aa r X i v : . [ m a t h . R A ] D ec On genuine infinite algebraic tensor products ∗ Chi-Keung NgOctober 16, 2018
Abstract
In this paper, we study genuine infinite tensor products of some algebraic structures. By agenuine infinite tensor product of vector spaces, we mean a vector space N i ∈ I X i whose linear mapscoincide with multilinear maps on an infinite family { X i } i ∈ I of vector spaces. After establishingits existence, we give a direct sum decomposition of N i ∈ I X i over a set Ω I ; X , through which weobtain a more concrete description and some properties of N i ∈ I X i . If { A i } i ∈ I is a family of unital ∗ -algebras, we define, through a subgroup Ω ut I ; A ⊆ Ω I ; A , an interesting subalgebra N ut i ∈ I A i . When all A i are C ∗ -algebras or group algebras, it is the linear span of the tensor products of unitary elementsof A i . Moreover, it is shown that N ut i ∈ I C is the group algebra of Ω ut I ; C . In general, N ut i ∈ I A i canbe identified with the algebraic crossed product of a cocycle twisted action of Ω ut I ; A . On the otherhand, if { H i } i ∈ I is a family of inner-product spaces, we define a Hilbert C ∗ (Ω ut I ; C )-module ¯ N mod i ∈ I H i ,which is the completion of a subspace N unit i ∈ I H i of N i ∈ I H i . If χ Ω ut I ; C is the canonical tracial state on C ∗ (Ω ut I ; C ), then ¯ N mod i ∈ I H i ⊗ χ Ωut I ; C C coincides with the Hilbert space ¯ N φ i ∈ I H i given by a very elementaryalgebraical construction and is a natural dilation of the infinite direct product Q ⊗ i ∈ I H i as definedby J. von Neumann. We will show that the canonical representation of N ut i ∈ I L ( H i ) on ¯ N φ i ∈ I H i isinjective (note that the canonical representation of N ut i ∈ I L ( H i ) on Q ⊗ i ∈ I H i is non-injective). Wewill also show that if { A i } i ∈ I is a family of unital Hilbert algebras, then so is N ut i ∈ I A i .MSC 2010: Primary: 15A69, 46M05; Secondary: 16G99, 16S35, 20C07, 46C05, 46L99, 47A80Keywords: infinite tensor products; unital ∗ -algebras; twisted crossed products; inner product spaces;representations In this paper, we study infinite tensor products of some algebraic structures. In the literature, infinitetensor products are often defined as inductive limit of finite tensor products (see e.g. [4], [5] [10], [15],[16]). As far as we know, the only alternative approach so far is the one by J. von Neumann, concerning infinite direct products of Hilbert spaces (see [21]). Some authors used this approach to define infinitetensor products of other functional analytic structures (see e.g. [3], [12] and [14]). The work of vonNeumann attracted the attention of many physicists who are interested in “quantum mechanics withinfinite degrees of freedom”, as well as mathematicians whose interest is in the field of operator algebras(see e.g. [1], [2], [3], [9], [13], [18], [20]).However, von Neumann’s approach is not appropriate for purely algebraic objects. The aim of thisarticle is to study “genuine infinite algebraic tensor products” (i.e. ones that are defined in terms ofmultilinear maps instead of through inductive limits) of some algebraic structures. There are severalmotivations behind this study. ∗ This work is supported by the National Natural Science Foundation of China (10771126)
1. Conceptually speaking, it is natural to define “infinite tensor products” as the object that producesa unique linear map from a multilinear map on a given infinite family of objects (see Definition 2.1). Asinfinite direct products of Hilbert spaces are important in both Physics and Mathematics, it is believedthat such infinite tensor products of algebraic structures are also important.2. We want to construct an infinite tensor product of Hilbert spaces that is easier for non-analyst tograsp (compare with the infinite direct product as defined by J. von Neumann; see Lemma 4.2 andRemark 4.7(d)) and is more natural (see Theorem 4.8, Example 4.10 and Example 5.6).3. Given a family of groups { G i } i ∈ I , it is well-known that the group algebra of the group M i ∈ I G i := (cid:8) [ g i ] i ∈ I ∈ Π i ∈ I G i : g i = e except for finite number of i ∈ I (cid:9) is an inductive limit of finite tensor products. However, if one wants to consider the group algebra C [Π i ∈ I G i ], one is forced to consider a “bigger version of tensor products” (see Example 3.1).In this article, the algebraic structures that we concern with are vector spaces, unital ∗ -algebras,inner-product spaces as well as ∗ -representations of unital ∗ -algebras on Hilbert spaces. In our study,we discovered some interesting phenomena of infinite tensor products that do not have counterparts inthe case of finite tensor products. Most of these phenomena related to certain object, Ω I ; X , defined asin Remark 2.4(d), which “encodes the asymptotic information” of a given family { X i } i ∈ I .In Section 2, we will begin our study by defining the infinite tensor product ( N i ∈ I X i , Θ X ) of afamily { X i } i ∈ I of vector spaces. Two particular concerns are bases of N i ∈ I X i as well as the relationshipbetween N i ∈ I X i and inductive limits of finite tensor products of { X i } i ∈ I (which depend on choicesof fixed elements in Π i ∈ I X i ). In order to do these, we obtain a direct sum decomposition of N i ∈ I X i indexed by a set Ω I ; X (see Theorem 2.5) with all the direct summand being inductive limits of finitetensor products (see Proposition 2.6(b)). From this, we also know that the canonical mapΨ : O i ∈ I L ( X i ; Y i ) → L ( O i ∈ I X i ; O i ∈ I Y i )is injective (but not surjective). As a consequence, N i ∈ I X i is automatically a faithful module over thebig unital commutative algebra N i ∈ I C (see Corollary 2.9 and Example 2.10). Moreover, one may regardthe canonical map Θ C : Π i ∈ I C → O i ∈ I C as a generalised multiplication (see Example 2.10(a)). In this sense, one can make sense of infiniteproducts like ( − I .Clearly, N i ∈ I A i is a unital ∗ -algebra if all A i are unital ∗ -algebras. We will study in Section 3, anatural ∗ -subalgebra N ut i ∈ I A i of N i ∈ I A i which is a direct sum over a subgroup Ω ut I ; A of the semi-groupΩ I ; A . The reasons for considering this subalgebra are that it has good representations (see the discussionafter Proposition 5.1), and it is big enough to contain C [Π i ∈ I G i ] when A i = C [ G i ] for all i ∈ I (seeExample 3.1(a)). Moreover, if all A i are generated by their unitary elements (in particular, if A i aregroup algebras or C ∗ -algebras), then N ut i ∈ I A i is the linear span of the tensor products of unitary elementsin A i . We will show that N ut i ∈ I A i can be identified with the crossed products of some twisted actionsin the sense of Busby and Smith (i.e., a cocycle action with a 2-cocycle) of Ω ut I ; A on N ei ∈ I A i (the unital ∗ -algebra inductive limit of finite tensor products of A i ). Moreover, it is shown that N ut i ∈ I C can beidentified with the group algebra of Ω ut I ; C (Corollary 3.4). We will also study the center of N ut i ∈ I A i inthe case when A i is generated by its unitary elements (for all i ∈ I ).In Section 4, we will consider tensor products of inner-product spaces. If { H i } i ∈ I is a family ofinner-product spaces, we define a natural inner-product on a subspace N unit i ∈ I H i of N i ∈ I H i (see Lemma4.2(b)). In the case of Hilbert spaces, the completion ¯ N φ i ∈ I H i of N unit i ∈ I H i is a “natural dilation” of the2nfinite direct product Q ⊗ i ∈ I H i as defined by J. von Neumann in [21] (see Remark 4.7(b)). Note thatthe construction for ¯ N φ i ∈ I H i is totally algebraical and is more natural (see Example 4.10 and Example5.6). Note also that one can construct Q ⊗ i ∈ I H i in a similar way as ¯ N φ i ∈ I H i (see Remark 4.7(d)). Onthe other hand, there is an inner-product C [Ω ut I ; C ]-module structure on N unit i ∈ I H i which produces ¯ N φ i ∈ I H i (see Theorem 4.8), as well as many other pre-inner-products on N unit i ∈ I H i (see Remark 4.9(a)).Section 5 will be devoted to the study of ∗ -representations of unital ∗ -algebras. More precisely, ifΨ i : A i → L ( H i ) is a unital ∗ -representations ( i ∈ I ), we define a canonical ∗ -representation O φ i ∈ I Ψ i : O ut i ∈ I A i → L (cid:0) ¯ O φ i ∈ I H i (cid:1) . We will show in Theorem 5.3(c) that if all Ψ i are injective, then N φ i ∈ I Ψ i is also injective. This is equiv-alent to the canonical ∗ -representations of N ut i ∈ I L ( H i ) on ¯ N φ i ∈ I H i being injective, and is related to the“strong faithfulness” of the canonical action of Ω ut I ; L ( H ) on Ω unit I ; H (see Remark 5.4(b)). Note however, thatthe corresponding tensor type representation of N ut i ∈ I L ( H i ) on Q ⊗ i ∈ I H i is non-injective. Consequently,if ( H i , π i ) is a unitary representation of a group G i that induced an injective ∗ -representation of C [ G i ]on H i ( i ∈ I ), then we obtain injective “tensor type” ∗ -representation of C [Π i ∈ I G i ] on ¯ N φ i ∈ I H i (seeCorollary 5.7). On the other hard, we will show that L ρ ∈ Π i ∈ I S ( A i ) (cid:0) ¯ N φ i ∈ I H ρ i , N φ i ∈ I π ρ i (cid:1) is an injective ∗ -representation of N ut i ∈ I A i when all A i are C ∗ -algebras (Corollary 5.9). Finally, we show that if all A i are unital Hilbert algebras, then so is N ut i ∈ I A i . Notation 1.1 i). In this article, all the vector spaces, algebras as well as inner-product spaces are overthe complex field C , although some results remain valid if one considers the real field instead.ii). Throughout this article, I is an infinite set, and F is the set of all non-empty finite subsets of I .iii). For any vector space X , we write X × := X \ { } and put X ∗ to be the set of linear functionalson X . If Y is another vector space, we denote by X ⊗ Y and L ( X ; Y ) respectively, the algebraic tensorproduct of X and Y , and the set of linear maps from X to Y . We also write L ( X ) := L ( X ; X ) .iv). If { X i } i ∈ I is a family of vector spaces and x ∈ Π i ∈ I X i , we denote by x i the “ i th -coordinate” of x (i.e. x = [ x i ] i ∈ I ). If x, y ∈ Π i ∈ I X i such that x i = y i except for a finite number of i ∈ I , we write x i = y i e.f.v). If V is a normed spaces, we denote by L ( V ) and V ′ the set of bounded linear operators and the setof bounded linear functionals respectively, on V . Moreover, we set S ( V ) := { x ∈ V : k x k = 1 } as wellas B ( V ) := { x ∈ V : k x k ≤ } .vi). If A is a unital ∗ -algebra, we denote by e A the identity of A and U A := { a ∈ A : a ∗ a = e A = aa ∗ } . In this section, { X i } i ∈ I and { Y i } i ∈ I are families of non-zero vector spaces. Definition 2.1
Let Y be a vector space. A map Φ : Π i ∈ I X i → Y is said to be multilinear if Φ is linearon each variable. Suppose that N i ∈ I X i is a vector space and Θ X : Π i ∈ I X i → N i ∈ I X i is a multilinearmap such that for any vector space Y and any multilinear map Φ : Π i ∈ I X i → Y , there exists a uniquelinear map ˜Φ : N i ∈ I X i → Y with Φ = ˜Φ ◦ Θ X . Then (cid:0)N i ∈ I X i , Θ X (cid:1) is called the tensor product of { X i } i ∈ I . We will denote ⊗ i ∈ I x i := Θ X ( x ) ( x ∈ Π i ∈ I X i ) and set X ⊗ I := N i ∈ I X i if all X i are equal tothe same vector space X . Example 2.2 (a) Let Π i ∈ I C := { β ∈ Π i ∈ I C : β i = 1 e.f. } and set ϕ ( β ) := ( Π i ∈ I β i if β ∈ Π i ∈ I C otherwise . It is not hard to check that ϕ is a non-zero multilinear map from Π i ∈ I C to C . If φ : N i ∈ I C → C isthe linear functional induced by ϕ (the existence of N i ∈ I C will be established in Proposition 2.3(a)),then φ is an involutive unital map.(b) Let Π i ∈ I C := { β ∈ Π i ∈ I C : P i ∈ I | β i − | < ∞} . For each β ∈ Π i ∈ I C , the net { Π i ∈ F β i } F ∈ F converges to a complex number, denoted by Π i ∈ I β i (see e.g. [21, 2.4.1]). We define ϕ ( β ) := Π i ∈ I β i whenever β ∈ Π i ∈ I C and set ϕ | Π i ∈ I C \ Π i ∈ I C ≡ . As in part (a), ϕ induces an involutive unital linearfunctional φ on N i ∈ I C . Clearly, infinite tensor products are unique (up to linear bijections) if they exist. The existence ofinfinite tensor products follows from a similar argument as that for finite tensor products, but we givean outline here for future reference.
Proposition 2.3 (a) The tensor product (cid:0)N i ∈ I X i , Θ X (cid:1) exists.(b) If { A i } i ∈ I is a family of algebras (respectively, ∗ -algebras), then N i ∈ I A i is an algebra (respectively,a ∗ -algebra) with ( ⊗ i ∈ I a i )( ⊗ i ∈ I b i ) := ⊗ i ∈ I a i b i (and ( ⊗ i ∈ I a i ) ∗ := ( ⊗ i ∈ I a ∗ i ) ) for a, b ∈ Π i ∈ I A i .(c) If Ψ i : A i → L ( X i ) is a homomorphism for each i ∈ I , there is a canonical homomorphism ˜ N i ∈ I Ψ i : N i ∈ I A i → L (cid:0)N i ∈ I X i (cid:1) such that (cid:0) ˜ N i ∈ I Ψ i (cid:1) ( ⊗ i ∈ I a i ) ⊗ i ∈ I x i = ⊗ i ∈ I Ψ i ( a i ) x i ( a ∈ Π i ∈ I A i and x ∈ Π i ∈ I X i ) .(d) If A = L ∞ n =0 A n is a graded algebra and L ∞ n =0 M n is a graded left A -module, then L ∞ n =0 N k ≥ n M k is a graded A -module with a m ( ⊗ k ≥ n x k ) = ⊗ k ≥ n a m x k ∈ N k ≥ m + n M k ( a m ∈ A m ; x ∈ Π k ≥ n M k ) . Proof:
Parts (b), (c) and (d) follow from the universal property of tensor products, and we will onlygive a brief account for part (a). Let V be the free vector space generated by elements in Π i ∈ I X i andΘ : Π i ∈ I X i → V be the canonical map. Suppose that W := span W , where W := (cid:8) λ Θ ( u ) + Θ ( v ) − Θ ( w ) : λ ∈ C ; u, v, w ∈ Π i ∈ I X i ; ∃ i ∈ I with λu i + v i = w i and u j = v j = w j , ∀ j ∈ I \ { i } (cid:9) . (2.1)If we put N i ∈ I X i := V /W , and set Θ X to be the composition of Θ with the quotient map from V to V /W , then they will satisfy the requirement in Definition 2.1. (cid:3)
In the following remark, we list some observations that may be used implicitly throughout this article.
Remark 2.4 (a) As Θ X is multilinear, N i ∈ I X i = span Θ X (cid:0) Π i ∈ I X × i (cid:1) .(b) If I and I are non-empty disjoint subsets of I with I = I ∪ I , it follows, from the universalproperty, that N i ∈ I X i ∼ = (cid:0) N i ∈ I X i (cid:1) ⊗ (cid:0) N j ∈ I X j (cid:1) canonically.(c) N i ∈ I ( X i ⊗ Y i ) ∼ = ( N i ∈ I X i ) ⊗ ( N i ∈ I Y i ) canonically.(d) For any x, y ∈ Π i ∈ I X × i , we denote x ∼ y if x i = y i e.f. bviously, ∼ is an equivalence relation on Π i ∈ I X × i , and we set [ x ] ∼ to be the equivalence class of x ∈ Π i ∈ I X × i . Let Ω I ; X be the collection of such equivalence classes. It is not hard to see that Ω I ; C is aquotient group of Π i ∈ I C × , and that it acts freely on Ω I ; X .(e) The element ⊗ i ∈ I ∈ C ⊗ I is non-zero. In fact, if ⊗ i ∈ I , then C ⊗ I = (0) (by Proposition 2.3(b)),and this implies the only multilinear map from Π i ∈ I C to C being zero, which contradicts Example 2.2. The “asymptotic object” Ω I ; X as defined in part (c) above is crucial in the study of genuine infinitetensor product, as can be seen in our next result. Let us first give some more notations here. For every u ∈ Π i ∈ I X × i , we setΠ ui ∈ I X i := { x ∈ Π i ∈ I X i : x ∼ u } and O ui ∈ I X i := span Θ X (Π ui ∈ I X i ) . If u ∼ v , then Π ui ∈ I X i = Π vi ∈ I X i , and we will also denote Π [ u ] ∼ i ∈ I X i := Π ui ∈ I X i as well as N [ u ] ∼ i ∈ I X i := N ui ∈ I X i . Theorem 2.5 N i ∈ I X i = L ω ∈ Ω I ; X N ωi ∈ I X i . Proof:
Suppose that x (1) , ..., x ( n ) ∈ Π i ∈ I X × i and 0 = n < · · · < n N = n is a sequence satisfying x ( n k +1) ∼ · · · ∼ x ( n k +1 ) for k ∈ { , ..., N − } , but x ( n k ) ≁ x ( n l ) whenever 1 ≤ k = l ≤ N . We first showthat if ν , ..., ν n ∈ C with P nl =1 ν l Θ X ( x ( l ) ) = 0, then X n k +1 l = n k +1 ν l Θ X ( x ( l ) ) = 0 ( k = 0 , ..., N − . In fact, by the proof of Proposition 2.3(a), there exist m ∈ N , µ , ..., µ m ∈ C and λ k Θ ( u ( k ) ) +Θ ( v ( k ) ) − Θ ( w ( k ) ) ∈ W ( k = 1 , ..., m ) such that X nl =1 ν l Θ ( x ( l ) ) = X mk =1 µ k (cid:0) λ k Θ ( u ( k ) ) + Θ ( v ( k ) ) − Θ ( w ( k ) ) (cid:1) . Observe that if one of the elements in { u ( k ) , v ( k ) , w ( k ) } is equivalent to x (1) (under ∼ ), then so are theother two (see (2.1)). After renaming, one may assume that u ( k ) ∼ v ( k ) ∼ w ( k ) ∼ x (1) for k = 1 , ..., m ,but none of u ( k ) , v ( k ) nor w ( k ) is equivalent to x (1) when k ∈ { m + 1 , ..., m } .Since the two sets { x ( n +1) , ..., x ( n ) } ∪ { u ( m +1) , ..., u ( m ) } ∪ { v ( m +1) , ..., v ( m ) } ∪ { w ( m +1) , ..., w ( m ) } and { x (1) , ..., x ( n ) } ∪ { u (1) , ..., u ( m ) } ∪ { v (1) , ..., v ( m ) } ∪ { w (1) , ..., w ( m ) } are disjoint and elements inΘ (Π i ∈ I X i ) are linearly independent in V , we have X n l =1 ν l Θ ( x ( l ) ) − X m k =1 µ k (cid:0) λ k Θ ( u ( k ) ) + Θ ( v ( k ) ) − Θ ( w ( k ) ) (cid:1) = 0 . This implies that P n l =1 ν l Θ X ( x ( l ) ) = 0. Similarly, P n k +1 l = n k +1 ν l Θ X ( x ( l ) ) = 0 for k = 1 , ..., N − (cid:0)N ω M i ∈ I X i (cid:1) ∩ (cid:16)P M − k =1 N ω k i ∈ I X i (cid:17) = { } whenever ω , ..., ω M are distinct elementsin Ω I ; X . On the other hand, for every x ∈ Π i ∈ I X × i , one has Θ X ( x ) ∈ N [ x ] ∼ i ∈ I X i . These give the requiredequality. (cid:3) For any F ∈ F and u ∈ Π i ∈ I X × i , one has a linear map J uF : O i ∈ F X i −→ O ui ∈ I X i J uF ( ⊗ i ∈ F x i ) := ⊗ j ∈ I ˜ x j ( x i ∈ X i ), where ˜ x j := x j when j ∈ F , and ˜ x j := u j when j ∈ I \ F .For any F, G ∈ F with F ⊆ G , a similar construction gives a linear map J uG ; F : N i ∈ F X i → N i ∈ G X i .It is clear that (cid:0)N i ∈ F X i , J uG ; F (cid:1) F ⊆ G ∈ F is an inductive system in the category of vector spaces with linearmaps as morphisms. Proposition 2.6 (a) J uF is injective for any u ∈ Π i ∈ I X × i and F ∈ F . Consequently, Θ X ( u ) = 0 .(b) The inductive limit of (cid:0)N i ∈ F X i , J uG ; F (cid:1) F ⊆ G ∈ F is (cid:0)N ui ∈ I X i , { J uF } F ∈ F (cid:1) . Proof: (a) Suppose that a ∈ ker J uF and ψ ∈ ( N i ∈ F X i ) ∗ . For each j ∈ I \ F , choose f j ∈ X ∗ j with f j ( u j ) = 1. Remark 2.4(b) and the universal property give a linear map ˇ ψ : N i ∈ I X i → C ⊗ I satisfyingˇ ψ ( ⊗ i ∈ I x i ) = ψ ( ⊗ i ∈ F x i ) (cid:0) ⊗ j ∈ I \ F f j ( x j ) (cid:1) ( x ∈ Π i ∈ I X i ) . Thus, ψ ( a )( ⊗ i ∈ I
1) = ˇ ψ ( J uF ( a )) = 0, which implies that a = 0 (as ψ is arbitrary) as required. On theother hand, if i ∈ I , then Θ X ( u ) = J u { i } ( u i ) = 0.(b) This follows directly from part (a). (cid:3) Part (b) of the above implies that Θ X ( C ω ) is a basis for N ωi ∈ I X i , where C ω is as defined in thefollowing result. Corollary 2.7 (a) Let c : Ω I ; X → Π i ∈ I X × i be a cross-section. For each ω ∈ Ω I ; X and i ∈ I , we pick abasis B ωi of X i that contains c ( ω ) i and set C ω := { x ∈ Π ωi ∈ I X i : x i ∈ B ωi , ∀ i ∈ I } . If C := S ω ∈ Ω I ; X C ω , then Θ X ( C ) is a basis for N i ∈ I X i .(b) If Φ i : X i → Y i is an injective linear map ( i ∈ I ), the induced linear map N i ∈ I Φ i : N i ∈ I X i → N i ∈ I Y i is injective. Proposition 2.8
The map
Ψ : N i ∈ I L ( X i ; Y i ) → L ( N i ∈ I X i ; N i ∈ I Y i ) (given by the universal property)is injective. Proof:
Suppose that T (1) , ..., T ( n ) ∈ Π i ∈ I L ( X i ; Y i ) × are mutually inequivalent elements (under ∼ ), F ∈ F , R (1) , ..., R ( n ) ∈ N i ∈ F L ( X i ; Y i ) with S ( k ) := J T ( k ) F ( R ( k ) ) ( k = 1 , ..., n ) satisfyingΨ (cid:0)X nk =1 S ( k ) (cid:1) = 0 . Using an induction argument, it suffices to show that S (1) = 0.If n = 1, we take any x ∈ Π i ∈ I X × i with T (1) i x i = 0 ( i ∈ I ). If n >
1, we claim that there is x ∈ Π i ∈ I X × i such that[ T (1) i x i ] i ∈ I ∈ Π i ∈ I Y × i and [ T ( k ) i x i ] i ∈ I ≁ [ T (1) i x i ] i ∈ I ( k = 2 , ..., n ) . In fact, let I k := { i ∈ I : T ( k ) i = T (1) i } , which is an infinite set for any k = 2 , ..., n . For any i ∈ I , weput N i := { k ∈ { , , .., n } : i ∈ I k } and take any x i ∈ X i \ (cid:0) S k ∈ N i ker( T ( k ) i − T (1) i ) ∪ ker T (1) i (cid:1) (note that X i cannot be a finite union of proper subspaces). Thus, T (1) i x i = 0 (for each i ∈ I ) and T ( k ) i x i = T (1) i x i (for k ∈ { , .., n } and i ∈ I k ). 6ow, we have Ψ( S (1) ) (cid:0)O xi ∈ I X i (cid:1) ∩ (cid:16)X nk =2 Ψ( S ( k ) ) (cid:0)O xi ∈ I X i (cid:1)(cid:17) = (0)by Theorem 2.5 and the fact that Ψ( S ( l ) ) (cid:0)N xi ∈ I X i (cid:1) ∈ N y ( l ) i ∈ I Y i , where y ( l ) i = T ( l ) i x i ( i ∈ I ; l = 1 , ..., n ).Consequently, Ψ( S (1) ) (cid:12)(cid:12) N xi ∈ I X i = 0. As T (1) i x i = 0 ( i ∈ I ), it is easy to see that R (1) = 0 as required. (cid:3) Note that Ψ is not surjective even if X i = Y i = C ( i ∈ I ) since in this case, Ψ is a homomorphismand N i ∈ I C is commutative while L ( N i ∈ I C ) is not.The following result follows from Proposition 2.3(c), Corollary 2.7(b) and Proposition 2.8, which saythat an infinite tensor product of vector spaces is automatically a faithful module over a big commutativealgebra. Corollary 2.9 If X i is a faithful A i -module ( i ∈ I ), then N i ∈ I X i is a faithful N i ∈ I A i -module. Inparticular, N i ∈ I Y i is a faithful unital C ⊗ I -module. Example 2.10 (a) If β ∈ Π i ∈ I C × , then N βi ∈ I C = C · ⊗ i ∈ I β i . In fact, for any F ∈ F and µ i ∈ C ( i ∈ F ), we have J βF ( ⊗ i ∈ F µ i ) = (Π i ∈ F µ i /β i ) ( ⊗ i ∈ I β i ) .(b) Let n ∈ N , I , ..., I n be infinite disjoint subsets of I with I = S nk =1 I k and β = ( β , ..., β n ) ∈ ( C × ) n .Define e β ∈ Π i ∈ I C × by e β i = β k whenever i ∈ I k . Then β [ e β ] ∼ is an injective group homomorphismfrom ( C × ) n to Ω I ; C .(c) Let G be a subgroup of T n ⊆ ( C × ) n (where T := { t ∈ C : | t | = 1 } ). If β (1) , ..., β ( m ) are distinctelements in G and g β (1) , ..., g β ( m ) ∈ Π i ∈ I C × are as in part (b), then ⊗ i ∈ I g β (1) i , ..., ⊗ i ∈ I g β ( m ) i are linearlyindependent in C ⊗ I . Therefore, the ∗ -subalgebra of C ⊗ I generated by {⊗ i ∈ I e β i : β ∈ G } is ∗ -isomorphicto the group algebra C [ G ] . As ⊗ i ∈ I α i = (Π i ∈ I α i )( ⊗ i ∈ I
1) if α i = 1 e.f., one may regard ⊗ i ∈ I α i as a generalisation of theproduct. In this case, one can consider infinite products like ( − I . ∗ -algebras Throughout this section, A i is a unital ∗ -algebra with identity e i ( i ∈ I ), and we set Ω ut I ; A := Π i ∈ I U A i / ∼ . Notice that in this case, Ω I ; A is a ∗ -semi-group with identity and Ω ut I ; A can be regarded as a subgroupof Ω I ; A with the inverse being the involution on Ω I ; A . Moreover, N i ∈ I A i is a Ω I ; A -graded ∗ -algebra inthe sense that for any ω, ω ′ ∈ Ω I ; A , (cid:16)O ωi ∈ I A i (cid:17) · (cid:16)O ω ′ i ∈ I A i (cid:17) ⊆ O ωω ′ i ∈ I A i and (cid:16)O ωi ∈ I A i (cid:17) ∗ ⊆ O ω ∗ i ∈ I A i . (3.1)By Proposition 2.6(b), N ei ∈ I A i can be identified with the unital ∗ -algebra inductive limit of finitetensor products of A i . We will study the following ∗ -subalgebra that contains N ei ∈ I A i : O ut i ∈ I A i := M ω ∈ Ω ut I ; A O ωi ∈ I A i . A i are linear spans of U A i (in particular, if they are C ∗ -algebras or group algebras), then N ut i ∈ I A i is the linear span of Θ A (Π i ∈ I U A i ). If A i = A for all i ∈ I , we denote A ⊗ I ut := N ut i ∈ I A i . Example 3.1 (a) Let G i be a group and C [ G i ] be its group algebra ( i ∈ I ). If Λ : Π i ∈ I G i → Π i ∈ I U C [ G i ] is the canonical map, then λ := Θ C [ G ] ◦ Λ gives a ∗ -isomorphism from C [Π i ∈ I G i ] to the ∗ -subalgebra O Λ(Π i ∈ I G i ) i ∈ I C [ G i ] := X t ∈ Π i ∈ I G i O Λ( t ) i ∈ I C [ G i ] ⊆ O ut i ∈ I C [ G i ] . In fact, λ induces a ∗ -homomorphism from C [Π i ∈ I G i ] to N ut i ∈ I C [ G i ] . Let q : Π i ∈ I G i → Π i ∈ I G i / ⊕ i ∈ I G i be the quotient map. For a fixed s ∈ Π i ∈ I G i , if we set M si ∈ I G i := (cid:8) t ∈ Π i ∈ I G i : q ( t ) = q ( s ) (cid:9) , then s − (cid:0) L si ∈ I G i (cid:1) = L i ∈ I G i . Thus, { λ ( t ) : t ∈ L si ∈ I G i } is a set of linearly independent elements in N i ∈ I C [ G i ] (as λ | C [ L i ∈ I G i ] is a bijection onto N ei ∈ I C [ G i ] ). On the other hand, if s (1) , ..., s ( n ) ∈ Π i ∈ I G i such that q ( s ( k ) ) = q ( s ( l ) ) whenever k = l , then λ ( s (1) ) , ..., λ ( s ( n ) ) are linearly independent in N i ∈ I C [ G i ] (see Theorem 2.5). Consequently, { λ ( t ) : t ∈ Π i ∈ I G i } form a basis for N Λ(Π i ∈ I G i ) i ∈ I C [ G i ] .(b) It is well-known that there is a twisted action ( α, u ) , in the sense of Busby and Smith, of Ω I ; G :=Π i ∈ I G i / ⊕ i ∈ I G i on C [ L i ∈ I G i ] ∼ = N ei ∈ I C [ G i ] (see [6, 2.1]) such that C [Π i ∈ I G i ] is ∗ -isomorphic to thealgebraic crossed-product N ei ∈ I C [ G i ] ⋊ α,u Ω I ; G . There is a canonical action Ξ of Π i ∈ I U A i on N ut i ∈ I A i given by inner-automorphisms, i.e.Ξ u ( a ) := ( ⊗ i ∈ I u i ) · a · ( ⊗ i ∈ I u ∗ i ) (cid:0) u ∈ Π i ∈ I U A i ; a ∈ O ut i ∈ I A i (cid:1) . This induces an action Ξ e of Π i ∈ I U A i on the subalgebra N ei ∈ I A i . The following result gives an identifi-cation of N ut i ∈ I A i as the algebraic crossed-product (see e.g. [17, p.166]) of a cocycle twisted action (i.e.a twisted action in the sense of Busby and Smith) of Ω ut I ; A on N ei ∈ I A i induced by Ξ e .Before we give this result, let us recall that an abelian group G is divisible if for any g ∈ G and n ∈ N , there is h ∈ G with g = h n . Theorem 3.2 (a) There is a cocycle twisted action (ˇΞ , m ) of Ω ut I ; A on N ei ∈ I A i such that N ut i ∈ I A i is Ω ut I ; A -graded ∗ -isomorphic to ( N ei ∈ I A i ) ⋊ ˇΞ ,m Ω ut I ; A .(b) Suppose that all A i are commutative. If N ei ∈ I A i is a unital ∗ -subalgebra of a commutative ∗ -algebra B with U B being divisible, N ut i ∈ I A i is Ω ut I ; A -graded ∗ -isomorphic to a unital ∗ -subalgebra of B ⊗ C [Ω ut I ; A ] .If U N ei ∈ I A i is itself divisible, N ut i ∈ I A i ∼ = ( N ei ∈ I A i ) ⊗ C [Ω ut I ; A ] as Ω ut I ; A -graded ∗ -algebras. Proof:
Let c : Ω ut I ; A → Π i ∈ I U A i be a cross-section with c ([ e ] ∼ ) = e .(a) For any µ, ν ∈ Ω ut I ; A , we setˇΞ µ := Ξ ec ( µ ) and m ( µ, ν ) := ⊗ i ∈ I c ( µ ) i c ( ν ) i c ( µν ) − i . As c ( µ ) c ( ν ) ∼ c ( µν ), we have m ( µ, ν ) ∈ N ei ∈ I A i . It is easy to check that (ˇΞ , m ) is a twisted action inthe sense of Busby and Smith. Furthermore, we define Ψ : ( N ei ∈ I A i ) ⋊ ˇΞ ,m Ω ut I ; A → N ut i ∈ I A i byΨ( f ) := X ω ∈ Ω ut I ; A f ( ω )( ⊗ i ∈ I c ( ω ) i ) (cid:0) f ∈ ( O ei ∈ I A i ) ⋊ ˇΞ ,m Ω ut I ; A (cid:1) .
8t is not hard to verify that Ψ is a bijective Ω ut I ; A -graded ∗ -homomorphism.(b) Let Π ei ∈ I U A i := Π ei ∈ I A i ∩ Π i ∈ I U A i . By the Baer’s theorem, Θ A | Π ei ∈ I U Ai can be extended to a grouphomomorphism ϕ : Π i ∈ I U A i → U B . Since ϕ ( c ( µ )) ϕ ( c ( ν )) ϕ ( c ( µν )) − = ⊗ i ∈ I c ( µ ) i c ( ν ) i c ( µν ) − i ( µ, ν ∈ Ω ut I ; A ) , the map Φ : N ut i ∈ I A i → B ⊗ C [Ω ut I ; A ] given byΦ( a ) := ( a · ⊗ i ∈ I c ( ω ) − i ) ϕ ( c ( ω )) ⊗ λ ( ω ) (cid:0) a ∈ O ωi ∈ I A i ; ω ∈ Ω ut I ; A (cid:1) (3.2)is a Ω ut I ; A -graded ∗ -homomorphism. If P ω ∈ Ω ut I ; A a ω ∈ ker Φ (with a ω ∈ N ωi ∈ I A i ), then for every ω ∈ Ω ut I ; A ,one has ( a ω · ⊗ i ∈ I c ( ω ) − i ) ϕ ( c ( ω )) = 0, which implies a ω = 0, and hence Φ is injective. The image of Φis the linear span of (cid:8) bϕ ( c ( ω )) ⊗ λ ( ω ) : b ∈ O ei ∈ I A i ; ω ∈ Ω ut I ; A (cid:9) , and it is clear that Φ is surjective if B = N ei ∈ I A i . (cid:3) Remark 3.3 (a) The cocycle twisted action (ˇΞ , m ) depend on the choice of a cross-section, and differentcross-sections may give different twisted actions (although their crossed-products are all isomorphic). Onthe other hand, the map Φ in part (b) also depends on the choice of a cross-section as well as the choiceof an extension of Θ A | Π ei ∈ I U Ai .(b) If S i is a set and A i is a ∗ -subalgebra of ℓ ∞ ( S i ) ( i ∈ I ), then by Theorem 3.2(b), N ut i ∈ I A i is a ∗ -subalgebra of ℓ ∞ (Π i ∈ I S i ) ⊗ C [Ω ut I ; A ] . Our first proof for this fact use [7, 18.4] and [8, 7.1].(c) If all A i are commutative, then N ut i ∈ I A i ∼ = ( N ei ∈ I A i ) ⊗ C [Ω ut I ; A ] as Ω ut I ; A -graded ∗ -algebras if and onlyif there is a group homomorphism π : Ω ut I ; A → U N ut i ∈ I A i such that π ( ω ) ∈ N ωi ∈ I A i ( ω ∈ Ω ut I ; A ). In fact,if such a π exists, one may replace ( a · ⊗ i ∈ I c ( ω ) − i ) ϕ ( c ( ω )) in (3.2) with aπ ( ω − ) and show that thecorresponding Φ is a ∗ -isomorphism. Clearly, the second statement of Theorem 3.2(b) applies to the case when A i = C n i for some n i ∈ N ( i ∈ I ). In particular, Theorem 3.2(b) and its argument give the following corollary. Corollary 3.4 If ϕ is as in Example 2.2(a) and ϕ : Π i ∈ I T → T is a group homomorphism that extends ϕ | Π i ∈ I T (it existence is guaranteed by the Baer’s theorem), then Φ( ⊗ i ∈ I α i ) := ϕ ( α ) λ ([ α ] ∼ ) ( α ∈ Π i ∈ I T )is a well-defined ∗ -isomorphism from C ⊗ I ut onto C [Ω ut I ; C ] . Conversely, it is clear that if ϕ : Π i ∈ I T → T is any map such that Φ as defined in the aboveis a well-defined ∗ -isomorphism, then ϕ is a group homomorphism extending ϕ | Π i ∈ I T . On the otherhand, there is a simpler proof for Corollary 3.4. In fact, for α, β ∈ Π i ∈ I T with α ∼ β , one has ϕ ( α ) − ·⊗ i ∈ I α i = ϕ ( β ) − ·⊗ i ∈ I β i . Thus, [ α ] ∼ ϕ ( α ) − ·⊗ i ∈ I α i is a well-defined group homomorphismfrom Ω ut I ; C to U C ⊗ I ut such that { ϕ ( α ) − · ⊗ i ∈ I α i : [ α ] ∼ ∈ Ω ut I ; C } is a basis for C ⊗ I ut . Example 3.5
For any subgroup G ⊆ T n , the algebra in Example 2.10(c) is a ∗ -subalgebras of C ⊗ I ut . In the remainder of this section, we will show that the center of N ut i ∈ I A i is the tensor product ofcenters of A i when A i = span U A i for all i ∈ I .If A is an algebra and G is a group, we denote by Z ( A ) and Z ( G ) the center of A and the center of G respectively. Clearly, the inclusion Π i ∈ I U Z ( A i ) ⊆ Π i ∈ I U A i induces an injective group homomorphismfrom Ω ut I ; Z ( A ) to Ω ut I ; A and we regard the former as a subgroup of the later.9 heorem 3.6 Suppose that there is F ∈ F with A i = span U A i for any i ∈ I := I \ F .(a) Z (Ω ut I ; A ) = Ω ut I ; Z ( A ) . Moreover, Z (Ω ut I ; A ) = Ω ut I ; A if and only if all but a finite number of A i arecommutative.(b) Every element in Ω ut I ; A \ Z (Ω ut I ; A ) has an infinite conjugacy class.(c) Z (cid:0)N ut i ∈ I A i (cid:1) = N ut i ∈ I Z ( A i ) . Proof: (a) It is obvious that Ω ut I ; Z ( A ) ⊆ Z (Ω ut I ; A ). Suppose u ∈ Π i ∈ I U A i with [ u ] ∼ / ∈ Ω ut I ; Z ( A ) . Thereis an infinite subset J ⊆ I such that u i / ∈ Z ( A i ) ( i ∈ J ). For each i ∈ J , one can find v i ∈ U A i such that u i v i = v i u i . For any i ∈ I \ J , we put v i = e i . Then [ v ] ∼ ∈ Ω ut I ; A and [ u ] ∼ [ v ] ∼ = [ v ] ∼ [ u ] ∼ .Consequently, [ u ] ∼ / ∈ Z (Ω ut I ; A ). This argument also shows that if the set { i ∈ I : Z ( A i ) = A i } is infinite,then Z (Ω ut I ; A ) = Ω ut I ; A . Conversely, it is clear that Ω ut I ; Z ( A ) = Ω ut I ; A if all but a finite numbers of A i arecommutative.(b) Suppose that [ u ] ∼ ∈ Ω ut I ; A \ Z (Ω ut I ; A ) and { i n } n ∈ N is a sequence of distinct elements in I such that u i n / ∈ Z ( A i n ) ( n ∈ N ). For each n ∈ N , choose v i n ∈ U A in with v i n u i n v ∗ i n = u i n . For any prime number p , we set w ( p ) i n := v i n ( n ∈ N p ), and w ( p ) i := e i if i ∈ I \ { i n : n ∈ N p } . If p and q are distinct primenumbers, then w ( q ) i n u i n ( w ( q ) i n ) ∗ = u i n = w ( p ) i n u i n ( w ( p ) i n ) ∗ ( n ∈ N p \ N q ) . Consequently, w ( q ) u ( w ( q ) ) ∗ ≁ w ( p ) u ( w ( p ) ) ∗ , and the conjugacy class of [ u ] ∼ is infinite.(c) Since Z ( N ut i ∈ I A i ) = N i ∈ F Z ( A i ) ⊗ Z ( N ut i ∈ I A i ), we may assume that A i = span U A i for all i ∈ I .In this case, Z ( N ut i ∈ I A i ) = (cid:0)N ut i ∈ I A i (cid:1) Ξ , where (cid:0)N ut i ∈ I A i (cid:1) Ξ is the fixed point algebra of the action Ξ asdefined above. Moreover, one has N ut i ∈ I Z ( A i ) ⊆ Z ( N ut i ∈ I A i ) and it remains to show that (cid:0)N ut i ∈ I A i (cid:1) Ξ ⊆ N ut i ∈ I Z ( A i ).Let v (1) , ..., v ( n ) ∈ Π i ∈ I U A i be mutually inequivalent elements, F ∈ F and b , ..., b n ∈ N i ∈ F A i \ { } such that a := P nk =1 J v ( k ) F ( b k ) ∈ (cid:0)N ut i ∈ I A i (cid:1) Ξ . We first claim that [ v ( k ) ] ∼ ∈ Ω ut I ; Z ( A ) ( k = 1 , ..., n ).Suppose on the contrary that [ v (1) ] ∼ / ∈ Ω ut I ; Z ( A ) = Z (Ω ut I ; A ). For every u ∈ Π i ∈ I U A i , one hasΞ u (cid:0) J v (1) F ( b k ) (cid:1) ∈ (cid:0)O [ uv (1) u ∗ ] ∼ i ∈ I A i (cid:1) \ { } . As Ξ u ( a ) = a , we see that [ uv (1) u ∗ ] ∼ ∈ { [ v (1) ] ∼ , ..., [ v ( n ) ] ∼ } , which contradicts the fact that { [ uv (1) u ∗ ] ∼ :[ u ] ∼ ∈ Ω ut I ; A } is an infinite set (by part (b)).By enlarging F , we may assume that v ( k ) ∈ Π i ∈ I U Z ( A i ) ( k = 1 , ..., n ). For each u ∈ Π i ∈ I U A i and k ∈ { , ..., n } , one has Ξ u ( J v ( k ) F ( b k )) = J v ( k ) F ( b k ) and so, b k ∈ Z ( N i ∈ F A i ). Therefore, a ∈ N ut i ∈ I Z ( A i ) asexpected. (cid:3) The readers should notice that N ut i ∈ I Z ( A i ) equals L ω ∈ Z (Ω ut I ; A ) N ωi ∈ I Z ( A i ) instead of L ω ∈ Ω ut I ; A N ωi ∈ I Z ( A i )(strictly speaking, the later object does not make sense). Example 3.7 (a) If n i ∈ N ( i ∈ I ), then Z (cid:0)N ut i ∈ I M n i ( C ) (cid:1) ∼ = C ⊗ I ut .(b) If G i are icc groups, then Z ( N ut i ∈ I C [ G i ]) ∼ = C ⊗ I ut canonically. We end this section with the following brief discussion on the non-unital case. Suppose that { A i } i ∈ I is a family of ∗ -algebras, not necessarily unital. If M ( A i ) is the double centraliser algebra of A i ( i ∈ I ),we define an ideal, N ut i ∈ I A i , of N ut i ∈ I M ( A i ) as follows: O ut i ∈ I A i := span (cid:8) J uF ( a ) : F ∈ F ; a ∈ O i ∈ F A i ; u ∈ Π i ∈ I U M ( A i ) (cid:9) .
10n general, N ut i ∈ I A i is not a subset of N i ∈ I A i . In a similar fashion, we define O ei ∈ I A i := span (cid:8) J uF ( a ) : F ∈ F ; a ∈ O i ∈ F A i ; u ∈ Π i ∈ I U M ( A i ) ; u ∼ e (cid:9) , which is an ideal of N ei ∈ I M ( A i ). By the proof of Theorem 3.2(a), one may identify N ut i ∈ I A i as the idealof ( N ei ∈ I M ( A i )) ⋊ ˇΞ ,m Ω ut I ; M ( A ) consisting of functions from Ω ut I ; M ( A ) to N ei ∈ I A i having finite supports. Throughout this section, ( H i , h· , ·i ) is a non-zero inner-product space ( i ∈ I ). Moreover, we denote Ω unit I ; H := Π i ∈ I S ( H i ) / ∼ . If B is a unital ∗ -algebra and X is a unital left B -module, a map h· , ·i B : X × X → B is called a (left)Hermitian B -form on X if h ax + y, z i B = a h x, z i B + h y, z i B and h x, y i ∗ B = h y, x i B ( x, y, z ∈ X ; a ∈ B ).It is easy to see that a Hermitian B -form on X can be regarded as a B -bimodule map θ : X ⊗ ˜ X → B satisfying θ ( x ⊗ ˜ y ) ∗ = θ ( y ⊗ ˜ x ) (where ˜ X is the conjugate vector spaces of X regarded as a unital right B -module in the canonical way). Consequently, part (a) of the following result follows readily from theuniversal property of tensor products, while part (b) is easily verified. Proposition 4.1 (a) There is a Hermitian C ⊗ I -form on N i ∈ I H i such that h⊗ i ∈ I x i , ⊗ i ∈ I y i i C ⊗ I := ⊗ i ∈ I h x i , y i i ( x, y ∈ Π i ∈ I H i ).(b) For a fixed µ ∈ Ω unit I ; H , one has h Θ H ( x ) , Θ H ( y ) i C ⊗ I = Π i ∈ I h x i , y i i ( ⊗ i ∈ I
1) ( x, y ∈ Π µi ∈ I H i ) . Thisinduces an inner-product on N µi ∈ I H i which coincides with the one given by the inductive limit of (cid:0) N i ∈ F H i , J µG ; F (cid:1) F ⊆ G ∈ F , in the category of inner-product spaces with isometries as morphisms. We want to construct a nice inner-product space from the above Hermitian C ⊗ I -form. A naivethought is to appeal to a construction in Hilbert C ∗ -modules that produces a Hilbert space from apositive linear functional on C ⊗ I . However, the difficulty is that there is no canonical order structureon C ⊗ I . Nevertheless, we will do a similar construction using the functional φ in Example 2.2(a). Inthis case, one can only consider a subspace of N i ∈ I H i (see Example 4.3 below). Lemma 4.2
Suppose that N ct i ∈ I H i := span Θ H (Π i ∈ I B ( H i )) , N unit i ∈ I H i := span Θ H (Π i ∈ I S ( H i )) and h ξ, η i φ := φ ( h ξ, η i C ⊗ I ) (cid:0) ξ, η ∈ O i ∈ I H i (cid:1) . (a) For any µ ∈ Ω unit I ; H , the restriction of h· , ·i φ on N µi ∈ I H i × N µi ∈ I H i coincides with the inner-productin Proposition 4.1(b).(b) h· , ·i φ is a positive sesquilinear form on N ct i ∈ I H i and is an inner-product on N unit i ∈ I H i . Moreover, if K := n y ∈ O ct i ∈ I H i : h x, y i φ = 0 , ∀ x ∈ O ct i ∈ I H i o , then N ct i ∈ I H i = K ⊕ N unit i ∈ I H i (as vector spaces).(c) If I = I ∪ I and I ∩ I = ∅ , then N unit i ∈ I H i = ( N unit i ∈ I H i ) ⊗ ( N unit j ∈ I H j ) as inner-product spaces. roof: (a) This part is clear.(b) It is obvious that h· , ·i φ is a sesquilinear form on N ct i ∈ I H i . Let E := (cid:8) x ∈ Π i ∈ I B ( H i ) : k x i k < i ∈ I (cid:9) and ˜ K := span Θ H ( E ). Clearly, N ct i ∈ I H i = ˜ K ⊕ N unit i ∈ I H i . Moreover, if u ∈ Π i ∈ I B ( H i ) and v ∈ E , then h u i , v i i 6 = 1 for an infinite number of i ∈ I , which implies that h⊗ i ∈ I u i , ⊗ i ∈ I v i i φ = 0. Consequently,˜ K ⊆ K .We claim that h ξ, ξ i φ ≥ ξ ∈ N ct i ∈ I H i . Suppose that ξ = P nk =1 λ k ⊗ i ∈ I u ( k ) i with λ , ..., λ n ∈ C and u (1) , ..., u ( n ) ∈ Π i ∈ I B ( H i ). Then h ξ, ξ i φ = X nk,l =1 λ k ¯ λ l φ (cid:0) ⊗ i ∈ I h u ( k ) i , u ( l ) i i (cid:1) . As in the above, φ (cid:0) ⊗ i ∈ I h u ( k ) i , u ( l ) i i (cid:1) = 0 if either u ( k ) or u ( l ) is in E . Thus, by rescaling, we may assumethat u (1) , ..., u ( n ) ∈ Π i ∈ I S ( H i ) . Furthermore, we assume that there exist 0 = n < · · · < n m = n such that u ( n p +1) ∼ · · · ∼ u ( n p +1 ) forall p ∈ { , ..., m − } , but u ( n p ) ≁ u ( n q ) whenever 1 ≤ p = q ≤ m . It is not hard to check that u ( k ) ∼ u ( l ) if and only if h u ( k ) i , u ( l ) i i = 1 e.f. (as k u ( k ) i k , k u ( l ) i k ≤ ≤ p = q ≤ m , φ (cid:0) ⊗ i ∈ I h u ( k ) i , u ( l ) i i (cid:1) = 0 when n p < k ≤ n p +1 and n q < l ≤ n q +1 . (4.1)Therefore, in order to show h ξ, ξ i φ ≥
0, it suffices to consider the case when u ( k ) ∼ u ( l ) for all k, l ∈{ , ..., n } , which is the same as ξ ∈ N u (1) i ∈ I H i . Thus, h ξ, ξ i φ ≥ h· , ·i φ is an inner-product on N unit i ∈ I H i . Suppose that ξ = P nk =1 λ k ⊗ i ∈ I u ( k ) i with λ , ..., λ n ∈ C and u (1) , ..., u ( n ) ∈ Π i ∈ I S ( H i ) such that h ξ, ξ i φ = 0. If n , ..., n m are as in theabove, then φ (cid:16)(cid:10)X n p +1 k = n p +1 λ k ⊗ i ∈ I u ( k ) i , X n q +1 l = n q +1 λ l ⊗ i ∈ I u ( l ) i (cid:11) C ⊗ I (cid:17) = 0 , because of (4.1) and the positivity of h· , ·i φ . Hence, we may assume u ( k ) ∼ u ( l ) for all k, l ∈ { , ..., n } ,and apply part (a) to conclude that ξ = 0.Finally, as h· , ·i φ is an inner-product on N unit i ∈ I H i and we have both N ct i ∈ I H i = ˜ K ⊕ N unit i ∈ I H i and˜ K ⊆ K , we obtain K ⊆ ˜ K as well.(c) Observe that the linear bijection Ψ : ( N i ∈ I H i ) ⊗ ( N j ∈ I H j ) → N i ∈ I H i as in Remark 2.4(b) re-stricts to a surjection from ( N unit i ∈ I H i ) ⊗ ( N unit j ∈ I H j ) to N unit i ∈ I H i . Moreover, for any u, u ′ ∈ Π i ∈ I S ( H i )and v, v ′ ∈ Π j ∈ I S ( H j ), we have ( u, u ′ ) ∼ ( v, v ′ ) as elements in Π i ∈ I S ( H i ) if and only if u ∼ u ′ and v ∼ v ′ . Thus, the argument is part (b) tells us that (cid:10) ( ⊗ i ∈ I u i ) ⊗ ( ⊗ j ∈ I v j ) , ( ⊗ i ∈ I u ′ i ) ⊗ ( ⊗ j ∈ I v ′ j ) (cid:11) φ = h⊗ i ∈ I u i , ⊗ i ∈ I u ′ i i φ h⊗ j ∈ I v j , ⊗ j ∈ I v ′ j i φ This shows that Ψ (cid:12)(cid:12) ( N unit i ∈ I H i ) ⊗ ( N unit j ∈ I H j ) is inner-product preserving. (cid:3) We denote by ¯ N µi ∈ I H i and ¯ N φ i ∈ I H i the completions of N µi ∈ I H i and N unit i ∈ I H i , respectively, underthe norms induced by h· , ·i φ . Example 4.3 If H i = C ( i ∈ I ), then the sesquilinear form h· , ·i φ is not positive on the whole space N i ∈ I H i since (cid:10) ( ⊗ i ∈ I / − ⊗ i ∈ I , ( ⊗ i ∈ I / − ⊗ i ∈ I (cid:11) φ = − . eu i ∈ I H i := { x ∈ Π i ∈ I H i : x i ∈ S ( H i ) except for a finite number of i } and K be an inner-product space. A multilinear map Φ : Π eu i ∈ I H i → K (i.e. Φ is coordinatewise linear) is said to be componentwise inner-product preserving if for any µ, ν ∈ Ω unit I ; H , h Φ( x ) , Φ( y ) i = δ µ,ν Π i ∈ I h x i , y i i ( x ∈ Π µi ∈ I H i ; y ∈ Π νi ∈ I H i )where δ µ,ν is the Kronecker delta. Theorem 4.4 (a) ¯ N φ i ∈ I H i ∼ = ¯ L ℓ µ ∈ Ω unit I ; H ¯ N µi ∈ I H i canonically as Hilbert spaces.(b) Θ H | Π eu i ∈ I H i : Π eu i ∈ I H i → N unit i ∈ I H i is a componentwise inner-product preserving multilinear map.For any inner-product space K and any componentwise inner-product preserving multilinear map Φ :Π eu i ∈ I H i → K , there is a unique isometry ˜Φ : N unit i ∈ I H i → K such that Φ = ˜Φ ◦ Θ H | Π eu i ∈ I H i . Proof: (a) Clearly, N unit i ∈ I H i = P µ ∈ Ω unit I ; H N µi ∈ I H i . Moreover, as in the proof of Lemma 4.2(b), the twosubspaces N µi ∈ I H i and N νi ∈ I H i are orthogonal if µ and ν are distinct elements in Ω unit I ; H . The rest of theargument is standard.(b) It is easy to see that Θ H | Π eu i ∈ I H i is componentwise inner-product preserving. The uniqueness of ˜Φfollows from the fact that Θ H (Π eu i ∈ I H i ) generates N unit i ∈ I H i . To show the existence of ˜Φ, we first define amultilinear map Φ : Π i ∈ I H i → K by setting Φ = Φ on Π eu i ∈ I H i and Φ = 0 on Π i ∈ I H i \ Π eu i ∈ I H i . Let˜Φ : N i ∈ I H i → K be the induced linear map and set ˜Φ := ˜Φ | N unit i ∈ I H i . Suppose that u, v ∈ Π i ∈ I S ( H i ), ξ ∈ N ui ∈ I H i and η ∈ N vi ∈ I H i . If u ≁ v , then h ξ, η i φ = 0 = h ˜Φ( ξ ) , ˜Φ( η ) i . Otherwise, there exist F ∈ F and ξ , η ∈ N i ∈ F H i such that ξ = J uF ( ξ ), η = J vF ( η ) and u i = v i if i ∈ I \ F . In this case, h ˜Φ( ξ ) , ˜Φ( η ) i = h ξ , η i = h ξ, η i φ . (cid:3) Example 4.5
Suppose that Φ and ϕ are as in Corollary 3.4, and { δ µ } µ ∈ Ω unit I ; C is the canonical orthonor-mal basis for ℓ (cid:0) Ω unit I ; C (cid:1) . Note that Ω ut I ; C = Ω unit I ; C and consider the linear bijection J : C [Ω ut I ; C ] → C [Ω unit I ; C ] given by J ( λ ([ α ] ∼ )) := δ [ α ] ∼ ( α ∈ Π i ∈ I T ). By Example 2.10(a) and Theorem 4.4(a), the map J ◦ Φ induces a Hilbert space isomorphism ˆΦ : ¯ N φ i ∈ I C → ℓ (cid:0) Ω unit I ; C (cid:1) such that ˆΦ( ⊗ i ∈ I β i ) = ϕ ( β ) δ [ β ] ∼ ( β ∈ Π i ∈ I T ). We would like to compare ¯ N φ i ∈ I H i with the infinite directed product as defined in [21], when { H i } i ∈ I is a family of Hilbert spaces. Let us first recall from [21, Definition 3.3.1] that x ∈ Π i ∈ I H i is a C -sequence if P i ∈ I (cid:12)(cid:12) k x i k − (cid:12)(cid:12) converges. As in [21, Definition 3.3.2], if x and y are C -sequences such that P i ∈ I (cid:12)(cid:12) h x i , y i i − (cid:12)(cid:12) converges, then we write x ≈ y . Denote by [ x ] ≈ the equivalence class of x under ≈ ,and by Γ I ; H the set of all such equivalence classes (see [21, Definition 3.3.3]).Let Q ⊗ i ∈ I H i be the infinite direct product Hilbert space as defined in [21], and Q ⊗ i ∈ I x i be theelement in Q ⊗ i ∈ I H i corresponding to a C -sequence x as in [21, Theorem IV]. Notice that if x ∈ Π eu i ∈ I H i ,then x is a C -sequence, and we have a multilinear mapΥ : Π eu i ∈ I H i −→ Y ⊗ i ∈ I H i . On the other hand, for any C ∈ Γ I ; H , we denote Q ⊗ C i ∈ I H i to be the closed subspace of Q ⊗ i ∈ I H i generated by { Q ⊗ i ∈ I x i : x ∈ C } (see [21, Definition 4.1.1]).13 roposition 4.6 Let { H i } i ∈ I be a family of Hilbert spaces.(a) [ x ] ∼ [ x ] ≈ ( x ∈ Π i ∈ I S ( H i ) ) gives a well-defined surjection κ H : Ω unit I ; H → Γ I ; H . Moreover, for any x, y ∈ Π i ∈ I S ( H i ) , there is a bijection between κ − H ([ x ] ≈ ) and κ − H ([ y ] ≈ ) .(b) There exists a linear map ˜Υ : N unit i ∈ I H i → Q ⊗ i ∈ I H i such that Υ = ˜Υ ◦ Θ H | Π eu i ∈ I H i and ˜Υ | N µi ∈ I H i extends to a Hilbert space isomorphism ˜Υ µ : ¯ N µi ∈ I H i → Q ⊗ κ H ( µ ) i ∈ I H i ( µ ∈ Ω unit I ; H ). Proof: (a) Clearly, if x ∼ z , then x ≈ z and κ H is well-defined. [21, Lemma 3.3.7] tells us that κ H is surjective. Furthermore, there exists a unitary u i ∈ L ( H i ) such that u i x i = y i ( i ∈ I ), and [ u i ] i ∈ I induces the required bijective correspondence in the second statement.(b) By the argument of Theorem 4.4(b), one can construct a linear map ˜Υ such that Υ = ˜Υ ◦ Θ H | Π eu i ∈ I H i .By the argument of part (a), we see that ˜Υ (cid:16)N [ u ] ∼ i ∈ I H i (cid:17) ⊆ Q ⊗ [ u ] ≈ i ∈ I H i ( u ∈ Π i ∈ I S ( H i )). Furthermore, byLemma 4.2(a), Proposition 4.1(b) and [21, Theorem IV], we see that ˜Υ | N [ u ] ∼ i ∈ I H i is an isometry. Finally,˜Υ | N [ u ] ∼ i ∈ I H i has dense range (by [21, Lemma 4.1.2]). (cid:3) Notice that ˜Υ is, in general, unbounded but Remark 4.7(b) below tells us that ¯ N φ i ∈ I H i is a “naturaldilation” of Q ⊗ i ∈ I H i . On the other hand, Remark 4.7(d) says that it is possible to construct Q ⊗ i ∈ I H i in a similar way as that for ¯ N φ i ∈ I H i . Note however, that the construction of ¯ N φ i ∈ I H i is totally algebraicaland ¯ N φ i ∈ I H i itself seems to be more natural (see Theorem 4.8 and Example 5.6 below). Remark 4.7
Suppose that { H i } i ∈ I is a family of Hilbert spaces.(a) ∼ and ≈ are different even in the case when I = N and H i = C ( i ∈ N ) because one can find x, y ∈ Π i ∈ N T with x i = y i for all i ∈ N but P ∞ i =1 (cid:12)(cid:12) h x i , y i i − (cid:12)(cid:12) converges. In fact, κ − H ([ x ] ≈ ) is an infiniteset.(b) By [21, Lemma 4.1.1], we have Y ⊗ i ∈ I H i = ¯ M ℓ C ∈ Γ I ; H Y ⊗ C i ∈ I H i . Therefore, Theorem 4.4(a) and Proposition 4.6 tell us that for a fixed γ ∈ Γ I ; H , one has a canonicalHilbert space isomorphism ¯ O φ i ∈ I H i ∼ = ℓ (cid:0) κ − H ( γ ) (cid:1) ¯ ⊗ (cid:0) Y ⊗ i ∈ I H i (cid:1) . (c) For each i ∈ I , let K i be an inner-product space such that H i is the completion of K i . Then ¯ N φ i ∈ I K i is, in general, not canonically isomorphic to ¯ N φ i ∈ I H i because Ω unit I ; K ( Ω unit I ; H if K i ( H i for aninfinite number of i ∈ I . On the other hand, if I is countable, for any x ∈ Π i ∈ I S ( H i ) , there exists y ∈ Π i ∈ I S ( K i ) such that x ≈ y . This shows that the restriction, κ H ; K , of κ H on Ω unit I ; K is also asurjection onto Γ I ; H . However, we do not know if the cardinality of κ − H ; K ( C ) are the same for different C ∈ Γ I ; H .(d) If φ is as in Example 2.2(b), it is easy to see that h Y ⊗ u i , Y ⊗ v i i = φ (cid:0) h⊗ i ∈ I u i , ⊗ i ∈ I v i i C ⊗ I (cid:1) ( u, v ∈ Π unit i ∈ I H i ) . Thus, the sesquilinear form φ (cid:0) h· , ·i C ⊗ I (cid:1) produces Q ⊗ H i . If one wants a self-contained alternativeconstruction for Q ⊗ H i , one needs to establish the positivity of φ (cid:0) h· , ·i C ⊗ I (cid:1) , which can be reduced toshowing the positivity when all H i are of the same finite dimension.
14n the remainder of this section, we show that N unit i ∈ I H i can be completed into a C ∗ (Ω ut I ; C )-module,which gives many pre-inner products on N unit i ∈ I H i including h· , ·i φ . In the following, we use the con-vention that the A -valued inner-product of an inner-product A -module is A -linear in the first variable(where A is a pre- C ∗ -algebra). On the other hand, we recall that if G is a group and λ g is the canonicalimage of g in C [ G ], the map P g ∈ G α g λ g α e ( α g ∈ C ), where e ∈ G is the identity, extends to a faithfultracial state χ G on C ∗ ( G ). Theorem 4.8 (a) There exists an inner-product C [Ω ut I ; C ] -module structure on N unit i ∈ I H i . If ¯ N mod i ∈ I H i isthe Hilbert C ∗ (Ω ut I ; C ) -module given by the completion of this C [Ω ut I ; C ] -module, we have a canonical Hilbertspace isomorphism ¯ O φ i ∈ I H i ∼ = (cid:0) ¯ O mod i ∈ I H i (cid:1) ¯ ⊗ χ Ωut I ; C C . (4.2) (b) If G ⊆ Ω ut I ; C is a subgroup and E G : C ∗ (Ω ut I ; C ) → C ∗ ( G ) is the canonical conditional expectation, thereis an inner-product C [ G ] -module structure on N unit i ∈ I H i , whose completion coincides with the Hilbert C ∗ ( G ) -module (cid:0) ¯ N mod i ∈ I H i (cid:1) ¯ ⊗ E G C ∗ ( G ) . Proof: (a) Clearly, N unit i ∈ I H i is a C ⊗ I ut -submodule of the C ⊗ I -module N i ∈ I H i (see Proposition 2.3(c)).Moreover, one has a linear “truncation” E from C ⊗ I = (cid:0)L ω ∈ Ω I ; C \ Ω ut I ; C N ωi ∈ I C (cid:1) ⊕ C ⊗ I ut to C ⊗ I ut sending( α, β ) to β . Define h ξ, η i C ⊗ I ut := E (cid:0) h ξ, η i C ⊗ I (cid:1) (cid:0) ξ, η ∈ O unit i ∈ I H i (cid:1) , which is a Hermitian C ⊗ I ut -form because by (3.1), we have E ( ab ) = E ( a ) b and E ( a ∗ ) = E ( a ) ∗ ( a ∈ C ⊗ I ; b ∈ C ⊗ I ut ) . For any u, v ∈ Π i ∈ I S ( H i ), we write u ∼ s v if there exists β ∈ Π i ∈ I T such that u i = β i v i e.f. Then ∼ s is an equivalence relation on Π i ∈ I S ( H i ) satisfying u ∼ s v if and only if h⊗ i ∈ I u i , ⊗ i ∈ I v i i C ⊗ I ∈ C ⊗ I ut . (4.3)Let Φ and ϕ be as in Corollary 3.4. Suppose that ξ = P nk =1 α k ⊗ i ∈ I u ( k ) i with α , ..., α n ∈ C and u (1) , ..., u ( n ) ∈ Π i ∈ I S ( H i ). We first show that Φ( h ξ, ξ i C ⊗ I ut ) ∈ C ∗ (Ω ut I ; C ) + . As in the proof of Lemma4.2(b), it suffices to consider the case when u ( k ) ∼ s u (1) for any k ∈ { , ..., n } (because of Relation(4.3)). Let F ∈ F and β (1) , ..., β ( n ) ∈ Π i ∈ I T such that u ( k ) i = β ( k ) i u (1) i ( i ∈ I \ F ; k = 1 , ..., n ). For any k, l ∈ { , ..., n } , we haveΦ (cid:0) (Π i ∈ F h u ( k ) i , u ( l ) i i i )( ⊗ i ∈ I \ F β ( k ) i β ( l ) i ) (cid:1) = h ˜ ϕ F ( u ( k ) ) , ˜ ϕ F ( u ( l ) ) i F , where ˜ ϕ F ( u ( k ) ) := (cid:0) ϕ ( β ( k ) )Π i ∈ F β ( k ) i (cid:1) − ( ⊗ i ∈ F u ( k ) i ) ⊗ λ [ β ( k ) ] ∼ and h· , ·i F is the canonical C [Ω ut I ; C ]-valuedinner-product on ( N i ∈ F H i ) ⊗ C [Ω ut I ; C ]. Therefore,Φ( h ξ, ξ i C ⊗ I ut ) = DX nk =1 α k ˜ ϕ F ( u ( k ) ) , X nk =1 α k ˜ ϕ F ( u ( k ) ) E F ≥ . Next, we show that χ Ω ut I ; C ◦ Φ ◦ E = φ . Let α ∈ Π i ∈ I C × . If α ≁
1, then χ Ω ut I ; C ◦ Φ ◦ E ( ⊗ i ∈ I α i ) = 0 (asΦ( E ( ⊗ i ∈ I α i )) / ∈ C · λ [1] ∼ \ { } , whether or not [ α ] ∼ ∈ Ω ut I ; C ) and we also have φ ( ⊗ i ∈ I α i ) = 0. If α ∼ ⊗ i ∈ I α i = (Π i ∈ I α i )( ⊗ i ∈ I
1) = (Π i ∈ I α i ) λ [1] ∼ , which implies that χ Ω ut I ; C (Φ( ⊗ i ∈ I α i )) = Π i ∈ I α i = φ ( ⊗ i ∈ I α i ). 15hus, we have χ Ω ut I ; C (cid:0) Φ( h ξ, η i C ⊗ I ut ) (cid:1) = h ξ, η i φ (cid:0) ξ, η ∈ O unit i ∈ I H i (cid:1) . (4.4)As a consequence, if Φ( h ξ, ξ i C ⊗ I ut ) = 0, we know from Lemma 4.2(b) that ξ = 0. This gives an inner-product C [Ω ut I ; C ]-module structure on N unit i ∈ I H i . Furthermore, the required isomorphism ¯ N φ i ∈ I H i ∼ =( ¯ N mod i ∈ I H i ) ¯ ⊗ χ Ωut I ; C C also follows from (4.4).(b) Since N unit i ∈ I H i is a C [ G ]-module (under the identification of C [ G ] with L ω ∈ G N ωi ∈ I C under the ∗ -isomorphism Φ in Corollary 3.4), every element in ( N unit i ∈ I H i ) ⊗ C [ G ] C [ G ] is of the form ξ ⊗ C [ G ] ξ ∈ N unit i ∈ I H i . Moreover, if ξ, η ∈ N unit i ∈ I H , then h ξ ⊗ C [ G ] , η ⊗ C [ G ] i ( ¯ N mod i ∈ I C ) ¯ ⊗ E G C ∗ ( G ) = E G (Φ( h ξ, η i C ⊗ I ut )) = Φ( E G ( h ξ, η i C ⊗ I )) , (4.5)where E G is the linear “truncation” map from C ⊗ I to L ω ∈ G N ωi ∈ I C defined as in part (a). Therefore,Φ( E G ( h· , ·i C ⊗ I )) is a positive Hermitian C [ G ]-form on N unit i ∈ I H i . Obviously, χ Ω ut I ; C = χ G ◦ E G , and by(4.4), χ G (Φ( E G ( h ξ, η i C ⊗ I ))) = χ Ω ut I ; C (Φ( h ξ, η i C ⊗ I ut )) = h ξ, η i φ (cid:0) ξ, η ∈ O unit i ∈ I H (cid:1) . This implies that Φ( E G ( h· , ·i C ⊗ I )) is non-degenerate (since h· , ·i φ is non-degenerate by Lemma 4.2(b)).Now, Equation (4.5) tells us that the Hilbert C ∗ ( G )-module (cid:0) ¯ N mod i ∈ I H i (cid:1) ¯ ⊗ E G C ∗ ( G ) is the completion of N unit i ∈ I H i under the norm induced by the C [ G ]-valued inner-product Φ( E G ( h· , ·i C ⊗ I )). (cid:3) Let { e } be the trivial subgroup of Ω ut I ; C . Since one can identify E { e } with φ (through the argumentof Theorem 4.8(b)), one has ¯ O φ i ∈ I H i ∼ = (cid:0) ¯ O mod i ∈ I H i (cid:1) ¯ ⊗ E { e } C . Remark 4.9 (a) For any subgroup G ⊆ Ω ut I ; C and any faithful state ϕ on C ∗ ( G ) , the Hilbert space (cid:16)(cid:0) ¯ O mod i ∈ I H i (cid:1) ¯ ⊗ E G C ∗ ( G ) (cid:17) ¯ ⊗ ϕ C induces an inner-product on N unit i ∈ I H i .(b) If x ∈ Π i ∈ I C (see Example 2.2(b)), then sup i ∈ I | x i | < ∞ . This, together with the surjectivity of κ C (see Proposition 4.6(a)), tells us that Γ I ; C is a group under the multiplication: [ x ] ≈ · [ y ] ≈ := [ xy ] ≈ (where ( xy ) i := x i y i for any i ∈ I ). Moreover, κ C : Ω ut I ; C = Ω unit I ; C → Γ I ; C is a group homomorphism,which induces a surjective ∗ -homomorphism ¯ κ C : C ∗ (Ω ut I ; C ) → C ∗ (Γ I ; C ) .(c) It is natural to ask whether (cid:0) ( ¯ N mod i ∈ I H i ) ¯ ⊗ ¯ κ C C ∗ (Γ I ; C ) (cid:1) ¯ ⊗ χ Γ I ; C C is isomorphic to Q ⊗ i ∈ I H i canonically.Unfortunately, it is not the case. In fact, for any x, y ∈ Π unit i ∈ I H i , we denote x ≈ T y if there exists α ∈ Π i ∈ I T with α ≈ such that x i = α i y i e.f. It is easy to check that ≈ T is an equivalent relationstanding strictly between ∼ and ≈ in general. Moreover, one has (cid:10) (( ⊗ i ∈ I x i ) ⊗ ¯ κ C ⊗ χ Γ I ; C , (( ⊗ i ∈ I y i ) ⊗ ¯ κ C ⊗ χ Γ I ; C (cid:11) = 0 whenever x T y, while (cid:10) Q ⊗ i ∈ I x i , Q ⊗ i ∈ I y i (cid:11) = 0 whenever x y . Note however, that if all H i = C , then ≈ T and ≈ coincide, and one can show that the two Hilbert spaces (cid:0) ( ¯ N mod i ∈ I C ) ¯ ⊗ κ C C ∗ (Γ I ; C ) (cid:1) ¯ ⊗ χ Γ I ; C C and Q ⊗ i ∈ I C coincide canonically. xample 4.10 (a) It is clear that ¯ N mod i ∈ I C = C ∗ (Ω ut I ; C ) . For any state ϕ on C ∗ (Ω ut I ; C ) , the Hilbert space ( ¯ N mod i ∈ I C ) ¯ ⊗ ϕ C is the GNS construction of ϕ .(b) If G is a subgroup of Ω ut I ; C , we have (cid:0) ¯ O mod i ∈ I C (cid:1) ¯ ⊗ E G C ∗ ( G ) ∼ = ℓ (Ω ut I ; C /G ) ¯ ⊗ C ∗ ( G ) . In fact, let q : Ω ut I ; C → Ω ut I ; C /G be the quotient map and σ : Ω ut I ; C /G → Ω ut I ; C be a cross-section. One hasa bijection from Ω ut I ; C to (Ω ut I ; C /G ) × G sending ω to ( q ( ω ) , σ ( q ( ω ) − ) ω ) . This induces a bijective linearmap ∆ : C [Ω ut I ; C ] → L Ω ut I ; C /G C [ G ] such that for any ω ∈ Ω ut I ; C and ε ∈ Ω ut I ; C /G , ∆( λ ω ) ε := ( λ σ ( ε − ) ω if q ( ω ) = ε otherwise . Let
Φ : N unit i ∈ I C = C ⊗ I ut → C [Ω ut I ; C ] and ϕ : Π i ∈ I T → T be as in Corollary 3.4. Suppose that α, β ∈ Π i ∈ I C × . If [ αβ − ] ∼ does not belong to G , then E G ( h⊗ i ∈ I α i , ⊗ i ∈ I β i i C ⊗ I ) = 0 , and (cid:10) ∆ ◦ Φ (cid:0) ⊗ i ∈ I α i (cid:1) , ∆ ◦ Φ( ⊗ i ∈ I β i (cid:1)(cid:11) L ℓ I ; C /G C [ G ] = 0 . On the other hand, if [ αβ − ] ∼ ∈ G , then (cid:10) ∆ ◦ Φ (cid:0) ⊗ i ∈ I α i (cid:1) , ∆ ◦ Φ( ⊗ i ∈ I β i (cid:1)(cid:11) L ℓ I ; C /G C [ G ] = ϕ ( αβ − ) λ [ αβ − ] ∼ = Φ( ⊗ i ∈ I α i β − i ) = Φ( E G ( h⊗ i ∈ I α i , ⊗ i ∈ I β i i C ⊗ I )) . This shows that ∆ ◦ Φ is an inner-product C [ G ] -module isomorphism from N unit i ∈ I C (equipped with theinner-product C [ G ] -module structure as in Theorem 4.8(b)) onto L ℓ Ω ut I ; C /G C [ G ] . ∗ -representations of ∗ -algebras In this section, { ( A i , H i , Ψ i ) } i ∈ I is a family of unital ∗ -representations, in the sense that A i is a unital ∗ -algebra, H i is a Hilbert space and Ψ i : A i → L ( H i ) is a unital ∗ -homomorphism ( i ∈ I ). Suppose that Ψ := ˜ N i ∈ I Ψ i : N i ∈ I A i → L ( N i ∈ I H i ) is the map as in Proposition 2.3(c). It is easyto check that (cid:10) Ψ ( a ) ξ, η (cid:11) C ⊗ I = (cid:10) ξ, Ψ ( a ∗ ) η (cid:11) C ⊗ I (cid:0) a ∈ O i ∈ I A i ; ξ, η ∈ O i ∈ I H i (cid:1) . (5.1)Furthermore, one has the following result (which is more or less well-known). Proposition 5.1
For any µ ∈ Ω unit I ; H , the map ˜ N i ∈ I Ψ i induces a unital ∗ -representation N µi ∈ I Ψ i : N ei ∈ I A i → L ( ¯ N µi ∈ I H i ) . If, in addition, all Ψ i are injective, then so is N µi ∈ I Ψ i . Consequently, one has a unital ∗ -representation of N ei ∈ I A i on the Hilbert space ¯ N φ i ∈ I H i . However,it seems impossible to extend it to a unital ∗ -representation of N i ∈ I A i on ¯ N φ i ∈ I H i . The biggest ∗ -subalgebra N i ∈ I A i that we can think of, for which such extension is possible, is the subalgebra N ut i ∈ I A i .Example 5.6(a) also tells us that it is probably the right subalgebra to consider.17et us digress a little bit and give another ∗ -representation of N ut i ∈ I A i , which is a direct consequenceof Proposition 5.1, Theorem 3.2(a) and [6, Theorem 4.1] (it is not hard to verify that the representationas given in [6, Theorem 4.1] is injective when N µi ∈ I Ψ i is injective). Note however, that such a ∗ -representation is not canonical since it depends on the choices a cross-section c : Ω ut I ; A → Π i ∈ I U A i (seeRemark 3.3(a)). Corollary 5.2
Suppose that Ψ i are injective. For any µ ∈ Ω unit I ; H , the injection N µi ∈ I Ψ i induces aninjective unital ∗ -representation of N ut i ∈ I A i on ( ¯ N µi ∈ I H i ) ⊗ ℓ (Ω ut I ; A ) . Let us now go back to the discussion of the tensor product type representation of N ut i ∈ I A i . Observethat { Ψ i } i ∈ I induces a canonical action α Ψ : Ω ut I ; A × Ω unit I ; H → Ω unit I ; H . For simplicity, we will denote α Ψ ω ( µ )by ω · µ ( ω ∈ Ω ut I ; A ; µ ∈ Ω unit I ; H ). Theorem 5.3 (a) The map ˜ N i ∈ I Ψ i induces a unital ∗ -representation N φ i ∈ I Ψ i : N ut i ∈ I A i → L (cid:0) ¯ N φ i ∈ I H i (cid:1) .(b) (cid:0) ¯ N φ i ∈ I H i , ( N φ i ∈ I Ψ i ) | N ei ∈ I A i (cid:1) = L µ ∈ Ω unit I ; H (cid:0) ¯ N µi ∈ I H i , N µi ∈ I Ψ i (cid:1) .(c) If all Ψ i are injective, then so is N φ i ∈ I Ψ i . Proof: (a) Set Ψ := ˜ N i ∈ I Ψ i . For any µ ∈ Ω unit I ; H , ω ∈ Ω ut I ; A and a ∈ Π ωi ∈ I A i , it is clear thatΨ ( ⊗ i ∈ I a i ) (cid:0)O µi ∈ I H i (cid:1) ⊆ O ω · µi ∈ I H i . (5.2)Suppose that u ∈ ω and F ∈ F such that a i = u i for i ∈ I \ F . If ξ = J xF ′ ( ξ ) where x ∈ µ , F ′ ∈ F with F ⊆ F ′ and ξ ∈ N i ∈ F ′ H i , then h Ψ ( ⊗ i ∈ I a i ) ξ, Ψ ( ⊗ i ∈ I a i ) ξ i C ⊗ I = (cid:10)(cid:0)O i ∈ F Ψ i ( a i ) ⊗ id (cid:1) ξ , (cid:0)O i ∈ F Ψ i ( a i ) ⊗ id (cid:1) ξ (cid:11) ( ⊗ i ∈ I . This means that Ψ ( ⊗ i ∈ I a i ) is bounded on (cid:0)N unit i ∈ I H i , h· , ·i φ (cid:1) (see Theorem 4.4(a) and Proposition4.1(b)) and produces a unital homomorphism N φ i ∈ I Ψ i : N ut i ∈ I A i → L (cid:0) ¯ N φ i ∈ I H i (cid:1) . Now, Relation (5.1)tells us that N φ i ∈ I Ψ i preserves the involution.(b) This part follows directly from the argument of part (a).(c) Set Ψ := N φ i ∈ I Ψ i . Suppose that v (1) , ..., v ( n ) ∈ Π i ∈ I U A i are mutually inequivalent elements, F ∈ F , b (1) , ..., b ( n ) ∈ N i ∈ F A i and a ( k ) := J v ( k ) F ( b ( k ) ) ( k = 1 , ..., n ) such thatΨ (cid:0)X nk =1 a ( k ) (cid:1) = 0 . By induction, it suffices to show that a (1) = 0.By replacing a ( k ) with ( v (1) ) − a ( k ) if necessary, we may assume that v (1) i = e i ( i ∈ I ). If n = 1, wetake an arbitrary ξ ∈ Π i ∈ I S ( H i ). If n >
1, we claim that there exists ξ ∈ Π i ∈ I S ( H i ) such that ξ ≁ [ V ( k ) i ξ i ] i ∈ I ( k = 2 , ..., n ) , (5.3)where V ( k ) i := Ψ i ( v ( k ) i ). In fact, if k ∈ { , ..., n } and i ∈ I k := { i ∈ I : v ( k ) i = e i } (which is an infinite set),the subset S ( H i ) ∩ ker( V ( k ) i − id H i ) is nowhere dense in S ( H i ) as ker( V ( k ) i − id H i ) is a proper closedsubspace of H i (note that Ψ i is injective). For any i ∈ I , we consider N i := { k ∈ { , ..., n } : i ∈ I k } .By the Baire Category Theorem, for every i ∈ I , one can choose ξ i ∈ S ( H i ) \ S k ∈ N i ker( V ( k ) i − id H i ).Now, ξ := [ ξ i ] i ∈ I will satisfy Relation (5.3). 18ince Ψ( a (1) ) (cid:0)N ξi ∈ I H i (cid:1) ⊆ N ξi ∈ I H i and O ξi ∈ I H i ∩ X nk =2 Ψ( a ( k ) ) (cid:0)O ξi ∈ I H i (cid:1) = { } (because of Theorem 2.5 as well as (5.2) and (5.3)), we have Ψ( a (1) ) | N ξi ∈ I H i = 0. Therefore, part (b)and Proposition 5.1 tells us that a (1) = 0. (cid:3) Remark 5.4 (a) By the argument of Theorem 5.3(c), if all Ψ i are injective, then α Ψ is strongly faithful in the sense that for any finite subset F ⊆ Ω ut I ; A \ { e } , there exists µ ∈ Ω unit I ; H with ω · µ = µ ( ω ∈ F ).(b) If y, z ∈ Π i ∈ I H i are C -sequences and u, v ∈ Π i ∈ I U A i , then y ≈ z if and only if [Ψ i ( u i ) y i ] i ∈ I ≈ [Ψ i ( u i ) z i ] i ∈ I (5.4) and [Ψ i ( u i ) y i ] i ∈ I ≈ [Ψ i ( v i ) y i ] i ∈ I whenever u ∼ v . Thus, { Ψ i } i ∈ I induces an action ˜ α Ψ : Ω ut I ; A × Γ I ; H → Γ I ; H . Again, we write ω · γ for ˜ α Ψ ω ( γ ) ( ω ∈ Ω ut I ; A ; γ ∈ Γ I ; A ). The map κ H in Proposition 4.6(a) is equivariant in the sense that κ H ◦ α Ψ ω = ˜ α Ψ ω ◦ κ H ( ω ∈ Ω ut I ; A ).(c) If all A i are C ∗ -algebras and all Ψ i are irreducible, then α Ψ is transitive. Corollary 5.5
There is a unital ∗ -representation Q ⊗ i ∈ I Ψ i : N ut i ∈ I A i → L ( Q ⊗ i ∈ I H i ) such that forany µ ∈ Ω unit I ; H , ω ∈ Ω ut I ; A and b ∈ N ωi ∈ I A i , (cid:0) Y ⊗ i ∈ I Ψ i (cid:1) ( b ) ◦ ˜Υ µ = ˜Υ ω · µ ◦ (cid:0)O φ i ∈ I Ψ i (cid:1) ( b ) (cid:12)(cid:12) ¯ N µi ∈ I H i , (5.5) where ˜Υ µ is as in Proposition 4.6(b). Proof:
By Proposition 4.6(b), there is a bounded linear map (cid:0) Y ⊗ i ∈ I Ψ i (cid:1) ( b ) : Y ⊗ κ H ( µ ) i ∈ I H i → Y ⊗ ω · κ H ( µ ) i ∈ I H i such that Equality (5.5) holds (see also Remark 5.4(b)). Since we have sup µ ∈ Ω unit I ; H (cid:13)(cid:13) ( N φ i ∈ I Ψ i )( b ) | ¯ N µi ∈ I H i (cid:13)(cid:13) < ∞ (because of Theorem 5.3(a)), we know from Proposition 4.6(a) and [21, Lemma 4.1.1] that ( Q ⊗ i ∈ I Ψ i )( b )induces an element in L ( Q ⊗ i ∈ I H i ), which clearly gives a ∗ -representation. (cid:3) It is natural to ask if Q ⊗ i ∈ I Ψ i is injective if all Ψ i are. However, Q ⊗ i ∈ I Ψ i is never injective ascan be seen in Example 5.6(b) and the discussion following it. Example 5.6
For any i ∈ I , let A i = C = H i and ι i : A i → L ( H i ) be the canonical map. Suppose that Φ , ϕ and ˆΦ are as in Example 4.5.(a) Let Λ : C [Ω ut I ; C ] → L ( ℓ (Ω ut I ; C )) be the left regular representation. For every α, β ∈ Π i ∈ I T , one has (cid:0) ˆΦ ∗ ◦ Λ( λ [ α ] ∼ ) ◦ ˆΦ (cid:1) ( ⊗ i ∈ I β i ) = ϕ ( α − ) ⊗ i ∈ I α i β i = (cid:0)O φ i ∈ I ι i (cid:1) (Φ − ( λ [ α ] ∼ ))( ⊗ i ∈ I β i ) . Consequently, N φ i ∈ I ι i can be identified with Λ (under Φ and ˆΦ ). b) Let α ∈ Π i ∈ I T such that α ≁ but α ≈ with Π i ∈ I α i = 1 . If β ∈ Π i ∈ I C is a C -sequence with k Q ⊗ i ∈ I β i k = 1 , one has k Q ⊗ i ∈ I α i β i k = 1 and (cid:10) Y ⊗ i ∈ I α i β i , Y ⊗ i ∈ I β i (cid:11) = 1 , which imply that Q ⊗ i ∈ I α i β i = Q ⊗ i ∈ I β i . Therefore, ( Q ⊗ i ∈ I ι i )( ⊗ i ∈ I α i ) = id but ⊗ i ∈ I α i = ⊗ i ∈ I .Consequently, Q ⊗ i ∈ I ι i is non-injective (actually, ( Q ⊗ i ∈ I ι i ) ◦ Φ − is non-injective as a group repre-sentation of Ω ut I ; C ). In general, even (cid:0) Q ⊗ i ∈ I Ψ i (cid:1) | N ut i ∈ I C e i is non-injective. In fact, suppose that α is as in the above. Forany C -sequence ξ ∈ Π i ∈ I H i , with k Q ⊗ i ∈ I ξ i k = 1, the same argument as Example 5.6(b) tells us that Q ⊗ i ∈ I α i ξ i = Q ⊗ i ∈ I ξ i . Thus, (cid:0) Q ⊗ i ∈ I Ψ i (cid:1) ( ⊗ i ∈ I e i − ⊗ i ∈ I α i e i ) = 0.On the other hand, by Theorem 5.3 and Corollary 5.5, there exist canonical ∗ -homomorphisms J φ : O ut i ∈ I L ( H i ) → L (cid:0) ¯ O φ i ∈ I H i (cid:1) and J Π : O ut i ∈ I L ( H i ) → L (cid:0) Y ⊗ i ∈ I H i (cid:1) . Notice that J φ is injective but J Π is never injective. Corollary 5.7
Let π i : G i → U L ( H i ) be a unitary representation of a group G i , for each i ∈ I .(a) There exist canonical unitary representations N φ i ∈ I π i and Q ⊗ i ∈ I π i of Π i ∈ I G i on ¯ N φ i ∈ I H i and Q ⊗ i ∈ I H i respectively.(b) If the induced ∗ -representation ˆ π i : C [ G i ] → L ( H i ) is injective for all i ∈ I , the induced ∗ -representation \ N φ i ∈ I π i of C [Π i ∈ I G i ] is also injective. Proof: (a) Let N ut i ∈ I π i := Θ L ( H ) ◦ Π i ∈ I π i : Π i ∈ I G i → N ut i ∈ I L ( H i ). Then O φ i ∈ I π i := J φ ◦ O ut i ∈ I π i and Y ⊗ i ∈ I π i := J Π ◦ O ut i ∈ I π i are the required representations.(b) By Theorem 5.3(c), N φ i ∈ I ˆ π i is injective. As \ N φ i ∈ I π i is the restriction of N φ i ∈ I ˆ π i on C [Π i ∈ I G i ] (seeExample 3.1(a)), it is also injective. (cid:3) Corollary 5.8 Q ⊗ i ∈ I Ψ i is never irreducible, and neither do N φ i ∈ I Ψ i . Proof:
Let τ i : C → A i be the canonical unital map and set ˇΨ i := Ψ i ◦ τ i ( i ∈ I ). Suppose that α, β ∈ Π i ∈ I T with α β and ξ ∈ Π unit i ∈ I H i . Then [ α i ξ i ] i ∈ I [ β i ξ i ] i ∈ I and the two unit vectors (cid:0) Y ⊗ i ∈ I ˇΨ i (cid:1) ( ⊗ i ∈ I α i ) (cid:0) Y ⊗ i ∈ I ξ i (cid:1) and (cid:0) Y ⊗ i ∈ I ˇΨ i (cid:1) ( ⊗ i ∈ I β i ) (cid:0) Y ⊗ i ∈ I ξ i (cid:1) are orthogonal. Consequently, dim ( Q ⊗ i ∈ I ˇΨ i )( C ⊗ I ut ) >
1. As ( Q ⊗ i ∈ I Ψ i ) ◦ ( N i ∈ I τ i ) = Q ⊗ i ∈ I ˇΨ i ,we have ( Q ⊗ i ∈ I ˇΨ i )( C ⊗ I ut ) ⊆ Z (cid:0) ( Q ⊗ i ∈ I Ψ i )( N ut i ∈ I A i ) (cid:1) and Q ⊗ i ∈ I Ψ i is not irreducible. A similar buteasier argument also shows that N φ i ∈ I Ψ i is not irreducible. (cid:3) For any C ∗ -algebra A , we denote by S ( A ) the state space of A and by ( H ρ , π ω , ξ ω ) the GNS con-struction of ω ∈ S ( A ). We would like to consider a natural injective ∗ -representation of N ut i ∈ I A i definedin terms of ( H ω i , π ω i ). 20f ρ ∈ Π i ∈ I S ( A i ) and ˇ ρ is defined asˇ ρ ( a ) := (cid:10)(cid:0)O φ i ∈ I π ρ i (cid:1) ( a )( ⊗ i ∈ I ξ ρ i ) , ( ⊗ i ∈ I ξ ρ i ) (cid:11) (cid:0) a ∈ O ut i ∈ I A i (cid:1) , then the closure of (cid:0)N φ i ∈ I π ρ i (cid:1) ( N ut i ∈ I A i )( ⊗ i ∈ I ξ ρ i ) will coincide with H ˇ ρ := ¯ L ω ∈ Ω ut I ; A ¯ N ω · [ ξ ρ ] ∼ i ∈ I H ρ i ⊆ ¯ N φ i ∈ I H ρ i . We set π ˇ ρ ( a ) := (cid:0)N φ i ∈ I π ρ i (cid:1) ( a ) | H ˇ ρ . Notice that if all ρ i are pure states, then H ˇ ρ = ¯ N φ i ∈ I H ρ i (see Remark 5.4(c)). Corollary 5.9
Let A i be a C ∗ -algebra ( i ∈ I ). The ∗ -representation Ψ A := L ρ ∈ Π i ∈ I S ( A i ) ( H ˇ ρ , π ˇ ρ ) isinjective. Consequently, the ∗ -representation Φ A := L ρ ∈ Π i ∈ I S ( A i ) (cid:0) ¯ N φ i ∈ I H ρ i , N φ i ∈ I π ρ i (cid:1) is also injective. Proof:
Suppose that ( H i , Ψ i ) is a universal ∗ -representation of A i ( i ∈ I ). Let F, u (1) , ..., u ( n ) , b (1) , ..., b ( n ) as well as a (1) , ..., a ( n ) be as in the proof of Theorem 5.3(c) with Ψ A (cid:16)P nk =1 a ( k ) (cid:17) = 0. Again, it sufficesto show that a (1) = 0, and we may assume that u (1) i = e i ( i ∈ I ). If n = 1, we take any x ∈ Π i ∈ I S ( H i ).If n >
1, we take an element x ∈ Π i ∈ I S ( H i ) satisfying x ≁ (cid:2) Ψ i (cid:0) u ( k ) i (cid:1) x i (cid:3) i ∈ I ( k = 2 , ..., n )(the argument of Theorem 5.3(c) ensures its existence). Let us set ρ i ( a ) := h Ψ i ( a ) x i , x i i when i ∈ I \ F ,and pick any ρ i ∈ S ( A i ) when i ∈ F . For every i ∈ I \ F , one may regard (cid:0) H ρ i , π ρ i (cid:1) as a subrepresentationof ( H i , Ψ i ) such that ξ ρ i ∈ H ρ i is identified with x i ∈ H i . Then x can be considered as an element in H ˇ ρ . Since x ≁ (cid:2) π ρ i (cid:0) u ( k ) i (cid:1) x i (cid:3) i ∈ I for all 2 ≤ k ≤ n , the argument of Theorem 5.3(c) tells us that (cid:0)O [ x ] ∼ i ∈ I π ρ i (cid:1) ( a (1) ) η = 0 (cid:0) η ∈ O xi ∈ I H ρ i (cid:1) . Consequently, (cid:0)N i ∈ F π ρ i (cid:1)(cid:0) b (1) (cid:1) = 0 and b (1) = 0 (as ρ i is arbitrary when i ∈ F ). The second statementfollows readily from the first one. (cid:3) Notice also that (cid:0) ¯ N φ i ∈ I H ρ i , N φ i ∈ I π ρ i (cid:1) is in general not a cyclic representation, and ( H ˇ ρ , π ˇ ρ ) can beregarded as a cyclic analogue of it.We end this paper with the following result concerning tensor product of Hilbert algebras. Corollary 5.10
Let { A i } i ∈ I is a family of unital Hilbert algebras (see e.g. [19, Definition VI.1.1]) suchthat k e i k = 1 ( i ∈ I ). Then A := N ut i ∈ I A i is also a unital Hilbert algebra with k⊗ i ∈ I e i k = 1 . Proof:
Note that since k e i k = 1, one has k u i k = 1 for any u i ∈ U A i . Thus, we have N ut i ∈ I A i ⊆ N unit i ∈ I A i ,which gives an inner product h· , ·i A on A . Observe that N ωi ∈ I A i is orthogonal to N ω ′ i ∈ I A i (in terms of h· , ·i A ) whenever ω and ω ′ are distinct elements in Ω ut I ; A . Thus, in order to show the involution on A being an isometry, it suffices to check that k x ∗ k = k x k whenever x ∈ N ωi ∈ I A i and ω ∈ Ω ut I ; A . In fact, forany u ∈ Π i ∈ I U A i , F ∈ F and a ∈ N i ∈ F A i , we have k J uF ( a ) ∗ k = k J u ∗ F ( a ∗ ) k = k a ∗ k = k a k = k J uF ( a ) k , because the involution on N i ∈ F A i is an isometry. Let H i be the completion of A i (with respect to theinner-product) and Ψ i : A i → L ( H i ) be the canonical unital ∗ -representation ( i ∈ I ). Since O φ i ∈ I Ψ i ( a ) b = ab ( a, b ∈ A ) , x ∈ A , one has h xy, z i A = h y, x ∗ z i A ( y, z ∈ A ) and sup k y k≤ k xy k < ∞ . Finally, as A is unital, we see that A is a Hilbert algebra (with k⊗ i ∈ I e i k = 1). (cid:3) Consequently, if all A i are weakly dense unital ∗ -subalgebras of finite von-Neumann algebras, thenso is N ut i ∈ I A i . References [1] H. Araki and Y. Nakagami, A remark on an infinite tensor product of von Neumann algebras, Publ.Res. Inst. Math. Sci. 8 (1972), 363-374.[2] E. B´edos and R. Conti, On infinite tensor products of projective unitary representations, RockyMountain J. Math. 34 (2004), 467-493.[3] W. Bergmann and R. Conti, On infinite tensor products of Hilbert C ∗ -bimodules, Operator algebrasand mathematical physics (Constanta 2001) , Theta, Bucharest (2003), 23-34.[4] B. Blackadar, Infinite tensor products of C ∗ -algebras, Pac. J. Math. 77 (1977), 313-334.[5] H. W. Bruce, Infinite Tensor Products of Commutative Subspace Lattices, Proc. Amer. Math. Soc.103 (1988), 429-437.[6] R. C. Busby and H. A. Smith, Representations of twisted group algebras, Trans. Amer. Math. Soc.149 (1970), 503-537.[7] L. Calabi, Sur les extensions des groupes topologiques, Ann. Mat. Pura Appl. 32 (1951), 295-370.[8] S. Eilenberg and S. MacLane, Group extensions and homology, Ann. of Math. 43 (1942), 757-831.[9] D. E. Evans and Y. Kawahigashi, Quantum Symmetries on Operator Algebras , Clarendon Press,Oxford (1998).[10] R. Floricel, Infinite tensor products of spatial product systems, Infin. Dimens. Anal. QuantumProbab. Relat. Top. 11 (2008), 447-465.[11] Y. P. Huo and C. K. Ng, Some algebraic structures related to the quantum system with infinitedegrees of freedom, Rep. in Math. Phy. 67 (2011), 97-107.[12] T. L. Gill, Infinite Tensor Products of Banach Spaces I, J. Funct. Anal. 30 (1978), 17-35.[13] T. Giordano and G. Skandalis, On infinite tensor products of factors of type I , Erg. Theory Dynam.Sys. 5 (1985), 565-586.[14] A. Guichardet, Tensor products of C ∗ -algebra II: Infinite Tensor products , Aarhus Univ. Lect. NoteSer. 13 (1969).[15] H. Grundling and K.-L. Neeb, Infinite Tensor products of C ( R ): Towards a Group Algebra for R ( N ) , preprint (2010), arXiv:1001.1012v1.[16] S. Power, Infinite tensor products of upper triangular matrix algebras, Math. Scand. 65 (1989),291-307.[17] I. Raeburn, A. Sims and D. P. Williams, Twisted actions and obstructions in group cohomology, C ∗ -algebras (M¨unster, 1999) , Springer (2000), 161-181.2218] E. Stormer, On Infinite Tensor Products of Von Neumann Algebras, Amer. J. Math. 93 (1971),810-818.[19] M. Takesaki, Theory of operator algebras II , Encyclopaedia of Mathematical Sciences 125, Springer-Verlag, Berlin (2003).[20] T. Thiemann and O. Winkler, Gauge Field Theory Coherent States (GCS): IV. Infinite TensorProduct and Thermodynamical Limit, Classical Quantum Gravity 18 (2001), 4997-5053.[21] J. von Neumann, On infinite direct products, Compositio Math. 6 (1937), 1-77.Chi-Keung Ng, Chern Institute of Mathematics, Nankai University, Tianjin 300071, China.