A classification of pure states on quantum spin chains satisfying the split property with on-site finite group symmetries
aa r X i v : . [ m a t h . OA ] A ug A classification of pure states on quantum spin chainssatisfying the split property with on-site finite groupsymmetries
Yoshiko Ogata ∗ August 26, 2019
Abstract
We consider a set
SP G ( A ) of pure split states on a quantum spin chain A which areinvariant under the on-site action τ of a finite group G . For each element ω in SP G ( A ) we canassociate a second cohomology class c ω,R of G . We consider a classification of SP G ( A ) whosecriterion is given as follows: ω and ω in SP G ( A ) are equivalent if there are automorphismsΞ R , Ξ L on A R , A L (right and left half infinite chains) preserving the symmetry τ , such that ω and ω ◦ (Ξ L ⊗ Ξ R ) are quasi-equivalent. It means that we can move ω close to ω withoutchanging the entanglement nor breaking the symmetry. We show that the second cohomologyclass c ω,R is the complete invariant of this classification. It is well-known that the pure state space P ( A ) of a quantum spin chain A (UHF-algebra, seesubsection 1.1) is homogeneous under the action of the asymptotically inner automorphisms [P],[B], [FKK]. In fact, the homogeneity is proven for much larger class, i.e., for all the separablesimple C ∗ -algebras [KOS]. In this paper, we focus on the subset SP ( A ) of P ( A ) consisting ofpure states satisfying the split property. (See Definition 1.4.) One equivalent condition for a state ω ∈ P ( A ) to satisfy the split property is that ω is quasi-equivalent to ω | A L ⊗ ω | A R . (See Remark1.5.) Here, ω | A L , ω | A R are restrictions of ω onto the left/right half-infinite chains. (See subsection1.1.) A product state on A = A L ⊗ A R has no entanglement between A L and A R by definition.In this sense, a state with the split property has small entanglement between A L and A R . Usingthe result of [P], [B], [FKK],[KOS], one can easily see that for any ω , ω ∈ SP ( A ), there existasymptotically inner automorphisms Ξ L , Ξ R on A L , A R such that ω | A L ∼ q.e. ω | A L ◦ Ξ L and ω | A R ∼ q.e. ω | A R ◦ Ξ R . (Here ∼ q.e. means quasi-equivalence.) From this and the split propertyof ω , ω , we see that ω and ω ◦ (Ξ L ⊗ Ξ R ) are quasi-equivalent. The product of automorphismsΞ L ⊗ Ξ R clearly does not create/destroy any entanglement between A L and A R . Hence any ω ∈ SP ( A ) can get ”close to” any ω ∈ SP ( A ) without changing the entanglement. In this sense,we may regard SP ( A ) to be ”homogeneous”.What we would like to show in this paper is that the situation changes when symmetry comesinto the game. This corresponds to the notion of symmetry protected topological phases in physics[O]. Let SP G ( A ) be the set of all states in SP ( A ) which are invariant under the onsite action τ of a finite group G . (See Definition 1.4.) We now require that the automorphisms Ξ L , Ξ R aboveto preserve the symmetry i.e., Ξ L ◦ τ L ( g ) = τ L ( g ) ◦ Ξ L and Ξ R ◦ τ R ( g ) = τ R ( g ) ◦ Ξ R for all g ∈ G .(See (3) for the definition of τ L and τ R .) For any ω , ω ∈ SP G ( A ), can we always find such ∗ Graduate School of Mathematical Sciences The University of Tokyo, Komaba, Tokyo, 153-8914, Japan Sup-ported in part by the Grants-in-Aid for Scientific Research, JSPS. ω ∼ q.e. ω ◦ (Ξ L ⊗ Ξ R )? We show that the answer is no in general. Theobstacle is given by the second cohomology class of the projective representation of G associatedto ω ∈ SP G ( A ). We show that this second cohomology class is the complete invariant of thisclassification. We consider the setting in this subsection throughout this paper. We use the basic notation insection A freely. We start by summarizing standard setup of quantum spin chains on the infinitechain [BR1, BR2]. Throughout this paper, we fix some 2 ≤ d ∈ N . We denote the algebra of d × d matrices by M d .For each subset Γ of Z , we denote the set of all finite subsets in Γ by S Γ . We use the notationΓ R = [0 , ∞ ) ∩ Z and Γ L = ( −∞ , − ∩ Z .For each z ∈ Z , let A { z } be an isomorphic copy of M d , and for any finite subset Λ ⊂ Z , weset A Λ = N z ∈ Λ A { z } . For finite Λ, the algebra A Λ can be regarded as the set of all boundedoperators acting on the Hilbert space N z ∈ Λ C d . We use this identification freely. If Λ ⊂ Λ ,the algebra A Λ is naturally embedded in A Λ by tensoring its elements with the identity. For aninfinite subset Γ ⊂ Z , A Γ is given as the inductive limit of the algebras A Λ with Λ ∈ S Γ . Wecall A Γ the quantum spin system on Γ. In particular, we use notation A := A Z , A R := A Γ R and A L := A Γ L . Occasionally, we call them quantum spin chain, right infinite chain, left infinite chain,respectively. Note that each of A Λ , A Γ can be regarded naturally as a subalgebra of A . We alsoset A loc , Γ = S Λ ∈ S Γ A Λ , for any Γ ⊂ Z .We denote the standard basis of C d by { e i } i =1 ,...,d , and denote the standard matrix unit of M d by { E i,j | i, j = 1 , . . . , d } . For each finite Λ ⊂ Z , we denote the tensor product N k ∈ Λ E i k ,j k of E i k ,j k along k ∈ Λ, by E (Λ) I,J with I := ( i k ) k ∈ Λ and J := ( j k ) k ∈ Λ . We also use the notation S Λ := n E (Λ) I,J | I, J ∈ { , . . . , d } × Λ o . (1)Furthermore, we set e (Λ) I := N k ∈ Λ e i k ∈ N Λ C d for I := ( i k ) k ∈ Λ .Throughout this paper we fix a finite group G and its unitary representation U on C d satisfying U ( g ) / ∈ CI C d , if g = e. (2)We denote the identity of G by e .Let Γ ⊂ Z be a non-empty subset. For each g ∈ G , there exists a unique automorphism τ Γ on A Γ such that τ Γ ( g ) ( a ) = Ad O I U ( g ) ! ( a ) , a ∈ A I , g ∈ G, (3)for any finite subset I of Γ. We call the group homomorphism τ Γ : G → Aut A Γ , the on-site actionof G on A Γ given by U . In particular, when Γ = Z , (resp. Γ = Γ R , Γ = Γ L ), we denote τ Γ by τ (resp. τ R , τ L ). For Γ ⊂ Z , we denote by A G Γ the fixed point subalgebra of A Γ with respect to τ Γ .For simplicity, also use the notation A GL := A G Γ L and A GR := A G Γ R . G A map σ : G × G → T is called a 2-cocycle of G if1. σ ( g, h ) σ ( gh, k ) = σ ( h, k ) σ ( g, hk ), for all g, h, k ∈ G ,2. σ ( g, e ) = σ ( e, g ) = 1 for all g ∈ G . 2efine the product of two 2-cocycles by their point-wise product. The set of all 2-cocycles of G then becomes an abelian group. The resulting group we denote by Z ( G, T ). The identity of Z ( G, T ) is given by 1 Z ( G, T ) ( g, h ) := 1, for g, h ∈ G . For an arbitrary function b : G → T suchthat b ( e ) = 1, σ b ( g, h ) = b ( gh ) − b ( g ) b ( h ) , g, h ∈ G (4)defines a 2-cocycle. The set of all 2-cocycles of this type forms a normal subgroup B ( G, T ) of Z ( G, T ). The quotient group H ( G, T ) := Z ( G, T ) /B ( G, T ) is called the second cohomologygroup of G . For each σ ∈ Z ( G, T ), we denote by [ σ ] H ( G, T ) the second cohomology class that σ belongs to.A projective unitary representation of G is a triple ( H , V, σ ) consisting of a Hilbert space H , amap V : G → U ( H ) and a 2-cocycle σ of G such that V ( g ) V ( h ) = σ ( g, h ) V ( gh ) for all g, h ∈ G .Note that we get V ( e ) = I H from the latter condition. We call σ , the 2-cocycle of G associatedto V , and call [ σ ] H ( G, T ) the second cohomology class of G associated to V . We occasionally say( H , V ) is a projective unitary representation with 2-cocycle σ . The character of a finite dimensionalprojective unitary representation ( H , V, σ ) is given by χ V ( g ) = Tr H V ( g ), for g ∈ G .We say a projective unitary representation ( H , V, σ ) of G is irreducible if H and 0 are theonly V -invariant subspaces of H . As G is a finite group, for any irreducible projective unitaryrepresentation ( H , V, σ ) of G , the Hilbert space H is finite dimensional. Projective unitary rep-resentations ( H , V , σ ) and ( H , V , σ ) are said to be unitarily equivalent if there is a unitary W : H → H such that W V ( g ) W ∗ = V ( g ), with g ∈ G . Clearly if ( H , V , σ ) and ( H , V , σ )are unitarily equivalent, the 2-cocycles σ and σ coincides. Schur’s Lemma holds: let ( H , V , σ )and ( H , V , σ ) be irreducible projective unitary representations of G , and W : H → H be alinear map such that W V ( g ) = V ( g ) W for all g ∈ G . Then either V = 0 or ( H , V , σ ) and( H , V , σ ) are unitarily equivalent. The proof is the same as that of the genuine representations(see [S] Theorem II.4.2 for example.)For σ ∈ Z ( G, T ), we denote by P σ , the set of all unitarily equivalence classes of irreducibleprojective representations with 2-cocycle σ . Note that P Z G, T ) is equal to ˆ G , the dual of G .For each α ∈ P σ , we fix a representative ( H α , V α , σ ). We denote the dimension of H α (whichis finite) by n α and fix an orthonormal basis { ψ ( α ) k } n α k =1 of H α . We introduce the matrix unit { f ( α ) k,j | k, j = 1 , . . . , n α } of B ( H α ) given by f ( α ) k,j ξ = D ψ ( α ) j , ξ E ψ ( α ) k , ξ ∈ H α . k, j = 1 , . . . , n α . (5)We will use the following vector later, in section 4Ω α := 1 √ n α n α X k =1 ψ ( α ) k ⊗ ψ ( α ) k ∈ H α ⊗ H α . (6)For each α ∈ P σ and k, j = 1 , . . . , n α , define a function ( V α ) k,j on G by( V α ) k,j ( g ) := D ψ ( α ) k , V α ( g ) ψ ( α ) j E , g ∈ G. (7)As in Theorem III.1.1 of [S], from Schur’s Lemma, we obtain the orthogonality relation:1 | G | X g ∈ G ( V α ) k,j ( g )( V β ) t,s ( g ) = δ α,β δ j,s δ k,t n α , (8)for all α, β ∈ P σ and k, j, t, s = 1 , . . . , n α . Here | G | denotes the number of elements in G . Inparticular, P σ is a finite set. We freely identify α and V α . For example, α ⊗ β ′ , α ⊗ V should beunderstood as V α ⊗ V β ′ , V α ⊗ V for α ∈ P σ , β ′ ∈ P σ ′ , and a projective unitary representation V .We repeatedly use the following fact. 3 emma 1.1. For any projective unitary representation ( H , V, σ ) , there are Hilbert spaces K α labeled by α ∈ P σ and a unitary W : H → L α ∈P σ H α ⊗ K α such that W V ( g ) W ∗ = M α ∈P σ V α ( g ) ⊗ I K α , g ∈ G. (9) Furthermore, the commutant V ( G ) ′ := { X ∈ B ( H ) | [ X, V ( g )] = 0 } of V ( G ) is of the form V ( G ) ′ = W ∗ M α ∈P σ I H α ⊗ B ( K α ) ! W (10) Proof.
For any V -invariant subspace of H , its orthogonal complement is V -invariant as well.Therefore, from Zorn’s Lemma, we may decompose ( H , V, σ ) as an orthogonal sum of irreducibleprojective unitary representations with 2-cocycle σ . This proves (9). The second statement (10)follows from the orthogonality relation (8). (cid:3) Notation 1.2.
When (9) holds, we say that V (or ( H , V, σ )) has an irreducible decompositiongiven by Hilbert spaces {K γ | γ ∈ P σ } . We say V (or ( H , V, σ )) contains all elements of P σ if K α = { } for all α ∈ P σ . We say V (or ( H , V, σ )) contains all elements of P σ with infinitemultiplicity if dim K α = ∞ for all α ∈ P σ . We hence force omit W in (9) and identify H and L α ∈P σ H α ⊗ K α freely. The Hilbert space H α ⊗ K α can be naturally regarded as a closed subspaceof H . We use this identification freely and call H α ⊗ K α the α -component of V (or ( H , V, σ )). Notation 1.3.
Let ( H , V, σ ) be a projective unitary representation. Let b : G → T be a map suchthat b ( e ) = 1. Setting σ b as in (4), we obtain σσ b ∈ Z ( G, T ). We also set ( b · V ) ( g ) := b ( g ) V ( g ),for g ∈ G . Then ( H , b · V, σσ b ) is a projective representation. Next let us introduce the split property.
Definition 1.4.
Let ω be a pure state on A . Let ω R be the restriction of ω to A R , and( H ω R , π ω R , Ω ω R ) be the GNS triple of ω R . We say ω satisfies the split property with respectto A L and A R , if the von Neumann algebra π ω R ( A R ) ′′ is a type I factor. We denote by SP ( A )the set of all pure states on A which satisfy the split property with respect to A L and A R . Wealso denote by SP G ( A ), the set of all states ω in SP ( A ), which are τ -invariant.Recall that a type I factor is ∗ -isomorphic to B ( K ), the set of all bounded operators on a Hilbertspace K . See [T]. Remark . Let ω be a pure state on A . Let ω L be the restriction of ω to A L . Then ω satisfies thesplit property if and only if ω L ⊗ ω R is quasi-equivalent to ω . ( See [M]. In Proposition 2.2 of [M],it is assumed that the state is translationally invariant because of the first equivalent condition(i). However, the proof for the equivalence between (ii) and (iii) does not require translationinvariance.) Therefore, by the symmetric argument, if ( H ω L , π ω L , Ω ω L ) is the GNS triple of ω L ,the the split property of ω implies that π ω L ( A L ) ′′ is also a type I factor.For each ω ∈ SP G ( A ), we may associate a second cohomology class of G . Proposition 1.6.
Let ω ∈ SP G ( A ) and ς = L, R . Then there exists an irreducible ∗ -representation ρ ω,ς of A ς on a Hilbert space L ω,ς that is quasi-equivalent to the GNS representation of ω | A ς . Foreach of such irreducible ∗ -representation ( L ω,ς , ρ ω,ς ) , there is a projective unitary representation u ω,ς of G on L ω,ς such that ρ ω,ς ◦ τ ς ( g ) = Ad ( u ω,ς ( g )) ◦ ρ ω,ς , (11)4 or all g ∈ G . Furthermore, if another triple ( ˜ L ω,ς , ˜ ρ ω,ς , ˜ u ω,ς ) satisfies the same conditions as ( L ω,ς , ρ ω,ς , u ω,ς ) above, then there is a unitary W : L ω,ς → ˜ L ω,ς and c : G → T such that (Ad W ) ◦ ρ ω,ς = ˜ ρ ω,ς , (12) c ( g ) · (Ad W ) ( u ω,ς ( g )) = ˜ u ω,ς ( g ) , g ∈ G. (13) In particular, for -cocycle σ ω,ς , ˜ σ ω,ς associated to u ω,ς , ˜ u ω,ς respectively, we have [ σ ω,ς ] H ( G, T ) =[˜ σ ω,ς ] H ( G, T ) . Proof.
Let ( H ω ς , π ω ς , Ω ω ς ) be the GNS triple of ω | A ς . The existence of irreducible ∗ -representation( L ω,ς , ρ ω,ς ) quasi-equivalent to π ω ς follows from the definition of the split property.To see the existence of u ω,ς satisfying (11) for such ( L ω,ς , ρ ω,ς ), let ι ω,ς : π ω ς ( A ς ) ′′ → B ( L ω,ς )be the ∗ -isomorphism such that ρ ω,ς = ι ω,ς ◦ π ω ς . By the τ ς -invariance of ω | A ς , the action τ ς of G can be extended to an action ˆ τ ς on π ω ς ( A ς ) ′′ , so that ˆ τ ς ( g ) ◦ π ω ς = π ω ς ◦ τ ς ( g ), for g ∈ G . By theWigner Theorem, the ∗ -automorphism ι ω,ς ◦ ˆ τ ς ( g ) ◦ ι − ω,ς on B ( L ω,ς ) is given by a unitary u ω,ς ( g )so that ι ω,ς ◦ ˆ τ ς ( g ) ◦ ι − ω,ς = Ad ( u ω,ς ( g )) , g ∈ G. (14)As ˆ τ ς is an action of G , u ω,ς is a projective unitary representation. We obtain (11) by ρ ω,ς ◦ τ ς ( g ) = ι ω,ς ◦ π ω ς ◦ τ ς ( g ) = ι ω,ς ◦ ˆ τ ς ( g ) ◦ π ω ς = ι ω,ς ◦ ˆ τ ς ( g ) ◦ ι − ω,ς ◦ ι ω,ς ◦ π ω ς = Ad ( u ω,ς ( g )) ◦ ρ ω,ς . (15)Suppose that ( ˜ L ω,ς , ˜ ρ ω,ς , ˜ u ω,ς ) satisfies the same conditions as ( L ω,ς , ρ ω,ς , u ω,ς ). Then by theWigner Theorem, there exists a unitary W : L ω,ς → ˜ L ω,ς satisfying (12). Note thatAd (˜ u ω,ς ( g )) ◦ ˜ ρ ω,ς = ˜ ρ ω,ς ◦ τ ς ( g ) = Ad W ◦ ρ ω,ς ◦ τ ς ( g ) = Ad W ◦ Ad ( u ω,ς ( g )) ◦ Ad W ∗ ◦ ˜ ρ ω,ς . (16)This implies that ˜ u ω,ς ( g ) ∗ Ad W ( u ω,ς ( g )) belongs to TI ˜ L ω,ς proving (13). (cid:3) Definition 1.7.
Let ω ∈ SP G ( A ) and ( L ω,ς , ρ ω,ς , u ω,ς ) be a triple satisfying the conditions inProposition 1.6. Let σ ω,ς be the 2-cocycle associated to u ω,ς . We call ( L ω,ς , ρ ω,ς , u ω,ς , σ ω,ς ) aquadruple associated to ( ω | A ς , τ ς ). Furthermore, we denote the second cohomology class [ σ ω,ς ] H ( G, T ) by c ω,ζ , and call it the second cohomology class of G associated to ( ω | A ς , τ ς ). Remark . For a quadruple ( L ω,ς , ρ ω,ς , u ω,ς , σ ω,ς ) associated to ( ω | A ς , τ ς ) and any map b : G → T ,( L ω,ς , ρ ω,ς , b · u ω,ς , σ b σ ω,ς ) is also a quadruple associated to ( ω | A ς , τ ς ). See (4) and Notation 1.3. Let us introduce AInn G ( A ς ). Definition 1.9.
Let ς = L, R . An automorphism Ξ ς of A ς is asymptotically inner in A Gς if thereis a norm continuous path w ς : [0 , ∞ ) → U (cid:0) A Gς (cid:1) with w ς (0) = I A ς thatΞ ς ( a ) = lim t →∞ Ad ( w ς ( t )) ( a ) , a ∈ A ς . (17)We denote by AInn G ( A ς ) the set of all automorphisms which are asymptotically inner in A Gς .In this paper, we consider the classification problem of SP G ( A ) with respect to the followingequivalence relation. 5 efinition 1.10. For ω , ω ∈ SP G ( A ), we write ω ∼ split ,τ ω if there exist automorphismsΞ L ∈ AInn G ( A L ) and Ξ R ∈ AInn G ( A R ) such that ω and ω ◦ (Ξ L ⊗ Ξ R ) are quasi-equivalent.Now we are ready to state our main theorem. Theorem 1.11.
For ω , ω ∈ SP G ( A ) , ω ∼ split ,τ ω if and only if c ω ,R = c ω ,R . The ”only if” part of the Theorem 1.11 is easy to prove. In order to prove ”if” part of theTheorem, we note that if c ω ,R = c ω ,R holds, ω and ω give covariant representations of a twisted C ∗ -dynamical systems Σ ( σ R )Γ R , Σ ( σ L )Γ L (see section 3), where σ R , σ L are 2-cocycles of G such that[ σ R ] H ( G, T ) = c ω ,R = c ω ,R and [ σ L ] H ( G, T ) = c ω ,L = c ω ,L . (See Remark 1.8 and Lemma 2.5.)One of the basic idea is to encode the information of these 2-cocycles σ R , σ L into C ∗ -algebraswe consider. Namely, instead of considering A R , A L ,we consider the the twisted crossed products C ∗ (Σ ( σ R )Γ R ), C ∗ (Σ ( σ L )Γ L ). We recall the twisted crossed product C ∗ (Σ ( σ )Γ ) of Σ ( σ )Γ in section 3. Insection 2, we show that for any ω ∈ SP G ( A ), and ς = L, R , u ω,ς contains all elements of P σ ω,ς .Therefore, for any fixed α ς ∈ P σ ς , both of u ω ,ς and u ω ,ς contains α ς . This fact allows us to regardthe problem as the homogeneity problem of B ( H α ς ) ⊗ C ∗ (Σ ( σ ς )Γ ς ), with symmetry (section 4). Theproof of the homogeneity relies on the machinery developed in [P], [B], [FKK],[KOS]. However,for our problem, we would like to take the path of unitaries in the fixed point algebras A GR , A GL .This requires some additional argument using the irreducible decompositions of u ω ,ς , u ω ,ς . Thisis given in section 4. u ω,ς In this section we show that u ω,ς contains all elements in P σ ω,ς with infinite multiplicity.As G is a finite group, its dual ˆ G is a finite set and we denote the number of the elements inˆ G by | ˆ G | . We use the following notation for any unitary/projective unitary representations V , V . We write V ≺ V if V is unitarily equivalent to a sub-representation of V . We also say V isincluded in V in this case. Clearly, ≺ is a preorder. We write V ∼ = V if V and V are unitarilyequivalent.For a unitary representation (resp. projective unitary representation) of G , we denote by ¯ V the complex conjugate representation (resp. projective representation) of V . (See [S] section II.6.) Lemma 2.1.
There is an l ∈ N such that for any l ≥ l , the tensor product U ⊗ l contains anyirreducible representation of G as its irreducible component. Proof.
Note that the character χ U ( g ) is the sum of the eigenvalues of a unitary U ( g ) acting on C d . Therefore, the maximal possible value of | χ U ( g ) | is d , which is equal to χ U ( e ). This value isattained only if U ( g ) ∈ TI C d . By the condition (2), for g ∈ G \ { e } , | χ U ( g ) | is strictly less than d .Now for any irreducible representation ( C m , V ) of G , for any l ∈ N , we have X g ∈ G χ V ( g ) χ U ⊗ l ( g ) = d l · m X g ∈ G \{ e } χ V ( g ) m (cid:18) χ U ( g ) d (cid:19) l . (18)Note that χ V ( g ) m (cid:18) χ U ( g ) d (cid:19) l (19)for g ∈ G \ { e } converges to 0 because of | χ U ( g ) | < d . Therefore, for l large enough, the left handside of (18) is non-zero. In other word, for l large enough, V is an irreducible component of U ⊗ l .As ˆ G is a finite set, this proves the Lemma. (cid:3) Lemma 2.2.
There is an l ∈ N such that α ≺ β ⊗ U ⊗ l holds for any σ ∈ Z ( G, T ) , α, β ∈ P σ ,and l ≤ l ∈ N . Proof.
Let l be the number given in Lemma 2.1. For any σ ∈ Z ( G, T ) and α, β ∈ P σ , α ⊗ ¯ β isa genuine representation of G . Let V ∈ ˆ G be an irreducible component of α ⊗ ¯ β . By Lemma 2.1,this V is realized as an irreducible component of U ⊗ l for l ≥ l . Therefore, for l ≥ l , we have X g ∈ G χ α ( g ) χ β ⊗ U ⊗ l ( g ) = X g ∈ G χ α ( g ) χ ¯ β ( g ) χ U ⊗ l ( g ) ≥ X g ∈ G χ V ( g ) χ U ⊗ l ( g ) > . (20)This means α ≺ β ⊗ U ⊗ l . (cid:3) Lemma 2.3.
Let σ ∈ Z ( G, T ) be a fixed -cocycle. For any m ∈ N , there exists an N ( σ ) m ∈ N satisfying the following: For any projective unitary representation ( H , u ) of G with -cocycle σ , α ∈ P σ , and N ∋ N ≥ N ( σ ) m , we have m · α ≺ U ⊗ N ⊗ u. (21) (Here m · α denotes the m direct sum of α . ) Proof.
First let us consider the case that P σ consists of a unique element α ∈ P σ . Then for any N ∈ N , and any projective representation ( H , u, σ ), the multiplicity of α in U ⊗ N ⊗ u is d N · dim H n α ,which is bigger or equal to a ( H , u )-independent value d N n α . The claim of Lemma 2.3 follows fromthis immediately for this case.Next let us consider the case that the number of elements |P σ | in P σ , is larger than 1. FromLemma 2.2, choose l ∈ N so that α ≺ β ⊗ U ⊗ l for all l ≥ l and α, β ∈ P σ . For any m ∈ N , choose M m ∈ N so that |P σ | M m > m . Here we use the condition that |P σ | >
1. We set N ( σ ) m := l ( M m +1).Let ( H , u ) be a projective unitary representation of G with 2-cocycle σ , α ∈ P σ , and N ∋ N ≥ N ( σ ) m .We would like to show that m · α ≺ U ⊗ N ⊗ u . By the choice of N ( σ ) m , N can be decomposed as N = k + k + · · · + k M m + k M m +1 with some l ≤ k j ∈ N , j = 1 , . . . , M m + 1. For each j = 1 , . . . , M m + 1 and β, γ ∈ P σ , we denote the multiplicity of γ in U ⊗ k j ⊗ β by n ( j ) β,γ . From thechoice of l , we have 1 ≤ n ( j ) β,γ for any j = 1 , . . . , M m + 1 and β, γ ∈ P σ . Fix some β ∈ P σ suchthat β ≺ u . From this, we get m · α ≺ |P σ | M m · α = M γ ,γ ,...,γ Mm α ≺ M γ ,γ ,...,γ Mm ,γ Mm +1 n (1) β ,γ n (2) γ ,γ · · · n ( M m +1) γ Mm ,γ Mm +1 · γ M m +1 ≺ U ⊗ k Mm +1 ⊗ U ⊗ k Mm ⊗ · · · U ⊗ k ⊗ U ⊗ k ⊗ β = U ⊗ N ⊗ β ≺ U ⊗ N ⊗ u. (22)This completes the proof. (cid:3) Now we are ready to show the main statement of this section. From the following Lemma, wesee that for any ω ∈ SP G ( A ), u ω,ς contains all elements of P σ ω,ς with infinite multiplicity. Theorem 2.4.
Let Γ be an infinite subset of Z . Let ( L , ρ, u, σ ) be a quadruple such that (i) ρ is a ∗ -representation of A Γ on a Hilbert space L , ii) u is a projective unitary representation of G on L with a -cocycle σ , (iii) for any g ∈ G , we have ρ ◦ τ Γ ( g ) = Ad ( u ( g )) ◦ ρ. (23) Then u contains all elements of P σ with infinite multiplicity. Proof.
Fix any α ∈ P σ and m ∈ N . We would like to show that m · α ≺ u . Let N ( σ ) m be thenumber given in Lemma 2.3 for this fixed m . Let Λ be a subset of Γ such that | Λ | = N ( σ ) m . Wemay factorize ( L , ρ, u ) to Λ-part and Γ \ Λ-part as follows: There exist a ∗ -representation ( ˜ L , ˜ ρ ) of A Γ \ Λ and a projective unitary representation ˜ u of G on ˜ L with 2-cocycle σ , implementing τ Γ \ Λ .There exists a unitary W : L → (cid:0)N Λ C d (cid:1) ⊗ ˜ L such that W ρ ( a ) W ∗ = (id A Λ ⊗ ˜ ρ ) ( a ) , a ∈ A Γ , (24)and W u ( g ) W ∗ = O Λ U ( g ) ! ⊗ ˜ u ( g ) , g ∈ G. (25)More precisely, set I = ( i k ) k ∈ Λ ∈ { , . . . , d } × Λ , with i k = 1 for all k ∈ Λ. We define theHilbert space ˜ L by ˜ L := ρ (cid:16) E (Λ) I ,I (cid:17) L , and the ∗ -representation ˜ ρ of A Γ \ Λ on ˜ L by˜ ρ ( a ) := ρ (cid:16) E (Λ) I ,I ⊗ a (cid:17) , a ∈ A Γ \ Λ . (26)The unitary W : L → (cid:0)N Λ C d (cid:1) ⊗ ˜ L is defined by W ξ := X I ∈{ ,...,n } × Λ e (Λ) I ⊗ ρ (cid:16) E (Λ) I ,I (cid:17) ξ, ξ ∈ L . (27)It is straight forward to check (24).By a straight forward calculation using (24), we can check that (cid:0) ( N Λ U ( g )) ∗ ⊗ I ˜ L (cid:1) W u ( g ) W ∗ with g ∈ G commute with any element of ( N Λ M d ) ⊗ CI ˜ L . Hence there exists a unitary ˜ u ( g ) on ˜ L such that (( N Λ U ( g ))) ∗ ⊗ I ˜ L ) W u ( g ) W ∗ = I N Λ C d ⊗ ˜ u ( g ). This gives (25). It is straight forwardto check that ˜ u is a projective unitary representation of G with 2-cocycle σ implementing τ Γ \ Λ .From (25) and Lemma 2.3, we have m · α ≺ U ⊗ N ( σ ) m ⊗ ˜ u = U ⊗ Λ ⊗ ˜ u ∼ = u. (28)This completes the proof. (cid:3) Recall Definition 1.7. We note that c ω,R and c ω,L are not independent. Lemma 2.5.
For any ω ∈ SP G ( A ) , we have c ω,R = c − ω,L . Proof.
Let ( H , π, Ω) be the GNS triple of ω . As ω satisfies the split property, there are Hilbertspaces H L , H R and a unitary W : H → H L ⊗ H R such that W π ( A R ) ′′ W ∗ = CI H L ⊗ B ( H R ) , W π ( A R ) ′ W ∗ = B ( H L ) ⊗ CI H R . (29)8See Theorem 1.31 V [T].) From (29), π ( A L ) ′′ ⊂ π ( A R ) ′ and π ( A L ) ′′ ∨ π ( A R ) ′′ = B ( H ), weobtain W π ( A L ) ′′ W ∗ = B ( H L ) ⊗ CI H R . (30)Hence we obtain irreducible representations ( H L , π L ) and ( H R , π R ) of A L , A R such that W π ( a ⊗ b ) W ∗ = π L ( a ) ⊗ π R ( b ) , a ∈ A L , b ∈ A R . (31)The triple ( H L ⊗ H R , π L ⊗ π R , W Ω) is a GNS triple of ω . Therefore, ω | A R is π R -normal. As π R ( A R ) ′′ is a factor, π R and the GNS representation of ω | A R are quasi-equivalent. Similarly, π L and the GNS representation of ω | A L are quasi-equivalent.By the τ -invariance of ω , there is a unitary representation V of G on H L ⊗ H R given by V ( g ) ( π L ⊗ π R ) ( a ) W Ω = ( π L ⊗ π R ) ( τ ( g ) ( a )) W Ω , g ∈ G, a ∈ A . (32)On the other hand, by Proposition 1.6, there are projective unitary representations u L , u R of G on H L , H R such that π L ◦ τ L ( g ) ( a ) = Ad ( u L ( g )) ◦ π L ( a ) , π R ◦ τ R ( g ) ( b ) = Ad ( u R ( g )) ◦ π R ( b ) , (33)for all a ∈ A L , b ∈ A R and g ∈ G . Note thatAd ( V ( g )) ◦ ( π L ⊗ π R ) ( x ) = ( π L ⊗ π R ) ◦ τ ( g )( x ) = Ad ( u L ( g ) ⊗ u R ( g )) ◦ ( π L ⊗ π R ) ( x ) , (34)for all x ∈ A . As ( π L ⊗ π R ) ( A ) ′′ = B ( H L ⊗ H R ), this means that there is a map b : G → T suchthat u L ( g ) ⊗ u R ( g ) = b ( g ) V ( g ) , g ∈ G. (35)Let σ L , σ R ∈ Z ( G, T ) be 2-cocycles of u L , u R respectively. From (35), we obtain σ L σ R = σ b (36)(Here σ b is defined by (4).) This means c ω,R = [ σ R ] H ( G, T ) = [ σ − L ] H ( G, T ) = c − ω,L . (37) (cid:3) C ∗ -dynamical system In this section we briefly recall basic facts about twisted C ∗ -crossed product. Throughout thissection, let Γ be an infinite subset of Z , and σ ∈ Z ( G, T ). The quadruple ( G, A Γ , τ Γ , σ ) is atwisted C ∗ -dynamical system which we denote by Σ ( σ )Γ . (This is a simple version of [BC].)A covariant representation of Σ ( σ )Γ is a triple ( H , π, u ) where π is a ∗ -representation of the C ∗ -algebra A Γ on a Hilbert space H and u is a projective unitary representation of G with 2-cocycle σ on H such that u ( g ) π ( a ) u ( g ) ∗ = π ( τ Γ ( g )( a )) , a ∈ A Γ , g ∈ G. (38)9n this paper, we say the covariant representation ( H , π, u ) is irreducible if π is an irreduciblerepresentation of A Γ . Note that for a quadruple( L ω,ς , ρ ω,ς , u ω,ς , σ ω,ς ) associated to ( ω | A ς , τ ς ) with ω ∈ SP G ( A ) (Definition 1.7), ( L ω,ς , ρ ω,ς , u ω,ς ) is an irreducible covariant representation of Σ ( σ ω,ς )Γ ς .Let C ( G, A Γ ) be the linear space of A Γ -valued functions on G . We equip C ( G, A Γ ) with aproduct and ∗ -operation as follows: f ∗ f ( h ) := X g ∈ G σ ( g, g − h ) · f ( g ) · τ Γ ( g ) (cid:0) f ( g − h ) (cid:1) , h ∈ G, (39) f ∗ ( h ) := σ ( h − , h ) τ Γ ( h ) (cid:0) f ( h − ) ∗ (cid:1) , h ∈ G, (40)for f , f , f ∈ C ( G, A Γ ). The linear space C ( G, A Γ ) which is a ∗ -algebra with these operations isdenoted by C (Σ ( σ )Γ ). We will omit the symbol ∗ for the multiplication (39).For a covariant representation ( H , π, u ) of Σ ( σ )Γ , we may introduce a ∗ -representation ( H , π × u )of C (Σ ( σ )Γ ) by ( π × u ) ( f ) := X g ∈ G π ( f ( g )) u ( g ) , f ∈ C (Σ ( σ )Γ ) . (41)The full twisted crossed product of Σ ( σ )Γ , denoted C ∗ (Σ ( σ )Γ ) is the completion of C (Σ ( σ )Γ ) withrespect to the norm k f k u := sup {k ( π × u ) ( f ) k | ( π, u ) : covariant representation } , f ∈ C (Σ ( σ )Γ ) . (42)From any representation ( H , π ) of A Γ , we can define a covariant representation ( H ⊗ l ( G ) ≃ l ( G, H ) , ˜ π, ˜ u π ) of Σ ( σ )Γ by(˜ π ( a ) ξ ) ( g ) := π (cid:0) τ Γ ( g − )( a ) (cid:1) ξ ( g ) , a ∈ A Γ , ξ ∈ l ( G, H ) , g ∈ G, (43)and ˜ u π = I H ⊗ u σr . Here, u σr is a projective unitary representation with 2-cocycle σ on l ( G ) definedby ( u σr ( g ) ξ ) ( h ) = σ ( g, g − h ) ξ ( g − h ) , g, h ∈ G, ξ ∈ l ( G ) . (44)Note that π is faithful because A Γ is simple. Therefore, the representation ˜ π × ˜ u π of C (Σ ( σ )Γ ) givenby ( H ⊗ l ( G ) ≃ l ( G, H ) , ˜ π, ˜ u π ) is faithful. We define a C ∗ -norm k·k r on C (Σ ( σ )Γ ) by k f k r := k ˜ π × ˜ u π ( f ) k B ( H⊗ l ( G )) , f ∈ C (Σ ( σ )Γ ) . (45)The completion C ∗ r (Σ ( σ )Γ ) of C (Σ ( σ )Γ ) with respect to this norm is the reduced twisted crossedproduct of Σ ( σ )Γ . As we are considering a finite group G , we have C (Σ ( σ )Γ ) = C ∗ r (Σ ( σ )Γ ) = C ∗ (Σ ( σ )Γ ),and k·k r = k·k u .For each a ∈ A Γ , ξ a : G ∋ g δ g,e a ∈ A Γ defines an element of C ∗ (Σ ( σ )Γ ). The map ξ : A Γ ∋ a ξ a ∈ C ∗ (Σ ( σ )Γ ) is a unital faithful ∗ -homomorphism. Note that ξ I A Γ is the identity of C ∗ (cid:16) Σ ( σ )Γ (cid:17) . Hence the C ∗ -algebra A Γ can be regarded as a subalgebra of C (Σ ( σ )Γ ) = C ∗ r (Σ ( σ )Γ ) = C ∗ (Σ ( σ )Γ ). Therefore, we simply write a to denote ξ a .From the condition (2), for any g ∈ G with g = e , the automorphism τ Γ ( g ) is properly outer.Therefore, by the argument in [E] Theorem 3.2, C (Σ ( σ )Γ ) = C ∗ r (Σ ( σ )Γ ) = C ∗ (Σ ( σ )Γ ) is simple.As A Γ is unital, we have unitaries λ g ∈ C ∗ (Σ ( σ )Γ ), g ∈ G , defined by G ∋ h δ g,h I A Γ ∈ A Γ such that λ g λ h = σ ( g, h ) λ gh , g, h ∈ G,λ g aλ ∗ g = τ Γ ( g ) ( a ) , a ∈ A Γ , g ∈ G. (46)10ote that λ e = ξ I A Γ is the identity of C ∗ (cid:16) Σ ( σ )Γ (cid:17) .We set C ∗ (cid:16) Σ ( σ )Γ (cid:17) G := n f ∈ C ∗ (cid:16) Σ ( σ )Γ (cid:17) | Ad ( λ g ) ( f ) = f, g ∈ G o . (47)Let ( H , π, u ) be an irreducible covariant representation of Σ ( σ )Γ . The projective unitary repre-sentation u has an irreducible decomposition given by some Hilbert spaces {K γ | γ ∈ P σ } (Lemma1.1 and Notation 1.2). Namely we have u ( g ) = M α ∈P σ V α ( g ) ⊗ I K α , g ∈ G, and u ( G ) ′ = M α ∈P σ I H α ⊗ B ( K α ) . (48)Note that ( π × u ) ( λ g ) = u ( g ) , g ∈ G. (49)From this we have ( π × u ) (cid:18) C ∗ (cid:16) Σ ( σ )Γ (cid:17) G (cid:19) ⊂ u ( G ) ′ = M α ∈P σ CI H α ⊗ B ( K α ) . (50)The following proposition is the immediate consequence of Theorem 2.4. Proposition 3.1.
Let ( H , π, u ) be an irreducible covariant representation of Σ ( σ )Γ . Then u containsall elements of P σ with infinite multiplicity. Throughout this section we fix Γ = Γ L , Γ R , σ ∈ Z ( G, T ), and α ∈ P σ . We use the followingnotation. Notation 4.1.
Let ( H , π, u ) be an irreducible covariant representation of Σ ( σ )Γ with an irreducibledecomposition of u given by a set of Hilbert spaces {K γ | γ ∈ P σ } . We use the symbol ˆ π to denotethe irreducible representation ˆ π = id B ( H α ) ⊗ ( π × u ) (51)of B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) on H α ⊗ H .For a unit vector ξ ∈ K α , we may define a state ˆ ϕ ξ on B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) byˆ ϕ ξ ( x ) := D ˜ ξ, ˆ π ( x ) ˜ ξ E , x ∈ B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) . (52)Here, ˜ ξ is an element of H α ⊗ H ,˜ ξ := Ω α ⊗ ξ ∈ H α ⊗ H α ⊗ K α ֒ → H α ⊗ H , (53)regarding H α ⊗ H α ⊗ K α as a subspace of H α ⊗ H . (See Notation 1.2.) Recall that Ω α is definedin (6). We call this ˆ ϕ ξ a state on B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) given by ( H , π, u, ξ ). By the irreducibility of π , ˆ π is irreducible and ˆ ϕ ξ is a pure state on B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ). Note that ( H α ⊗ H , ˆ π, ˜ ξ ) is a GNStriple of ˆ ϕ ξ .The goal of this section is to prove the following Proposition.11 roposition 4.2. Let ( H i , π i , u i ) with i = 0 , be irreducible covariant representations of Σ ( σ )Γ with irreducible decomposition of u i given by a set of Hilbert spaces {K γ,i | γ ∈ P σ } . Let ξ i ∈ K α,i be unit vectors in K α,i for i = 0 , . (Recall Proposition 3.1 for existence of such vectors.) Let ˆ ϕ ξ i be a state on B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) given by ( H i , π i , u i , ξ i ) , for each i = 0 , . Let ϕ i be the restrictionof ˆ ϕ ξ i onto A Γ . Then there exists a norm-continuous path w : [0 , ∞ ) → U ( A G Γ ) with w (0) = I ,such that1. for each a ∈ A Γ , the limit lim t →∞ Ad ( w ( t )) ( a ) =: Ξ Γ ( a ) (54) exists and defines an automorphism Ξ Γ on A Γ , and2. the automorphism Ξ Γ in 1. satisfies ϕ = ϕ ◦ Ξ Γ .Remark . Basically, what we would like to do is to connect some π -normal state ϕ and some π -normal state ϕ via some Ξ Γ ∈ AInn G ( A Γ ). Without symmetry, ϕ and ϕ can be taken to bepure states. When the symmetry comes into the game, to guarantee that Ξ Γ commutes with τ Γ ( g ),we would like to assume that ϕ and ϕ are τ Γ -invariant. If σ is trivial, there is a u i -invariantnon-zero vector that we may find such pure τ Γ -invariant states ϕ and ϕ . But if the cohomologyclass of σ is not trivial, u i does not have a non-zero invariant vector. However, there is still a rank n α u i -invariant density matrix. That is the reason why we consider B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ). Note thatthe density matrix of ϕ i is a rank n α operator which commutes with u i .For the proof of Proposition 4.2, we use the machinery used in [FKK] and [KOS]. (See AppendixB.) However, as we would like to have a path in the fixed point algebra A G Γ , we need additionalarguments. For that purpose, the following Lemma plays an important role. Lemma 4.4.
Let Γ be an infinite subset of Z . Let ( H , π, u ) be an irreducible covariant represen-tation of Σ ( σ )Γ with an irreducible decomposition of u given by a set of Hilbert spaces {K γ | γ ∈ P σ } .Then there exist irreducible ∗ -representations ( K γ , π γ ) , γ ∈ P σ of A G Γ such that π ( a ) = M γ ∈P σ I H γ ⊗ π γ ( a ) , a ∈ A G Γ . (55) Furthermore, we have π (cid:0) A G Γ (cid:1) ′′ = M γ ∈P σ I H γ ⊗ B ( K γ ) . (56) Notation 4.5.
We call { ( K γ , π γ ) | γ ∈ P σ } , the family of representations of A G Γ associated to( H , π, u ). Proof.
For any a ∈ A G Γ , we have π ( a ) ∈ u ( G ) ′ . Therefore, from Lemma 1.1, each π ( a ) with a ∈ A G Γ has a form π ( a ) = M γ ∈P σ I H γ ⊗ π γ ( a ) , (57)with uniquely defined π γ ( a ) ∈ B ( K γ ), for each γ ∈ P σ .As π is a ∗ -representation, for each γ ∈ P σ , the map π γ : A G Γ ∋ a π γ ( a ) ∈ B ( K γ ) is a ∗ -representation and we have π (cid:0) A G Γ (cid:1) ′′ ⊂ M γ ∈P σ I H γ ⊗ B ( K γ ) . (58)12e claim that each π γ is an irreducible representation of A G Γ , and (56) holds. To see this,note that for any x ∈ B ( K γ ), there exists a bounded net { a λ } λ ∈ A such that π ( a λ ) convergesto I H γ ⊗ x ∈ B ( H γ ) ⊗ B ( K γ ) ⊂ B ( H ) in the σ -strong topology, by the irreducibility of π and theKaplansky density theorem. For this { a λ } λ , we have1 | G | X g ∈ G u g π ( a λ ) u ∗ g = π | G | X g ∈ G τ Γ ( g ) ( a λ ) = M γ ∈P σ I H γ ⊗ π γ | G | X g ∈ G τ Γ ( g ) ( a λ ) ∈ π (cid:0) A G Γ (cid:1) ′′ (59)because | G | P g ∈ G τ Γ ( g ) ( a λ ) is τ Γ -invariant. Since the left hand side of (59) converges to I H γ ⊗ x in the σ -strong topology, we conclude that I H γ ⊗ x ∈ π (cid:0) A G Γ (cid:1) ′′ . Hence (56) holds. Looking at the γ -component of (59), we see that π γ (cid:0) A G Γ (cid:1) ′′ = B ( K γ ). Hence π γ is irreducible. This completes theproof. (cid:3) For each Lemma below, we use the machinery used in [FKK] and [KOS]. We remark argumentsrequired to get a path inside of A G Γ . Lemma 4.6.
Let ( H i , π i , u i ) with i = 0 , be irreducible covariant representations of Σ ( σ )Γ withirreducible decomposition of u i given by a set of Hilbert spaces {K γ,i | γ ∈ P σ } , i = 0 , . Let ξ i ∈ K α,i , i = 0 , be unit vectors. Let ˆ ϕ ξ i be a state on B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) given by ( H i , π i , u i , ξ i ) ,for each i = 0 , . (Recall Notation 4.1.) Then for any F ⋐ B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) and any ε > , thereexists a self-adjoint h ∈ A G Γ such that (cid:12)(cid:12) ˆ ϕ ξ ( x ) − ˆ ϕ ξ ◦ Ad( e ih )( x ) (cid:12)(cid:12) < ε, x ∈ F . (60) Proof.
First we prepare some notations. We denote by ˜ ξ i , ˆ π i the vector and the representation˜ ξ , ˆ π defined in Notation 4.1 with ( H , π, u, ξ ) replaced by ( H i , π i , u i , ξ i ). (See (53) and (51).) Thetriple ( H α ⊗ H i , ˆ π i , ˜ ξ i ) is a GNS-triple of ˆ ϕ ξ i . As B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) is simple, kernel of ˆ π i is zerofor each i = 0 ,
1. For each k, j = 1 , . . . , n α , we define an element Q ( α ) k,j := n α | G | X g ∈ G D ψ ( α ) k , V α ( g ) ψ ( α ) j E λ g ∈ C ∗ (Σ ( σ )Γ ) , (61)with λ g ∈ C ∗ (Σ ( σ )Γ ) introduced in section 3 (46). We also set R ( α ) := 1 n α n α X k,j =1 (cid:12)(cid:12)(cid:12) ψ ( α ) k E D ψ ( α ) j (cid:12)(cid:12)(cid:12) ⊗ Q ( α ) k,j ∈ B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) . (62)We claim ( π i × u i ) (cid:16) Q ( α ) k,j (cid:17) = (cid:12)(cid:12)(cid:12) ψ ( α ) k E D ψ ( α ) j (cid:12)(cid:12)(cid:12) ⊗ I K α,i ∈ B ( H α ⊗ K α,i ) ⊂ B ( H i ) , (63)for each k, j = 1 , . . . , n α and i = 0 ,
1. From this we haveˆ π i (cid:16) R ( α ) (cid:17) = | Ω α i h Ω α | ⊗ I K α,i =: P ( α,i ) , i = 0 , . (64)From (64), we obtain ˆ ϕ ξ i ( R ( α ) ) = 1 , i = 0 , . (65)13o see (63), recall the orthogonality relation (8), the irreducible decomposition of u i given by {K γ,i | γ ∈ P σ } and that ( π i × u i )( λ g ) = u i ( g ) (49). Then we have( π i × u i ) (cid:16) Q ( α ) k,j (cid:17) = n α | G | X g ∈ G D ψ ( α ) k , V α ( g ) ψ ( α ) j E u i ( g )= n α | G | X g ∈ G D ψ ( α ) k , V α ( g ) ψ ( α ) j E M γ ∈P σ V γ ( g ) ⊗ I K γ,i = M γ ∈P σ (cid:16) δ α,γ (cid:12)(cid:12)(cid:12) ψ ( α ) k E D ψ ( α ) j (cid:12)(cid:12)(cid:12) ⊗ I K α,i (cid:17) . (66)We now start the proof of Lemma. We fix an arbitrary F ⋐ B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) and ε >
0. Wethen choose 0 < ˜ ε small enough so that˜ ε < min { , ε } , and 4 max a ∈F k a k ˜ ε < ε . (67)We also set ˜ F := F ∪ n R ( α ) o ⋐ B ( H ( α ) ) ⊗ C ∗ (Σ ( σ )Γ ) . (68)Applying Lemma B.1 to this ˜ ε and ˜ F , and pure states ˆ ϕ ξ i , i = 0 , C ∗ -algebra B ( H ( α ) ) ⊗ C ∗ (Σ ( σ )Γ ), we obtain an f ∈ (cid:16) B ( H ( α ) ) ⊗ C ∗ (Σ ( σ )Γ ) (cid:17) + , and a unit vector ζ ∈ H ( α ) ⊗ H such that ˆ π ( f ) ζ = ζ, k f ( a − ˆ ϕ ξ ( a ) I ) f k < ˜ ε, for all a ∈ ˜ F . (69)For P ( α, in (64) and the ζ in (69), we have (cid:13)(cid:13)(cid:13)(cid:16) I − P ( α, (cid:17) ζ (cid:13)(cid:13)(cid:13) = D ζ, ˆ π ( f ) (cid:16) I − ˆ π ( R ( α ) ) (cid:17) ˆ π ( f ) ζ E = D ζ, ˆ π ( f ) (cid:16) ˆ ϕ ξ ( R ( α ) ) I − ˆ π ( R ( α ) ) (cid:17) ˆ π ( f ) ζ E ≤ (cid:13)(cid:13)(cid:13) f (cid:16) ˆ ϕ ξ ( R ( α ) ) I − R ( α ) (cid:17) f (cid:13)(cid:13)(cid:13) < ˜ ε. (70)Here we used (65), for the second equality. For the inequality we used (69) and R ( α ) ∈ ˜ F (68).Therefore, P ( α, ζ is not zero, and we may define a unit vector˜ ζ := 1 (cid:13)(cid:13) P ( α, ζ (cid:13)(cid:13) P ( α, ζ ∈ H α ⊗ H . (71)Furthermore, it satisfies (cid:13)(cid:13)(cid:13) ζ − ˜ ζ (cid:13)(cid:13)(cid:13) ≤ ε . (72)From this and two properties in (69) for any a ∈ F , we have (cid:12)(cid:12)(cid:12) ˆ ϕ ξ ( a ) − D ˜ ζ, ˆ π ( a )˜ ζ E(cid:12)(cid:12)(cid:12) ≤ |h ˆ π ( f ) ζ, ( ˆ ϕ ξ ( a ) − ˆ π ( a )) ˆ π ( f ) ζ i| + (cid:12)(cid:12)(cid:12) h ζ, ˆ π ( a ) ζ i − D ˜ ζ, ˆ π ( a )˜ ζ E(cid:12)(cid:12)(cid:12) ≤ k f ( a − ˆ ϕ ξ ( a ) I ) f k + 2 max a ∈F k a k (cid:13)(cid:13)(cid:13) ζ − ˜ ζ (cid:13)(cid:13)(cid:13) < ˜ ε + 2 max a ∈F k a k ε < ε, (73)by the choice of ˜ ε (67).Since P ( α, ˜ ζ = ˜ ζ , there exists a unit vector η ∈ K α, such that ˜ ζ = Ω α ⊗ η . By Lemma 4.4,for each γ ∈ P σ , there exists an irreducible representation π γ, of A G Γ on K γ, such that π ( a ) = M γ ∈P σ I H γ ⊗ π γ, ( a ) , a ∈ A G Γ . (74)14pplying the Kadison transitivity theorem for unit vectors ξ , η ∈ K α, and an irreducible repre-sentation ( K α, , π α, ) of A G Γ , we obtain a self-adjoint h ∈ A G Γ such that π α, ( e − ih ) ξ = η . Withthis h , we can write ˜ ζ as ˜ ζ = ˆ π (cid:0) I B ( H α ) ⊗ e − ih (cid:1) ˜ ξ . (75)Hence we obtain ˆ ϕ ξ ◦ Ad (cid:0) e ih (cid:1) = D ˜ ζ, ˆ π ( · ) ˜ ζ E . (76)Combining this with (73), we see that (60) holds. (cid:3) Remark . The main difference of the proof of Lemma 4.6 from [KOS], [FKK] is that in orderto find h in A G Γ , we add R ( α ) to F . This allows us to replace ζ with ˜ ζ = Ω α ⊗ η . From thiscombined with Lemma 4.4, the problem is reduced to the Kadison transitivity for the irreducible( K α, , π α, ( A G Γ )). Note that R ( α ) belongs to B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) but not in A Γ . By extending the C ∗ -algebra we consider, we are allowed to have the projection P ( α,i ) (64) corresponding to theirreducible component of u i in the C ∗ -algebra. Notation 4.8.
For Λ ⋐ Γ, we introduce a finite subset of B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) given by G Λ := ( p | G | (cid:12)(cid:12)(cid:12) ψ ( α ) j E D ψ ( α )1 (cid:12)(cid:12)(cid:12) ⊗ λ g E (Λ) I,I | j = 1 , . . . , n α , I ∈ { , . . . , d } × Λ , g ∈ G ) . (77)Here, we set I := ( i k ) k ∈ Λ ∈ { , . . . , d } × Λ , with i k = 1 for all k ∈ Λ. Notation 4.9.
We say an irreducible covariant representation ( H , π, u ) of Σ ( σ )Γ and unit vec-tors ξ, η ∈ H α ⊗ H satisfy Condition 1 for a pair δ >
0, Λ ⋐ Γ, if the representation ˆ π :=id B ( H α ) ⊗ ( π × u ) of B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) satisfies the following:1. For any x, y ∈ G Λ , ˆ π ( x ) ∗ ξ and ˆ π ( y ) ∗ η are orthogonal.2. For any x, y ∈ G Λ , |h ξ, ˆ π ( xy ∗ ) ξ i − h η, ˆ π ( xy ∗ ) η i| < δ. (78)Let δ ,a be the function given in Lemma B.4. Lemma 4.10.
For any ε > and Λ ⋐ Γ , there exists a δ ( ε, Λ) > satisfying the following: Forany irreducible covariant representation ( H , π, u ) of Σ ( σ )Γ and unit vectors ξ, η ∈ H α ⊗ H satisfyingCondition 1 for a pair δ ( ε, Λ) > , Λ ⋐ Γ , there exists a positive element h of ( A G Γ \ Λ ) such that (cid:13)(cid:13)(cid:13) e iπ ˆ π ( h ) ξ − η (cid:13)(cid:13)(cid:13) < √ δ ,a (cid:16) ε (cid:17) . (79) Proof.
Recall Lemma B.6. We set δ ( ε, Λ) := δ ,a (cid:16) ε, n α d | Λ | | G | (cid:17) , (80)with δ ,a ( · , · ) in Lemma B.6. We prove that this δ satisfies the condition above.Let us consider an arbitrary irreducible covariant representation ( H , π, u ) of Σ ( σ )Γ and unitvectors ξ, η ∈ H α ⊗ H satisfying Condition 1 for a pair δ ( ε, Λ) >
0, Λ ⋐ Γ. We again use thenotation ˆ π (51) for this π . 15e apply Lemma B.6, to an infinite dimensional Hilbert space H α ⊗ H , a unital C ∗ -algebraˆ π (cid:16) B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) (cid:17) acting irreducibly on H α ⊗H , a finite subset ˆ π ( G Λ ) of ˆ π (cid:16) B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) (cid:17) ,and unit vectors ξ, η ∈ H α ⊗ H . Note that P x ∈ ˆ π ( G Λ ) xx ∗ = I by the definition of G Λ . From Condi-tion 1 , ξ, η satisfy the required conditions in Lemma B.6. By Lemma B.6 , there exists a positive˜ h ∈ (cid:16) B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) (cid:17) + , such that (cid:13)(cid:13) ˆ π (cid:0) ¯ h (cid:1) ( ξ + η ) (cid:13)(cid:13) < √ δ ,a (cid:16) ε (cid:17) e − π , and (cid:13)(cid:13)(cid:0) I − ˆ π (¯ h ) (cid:1) ( ξ − η ) (cid:13)(cid:13) < √ δ ,a (cid:16) ε (cid:17) e − π , (81)for ¯ h := X x ∈G Λ x ˜ hx ∗ . (82)Here the function δ ,a is given in Theorem B.4. By this definition of ¯ h , we see that¯ h ∈ ( B ( H α ) ⊗ A Λ ) ′ ∩ { λ g | g ∈ G } ′ ∩ (cid:16) B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) (cid:17) + , . (83)We would like to replace ¯ h in (81) to some positive element h ∈ ( A G Γ \ Λ ) + , . In order to do so,we factorize ( H , π, u ) to Λ-part and Γ \ Λ-part: As in the proof of Theorem 2.4, there exists anirreducible covariant representation ( ˜ H , ˜ π, ˜ u ) of Σ ( σ )Γ \ Λ and a unitary W : H → (cid:0)N Λ C d (cid:1) ⊗ ˜ H suchthat W π ( a ) W ∗ = (id A Λ ⊗ ˜ π ) ( a ) , a ∈ A Γ , (84)and W u ( g ) W ∗ = O Λ U ( g ) ! ⊗ ˜ u ( g ) , g ∈ G. (85)By Lemma 1.1, ˜ u has an irreducible decomposition of given by a set of Hilbert spaces {K γ | γ ∈ P σ } .By Lemma 4.4 and Lemma 1.1 we have˜ π (cid:16) A G Γ \ Λ (cid:17) ′′ = M γ ∈P σ I H γ ⊗ B ( K γ ) = ˜ u ( G ) ′ . (86)Recall (81). Choose δ > (cid:13)(cid:13) ˆ π (cid:0) ¯ h (cid:1) ( ξ + η ) (cid:13)(cid:13) + δ < √ δ ,a (cid:16) ε (cid:17) e − π , and (cid:13)(cid:13)(cid:0) I − ˆ π (¯ h ) (cid:1) ( ξ − η ) (cid:13)(cid:13) + δ < √ δ ,a (cid:16) ε (cid:17) e − π . (87)As ¯ h is in ( B ( H α ) ⊗ A Λ ) ′ , from (84), we see that there exists a positive y ∈ B ( ˜ H ) such that( I H α ⊗ W ) ˆ π (¯ h ) ( I H α ⊗ W ∗ ) = I H α ⊗ I N Λ C d ⊗ y. (88)Furthermore, as ¯ h is in { λ g | g ∈ G } ′ , from (85), y belongs to ˜ u ( G ) ′ = ˜ π ( A G Γ \ Λ ) ′′ by (86). By theKaplansky density theorem, there exists a positive h ∈ (cid:16) A G Γ \ Λ (cid:17) + , such that (cid:13)(cid:13)(cid:0) ˆ π ( h ) − ˆ π (¯ h ) (cid:1) ( ξ ± η ) (cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:16) I H α ⊗ I N Λ C d ⊗ (˜ π ( h ) − y ) (cid:17) ( I H α ⊗ W ) ( ξ ± η ) (cid:13)(cid:13)(cid:13) < δ. (89)16his h satisfies k ˆ π ( h ) ( ξ + η ) k ≤ (cid:13)(cid:13)(cid:0) ˆ π ( h ) − ˆ π (¯ h ) (cid:1) ( ξ + η ) (cid:13)(cid:13) + (cid:13)(cid:13) ˆ π (¯ h ) ( ξ + η ) (cid:13)(cid:13) < √ δ ,a (cid:16) ε (cid:17) e − π , and k ( I − ˆ π ( h )) ( ξ − η ) k ≤ (cid:13)(cid:13)(cid:0) ˆ π ( h ) − ˆ π (¯ h ) (cid:1) ( ξ − η ) (cid:13)(cid:13) + (cid:13)(cid:13)(cid:0) I − ˆ π (¯ h ) (cid:1) ( ξ − η ) (cid:13)(cid:13) < √ δ ,a (cid:16) ε (cid:17) e − π , (90)from the choice of δ , (87). We then obtain the required property of h : (cid:13)(cid:13)(cid:13) e iπ ˆ π ( h ) ξ − η (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13) (cid:16) e iπ ˆ π ( h ) ( ξ + η ) − ( ξ + η ) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13) (cid:16) e iπ ˆ π ( h ) ( ξ − η ) + ( ξ − η ) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) ≤ e π k ˆ π ( h ) ( ξ + η ) k + e π k ( I − ˆ π ( h )) ( ξ − η ) k < √ δ ,a (cid:16) ε (cid:17) . (91) (cid:3) Remark . Note that an average over G is contained in (82). Because of this, we could take ¯ h to be Ad λ g -invariant. This is possible because λ g is included in the C ∗ -algebra we consider, i.e.,in B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ).The main difference of Lemma 4.10 compared to [KOS] is replacing ¯ h with h ∈ A G Γ \ Λ . To carryit out, the decomposition (86) given from Lemma 1.1 Lemma 4.4 is used. This decompositionreduces the problem to the Kaplansky density Theorem for ˆ π ( A G Γ \ Λ ). Notation 4.12.
For any ε > F ⋐ A , there exists a Λ( ε, F ) ⋐ Γ such thatinf (cid:8) k a − b k | b ∈ A Λ( ε, F ) (cid:9) < ε , for all a ∈ F . (92)For each ε > F ⋐ A , we fix such Λ( ε, F ). If F is included in A Λ for some Λ ⋐ Γ, we chooseΛ( ε, F ) so that Λ( ε, F ) ⊂ Λ. For any ε > F ⋐ A , set δ ( ε, F ) := 12 δ (cid:16) ε , Λ( ε, F ) (cid:17) . (93)Here we used the function δ introduced in Lemma 4.10. Lemma 4.13.
Let ε > , and F ⋐ ( A Γ ) . Let ( H , π, u ) be an irreducible covariant representationof Σ ( σ )Γ with an irreducible decomposition of u given by a set of Hilbert spaces {K γ | γ ∈ P σ } . Let ξ, η be unit vectors in K α . Suppose that unit vectors ˜ ξ := Ω α ⊗ ξ, ˜ η := Ω α ⊗ η ∈ H α ⊗ H (94) satisfy (cid:12)(cid:12)(cid:12) h ˜ η, ˆ π ( xy ∗ )˜ η i − D ˜ ξ, ˆ π ( xy ∗ ) ˜ ξ E(cid:12)(cid:12)(cid:12) < δ ( ε, F ) , for all x, y ∈ G Λ( ε, F ) . (95) (Recall Notation 4.1 and Notation 4.8.) Then there exists a norm-continuous path of unitaries v : [0 , → U ( A G Γ ) such that v (0) = I A Γ , ˜ η = ( I H α ⊗ π ( v (1))) ˜ ξ, (96) and sup t ∈ [0 , k Ad v ( t )( a ) − a k < ε, for all a ∈ F . (97)17 roof. We denote by N , the finite dimensional subspace spanned by { ˆ π ( xy ∗ ) ˜ ξ, ˆ π ( xy ∗ )˜ η | x, y ∈G Λ( ε, F ) } . Then there exists a unit vector ζ in N ⊥ , the orthogonal complement of N , such that (cid:12)(cid:12)(cid:12) h ζ, ˆ π ( xy ∗ ) ζ i − D ˜ ξ, ˆ π ( xy ∗ ) ˜ ξ E(cid:12)(cid:12)(cid:12) < δ ( ε, F ) ≤ δ (cid:16) ε , Λ( ε, F ) (cid:17) , x, y ∈ G Λ( ε, F ) . (98)To see this, note that the intersection of the set of all compact operators on H α ⊗ H andˆ π (cid:16) B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) (cid:17) is 0 because B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) is simple. Applying Glimm’s Lemma (The-orem B.5) to δ ( ε, F ) >
0, a pure state D ˜ ξ, · ˜ ξ E on ˆ π (cid:16) B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) (cid:17) , a finite dimensionalsubspace N of H α ⊗ H and a finite subset ˆ π (cid:16) G Λ( ε, F ) G ∗ Λ( ε, F ) (cid:17) , we obtain ζ above.Combining (98) with (95)we also get |h ζ, ˆ π ( xy ∗ ) ζ i − h ˜ η, ˆ π ( xy ∗ )˜ η i| ≤ (cid:12)(cid:12)(cid:12) h ζ, ˆ π ( xy ∗ ) ζ i − D ˜ ξ, ˆ π ( xy ∗ ) ˜ ξ E(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) h ˜ η, ˆ π ( xy ∗ )˜ η i − D ˜ ξ, ˆ π ( xy ∗ ) ˜ ξ E(cid:12)(cid:12)(cid:12) < δ ( ε, F ) = δ (cid:16) ε , Λ( ε, F ) (cid:17) , x, y ∈ G Λ( ε, F ) . (99)Hence ( H , π, u ) and unit vectors ˜ ξ, ζ (resp. ˜ η, ζ ) satisfy Condition 1. (Notation 4.9) for a pair δ (cid:0) ε , Λ( ε, F ) (cid:1) >
0, Λ( ε, F ) ⋐ Γ. Therefore, from Lemma 4.10, there exist positive elements h , h in ( A G Γ \ Λ( ε, F ) ) such that (cid:13)(cid:13)(cid:13) e iπ ˆ π ( h ) ˜ ξ − ζ (cid:13)(cid:13)(cid:13) < √ δ ,a (cid:16) ε (cid:17) , and (cid:13)(cid:13)(cid:13) e iπ ˆ π ( h ) ˜ η − ζ (cid:13)(cid:13)(cid:13) < √ δ ,a (cid:16) ε (cid:17) . (100)Here δ ,a is given in Theorem B.4. By the definition of ˜ ξ (94) and the decomposition π ( a ) = M γ ∈P σ I H γ ⊗ π γ ( a ) , a ∈ A G Γ (101)((55) of Lemma 4.4), with irreducible ∗ -representations ( K γ , π γ ) of A G Γ , we have e iπ ˆ π ( h ) ˜ ξ = Ω α ⊗ e iππ α ( h ξ . Similarly, we have e iπ ˆ π ( h ) ˜ η = Ω α ⊗ e iππ α ( h η . Combining this with (100), we see thatthe unit vectors e iππ α ( h ξ, e iππ α ( h η in K α satisfies (cid:13)(cid:13)(cid:13) e iππ α ( h ) ξ − e iππ α ( h ) η (cid:13)(cid:13)(cid:13) < √ δ ,a (cid:16) ε (cid:17) . (102)Then from Lemma B.2, there exists a unitary v on K α such that v e iππ α ( h ) ξ = e iππ α ( h ) η, and k v − I K α k < δ ,a (cid:16) ε (cid:17) < δ ,a (cid:16) ε (cid:17) . (103)From this and the fact that π α is an irreducible representation of A G Γ , applying Theorem B.4, weobtain a self-adjoint k ∈ A G Γ such that e iπ α ( k ) e iππ α ( h ) ξ = e iππ α ( h ) η, and k k k ≤ δ ,a (cid:16) ε (cid:17) . (104)Here the function δ ,a is given in Notation B.3.Now we define a continuous path of unitaries v : [0 , → U ( A G Γ ). Set v ( t ) := e itπh ∈ U (cid:16) A G Γ \ Λ( ε, F ) (cid:17) , v ( t ) := e itk ∈ U (cid:0) A G Γ (cid:1) , v ( t ) := e − itπh ∈ U (cid:16) A G Γ \ Λ( ε, F ) (cid:17) (105)for each t ∈ [0 , i = 1 ,
3, as v i takes value in U (cid:16) A G Γ \ Λ( ε, F ) (cid:17) , v i ( t ) commutes with elements in A Λ( ε, F ) . Fromthis and the fact that the distance between F and A Λ( ε, F ) is less than ε (Notation 4.12 (92)), weget k Ad v i ( t )( a ) − a k < ε , for all a ∈ F , t ∈ [0 , i = 1 ,
3. For i = 2, from k k k ≤ δ ,a (cid:0) ε (cid:1) ,recalling the definition of δ ,a in Notation B.3, we obtain k Ad v ( t )( a ) − a k ≤ k v ( t ) − I k ≤ ε ,for all a ∈ F ⊂ ( A Γ ) and t ∈ [0 , v : [0 , → U (cid:0) A G Γ (cid:1) by v ( t ) := v (3 t ) , t ∈ (cid:20) , (cid:21) ,v (cid:18) (cid:18) t − (cid:19)(cid:19) v (1) , t ∈ (cid:20) , (cid:21) ,v (cid:18) (cid:18) t − (cid:19)(cid:19) v (1) v (1) , t ∈ (cid:20) , (cid:21) . (106)Clearly v (0) = I A Γ and v is norm-continuous, and it takes values in U (cid:0) A G Γ (cid:1) . From the aboveestimates on k Ad v i ( t )( a ) − a k for a ∈ F and i = 1 , ,
3, we also get (97). Furthermore, we have( I H α ⊗ π ( v (1))) ˜ ξ = Ω α ⊗ π α ( v (1)) ξ = Ω α ⊗ π α (cid:0) e − iπh e ik e iπh (cid:1) ξ = Ω α ⊗ η = ˜ η. (107)Here, for the first equality, we used the fact that v (1) is in in A G Γ and (101). The third equality isfrom (104). (cid:3) Remark . By Lemma 4.10, we can take h , h in the fixed point algebra A G Γ \ Λ ( ε, F ) . Withthe special form of ˜ ξ, ˜ η , in (94), the problem is reduced to the Kadison transitivity theorem for( K α , π α ( A G Γ )). The irreducibility of π α is used there. From this we may obtain k interpolating e iπ ˆ π ( h ) ˜ ξ and e iπ ˆ π ( h ) ˜ η , from A G Γ . Lemma 4.15.
For any ε > and F ⋐ ( A Γ ) , the following holds: Let ( H i , π i , u i ) with i = 0 , beirreducible covariant representations of Σ ( σ )Γ with irreducible decomposition of u i given by a set ofHilbert spaces {K γ,i | γ ∈ P σ } . Let ξ i ∈ K α,i be a unit vector in K α,i for i = 0 , . Suppose that therepresentation ˆ π i := id H α ⊗ ( π i × u i ) , i = 0 , of B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) and unit vectors ˜ ξ i := Ω α ⊗ ξ i in H α ⊗ H i , i = 0 , satisfy (cid:12)(cid:12)(cid:12)D ˜ ξ , ˆ π ( xy ∗ ) ˜ ξ E − D ˜ ξ , ˆ π ( xy ∗ ) ˜ ξ E(cid:12)(cid:12)(cid:12) < δ ( ε, F ) , for all x, y ∈ G Λ( ε, F ) . (108) (Recall Notation 4.12 for δ .) Then for any ε ′ > and F ′ ⋐ B ( H α ) ⊗ C ∗ (Σ ( σ )Γ ) , there exists anorm-continuous path v : [0 , → U ( A G Γ ) with v (0) = I A Γ such that (cid:12)(cid:12)(cid:12)D ˜ ξ , ˆ π ( a ) ˜ ξ E − D ˜ ξ , (ˆ π ◦ Ad ( v (1))) ( a ) ˜ ξ E(cid:12)(cid:12)(cid:12) < ε ′ , for all a ∈ F ′ , (109) and k Ad v ( t )( y ) − y k < ε, for all y ∈ F , and t ∈ [0 , . (110) Proof.
From Lemma 4.6, there exists a self-adjoint h ∈ A G Γ such that (cid:12)(cid:12)(cid:12)D ˜ ξ , ˆ π ( a ) ˜ ξ E − D ˜ ξ , ˆ π ◦ Ad (cid:0) e ih (cid:1) ( a ) ˜ ξ E(cid:12)(cid:12)(cid:12) < min (cid:26) ε ′ , δ ( ε, F ) (cid:27) for all a ∈ F ′ ∪ G Λ( ε, F ) (cid:0) G Λ( ε, F ) (cid:1) ∗ . (111)19rom this and (108), we have (cid:12)(cid:12)(cid:12)D ˜ ξ , ˆ π ◦ Ad (cid:0) e ih (cid:1) ( xy ∗ ) ˜ ξ E − D ˜ ξ , ˆ π ( xy ∗ ) ˜ ξ E(cid:12)(cid:12)(cid:12) < δ ( ε, F ) for all x, y ∈ G Λ( ε, F ) . (112)Recall from Lemma 4.4 that π ( a ) = M γ ∈P σ I H γ ⊗ π γ, ( a ) , a ∈ A G Γ (113)with irreducible ∗ -representations ( K γ, , π γ, ) of A G Γ . From this and h ∈ A G Γ , we see that ˆ π (cid:0) e − ih (cid:1) ˜ ξ =Ω α ⊗ π α, ( e − ih ) ξ By (112), ( H , π , u ), ξ , π α, ( e − ih ) ξ satisfies the required condition in Lemma4.13.Applying Lemma 4.13 for ( H , π , u ) and ξ , π α, ( e − ih ) ξ , we obtain a norm-continuous pathof unitaries v : [0 , → U ( A G Γ ) such that v (0) = I A Γ ,ˆ π (cid:0) e − ith (cid:1) ˜ ξ = ˆ π ( v (1) ∗ ) ˜ ξ , (114)and sup t ∈ [0 , k Ad v ( t )( a ) − a k < ε, for all a ∈ F . (115)From (111) and (114), we obtain (109). (cid:3) Remark . As in [KOS], we replace e ith with v ( t ) which satisfy (115). We may do so with v ( t )in A G Γ because of Lemma 4.13.After these preparation, the proof of Proposition 4.2 is the same as proof of Theorem 2.1 of[KOS]. We give it here for the reader’s convenience. Proof of Proposition 4.2.
We fix an increasing sequence Λ n , n = 0 , , , . . . of non-empty finitesubsets of Γ such that Λ n ր Γ.For each i = 0 ,
1, we use the notation ˆ π i , ˜ ξ i , ˆ ϕ ξ i given in Notation 4.1, replacing ( H , π, u ) and ξ ∈ K α with ( H i , π i , u i ) and ξ i ∈ K α,i . Let ( K α,i , π α,i ), i = 0 , ∗ -representationof A G Γ obtained in Lemma 4.4 (55) with ( H , π, u ), Γ replaced by ( H i , π i , u i ), Γ.Set F := S Λ . (Recall (1).) Fix ε > ε = 1. Set G := G Λ( ε, F ) G ∗ Λ( ε, F ) . From Lemma4.6, there exists a self-adjoint h ∈ A G Γ . such that (cid:12)(cid:12) ˆ ϕ ξ ◦ Ad (cid:0) e ih (cid:1) ( a ) − ˆ ϕ ξ ( a ) (cid:12)(cid:12) < min (cid:26) δ ( ε, F ) , ε (cid:27) , a ∈ G ∪ F . (116)(Recall Notation 4.12 for δ .) We define v := [0 , → U ( A G Γ ) by v ( t ) = e ith , t ∈ [0 , . (117)We consider the following proposition [ P n ] for each n ∈ N :[ P n ] There exist norm-continuous paths v k : [0 , → U ( A G Γ ), k = 0 , . . . , n with v k (0) = I A Γ satisfying the following: Set F j := (cid:8) x, Ad ( v j − (1) ∗ v j − (1) ∗ · · · v (1) ∗ v (1) ∗ ) ( x ) | x ∈ S Λ j (cid:9) , j = 1 , . . . , n, (118)and F j − := (cid:8) x, Ad ( v j − (1) ∗ v j − (1) ∗ · · · v (1) ∗ v (1) ∗ v (1) ∗ ) ( x ) | x ∈ S Λ j − (cid:9) , j = 1 , . . . , n. (119)(Recall (1).) We also denote the finite subset G Λ ( ε k , F k ) G ∗ Λ ( ε k , F k ) by G k , for each k = 0 , , . . . , n . Then the following three inequalities hold.20. For all a ∈ G n ∪ F n , | ˆ ϕ ξ ◦ Ad ( v (1) v (1) · · · v n (1)) ( a ) − ˆ ϕ ξ ◦ Ad ( v (1) v (1) · · · v n − (1)) ( a ) | < min (cid:26) δ (cid:16) ε n , F n (cid:17) , ε n (cid:27) (120)2. For all a ∈ G n − ∪ F n − , | ˆ ϕ ξ ◦ Ad ( v (1) v (1) · · · v n − (1)) ( a ) − ˆ ϕ ξ ◦ Ad ( v (1) v (1) · · · v n − (1)) ( a ) | < min (cid:26) δ (cid:16) ε n − , F n − (cid:17) , ε n − (cid:27) (121)3. For all t ∈ [0 , k = 1 , , . . . , n with k ≤ n and x ∈ F k − , we have k Ad v k ( t )( x ) − x k < ε k − . (122)Let us check that [ P ] with v given in (117) holds. Set F as in (119) with j = 1 and this v .Set G := G Λ ( ε , F ) G ∗ Λ ( ε , F ). From (116), applying Lemma 4.15 for vectors ξ and π α, ( v (1) ∗ ) ξ ,there exists a norm-continuous path v : [0 , → U (cid:0) A G Γ (cid:1) with v (0) = I such that | ˆ ϕ ξ ◦ Ad ( v (1)) ( a ) − ˆ ϕ ξ ◦ Ad ( v (1)) ( a ) | < min (cid:26) δ (cid:16) ε , F (cid:17) , ε (cid:27) , for all a ∈ G ∪ F (123)and k Ad v ( t )( y ) − y k < ε, for all y ∈ F , and t ∈ [0 , . (124)Set F as in (118) with j = 1 for this v . And set G := G Λ ( ε , F ) G ∗ Λ ( ε , F ). From (123), applyingLemma 4.15 again to vectors π α, ( v (1) ∗ ) ξ and π α, ( v (1) ∗ ) ξ , we obtain a norm-continuouspath v : [0 , → U (cid:0) A G Γ (cid:1) with v (0) = I such that | ˆ ϕ ξ ◦ Ad ( v (1) v (1)) ( a ) − ˆ ϕ ξ ◦ Ad ( v (1)) ( a ) | < min (cid:26) δ (cid:16) ε , F (cid:17) , ε (cid:27) , for all a ∈ G ∪ F (125)and k Ad v ( t )( y ) − y k < ε , for all y ∈ F , and t ∈ [0 , . (126)Hence we have proven [ P ] with v given in (117). The proof that [ P n ] implies [ P n +1 ] with thesame v , v , . . . , v n as in [ P n ] can be carried out in the same way, by the repeated use of Lemma4.15. Hence we obtain a sequence { v n } ∞ n =0 of norm-continuous paths v n : [0 , → U ( A G Γ ) with v n (0) = I A Γ satisfying (120) (121) (122).We define norm continuous paths y, z : [0 , ∞ ) → U ( A G Γ ) by y ( t ) := v ( t ) v ( t ) · · · v j − (1) v j +1 ( t − [ t ]) , j ≤ t < j + 1 , j = 0 , , , . . . ,z ( t ) := v ( t ) v ( t ) · · · v j − (1) v j ( t − [ t ]) , j ≤ t < j + 1 j = 0 , , , . . . . (127)Here [ t ] denotes the largest integer less than or equal to t . Then as in section 2 of [KOS], for any a ∈ A loc , Γ , the limit γ ( a ) := lim t →∞ Ad ( z ( t )) ( a ) , γ ( a ) := lim t →∞ Ad ( y ( t )) ( a ) (128)21xist because of (122) and the fact that S Λ n ⊂ F n . These limit define endomorphisms γ , γ on A Γ . Furthermore, because of (122) and the fact that Ad ( v n − (1) ∗ v n − (1) ∗ · · · ) ( S Λ n ) ⊂ F n , by thedefinition (118) and (119), for any x ∈ A loc , Γ , the limitlim j →∞ Ad ( v j (1) ∗ v j − (1) ∗ · · · v (1) ∗ ) ( x ) =: a x (129)lim j →∞ Ad ( v j − (1) ∗ v j − (1) ∗ · · · v (1) ∗ ) ( x ) =: b x (130)exist. For these limits, we have γ ( a x ) = x , and γ ( b x ) = x , for all x ∈ A loc , Γ . Therefore, γ and γ are automorphisms. By , of [ P n ], we also have ϕ ◦ γ = ˆ ϕ ξ | A Γ ◦ γ = ˆ ϕ ξ | A Γ ◦ γ = ϕ ◦ γ . (131)Let Ξ Γ be an automorphism given by Ξ Γ := γ ◦ γ − on A . Define a norm-continuous path w : [0 , ∞ ) → U ( A G Γ ) by w ( t ) := z ( t ) y ( t ) ∗ , t ∈ [0 , ∞ ) . (132)We have Ξ Γ ( x ) = γ ◦ γ − ( x ) = lim t →∞ Ad( w ( t ))( x ) , x ∈ A Γ , (133)and w (0) = I . From (131), we have ϕ ◦ Ξ Γ = ϕ . This completes the proof. (cid:3) Now we are ready to prove Theorem 1.11. Let ω and ω be elements of SP G ( A ). Proof of ”if ” part of Theorem 1.11.
Suppose that c ω ,R = c ω ,R . From Lemma 2.5, we have c ω ,L = c ω ,L . For each ζ = L, R and i = 0 ,
1, let ( L ω i ,ς , ρ ω i ,ς , u ω i ,ς , σ ω i ,ς ) be a quadruple associatedto ( ω i | A ς , τ ς ). By Remark 1.8, we may assume that σ R := σ ω ,R = σ ω ,R and σ L := σ ω ,L = σ ω ,L . For each ζ = L, R and i = 0 ,
1, the triple ( L ω i ,ς , ρ ω i ,ς , u ω i ,ς ) is an irreducible covariantrepresentations of the twisted C ∗ -dynamical system Σ ( σ ς )Γ ς . By Lemma 1.1, u ω i ,ς has an irreducibledecomposition given by a set of Hilbert spaces {K γ,i,ς | γ ∈ P σ ς } . For each ς = L, R , fix some α ς ∈ P σ ς . The spaces K α ς ,i,ς i = 0 , ξ i,ς ∈ K α ς ,i,ς for each ς = L, R and i = 0 , ς = L, R and i = 0 ,
1, let ˆ ϕ ξ i,ς be a state on B ( H α ς ) ⊗ C ∗ (Σ ( σ ς ) ς ) given by ( L ω i ,ς , ρ ω i ,ς , u ω i ,ς , σ ω i ,ς )(defined in Notation 4.1 (52) with H , π, u, ξ replaced by L ω i ,ς , ρ ω i ,ς , u ω i ,ς , ξ i,ς ). Let ϕ i,ς be the re-striction of ˆ ϕ ξ i,ς onto A Γ ς . By the definition, ϕ ,ς , ϕ ,ς are quasi-equivalent to ω | A ς , ω | A ς ,respectively.By Proposition 4.2, there exist Ξ ς ∈ AInn G ( A ς ) such that ϕ ,ς = ϕ ,ς ◦ Ξ ς , ς = L, R . Recallthat ω , ω are quasi-equivalent to ω | A L ⊗ ω | A R and ω | A L ⊗ ω | A R respectively from the splitproperty. (Remark 1.5.) Hence we obtain ω ∼ q . e . ω | A L ⊗ ω | A R ∼ q . e . ϕ ,L ⊗ ϕ ,R = ( ϕ ,L ◦ Ξ L ) ⊗ ( ϕ ,R ◦ Ξ R ) = ( ϕ ,L ⊗ ϕ ,R ) ◦ (Ξ L ⊗ Ξ R ) ∼ q . e . ( ω | A L ⊗ ω | A R ) ◦ (Ξ L ⊗ Ξ R ) ∼ q . e . ω ◦ (Ξ L ⊗ Ξ R ) . (134)This completes the proof. (cid:3) roof of ”only if ” part of Theorem 1.11. Suppose that ω ∼ split ,τ ω . Then there existautomorphisms Ξ L ∈ AInn G ( A L ) and Ξ R ∈ AInn G ( A R ) such that ω and ω ◦ (Ξ L ⊗ Ξ R ) are quasi-equivalent. From the split property, we have ω ∼ q . e . ω | A L ⊗ ω | A R and ω ◦ (Ξ L ⊗ Ξ R ) ∼ q . e . ( ω | A L ◦ Ξ L ) ⊗ ( ω | A R ◦ Ξ R ). Combining these, we see that ω | A R and ω | A R ◦ Ξ R are quasi-equivalent.For each ζ = L, R and i = 0 ,
1, let ( L ω i ,ς , ρ ω i ,ς , u ω i ,ς , σ ω i ,ς ) be a quadruple associated to( ω i | A ς , τ ς ). From ω | A R ∼ q . e . ω | A R ◦ Ξ R , ρ ω ,R ◦ Ξ R is an irreducible ∗ -representation of A R on L ω ,R , which is quasi-equivalent to the GNS representation of ω | A R . Furthermore, the projectiveunitary representation u ω ,R of G with 2-cocycle σ ω ,R satisfies ρ ω ,R ◦ Ξ R ◦ τ R ( g ) ( a ) = ρ ω ,R ◦ τ R ( g ) ◦ Ξ R ( a ) = Ad ( u ω ,R ( g )) ◦ ρ ω ,R ◦ Ξ R ( a ) , a ∈ A R , g ∈ G. (135)From this, ( L ω ,R , ρ ω ,R ◦ Ξ R , u ω ,R , σ ω ,R ) is a quadruple associated to ( ω | A R , τ R ). Hence weobtain c ω ,R = c ω ,R . This proves the claim. (cid:3) Acknowledgment.
This work was supported by JSPS KAKENHI Grant Number 16K05171 and 19K03534.
A Basic Notation
For a Hilbert space H , B ( H ) denotes the set of all bounded operators on H . If V : H → H is a linear/anti-linear map from a Hilbert space H to another Hilbert space H , then Ad( V ) : B ( H ) → B ( H ) denotes the map Ad( V )( x ) := V xV ∗ , x ∈ B ( H ).For a set X , F ⋐ X means that F is a finite subset of X . For a finite set S , | S | indicates thenumber of elements in S .For a C ∗ -algebra B , we denote by B the set of all elements in B with norm less than or equalto 1 and by B + , the set of all positive elements in B . For a state ω , ϕ on a C ∗ -algebra B , wewrite ω ∼ q . e . ϕ when they are quasi-equivalent. We denote by Aut B the group of automorphismson a C ∗ -algebra B . For a unital C ∗ -algebra B , the unit of B is denoted by I B . For a Hilbertspace we write I H = I B ( H ) . For a unital C ∗ -algebra B , by U ( B ), we mean the set of all unitaryelements in B . For a Hilbert space we write U ( H ) for U ( B ( H )). For a C ∗ -algebra B and v ∈ B , weset Ad( v )( x ) := vxv ∗ , x ∈ B . For a state ϕ on B and a C ∗ -subalgebra C of B , ϕ | C indicates therestriction of ϕ to C . B Facts from [KOS] and [FKK]
In this section, we list up facts used/proven in [KOS] and [FKK].
Lemma B.1.
Let ϕ i , i = 0 , be pure states on a simple unital C ∗ -algebra A with GNS triple ( H i , π i , Ω i ) . Then for all F ⋐ A and ε > there exists an f ∈ A + , and a unit vector ζ ∈ H suchthat π ( f ) ζ = ζ, k f ( a − ϕ ( a ) I A ) f k < ε, for all a ∈ F . (136) Proof.
See proof of Lemma 2.3 of [FKK]. (cid:3)
By a basic consideration of 2-dimensional Hilbert space, we obtain the following.23 emma B.2.
For any ε > , a Hilbert space H , and unit vectors ξ , ξ ∈ H with k ξ − ξ k < √ ε ,there exists a unitary V on H such that ξ = V ξ and k V − I H k < ε . Notation B.3.
For any ε >
0, there exists a δ ,a ( ε ) > t ∈ R with | t | ≤ δ ,a ( ε ), we have (cid:12)(cid:12) e it − (cid:12)(cid:12) < ε . We will fix such δ ,a ( ε ) > ε > Theorem B.4.
For any ε > , there exists a δ ,a ( ε ) > satisfying the following: For any Hilbertspace H , unital C ∗ -algebra A acting irreducibly on H , and ξ, η ∈ H , if there is a unitary operator v on H satisfying k v − I k < δ ,a ( ε ) and η = vξ , there exists a self-adjoint h ∈ A such that e ih ξ = η and k h k ≤ δ ,a ( ε ) . Proof.
This is a quantitative version of the Kadison transitivity theorem. It can be obtained byprecise estimation of approximation in each step of the proof of the Kadison transitivity theorem. (cid:3)
The following Theorem is called Glimm’s Lemma
Theorem B.5.
Let A be a unital C ∗ -algebra acting on a Hilbert space H . Suppose that theintersection of A and the set of compact operators on H is { } . Then for any ε > , a pure state ϕ of A , a finite dimensional subspace K of H and a finite subset F of A , there exists a unit vector ξ in the orthogonal complement of K such that | ϕ ( x ) − h ξ, xξ i| < ε, for all x ∈ F . (137)Let us recall the following fact from [KOS]. Lemma B.6.
For any ε > and n ∈ N , there exists a δ ,a ( ε, n ) > satisfying the following.: Let H be an infinite dimensional Hilbert space and A a unital C ∗ -algebra acting irreducibly on H . Let { x i } ni =1 ⊂ A be a finite sequence satisfying P ni =1 x i x ∗ i = 1 . Let ξ, η ∈ H be unit vectors such that x ∗ i ξ and x ∗ j η are orthogonal for any i, j = 1 , . . . , d and (cid:12)(cid:12)(cid:10) ξ, x i x ∗ j ξ (cid:11) − (cid:10) η, x i x ∗ j η (cid:11)(cid:12)(cid:12) < δ ,a ( ε, n ) , i, j = 1 , . . . , n. (138) Then there exists a positive element h ∈ A + , such that (cid:13)(cid:13) ¯ h ( ξ + η ) (cid:13)(cid:13) < √ δ ,a (cid:16) ε (cid:17) e − π , and (139) (cid:13)(cid:13)(cid:0) I − ¯ h (cid:1) ( ξ − η ) (cid:13)(cid:13) < √ δ ,a (cid:16) ε (cid:17) e − π (140) hold for ¯ h := n X j =1 x j hx ∗ j . (141) Here the function δ ,a is given in Theorem B.4. Proof.
See section 3 of [KOS] for the proof. (cid:3) eferences [BC] E. B´edos and R. Conti. On discrete twisted C ∗ -dynamical systems, Hilbert C ∗ -modules andregularity. M¨unster J. Math. C ∗ -algebras. Trans. Amer. Math. Soc. Operator Algebras and Quantum Statistical Mechanics 1.
Springer-Verlag. (1986).[BR2] O. Bratteli and D. W. Robinson.
Operator Algebras and Quantum Statistical Mechanics 2.
Springer-Verlag. (1996).[E] G. Elliott. A Some simple C ∗ -algebras constructed as crossed products with discrete outerautomorphism groups. Publ. Res. Inst. Math. Sci. C ∗ algebras. Internat. J. Math. Z -Index of Symmetry Protected Topological Phases with Time Reversal Symme-try for Quantum Spin Chains Commun. Math. Phys. (2019). https://doi.org/10.1007/s00220-019-03521-5.[P] R. T. Powers. Representations of uniformly hyperfinite algebras and their associated vonNeumann rings. Ann. Math. Representations of Finite and Compact Groups.
AMS. (1995).[T] M. Takesaki.