The Picard groups of unital inclusions of unital C ∗ -algebras induced by involutive equivalence bimodules
aa r X i v : . [ m a t h . OA ] J un THE PICARD GROUPS OF UNITAL INCLUSIONS OF UNITAL C ∗ -ALGEBRAS INDUCED BY INVOLUTIVE EQUIVALENCEBIMODULES KAZUNORI KODAKA
Abstract.
Let A be a unital C ∗ -algebra and X an invulutive A − A -equivalencebimodule. Let A ⊂ C X be the unital inclusion of unital C ∗ -algebras inducedby X . We suppose that A ′ ∩ C X = C
1. We shall compute the Picard groupof the unital inclusion A ⊂ C X . Introduction
Let A be a unital C ∗ -algebra and X an A − A -equivalence bimodule. Following[7], we say that X is involutive if there exists a conjugate linear map x x ♮ on X such that(1) ( x ♮ ) ♮ = x , x ∈ X ,(2) ( a · x · b ) ♮ = b ∗ · x ♮ · a ∗ , x ∈ X , a, b ∈ A ,(3) A h x, y ♮ i = h x ♮ , y i A , x, y ∈ X ,where A h− , −i and h− , −i A are the left and the right A -valued inner products on X ,respectively. We call the above conjugate linear map an involution on X . For each A − A -equivalence bimodule X , e X denotes its dual A − A -equivalence bimodule.For each x ∈ X , e x denotes the element in e X induced by x . For each involutive A − A -equivalence bimodule X , let L X be the linking C ∗ -algebra for X defined inBrown, Green and Rieffel [1]. Following [7], we define the C ∗ -subalgebra C X of L X by C X = { (cid:20) a x e x ♮ a (cid:21) | a ∈ A, x ∈ X } . We regard A as a C ∗ -subalgebra of C X , that is, A = { (cid:20) a a (cid:21) | a ∈ A } .In [5], we defined the Picard of unital inclusion of unital C ∗ -algebras A ⊂ C .We denote it by Pic( A, C ). In this paper, we shall compute Pic(
A, C X ) underthe assumption that A ′ ∩ C X = C
1. Let us explain the strategy of computingPic(
A, C X ). Let f A be the homomorphism of Pic( A, C X ) to Pic( A ) defined in [5],where Pic( A ) is the Picard group of A . We compute Ker f A and Im f A , the kernelof f A and the image of f A , respectively and we construct a homomorphism g A ofIm f A to Pic( A, C X ) with f A ◦ g A = id Pic( A ) . We can compute Pic( A, C X ) in theabove way. 2. Preliminaries
We recall the definition of the Picard group for a unital inclusion of unital C ∗ -algebras A ⊂ C . Let Y be a C − C -equivalence bimodule and X its closed subspacesatisfying Conditions (1), (2) in [11, Definition 2.1]. Let Equi( A, C ) be the set ofall such pairs (
X, Y ) as above. We define an equivalence relation “ ∼ ” as follows: Mathematics Subject Classification.
Primary 46L05, Secondary 46L08.
Key words and phrases. inclusions of C ∗ -algebras, involutive equivalence bimodules, Picardgroups, strong Morita equivalence. or ( X, Y ) , ( Z, W ) ∈ Equi(
A, C ), (
X, Y ) ∼ ( Z, W ) in Equi(
A, C ) if and only ifthere is a C − C -equivalence bimodule isomorphism Φ of Y onto W such thatthe restriction of Φ to X , Φ | X is an A − A -equivalence bimodule isomorphism X onto Z . We denote by [ X, Y ], the equivalence class of (
X, Y ) in Equi(
A, C ). LetPic(
A, C ) = Equi(
A, C ) / ∼ . We define the product in Pic( A, C ) as follows: For(
X, Y ) , ( Z, W ) ∈ Pic(
A, C )[ X, Y ][ Z, W ] = [ X ⊗ A Z , Y ⊗ C W ] , where the A − A -equivalence bimodule X ⊗ A Z is identified with the closed subspace“ X ⊗ C Z ” of Y ⊗ C W by [5, Lemma 3.1] and “ X ⊗ C Z ” is defined by the closureof linear span of the set { x ⊗ z ∈ Y ⊗ C W | x ∈ X, z ∈ Z } by [5] and easy computations, Y ⊗ C W and its closed subspace X ⊗ A Z satisfy Condi-tions (1), (2) in [11, Definition 2.1] and Pic( A, C ) is a group. We regard (
A, C ) as anelement in Equi(
A, C ) in the evident way. Then [
A, C ] is unit element in Equi(
A, C )in Pic(
A, C ). For any element (
X, Y ) ∈ Equi(
A, C ), ( e X, e Y ) ∈ Equi(
A, C ) and[ e X, e Y ] is the inverse element of [ X, Y ] in Pic(
A, C ). We call the group Pic(
A, C )defined in the above, the Picard group of the unital inclusion of unital C ∗ -algebras A ⊂ C .Let f A be the homomorphism of Pic( A, C ) to Pic( A ) defined by f A ([ X, Y ]) = [ X ]for any ( X, Y ) ∈ Equi(
A, C ). 3.
Kernel
Let A be a unital C ∗ -algebra and X an involutive A − A -equivalence bimodule.Let A ⊂ C X be the unital inclusion of unital C ∗ -algebras induced by X and wesuppose that A ′ ∩ C X = C
1. Let f A be the homomorphism of Pic( A, C X ) to Pic( A )defined by f A ([ M, N ]) = [ M ]for any ( M, N ) ∈ Equi(
A, C X ). In this section, we compute Ker f A . Let ( M, N ) ∈ Equi(
A, C X ). We suppose that [ M, N ] ∈ Ker f A . Then [ M ] = [ A ] in Pic( A ) and by[5, Lemma 7.5], there is a β ∈ Aut ( A, C X ) such that[ M, N ] = [
A, N β ]in Pic( A, C X ) where Aut ( A, C X ) is the group of all automorphisms β such that β ( a ) = a for any a ∈ A and N β is the C X − C X -equivalence bimodule induced by β which is defined in [5, Section 2]. By the above discussions, we obtain the followinglemma. Lemma 3.1.
With the above notation,
Ker f A = { [ A, N β ] ∈ Pic(
A, C X ) | β ∈ Aut ( A, C X ) } . Let Aut(
A, C X ) be the group of all automorphisms α of C X such that the re-striction of α to A , α | A is an automorphism of A . Then Aut ( A, C X ) is a normalsubgroup of Aut( A, C X ). Let π be the homomorphism of Aut( A, C X ) to Pic( A, C X )defined by π ( α ) = [ M α , N α ]for any α ∈ Aut(
A, C X ), where ( M α , N α ) is the element in Equi( A, C X ) inducedby α ∈ Aut(
A, C X ) (See [5, Section 3]). By Lemma 3.1, π (Aut ( A, C X )) = Ker f A and [5, Lemma 3.4],Ker π ∩ Aut ( A, C X ) = Int( A, C X ) ∩ Aut ( A, C X ) , here Int( A, C X ) is the group of all Ad( u ) such that u is a unitary element in A .HenceKer π ∩ Aut ( A, C X ) = { Ad( u ) ∈ Aut ( A, C X ) | u is a unitary element in A } = { Ad( u ) ∈ Aut ( A, C X ) | u is a unitary element in A ′ ∩ A } . Since A ′ ∩ C X = C A ′ ∩ A = C
1. Thus we can see that Ker π ∩ Aut ( A, C X ) = { } .It follows that we can obtain that the following lemma. Lemma 3.2.
With the above notation,
Ker f A ∼ = Aut ( A, C X ) . Let A Aut ♮A ( X ) be the group of all involutive A − A -equivalence bimodule auto-morphisms of X . Let E A be the conditional expectation from C X onto A definedby E A ( (cid:20) a x e x ♮ a (cid:21) ) = (cid:20) a a (cid:21) for any a ∈ A , x ∈ X . Then E A is of Watatani index-finite type by [7, Lemma 3.4]. Lemma 3.3.
Withe the above notation, E A = E A ◦ β for any β ∈ Aut ( A, C X ) .Proof. Let β ∈ Aut ( A, C X ). Then E A ◦ β is also a conditional expectation from C X onto A . Since A ′ ∩ C X = C
1, by Watatani [17, Proposition 1.4.1], E A = E A ◦ β . (cid:3) Lemma 3.4.
With the above notation, for any β ∈ Aut ( A, C X ) , there is theunique θ ∈ Aut ♮ ( X ) such that β ( (cid:20) a x e x ♮ a (cid:21) ) = " a θ ( x ) ] θ ( x ♮ ) a for any a ∈ A , x ∈ X .Proof. For any x ∈ X , let β ( (cid:20) x e x ♮ (cid:21) ) = (cid:20) b y e y ♮ b (cid:21) , where b ∈ A , y ∈ X . Then by Lemma 3.3, (cid:20) b b (cid:21) = ( E A ◦ β )( (cid:20) x e x ♮ (cid:21) ) = E A ( (cid:20) x e x ♮ (cid:21) ) = (cid:20) (cid:21) . Hence b = 0. Thus β ( (cid:20) x e x ♮ (cid:21) ) = (cid:20) y e y ♮ (cid:21) . We define a map θ on X by θ ( x ) = y, where y is the element in X defined as above. Then clearly θ is linear and since β ( (cid:20) x ♮ e x (cid:21) ) = β ( (cid:20) x e x ♮ (cid:21) ) ∗ = (cid:20) y ♮ e y (cid:21) , we obtain that θ ( x ♮ ) = y ♮ = θ ( x ) ♮ . Hence θ preserves the involution ♮ . Also, for any a ∈ A , x ∈ X , " θ ( a · x ) θ ( ^ ( a · x ) ♮ ) 0 = β ( (cid:20) a · xa · e x ♮ (cid:21) ) = β ( (cid:20) a a (cid:21) (cid:20) x e x ♮ (cid:21) )= (cid:20) a a (cid:21) (cid:20) θ ( x ) θ ( e x ♮ ) 0 (cid:21) = (cid:20) a · θ ( x ) a · θ ( e x ♮ ) 0 (cid:21) . ence θ ( a · x ) = a · θ ( x ) for any a ∈ A , x ∈ X . Similarly θ ( x · a ) = θ ( x ) · a for any a ∈ A , x ∈ X . Furthermore, for any x, y ∈ X , (cid:20) A h θ ( x ) , θ ( y ) i A h θ ( x ) , θ ( y ) i (cid:21) = " θ ( x ) ] θ ( x ) ♮ θ ( y ) ♮ g θ ( y ) 0 (cid:21) = β ( (cid:20) x e x ♮ (cid:21) (cid:20) y ♮ e y (cid:21) )= β ( (cid:20) A h x , y i A h x , y i (cid:21) )= (cid:20) A h x , y i A h x , y i (cid:21) . Hence A h θ ( x ) , θ ( y ) i = A h x , y i for any x, y ∈ X . Similarly for any x, y ∈ X , (cid:20) h θ ( x ) , θ ( y ) i A h θ ( x ) , θ ( y ) i A (cid:21) = (cid:20) h x , y i A h x , y i A (cid:21) . Hence h θ ( x ) , θ ( y ) i A = h x , y i A for any x, y ∈ X . Thus θ ∈ A Aut ♮A ( X ). Next, let θ ∈ A Aut ♮A ( X ). Then let β be a map on C X defined by β ( (cid:20) a x e x ♮ a (cid:21) ) = " a θ ( x ) ] θ ( x ) ♮ a for any a ∈ A , x ∈ X . Then by easy computations, β ∈ Aut ( A, C X ). Therefore,we obtain the conclusion. (cid:3) Corollary 3.5.
With the above notation,
Aut ( A, C X ) ∼ = A Aut ♮A ( X ) .Proof. This is immediate by Lemma 3.4. (cid:3)
Let A Aut A ( X ) be the group of all A − A -equivalence bimodule automorphisms.Since X is an A − A -equivalence bimodule, A Aut A ( X ) is isomorphic to U ( A ′ ∩ A ),the group of all unitary elements in A ′ ∩ A . Since A ′ ∩ C X = C U ( A ′ ∩ A ) = T T is the 1-dimensional torus. Hence A Aut ♮A ( X ) is isomorphic to a subgroupof T . But since λ ♮ for any λ ∈ T , A Aut ♮A ( X ) ∼ = T
1. Bythe above discussions, we can obtain the following proposition.
Proposition 3.6.
With the above notation,
Ker f A ∼ = T .Proof. This is immediate by Lemma 3.2, Corollary 3.5 and the above discussions. (cid:3) A result on strongly Morita equivalent unital inclusions ofunital C ∗ -algebras In this section, we shall prove the following result: Let H be a finite dimensional C ∗ -Hopf algebra and H its dual C ∗ -Hopf algebra. Let ( ρ, u ) and ( σ, v ) be twistedcoactions of H on unital C ∗ -algebras A and B , respectively. Let A ⊂ A ⋊ ρ,u H and B ⊂ B ⋊ σ,v H be unital inclusions of unital C ∗ -algebras. We suppose that theyare strongly Morita equivalent with respect to an A ⋊ ρ,u H − B ⋊ σ,v H -equivalencebimodule Y and its closed subspace X . And we suppose that A ′ ∩ ( A ⋊ ρ,u H ) = C γ, w ) of H on B and a twisted coaction λ of H on X satisfying the following:(1) ( ρ, u ) and ( γ, w ) are strongly Morita equivalent with respect to λ ,(2) B ⋊ σ,v H = B ⋊ γ,w H ,(3) Y ∼ = X ⋊ λ H as A ⋊ ρ,u H − B ⋊ σ,v H - equivalence bimodules. n the next section, we shall use this result in the case of Z -actions, where Z = Z / Z . We shall use the results in [12] in order to prove the above result.First we recall [12].Let H be a finite dimensional C ∗ -Hopf algebra. We denote its comultiplication,counit and antipode by ∆, ǫ , and S , respectively. We shall use Sweedler’s notation∆( h ) = h (1) ⊗ h (2) for any h ∈ H which surppresses a possible summation whenwe write comultiplications. We denote by N the dimension of H . Let H be thedual C ∗ -Hopf algebra of H . We denote its comultiplication, counit and antipode by∆ , ǫ and S , respectively. There is the distinguished projection e in H . We notethat e is the Haar trace on H . Also, there is the distinguished projection τ in H which is the Haar trace on H . Since H is finite dimensional, H ∼ = ⊕ Kk =1 M d k ( C )as C ∗ -algebras, where M n ( C ) is the n × n - matrix algebra over C . Let { w kij | k = 1 , , . . . , K, i, j = 1 , , . . . , d k } be a basis of H satisfying Szyma´nski and Peligrad’s [16, Theorem 2.2,2], which iscalled a system of comatrix units of H , that is, the dual basis of a system of matrixunits of H .Let A be a unital C ∗ -algebra and ( ρ, u ) a twisted coaction of H on A , that is, ρ is a weak coaction of H on A and u is a unitary element in A ⊗ H ⊗ H satisfyingthat(1) ( ρ ⊗ id) ◦ ρ = Ad( u ) ◦ (id ⊗ ∆ ) ◦ ρ ,(2) ( u ⊗ )(id ⊗ ∆ ⊗ id)( u ) = ( ρ ⊗ id ⊗ id)( u )(id ⊗ id ⊗ ∆ )( u ),(3) (id ⊗ h ⊗ ǫ )( u ) = (id ⊗ ǫ ⊗ h )( u ) = ǫ ( h )1 for any h ∈ H .For a twisted coaction ( ρ, u ) of H on A , we can consider the twisted action of H on A and its unitary element b u defined by h · ρ,u x = (id ⊗ h )( ρ ( x )) , b u ( h, l ) = (id ⊗ h ⊗ l )( u )for any x ∈ A , h, l ∈ H . We call it the twisted action of H on A induced by( ρ, u ). Let A ⋊ ρ,u H be the twisted crossed product of A by the twisted action of H induced by ( ρ, u ). Let x ⋊ ρ,u h be the element in A ⋊ ρ,u H induced by x ∈ A and h ∈ H . Let b ρ be the dual coaction of H on A ⋊ ρ,u H defined by b ρ ( x ⋊ ρ,u h ) = ( x ⋊ ρ,u h (1) ) ⊗ h (2) for any x ∈ A , h ∈ H . Let E ρ,u be the canonical conditional expectation from A ⋊ ρ,u H onto A defined by E ρ,u ( x ⋊ ρ,u h ) = τ ( h ) x for any x ∈ A , h ∈ H . Let Λ be the set of all triplets ( i, j, k ), where i, j =1 , , . . . , d k and k = 1 , , . . . , K with P Kk =1 d k = N . Let W ρI = √ d k ⋊ ρ,u w kij foany I = ( i, j, k ) ∈ Λ. By [9, Proposition 3.18], { ( W ρ ∗ I , W ρI ) } I ∈ Λ is a quasi-basis for E ρ,u .Let A and B be unital C ∗ -algebras and let ( ρ, u ) and ( σ, v ) be twisted coactionsof H on A and B , respectively. Let A ⋊ ρ,u H and B ⋊ σ,v H be the twistedcrossed products of A and B by ( ρ, u ) and ( σ, v ), respectively. We denote themby C and D , respectively. Then we obtain unital inclusions of unital C ∗ -algebras, A ⊂ C and B ⊂ D . We suppose that A ⊂ C and B ⊂ D are strongly Moritaequivalent with respect to a C − D -equivalence bimodule Y and its closed subspace X . We also suppose that A ′ ∩ C = C
1. Then B ′ ∩ D = C F B from D onto B and a conditional expectation E X from Y onto X with respect to E ρ,u and F B satisfying Conditions (1)–(6) in [11, Definition 2.4]. Since B ′ ∩ D = C
1, byWatatani [17, Proposition 1.4.1], F B = E σ,v , the canonical conditional expectationfrom D onto B . Furthermre, by [11, Section 6], we can see that the unital inclusions f unital C ∗ -algebras, C ⊂ C and D ⊂ D are strongly Morita equivalent withrespect to the C − D -equivalence bimodule Y and its closed subspace Y , where C = C ⋊ b ρ H and D = D ⋊ b σ H and b ρ and b σ are the dual coactions of ( ρ, u ) and( σ, v ), respectively. And Y is defined as follows: We regard C and D as a C − A -equivalence bimodule and a D − B -equivalence bimodule in the usual way as in[11, Section 4], respectively. Let Y = C ⊗ A X ⊗ B e D . Let E Y be the conditionalexpectation from Y onto Y with respect to E ρ,u and E σ,v defined by E Y ( c ⊗ x ⊗ e d ) = 1 N c · x · d ∗ for any c ∈ C , d ∈ D , x ∈ X , where E ρ,u and E σ,v are the canonical conditionalexpectations from C and D onto C and D , respectively. We regard Y as a closedsubspace of Y by the injective linear map φ from Y into Y defined by φ ( y ) = X I,J ∈ Λ W ρ ∗ I ⊗ E X ( W ρI · y · W σ ∗ J ) ⊗ ] W σ ∗ J for any y ∈ Y . By [12, Sections 3 and 4], there are a coaction β of H on D and acoaction µ of H on Y such that ( C, D, Y, b ρ, β, H ) is a covariant system, that is, µ is a coaction of H on Y with respect to ( C, D, b ρ, β ). We define the action of H on Y induced by µ as follows: For any ψ ∈ H , y ∈ Y , ψ · µ y = N E Y ((1 ⋊ ρ,u ⋊ b ρ ψ ) · φ ( y ) · (1 ⋊ σ,v ⋊ b σ τ ))By [12, Remark 3.1], for any ψ ∈ H , y ∈ Y , ψ · µ y = X I ∈ Λ [ ψ · b ρ W ρ ∗ I ] · E X ( W ρI · y ) . Also, we define the action of H on D induced by β as follows: For any ψ ∈ H , y, z ∈ Y , ψ · β h y, z i D = h S ( ψ ∗ (1) ) · µ y , ψ (2) · µ z i D , where we regard D as the linear span of the set {h y, z i D | y, z ∈ Y } . Lemma 4.1.
For any y ∈ Y , τ · µ y = E X ( y ) .Proof. By routine computations, we obtain the lemma. Indeed, by [12, Remark3.1], for any y ∈ Y , τ · µ y = X I ∈ Λ [ τ · b ρ W ρ ∗ I ] · E X ( W ρI · y )= X i,j,k [ τ · b ρ ( p d k ⋊ ρ,u w kij ) ∗ ] · E X (( p d k ⋊ ρ,u w kij ) · y )= X i,j,j ,j ,j ,k [ τ · b ρ ( b u ( S ( w kj j ) , w kij ) ∗ [ w k ∗ j j · b ρ p d k ] ⋊ ρ,u w k ∗ j j )] · E X (( p d k ⋊ ρ,u w kij ) · y )= X i,j,j ,j ,k p d k [ τ · b ρ ( b u ( S ( w kj j ) , w kij ) ∗ ⋊ ρ,u w k ∗ j j )] · E X (( p d k ⋊ ρ,u w kij ) · y )= X i,j,j ,j ,j ,k p d k ( b u ( S ( w kj j ) , w kij ) ∗ ⋊ ρ,u w k ∗ j j ) τ ( w k ∗ j j ) · E X (( p d k ⋊ ρ,u w kij ) · y )= X i,j,j ,j ,k p d k ( b u ( S ( w kj j ) , w kij ) ∗ ⋊ ρ,u τ ( w k ∗ j j )) · E X (( p d k ⋊ ρ,u w kij ) · y ) . ince τ ◦ S = τ , τ ( w k ∗ j j ) = τ ( S ( w kjj )) = ( τ ◦ S )( w kjj ) = τ ( w kjj ). Hence τ · µ y = X i,j,j ,j ,k p d k ( b u ( S ( w kj j ) , w kij ) ∗ ⋊ ρ,u · E X (( p d k ⋊ ρ,u w kij τ ( w kjj )) · y )= X i,j ,j ,k p d k ( b u ( S ( w kj j ) , w kij ) ∗ ⋊ ρ,u · E X (( p d k ⋊ ρ,u τ ( w kij )) · y )= X i,j ,j ,k d k ( b u ( S ( w kj j ) , w kij τ ( w kij )) ∗ ⋊ ρ,u · E X ( y ) . Since τ = τ ∗ , τ ( w kij ) = τ ∗ ( w kij ) = τ ( S ( w k ∗ ij )) = τ ( w kj i ). Hence τ · µ y = X i,j ,j ,k d k ( b u ( S ( w kj j ) , w kij τ ( w kj i )) ∗ ⋊ ρ,u · E X ( y )= X j ,j ,k d k ( b u ( S ( w kj j ) , τ ( w kj j )) ∗ ⋊ ρ,u · E X ( y )= X j ,k d k ( b u ( S ( τ ( w kj j )) , ∗ ⋊ ρ,u · E X ( y )= X j ,k d k ǫ ( S ( τ ( w kj j ))) · E X ( y ) . Since ǫ ◦ S = ǫ , ǫ ( S ( τ ( w kj j ))) = τ ( w kj j )1. Hence τ · µ y = X j k d k τ ( w kj j ) E X ( y ) = X j k d k τ ( w kj j ) E X ( y ) = N τ ( e ) E X ( y )= E X ( y )since e = N P j,k d k w kjj . Therefore, we obtain the conclusion. (cid:3) We recall that the unital inclusions of unital C ∗ -algebras, C ⊂ C and D ⊂ D are strongly Morita equivalent with respect to Y and its closed subspace Y . Also, C ⊂ C and D ⊂ D ⋊ β H are strongly Morita equivalent with respect to the C − D ⋊ β H -equivalence bimodule Y ⋊ µ H and its closed subspace Y , where Y ⋊ µ H is the crossed product of Y by the coaction µ and it is a C − D ⋊ β H -equivalencebimodule (See [10]). Hence the unital inclusions D ⊂ D and D ⊂ D ⋊ β H arestrongly Morita equivalent with respect to the D − D ⋊ β H -equivalence bimodule f Y ⊗ C ( Y ⋊ µ H ) and its closed subspace e Y ⊗ C Y . Then since e Y ⊗ C Y is isomorphicto D as D − D -equivalence bimodule, we can see that there is an isomorphism Ψof D onto D ⋊ β H which is defined as follows: Since Y is a C − D - equivalencebimodule, there are elements y , . . . , y n ∈ Y such that P ni =1 h y i , y i i D = 1. Let Ψbe the map from D to D ⋊ β H defined byΨ( d ) = X i,j h d · e y i ⊗ y i , e y j ⊗ y j i D ⋊ β H for any d ∈ D . By [12, Section 5], Ψ is an isomorphism of D onto D ⋊ β H satisfying that Ψ( d ) = d for any d ∈ D and that E β ◦ Ψ = E σ,v , where E β is acanonical conditional expectation from D ⋊ β H onto D and E σ,v is the canonicalconditional expectation from D onto D . Remark . e Y is a closed subspace of f Y by the inclusion e φ defined by e φ ( e y ) = g φ ( y )for any y ∈ Y . Also, Y is a closed subspace Y ⋊ µ H by the inclusion defined by Y −→ Y ⋊ µ H : y y ⋊ µ . et e B be the Jones projection in D for the canonical conditional expectation E σ,v from D onto B . We identify e B with the projection 1 ⋊ b σ τ in D . Lemma 4.3.
With the above notation, Ψ( e B ) = Ψ(1 ⋊ b σ τ ) = 1 ⋊ β τ .Proof. The lemma can be proved by routine computations. Indeed, we note that e Y is regarded as a closed subspace of f Y by the inclusion e φ and Y is a regarded asa closed subspace Y ⋊ β of Y ⋊ β H . ThenΨ( e B ) = Ψ(1 ⋊ b σ τ ) = X i,j h (1 ⋊ b σ τ ) · ( e y i ⊗ y i ) , e y j ⊗ y j i D ⋊ β H = X i,j h [ y i · (1 ⋊ b τ τ )] e ⊗ y i , e y j ⊗ y j i D ⋊ β H = X i,j h y i , h [ y i · (1 ⋊ b σ τ )] e , e y j i C · y j i D ⋊ β H . We note that e y i , e y j ∈ e Y ⊂ f Y and that y i , y j ∈ Y = Y ⋊ µ ⊂ Y ⋊ µ H . HenceΨ( e B ) = X i,j h y i ⋊ µ , C h φ ( y i ) · (1 ⋊ b σ τ ) , φ ( y j ) i · y j ⋊ µ i D ⋊ β H . Furthermore, let { ( u k , u ∗ k ) } and { ( v l , v ∗ l ) } be quasis-bases for E ρ,u and E σ,v , re-spectively. Then φ ( y i ) · (1 ⋊ b σ τ ) = φ ( y i ) · e B = X k,l u k ⊗ E X ( u ∗ k · y i · v l ) ⊗ e v l · e B = X k,l u k ⊗ E X ( u ∗ k · y i · v l ) ⊗ g E B ( v l )= X k.l u k ⊗ E X ( u ∗ k · y i · v l E B ( v ∗ l )) ⊗ f D = X k u k ⊗ E X ( u ∗ k · y i ) ⊗ f D . Hence since 1 ⋊ b σ τ is a projection in D , C h y i · (1 ⋊ b σ τ ) , y j i = C h y i · (1 ⋊ b σ τ ) , y j · (1 ⋊ b σ τ ) i = X k,l C h u k ⊗ E X ( u ∗ k · y i ) ⊗ f D , u l ⊗ E X ( u ∗ l · y j ) ⊗ f D i = X k,l C h u k · A h E X ( u ∗ k · y i ) ⊗ f D , E X ( u ∗ l · y j ) ⊗ f D i , u l i = X k,l C h u k · A h E X ( u ∗ k · y i ) · B h f D , f D i , E X ( u ∗ l · y j ) i , u l i = X k,l C h u k · A h E X ( u ∗ k · y i ) , E X ( u ∗ l · y j ) i , u l i = X k,l C h u k E A ( C h u ∗ k · y i , E X ( u ∗ l · y j ) i ) , u l i = X k,l C h u k E A ( u ∗ kC h y i , E X ( u ∗ l · y j ) i ) , u l i = X l C h C h y i , E X ( u ∗ l · y j ) i , u l i = X l C h y i , E X ( u ∗ l · y j ) i e A u ∗ l . hus Ψ( e B ) = X i,j,l h y i ⋊ µ , C h y i , E X ( u ∗ l · y j ) i e A u ∗ l · ( y j ⋊ µ ) i D ⋊ β H = X i,j,l h C h E X ( u ∗ l · y j ) , y i i · ( y i ⋊ µ ) , e A u ∗ l · ( y j ⋊ µ ) i D ⋊ β H = X i,j,l h E X ( u ∗ l · y j ) · h y i , y i i D , e A u ∗ l · y j i D ⋊ β H = X j,l h E X ( u ∗ l · y j ) , e A u ∗ l · y j i D ⋊ β H = X j,l h u l e A · E X ( u ∗ l · y j ) , y j i D ⋊ β H . Since we identify e A with 1 ⋊ b ρ τ , we obtain that u l e A · E X ( u ∗ l · y j ) = ( u l ⋊ b ρ )(1 ⋊ b ρ τ ) · ( E X ( u ∗ l · y j ) ⋊ µ )= ( u l ⋊ b ρ τ ) · ( E X ( u ∗ l · y j ) ⋊ µ ) . By Lemma 4.1, E X ( u ∗ l · y j ) = τ ′ · µ ( u ∗ l · y j ), where τ ′ = τ . Thus u l e A · E X ( u ∗ l · y j ) = ( u l ⋊ b ρ τ ) · [ τ ′ · µ ( u ∗ l · y j )] ⋊ µ = u l [ τ (1) · µ [ τ ′ · µ ( u ∗ l · y j )] ⋊ µ τ (2) ]= u l [ τ ′ · µ ( u ∗ l · y j )] ⋊ µ τ = u l E X ( u ∗ l · y j ) ⋊ µ τ. It follows by [11, Lemma 5.4] thatΨ( e B ) = X j,l h u l E X ( u ∗ l · y j ) ⋊ µ τ , y j i D ⋊ β H = X j h y j ⋊ µ τ , y j ⋊ µ i D ⋊ β H = X j τ ∗ (1) · µ h y j , y j i D ⋊ β τ ∗ (2) = [ τ ∗ (1) · µ D ] ⋊ β τ ∗ (2) = 1 ⋊ β τ ∗ = 1 ⋊ β τ. Therefore we obtain the conclusion. (cid:3)
Let ( Y ⋊ µ H o ) Ψ be the C − D -equivalence bimodule induced by the C − D ⋊ β H -equivalence bimodule Y ⋊ µ H and the isomorphism Ψ of D onto D ⋊ β H .Let E µ be the linear map from Y ⋊ µ H onto Y defined by E µ ( y ⋊ µ ψ ) = ψ ( e ) y for any y ∈ Y , ψ ∈ H , where y ⋊ µ ψ is the element in Y ⋊ µ H induced by y ∈ Y , ψ ∈ H . Then E µ is a conditional expectation from Y ⋊ µ H onto Y with respectto E ρ,u and E β , the canonical conditional expectation from D ⋊ β H onto D by[11, Proposition 4.1]. Let E µ, Ψ1 be the linear map from ( Y ⋊ µ H ) Ψ onto Y inducedby E µ and Ψ. Lemma 4.4.
With the above notation, E µ, Ψ1 is a conditional expectation from ( Y ⋊ µ H ) Ψ onto Y with respect to E ρ,u and E σ,v .Proof. We shall show that Conditions (1)-(6) in [11, Definition 2.4] hold. Let y, z ∈ Y , c ∈ C , d ∈ D and ψ ∈ H .(1) E µ, Ψ1 (( c ⋊ b ρ ψ ) · y ) = E µ, Ψ1 (( c ⋊ b ρ ψ ) · ( y ⋊ µ )) = E µ, Ψ1 ( c · [ ψ (1) · µ y ] ⋊ µ ψ (2) )= c · [ ψ (1) · µ y ] ψ (2) ( e ) = c · ψ ( e ) y = ψ ( e ) c · y. n the other hand, E ρ,u ( c ⋊ b ρ ψ ) · y = ψ ( e ) c · y. Hence Condition (1) holds.(2) E µ, Ψ1 ( c · ( y ⋊ µ ψ )) = E µ, Ψ1 (( c · y ) ⋊ µ ψ ) = c · yψ ( e ) = ψ ( e ) c · y. On the other hand, c · E µ, Ψ1 ( y ⋊ µ ψ ) = c · ψ ( e ) y = ψ ( e ) c · y. Hence Condition (2) holds.(3) E ρ,u ( C ⋊ b ρ H h y ⋊ µ ψ , z i ) = E ρ,u ( C ⋊ b ρ H h y ⋊ µ ψ , z ⋊ µ i )= E ρ,u ( C h y , [ S ( ψ ∗ (1) ) · µ z ] i ⋊ b ρ ψ (2) )= C h y , [ S ( ψ ∗ (1) ) · µ z ] i ψ (2) ( e )= C h y , [ ψ (2) ( e ) S ( ψ ∗ (1) ) · µ z ] i = C h y , [ S ( ψ ∗ (2) )( e ) S ( ψ ∗ (1) ) · µ z ] i = C h y , [ e ( S ( ψ ∗ ) · µ z ] i = C h y , ψ ( e ) z i = ψ ( e ) C h y, z i . On the other hand, C h E µ, Ψ1 ( y ⋊ µ ψ ) , z i = C h ψ ( e ) y , z i = ψ ( e ) C h y , z i . Hence Condition (3) holds.(4) E µ, Ψ1 ( y · ( d ⋊ b σ ψ )) = E µ ( y · Ψ( d × b σ ψ )) = y · E β (Ψ( d ⋊ b σ ψ ))= y · E σ,v ( d ⋊ b σ ψ ) . Hence Condition (4) holds.(5) E µ, Ψ1 (( y ⋊ µ ψ ) · d ) = E µ (( y ⋊ µ ψ ) · Ψ( d )) = E µ ( y ⋊ µ ψ ) · Ψ( d )= E µ, Ψ1 ( y ⋊ µ ψ ) · d. Hence Condition (5) holds.(6) E σ,v ( h y ⋊ µ ψ , z i D ⋊ b σ H ) = E σ,v (Ψ − ( h y ⋊ µ ψ , z ⋊ µ i D ⋊ β H ))= E β ( h y ⋊ µ ψ , z ⋊ µ i D ⋊ β H )= E β ([ ψ ∗ (1) · β h y , z i D ] ⋊ β ψ ∗ (2) )= ψ ( e ) h y , z i D . On the other hand, h E µ, Ψ1 ( y ⋊ µ ψ ) , z ⋊ µ i D ⋊ b σ H = Ψ − ( h ψ ( e ) y , z i D ⋊ β H ) = ψ ( e ) h y , z i D . Hence Condition (6) holds. Therefore, we obtain the conclusion. (cid:3)
Lemma 4.5.
With the above notation, for any y ∈ Y , E µ, Ψ1 ( e A · y · e B ) = 1 N E X ( y ) . roof. By the definition of E µ, Ψ1 and Lemma 4.3, E µ, Ψ1 ( e A · y · e B ) = E µ ((1 ⋊ b ρ τ ) · y · Ψ( e B )) = E µ ((1 ⋊ b ρ τ ) · y · (1 ⋊ β τ )) . Also,(1 ⋊ b ρ τ ) · y · (1 ⋊ β τ ) = (1 ⋊ b ρ τ ) · ( y ⋊ µ ) · (1 ⋊ β τ ) = (1 ⋊ b ρ τ ) · ( y ⋊ µ τ )= [ τ (1) · µ y ] ⋊ µ τ (2) τ ′ = [ τ · µ y ] ⋊ µ τ, where τ ′ = τ . Hence E µ, Ψ1 ( e A · y · e B ) = E µ ([ τ · µ y ] ⋊ µ τ ) = [ τ · µ y ] τ ( e ) = 1 N E X ( y )by Lemma 4.1. (cid:3) Proposition 4.6.
With the above notation, there is a C − D -equivalence bimoduleisomorphism θ of Y onto ( Y ⋊ µ H ) Ψ such that E µ, Ψ1 = E Y ◦ θ .Proof. This is immediate by Lemma 4.5 and [11, Theorem 6.13]. (cid:3)
Next, modifying the discussions of [12, Section 5], we shall show that there is a C ∗ -Hopf algebra automorphism f of H such that b β ◦ Ψ = (Ψ ⊗ f ) ◦ bb σ, where b β is the dual coaction of β and bb σ is the second dual coaction of ( σ, v ). Lemma 4.7.
With the above notation, Ψ | B ′ ∩ D , the restriction of Ψ to B ′ ∩ D isan isomorphism of B ′ ∩ D onto B ′ ∩ ( D ⋊ β H ) .Proof. It suffices to show that Ψ( d ) ∈ B ′ ∩ ( D ⋊ β H ) for any d ∈ B ′ ∩ D . Forany d ∈ B ′ ∩ D , b ∈ B ,Ψ( d ) b = Ψ( d )Ψ( b ) = Ψ( db ) = Ψ( b )Ψ( d ) = b Ψ( d ) . Hence Ψ( d ) ∈ B ′ ∩ ( D ⋊ β H ) for any d ∈ B ′ ∩ D . (cid:3) By Lemma [12, Lemma 5.8], B ′ ∩ D = 1 ⋊ σ,v ⋊ b σ H . Also, we have the nextlemma. Lemma 4.8.
With the above notation, B ′ ∩ ( D ⋊ β H ) = 1 ⋊ σ,v ⋊ β H .Proof. We note that ψ · µ x = ǫ ( ψ ) x for any ψ ∈ H , x ∈ X by [12, Lemma 3.2].Thus by the definition of β , ψ · β ( b ⋊ σ,v
1) = ǫ ( ψ )( b ⋊ σ,v
1) for any ψ ∈ H , b ∈ B (See [12, Sectoin 4]). Hence in the same way as in the proof of [12, Lemma 5.8], weobtain the conclusion. (cid:3) Since Ψ(1 ⋊ b σ τ ) = 1 ⋊ β τ by Lemma 4.3 and Ψ( d ) = d for any d ∈ D , in thesame way as in [12, Lemma 5.6], we can see that there is an isomorphism b Ψ of D onto D ⋊ β H ⋊ b β H satisfying that b Ψ | D = Ψ , E β ◦ b Ψ = Ψ ◦ E σ,v , b Ψ(1 D ⋊ b σ ⋊ bb σ e ) = 1 D ⋊ β ⋊ b β e, where bb σ is the second dual coaction of ( σ, v ), b β is the dual coaction of β , D = D ⋊ bb σ H and E σ,v and E β are the canonical conditional expectations from D and D ⋊ β H ⋊ b β H onto D and D ⋊ β H , respectively. Furthermore, in the sameway as in the above or [12, Section 5], b Ψ | D ′ ∩ D is an isomorphism of D ′ ∩ D onto D ′ ∩ ( D ⋊ β H ⋊ b β H ). Since B ′ ∩ D = B ′ ∩ ( D ⋊ β H ) = 1 ⋊ σ,v ⋊ β H y Lemma 4.8, we identify B ′ ∩ D and B ′ ∩ ( D ⋊ β H ) with H . Let f = Ψ | B ′ ∩ D and we regard f as a C ∗ -algebra automorphism of H . By the proof of [12, Lemma5.9], we can see that N ( E σ,v ◦ E σ,v )((1 ⋊ b σ ψ ⋊ bb σ ⋊ b σ ⋊ bb σ e )(1 ⋊ b σ τ ⋊ bb σ ⋊ b σ ⋊ bb σ h )) = ψ ( e ) ,N ( E β ◦ E β )((1 ⋊ β ψ ⋊ b β ⋊ β ⋊ b β e )(1 ⋊ β τ ⋊ b β ⋊ β ⋊ b β h )) = ψ ( e ) , for any h ∈ H , ψ ∈ H . Hence in the same way as in the proof of [12, Lemma 5.9],we can see that f is a C ∗ -Hopf algebra automorphism of H . Lemma 4.9.
With the above notation, b β ◦ Ψ = (Ψ ⊗ f ) ◦ bb σ .Proof. This can be proved in the same way as in the proof of [12, Lemma 5.10]. (cid:3)
Lemma 4.10.
With the above notation, b β (1 D ⋊ β τ ) is Murray-von Neumann equiv-alent to (1 D ⋊ β τ ) ⊗ in ( D ⋊ β H ) ⊗ H .Proof. By Lemmas 4.3, 4.9, b β (1 D ⋊ β τ ) = b β (Ψ(1 D ⋊ b σ τ )) = (Ψ ⊗ f )( bb σ (1 D ⋊ b σ τ )) . By [9, Proposition 3.19], bb σ (1 D ⋊ b σ τ ) is Murray-von Neumann equivalent to (1 ⋊ b σ τ ) ⊗ in D ⊗ H . Hence we obtain the conclusion by Lemma 4.3. (cid:3) Lemma 4.11.
With the above notation, β is saturated, that is, the action of H on D induced by β is saturated in the sense of Szyma´nski and Peligrad [16] .Proof. By the definition of b σ , D (1 D ⋊ b σ τ ) D = D . Since Ψ is an isomorphism of D onto D ⋊ β H ,( D ⋊ β H )(1 D ⋊ β τ )( D ⋊ β H ) = Ψ( D (1 D ⋊ b σ τ ) D ) = Ψ( D ) = D ⋊ β H by Lemma 4.1. Hence β is saturated. (cid:3) Since β is saturated by Lemma 4.11, there is the conditional expectation E D β from D onto D β defined by E D β ( d ) = τ · β d for any d ∈ D (See [16, Proposition 2.12]), where D β is the fixed-point C ∗ -subalgebra of D for β . Also, since b β (1 ⋊ β τ ) is Murray-von Neumann equivalent to(1 ⋊ β τ ) ⊗ in ( D ⋊ β H ) ⊗ H by Lemma 4.10, there is a twisted coaction ( γ, w )of H on D β and an isomorphism π D of D onto D β ⋊ γ,w H satisfying E σ,v = E D β ◦ π D , ψ · b γ π D ( d ) = π D ( ψ · β d )for any d ∈ D , ψ ∈ H by [9, Proposition 6.1, 6.4 and Theorem 6.4]. We identify D β ⋊ γ,w H and E γ,w with D and E D β by the above isomorphism π D , respectively.We show that B = D β . By the definition of β , B ⊂ D β . Let F be the conditionalexpectation of D β onto B defined by F = E σ,v | D β , the restriction of E σ,v to D β .Since E σ,v is of Watatani index-finite type, there is a quasi-basis { ( d i , d ∗ i ) } ni =1 for E σ,v . Then F ◦ E γ,w is also a conditional expectation from D onto B . Since B ′ ∩ D = C
1, by [17, Proposition 1.4.1], E σ,v = F ◦ E γ,w . Lemma 4.12.
With above notation, F is of Watatani index-finite type and itsWatatani index, Ind W ( F ) ∈ C . roof. We claim that { ( E γ,w ( d i ) , E γ,w ( d ∗ i )) } ni =1 is a qusi-basis for F . Indeed, forany d ∈ D β , n X i =1 E γ,w ( d i ) F ( E γ,w ( d ∗ i ) d ) = n X i =1 E γ,w ( d i ) F ( E γ,w ( d ∗ i ) E γ,w ( d ))= n X i =1 E γ,w ( d i )( F ◦ E γ,w )( d ∗ i E γ,w ( d ))= n X i =1 E γ,w ( d i ) E σ,v ( d ∗ i E γ,w ( d ))= n X i =1 E γ,w ( d i E σ,v ( d ∗ i E γ,w ( d )))= E γ,w ( E γ,w ( d )) = d since E σ,v = F ◦ E γ,w and E γ,w ( d ) = d for any d ∈ D β . Hence F is of Watataniindex-finite type. Also, Ind W ( F ) ∈ ( D β ) ′ ∩ D β ⊂ B ′ ∩ D = C (cid:3) Lemma 4.13.
With the above notation, B = D β .Proof. It suffices to show that Ind W ( F ) = 1. By [17, Proposition 1.7.1],Ind W ( E σ,v ) = Ind W ( F )Ind W ( E γ,w ) . By [9, Proposition 3.18] Ind W ( E σ,v ) = Ind W ( E γ,w ) = N . Hence Ind W ( F ) = 1.Therefore, we obtain the conclusion by [17]. (cid:3) Let Y µ = { y ∈ Y | µ ( y ) = y ⊗ } . By [4, Theorem 4.9], there are a twistedcoaction λ of H on Y µ and a Hilbert A ⋊ ρ,u H − B ⋊ γ,w H -bimodule isomorphism π Y of Y µ ⋊ λ H onto Y such that ψ · µ π Y ( x ⋊ λ h ) = π Y ( ψ · b λ ( x ⋊ λ h ))for any x ∈ Y µ , h ∈ H , ψ ∈ H . Furthermore, by [4, Lemma 3.10], Y µ is an A − B -equivalence bimodule and hence π Y is an A ⋊ ρ,u H − B ⋊ γ,w H -equivalencebimodule isomorphism. We identify Y with Y µ ⋊ λ H by the isomorphism π Y . Thusthe twisted coactions ( ρ, u ) and ( γ, w ) are strongly Morita equivalent with respectto the twisted coaction λ of H on the A − B -equivalence bimodule Y µ . We showthat Y µ = X . Lemma 4.14.
With the above notation, Y µ = X .Proof. By [12, Lemma 3.2], X ⊂ Y µ . Also, for any y ∈ Y µ , τ · µ y = ǫ ( τ ) y = y .On the other hand, by Lemma 4.1, τ · µ y = E X ( y ). Hence y = E X ( y ) ∈ X . Thuswe obtain that Y µ ⊂ X . (cid:3) By the above discussions, we obtain the following theorem:
Theorem 4.15.
Let H be a finite dimensional C ∗ -Hopf algebra and H its dual C ∗ -Hopf algebra. Let ( ρ, u ) and ( σ, v ) be twisted coactions of H on unital C ∗ -algebras A and B , respectively. Let A ⊂ A ⋊ ρ,u H and B ⊂ B ⋊ σ,v H be unital inclusionsof unital C ∗ -algebras. We suppose that they are strongly Morita equivalent withrespect to an A ⋊ ρ,u H − B ⋊ σ,v H -equivalence bimodule Y and its closed subspace X . And we suppose that A ′ ∩ ( A ⋊ ρ,u H ) = C . Then there are a twisted coaction ( γ, w ) of H on B and a twisted coaction λ of H on X satisfying the following: (1) ( ρ, u ) and ( γ, w ) are strongly Morita equivalent with respect to λ , (2) B ⋊ σ,v H = B ⋊ γ,w H , (3) Y ∼ = X ⋊ λ H as A ⋊ ρ,u H − B ⋊ σ,v H - equivalence bimodules. . Image
Let A be a unital C ∗ -algebra and X an involutive A − A -equivalence bimodule.Let A ⊂ C X be the unital inclusion of unital C ∗ -algebras induced by X . Wesuppose that A ′ ∩ C X = C
1. Let f A be the homomorphism of Pic( A, C X ) ontoPic( A ) defined in Preliminaries, that is, f A ([ M, N ]) = [ M ]for any ( M, N ) ∈ Equi(
A, C X ). In this section, we shall compute Im f A , the imageof f A .Let E A be the conditional expectation from C X onto A defined in Section 3 andlet e A be the Jones projection for E A . Since E A is of Watatani index-finite typeby [7, Lemma 3.4], there is the C ∗ -basic construction of the inclusion A ⊂ C X for E A , which is the linking C ∗ -algebra L X for X , that is, L X = { (cid:20) a x e y b (cid:21) | a, b ∈ A, x, y ∈ X } . By [7, Lemma 2.6], we can see that there is the action α X of Z , the group of ordertwo, on C X defined by α X ( (cid:20) a x e x ♮ a (cid:21) ) = (cid:20) a − x − e x ♮ a (cid:21) for any (cid:20) a x e x ♮ a (cid:21) ∈ C X and that L X ∼ = C X ⋊ α X Z as C ∗ -algebras. We note that weregard an action β of Z on a unital C ∗ -algebra B as the automorphism β of B with β = id on B . We identify L X with C X ⋊ α X Z . Let M be an A − A -equivalencebimodule satisfying that f M ⊗ A X ⊗ A M ∼ = X as involutive A − A -equivalence bimodules. Then by the proof of [6, Lemma 5.11], wecan see that there is an element ( M, C M ) ∈ Equi(
A, C X ), where C M is a C X − C X -equivalence bimodule induced by M , which is defined in [6, Section 5.]. Next, weshow that f M ⊗ A X ⊗ A M ∼ = X as involutive A − A -equivalence bimodules for any ( M, N ) ∈ Equi(
A, C X ). Let( M, N ) be any element in Equi(
A, C X ). Since A ′ ∩ C X = C
1, by [5, Lemma 4.1]there is the unique conditional expectation E M from N onto M with respect to E A and E A . Let N be the upward basic construction of N for E M (See [11, Definition6.5]). Then by [11, Corollary 6.3], the unital inclusion C X ⊂ L X is strongly Moritaequivalent to itself with respect to N and its closed subspace N . Hence by Theorem4.15, there are an action γ of Z on C X and an action λ of Z on N satisfying thefollowing:(1) The actions α X and γ of Z on C X are strongly Morita equivalent with respectto the action λ of Z on N ,(2) L X = C X ⋊ α X Z = C X ⋊ γ Z ,(3) N ∼ = N ⋊ λ Z as L X − L X -equivalence bimodules.We identify N with N ⋊ λ Z . Let b α X be the dual action of α X , which is an actionof Z on L X . We regard b α X as an automorphism of L X with ( b α X ) = id on L X , hich is defined by b α X ( (cid:20) a x e x ♮ a (cid:21) ) = (cid:20) a x e x ♮ a (cid:21) for any (cid:20) a x e x ♮ a (cid:21) ∈ C X , b α X ( (cid:20) (cid:21) ) = (cid:20) (cid:21) . Let ( L X ) b α X be the involutive L X − L X -equivalence bimodule induced by b α X , thatis, ( L X ) b α X = L X as vector spaces over C and the left L X -action and the left L X -valued inner product on ( L X ) b α X are defined in the usual way. The right L X -actionand the right L X -valued inner product on ( L X ) b α X are defined as follows: For any a ∈ L X , x, y ∈ ( L X ) b α X , x · a = x b α X ( a ) , h x, y i L X = b α X ( x ∗ y ) . Furthermore, we define the involution ♮ as follows: For any x ∈ ( L X ) b α X , x ♮ = b α X ( x ) ∗ . Then by easy computations ( L X ) b α X is an involutive L X − L X -equivalence bimodule.Let b λ be the dual action of λ , which is an action of Z by linear automorphisms of N = N ⋊ λ Z such that b α X ( L X h m, n i ) = L X h b λ ( m ) , b λ ( n ) i , b γ ( h m, n i L X ) = h b λ ( m ) , b λ ( n ) i L X for any m, n ∈ N , where we regard the action b λ as a linear automorphism of N with b λ = id on N . We note that b λ ( x · m ) = b α X ( x ) · b λ ( m ) , b λ ( m · x ) = b λ ( m ) · b γ ( x )for any m ∈ N , x ∈ L X . Since L X = C X ⋊ α X Z = C X ⋊ γ Z and N = N ⋊ λ Z ,in the same way as after the proof of Lemma 4.1 and in the proof of Lemma 4.9 orby the discussions of [12, Section 5], there is an automorphism κ of L X satisfyingthe following: b γ ◦ κ = κ ◦ b α X , κ | C X = id C X . Then κ | A ′ ∩ L X is an automorphism of A ′ ∩ L X . And by [8], [6], A ′ ∩ L X ∼ = C . Since e A ∈ A ′ ∩ L X , κ ( e A ) = e A or 1 − e A . If κ ( e A ) = e A , κ = id L X since κ | C X = id C X .Hence b γ = b α X . If κ ( e A ) = 1 − e A , κ = b α X since κ = b α X = id on C X . Hence b γ ◦ b α X = b α X ◦ b α X = id L X . Thus b γ = ( b α X ) − = b α X . Then, we obtain the following: Lemma 5.1.
With the above notation, f N ⊗ L X ( L X ) b α X ⊗ L X N ∼ = ( L X ) b α X as L X − L X -equivalence bimodules.Proof. We note that N = N ⋊ λ Z . Let π be the linear map from f N ⊗ L X ( L X ) b α X ⊗ L X N to ( L X ) b α X defined by π ( e m ⊗ x ⊗ n ) = h x ∗ · m , b λ ( n ) i L X for any m, n ∈ N , x ∈ ( L X ) b α X , where we regard h x ∗ · m , b λ ( n ) i L X as an elementin ( L X ) b α X . We show that π is an involutive L X − L X -equivalence bimodule iso-morphism of f N ⊗ L X ( L X ) b α X ⊗ L X N onto ( L X ) b α X . By routine computations, wecan see that π is well-defined. Since L X · N = N by Brown, Mingo and Shen
2, Proposition 1.7] and ( L X ) b α X is full with respect to the right L X -valued innerproduct, π is surjective. For any m, n, m , n ∈ N , x, x ∈ L X , h π ( e m ⊗ x ⊗ n ) , π ( f m ⊗ x ⊗ n ) i L X = hh x ∗ · m , b λ ( n ) i L X , h x ∗ · m , b λ ( n ) i L X i L X = b α X ( h b λ ( n ) , x ∗ · m i L X h x ∗ · m , b λ ( n ) i L X )= h n , b λ ( x ∗ · m ) i L X h b λ ( x ∗ · m ) , n i L X . On the other hand, h e m ⊗ x ⊗ n , f m ⊗ x ⊗ n i L X = h n , h e m ⊗ x , f m ⊗ x i L X · n i L X = h n , h x , h e m , f m i L X · x i L X · n i L X = h n , h x , L X h m , m i · x i L X · n i L X = h n , b α X ( x ∗ L X h m , m i x ) · n i L X = h n , b α X ( L X h x ∗ · m , x ∗ · m i ) · n i L X = h n , b λ ( x ∗ · m ) · h b λ ( x ∗ · m ) , n i L X i L X = h n , b λ ( x ∗ · m ) i L X h b λ ( x ∗ · m ) , n i L X . Hence π preserves the right L X -valued inner products. Similarly, we can see that π preserves the left L X -valued inner products. Thus we can obtain that π is an L X − L X -equivalence bimodule isomorphism by the remark after [3, Definition1.1.18]. Furthermore, π ( e m ⊗ x ⊗ n ) ♮ = h x ∗ · m , b λ ( n ) i ♮L X = b α X ( h x ∗ · m , b λ ( n ) i ∗ L X )= b α X ( h b λ ( n ) , x ∗ · m i L X ) = h n , b λ ( x ∗ · m ) i L X . On the other hand, π (( e m ⊗ x ⊗ n ) ♮ ) = π ( e n ⊗ x ♮ ⊗ m ) = h ( x ♮ ) ∗ · n , b λ ( m ) i L X = h b α X ( x ) · n , b λ ( m ) i L X = h n , b α X ( x ) ∗ · b λ ( m ) i L X = h n , b λ ( x ∗ · m ) i L X . Hence π preserves involutions ♮ . Therefore, we obtain the conclusion. (cid:3) We regard e A L X as an A − L X -equivalence bimodule in the usual way, wherewe identify e A L X e A with A . Also, we regard L X e A as an L X − A - equivalencebimodule in the usual way. We note that L X e A ∼ = ^ e A L X as L X − A -equivalencebimodules by the map xe A ∈ L X e A e A x ∗ ∈ ^ e A L X . In the same way as in [7,Section 3], we regard e A L X (1 − e A ) as an involutive A − A -equivalence bimodule. Lemma 5.2.
With the above notation, e A L X ⊗ L X ( L X ) b α X ⊗ L X L X e A ∼ = e A L X (1 − e A ) ∼ = X as involutive A − A -equivalence bimodules.Proof. By [7, Theorem 3.11], we can see that e A L X (1 − e A ) ∼ = X as involutive A − A -equivalence bimodules. Let π be the linear map from e A L X ⊗ L X ( L X ) b α X ⊗ L X L X e A to e A L X (1 − e A ) defined by π ( e A x ⊗ y ⊗ ze A ) = e A xy b α X ( ze A ) = e A xy b α X ( z )(1 − e A ) or any x, y, z ∈ L X . We note that b α X ( e A ) = 1 − e A by [7, Remark 2.7]. Clearly π is surjective. For any x, y, z, x , y , z ∈ L X , A h π ( e A x ⊗ y ⊗ ze A ) , π ( e A x ⊗ y ⊗ z e A ) i = A h e A xy b α X ( z )(1 − e A ) , e A x y b α X ( z )(1 − e A ) i = e A xy b α X ( z )(1 − e A ) b α X ( z ∗ ) y ∗ x ∗ e A . On the other hand, A h e A x ⊗ y ⊗ ze A , e A x ⊗ y ⊗ z e A i = A h e A x · L X h y ⊗ ze A , y ⊗ z e A i , e A x i = [ e A x · L X h y ⊗ ze A , y ⊗ z e A i ] x ∗ e A = e A x L X h y ⊗ ze A , y ⊗ z e A i x ∗ e A = e A x L X h y · L X h ze A , z e A i , y i x ∗ e A = e A x L X h y · ze A z ∗ , y i x ∗ e A = e A x L X h y b α X ( ze A z ∗ ) , y i x ∗ e A = e A xy b α X ( ze A z ∗ ) y ∗ x ∗ e A = e A xy b α X ( z )(1 − e A ) b α X ( z ∗ ) y ∗ x ∗ e A . Hence π preserves the left A -valued inner products. Similarly, we can see that π preserves the right A -valued inner products. Thus we can obtain that π is an A − A -equivalence bimodule isomorphism by the remark after [3, Definition 1.1.18].Furthermore, π ( e A x ⊗ y ⊗ ze A ) ♮ = ( e A xy b α X ( z )(1 − e A )) ♮ = b α X (1 − e A ) z ∗ b α X ( y ∗ x ∗ e A )= e A z ∗ b α X ( y ∗ x ∗ )(1 − e A ) . On the other hand, π (( e A x ⊗ y ⊗ ze A ) ♮ ) = π ( e A z ∗ ⊗ b α X ( y ∗ ) ⊗ x ∗ e A ) = e A z ∗ b α X ( y ∗ x ∗ )(1 − e A ) . Hence π preserves the involutions ♮ . Therefore, we obtain the conclusion. (cid:3) Lemma 5.3.
With the above notation, e A L X ⊗ L X C X ∼ = A as A − A -equivalencebimodules, where C X is regarded as an L X − A -equivalence bimodule in the usualway and A is regarded as the trivial A − A -equivalence bimodule.Proof. Let π be the linear map from e A L X ⊗ L X C X to A defined by π ( e A ae A b ⊗ c ) = e A ae A b · c = E A ( a ) e A b · c = E A ( a ) E A ( bc )for any a, b, c ∈ C X . Clearly π is surjective. For any a, b, c, a , b , c ∈ C X , A h π ( e A ae A b ⊗ c ) , π ( e A a e A b ⊗ c ) i = A h E A ( a ) E A ( bc ) , E A ( a ) E A ( b c ) i = E A ( a ) E A ( bc ) E A ( c ∗ b ∗ ) E A ( a ∗ ) . On the other hand, A h e A ae A b ⊗ c , e A a e A b ⊗ c i = A h e A ae A b · L X h c , c i , e A a e A b i = A h e A ae A b · ce A c ∗ , e A a e A b i = A h e A ae A bce A c ∗ , e A a e A b i = A h e A E A ( a ) E A ( bc ) c ∗ , e A E A ( a ) b i = e A E A ( a ) E A ( bc ) c ∗ b ∗ E A ( a ∗ ) e A = E A ( a ) E A ( bc ) E A ( c ∗ b ∗ ) E A ( a ∗ ) e A . ince we identify A with Ae A by the map a ∈ A ae A ∈ Ae A , π preserves theleft A -valued inner products. Similarly, we can see that π preserves the right A -valued inner products. Thus by the remark after [3, Definition 1.1.18], we obtainthe conclusion. (cid:3) Proposition 5.4.
For any ( M, N ) ∈ Equi(
A, C X ) , X ∼ = f M ⊗ A X ⊗ A M as involutive A − A -equivalence bimodules.Proof. By Lemmas 5.2, 5.1, X ∼ = e A L X ⊗ L X ( L X ) b α X ⊗ L X L X e A ∼ = e A L X ⊗ L X f N ⊗ L X ( L X ) b α X ⊗ L X N ⊗ L X L X e A ∼ = e A L X ⊗ L X f N ⊗ L X L X e A ⊗ A X ⊗ A e A L X ⊗ L X N ⊗ L X L X e A . as involutive A − A -equivalence bimodules. Since N = C X ⊗ A M ⊗ A g C X , e A L X ⊗ L X N ⊗ L X L X e A = e A L X ⊗ L X C X ⊗ A M ⊗ A g C X ⊗ L X L X e A , where C X is regarded as an L X − A -equivalence bimodule. Hence by Lemma 5.3, e A L X ⊗ L X N ⊗ L X L X e A ∼ = e A L X ⊗ L X C X ⊗ A M ⊗ A [ e A L X ⊗ L X C X ] e ∼ = A ⊗ A M ⊗ A A ∼ = M as A − A -equivalence bimodules. Therefore, X ∼ = [ e A L X ⊗ L X N ⊗ L X L X e A ] e ⊗ A X ⊗ A [ e A L X ⊗ L X N ⊗ L X L X e A ] ∼ = f M ⊗ A X ⊗ A M as involutive A − A -equivalence bimodules. (cid:3) Theorem 5.5.
Let A be a unital C ∗ -algebra and X an involutive A − A -equivalencebimodule. Let A ⊂ C X be the unital inclusion of unital C ∗ -algebras induced by X .We suppose that A ′ ∩ C X = C . Let f A be the homomorphism of Pic(
A, C X ) to Pic( A ) defined by f A ([ M, N ]) = [ M ] for any ( M, N ) ∈ Equi(
A, C X ) . Then the image of f A is: Im f A = { [ M ] ∈ Pic( A ) | M is an A − A -equivalence bimodule with X ∼ = f M ⊗ A X ⊗ A M as involutive A − A -equivalence bimodules } . Proof.
This is immediate by Proposition 5.4 and the proof of [6, Lemma 5.11]. (cid:3) A homomorphism
In this section, we shall construct a homomorphism g of Im f A to Pic( A, C X )with f A ◦ g = id on Im f A . Let M be an A − A -equivalence bimodule with X ∼ = f M ⊗ A X ⊗ A M as involutive A − A -equivalence bimodules. Let Φ M be an involutive A − A -equivalence bimodule isomorphism of f M ⊗ A X ⊗ A M onto X and let g Φ M bethe involutive A − A -equivalence bimodule isomorphism of f M ⊗ A e X ⊗ A M onto e X induced by Φ M (See [6, Section 5]). Let Ψ M and g Ψ M be the A − A - equivalencebimodule isomorphism of X ⊗ A M onto M ⊗ A X and the A − A -equivalence bimodule somorphism of e X ⊗ A M onto M ⊗ A e X indced by Φ M and g Φ M , which are definedin [6, Section 5], respectively. Let C M be the linear span of the set C XM = { (cid:20) m m ⊗ xm ⊗ e x ♮ m (cid:21) | m , m ∈ M, x ∈ X } . Also, let C M be the linear span of the set X C M = { (cid:20) m x ⊗ m e x ♮ ⊗ m m (cid:21) | m , m ∈ M, x ∈ X } . As mentioned in [6, Section 5], we identify C M with C M by Ψ M and g Ψ M . In thesame way as in [6, Section 5], we define the left C X -action and the right C X -actionon C M as follows: (cid:20) a x e x ♮ a (cid:21) · (cid:20) m m ⊗ ym ⊗ e y ♮ m (cid:21) = (cid:20) a ⊗ m + x ⊗ m ⊗ e y ♮ a ⊗ m ⊗ y + x ⊗ m e x ♮ ⊗ m + a ⊗ m ⊗ e y ♮ e x ♮ ⊗ m ⊗ y + a ⊗ m (cid:21) , (cid:20) m m ⊗ ym ⊗ e y ♮ m (cid:21) · (cid:20) a x e x ♮ a (cid:21) = (cid:20) m ⊗ a + m ⊗ y ⊗ e x ♮ m ⊗ x + m ⊗ y ⊗ am ⊗ e y ♮ ⊗ a + m ⊗ e x ♮ m ⊗ e y ♮ ⊗ x + m ⊗ a (cid:21) . for any a ∈ A , m , m ∈ M , x, y ∈ X . But we identify A ⊗ A M , M ⊗ A A and X ⊗ A e X , e X ⊗ A X with M and A by the isomorphisms defined by a ⊗ m ∈ A ⊗ A M a · m ∈ M,m ⊗ a ∈ M ⊗ A A m · a ∈ M,x ⊗ e y ∈ X ⊗ A e X A h x, y i ∈ A, e x ⊗ y ∈ e X ⊗ A X
7→ h x, y i A ∈ A, respectively and we identify X ⊗ A M and e X ⊗ A M with M ⊗ A X and M ⊗ A e X byΨ M and g Ψ M , respectively. By the above identifications, the right hand-sides of theabove equations are in C M . Before we define a left C X -valued inner product and aright C X -valued inner product on C M , we define a conjugate linear map on C M , (cid:20) m m ⊗ xm ⊗ e x ♮ m (cid:21) ∈ C m (cid:20) m m ⊗ xm ⊗ e x ♮ m (cid:21) e ∈ C m by (cid:20) m m ⊗ xm ⊗ e x ♮ m (cid:21) e = (cid:20) f m x ♮ ⊗ f m e x ⊗ f m f m (cid:21) or any m , m ∈ M , x ∈ X . We define the left C X -valued inner product and theright C X -valued inner product as follows: C X h (cid:20) m m ⊗ xm ⊗ e x ♮ m (cid:21) , (cid:20) n n ⊗ yn ⊗ e y ♮ n (cid:21) i = (cid:20) m m ⊗ xm ⊗ e x ♮ m (cid:21) · (cid:20) n n ⊗ yn ⊗ e y ♮ n (cid:21) e = (cid:20) m ⊗ f n + m ⊗ x ⊗ e y ⊗ f n m ⊗ y ♮ ⊗ f n + m ⊗ x ⊗ f n m ⊗ e x ♮ ⊗ f n + m ⊗ e y ⊗ f n m ⊗ e x ♮ ⊗ y ♮ ⊗ f n + m ⊗ f n (cid:21) , h (cid:20) m m ⊗ xm ⊗ e x ♮ m (cid:21) , (cid:20) n n ⊗ yn ⊗ e y ♮ n (cid:21) i C X = (cid:20) m m ⊗ xm ⊗ e x ♮ m (cid:21) e · (cid:20) n n ⊗ yn ⊗ e y ♮ n (cid:21) = (cid:20) f m ⊗ n + x ♮ ⊗ f m ⊗ n ⊗ e y ♮ f m ⊗ n ⊗ y + x ♮ ⊗ f m ⊗ n e x ⊗ f m ⊗ n + f m ⊗ n ⊗ e y ♮ e x ⊗ f m ⊗ n ⊗ y + f m ⊗ n (cid:21) , for any m , m , n , n ∈ M , x, y ∈ X , where we regard the tensor product as aproduct on C M in the formal manner. We denote it by “ · ”. Also, we identify A ⊗ A M , M ⊗ A A and X ⊗ A e X , e X ⊗ A X with M and A by the same isomorphismsas above and we identify X ⊗ A M and e X ⊗ A M with M ⊗ A X and M ⊗ A e X byΨ M and g Ψ M . By the above identifications, we can define the left C X -valued andthe right C X -valued inner products. In the same way as above, we can define theleft C X -action and the right C X -valued action on C M and the left C X -valued innerproduct and the right C X -valued inner product on C M . Since we identify C M with C M by Ψ M and g Ψ M , we can see that C M and C M are C X − C X -equivalencebimodules by [6, Lemma 5.10] and that each of them agrees with the other byroutine computations (See [6, Section 5]. We identify C M with C M as C X − C X -equivalence bimodules by the isomorphisms Ψ M and g Ψ M and we denote them bythe same symbol C M . Furthermore, by [6, Lemma 5.11], ( M, C M ) ∈ Equi(
A, C C ).Let Φ ′ M be another involutive A − A -equivalence bimodule isomorphism of f M ⊗ A X ⊗ A M onto X and let g Φ M ′ be the involutive A − A -equivalence bimodule isomor-phism of f M ⊗ A e X ⊗ A M onto e X induced by Φ ′ M . Let Ψ ′ M be the A − A -equivalencebimodule isomorphism of X ⊗ A M onto M ⊗ A X induced by Φ ′ M and let g Ψ M ′ bethe A − A -equivalence bimodule isomorphism of e X ⊗ A M onto M ⊗ A e X inducedby g Φ M ′ . Then we can identify C M with C M by the isomorphisms Ψ ′ M and g Ψ ′ M .Hence we can obtain an element in Equi( M, C X ) by the above identification. Wedenote the element by ( M, C ′ M ). Lemma 6.1.
With the above notation, [ M, C M ] = [ M, C ′ M ] in Pic(
A, C X ) .Proof. We can construct a C X − C X -equivalence bimodule isomorphism using the A − A - equivalence isomorphisms Ψ M , g Ψ M , Ψ ′ M , g Ψ M ′ . Hence C M and C ′ M are iso-morphic as C X − C X -equivalence bimodules by the C X − C X -equivalence bimoduleisomorphism, which leaves the diagonal elements in C M and C ′ M invariant. Thus[ M, C M ] = [ M, C ′ M ] in Pic( A, C X ). (cid:3) Let M be another A − A -equivalence bimodule with f M ⊗ A X ⊗ A M ∼ = X asinvolutive A − A -equivalence bimodules. Let [ M , C M ] be the element in Pic( A, C X )induced by M in the above. Lemma 6.2.
With the above notation, we suppose that M and M are isomorphicas A − A -equivalence bimodules. Then [ M, C M ] = [ M , C M ] in Pic(
A, C X ) . roof. Since M ∼ = M as A − A -equivalence bimodules, there is an A − A -equivalencebimodule isomorphism π of M onto M . Let Φ M be an involutive A − A -equivalencebimodules isomorphism of f M ⊗ A X ⊗ A M onto X . Then Φ M ◦ ( e π ⊗ id X ⊗ π ) is aninvolutive A − A -equivalence bimodule isomorphism of f M ⊗ A X ⊗ A M onto X ,where e π is the A − A -equivalence bimodule isomorphism of f M onto M defined by e π ( e m ) = ] π ( m )for any m ∈ M . Let [ M, C M ] and [ M , C M ] be the element in Pic( A, C X ) inducedby Φ M and Φ M ◦ ( e π ⊗ id X ⊗ π ). Let ( M, C M ) be the element in Equi( A, C X ) obtainedby using the isomorphism Φ M and let ( M , C M ) be the element in Equi( A, C X )obtained by using the isomorphism Φ M ◦ ( e π ⊗ id ⊗ π ). Then by the definitionsof ( M, C M ), ( M , C M ) and Lemma 6.1, we obtain that [ M, C M ] = [ M , C M ] inPic( A, C X ). (cid:3) Let g be the map from Im f A to Pic( A, C X ) defined by g ([ M ]) = [ M, C M ]for any [ M ] ∈ Im f A . By Lemmas 6.1 and 6.2, g is well-defined.Let M and K be A − A -equivalence bimodules with f M ⊗ A X ⊗ A M ∼ = X and e K ⊗ A X ⊗ A K ∼ = X as A − A -equivalence bimodules, respectively. Let ( M, C M )and ( K, C K ) be the elements in Equi( A, C X ) induced by M and K , respectively.Also, let ( M ⊗ A K, C M ⊗ A K ) be the element in Equi( A, C X ) induced by M ⊗ A K . Lemma 6.3.
With the above notation, C M ⊗ C X C K ∼ = C M ⊗ A K as C X − C X -equivalence bimodules.Proof. Let π be the linear map from C M ⊗ C X C K onto C M ⊗ A K defined by π ( (cid:20) m m ⊗ xm ⊗ e x ♮ m (cid:21) ⊗ (cid:20) k y ⊗ k e y ♮ ⊗ k k (cid:21) )= (cid:20) m m ⊗ xm ⊗ e x ♮ m (cid:21) · (cid:20) k y ⊗ k e y ♮ ⊗ k k (cid:21) = (cid:20) m ⊗ k + m ⊗ x ⊗ e y ♮ ⊗ k m ⊗ y ⊗ k + m ⊗ x ⊗ k m ⊗ e x ♮ ⊗ k + m ⊗ e y ♮ ⊗ k m ⊗ e x ♮ ⊗ y ⊗ k + m ⊗ k (cid:21) , for any (cid:20) m m ⊗ xm ⊗ e x ♮ m (cid:21) ∈ C XM and (cid:20) k y ⊗ k e y ♮ ⊗ k k (cid:21) ∈ X C K , where we regardthe tensor product as a product on C M in the formal manner. But we identify A ⊗ A K and X ⊗ A e X , e X ⊗ A X with K and A by the isomorphisms defined by a ⊗ k ∈ A ⊗ A K a · k ∈ K,x ⊗ e y ∈ X ⊗ A e X A h x, y i ∈ A, e x ⊗ y ∈ e X ⊗ A X
7→ h x, y i A ∈ A, respectively. Furthermore, we identify X ⊗ A K and e X ⊗ A K with K ⊗ A X and K ⊗ A e X as A − A -equivalence bimodules by Ψ K and g Ψ K , which are defined asabove, respectively. Thus, π ( (cid:20) m m ⊗ xm ⊗ e x ♮ m (cid:21) ⊗ (cid:20) k y ⊗ k e y ♮ ⊗ k k (cid:21) )= (cid:20) m ⊗ k + m ⊗ A h x, y ♮ i · k m ⊗ Ψ K ( y ⊗ k ) + m ⊗ Ψ K ( x ⊗ k ) m ⊗ g Ψ K ( e x ♮ ⊗ k ) + m ⊗ g Ψ K ( e y ♮ ⊗ k ) m ⊗ h x ♮ , y i A · k + m ⊗ k (cid:21) . hen by routine computations, A h x, y ♮ i = h x ♮ , y i A , g Ψ K ( e x ♮ ⊗ k ) = n X i =1 u i ⊗ Φ K ( e u i ⊗ x ⊗ k ) e ♮ , Ψ K ( x ⊗ k ) = n X i =1 u i ⊗ Φ K ( e u i ⊗ x ⊗ k ) , g Ψ K ( e y ♮ ⊗ k ) = n X i =1 u i ⊗ Φ K ( e u i ⊗ y ⊗ k ) e ♮ , Ψ K ( y ⊗ k ) = n X i =1 u i ⊗ Φ K ( e u i ⊗ y ⊗ k ) , where { u i } ni =1 is a finite subset of K with P ni =1 A h u i , u i i = 1 and Φ K and g Φ K areas defined in the above. Hence π is a linear map from C M ⊗ C X C K to C M ⊗ A K .Next, we show that π is surjective. We take elements (cid:20) m m ⊗ xm ⊗ e x ♮ m (cid:21) ∈ C XM , (cid:20) k k (cid:21) ∈ X C K . Then π ( (cid:20) m m ⊗ xm ⊗ e x ♮ m (cid:21) ⊗ (cid:20) k k (cid:21) )= (cid:20) m ⊗ k m ⊗ Ψ K ( x ⊗ k ) m ⊗ g Ψ K ( e x ♮ ⊗ k ) m ⊗ k (cid:21) . We also take elements (cid:20) m ⊗ ym ⊗ e y ♮ (cid:21) ∈ C XM , (cid:20) k k (cid:21) ∈ X C K . Then π ( (cid:20) m ⊗ ym ⊗ e y ♮ (cid:21) ⊗ (cid:20) k k (cid:21) )= (cid:20) m ⊗ Ψ K ( y ⊗ k ) m ⊗ g Ψ K ( e y ♮ ⊗ k ) 0 (cid:21) . Thus π ( (cid:20) m m ⊗ xm ⊗ e x ♮ m (cid:21) ⊗ (cid:20) k k (cid:21) + (cid:20) m ⊗ ym ⊗ e y ♮ (cid:21) ⊗ (cid:20) k k (cid:21) )= (cid:20) m ⊗ k m ⊗ Ψ K ( x ⊗ k + y ⊗ k ) m ⊗ g Ψ K ( e x ♮ ⊗ k + e y ♮ ⊗ k ) m ⊗ k (cid:21) . Since Ψ K and g Ψ K are isomorphisms of X ⊗ A K and e X ⊗ A K onto K ⊗ A X and K ⊗ A e X , respectively, we can see that π is surjective. Furthermore, by the definitionsof π and the left and the right A -valued inner products on C M , C K and C M ⊗ K ,we can easily see that π preserves the left and the right A -valued inner products.Indeed, let M , M ∈ C M and K , K ∈ C K . Then C X h π ( M ⊗ K ) , π ( M ⊗ K ) i = C X hM · K , M · K i = M · K · ( M · K ) e = M · K · f K · g M . lso, C X hM ⊗ K , M ⊗ K i = C X hM · C X hK , K i , M i = C A hM · K · f K , M i = M · K · f K · g M . Hence π preserves the left C X -valued inner products. Similarly, we can see that π preserves the right C X -valued inner products. Therefore, π is a C X − C X -equivalence bimodule isomorphism of C M ⊗ C X C K onto C M ⊗ A K by the remarkafter [3, Definition 1.1.18]. (cid:3) Proposition 6.4.
With the above notation, g is a homomorphism of Im f A to Pic(
A, C X ) with f A ◦ g = id on Im f A .Proof. This is immediate by Lemma 6.3 and the definition of g . (cid:3) We give the main result of this paper.
Theorem 6.5.
Let A be a unital C ∗ -algebra and X an involutive A − A -equivalencebimodule. Let A ⊂ C X be the unital inclusion of unital C ∗ -algebras induced by X .We suppose that A ′ ∩ C X = C . Let f A be the homomorphism of Pic(
A, C X ) to Pic( A ) defined by f A ([ M, N ]) = [ M ] for any ( M, N ) ∈ Equi(
A, C X ) . Then Pic(
A, C X ) is isomorphic to a semi-directproduct group of T by the group { [ M ] ∈ Pic( A ) | M is an A − A -equivalence bimodule with X ∼ = f M ⊗ A X ⊗ A M as involutive A − A -equivalence bimodules } . Proof.
This is immediate by Proposition 3.6, Theorem 5.5 and Proposition 6.4. (cid:3)
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E-mail address: [email protected]@math.u-ryukyu.ac.jp