A classification of pure states on quantum spin chains satisfying the split property with on-site finite group symmetries

We consider a set $SPG(\mathcal{A})$ of pure split states on a quantum spin chain $\mathcal{A}$ which are invariant under the on-site action $\tau$ of a finite group $G$. For each element $\omega$ in $SPG(\mathcal{A})$ we can associate a second cohomology class $c_{\omega,R}$of $G$. We consider a classification of $SPG(\mathcal{A})$ whose criterion is given as follows: $\omega_{0}$ and $\omega_{1}$ in $SPG(\mathcal{A})$ are equivalent if there are automorphisms $\Xi_{R}$, $\Xi_L$ on $\mathcal{A}_{R}$, $\mathcal{A}_{L}$ (right and left half infinite chains) preserving the symmetry $\tau$, such that $\omega_{1}$ and $\omega_{0}\circ( \Xi_{L}\otimes \Xi_{R})$ are quasi-equivalent. It means that we can move $\omega_{0}$ close to $\omega_{1}$ without changing the entanglement nor breaking the symmetry. We show that the second cohomology class $c_{\omega,R}$ is the complete invariant of this classification.


Introduction
It is well-known that the pure state space P (A) of a quantum spin chain A (UHFalgebra, see subsection 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 separable simple C * -algebras [KOS].
In this paper, we focus on the subset SP (A) of P (A) consisting of pure 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 Remark 1.5.) Here, ω| A L , ω| A R are restrictions of ω onto the left/right half-infinite chains. (See subsection 1.1.) Recall that two state being quasi-equivalent can be understood physically that they are "macroscopically same", because it means that one state can be represented as a local perturbation of the other and vice versa. On the other hand, a product state on A = A L ⊗ A R has no entanglement between A L and A R by definition. Therefore, a state with the split property can be physically understood as a state without macroscopic entanglement between the left half and the right half of the chain. Using the result of [P], [B], [FKK], [KOS], one can easily see that for any ω 0 , ω 1 ∈ SP (A), there exist asymptotically inner automorphisms Ξ L , Ξ R on A L , A R such that ω 1 | A L ∼ q.e. ω 0 | A L • Ξ L and ω 1 | A R ∼ q.e. ω 0 | A R • Ξ R . (Here ∼ q.e. means quasi-equivalence.) From this and the split property of ω 0 , ω 1 , we see that ω 1 and ω 0 •(Ξ L ⊗ Ξ R ) are quasi-equivalent. The product of automorphisms Ξ L ⊗ Ξ R clearly does not create/destroy any entanglement between the left half and the right half of the chain. Hence any ω 0 ∈ SP (A) 40 YOSHIKO OGATA can get "close to" any ω 1 ∈ 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 comes into 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 above to 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 0 , ω 1 ∈ SP G(A), can we always find such automorphisms giving ω 1 ∼ q.e. ω 0 • (Ξ L ⊗ Ξ R )? We show that the answer is no in general. The obstacle is given by the second cohomology class of the projective representation of G associated to ω ∈ SP G(A). We show that this second cohomology class is the complete invariant of this classification.
1.1. Setting. We consider the setting in this subsection throughout this paper. We use the basic notation in Appendix A freely. We start by summarizing standard setup of quantum spin chains on the infinite chain [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 notation Γ R = [0, ∞) ∩ Z and Γ L = (−∞, −1] ∩ Z.
For each z ∈ Z, let A {z} be an isomorphic copy of M d , and for any finite subset Λ ⊂ Z, we set A Λ = z∈Λ A {z} . For finite Λ, the algebra A Λ can be regarded as the set of all bounded operators acting on the Hilbert space z∈Λ C d . We use this identification freely. If Λ 1 ⊂ Λ 2 , the algebra A Λ 1 is naturally embedded in A Λ 2 by tensoring its elements with the identity. For an infinite subset Γ ⊂ Z, A Γ is given as the inductive limit of the algebras A Λ with Λ ∈ S Γ . We call 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 also set A loc,Γ = 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}. Namely, E i,j is a matrix such that E i,j e k = δ j,k e i . For each finite Λ ⊂ Z, we denote the tensor product k∈Λ I,J with I := (i k ) k∈Λ and J := (j k ) k∈Λ . We also use the notation Furthermore, we set e (Λ) I := k∈Λ e i k ∈ 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 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 for any finite subset I of Γ. We call the group homomorphism τ Γ : G → Aut A Γ , the on-site action of 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 (2) σ(g, e) = σ(e, g) = 1 for all g ∈ G. Define 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 2 (G, T). The identity of Z 2 (G, T) is given by 1 Z 2 (G,T) (g, h) := 1, for g, h ∈ G. For an arbitrary function b : G → T such that b(e) = 1, defines a 2-cocycle. The set of all 2-cocycles of this type forms a subgroup B 2 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, a map 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 associated to V , and call [σ] H 2 (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 dimensional projective 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 the only V -invariant subspaces of H. As G is a finite group, for any irreducible projective unitary representation (H, V, σ) of G, the Hilbert space H is finite dimensional. Projective unitary representations (H 1 , V 1 , σ 1 ) and (H 2 , V 2 , σ 2 ) are said to be unitarily equivalent if there is a unitary W : H 1 → H 2 such that W V 1 (g)W * = V 2 (g), with g ∈ G. Clearly if (H 1 , V 1 , σ 1 ) and (H 2 , V 2 , σ 2 ) are unitarily equivalent, the 2-cocycles σ 1 and σ 2 coincides. Schur's Lemma holds: let (H 1 , V 1 , σ 1 ) and (H 2 , V 2 , σ 2 ) be irreducible projective unitary representations of G, and W : H 1 → H 2 be a linear map such that W V 1 (g) = V 2 (g)W for all g ∈ G.
Then either W = 0 or (H 1 , V 1 , σ 1 ) and (H 2 , V 2 , σ 2 ) 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 2 (G, T), we denote by P σ , the set of all unitarily equivalence classes of irreducible projective representations with 2-cocycle σ. Note that P 1 Z 2 (G,T) is equal toĜ, the dual of G i.e. the set of equivalence classes of irreducible representations.
For each α ∈ P σ , we fix a representative (H α , V α , σ). We denote the dimension of H α (which is finite) by n α and fix an orthonormal basis {ψ

YOSHIKO OGATA
We will use the following vector later, in section 4 For each α ∈ P σ and k, j = 1, . . . , n α , define a function (V α ) k,j on G by As in Theorem III.1.1 of [S], from Schur's Lemma, we obtain the orthogonality relation: for all α, β ∈ P σ and k, j, t, s = 1, . . . , n α . Here |G| denotes the number of elements in G. In particular, P σ is a finite set. We freely identify α and V α . For example, and a projective unitary representation V . We repeatedly use the following fact.
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 irreducible projective unitary representations with 2-cocycle σ. This proves (9). The second statement (10) follows from the orthogonality relation (8).
Notation 1.2. When (9) holds, we say that V (or (H, V, σ)) has an irreducible decomposition given by Hilbert spaces {K γ | γ ∈ P σ }. We say V (or (H, V, σ)) contains all elements of P σ if K α = {0} for all α ∈ P σ . We say V (or (H, V, σ)) contains all elements of P σ with infinite multiplicity if dim K α = ∞ for all α ∈ P σ . We hence force omit W in (9) and identify H and α∈P σ H α ⊗ K α freely. The Hilbert space H α ⊗ K α can be naturally regarded as a closed subspace of H. We use this identification freely and call H α ⊗ K α the α-component of V (or (H, V, σ)).  (4), we obtain σσ b ∈ Z 2 (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.
1.3. The split property and projective representations. 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 respect to 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 . We also 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 Hilbert space K. See [T].
Remark 1.5. Let ω be a pure state on A. Let ω L be the restriction of ω to A L . Then ω satisfies the split property if and only if ω L ⊗ ω R is quasi-equivalent to ω. ( See [M1]. In Proposition 2.2 of [M1], 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 translation invariance.) Therefore, by the symmetric argument, if (H ω L , π ω L , Ω ω L ) is the GNS triple of ω L , then 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 . For each 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 for all g ∈ G. Furthermore, if another triple (L ω,ς ,ρ ω,ς ,ũ 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 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 By the Wigner Theorem, the * -automorphism

Main theorem. Let us introduce AInn
We denote by AInn G (A ς ) the set of all automorphisms which are asymptotically inner in A G ς .
For any pure states ϕ 1 and ϕ 2 on A, there is an asymptotically inner automorphism Ξ such that ϕ 2 = ϕ 1 •Ξ. This fact, which can be understood as a homogeneity of the pure states space, has been well-known from old time [P], and after further development of the techniques and results, [B], [FKK], the homogeneity is now proven for all the separable simple C * -algebras in [KOS].
In this paper, we consider the classification problem of SP G(A) with respect to the following equivalence relation.
Now we are ready to state our main theorem.
The "only if" part of the Theorem 1.11 is easy to prove. In order to prove "if" part of the Theorem, we note that if c ω 1 ,R = c ω 0 ,R holds, ω 0 and ω 1 give covariant representations of a twisted C * -dynamical systems Σ (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 * -algebras we consider. Namely, instead of considering A R , A L ,we consider the twisted crossed products C * (Σ In section 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 0 ,ς and u ω 1 ,ς contains α ς . This fact allows us to regard the problem as the homogeneity Γ ς ), with symmetry (section 4). The proof 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 G R , A G L . This requires some additional argument using the irreducible decompositions of u ω 0 ,ς , u ω 1 ,ς . This is given in section 4.
Before closing the introduction, let us remark on the relation between our result and the classification of SPT-phases. The notion of symmetry protected topological (SPT) phases was introduced by Gu and Wen [GW]. It is defined as follows: we consider the set of all Hamiltonians with some symmetry, which have a unique gapped ground state in the bulk. We regard two of such Hamiltonians are equivalent, if there is a smooth path inside that family, connecting them. By this equivalence relation, we may classify the Hamiltonians in this family. A Hamiltonian which has only on-site interaction can be regarded as a trivial one. The set of Hamiltonians equivalent to such trivial ones represents a trivial phase. If a phase is nontrivial, it is a SPT phase. A basic question to ask is how to show that a given Hamiltonian belongs to a SPT phase. A mathematically natural approach for this problem is to define an invariant of the classification. An important fact for us in that context, which was proven by Matsui in [M2], is that the unique gapped ground states satisfy the split property. Hence there is a second cohomology class associated to symmetric gapped Hamiltonians. In [O], we showed that the second cohomology class is actually an invariant of the classification of SPT-phases. The question if it is a complete invariant is still open. Our classification criteria in this paper is coarser than the criterion of the classification of SPT-phases. However, it still follows the same philosophy as the original classification. The ground states in SPT-phases are states with short range entanglement, and as we saw in the beginning, the split states can be seen as states without macroscopic entanglement. And the operations we consider in our classification are the ones which preserves the entanglement property and the symmetry.

Irreducible components in 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 dualĜ is a finite set and we denote the number of the elements inĜ by |Ĝ|. We use the following notation for any unitary/projective For a unitary representation (resp. projective unitary representation) of G, we denote byV the complex conjugate representation (resp. projective representation) of V . (See [S] section II.6.) Lemma 2.1. There is an l 0 ∈ N such that for any l ≥ l 0 , the tensor product U ⊗l contains any irreducible 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 is attained only if U (g) ∈ TI C d . By the condition (2), Note that for g ∈ G \ {e} converges to 0 because of |χ U (g)| < d. Therefore, for l large enough, the left hand side of (18) is non-zero. In other word, for l large enough, V is an irreducible component of U ⊗l . AsĜ is a finite set, (taking maximal such l over all V ∈Ĝ), this proves the Lemma.
From this we obtain the following.
Proof. Let l 0 be the number given in Lemma 2.1. For any σ ∈ Z 2 (G, T) and α, β ∈ P σ , α ⊗β is a genuine representation of G. Let V ∈Ĝ 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 0 . Therefore, for l ≥ l 0 , we have This means α ≺ β ⊗ U ⊗l .
(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 from this immediately for this case.
Next let us consider the case that the number of elements |P σ | in P σ , is larger than 1. From Lemma 2.2, choose l 0 ∈ N so that α ≺ β ⊗ U ⊗l for all l ≥ l 0 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 0 (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 1 + k 2 + · · · + k M m + k M m +1 with some l 0 ≤ 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 the choice of l 0 , we have 1 ≤ n (j) β,γ for any j = 1, . . . , M m + 1 and β, γ ∈ P σ . Fix some β 0 ∈ P σ such that β 0 ≺ u. From this, we get This completes the proof.
Now we are ready to show the main statement of this section. From the following Lemma, we see that for any ω ∈ SP G(A), u ω,ς contains all elements of P σ ω,ς with infinite multiplicity.
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 the number given in Lemma 2.3 for this fixed m. Let Λ be a subset of Γ such that |Λ| = N (σ) m . We may 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 representationũ of G onL with 2-cocycle σ, implementing τ Γ\Λ . There exists a unitary W : and More precisely, set I 0 = (i k ) k∈Λ ∈ {1, . . . , d} ×Λ , with i k = 1 for all k ∈ Λ. We define the Hilbert spaceL byL := ρ E It is straight forward to check (24).
By a straight forward calculation using (24), we can check that with g ∈ G commute with any element of ( Λ M d ) ⊗ CIL. Hence there exists a unitaryũ(g) onL such that (( Λ U (g))) * ⊗ IL) W u(g)W * = I Λ C d ⊗ũ(g). This gives (25). It is straight forward to check thatũ is a projective unitary representation of G with 2-cocycle σ implementing τ Γ\Λ .
From (25) and Lemma 2.3, we have This completes the proof.
Recall Definition 1.7. We note that c ω,R and c ω,L are not independent.
(See Theorem 1.31 V [T].) From (29), π (A L ) ⊂ π (A R ) and π (A L ) ∨ π (A R ) = B(H), we obtain Hence we obtain irreducible representations (H L , π L ) and (H R , π R ) of A L , A R such that 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 quasiequivalent.
By the τ -invariance of ω, there is a unitary representation V of G on H L ⊗ H R given by On the other hand, by Proposition 1.6, there are projective unitary representations for all a ∈ A L , b ∈ A R and g ∈ G. Note that 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 such that Let σ L , σ R ∈ Z 2 (G, T) be 2-cocycles of u L , u R respectively. From (35), we obtain (Here σ b is defined by (4).) This means 3. Twisted C * -dynamical system In this section we briefly recall basic facts about twisted C * -crossed product. Throughout this section, let Γ be an infinite subset of Z, and σ ∈ Z 2 (G, T). The quadruple (G, A Γ , τ Γ , σ) is a twisted C * -dynamical system which we denote by Σ (This is a simple version of [BC].) A covariant representation of 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 In this paper, we say the covariant representation (H, π, u) is irreducible if π is an irreducible representation 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 a product and * -operation as follows: for f 1 , f 2 , f ∈ C(G, A Γ ). The linear space C(G, A Γ ) which is a * -algebra with these operations is denoted by C(Σ (σ) Γ ). We will omit the symbol * for the multiplication (39).
For each a ∈ A Γ , ξ a : G g → δ g,e a ∈ A Γ defines an element of C * (Σ . Hence the C * -algebra A Γ can be regarded as a . 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] Note that λ e = ξ I A Γ is the identity of C * Σ (σ) Γ . We set Let (H, π, u) be an irreducible covariant representation of Σ (σ) Γ . The projective unitary representation u has an irreducible decomposition given by some Hilbert spaces {K γ | γ ∈ P σ } (Lemma 1.1 and Notation 1.2). Namely we have Note that (π × u) (λ g ) = u(g), g ∈ G.
(49) From this we have 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
Then u contains all elements of P σ with infinite multiplicity.
Notation 4.1. Let (H, π, u) be an irreducible covariant representation of Σ (σ) Γ with an irreducible decomposition of u given by a set of Hilbert spaces {K γ | γ ∈ P σ }. We use the symbolπ to denote the irreducible representation Here,ξ is an element of H α ⊗ H, given by (H, π, u, ξ). By the irreducibility of π,π is irreducible andφ ξ is a pure state on The goal of this section is to prove the following Proposition. (1) 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 , and (2) the automorphism Ξ Γ in 1. satisfies ϕ 1 = ϕ 0 • Ξ Γ . Remark 4.3. Basically, what we would like to do is to connect some π 0 -normal state ϕ 0 and some π 1 -normal state ϕ 1 via some Ξ Γ ∈ AInn G (A Γ ). Without symmetry, ϕ 0 and ϕ 1 can be taken to be pure 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 0 and ϕ 1 are τ Γ -invariant. If σ is trivial, there is a u i -invariant non-zero vector that we may find such pure τ Γ -invariant states ϕ 0 and ϕ 1 . But if the cohomology class 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 that the density matrix of ϕ i is a rank n α operator which commutes with u i .

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For the proof of Proposition 4.2, we use the machinery used in [FKK] and [KOS]. (See Appendix B.) However, as we would like to have a path in the fixed point algebra A G Γ , we need additional arguments. For that purpose, the following Lemma plays an important role.
As π is a * -representation, for each γ ∈ P σ , the map π γ : A G Γ 0 a → π γ (a) ∈ B(K γ ) is a * -representation and we have We claim that each π γ is an irreducible representation of A G Γ 0 , 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 ) converges to I H γ ⊗ x ∈ B(H γ ) ⊗ B(K γ ) ⊂ B(H) in the σ-strong topology, by the irreducibility of π and the Kaplansky density theorem. For this Since the left hand side of (59) converges to I H γ ⊗ x in the σ-strong topology, we conclude that Hence (56) holds. Looking at the γ-component of (59), we see that π γ A G Γ 0 = B(K γ ). Hence π γ is irreducible. This completes the proof.
We also setF 54 YOSHIKO OGATA Applying Lemma B.1 to thisε andF , and pure statesφ ξ i , i = 0, 1 of a simple unital C * -algebra B(H (α) and a unit vector ζ ∈ H (α) ⊗ H 1 such that For P (α,1) in (64) and the ζ in (69), we have Here we used (65), for the second equality. For the inequality we used (69) and R (α) ∈F (68). Therefore, P (α,1) ζ is not zero, and we may define a unit vector From this and two properties in (69) for any a ∈ F, we have by the choice ofε (67).
Remark 4.7. The main difference of the proof of Lemma 4.6 from [KOS], [FKK] is that in order to find h in A G Γ , we add R (α) to F. This allows us to replace ζ with ζ = Ω α ⊗ η. From this combined with Lemma 4.4, the problem is reduced to the Kadison transitivity for the irreducible (K α,1 , π α,1 (A G Γ )). Note that R (α) belongs By extending the C * -algebra we consider, we are allowed to have the projection P (α,i) (64) corresponding to the irreducible component of u i in the C * -algebra.
Let δ 2,a be the function given in Lemma B.4.
Here we used the function δ 1 introduced in Lemma 4.10.
. Combining these, we see that ω 1 | A R and ω 0 | A R • Ξ R are quasi-equivalent.
From this, (L ω 0 ,R , ρ ω 0 ,R •Ξ R , u ω 0 ,R , σ ω 0 ,R ) is a quadruple associated to (ω 1 | A R , τ R ). Hence we obtain c ω 1 ,R = c ω 0 ,R . This proves the claim. For a set X , F X means that F is a finite subset of X . For a finite set S, |S| indicates the number of elements in S.
For a C * -algebra B, we denote by B 1 the set of all elements in B with norm less than or equal to 1 and by B +,1 the set of all positive elements in B 1 . For a state ω, ϕ on a C * -algebra B, we write ω ∼ q.e. ϕ when they are quasi-equivalent. We denote by Aut B the group of automorphisms on a C * -algebra B. For a unital C * -algebra B, the unit of B is denoted by I B . For a Hilbert space we write I H = I B(H) . For a unital C * -algebra B, by U (B), we mean the set of all unitary elements in B. For a Hilbert space we write U(H) for U (B(H)). For a C * -algebra B and v ∈ B, we set Ad(v)(x) := vxv * , x ∈ B. For a state ϕ on B and a C * -subalgebra C of B, ϕ| C indicates the restriction of ϕ to C.