A class of AF-algebras up to universal UHF-algebra stability
aa r X i v : . [ m a t h . OA ] S e p A CLASS OF AF-ALGEBRAS UP TO UNIVERSAL UHF-ALGEBRASTABILITY
SAEED GHASEMI
Abstract.
We will show that separable unital AF-algebras whose Bratteli diagrams do not allowconverging two nodes into one node, can be classified up to the tensor product with the universalUHF-algebra Q only by their trace spaces. That is, if A and B are such AF-algebras, then T ( A ) = T ( B ) if and only if A ⊗ Q ∼ = B ⊗ Q . Introduction
UHF-algebras were first studied and classified by Glimm [4]. To any UHF-algebra one assigns aunique a “supernatural number” (and vise versa) which is a complete isomorphism invariant. Moregenerally, later separable AF-algebras were classified by Elliott [2] using their K -groups. In the uni-tal case, Elliott’s theorem assigns to a unital separable AF-algebra A a unique “dimension groupwith order-unit” h K ( A ) , K ( A ) + , [1 A ] i (and vice versa), as a complete isomorphism invariant.Since then, the aim of Elliott’s classification program has been to classify more classes of separablenuclear C*-algebras. The so called “strongly self-absorbing” C*-algebras play a particularly im-portant role in the classification program. In fact, the classification results are usually obtainedup to tensoring with one of the C*-algebras in the short list of known strongly self-absorbing C*-algebras (refer to [9] or [8]). In the classification of separable nuclear C*-algebras in addition tothe K-theoretic data, the “trace space” of a C*-algebra, which is tightly related to the state spaceof the K -group, is also usually used as an invariant. In these notes we show that a fairly richclass of AF-algebras can be classified up to tensoring with the “universal UHF-algebra” Q (up to Q -stability) using only the trace space. The universal UHF-algebra is strongly self-absorbing andis the UHF-algebras with K -group isomorphic to the additional group of all rational numbers Q with the usual ordering and 1 as order-unit.The AF-algebras that we consider are unital and their Bratteli diagrams do not allow edges ofthe form • •• Equivalently, these separable unital AF-algebras are precisely the ones that arise as the limit ofa direct sequence of finite-dimensional C*-algebras where the connecting maps have exactly onenon-zero entry in each row in their representing matrices. We will denote this class of unital AF-algebras by −→ D . It contains all the UHF-algebras (for which Theorem 1.1 is a tautology), all thecommutative C*-algebras C ( X ) for compact, second countable and totally disconnected spaces X This research is supported by the GAˇCR project 19-05271Y and RVO: 67985840. (because every such X is the inverse limit of an inverse sequence of finite sets and surjective maps).Also −→ D is closed under direct sums and tensor products.For a unital C*-algebra A , the space of all tracial states of A is denoted by T ( A ). Since Q has aunique tracial state, it is known and easy to check that T ( A ) ∼ = T ( A ⊗ Q ) for any unital C*-algebra A . In particular, for unital AF-algebras A , B , if A ⊗ Q ∼ = B ⊗ Q then we always have T ( A ) ∼ = T ( B )(i.e. they are affinely homeomorphic simplices). Theorem 1.1.
Suppose A and B are unital AF-algebras in −→ D . Then T ( A ) ∼ = T ( B ) if and only if A ⊗ Q ∼ = B ⊗ Q . For any unital AF-algebra A , its trace space T ( A ) is affinely homeomorphic to “state space” ofthe dimension group with order-unit h K ( A ) , K ( A ) + , [1 A ] i via the map which sends τ to K ( τ )(see [8, Proposition 1.5.5]). Therefore Theorem 1.1 (as well as the whole theory of unital AF-algebras) can be restated in the language of dimension groups with order-units and their statespaces (Theorem 2.7). Preliminaries
An (partially) ordered abelian group G with a fixed order-unit u is denoted by h G, u i . Suppose G , G , . . . G k are ordered abelian groups. The tensor product G = G ⊗ G ⊗ · · · ⊗ G k , as definedin [6], is an ordered abelian group with the positive cone G + defined as the collection of all finitesums of the elements of the set { x ⊗ · · · ⊗ x k : x i ∈ G + i } . If u i is an order-unit of G i then u ⊗ · · · ⊗ u k is an order-unit of G ([6, Lemma 2.4]). Tensorproducts of positive homomorphisms are positive homomorphisms and if G and H are respectivedirect limits of sequences ( G i , α ji ) and ( H i , β ji ) of ordered abelian groups, then G ⊗ H is the directlimit of the sequence ( G i ⊗ H i , α ji ⊗ β ji ) ([6, Lemma 2.2]).We always consider the abelian group of rational numbers Q with its usual ordering and the order-unit 1. In the following write simply Q instead of h Q , i . There is a one-to-one correspondencebetween the subgroups of Q which contain 1 and the supernatural numbers (see [7, Proposition7.4.3]). For a supernatural number n we denote the corresponding subgroup of Q by Q n . Mul-tiplication of supernatural numbers is defined naturally as an extension of the multiplication ofthe natural numbers. If m is also a supernatural number then Q n ⊗ Q m ∼ = Q nm , i.e., they areisomorphic as ordered groups with order-units. In particular, Q n ⊗ Q ∼ = Q for every supernaturalnumber n .Recall that an ordered abelian group is archimedean if nx ≥ y for every n ∈ N implies that x ≤ Proposition 1.2.
Suppose G is an ordered abelian group and n is a supernatural number. Then G is archimedean if and only if G ⊗ Q n is archimedean.Proof. The map x → x ⊗ G into G ⊗ Q n . Therefore if G is notarchimedean it is clear that G ⊗ Q n is not archimedean. Assume G is archimedean. Suppose x, y ∈ G ⊗ Q n and nx ≤ y for every natural number n . Suppose x = P ri =1 x i ⊗ a i b i and y = P sj =1 y j ⊗ c j d j .By changing the sign of the integers a i and c j , if necessary, we may assume b i and d j are positivenatural numbers for every i ≤ r and j ≤ s . Let b = b . . . b r > d = d . . . d s > k i = a i b/b i ∈ N CLASS OF AF-ALGEBRAS UP TO UNIVERSAL UHF-ALGEBRA STABILITY 3 and l j = c j d/d j ∈ N . Then we have x = ( P i k i x i ) ⊗ b and y = ( P j l j y j ) ⊗ d . For every ny − nx = ( X j l j y j ) ⊗ d − n ( X i k i x i ) ⊗ b = ( b X j l j y j − nd X i k i x i ) ⊗ bd ≥ . Since bd > b P j l j y j − nd P i k i x i ≥ n . Since G is archimedean thisimplies that d P i k i x i ≤
0. As d is a positive integer, we have P i k i x i ≤ x = ( P i k i x i ) ⊗ b ≤ (cid:3) For any non-negative integer r , the ordered abelian group Z r equipped with the positive cone( Z r ) + = { ( x , x , . . . , x r ) ∈ Z r : x i ≥ } . is called a simplicial group . A (not necessarily countable) partially ordered abelian group h G, G + i iscalled a dimension group if it is directed, unperforated interpolation group. We refer the reader to[5] for definitions and more on dimension groups. By a well-known result of Effros, Handelman andShen [1] any dimension group (with order-unit) is isomorphic to a direct limit of a direct systemof simplicial groups (with order-units) and positive (order-unit preserving) homomorphisms, inthe category of ordered abelian groups (with order-units). In particular, a countable dimensiongroup h G, G + , u i with order-unit u (we would simply write h G, u i in the following) is isomorphicto the direct limit of a sequence simplicial groups and normalized (order-unit preserving) positivehomomorphisms h Z r , u i α −→ h Z r , u i α −→ h Z r , u i α −→ . . . h G, u i . Tensor product of two dimension groups with order-units is again a dimension group with order-unit. In fact, their tensor products correspond to the tensor products of the corresponding AF-algebras. That is, if A and B are AF-algebras, then h K ( A ⊗ B ) , [1] A⊗B i ∼ = h K ( A ) , [1] A i ⊗h K ( B ) , [1] B i ([6, Proposition 3.4]).We are only concerned here with countable dimension groups and therefore by a dimension groupwe mean a countable one. Clearly Q and its subgroups (with the inherited ordering) are dimensiongroups. Every ordered subgroup of Q containing 1 is isomorphic as an ordered abelian group withorder-unit to the limit of a sequence of simplicial groups and normalized positive homomorphisms h Z , n i → h Z , n i → . . . for some sequence { n i } of natural numbers, such that n i | n i +1 for each i . Wesay such a sequence { n i } of natural numbers is associated to the supernatural number n if the limitof the directed sequence h Z , n i → h Z , n i → . . . is isomorphic to Q n . Note that this is differentfrom the usual unique representation of n as extended powers of prime numbers, since it is not evenuniquely associated to n . However, it is convenient to use for our purpose. For a natural number m and x = ( x , x , . . . , x r ) ∈ Z r we write mx for ( mx , mx , . . . , mx r ). If h G, u i = lim −→ ( Z r i , u i , α ji ) isa dimension group with order-unit and { n i } is any sequence associated to the supernatural number n , then h G, u i ⊗ Q n = lim −→ ( Z r i , n i u i , n j n i α ji ) (cf. [6, Lemma 2.2]). Definition 1.3.
Define the equivalence relation ∼ Q on the set of all dimension groups with order-units by h G, u i ∼ Q h H, v i ⇔ h
G, u i ⊗ Q ∼ = h H, v i ⊗ Q . Equivalently, h G, u i ∼ Q h H, v i if and only if there are supernatural numbers n, m such that h G, u i ⊗ Q m ∼ = h H, v i ⊗ Q n as ordered abelian groups with order-units. SAEED GHASEMI Dimension groups with positive non-mixing connecting maps
We say a homomorphism α : Z r → Z s is non-mixing if α ( x , . . . , x r ) = ( k x i , . . . , k s x i s ) where i j ∈ { , . . . r } for each j ≤ s . A non-mixing homomorphism is positive if and only if each k j is apositive integer. Let −→ D denote the class of all dimension groups G that arise as the limit of a directsequence of simplicial groups and positive non-mixing homomorphisms ( G i , α ji ). Since positivenon-mixing homomorphisms send order-units to order-units, the image of each order-unit of each G i in G is an order-unit of G . Note that positive non-mixing homomorphisms are not necessarilyembeddings. A dimension groups that is inductive limit of a sequence of simplicial groups withinjective connecting maps is called ultrasimplicial [3]. Not all dimension groups are ultrasimplicial[3, Example 2.7]. Proposition 2.1.
Suppose G is a dimension group in −→ D (1) G is ultrasimplicial.(2) G is archimedean.Proof. (1) Suppose G is the limit of a sequence ( G i , α ji ) of simplicial groups and positive non-mixinghomomorphisms. Restricting each α i +1 i to G i / ker( α i +1 i ) and projecting its image to G i +1 / ker( α i +2 i +1 )yields a sequence with injective maps with the same limit G .(2) By (1) we can assume α ji are injective positive non-mixing homomorphisms. If z ∈ G i and z α ji ( z ) j ≥ i and therefore α ∞ i ( z )
0. Suppose x, y ∈ G are such that nx ≤ y for every n ∈ N , and x i , y i are such that α ∞ i ( x i ) = x and α ∞ i ( y i ) = y . Then nx − y ≤ nx i − y i ≤ n . Since G i is archimedean we have x i ≤ x ≤ (cid:3) Next we will show that for dimension groups in −→ D changing the order-unit results in ∼ Q -equivalent (Definition 1.3) dimension groups. First we need the following fairly trivial lemma. Lemma 2.2.
Suppose α : Z r → Z s is a positive non-mixing homomorphism of simplicial groupsand γ : Z r → Z r is a homomorphism defined by γ ( x , . . . , x r ) = ( l x , . . . , l r x r ) for positive integers l , . . . l r . Then there are natural numbers n , positive integers l ′ , . . . , l ′ s such that if η : Z s → Z s isdefined by η ( x , . . . , x s ) = ( l ′ x , . . . , l ′ s x s ) , then η ◦ α ◦ γ = nα , i.e., the following diagram commutes. Z r Z s Z r Z sγ nαα η Proof.
Suppose α ( x , . . . , x r ) = ( k x i , . . . , k s x i s ) for i j ∈ { , . . . r } . Let n = k . . . k s l i . . . l i s anddefine η ( x , . . . , x s ) = ( nk l i x , . . . , nk s l is x s ). Then η ◦ α ◦ γ = nα . (cid:3) Proposition 2.3. If G ∈ −→ D and u, w are order-units of G then h G, u i ∼ Q h G, w i .Proof. Suppose G = lim −→ ( G i , α ji ) where ( G i , α ji ) is a direct sequence of simplicial groups and positivenon-mixing homomorphisms. Without loss of generality assume that there are u , w ∈ G suchthat α ∞ ( u ) = u and α ∞ ( w ) = w . Set u i = α i ( u ) and w i = α i ( w ). Since each u i and w i areorder-units of G i (because α i is order-unit preserving) the convex subgroups (ideals) of G i generatedby u i and w i are all of G i . Therefore we have h G, u i = lim −→ ( G i , u i , α ji ) and h G, w i = lim −→ ( G i , w i , α ji );cf. [5, Corollary 3.18]. CLASS OF AF-ALGEBRAS UP TO UNIVERSAL UHF-ALGEBRA STABILITY 5
Suppose G = Z r and let m = u . . . u r where u = ( u , . . . , u r ). We can find γ : h G , u i → h G , m w i given by ( x . . . , x r ) → ( k x , . . . , k r x r ) where k i = u ...u r u i w i . Start-ing wit γ use Lemma 2.2 recursively to find sequences of natural numbers { n i } and { m i } andnormalized positive homomorphism γ i and η i such that the following diagram commutes h G , u i h G , n u i h G , n u i h G , n n u i . . . h G , m w i h G , m w i h G , m m w i h G , m m w i . . . n α γ α n α γ α m α η α η This intertwining provides an isomorphism between h G, u i ⊗ Q n and h G, w i ⊗ Q m , where n and m are the supernatural numbers associated to the sequences { n i . . . n } i ∈ N and { m i . . . m } i ∈ N ,respectively. (cid:3) Corollary 2.4.
Suppose
G, H ∈ −→ D and h G, w i and h H, z i are isomorphic for some order-units w, z . Then for every u order-unit of G and v order-unit of H we have h G, u i ∼ Q h H, v i .Proof. By Proposition 2.3 and our assumption we have h G, u i ∼ Q h G, w i ∼ = h H, z i ∼ Q h H, v i . (cid:3) State spaces.
Let h G, u i be an ordered abelian group with order-unit u . The set S ( G, u ) of allstates on h G, u i , called the state space of h G, u i , when equipped with the relative topology endowedfrom R G , is a compact convex subset of the locally convex topological vector space R G . Given anormalized positive homomorphism α : h G, u i → h
H, v i , one associates an affine and continuousmap S ( α ) : S ( H, v ) → S ( G, u ) defined by S ( α )( s ) = s ◦ α , for every s ∈ S ( H, v ). In fact, S isa contravariant and continuous functor from the category of ordered abelian groups with order-unit into the category of compact convex sets. If h G, u i is a dimension group with an order-unit,then S ( G, u ) is a Choquet simplex ([5, Theorem 10.17]). Conversely, a Choquet simplex is affinelyhomeomorphic to the state space of a (countable) dimension group with an order-unit ([5, Corollary14.9]).If G is a ordered abelian group and u , u ∈ G + are order-units of G , then the map φ : S ( G, u ) → S ( G, u ) defined by φ ( s )( x ) = s ( u ) s ( x ) is a (not necessarily affine) homeomorphism. In fact, thereare dimension groups G with order-units u and v such that S ( G, u ) is not affinely homeomorphicto S ( G, v ) (see [5, Example 6.18]).
Proposition 2.5.
For every dimension group with order-unit h G, u i and supernatural number n we have S ( G, u ) ∼ = S ( h G, u i ⊗ Q n ) . If h G, u i ∼ Q h H, v i , then S ( G, u ) ∼ = S ( H, v ) .Proof. If h G, u i = lim −→ ( G i , u i , α ji ) where each h G i , u i i is a simplicial group with order-unit, then forany supernatural number n and an associated sequence { n i } , we have h G, u i⊗ Q n = lim −→ ( G i , n i u i , n j n i α ji ).The following diagram commutes. h G , u i h G , u i h G , u i . . . h G, u ih G , n u i h G , n u i h G , n u i . . . h G, u i ⊗ Q nα n α n α n n n α n n α n n α After applying the state functor S to the above diagram we get a commuting diagram where theinverse limit of the first row is S ( G, u ) and the inverse limit of the second row is S ( h G, u i ⊗ Q n )(cf. [5, Proposition 6.14]). Clearly S ( n i ) : S ( G i , n i u i ) → S ( G i , u i ) is an affine homeomorphism andtherefore S ( G, u ) ∼ = S ( h G, u i ⊗ Q n ). The second statement follows immediately. (cid:3) SAEED GHASEMI
Corollary 2.6.
Suppose u and v are order-units of a dimension group G ∈ −→ D . Then S ( G, u ) isaffinely homeomorphic to S ( G, v ) .Proof. Follows from Proposition 2.3 and Proposition 2.5. (cid:3)
In general S ( G, u ) ∼ = S ( H, v ) does not imply h G, u i ∼ Q h H, v i , not even in the finite-dimensionalcase. For example, simply let h G, u i denote the simplicial group h Z , i and h H, v i denote thegroup Q with the strict ordering ≪ and order-unit 1. Note that h H, v i is a dimension groupsince it is a directed and unperforated interpolation group. Clearly S ( G, u ) ∼ = S ( H, v ) ∼ = ∆ but h G, u i 6∼ Q h H, v i , since the left side is archimedean while the right side is not, and tensoring with Q does not change the state of being archimedean (Proposition 1.2). However, we will show thatfor dimension groups G, H in −→ D we have S ( G, u ) ∼ = S ( H, v ) implies that h G, u i ∼ Q h H, v i .Given a compact convex set K in a linear topological space Aff( K ) denotes the collection of allaffine continuous real-valued functions on K . Let 1 K denote the constant function with value 1 on K . Equipped with the pointwise ordering, h Aff( K ) , k i is an ordered real vector space with order-unit. With the supremum norm Aff( K ), as a norm-closed subspace of C ( K, R ), is an “ordered realBanach space”. A compact convex K is a Choquet simplex if and only if Aff( K ) is a (uncountable)dimension group (cf. [5, Theorem 11.4]).For any affine continuous map ν : K → L between compact convex sets, the map A ( ν ) : h Aff( L ) , L i → h Aff( K ) , K i defined by A ( ν )( f ) = f ◦ ν , for f ∈ Aff( L ) is an order-unit preservingpositive homomorphism.2.2. The main result.
Suppose h G, u i is an ordered abelian group with order-unit. Then thereis a natural normalized positive homomorphism ϕ : h G, u i → h
Aff( S ( G, u )) , i defined by x → b x where b x ( s ) = s ( x ) for any state s . The map ϕ is called the natural affine representation of h G, u i and it is an embedding if and only if G is archimedean (cf. [5, Theorem 7.7]).Note that Aff( S ( Z r , u )) is isomorphic to h R r , i as ordered real Banach space. Any normalizedpositive homomorphism β : h R r , i → h R s , i is of the form β ( x , . . . , x r ) = ( r X i a i x i , . . . , r X i a is x i )where a ij ∈ R and P ri a ij = 1 for every 1 ≤ j ≤ s . We say β is rational if each a ij is a rationalnumber. Theorem 2.7.
Suppose
G, H are in −→ D and u ∈ G and v ∈ H are order-units. Then S ( G, u ) ∼ = S ( H, v ) if and only if h G, u i ∼ Q h H, v i .Proof. We only need to prove the forward direction. First assume S ( G, u ) and S ( H, v ) are finite-dimensional simplices. If S ( G, u ) ∼ = S ( H, v ) then Aff( S ( G, u )) ∼ = Aff( S ( H, v )) ∼ = R k for some k .Since G and H are archimedean (Proposition 2), they are isomorphic to countable subgroups of R k via positive homomorphisms which send u and v to 1. This together with the fact that everycountable additive subgroup of R is isomorphic to Z or Q implies that h G, u i ⊗ Q ∼ = h H, v i ⊗ Q .‘ Now assume S ( G, u ) and S ( H, v ) are infinite-dimensional. Suppose h G, u i = lim −→ ( G i , u i , α ji )and h H, v i = lim −→ ( H i , v i , β ji ), where h G i , u i i and h H i , v i i are simplicial groups. Without loss ofgenerality we can assume that G i ∼ = H i ∼ = Z i . Let A = h Aff( S ( G, u )) , i , B = h Aff( S ( H, v )) , i , A i = h Aff( S ( G i , u i )) , i and B i = h Aff( S ( H i , v i )) , i . Let e α ji : A i → A j denote the map A ( s ( α ji ))and e β ji : B i → B j denote the map A ( s ( β ji )). Also let ϕ : h G, u i → A , ψ : h H, v i → B , ϕ i : h G i , u i i → A i and ψ : h H i , v i i → B i be the respective natural affine representations. Since h G, u i and h H, v i CLASS OF AF-ALGEBRAS UP TO UNIVERSAL UHF-ALGEBRA STABILITY 7 are archimedean, the natural maps ϕ and ψ are embeddings and therefore we may identify h G, u i with h e G, A i and h H, v i with h e H, B i as ordered subgroups of A and B , respectively. We may alsoidentify each h G i , u i i with h e G i , A i i , its image under the map ϕ i which sends ( x , . . . , x i ) ∈ Z i to ( x /u i , . . . , x i /u ii ) ∈ Q i , where u i = ( u i , . . . , u ii ), as an ordered subgroup of A i of rationalcoordinates. It is easy to check that the diagram h G , u i . . . h G i , u i i h G i +1 , u i +1 i . . . h G, u ih e G , A i . . . h e G i , A i i h e G i +1 , A i +1 i . . . h e G, A i α ϕ α i +1 i ϕ i ϕ i +1 ϕ e α e α i +1 i commutes, i.e. e α i +1 i ( b x ) = \ α i +1 i ( x ) and e α ∞ i ( b x ) = \ α ∞ i ( x ) for every i and x ∈ G i (this is true for anypositive homomorphism and not just the non-mixing ones). Therefore( ∗ ) h G, u i ∼ = h e G, A i ∼ = lim −→ ( e G i , A i , e α ji )and similarly,( ∗∗ ) h H, v i ∼ = h e H, B i ∼ = lim −→ ( e H i , B i , e β ji ) , where e H i is a subgroup of Q i in B i . Since S ( G, u ) ∼ = S ( H, v ), the ordered real Banach spaces h A, i and h B, i are isomorphic, which induces an approximate intertwining between the systems( A i , A i , e α ji ) and ( B i , B i , e β ji ) as follows. Let X i denote the natural set of generators of e G i and Y i denote the natural set of generators of e H i as subgroups of Q i (note that | X i | = | Y i | = i ). Findincreasing sequences of natural numbers { k i } and { l i } and normalized rational positive homomor-phisms γ i : A k i → B l i and η i : B l i → A k i +1 and finitely generated subgroups of rational coordinates E i ⊆ Q k i ⊆ A k i and F i ⊆ Q l i ⊆ B k i such that for every i we have • X k i ⊆ E i and Y l i ⊆ F i , • e α k i +1 k i [ E i ] ⊆ E i +1 , e β l i +1 l i [ F i ] ⊆ F i +1 , γ i [ E i ] ⊆ F i and η i [ F i ] ⊆ E i +1 , • k η i ◦ γ i ( x ) − α k i +1 k i ( x ) k < / i and k γ i +1 ◦ η i ( y ) − β l i +1 l i ( y ) k < / i for every x ∈ E i and y ∈ F i .Finding such an approximate intertwining between ( E i , A ki , e α ji ) and ( F i , B li , e β ji ) is an standardargument and it implies that the two sequences have isomorphic limits. Since E i and F i are finitelygenerated and X k i ⊆ E i and Y l i ⊆ F i , we have h E i , A ki i ∼ = h Z k i , w i i and h F i , B li i ∼ = h Z l i , z i i forsome order-units w i , z i . Note that in particular, E i ∼ = e G k i and F i ∼ = e H l i . Hence by ( ∗ ) and ( ∗∗ ) forsome order-units w ∈ G and z ∈ H we have h G, w i = lim −→ ( E i , A ki , e α k j k i ) = lim −→ ( Z k i , w i , α k j k i )and h H, z i = lim −→ ( F i , B li , e β l j l i ) = lim −→ ( Z l i , z i , β l j l i ) . Therefore h G, w i ∼ = h H, z i . By Corollary 2.4 we have h G, u i ∼ Q h H, v i . (cid:3) Let −→ D also denote the class of unital AF-algebras A such that the dimension group K ( A )belongs to −→ D . Corollary 2.8.
Suppose A and B are unital AF-algebras in −→ D . Then T ( A ) ∼ = T ( B ) if and only if A ⊗ Q ∼ = B ⊗ Q . SAEED GHASEMI
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