H π and, at the same time, the following conditionshold:(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 ).Concerning (ii), note that for a ∈ A and x ∈ A , one also has that π ( a )is a right multiplier of π ( x ) and π ( x ) (cid:3) π ( a ) = π ( xa ).A *-representation π , as before, is faithful if a = 0 implies π ( a ) = 0and it is cyclic if π ( A ) ξ is dense in H π , for some ξ ∈ D π . If ( A , A ) hasan identity element e , we suppose that π ( e ) = I D π , the latter denotingthe identity operator from H π on H π , restricted on D π .The closure e π of a *-representation π of a quasi *-algebra ( A , A ) in L † ( D π , H π ) is defined as follows 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,defined by the seminorms η ∈ D π
7→ k π ( a ) η k , ∀ a ∈ A . A *-representation π is said to be closed if π = e π . Definition 2.9.
Let ( A , A ) be a quasi *-algebra. A linear functional ω on A is said to be representable if it satisfies the following conditions:(L.1) ω ( x ∗ x ) ≥ , ∀ x ∈ A ; ENSOR PRODUCTS 7 (L.2) ω ( y ∗ a ∗ x ) = ω ( x ∗ ay ) , ∀ x, y ∈ A , a ∈ A ;(L.3) for all a ∈ A , there exists γ a >
0, such that | ω ( a ∗ x ) | ≤ γ a ω ( x ∗ x ) / , ∀ x ∈ A . The set of all representable linear functionals is denoted by R ( A , A ).Given a quasi *-algebra ( A , A ), we denote by Q A ( A ) the set of allsesquilinear forms on A × A , such that (see [49, Definition 2.1])(i) ϕ ( a, a ) ≥
0, for every a ∈ A .(ii) ϕ ( ax, y ) = ϕ ( x, a ∗ y ), for every a ∈ A and x, y ∈ A .If ( A [ k · k ] , A ) is a normed quasi *-algebra, denote by R c ( A , A )the subset of R ( A , A ) consisting of all continuous representable linearfunctionals on ( A , A ).As shown in [23, Proposition 2.7] if ω ∈ R c ( A , A ) for a given normedquasi *-algebra ( A [ k · k ] , A ), then the sesquilinear form ϕ ω defined on A × A by ϕ ω ( x, y ) := ω ( y ∗ x ) , ∀ x, y ∈ A , (2.3)is closable ; that is, for a sequence { x n } in A , one has that k x n k → ϕ ω ( x n − x m , x n − x m ) → ϕ ω ( x n , x n ) → . In this case, ϕ ω has a closed extension ϕ ω to a dense domain D ( ϕ ω ) ×D ( ϕ ω ) containing A × A , where D ( ϕ ω ) = { a ∈ A : ∃ { x n } ⊂ A : with x n → k·k a and ϕ ω ( x n − x m , x n − x m ) → } , so that if ( a, a ′ ) ∈ D ( ϕ ω ) × D ( ϕ ω ), we put ϕ ω ( a, a ′ ) := lim n ϕ ω ( x n , x ′ n ) . (2.4)In this regard, having a normed quasi *-algebra ( A [ k · k ] , A ) and D ( ϕ )a dense subspace of A [ k · k ], we shall say that a sesquilinear form ϕ : D ( ϕ ) ×D ( ϕ ) → C is closed [19, Definition 53.12], if whenever { v n } ∞ n =1 ⊂D ( ϕ ) is a sequence, such that v n → v in A [ k · k ] and ϕ ( v n − v m , v n − v m ) → , as n, m → ∞ one has v ∈ D ( ϕ ) and lim n →∞ ϕ ( v − v n , v − v n ) = 0.Coming back to ϕ ω , note that in [3, Proposition 3.6] is proved that in every Banach quasi *-algebra one has that D ( ϕ ω ) = A .Consider now the set A R := \ ω ∈R c ( A , A ) D ( ϕ ω ) . (2.5) MARIA STELLA ADAMO AND MARIA FRAGOULOPOULOU 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 norm k · k of A ,we obtain A R = A . In this regard, see also [3, Proposition 3.6].Furthermore, we put 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 [23, beginning of Section 3], we call positiveelements of A the elements of A + k·k . We set A + := A + k·k and for anelement a ∈ A + we shall write a ≥ ω on A is positive if ω ( a ) ≥ a ∈ A + . Definition 2.10. [23, Definition 3.7] A family of positive linear func-tionals F on a normed quasi *-algebra ( A [ k · k ] , A ) is called sufficient ,if for every a ∈ A + , a = 0, there exists ω ∈ F , such that ω ( a ) > Definition 2.11. [23, Definition 4.1] A normed quasi *-algebra ( A [ k ·k ] , A ) is called fully representable if R c ( A , A ) is sufficient and A R = A .It is clear from the discussion before (2.5) that every Banach quasi*-algebra is fully representable if R c ( A , A ) is sufficient . In fact, by[3, Theorem 3.9], sufficiency of R c ( A , A ) is a necessary and sufficientcondition, in such a way that a Banach quasi *-algebra is fully repre-sentable.Observe that a Hilbert quasi *-algebra , by its very definition, is *-semisimple (cf. Definition 2.13), therefore by [3, Theorem 3.9] is fullyrepresentable .Further examples of fully representable topological quasi *-algebrascan be found in [23, Section 4]. Remark . Let ( A [ k · k ] , A ) be a normed quasi *-algebra and ω ∈R c ( A , A ). For x ∈ A define ω x ( a ) := ω ( x ∗ ax ), for every a ∈ A .Then ω x ∈ R c ( A , A ). Note that the condition of sufficiency requiredin Definition 2.11 together with the following condition a ∈ A and ω x ( a ) ≥ , for all ω ∈ R c ( A , A ) and x ∈ A , implies a ≥ , says that , if a ∈ A , with ω ( a ) = 0, for every ω ∈ R c ( A , A ), then a = 0.Denote by S A ( A ) the subset of Q A ( A ) consisting of all continuoussesquilinear forms Ω : A × A → C , such that | Ω( a, b ) | ≤ k a kk b k , ∀ a, b ∈ A . ENSOR PRODUCTS 9
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 2.13.
A normed quasi *-algebra ( A [ k · k ] , A ) is called *-semisimple if, for every 0 = a ∈ A , there exists Ω ∈ S A ( A ), suchthat Ω( a, a ) > ω ∈ R ( A , A ), one may associate to ω two sesquilinear forms. One is already defined by (2.3) and the secondone is given as follows:Ω ω ( a, b ) := h π ω ( a ) ξ ω | π ω ( b ) ξ ω i , a, b ∈ A , (2.6)where Ω ω ( a, e ) = ω ( a ) , for every a ∈ A ; ξ ω is the cyclic vector of theGNS representation π ω associated to ω (see [49, Theorem 3.5]). PART II:
TENSOR PRODUCTSFor algebraic tensor products the reader is referred to [16, 30]; fortopological tensor products to [18, 20, 25, 36, 43, 45].Suppose that A , B are *-algebras and ( x, y ) ∈ A × B . The element x ⊗ y is called elementary tensor of the vector space tensor product A ⊗ B . An arbitrary element z in A ⊗ B has the form z = P ni =1 x i ⊗ y i .Let z ′ = P mj =1 x j ⊗ y j be another arbitrary element in A ⊗ B ; set zz ′ := n X i =1 m X j =1 x i x ′ j ⊗ y i y ′ j . Then, zz ′ is a well defined (associative) product on A ⊗ B , underwhich A ⊗ B becomes a complex algebra (see, for instance, [36, p. 361,Lemma 1.4], [37, pp.188,189]).Using the involutions of A , B , an involution is defined on A ⊗ B ,in a natural way: A ⊗ B ∋ z = n X i =1 x i ⊗ y i z ∗ := n X i =1 x ∗ i ⊗ y ∗ i ∈ A ⊗ B . (2.7)Thus, A ⊗ B becomes a *-algebra. If A , B are ∗ -subalgebras of A , B respectively, we may obviously regard A ⊗ B as a ∗ -subalgebraof A ⊗ B Instead of the *-algebras A and B , consider now two locally convexspaces E[ τ E ], F[ τ F ] and let E ⊗ F be their vector space tensor product.
Definition 2.14. [25, pp. 88, 89] A topology τ on E ⊗ F is called compatible (with the tensor product vector space structure of E ⊗ F) ifthe following conditions are satisfied:(1) The vector space E ⊗ F equipped with τ is a locally convexspace, that will be denoted by E ⊗ τ F.(2) The tensor map Φ : E × F → E ⊗ τ F : ( x, y ) x ⊗ y is separatelycontinuous (that is, continuous in each variable).(3) For any equicontinuous subset M of E ∗ and N of F ∗ , the set M ⊗ N ≡ { f ⊗ g : f ∈ M, g ∈ N } is an equicontinuous subsetof (cid:0) E ⊗ τ F (cid:1) ∗ ; E ∗ and F ∗ denote the topological dual of E[ τ E ] andF[ τ F ] respectively.The completion of the locally convex space E ⊗ τ F is denoted byE b ⊗ τ F. For *-compatibility, see beginning of Section 6.Let now E [ k · k ], E [[ k · k ] be Banach spaces. A norm k · k on thetensor product space E ⊗ E that satisfies the equality k x ⊗ x k = k x k k x k , ∀ x ∈ E , x ∈ E , (2.8)is called a cross-norm on E ⊗ E . The injective cross-norm on E ⊗ E Taking an arbitrary element z = P ni =1 x i ⊗ y i in E ⊗ E , we put(2.8) k z k λ := sup n(cid:12)(cid:12)(cid:12) n X i =1 f ( x i ) g ( y i ) (cid:12)(cid:12)(cid:12) : f ∈ E ∗ , k f k ≤ g ∈ E ∗ , k g k ≤ o . The quantity k · k λ is a well-defined cross-norm on E ⊗ E , called in-jective cross-norm . It is also a compatible topology, fulfilling Definition2.14, and it is the least cross-norm on E ⊗ E .The normed space induced by (E ⊗ E )[ k · k λ ], will be denoted asE ⊗ λ E ; its respective completion, which is a Banach space, will bedenoted by E b ⊗ λ E . Grothendieck’s notation for the latter Banachspace, used also by many authors, is E bb ⊗ E .When A [ τ ] , A [ τ ] are two locally convex *-algebras (in this case, weshall always assume that involution is continuous and multiplication isjointly continuous ), then Definition 2.14 can be modified as follows Definition 2.15. [20] Let A [ τ ] , A [ τ ] be as before, where the topolo-gies τ , τ , are defined by upwards directed families of seminorms, say { p } and { q } , respectively. Let A ⊗ A be their corresponding tensorproduct *-algebra. A topology τ on A ⊗ A is called ∗ -admissible (that ENSOR PRODUCTS 11 is, compatible with the tensor product *-algebra structure of A ⊗ A ),if the following conditions are satisfied:(1) A ⊗ A endowed with τ is a locally convex ∗ -algebra, denotedby A ⊗ τ A ;(2) the tensor map Φ : A [ τ ] × A [ τ ] → A ⊗ τ A is continuous, inthe sense that if τ is determined by the family of ∗ -seminorms { r } , then for every r there exist p, q , such that r ( x ⊗ y ) ≤ p ( x ) q ( y ) , ∀ ( x, y ) ∈ A × A ;(3) for any equicontinuous subsets M of A ∗ and N of A ∗ , the set M ⊗ N = { f ⊗ g : f ∈ M, g ∈ N } is an equicontinuous subsetof (cid:0) A ⊗ τ A (cid:1) ∗ ; A ∗ , A ∗ denote respectively the dual of A , A .The completion of A ⊗ τ A is a complete locally convex *-algebradenoted by A b ⊗ τ A .Let us now assume that A [ k · k ], A [ k · k ] are normed *-algebraswith isometric involution. We shall define the projective cross-norm onthe tensor product *-algebra A ⊗ A (see [45, p. 189]). The projective cross-norm on A ⊗ A Let z = P ni =1 x i ⊗ y i be an arbitrary element in A ⊗ A . Put k z k γ = inf ( n X i =1 k x i k k y i k ) , (2.9)where the infimum is taken over all representations P ni =1 x i ⊗ y i of z . The quantity k · k γ is a well-defined cross-norm that majorizes allother cross-norms on A ⊗ A ; it is called projective cross-norm . Thenormed *-algebra induced by ( A ⊗ A )[ k·k γ ], will be denoted as A ⊗ γ A and its respective completion, which is a Banach *-algebra, will bedenoted by A b ⊗ γ A . Grothendieck’s notation for the latter, used alsoby many authors, is A b ⊗ A . Note that the cross-norm k · k γ satisfiesDefinition 2.15, therefore is a *-admissible (hence, compatible) cross-norm, whereas the injective cross-norm k · k λ is not *-admissible ingeneral.In particular, any compatible cross-norm k · k on E ⊗ E lies betweenthe injective and projective cross-norm , i.e., k · k λ ≤ k · k ≤ k · k γ . (2.10)Even more, a cross-norm k · k on E ⊗ E is compatible, if and only if,the inequality (2.10) is valid . For the specific case, when the topology τ is generated by a cross-norm k·k , the condition (2) in Definition 2.15 always holds by the cross-norm property. Condition (3) clearly implies that the tensor productof continuous linear functionals is bounded.The situation is different for operators, i.e., the tensor product ofcontinuous operators is bounded only for certain cross-norms, thosethat are uniform . For further reading in this aspect, see [42].More precisely, if E [ k · k ], E [ k · k ] are Banach spaces, let E b ⊗ k·k E be their respective tensor product Banach space, under a cross-norm k · k . If T : E → E , T : E → E are linear operators, then the map T ⊗ T : E ⊗ E → E ⊗ E is uniquely defined by the linearization ofthe bilinear map ( x, y ) ∈ E ⊗ E T ( x ) ⊗ T ( y ) ∈ E ⊗ E . Hence T ⊗ T is a linear operator, such that ( T ⊗ T ) n X i =1 x i ⊗ y i ! := n X i =1 T ( x i ) ⊗ T ( y i ) , ∀ n X i =1 x i ⊗ y i ∈ E ⊗ E . If T and T are bounded operators, we would like T ⊗ T to be a boundedoperator too. This is true if k · k is a uniform cross-norm, in sense of thefollowing Definition 2.16.
Let E b ⊗ k·k E and T , T be exactly as before. If thetensor product operator T ⊗ T : E ⊗ k·k E → E ⊗ k·k E is continuous and itsextension T b ⊗ T : E b ⊗ k·k E → E b ⊗ k·k E satisfies the relation k T b ⊗ T k ≤k T kk T k , then the cross-norm k · k is said to be uniform . Remark . The injective and projective cross norms λ and γ are examplesof uniform cross-norms.Observe that the condition k T b ⊗ T k ≤ k T kk T k automatically impliesthe equality. Indeed, we have the following Proposition 2.18.
Under the hypotheses of Definition , the operator T b ⊗ T : E b ⊗ k·k E → E b ⊗ k·k E verifies the equality k T b ⊗ T k = k T kk T k .Proof. To prove the claim, we have to show that k T b ⊗ T k ≥ k T kk T k . For z ∈ E b ⊗ k·k E and x ∈ E , y ∈ E , what we have is k T b ⊗ T k = sup k z k≤ k T b ⊗ T ( z ) k ≥ sup k x ⊗ y k≤ k ( T ⊗ T )( x ⊗ y ) k = sup k x k k y k ≤ k T ( x ) k k T ( y ) k ≥ sup k x k ≤ , k y k ≤ k T ( x ) k k T ( y ) k = sup k x k ≤ k T ( x ) k sup k y k ≤ k T ( y ) k = k T kk T k . This completes the proof. (cid:3)◮
Let A [ k · k ], A [ k · k ] and k · k be again as above. If B and B are subspaces of A and A respectively, then B ⊗ B is a subspace of A ⊗ A ENSOR PRODUCTS 13 and in this paper, if not explicitly said, it will be endowed with the topologyinduced by that of A ⊗ k·k A .3. Algebraic tensor product of quasi *-algebras
Let ( A , A ) , ( B , B ) be given quasi *-algebras. It is then known that thealgebraic tensor product A ⊗ B of the *-algebras A , B is again a *-algebra(see Section 2, beginning of PART II) contained as a vector subspace in thevector space tensor product A ⊗ B .Since A and B carry an involution a a ∗ , a ∈ A , respectively b b ∗ , b ∈ B , extending those of A , B respectively, then A ⊗ B attains an involutionsuch that a ⊗ b ( a ⊗ b ) ∗ := a ∗ ⊗ b ∗ , a ∈ A , b ∈ B , extending the involutionof A ⊗ B (see (2.7)).As stated in Definition 2.1, the vector space tensor product of A and B for the given above quasi *-algebras has to be endowed with the left andright multiplications by elements of a *-algebra contained in it, verifyingcertain properties. A natural candidate for this is A ⊗ B as a *-algebraand a subspace of A ⊗ B .Define now the following actions on A ⊗ B , with respect to A ⊗ B : ( A ⊗ B ) × ( A ⊗ B ) → A ⊗ B , ( a ⊗ b ) · ( x ⊗ y ) := R x ⊗ y ( a ⊗ b ) = ( ax ) ⊗ ( by ) , ( A ⊗ B ) × ( A ⊗ B ) → A ⊗ B , ( x ⊗ y ) · ( a ⊗ b ) := L x ⊗ y ( a ⊗ b ) = ( xa ) ⊗ ( yb ) , with ( x, y ) in A × B and ( a, b ) in A × B . By the universal property ofthe vector space tensor product, both actions are well defined and extend tobilinear maps, extending the multiplication of A ⊗ B (see [37, p. 188, 189],for similar arguments). Routine calculations show that using the laws ofDefinition 2.1(ii) for A and B , we obtain the corresponding laws for A ⊗ B .If e A , e B are the identities of our given quasi *-algebras respectively, then e A ⊗ e B is an identity element for A ⊗ B , i.e., (see discussion after Definition2.1) ( a ⊗ b ) · ( e A ⊗ e B ) = a ⊗ b = ( e A ⊗ e B ) · ( a ⊗ b ) , for all a ∈ A and b ∈ B . ◮ From now on we shall simply write ( x ⊗ y )( a ⊗ b ) instead of ( x ⊗ y ) · ( a ⊗ b ) . Similarly, of course, for ( a ⊗ b ) · ( x ⊗ y ).Concerning the extension of the involution * of A ⊗ B on A ⊗ B , weclearly have the property (iii) of Definition 2.1 for the extension of the in-volutions of A , B on A , B respectively, i.e., (cid:0) ( a ⊗ b )( x ⊗ y ) (cid:1) ∗ = ( ax ) ∗ ⊗ ( by ) ∗ = x ∗ a ∗ ⊗ y ∗ b ∗ = ( x ⊗ y ) ∗ ( a ⊗ b ) ∗ , for all ( x, y ) in A × B and ( a, b ) in A × B .In conclusion, all properties of Definition 2.1 are fulfilled, therefore A ⊗ B is a quasi*-algebra over A ⊗ B . Definition 3.1.
The algebraic tensor product A ⊗ B that was constructedabove from two quasi *-algebras ( A , A ), ( B , B ), will be called tensor prod-uct quasi *-algebra over A ⊗ B , or we shall just say that ( A ⊗ B , A ⊗ B ) is a tensor product quasi *-algebra .4. Topological tensor products of normed and Banach quasi*-algebras
Given two normed (resp. Banach) quasi *-algebras ( A [ k · k A ] , A ) , ( B [ k ·k B ] , B ), we shall construct their tensor product normed (resp. Banach)quasi *-algebra.We have already seen in the preceding Definition 3.1 that A ⊗ B is aquasi *-algebra over A ⊗ B . Hence, according to Definition 2.3, we stillhave to show that A ⊗ B becomes a normed (resp. Banach) space, under asuggesting tensor norm that fulfills the conditions of Definition 2.3.First we consider on A ⊗ B the injective cross-norm (2.8) and as we havealready said in Section 2, A b ⊗ λ B is the Banach space, completion of therespective normed space A ⊗ λ B ≡ ( A ⊗ B )[ k · k λ ]. ◮ In the sequel, for distinction , we shall often denote by k · k A , k · k B , thegiven norms on A , B , respectively.By Definition 2.3(i), the (extended) involution on A and B from that of A and B respectively, is isometric, therefore by Remark 2.17 the map (2.7)defines a continuous involution on A ⊗ λ B , which is continuously extendedon A b ⊗ λ B . Applying Proposition 2.18 for the operators ∗ : A → A , with ∗ ( a ) = a ∗ , for all a ∈ A and ∗ : B → B , with ∗ ( b ) = b ∗ , for all b ∈ B , weobtain that k ∗ b ⊗ ∗ k = k ∗ kk ∗ k = 1. Using, in addition, the continuityof ∗ b ⊗∗ , we conclude that k z ∗ k λ = k z k λ , ∀ z ∈ A ⊗ λ B . Hence by continuity, we pass to limits, having thus that A b ⊗ λ B has anisometric involution.We prove now that A ⊗ B is dense in A ⊗ λ B and A b ⊗ λ B . Without lossof generality, take an elementary tensor a ⊗ b in A ⊗ λ B .By Definition 2.3, A is dense in A [ k · k A ] and B in B [ k · k B ]. Thus, since a is in A and b in B there exist sequences { x n } in A and { y n } in B , suchthat k x n − a k A → k y n − b k B → . Then the sequence { x n ⊗ y n } in A ⊗ B is k · k λ -converging to a ⊗ b . Indeed,from (2.8), we have k x n ⊗ y n − x m ⊗ y m k λ = sup (cid:8) | f ( x n ) g ( y n ) − f ( x m ) g ( y m ) | : f ∈ A ∗ , k f k ≤ , g ∈ B ∗ , k g k ≤ (cid:9) → , The above argument shows that A ⊗ B is dense in A ⊗ λ B and consequentlyalso in A b ⊗ λ B . ENSOR PRODUCTS 15
It remains to show that for every z = P i ∈ F x i ⊗ y i in A ⊗ B , F a finitesubset in N , the (right) multiplication operator R z : A ⊗ λ B → A ⊗ λ B : c cz (4.1)is continuous.First recall that for x ∈ A and y ∈ B the operators R x : A → A , R y : B → B with R x ( a ) := ax and R y ( b ) := by , a ∈ A , b ∈ B are continuous andthe operator R x ⊗ R y is uniquely defined on A ⊗ λ B into itself, such that( R x ⊗ R y )( a ⊗ b ) = R x ( a ) ⊗ R y ( b ) , a ∈ A , b ∈ B . In particular, R x ⊗ R y is continuous since R x , R y are continuous and k · k λ is a uniform cross-norm (see Remark 2.17). Yet, it is easily seen that definingthe map R x ⊗ y : A ⊗ λ B → A ⊗ λ B as R x ⊗ y ( a ⊗ b ) := ( a ⊗ b )( x ⊗ y ) , a ∈ A , b ∈ B , we obviously obtain that R x ⊗ y = R x ⊗ R y , so that the operator R x ⊗ y is alsocontinuous.In conclusion, R z in (4.1) is well-defined and continuous, such that R z = X i ∈ F R x i ⊗ y i = X i ∈ F R x i ⊗ R y i . By linearity and continuity we infer that R z , z in A ⊗ B , is uniquelyextended to a continuous operator on A b ⊗ λ B too. Hence, we conclude thatleft and right multiplications by elements in A ⊗ B , as well as the involution*, are well defined on the completion A b ⊗ λ B and verify all the requiredproperties.Summing up, according to Definition 2.3, we have that • ( A ⊗ λ B , A ⊗ B ) is a tensor product normed quasi *-algebra and • ( A b ⊗ λ B , A ⊗ B ) is a tensor product Banach quasi *-algebra .All the above arguments, as well as the fixed notation, can be equally well applied for the projective cross-norm k · k γ (see (2.9)), to give that the pair ( A ⊗ γ B , A ⊗ B ) is a normed quasi *-algebra and its ‘completion’, i.e., thepair ( A b ⊗ γ B , A ⊗ B ) is a Banach quasi *-algebra .The same is true for any uniform cross-norm on A ⊗ B by Definition 2.16and Proposition 2.18.5. Examples of tensor product Banach quasi *-algebras
Example . Take I to be the unit interval in the real line and consider theBanach quasi *-algebra ( L ( I ) , C ( I )) (see discussion before Definition 2.3,as well as Example 2.4). Consider the tensor product of ( L ( I ) , C ( I )) withitself. Then we obtain the Banach quasi *-algebra( L ( I ) b ⊗ γ L ( I ) , C ( I ) ⊗ C ( I )) = ( L ( I × I ) , C ( I ) ⊗ C ( I ))(see discussion at the end of Section 4). It is known that the tensor product L ( I ) b ⊗ γ L ( I ) is linearly and topo-logically isomorphic to L ( I × I ) ≡ L ( I × I, λ × λ ) (with λ the Lebesquemeasure on I and λ × λ the product measure on I × I ); see [25]. Example . Recently A.Ya. Helemskii [28] working in the context of L − quantizations with L ≡ L p ( Z, ζ ), 1 < p < ∞ , ( Z, ζ ) a “convenient”measure space (i.e., Z has either no atoms or an infinite set of atoms) and p -convex tensor products of the spaces L q ( · ), q ∈ (1 , ∞ ) (ibid., Section 6),showed a general result (ibid., Theorem 6.4) and gave the Banach versionof it, from which one obtains the following:Let X, Y be measure spaces with countable bases, p ∈ (1 , ∞ ) and q = p/ ( p −
1) the conjugate number of p . Then, L q ( X ) b ⊗ p L L q ( Y ) = L q ( X × Y ) , (5.1)with respect to a well-defined L -isometric isomorphism [28, Remark 6.5],such that f ⊗ g h , with h ( s, t ) := f ( s ) g ( t ), f ∈ L q ( X ) , g ∈ L q ( Y ), h ∈ L q ( X × Y ) and ( s, t ) ∈ X × Y . The notation b ⊗ p L means completionwith respect to the norm k · k p L ; here, the index, p ∈ (1 , ∞ ) comes fromthe base space L ≡ L p ( Z, ζ ), involved in the L -quantization theory of [28],where a consequence of the latter, (5.1) is. For the definition of k · k p L ,see [28, (5.3)], where (the L -norm) k · k p L is, in a sense, an analogue of theprojective norm k · k γ .For the term countable base , see [46, p. 195]. For some ‘similar’ resultsto the preceding one, see also [17, 18, 46]. Note that the preceding term isequivalent with the fact that the measure µ on X is separable [14, Vol. II,p. 132, 7.14(iv)].Consider the Banach quasi *-algebras ( L q ( X, µ ) , C ( X )) and ( L q ( Y, ν ) , C ( Y )) with ( X, µ ) , ( Y, ν ) metric compact measure spaces and q ∈ (1 , ∞ )(see, for instance, [10] or [11, Example 2.7]). Suppose that µ, ν are Borelprobability measures, that are also diffused [14, Vol. II, Definition 7.14.14].We want to apply (5.1) in this case. For this we must have on X (resp. Y ) an atomless (ibid., Vol. II, Definition 7.14.15), separable measure. It isknown that a Borel probability measure on a compact metric space is regularand (trivially) locally finite, therefore a Radon measure. But, every Radonmeasure is τ − additive (or τ − regular) (see [14, Vol. II, Definition 7.2.1 andProposition 7.2.2(i)]) and since it is also diffused it follows that it is atomless(ibid., Lemma 7.14.16). Finally, our Radon measure is also separable, sinceevery compact subset of X (resp. Y ) is metrizable (ibid., Example 7.14.13).Thus, applying (5.1), we have that L q ( X, µ ) b ⊗ p L L q ( Y, ν ) = L q ( X × Y, µ × ν );therefore, ( L q ( X, µ ) b ⊗ p L L q ( Y, ν ) , C ( X ) ⊗ C ( Y )) is a Banach quasi *-algebra. Example . Take two Hilbert quasi *-algebras ( H , A ), ( H , B ), (seeDefinition 2.7), with H and H the Hilbert space completions of A and B ,with inner product h·|·i and h·|·i , respectively. Then, as it has been shown ENSOR PRODUCTS 17 in [1], ( H b ⊗ h H , A ⊗ B ) is a Banach quasi *-algebra, where H b ⊗ h H is, infact, the Hilbert space completion of the pre-Hilbert space (and *-algebra) A ⊗ B , under the norm k · k h induced by the inner product h ξ | ξ ′ i := n X i =1 m X j =1 h ξ i | ξ ′ j i h η i | η ′ j i , ∀ ξ, ξ ′ ∈ H ⊗ h H , with ξ = P ni =1 ξ i ⊗ η i and ξ ′ = P mj =1 ξ ′ j ⊗ η ′ j (cf., e.g., [43] and/or [20,p. 371]). The left and right outer multiplications on H b ⊗ h H are defined inthe usual way:( x ⊗ y )( a ⊗ b ) := ( xa ) ⊗ ( yb ) , ( ax ) ⊗ ( by ) =: ( a ⊗ b )( x ⊗ y ) , for any x ⊗ y in A ⊗ B and a ⊗ b in H ⊗ h H ; see also Section 3. Example . A concrete realization of the Example 5.3 arises naturallyfrom quantum physics. In quantum mechanics, for two quantum systems S and S both described by the Hilbert space L ( R ) of all square integrablecomplex functions of two variables, the joint system S will be well describedby the tensor product Hilbert space of L ( R ) with itself, that is known tobe isomorphic to L ( R ); for further reading, in this aspect, see for instance,[4, 41]. Considering more abstract measure spaces ( X, µ ) and (
Y, ν ), thereexists an isometric isomorphism, such that L ( X, µ ) b ⊗ h L ( Y, ν ) = L ( X × Y, µ × ν ) , where ( X × Y, µ × ν ) denotes the product measure space. In particular,taking X and Y to be the real line R endowed with the Lebesgue measure,say λ , we obtain L ( R , λ ) b ⊗ h L ( R , λ ) = L ( R , λ × λ )) , thus L ( R , λ ) b ⊗ h L ( R , λ ) is a Banach quasi *-algebra over, for instance, C ∞ c ( R ) ⊗ C ∞ c ( R ), where C ∞ c ( R ) denotes the *-algebra of smooth functions on R with compact support.6. Representations of tensor product normed and Banachquasi *-algebras
If ( A [ k · k A ] , A ) and ( B [ k · k B ] , B ) are normed quasi *-algebras, a com-patible tensor norm ¯ n on A ⊗ B that respects the involutive structure of A ⊗ ¯ n B (i.e., ¯ n makes A ⊗ ¯ n B into a normed *-space, meaning a normedspace endowed with a continuous involution * (see Definition 2.3)) is called *-compatible (cf. Definitions 2.14 and 2.15). If moreover ¯ n is a cross-norm(see (2.8)), then we speak about a *-compatible cross-norm . Note that uni-form cross-norms are *-compatible. ◮ For the calculations , we shall use the symbol k·k ¯ n , instead of the symbol¯ n , in order to be in accordance to (2.8) and (2.9).For two given normed quasi *-algebras ( A [ k · k A ] , A ), ( B [ k · k B ] , B ) anda uniform cross-norm ¯ n (e.g., the injective or projective cross-norm), the tensor product normed quasi *-algebra A ⊗ ¯ n B and its completion A b ⊗ ¯ n B ,have been studied in Sections 3 and 4.We can assume, without loss of generality, that our normed, respectivelyBanach quasi *-algebras ( A [ k · k A ] , A ) and ( B [ k · k B ] , B ) are unital. Indeed,if they are not, we can add a unit in a very standard way as in the Banach al-gebra case and obtain a unital normed, respectively Banach quasi *-algebra.Recall that the unit will be an element of the underlying *-algebra A or B respectively (see discussion before Example 2.2). Denote by ( A e A , A ) and ( B e B , B ) the respective unitizations of ( A , A )and ( B , B ), which are normed (resp. Banach) quasi *-algebras, under thenorm k ( a, λ ) k e A := k a k A + | λ | , for every ( a, λ ) ∈ A e A , with k · k e A ↿ A = k · k A .In the same way, the norm k · k e B is defined on B e B .Taking the tensor product A e A ⊗ B e B , it is obvious that the *-algebra A ⊗ B , as well as the space A ⊗ B can be regarded as subspaces of A e A ⊗ B e B .If ¯ n is a uniform cross-norm on A ⊗ B , it is technical to show that it liftsto a uniform cross-norm, say ¯ n , on the tensor product A e A ⊗ B e B of theunitizations of A and B respectively. ◮ From now on, we shall assume that our quasi *-algebras will be unital ,unless otherwise specified.
Remark . Observe that the map A [ k · k A ] → A ⊗ ¯ n B : a a ⊗ e B is an isometric *-isomorphism if we restrict to its image A ⊗ ¯ n { e B } . In thesame way, B [ k·k B ] is isometrically *-isomorphic to { e A }⊗ ¯ n B . In conclusion,we have the following identifications A [ k · k A ] = A ⊗ ¯ n { e B } and B [ k · k B ] = { e A } ⊗ ¯ n B . Representable functionals and *-representations on tensor prod-uct topological quasi *-algebras.
We are interested in studying prop-erties concerning the representability of a tensor product normed (and/orBanach) quasi *-algebra. For this aim, we first begin stating and provingresults connecting to the manner a *-representation on a tensor product,as before, is related to the *-representations on the tensor factors and viceversa.
Proposition 6.2.
Let ( A [ k · k A ] , A ) , ( B [ k · k B ] , B ) be Banach quasi *-algebras. Let ¯ n be a uniform cross-norm on A ⊗ B . Let π : A b ⊗ ¯ n B →L † ( D π , H π )[ τ w ] be a continuous *-representation of the tensor product Ba-nach quasi *-algebra A b ⊗ ¯ n B . Then there exist unique continuous *-represen-tations π : A [ k · k A ] → L † ( D π , H π )[ τ w ] and π : B [ k · k B ] → L † ( D π , H π )[ τ w ] ,such that for any x ∈ A , y ∈ B and a ∈ A , b ∈ B , we have (6.1) π ( x ⊗ b ) = π ( x ) (cid:3) π ( b ) = π ( b ) (cid:3) π ( x ) ,π ( a ⊗ y ) = π ( a ) (cid:3) π ( y ) = π ( y ) (cid:3) π ( a ) . ENSOR PRODUCTS 19
The *-representations π , π are restrictions of the *-representation π to A [ k · k A ] , B [ k · k B ] respectively.Proof. Using the given continuous *-representation π of A b ⊗ ¯ n B (see Defini-tion 2.8), we define a map π on A in the following way π ( a ) ξ := π ( a ⊗ e B ) ξ, ∀ a ∈ A , ξ ∈ D π . It is easily seen that the map π is a *-representation of A in L † ( D π , H π )with D π = D π . In a similar way, a *-representation π of B in L † ( D π , H π )is defined, with D π = D π , i.e., π ( b ) ξ := π ( e A ⊗ b ) ξ, ∀ b ∈ B , ξ ∈ D π . Since π : A b ⊗ ¯ n B → L † ( D π , H π ) is ( k · k ¯ n - τ w )-continuous, π and π are( k · k A - τ w ), ( k · k B - τ w )-continuous *-representations of ( A [ k · k A ] , A ) and( B [ k · k B ] , B ) respectively. Indeed, let { a n } be a sequence of elements in A [ k · k A ] and a ∈ A , such that k a n − a k A →
0, as n → ∞ . By Remark 6.1,we have k a n ⊗ e B − a ⊗ e B k ¯ n → . Hence by ( k · k ¯ n - τ w )-continuity of π , we obtain (see discussion before Defi-nition 2.8) h π ( a n ) ξ | η i = h π ( a n ⊗ e B ) ξ | η i → h π ( a ⊗ e B ) ξ | η i = h π ( a ) ξ | η i , for all ξ, η ∈ D π = D π . Hence, π ( a n ) τ w -converges to π ( a ). The sameargument is valid for π .Let us now show the equalities (6.1). Take x ∈ A and b ∈ B , then π ( x ⊗ b ) = π (cid:0) ( x ⊗ e B )( e A ⊗ b ) (cid:1) = π ( x ⊗ e B ) (cid:3) π ( e A ⊗ b )= π ( x ) (cid:3) π ( b ) . In an absolutely similar way, we obtain the 2nd line equalities of (6.1), forevery y ∈ B and a ∈ A .The uniqueness of π , π is a direct consequence of their definition. (cid:3) Remark . Observe that in Proposition 6.2 the two Banach quasi *-algebras( A [ k · k A ] , A ) and ( B [ k · k B ] , B ) have been represented in the same family L † ( D π , H π ) of unbounded operators as their tensor product Banach quasi*-algebra ( A b ⊗ ¯ n B , A ⊗ B ), i.e., D π = D π = D π and H π = H π = H π .Moreover, by (6.1) the image of A under π (resp. the image of B under π )commutes with π ( B ) (resp. π ( A )). We say that the *-representations π and π of the (Banach) quasi *-algebras ( A , A ) and ( B , B ) respectivelyhave quasi-commuting ranges . Proposition 6.4.
Under the assumptions of Proposition , for every fixed ξ ∈ D π , the linear functionals ω ( a ) := h π ( a ⊗ e B ) ξ | ξ i , for all a ∈ A , and ω ( b ) = h π ( e B ⊗ b ) ξ | ξ i , for all b ∈ B , are representable and continuousrespectively on ( A [ k · k A ] , A ) and ( B [ k · k B ] , B ) . Proof.
Let π be the *-representation of ( A , A ) defined by π as in the proofof Proposition 6.2. For every fixed ξ ∈ D π = D π define a linear functional ω : A → C as follows: ω ( a ) := h π ( a ⊗ e B ) ξ | ξ i = h π ( a ) ξ | ξ i , ∀ a ∈ A . We show that ω is representable. The conditions (L.1) and (L.2) of Def-inition 2.9 are easily verified. To show (L.3), consider a ∈ A and x ∈ A .Then | ω ( a ∗ x ) | = |h π ( a ∗ x ) ξ | ξ i| = (cid:12)(cid:12)(cid:12) h π ( a ) † (cid:3) π ( x ) ξ | ξ i (cid:12)(cid:12)(cid:12) = |h π ( x ) ξ | π ( a ) ξ i|≤ k π ( a ) ξ kk π ( x ) ξ k ≤ ( γ a + 1) h π ( x ∗ x ) ξ | ξ i = ( γ a + 1) ω ( x ∗ x ) , where γ a = k π ( a ) ξ k ≥ ω : B → C defined as ω ( b ) := h π ( e A ⊗ b ) ξ | ξ i = h π ( b ) ξ | ξ i , ∀ b ∈ B is a representable linear functional on B .We can prove now that ω is continuous. For a ∈ A , consider a sequence { a n } in A [ k · k A ] such that k a n − a k A →
0, as n → ∞ . Then | ω ( a n − a ) | = |h π ( a n − a ) ξ | ξ i| ≤ γ ξ k a n − a k A , for a positive constant γ ξ , since π is ( k·k A − τ w )-continuous *-representationof ( A , A ).In exactly the same way, it is shown that ω is continuous on ( B [ k ·k B ] , B ). (cid:3) If we now consider two *-representations π , π of the (Banach) quasi *-algebras ( A [ k · k A ] , A ) and ( B [ k · k B ] , B ) respectively with quasi-commutingranges, it is unclear how to define a *-representation on the tensor productnormed quasi *-algebra A ⊗ ¯ n B , since a priori the range π ( A ) (resp. π ( B ))commutes only with the range π ( B ) (resp π ( A )).Let us see what we can do in this case. If π : A → L † ( D π , H π ), π : B → L † ( D π , H π ) are two *-representations of the quasi *-algebras( A , A ), ( B , B ) respectively, then there is a unique *-representation π : A ⊗ B → L † ( D π ⊗ D π , H π b ⊗ h H π ) on the tensor product quasi *-algebra( A ⊗ B , A ⊗ B ) defined as follows π ( c ) := n X i =1 π ( a i ) ⊗ π ( b i ) , ∀ c = n X i =1 a i ⊗ b i ∈ A ⊗ B , where H π b ⊗ h H π is the Hilbert space completion of H π ⊗ h H π with respectto the norm induced by the inner product of H π ⊗ h H π (see Example 5.3).Moreover π ( a i ) ⊗ π ( b i ), i = 1 , ..., n , are uniquely defined linear operatorsfrom D π ⊗ D π in H π b ⊗ h H π as in the discussion before Definition 2.16. The *-representation π will be denoted by π ⊗ π . ENSOR PRODUCTS 21
Notice that D π ⊗ D π is a dense subspace in H π b ⊗ h H π . The linearoperator ( π ⊗ π )( c ) belongs to L † ( D π ⊗ D π , H π b ⊗ h H π ), for every c ∈ A ⊗ B , thus the map π ⊗ π is well-defined and it is linear. Moreover, forevery c ∈ A ⊗ B , z ∈ A ⊗ B we have ( π ⊗ π )( c ∗ ) = (cid:0) ( π ⊗ π )( c ) (cid:1) † and π ( cz ) = π ( c ) (cid:3) π ( z ). Hence, all the requirements of Definition 2.8 are verifiedand π is indeed a *-representation of ( A ⊗ B , A ⊗ B ). Furthermore, wehave the following Proposition 6.5.
Let ( A [ k · k A ] , A ) , ( B [ k · k B ] , B ) be Banach quasi *-algebras. Let ¯ n be a uniform cross-norm on A ⊗ B . Let π : A [ k · k A ] →L † ( D π , H π )[ τ w ] and π : B [ k · k B ] → L † ( D π , H π )[ τ w ] be continuous *-representations of ( A [ k · k A ] , A ) and ( B [ k · k B ] , B ) , respectively. Then π ⊗ π : A ⊗ ¯ n B → L † ( D π ⊗D π , H π b ⊗ h H π )[ τ w ] is a continuous *-representationof A ⊗ ¯ n B .Proof. We have seen that π ⊗ π : A ⊗ ¯ n B → L † ( D π ⊗ D π , H π b ⊗ h H π )[ τ w ]is a well defined *-representation of A ⊗ ¯ n B . What remains to be shownis that π ⊗ π is weakly continuous. Let c = P ni =1 a i ⊗ b i ∈ A ⊗ ¯ n B andΨ = P mj =1 ξ j ⊗ η j , Λ = P lk =1 ζ k ⊗ χ k ∈ D π ⊗ D π . Then, h ( π ⊗ π )( c )Ψ | Λ i = m X j =1 l X k =1 n X i =1 h ( π ( a i ) ⊗ π ( b i ))( ξ j ⊗ η j ) | ζ k ⊗ χ k i = m X j =1 l X k =1 n X i =1 h π ( a i ) ξ j ⊗ π ( b i ) η j | ζ k ⊗ χ k i = m X j =1 l X k =1 n X i =1 h π ( a i ) ξ j | ζ k ih π ( b i ) η j | χ k i = m X j =1 l X k =1 n X i =1 f ξ j ,ζ k ( a i ) g η j ,χ k ( b i )= m X j =1 l X k =1 ( f ξ j ,ζ k ⊗ g η j ,χ k ) n X i =1 a i ⊗ b i ! , where, for a ∈ A , b ∈ B , we define the linear functionals f ξ j ,ζ k ( a ) := h π ( a ) ξ j | ζ k i and g η j ,χ k ( b ) := h π ( b ) ξ j | χ k i on A and B respectively, for j =1 , . . . , m , k = 1 , . . . , l . Since π and π are weakly continuous, these function-als are norm continuous, so the same is also true for their tensor product andtheir sum. This implies that π ⊗ π : A ⊗ ¯ n B → L † ( D π ⊗D π , H π b ⊗ h H π )[ τ w ]is continuous. (cid:3) Proposition 6.6.
With the same assumptions as in Proposition , wehave the following: for a fixed ξ ∈ D π and ξ ∈ D π , the linear functionals ω ( a ) := h π ( a ) ξ | ξ i , for all a ∈ A , ω ( b ) := h π ( b ) ξ | ξ i , for all b ∈ B , are representable and continuous on ( A [ k · k A ] , A ) and ( B [ k · k B ] , B ) , respectively. Moreover, their tensor product ω ⊗ ω is a continuous andrepresentable linear functional on A ⊗ ¯ n B , represented by the *-representation π ⊗ π .Proof. From Proposition 6.4, the linear functionals ω and ω are repre-sentable and continuous on the Banach quasi *-algebras ( A [ k · k A ] , A ) and( B [ k · k B ] , B ), respectively and their tensor product is uniquely defined as( ω ⊗ ω )( c ) := n X i =1 ω ( a i ) ω ( b i ) , ∀ c = n X i =1 a i ⊗ b i ∈ A ⊗ ¯ n B . We show that for some ξ ⊗ ξ ∈ D π ⊗ D π we have( ω ⊗ ω )( c ) = h ( π ⊗ π )( c )( ξ ⊗ ξ ) | ξ ⊗ ξ i , for all c = P ni =1 a i ⊗ b i ∈ A ⊗ ¯ n B . Indeed,( ω ⊗ ω ) n X i =1 a i ⊗ b i ! = n X i =1 ω ( a i ) ω ( b i )= n X i =1 h π ( a i ) ξ | ξ ih π ( b i ) ξ | ξ i = h ( π ⊗ π )( c )( ξ ⊗ ξ ) | ξ ⊗ ξ i , where the *-representation π ⊗ π is continuous from Proposition 6.5 andthis implies that the linear functional ω ⊗ ω is continuous too. Since ω ⊗ ω is represented by π ⊗ π , its representability can be deduced fromProposition 2.9 of [3]. (cid:3) Notice that, in Proposition 6.6, π ⊗ π may be not the *-representationobtained by the GNS-like triple (see, [24, Theorem 2.4.8]) for the linear func-tional ω ⊗ ω . Remark . In Proposition 6.6, since ω ⊗ ω is continuous, we can considerits extension ω b ⊗ ω to A b ⊗ ¯ n B . We don’t know whether the aforementionedextension is still representable . However in [1, 2] it has been proved that thisis the case if ( A , A ) and ( B , B ) are both Hilbert quasi *-algebras . Later on (Proposition 6.10), we shall show other cases in the Banach quasi *-algebrasframework in which this extension is representable . Remark . If ω and ω are representable linear functionals on the quasi*-algebras ( A , A ) and ( B , B ) respectively, then we can consider the tensorproduct of their GNS *-representations π ω and π ω , denoted by π ω ⊗ π ω . The tensor product ω ⊗ ω is represented by π ω ⊗ π ω , hence it is repre-sentable on A ⊗ B (see [3, Proposition 2.9]). However, it remains unclear if π ω ⊗ π ω is unitary equivalent to π ω ⊗ ω . ENSOR PRODUCTS 23 *-Semisimplicity and full representability.
We now want to inves-tigate how *-semisimplicity and full representability behave with the con-struction of a tensor product normed, respectively Banach quasi *-algebra.For the normed case, we have to assume some extra properties on the con-sidered topological quasi *-algebras (see Theorems 6.11 and 6.18). For theBanach case, the question remains, at the moment, open.
Proposition 6.9.
Let ( A [ k · k A ] , A ) and ( B [ k · k B ] , B ) be Banach quasi*-algebras. Let A b ⊗ ¯ n B be their tensor product Banach quasi *-algebra for auniform cross-norm ¯ n . • If Ω is a representable and continuous linear functional on A b ⊗ ¯ n B ,then the linear functionals ω ( a ) := Ω( a ⊗ e B ) , a ∈ A , ω ( b ) :=Ω( e A ⊗ b ) , b ∈ B , are representable and continuous on ( A [ k · k A ] , A ) and ( B [ k · k B ] , B ) , respectively. • If Φ is an element in S A ⊗ B ( A b ⊗ ¯ n B ) , then φ ( a , a ) := Φ( a ⊗ e B , a ⊗ e B ) is in S A ( A ) , for any a , a ∈ A and φ ( b , b ) := Φ( e A ⊗ b , e A ⊗ b ) is in S B ( B ) , for any b , b ∈ B .Proof. Let Ω and ω be as above. By Remark 6.1, A [ k · k A ] = A ⊗ ¯ n { e A } ,with respect to a *-isometric isomorphism, therefore ω is continuous andrepresentable on A [ k · k A ]. Analogously, the same is true for ω .Consider now Φ ∈ S A ⊗ B ( A b ⊗ ¯ n B ) and define φ ( a , a ) := Φ( a ⊗ e B , a ⊗ e B ) , ∀ a , a ∈ A ,φ ( b , b ) := Φ( e A ⊗ b , e A ⊗ b ) , ∀ b , b ∈ B . Then, again by Remark 6.1, φ ∈ S A ( A ) and φ ∈ S B ( B ) as restrictionsof Φ on A × A and B × B respectively. (cid:3) We would like to know whether the two assertions of Proposition 6.9 havea kind of converse. More precisely, If ω ∈ R c ( A , A ), ω ∈ R c ( B , B ), thentheir tensor product ω ⊗ ω , defined as( ω ⊗ ω ) n X i =1 a i ⊗ b i ! := n X i =1 ω ( a i ) ω ( b i ) , ∀ n X i =1 a i ⊗ b i ∈ A ⊗ B , is a well-defined linear functional on A ⊗ ¯ n B . Moreover, it is continuous bythe uniform cross-norm property and representable by Remark 6.8. Thus , ω ⊗ ω ∈ R c ( A ⊗ ¯ n B , A ⊗ B ). Concerning its extension to A b ⊗ ¯ n B , seeRemark 6.7.The situation is different when considering φ ∈ S A ( A ), φ ∈ S B ( B ).Indeed, their tensor product φ ⊗ φ , defined as( φ ⊗ φ )( c, c ′ ) := n X i,j =1 φ ( a i , c j ) φ ( b i , d j ) . (6.1)for any c = P ni =1 a i ⊗ b i , c ′ = P nj =1 c j ⊗ d j in A ⊗ ¯ n B , is a sesquilinear form,but only separately continuous . To get its continuity, we have to assume an extra condition (see Proposition 6.10). Moreover, as in the case of continuousand representable linear functionals, discussed above, if ( A , A ) and ( B , B )are Hilbert quasi *-algebras, then for φ ∈ S A ( A ) and φ ∈ S B ( B ), onehas that φ b ⊗ φ belongs to S A ⊗ B ( A b ⊗ h B ); see [2]. ◮ In the sequel , we shall assume barrelledness (cf. [40, Definition 4.1.1]) for our normed space A ⊗ ¯ n B . Note that all Banach spaces are barrelled , soby [40, Corollary 11.3.8], A ⊗ ¯ n B is barrelled when ¯ n = γ . Proposition 6.10.
Let ( A [ k · k A ] , A ) , ( B [ k · k B ] , B ) be Banach quasi *-algebras. Let ¯ n be a uniform cross-norm, such that A ⊗ ¯ n B is a barrellednormed space. The following hold: (1) if φ ∈ S A ( A ) , φ ∈ S B ( B ) , then φ ′ = ( φ ⊗ φ ) / k φ ⊗ φ k can beextended to an element Φ ∈ S A ⊗ B ( A b ⊗ ¯ n B ) ; (2) if ω ∈ R c ( A , A ) , ω ∈ R c ( B , B ) , then ω ⊗ ω can be extended toan element of R c ( A b ⊗ ¯ n B , A ⊗ B ) .Proof. (1) Let 0 = φ ∈ S A ( A ), 0 = φ ∈ S B ( B ) be given. Then by (6.1)the map φ ⊗ φ is non-zero and well-defined. We show that it is positive,i.e., for c = P ni =1 a i ⊗ b i ∈ A ⊗ ¯ n B , one has ( φ ⊗ φ )( c, c ) ≥ . Indeed,by [3, Proposition 2.9] (see also (2.6) and proofs of [3, Proposition 3.6 andTheorem 3.9], we have φ ( a i , a j ) = h π ω φ ( a i ) ξ ω φ | π ω φ ( a j ) ξ ω φ i ,φ ( b i , b j ) = h π ω φ ( b i ) ξ ω φ | π ω φ ( b j ) ξ ω φ i , for all i, j = 1 , . . . , n , where π ω φ and π ω φ are the GNS *-representationsassociated to ω φ and ω φ , respectively. Therefore,( φ ⊗ φ )( c, c ) = n X i,j =1 φ ( a i , a j ) φ ( b i , b j )= n X i,j =1 h π ω φ ( a i ) ξ ω φ | π ω φ ( a j ) ξ ω φ ih π ω φ ( b i ) ξ ω φ | π ω φ ( b j ) ξ ω φ i = n X i,j =1 h ( π ω φ ⊗ π ω φ )( a i ⊗ b i )( ξ ω φ ⊗ ξ ω φ ) | ( π ω φ ⊗ π ω φ )( a j ⊗ b j )( ξ ω φ ⊗ ξ ω φ ) i = k ( π ω φ ⊗ π ω φ )( c )( ξ ω φ ⊗ ξ ω φ ) k ≥ φ , φ , we obtain( φ ⊗ φ )( cz, z ′ ) = ( φ ⊗ φ )( z, c ∗ z ′ ) , ∀ c ∈ A ⊗ ¯ n B , z, z ′ ∈ A ⊗ B . What remains to be shown is that φ ⊗ φ is continuous. For this aim, weassociate an operator T : A → A ∗ , where A ∗ is the topological dual of A ,to φ , in the following way T : A → A ∗ , T ( a ) = φ ( · , a ) , ∀ a ∈ A , ENSOR PRODUCTS 25 where φ ( · , a ) is a continuous linear functional on A [ k·k A ] defined as φ ( a , a ),for every a ∈ A . Similarly, we can define T : B → B ∗ . Then for c, c ′ asabove we have (cid:12)(cid:12) ( φ ⊗ φ )( c, c ′ ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 m X j =1 φ ( a i , c j ) φ ( b i , d j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X j =1 T ( c j ) ⊗ T ( d j ) n X i =1 a i ⊗ b i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =1 T ( c j ) ⊗ T ( d j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 a i ⊗ b i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , since T ( c j ), T ( d j ) are continuous for j = 1 , . . . , m and ¯ n is a uniformcross-norm. On the other hand, by the fact that φ and φ are hermitian(i.e., Φ ( a, b ) = Φ ( b, a ) , ( a, b ) ∈ A × A , similarly for Φ ), using the previousinequality, we obtain (cid:12)(cid:12) ( φ ⊗ φ )( c, c ′ ) (cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 T ( a i ) ⊗ T ( b i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X j =1 c j ⊗ d j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Thus we have shown that φ ⊗ φ is separately continuous on each compo-nent. By hypothesis, A ⊗ ¯ n B is a barrelled normed space and the same istrue for its product with itself, hence by [29, Theorem 1, p. 357], φ ⊗ φ :( A ⊗ ¯ n B ) × ( A ⊗ ¯ n B ) → C is continuous and thus it can be extended to A b ⊗ ¯ n B × A b ⊗ ¯ n B , in the following way( φ b ⊗ φ )( u, v ) = lim n →∞ ( φ ⊗ φ )( c n , d n ) , u, v ∈ A b ⊗ ¯ n B , where { c n } and { d n } are sequences in A ⊗ ¯ n B , k · k ¯ n -converging to u and v , respectively. Notice that φ b ⊗ φ belongs to Q A ⊗ B ( A b ⊗ ¯ n B ) (see af-ter Definition 2.9) and if we consider Φ := ( φ b ⊗ φ ) / k φ b ⊗ φ k , then Φ ∈S A ⊗ B ( A b ⊗ ¯ n B ).(2) Let ω , ω be representable and continuous linear functionals on A [ k ·k A ] and B [ k · k B ], respectively. Then by [3, Proposition 3.6], the associatedsesquilinear forms ϕ ω and ϕ ω , defined as ϕ ω ( a, a ) := lim n →∞ ϕ ω ( x n , x n ) , ϕ ω ( b, b ) := lim n →∞ ϕ ω ( y n , y n ) , whenever { x n } in A and { y n } in B are sequences converging respectivelyto a ∈ A and b ∈ B , are bounded. Hence, employing the same argumentsas in (1), ϕ ω ⊗ ϕ ω is continuous on A ⊗ ¯ n B × A ⊗ ¯ n B and thus it can beextended to the completion A b ⊗ ¯ n B × A b ⊗ ¯ n B . Let us denote this extension as ϕ ω b ⊗ ϕ ω .Define now the linear functional Ω( u ) := ( ϕ ω b ⊗ ϕ ω )( u, e A ⊗ e B ), for u in A b ⊗ ¯ n B . Ω is continuous and representable, since as in the proof of (1) ϕ ω b ⊗ ϕ ω belongs to Q A ⊗ B ( A b ⊗ ¯ n B ) and it is continuous. Notice that ω ⊗ ω is continuous (see comments after Proposition 6.9),hence it can be extended to A b ⊗ ¯ n B . We want to show that its (continuous)extension ω b ⊗ ω corresponds to Ω. For this aim, it suffices to show thatthey agree on the dense subspace A ⊗ ¯ n B of A b ⊗ ¯ n B .Let c = P ni =1 a i ⊗ b i ∈ A ⊗ ¯ n B . ThenΩ( c ) = ( ϕ ω ⊗ ϕ ω )( c, e A ⊗ e B ) = n X i =1 ϕ ω ( a i , e A ) ϕ ω ( b i , e B )= n X i =1 ω ( a i ) ω ( b i ) = ( ω ⊗ ω )( c ) . We conclude that Ω = ω b ⊗ ω and therefore ω b ⊗ ω is representable. (cid:3) In the rest of this section we investigate whether full representability and *-semisimplicity (see Definitions 2.11, 2.13, resp.) of a tensor product normedquasi *-algebra passes to the normed quasi *-algebras, factors of the tensorproduct under consideration and vice versa . Note that full representability isclosely related with *-semisimplicity, but also with the existence of faithfulcontinuous *-representations on topological quasi *-algebras [13, Theorem7.3]. We begin with an answer to the question concerning *-semisimplicity.
Theorem 6.11.
Let ( A [ k·k A ] , A ) , ( B [ k·k B ] , B ) be Banach quasi *-algebrasand let ¯ n be a uniform cross-norm. Consider the following statements: (1) ( A ⊗ ¯ n B , A ⊗ B ) is *-semisimple; (2) ( A [ k · k A ] , A ) and ( B [ k · k B ] , B ) are *-semisimple;Then (1) ⇒ (2) and when A ⊗ ¯ n B is barrelled, one also has that (2) ⇒ (1) .Proof. (1) ⇒ (2) Let a = 0. We show that A [ k · k A ] is *-semisimple. ByRemark 6.1, a ⊗ e B = 0. Since A ⊗ ¯ n B is *-semisimple, there exists Φ in S A ⊗ B ( A ⊗ ¯ n B ), such that Φ( a ⊗ e B , a ⊗ e B ) > φ ∈ S A ( A ) definedas restriction of Φ on A × A . Thus we have φ ( a, a ) = Φ( a ⊗ e B , a ⊗ e B ) > , ∀ a ∈ A \ { } , which implies that ( A [ k · k A ] , A ) is *-semisimple. In the same way, we havethat ( B [ k · k B ] , B ) is *-semisimple.(2) ⇒ (1) Suppose now that A ⊗ ¯ n B is barrelled. Let 0 = P ni =1 a i ⊗ b i ∈ A ⊗ ¯ n B . Then for some index i , a i ⊗ b i = 0, so that a i = 0and b i = 0. Since ( A [ k · k A ] , A ) and ( B [ k · k B ] , B ) are *-semisimple,there will exist φ ∈ S A ( A ) and φ ∈ S B ( B ), such that φ ( a i , a i ) > φ ( b i , b i ) >
0. Take now Φ ∈ S A ⊗ B ( A ⊗ ¯ n B ) defined by the pair( φ , φ ) ∈ S A ( A ) × S B ( B ), as in Proposition 6.10(1). ThenΦ n X i =1 a i ⊗ b i , n X j =1 a j ⊗ b j = γ φ ⊗ φ n X i,j =1 φ ( a i , a j ) φ ( b i , b j ) ENSOR PRODUCTS 27 = γ φ ⊗ φ X i ∧ j = i φ ( a i , a i ) φ ( b i , b i ) + φ ( a i , a i ) φ ( b i , b i ) > , where γ φ ⊗ φ = k φ ⊗ φ k >
0. Thus, ( A ⊗ ¯ n B , A ⊗ B ) is *-semisimple. (cid:3) The property of *-semisimplicity for a normed quasi *-algebra A can alsobe characterized through the existence of faithful ( k · k A - τ s ∗ )-continuous *-representations. More precisely, one has the following (on the same lines ofproof of the indicated result in [13]) Theorem 6.12. [13, Theorem 7.3]
Let ( A [ k · k A ] , A ) be a normed quasi*-algebra. Then the following statements are equivalent: (1) there exists a faithful ( k · k A - τ s ∗ )- continuous *-representation π of ( A [ k · k A ] , A ) ; (2) ( A [ k · k A ] , A ) is *-semisimple. As a direct consequence of Theorem 6.11 and Theorem 6.12, we obtain
Theorem 6.13.
Let ( A [ k·k A ] , A ) , ( B [ k·k B ] , B ) be Banach quasi *-algebrasand let ¯ n be a uniform cross-norm on A ⊗ B . Consider the following state-ments: (1) there exists a faithful ( k · k ¯ n - τ s ∗ ) continuous *-representation π of ( A ⊗ ¯ n B , A ⊗ B ) ; (2) there exist faithful ( k·k - τ s ∗ ) continuous *-representations π of ( A [ k·k ] , A ) and π of ( B [ k · k ] , B ) , with k · k being k · k A and k · k B ,respectively.Then (1) ⇒ (2) and when A ⊗ ¯ n B is barrelled, one also has that (2) ⇒ (1).Theorems 6.14 and 6.18 below give an answer to the second questionposed right before Theorem 6.11 and concerns full representability. Theorem 6.14.
Let ( A [ k · k A ] , A ) and ( B [ k · k B ] , B ) be Banach quasi *-algebras. Let ¯ n be a uniform cross-norm. If ( A ⊗ ¯ n B , A ⊗ B ) is fullyrepresentable, then ( A [ k · k A ] , A ) and ( B [ k · k B ] , B ) are fully representabletoo.Proof. We show that ( A [ k · k A ] , A ) is fully representable. By [3, Theorem3.9], it suffices to show equivalently that R c ( A , A ) is sufficient. Let a ∈ A + ,with a = 0. Then, by Remark 6.1, a ⊗ e B is positive and nonzero in A ⊗ ¯ n B .So by full representability of A ⊗ ¯ n B , there exists Ω ∈ R c ( A ⊗ ¯ n B , A ⊗ B ),such that Ω( a ⊗ e B ) > ω on A defined as restriction of Ω on A × A . Thus ω ( a ) =Ω( a ⊗ e B ) >
0. Therefore, we conclude that R c ( A , A ) is sufficient. Thesame argument applies for the Banach quasi *-algebra ( B [ k · k B ] , B ). (cid:3) For the opposite direction of Theorem 6.14, we shall assume the condition(of positivity) ( P ), which reads as follows:(6.4) a ∈ A with ω ( x ∗ ax ) ≥ , ∀ ω ∈ R c ( A , A )and ∀ x ∈ A implies a ∈ A + , see e.g., [23, Section 3] and [3, Remark 2.18].For the type of converse to Theorem 6.14 mentioned before we shall firstprove a series of lemmas. We begin with two Banach quasi *-algebras ( A [ k ·k A ] , A ) and ( B [ k · k B ] , B ). Let ( A ⊗ ¯ n B , A ⊗ B ) be the correspondingtensor product normed quasi *-algebra with respect to the uniform cross-norm ¯ n as above. Consider Ω ∈ R c ( A ⊗ ¯ n B , A ⊗ B ). Then a sesquilinearform ϕ Ω is defined on A ⊗ B × A ⊗ B by (2.3). Employing the GNSrepresentation π Ω of Ω, as well as the corresponding cyclic vector ξ Ω (see,e.g., [24, Theorem 2.4.8]), we have thatΩ( c ) = h π Ω ( c ) ξ Ω | ξ Ω i , ∀ c ∈ A ⊗ ¯ n B . Consider the sesquilinear form φ Ω : A ⊗ ¯ n B × A ⊗ ¯ n B → C defined by(6.5) φ Ω ( c, c ′ ) = h π Ω ( c ) ξ Ω | π Ω ( c ′ ) ξ Ω i , ∀ c, c ′ ∈ A ⊗ ¯ n B , where π Ω and ξ Ω are as before and note that φ Ω = ϕ Ω on A ⊗ B × A ⊗ B .Then we have the following Lemma 6.15.
The sesquilinear form φ Ω , defined everywhere on A ⊗ ¯ n B × A ⊗ ¯ n B → C , is closed.Proof. If { v n } is a sequence in A ⊗ ¯ n B , such that v n → v in A ⊗ ¯ n B and(6.6) φ Ω ( v n − v m , v n − v m ) → , as n, m → ∞ , we must show that φ Ω ( v n − v, v n − v ) →
0, as n → ∞ (see, for instance [19,Definition 53.12]). From (6.6), we obtain φ Ω ( v n − v m , v n − v m ) = h π Ω ( v n − v m ) ξ Ω | π Ω ( v n − v m ) ξ Ω i = k π Ω ( v n − v m ) ξ Ω k = k π Ω ( v n ) ξ Ω − π Ω ( v m ) ξ Ω k → . This proves that { π Ω ( v n ) ξ Ω } is a Cauchy sequence in H Ω . Thus there exists ζ ∈ H Ω , such that k π Ω ( v n ) ξ Ω − ζ k →
0. The weak continuity of π Ω gives h π Ω ( v n ) ξ Ω | η i → h π Ω ( v ) ξ Ω | η i , ∀ η ∈ D π Ω . Therefore, h ζ | η i = h π Ω ( v ) ξ Ω | η i , for every η ∈ D π Ω . We conclude that ζ = π Ω ( v ) ξ Ω , v ∈ D ( φ Ω ) = A ⊗ ¯ n B and that k π Ω ( v n ) ξ Ω − π Ω ( v ) ξ Ω k = φ Ω ( v n − v, v n − v ) → , i.e., φ Ω is a closed sesquilinear form. (cid:3) ENSOR PRODUCTS 29
On the quasi *-algebra A ⊗ B define the norm [38, Subsection 1.2](6.7) k c k φ Ω := p k c k ¯ n + φ Ω ( c, c ) = p k c k ¯ n + k π Ω ( c ) ξ Ω k , ∀ c ∈ A ⊗ B . The normed space A ⊗ B [ k · k φ Ω ] will be denoted, for short, by A ⊗ φ Ω B andits respective completion by A b ⊗ φ Ω B .In this regard, we have the following Lemma 6.16.
The correspondence j : A ⊗ ¯ n B → A b ⊗ φ Ω B : j ( c ) = c ∈ A b ⊗ φ Ω B , ∀ c ∈ A ⊗ ¯ n B , is a well defined closed linear operator.Proof. We first prove that j is well defined. Indeed, let c ∈ A ⊗ ¯ n B with c = P ni =1 a i ⊗ b i = 0. Then k c k ¯ n = 0 and π Ω ( c ) = 0. Hence k π Ω ( c ) ξ Ω k = 0.Therefore, k c k φ Ω = 0, i.e., j ( c ) = 0. Clearly, the map j is the identity mapand it is linear.We know that j will be closed if, and only if, its graph G j := { ( c, j ( c )) : c ∈ A ⊗ ¯ n B } is closed in A b ⊗ ¯ n B × A b ⊗ φ Ω B . To show the closedness of the operator j means that for any sequence { c n } in A ⊗ ¯ n B , such that k c n − c k ¯ n → k j ( c n ) − d k φ Ω →
0, for some d ∈ A b ⊗ φ Ω B , it holds that c ∈ A ⊗ ¯ n B and j ( c ) = d .The sequences { c n } and { j ( c n ) } are k · k ¯ n -, respectively k · k φ Ω -Cauchy, so k j ( c n ) − j ( c m ) k φ Ω = k c n − c m k ¯ n + φ Ω ( c n − c m , c n − c m ) → , which implies that φ Ω ( c n − c m , c n − c m ) →
0. Since k c n − c k ¯ n → φ Ω isclosed (see Lemma 6.15), we have that c ∈ A ⊗ ¯ n B and φ Ω ( c n − c, c n − c ) → k j ( c n ) − j ( c ) k φ Ω = k c n − c k ¯ n + φ Ω ( c n − c, c n − c ) → , consequently j ( c ) = d . (cid:3) Lemma 6.17.
Let ( A [ k · k A ] , A ) , ( B [ k · k B ] , B ) be Banach quasi *-algebras,such that their ¯ n -tensor product normed quasi *-algebra ( A ⊗ ¯ n B , A ⊗ B ) is barrelled. Then if Ω ∈ R c ( A ⊗ ¯ n B , A ⊗ B ) , the sesquilinear form φ Ω defined in (6.5) is continuous.Proof. Consider the closed identity operator j of Lemma 6.16. Notice thatits domain A ⊗ ¯ n B is a barrelled space and its range A b ⊗ φ Ω B is a Pt´ak space,as a Banach space (see [29, p. 299, Definition 2 and Proposition 3(a)]).Moreover, the identity operator j being closed has a closed graph, so bythe closed graph theorem [29, p. 301, Theorem 4] is continuous. Therefore,there exists a non-negative constant γ , such that k j ( c ) k φ Ω ≤ γ k c k ¯ n , ∀ c ∈ A ⊗ ¯ n B . From (6.7), we now obtain φ Ω ( c, c ) ≤ γ k c k ¯ n , ∀ c ∈ A ⊗ ¯ n B , that yields continuity of φ Ω . (cid:3) We are now ready to state and prove the type of converse to Theorem6.14 announced after the proof of the latter.
Theorem 6.18.
Let ( A [ k·k A ] , A ) , ( B [ k·k B ] , B ) be Banach quasi *-algebrasthat are fully representable and satisfy the condition (P) (see (6.4)) . Supposealso that the normed quasi *-algebra ( A ⊗ ¯ n B , A ⊗ B ) is barrelled. Then ( A ⊗ ¯ n B , A ⊗ B ) is fully representable.Proof. Following the same argument as in [3, Theorem 3.9], we can showthat fully representability and condition (P) for ( A [ k · k A ] , A ) and ( B [ k ·k B ] , B ) implies their *-semisimplicity. From Theorem 6.11, this gives that( A ⊗ ¯ n B , A ⊗ B ) is *-semisimple. Hence the family R c ( A ⊗ ¯ n B , A ⊗ B )is sufficient. We still have to show that D ( ϕ Ω ) = A ⊗ ¯ n B , for every Ω ∈R c ( A ⊗ ¯ n B , A ⊗ B ); for the definition of ϕ Ω , see (2.4).For this aim, consider now the sesquilinear form φ Ω : A ⊗ ¯ n B × A ⊗ ¯ n B → C defined in (6.5). Observe that the restriction of φ Ω on A ⊗ B × A ⊗ B is ϕ Ω (see discussion after (6.4)) and that A ⊗ B is dense in A ⊗ ¯ n B . Sinceby Lemma 6.17, φ Ω is continuous, we conclude that φ Ω = ϕ Ω on the wholeof A ⊗ ¯ n B × A ⊗ ¯ n B ; thus D ( ϕ Ω ) = A ⊗ ¯ n B and this completes the proof. (cid:3) An immediate consequence of Theorem 6.14 and Theorem 6.18 is thefollowing
Corollary 6.19.
Let ( A [ k·k A ] , A ) , ( B [ k·k B ] , B ) be Banach quasi *-algebrassatisfying condition (P) . Consider on A ⊗ B the projective tensorial topology γ . Then the following are equivalent: (1) both of ( A [ k · k A ] , A ) and ( B [ k · k B ] , B ) are fully representable; (2) the tensor product normed quasi *-algebra A ⊗ γ B is fully repre-sentable. Note that in all the results of Section 6, the uniform cross-norm ¯ n canbe replaced by the projective cross-norm γ . The tensor product Banachquasi *-algebra defined in Example 5.3 is, in particular, a tensor productHilbert quasi *-algebra (see [1, Theorem 3.3]); therefore it is automatically*-semisimple and fully representable (see [3, Theorem 3.9]). So after Theo-rems 6.11 and 6.18, as well Corollary 6.19, one naturally asks ‘under whichconditions a tensor product Banach quasi *-algebra becomes *-semisimpleand fully representable, when its tensor factors have this property, and viceversa’ . Both questions stated in this paper concerning the preceding twoconcepts do not look easy (see also beginning of Subsection 6.2 amd com-ments before Theorem 6.11). Thus there is still a lot of work to be done.Since both (topological) tensor products and (topological) quasi *-algebrashave applications to quantum dynamics and quantum statistics (for moreinformation, in this aspect, see [24]) it is certainly worth continuing thisproject. ENSOR PRODUCTS 31
Acknowledgments : This work has been done in the framework of theProject “Alcuni aspetti di teoria spettrale di operatori e di algebre; frames inspazi di Hilbert rigged”, INDAM-GNAMPA 2018. The first author (M-S.A.)wishes to thank the Department of Mathematics of National and Kapodis-trian University of Athens in Greece, where part of this work has been done.In the final stage of this project, MSA has been supported by the ERC Ad-vanced Grant QUEST “Quantum Algebraic Structures and Models”.
References [1] M. S. Adamo,
About tensor products of Hilbert quasi *-algebras and theirrepresentability , submitted.[2] M. S. Adamo,
On some applications of representable and continuous func-tionals of Banach quasi *-algebras , submitted.[3] M. S. Adamo and C. Trapani,
Representable and Continuous Functionals onBanach Quasi *-Algebras , Mediterr. J. Math. (2017) 14:157.[4] D. Aerts and I. Daubechies,
Physical justification for using the tensor prod-uct to describe two quantum systems as one joint system , Helv. Phys. Acta51(1978), 661-675.[5] G.R. Allan,
On a class of locally convex algebras , Proc. London Math. Soc.17(1967), 91-114.[6] J-P. Antoine, W. Karwowski, in: B. Jancewitz, J. Lukerski (Eds.),
Partial*-Algebras of Closed Operators in Quantum Theory of Particles and Fields ,World Scientific, Singapore, 1983, pp. 13-30.[7] J-P. Antoine, W. Karwowski,
Partial *-algebras of closed linear operators inHilbert space , Publ. Res. Inst. Math. Sci. 21 (1985) 205-236; Publ. Res. Inst.Math. Sci. 22 (1986), 507-511 (Addendum/Erratum).[8] J-P. Antoine, A. Inoue and C. Trapani,
Partial *-Algebras and their OperatorRealizations , Math. Appl. 553, Kluwer Academic, Dordrecht, 2002.[9] F. Bagarello, C. Trapani,
The Heisenberg dynamics of spin systems: a quasi*-algebra approach , J. Math. Phys. 37(1996), 4219-4234.[10] F. Bagarello and C. Trapani, L p − spaces as quasi *-algebras , J. Math. Anal.Appl. 197(1996), 810-824.[11] F. Bagarello, C. Trapani, CQ*-algebras: structure properties , Publ. RIMSKyoto Univ. 32(1996), 85-116.[12] F. Bagarello, C. Trapani, S. Triolo,
Quasi *-algebras of measurable operators ,Stud. Math. 172(2006), 289-305.[13] F. Bagarello, M. Fragoulopoulou, A. Inoue, C. Trapani,
Structure of locallyconvex quasi C*-algebras , J. Math. Soc. Japan 60(2008), 511-549.[14] V.I. Bogachev,
Measure Theory , Vol. I, II, Springer, 2007.[15] A. B¨ohm,
Quantum Mechanics , Springer, New York, 1979.[16] L. Chambadal, J.L. Ovaert,
Alg`ebre Lin´eaire et Alg`ebre Tensorielle , DunodUniversit´e, Paris, 1968.[17] S. Chevet,
Sur certains produits tensoriels topologiques d’ espaces de Banach ,Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 11(1969), 120-138.[18] A. Defant, K. Floret,
Tensor Norms and Operator Ideals , North-Holland,Amsterdam, 1993.[19] B.K. Driver,
Analysis Tools with Applications , Draft, 2003. [20] M. Fragoulopoulou,
Topological Algebras with Involution , North-Holland, Am-sterdam, 2005.[21] M. Fragoulopoulou, A. Inoue and M. Weigt,
Tensor products of generalized B ∗ -algebras , J. Math. Anal. Appl. 420(2014), 1787-1802.[22] M. Fragoulopoulou, A. Inoue and M. Weigt, Tensor products of unboundedoperator algebras , Rocky Mount. 44(2014), 895-912.[23] M. Fragoulopoulou, C. Trapani, S. Triolo,
Locally convex quasi *-algebras withsufficiently many *-representations , J. Math. Anal. Appl. 388(2012), 1180-1193.[24] M. Fragoulopoulou and C. Trapani,
Locally Convex Quasi *-Algebras andtheir Representations , Lecture Notes in Mathematics, Vol. 2257, Springer,2020.[25] A. Grothendieck,
Produits Tensoriels Topologiques et Espaces Nucl´eaires ,Mem. Amer. Math. Soc. No 16, 1955.[26] S.J. Gustafson and I.M. Sigal,
Mathematical Concepts of Quantum Mechanics ,Springer (2nd edition), 2006.[27] W-D. Heinrichs,
Topological tensor products of unbounded operator algebrason Fr`echet domains , Publ. RIMS, Kyoto Univ. 33(1997), 241-255.[28] A.Ya. Helemskii,
Multi-normed spaces based on non-discrete measures andtheir tensor products , Izv. RAN. Ser. Mat., 82:2 (2018), 194-216; Izv. Math.,82:2 (2018), 428-449.[29] J. Horv´ath,
Topological Vector Spaces and Distributions , Vol. I, Addison–Wesley Publ. Co., Reading Massachusetts, 1966.[30] Th.W. Hungerford,
Algebra , Holt, Rinehart and Winston, Inc., New York,1974.[31] A. Inoue,
Unbounded generalizations of standard von Neumann algebras , Rep.Math. Phys. 13(1978), 25-35.[32] A. Inoue,
Tomita-Takesaki Theory in Algebras of Unbounded Operators , Lec-ture Notes Math. 1699, Springer-Verlag, Berlin, 1998.[33] G. Lassner, Topological algebras and their applications in quantum statistics,Wiss. Z. KMU-Leipzig, Math. Naturwiss. R. 30(1981) 572-595.[34] G. Lassner,
Algebras of unbounded operators and quantum dynamics , Phys.A 124(1984) 471-480.[35] N. Linden and S. Popescu,
On multi-particle entanglement , Newton Institute,arXiv:quant-ph/9711016v1.[36] A. Mallios,
Topological Algebras. Selected topics , North-Holland, Amsterdam,1986.[37] G.J. Murphy
C*-Algebras and Operator Theory , Academic Press, 1990.[38] El M. Ouhabaz,
Analysis of Heat Equations on Domains , London Mathemat-ical Society Monographs (LMS-31), 2009.[39] Th.W. Palmer,
Banach Algebras and the General Theory of *-algebras , Vol.II, Cambridge University Press, 2001.[40] P. P´erez Carreras, J. Bonet,
Barrelled Locally Convex Spaces , North-Holland,Amsterdam, 1987.[41] M. Reed and B. Simon,
Functional Analysis , Vol. I: Methods of ModernMathematical Physics, Revised and Enlarged Edition, Academic Press, NewYork, 1980.[42] R.A. Ryan,
Introduction to tensor products of Banach spaces , Springer, 2002.
ENSOR PRODUCTS 33 [43] R. Schatten,
A Theory of Cross-Spaces , Princeton Univ. Press, Princeton,1950.[44] K. Schm¨udgen,
Unbounded Operator Algebras and Representation Theory ,Birkh¨auser-Verlag, Basel, 1990.[45] M. Takesaki,
Theory of Operator Algebras I , Springer-Verlag, New York, 1979.[46] S.J. Taylor,
Introduction to Measure and Integration , Cambridge UniversityPress, 1973.[47] C. Trapani,
States and derivations on quasi *-algebras , J. Math. Phys.29(1988), 1885-1890.[48] C. Trapani,
Quasi *-algebras and their applications , Rev. Math. Phys.7(1995), 1303-1332.[49] C. Trapani, *-Representations, seminorms and structure properties of normedquasi *-algebras , Studia Math. 186(2008), 47-75.
Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”,I-00133 Roma, Italy
E-mail address : [email protected]; [email protected] Department of Mathematics, University of Athens, Panepistimiopo-lis, Athens 15784, Greece
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