The construction of observable algebra in field algebra of G -spin models determined by a normal subgroup
aa r X i v : . [ m a t h . OA ] M a y The construction of observable algebra in field algebra of G -spinmodels determined by a normal subgroup ∗ Xin Qiaoling, Jiang Lining † School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P. R. China
Abstract : Let G be a finite group and H a normal subgroup. Starting from G -spin models, in whicha non-Abelian field F H w.r.t. H carries an action of the Hopf C ∗ -algebra D ( H ; G ), a subalgebra of thequantum double D ( G ), the concrete construction of the observable algebra A ( H,G ) is given, as D ( H ; G )-invariant subspace. Furthermore, using the iterated twisted tensor product, one can prove that theobservable algebra A ( H,G ) = · · · ⋊ H ⋊ b G ⋊ H ⋊ b G ⋊ H ⋊ · · · , where b G denotes the algebra of complexfunctions on G , and H the group algebra. Keywords : twisted tensor product, field algebra, observable algebra, C ∗ -inductive limitMathematics Subject Classification (2010): 46N50, 46L40, 16T05 Let G be a finite group with a unit e . The G -valued spin configuration on the 2-dimensional squarelattices is the map σ : Z → G with Euclidean action functional S ( σ ) = P ( x,y ) f ( σ − x σ y ) , where the summation runs over the nearest neighbor pairs in Z and f : G → R is a function of thepositive type. This kind of classical statistical systems or the corresponding quantum field theoriesare called G -spin models, see [5, 9, 16]. Such models provide the simplest examples of latticefield theories exhibiting quantum symmetry. Generally, G -spin models with an Abelian group G have a symmetry structure of G × e G , where e G is the Pontryagin dual of G . If G is non-Abelian,the Pontryagin dual loses its meaning, and one usually considers the quantum double D ( G ) of G ,[4, 12]. Here D ( G ) is defined as the crossed product of C ( G ), the algebra of complex functions on G , and group algebra C G with respect to the adjoint action of the latter on the former. Then D ( G )becomes a Hopf *-algebra of finite dimension [2, 10, 14]. As in the traditional quantum field theory,one can define a field algebra F associated with this models, which is a C ∗ -algebra generated by { δ g ( x ) , ρ h ( l ) : g ∈ G, h ∈ G, x ∈ Z , l ∈ Z + } subject to some relations [16]. There is a natural actionof D ( G ) on F so that F becomes a D ( G )-module algebra. Under this action on F , the observable ∗ This work is supported by National Science Foundation of China (10971011,11371222) † E-mail address: [email protected] A G is obtained. In [14], Nill and Szlach´anyi pointed out A G = · · · ⋊ G ⋊ b G ⋊ G ⋊ b G ⋊ G ⋊ · · · ,where the crossed product is taken with respect to the natural left action of the latter factor onthe former one. In this paper, we extend the result to a general situation.Assume that G is a finite group and H is a normal subgroup of G . In our previous paper[17], we define a Hopf C ∗ -algebra D ( H ; G ), which is only a subalgebra of D ( G ). Subsequently, weconstruct an algebra F H in the field algebra F of G -spin models, which is a C ∗ -algebra generated by { δ g ( x ) , ρ h ( l ) : g ∈ G, h ∈ H ; x ∈ Z , l ∈ Z + } , called the field algebra of G -spin models determinedby H . There also exists a natural action of D ( H ; G ) on F H , such that F H is a D ( H ; G )-modulealgebra whereas F is not. Then the observable algebra A ( H,G ) , which is the set of fixed points of F H under the action of D ( H ; G ) is obtained. In Section 2, we point out the concrete constructionof the observable algebra A ( H,G ) .In Section 3, we identify H with the group algebra C H , and b G the set of complex func-tions on G . We firstly construct iterated twisted tensor product algebras of three factors, i.e. H N R , b G N R , H and b G N R , H N R , b G by means of twisting maps R , and R , , and thenby induction build an iterated twisted product of any number of factors A n,m = A n N R n,n +1 A n +1 N R n +1 ,n +2 · · · A m − N R m − ,m A m , where n, m ∈ Z with n < m and A i = (cid:26) H, if i is even b G, if i is odd , which is a C ∗ -algebra of finite dimension.Let n < n ′ and m ′ < m , one can show that A n ′ ,m ′ ⊆ A n,m , and then by the C ∗ -inductive limit of A n,m , we obtain A = ( C ∗ ) lim n Definition 2.2. [ ] The local field F H, loc determined by H is an associative algebra with a unit I generated by { δ g ( x ) , ρ h ( l ) : g ∈ G, h ∈ H ; x ∈ Z , l ∈ Z + } subject to P g ∈ G δ g ( x ) = I = ρ e ( l ) ,δ g ( x ) δ g ( x ) = δ g ,g δ g ( x ) ,ρ h ( l ) ρ h ( l ) = ρ h h ( l ) ,δ g ( x ) δ g ( x ′ ) = δ g ( x ′ ) δ g ( x ) ,ρ h ( l ) δ g ( x ) = (cid:26) δ hg ( x ) ρ h ( l ) , l < x,δ g ( x ) ρ h ( l ) , l > x,ρ h ( l ) ρ h ( l ′ ) = (cid:26) ρ h ( l ′ ) ρ h − h h ( l ) , l > l ′ ,ρ h h h − ( l ′ ) ρ h ( l ) , l < l ′ for x, x ′ ∈ Z ; l, l ′ ∈ Z + and h , h ∈ H, g , g ∈ G .The *-operation is defined on the generators as δ ∗ g ( x ) = δ g ( x ) , ρ ∗ h ( l ) = ρ h − ( l ) and is extended toa involution on F H, loc . In this way, F H, loc becomes a unital *-algebra. Using the C ∗ -inductive limit, F H, loc can be extended to a C ∗ -algebra F H , called the field algebra of G -spin models determinedby a normal subgroup H . There is an action γ of D ( H ; G ) on F H in the following. For x ∈ Z ; l ∈ Z + and h ∈ H, g ∈ G , set( h, g ) δ f ( x ) = δ h,e δ gf ( x ) , ∀ f ∈ G, ( h, g ) ρ t ( l ) = δ h,gtg − ρ h ( l ) , ∀ t ∈ H. One can check that F H is a D ( H ; G )-module algebra [17].Set A ( H,G ) = { F ∈ F H : a ( F ) = ε ( a )( F ) , ∀ a ∈ D ( H ; G ) } . 3e call it an observable algebra related to H in the field algebra F of G -spin models. Furthermore,one can show that A ( H,G ) is a nonzero C ∗ -subalgebra of F H , and A ( H,G ) = { F ∈ F H : z ( H,G ) ( F ) = F } ≡ z ( H,G ) ( F H ) . Now, we will discuss the concrete construction of A ( H,G ) . In order to do this, for g ∈ G , x ∈ Z ,and l ∈ Z + , put v g ( x ) = P h ∈ G ̺ hg − h − ( x − ) δ h ( x ) ̺ hgh − ( x + ) ,w g ( l ) = P h ∈ G δ h ( l − ) δ hg ( l + ) . Theorem 2.1. The observable algebra A ( H,G ) related to H is a unital C ∗ -subalgebra of F H gen-erated by (cid:26) v h ( x ) , w g ( l ) : h ∈ H, g ∈ G, x ∈ Z , l ∈ Z + 12 (cid:27) . Proof. Since z ( G,G ) = z ( G,G ) z ( G, { e } ) , where z ( G, { e } ) is the unique integral of Hopf algebra D ( G, { e } ),we have that γ z ( G,G ) ( F H ) = γ z ( G,G ) z ( G, { e } ) ( F H ) = γ z ( G,G ) ( γ z ( G, { e } ) ( F H )) .γ z ( G, { e } ) is the projection to operators with trivial twist ([16]). Hence, γ z ( G, { e } ) ( F H ) is generated by { δ g ( x ) , v h ( x ) : g ∈ G, h ∈ H, x ∈ Z } . Moreover, we can obtain that γ z ( G,G ) ( F H ) is generated by (cid:26) w g ( l ) , v h ( x ) : g ∈ G, h ∈ H, x ∈ Z , l ∈ Z + 12 (cid:27) , since γ z ( G,G ) ( δ g (1) δ g (2) · · · δ g n ( n )) = 1 | G | w g − g ( 32 ) w g − g ( 52 ) · · · w g n − − g n ( n − 12 ) . Now let us consider γ z ( G,G ) | F H , the restriction of γ z ( G,G ) on F H , and γ z ( H,G ) as projections on F H , we have that γ z ( G,G ) | F H γ z ( H,G ) = γ z ( H,G ) γ z ( G,G ) | F H = γ z ( H,G ) , then γ z ( H,G ) ≤ γ z ( G,G ) | F H , which implies γ z ( H,G ) ( F H ) ⊆ γ z ( G,G ) ( F H ). Again, for g ∈ G, h ∈ H, x ∈ Z , l ∈ Z + ,γ z ( H,G ) ( w g ( l )) = w g ( l ) , γ z ( H,G ) ( v h ( x )) = v h ( x ) . Hence, γ z ( H,G ) ( F H ) is generated by (cid:26) v h ( x ) , w g ( l ) : h ∈ H, g ∈ G, x ∈ Z , l ∈ Z + 12 (cid:27) . The characterization of the observable algebra A ( H,G ) In this section, we identify H with the group algebra C H , and b G the set of complex functions on G . We will construct iterated twisted tensor product algebras of any number of factors. To do this,let us recall briefly the concept of a twisted tensor product of algebras ([3]). Definition 3.1. Let A and B be two unital associative algebras. Suppose that R : B N A → A N B is a linear map such that R ◦ ( id B N m A ) = ( m A N id B ) ◦ ( id A N R ) ◦ ( R N id A ) ,R ◦ ( m B N id A ) = ( id A N m B ) ◦ ( R N id B ) ◦ ( id B N R ) , then m R = ( m A N m B ) ◦ ( id A N R N id B ) is an associative product on A N B . Here, m A and m B denote the multiplication in algebras A and B , respectively. In this case, we call R a twisting map,and ( A N B, m R ) a twisted tensor product of A and B , which has A N B as underlying vectorspace endowed with the multiplication m R , simply denoted by A N R B . Using a Sweedler-typenotation, we denote by R ( b ⊗ a ) = a R ⊗ b R for a ∈ A and b ∈ B .Observe that the multiplication m R in the twisted product A N R B of algebras A and B canbe given in the following form: ( a ⊗ b )( a ′ ⊗ b ′ ) = aa ′ R ⊗ b R b ′ , where, as already mentioned, the Sweedler-type notation for the twisting map R has been used,i.e., R ( b ⊗ a ) = a R ⊗ b R for a ∈ A and b ∈ B . Example 3.1. (1) Consider the usual flip τ : B N A → A N B defined by τ ( b ⊗ a ) = a ⊗ b. It is obvious that τ satisfies all conditions for the twisting map, and then it give rise to the standardtensor product of algebras A N B .(2) Suppose that M is a Hopf algebra over C , and B is a (left) M -module algebra, that is, B isan algebra which is a left M -module such that m · ( ab ) = P ( m ) ( m (1) · a )( m (2) · b ) and m · B = ε ( m )1 B ,for all a, b ∈ B , m ∈ M .Let the map R : M N B → B N M be defined by R ( m ⊗ b ) = P ( m ) ( m (1) · b ) ⊗ m (2) . One can show that R is a twisting map, and then obtain the algebra B N R M , which is B N M as a vector space with multiplication( a ⊗ m )( b ⊗ n ) = P ( m ) a ( m (1) · b ) ⊗ m (2) n. From the definition of the smash product, it is easy to see that B N R M coincides with the ordinarysmash product B M introduced in [6]. 5n order to study the construction of iterated twisted tensor products, we consider three twistedtensor products A N R B , B N R C and A N R C , and the maps T : C N ( A N R B ) → ( A N R B ) N C defined by T = ( id A N R ) ◦ ( R N id B ) and T : ( B N R C ) N A → A N ( B N R C )defined by T = ( R N id C ) ◦ ( id B N R ) associated to R , R and R . The following lemma statesa sufficient and necessary condition ensuring that both T and T are twisting maps. Lemma 3.1. [ ] The following statements are equivalent:(1) T is a twisting map.(2) T is a twisting map.(3) The maps R , R , R satisfy the following compatibility condition (called the hexagon equa-tion): ( id A N R ) ◦ ( R N id B ) ◦ ( id C N R ) = ( R N id C ) ◦ ( id B N R ) ◦ ( R N id A ) . Moreover, if all the three conditions are satisfied, then A N T ( B N R C ) and ( A N R B ) N T C are equal. In this case, we will denote this algebra by A N R B N R C , which is called the iteratedtwisted tensor product.Now, we consider the group algebra C G , endowed with a comultiplication △ ( g ) = g ⊗ g , a counit ε ( g ) = 1, antipode S ( g ) = g − and g ∗ = g − for all g ∈ G , such that C G is a C ∗ -Hopf algebra.From now on, we use G for C G . The dual of G is b G , with △ ( δ g ) = P t ∈ G δ t ⊗ δ t − g , ε ( δ g ) = δ g,e , S ( δ g ) = δ g − and δ ∗ g = δ g for all g ∈ G . There is a natural pairing between G and b G given by h , i : G N b G → C , g ⊗ δ s 7→ h g, δ s i : = δ s ( g ) , h , i : b G N G → C , δ s ⊗ g 7→ h δ s , g i : = δ s ( g ) . Associated to this pairing we have the natural action of G on b G and that on b G on G given by theSweedler’s arrows: g → δ s : = P ( δ s ) δ s (1) h g, δ s (2) i = δ sg − ,δ s → g : = P ( g ) g (1) h δ s , g (2) i = gδ s ( g ) , where △ ( δ s ) = P ( δ s ) δ s (1) ⊗ δ s (2) = P t ∈ G δ t ⊗ δ t − s , and △ ( g ) = P ( g ) g (1) ⊗ g (2) = g ⊗ g .For every n ∈ Z , take A n : = H if n is even and A n : = b G if n is odd, and define the maps: R n, n +1 : A n +1 N A n −→ A n N A n +1 δ g ⊗ h P ( δ g ) ( δ g (1) → h ) ⊗ δ g (2) = h ⊗ δ h − g ,R n − , n : A n N A n − −→ A n − N A n h ⊗ δ g P ( h ) ( h (1) → δ g ) ⊗ h (2) = δ gh − ⊗ h,R i,j : A j N A i −→ A i N A j x j ⊗ x i x i ⊗ x j , if j − i ≥ . 6t is clear that all of them are twisting maps. Proposition 3.1. R , , R , , R , and R , , R , , R , are compatible, respectively. Proof. It suffices to show that R , , R , , R , are compatible. To do this, apply the left-hand sideof the hexagon equation to a generator h ⊗ δ g ⊗ h of A N A N A , we get( id H ⊗ R , ) ◦ ( R , ⊗ id b G ) ◦ ( id H ⊗ R , )( h ⊗ δ g ⊗ h )= ( id H ⊗ R , ) ◦ ( R , ⊗ id b G )( h ⊗ h ⊗ δ h − g )= ( id H ⊗ R , )( h ⊗ h ⊗ δ h − g )= h ⊗ δ h − gh − ⊗ h . On the other hand, for the right hand side we obtain that( R , ⊗ id H ) ◦ ( id b G ⊗ R , ) ◦ ( R , ⊗ id H )( h ⊗ δ g ⊗ h )= ( R , ⊗ id H ) ◦ ( id b G ⊗ R , )( δ gh − ⊗ h ⊗ h )= ( R , ⊗ id H )( δ gh − ⊗ h ⊗ h )= h ⊗ δ h − gh − ⊗ h . Now, we have shown R , , R , and R , are compatible. Similarly, one can prove R , , R , and R , are compatible.It follows from Proposition 3.1 and Lemma 3.1 that one can construct the algebras A N R , A N R , A and A N R , A N R , A , in which the multiplications can be given, respectively, by the formulas( h ⊗ δ g ⊗ f )( h ⊗ δ g ⊗ f ) = h h ⊗ δ h − g δ g f − ⊗ f f , ( δ g ⊗ h ⊗ δ s )( δ g ⊗ h ⊗ δ s ) = δ g δ g h − ⊗ h h ⊗ δ h − s δ s . Moreover, we can define a map θ : A N R , A N R , A −→ A N R , A N R , A h ⊗ δ g ⊗ h −→ h − ⊗ δ h gh ⊗ h − . Then θ satisfies the following properties: for any x, y ∈ A N R , A N R , A , θ ( θ ( x )) = x, θ ( xy ) = θ ( y ) θ ( x ) . Thus A N R , A N R , A is a *-algebra by means of( h ⊗ δ g ⊗ h ) ∗ = θ ( h ⊗ δ g ⊗ h )for h ⊗ δ g ⊗ h ∈ A N R , A N R , A .Similarly, for δ g ⊗ h ⊗ δ s ∈ A N R , A N R , A , set( δ g ⊗ h ⊗ δ s ) ∗ = δ gh ⊗ h − ⊗ δ hs . Then A N R , A N R , A is also a *-algebra.Moreover, we have the following proposition.7 roposition 3.2. A N R , A N R , A and A N R , A N R , A are C ∗ -algebras. Proof. Let H = L ( b G, h ) be a Hilbert space with inner product h ϕ, φ i = | G | P g ∈ G ϕ ( g ) φ ( g ) , and H , , H ⊗ H .Consider the map π , : A N R , A N R , A → End H , be given by( π , ( g (0) ) ψ )( g , g ) = ψ ( g g (0) , g )( π , ( g (2) ) ψ )( g , g ) = ψ ( g , g g (2) )( π , ( δ g ) ψ )( g , g ) = δ g ( g − g ) ψ ( g , g ) , where δ g ∈ A , g (0) ∈ A , g (2) ∈ A and g , g ∈ G . One can show that ( π , , H , ) is a faithful*-representation of A , .For h ⊗ δ g ⊗ h ∈ A N R , A N R , A , set k h ⊗ δ g ⊗ h k = k π , ( h ⊗ δ g ⊗ h ) k , then ( A N R , A N R , A , k · k ) is a C ∗ -algebra of finite dimension.As to A , , we define faithful *-representation ( π , , H , ) of A , , where H , = e H⊗ e H with e H = L ( G, δ e ), δ e ∈ b G being the Haar measure on the group algebra G . Hence, A N R , A N R , A is a C ∗ -algebra.In the following, by induction, we will construct an iterated twisted tensor product of anynumber of factors. Proposition 3.3. The three maps R i,j , R j,k and R i,k are compatible for any i < j < k . Proof. Let us distinguish among several cases:If j − i ≥ k − j ≥ 2, all three maps are just usual flips, and thus R i,j , R j,k and R i,k arecompatible.If j − i = 1 and k − j ≥ 2, then we have that both R i,k and R j,k are usual flips. Hence, R i,j , R j,k and R i,k are compatible. Indeed, for any x ⊗ y ⊗ z ∈ A k N A j N A i , we have( id A i ⊗ R j,k ) ◦ ( R i,k ⊗ id A j ) ◦ ( id A k ⊗ R i,j )( x ⊗ y ⊗ z )= ( id A i ⊗ R j,k ) ◦ ( R i,k ⊗ id A j )( x ⊗ z Ri,j ⊗ y Ri,j )= ( id A i ⊗ R j,k )( z Ri,j ⊗ x ⊗ y Ri,j )= z Ri,j ⊗ y Ri,j ⊗ x, and ( R i,j ⊗ id A k ) ◦ ( id A j ⊗ R i,k ) ◦ ( R j,k ⊗ id A i )( x ⊗ y ⊗ z )= ( R i,j ⊗ id A k ) ◦ ( id A j ⊗ R i,k )( y ⊗ x ⊗ z )= ( R i,j ⊗ id A k )( y ⊗ z ⊗ x )= z Ri,j ⊗ y Ri,j ⊗ x. R i,j , R j,k and R i,k are compatible for j − i ≥ k − j = 1.If j − i = 1 and k − j = 1, then R i,j , R j,k and R i,k are compatible, the proof of which is thesame as that of R , , R , , R , and R , , R , , R , are compatible.Assume that we have n algebras A , A , · · · , A n with a twisting map R i,j : A j N A i → A i N A j for any i < j , and such that for every i < j < k the maps R i,j , R j,k and R i,k are compatible. Definenow for any i < n − T in − ,n : ( A n − N R n − ,n A n ) N A i → A i N ( A n − N R n − ,n A n )given by T in − ,n = ( R i,n − ⊗ A n ) ◦ ( A n − ⊗ R i,n ), which are twisting maps for each i ∈ Z , asthe maps R i,n − , R i,n and R n − ,n are compatible (Lemma 3.1). Moreover, for every i < j 1, the compatibility for R i,j , R i,n − , R j,n − and R i,j , R i,n , R j,n implies that for R i,j , T in − ,n and T jn − ,n . Hence, we can apply the induction hypothesis to the n − A , A , · · · , A n − and A n − N R n − ,n A n , and can construct the twisted product of these n − C ∗ -algebra A N R , · · · N R n − ,n − A n − N T n − n − ,n ( A n − N R n − ,n A n ) . Notice that the maps R n − ,n − , R n − ,n and R n − ,n − are compatible, which implies that A n − N T n − n − ,n ( A n − N R n − ,n A n ) = ( A n − N R n − ,n − A n − ) N T nn − ,n − A n . Now, we get the C ∗ -algebra A N R , · · · N R n − ,n − A n − N R n − ,n − A n − N R n − ,n A n . In particular, for any n, m ∈ Z with n < m , we can define the C ∗ -algebras of finite dimension A n,m : = A n N R n,n +1 A n +1 N R n +1 ,n +2 · · · A m − N R m − ,m A m . If n < n ′ and m ′ < m , then one can check that A n ′ ,m ′ ⊆ A n,m , with the map i : A n ′ ,m ′ → A n,m defined by i ( x n ′ ⊗ x n ′ +1 ⊗ · · · ⊗ x m ′ ) = 1 A n ⊗ · · · ⊗ A n ′− ⊗ x n ′ ⊗ x n ′ +1 ⊗ · · · ⊗ x m ′ ⊗ A m ′ +1 ⊗ · · · ⊗ A m is a C ∗ -algebra homomorphism preserving the norm, where we use 1 A i for the identity in A i for i ∈ Z .We write A for the C ∗ -inductive limit of A n,m with n, m ∈ Z : A : = S n For a finite interval Λ, let A H (Λ) = (cid:10) v h ( x ) , w g ( l ) : h ∈ H, g ∈ G, x ∈ Λ ∩ Z , l ∈ Λ ∩ ( Z + ) (cid:11) . It follows from Theorem 2.1 that A ( H,G ) = S Λ A H (Λ) , where the union is taken over finite intervals Λ and the bar denotes uniform closure.Consider the mapΦ , : A , = H N R , b G N R , H −→ A H (Λ , ) = h v h (0) , w g ( ) , v f (1) : h, f ∈ H, g ∈ G i defined on a generator h ⊗ δ g ⊗ f of H N R , b G N R , H byΦ , ( h ⊗ δ g ⊗ f ) = v h (0) w g ( ) v f (1)Observe that v h (0) w g ( ) v f (1) v h (0) w g ( ) v f (1) = v h (0) w g ( ) v h (0) v f (1) w g ( ) v f (1)= v h (0) v h (0) w h − g ( ) v f (1) w g ( ) v f (1)= v h h (0) w h − g ( ) w g f − ( ) v f (1) v f (1)= v h h (0) w h − g ( ) w g f − ( ) v f f (1) , where we use the commutation relations of the v, w generators v h ( x ) v h ( x ) = v h h ( x ) ,w g ( l ) w g ( l ) = δ g ,g w g ( l ) ,v h ( x ) w g ( x + ) = w hg ( x + ) v h ( x ) ,v h ( x ) w g ( x − ) = w gh − ( x − ) v h ( x ) , other pairs of v and / or w fields commute.On the other hand,( h ⊗ δ g ⊗ f )( h ⊗ δ g ⊗ f ) = h h ⊗ δ h − g δ g f − ⊗ f f . Thus, Φ , is an algebra homomorphism. And we have that(Φ , ( h ⊗ δ g ⊗ f )) ∗ = ( v h (0) w g ( ) v f (1)) ∗ = v f − (1) w g ( ) v h − (0)= v h − (0) v f − (1) w hg ( )= v h − (0) w hgf ( ) v f − (1)= Φ , ( h − ⊗ δ hgf ⊗ f − )= Φ , (( h ⊗ δ g ⊗ f ) ∗ ) , where we use the properties: w g ( l ) is a self-adjoint projection in A H (Λ), and v h ( x ) is a unitaryelement in A H (Λ) for any l ∈ Λ ∩ ( Z + ) , x ∈ Λ ∩ Z . Hence, Φ , is a C ∗ -homomorphism. ByTheorem 2.1.7 in [13], we know Φ , is norm-decreasing.10lso, Φ , is bijective, which together with the open mapping theorem yields that Φ , is a C ∗ -isomorphism between C ∗ -algebras A , and A H (Λ , ).By induction, we can build a C ∗ -isomorphism Φ − n,m between A − n, m and A H (Λ − n,m ), for any n, m ∈ Z .Now, because of the last relation we can define a C ∗ -isomorphismΦ : S n From Theorem 3.1, one can see that for a normal subgroup H of G the observablealgebra related to H in the field algebra F of G -spin models A ( H,G ) can be defined as A ( H,G ) = · · · H N R − , − b G N R − , H N R , b G N R , H N R , b G · · · . In particular, take G as a normal subgroup of G , then we have the observable algebra A G ([14])can also be expressed as A G = · · · G N R − , − b G N R − , G N R , b G N R , G N R , b G · · · . What is more, we can get A ( H,G ) ⊆ A G , from the above expressions of A ( H,G ) and A G , which isdifferent from A G ⊆ A ( G,H ) ([8]). Remark 3.2. Notice that the linear map ϕ : G N b H → b H defined naturally by ϕ g ( δ h ) = g → δ h = P t ∈ H δ t δ t − h ( g ) = ( δ hg − , if g ∈ H , if g ∈ G/H is not a left action of G on b H . Thus, b H N R , G can not be defined for any subgroup H of G , andthen · · · G N R − , − b H N R − , G N R , b H N R , G N R , b H · · · can not be defined naturally. However, the observable algebra A ( G,H ) in the field algebra F is welldefined [8], which can be obtained as the fixed point algebra A ( G,H ) = F D ( G ; H ) ≡ { F ∈ F : a ( F ) = ε ( a ) F, ∀ a ∈ D ( G ; H ) } . Here D ( G ; H ) denotes the crossed product of C ( G ) and C G with respect to the adjoint to theaction of the latter on the former, ε is the counit of D ( G ; H ), and F is the field algebra of G -spinmodels. 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