aa r X i v : . [ m a t h . OA ] S e p THE PICARD GROUPS FOR CONDITIONAL EXPECTATIONS
KAZUNORI KODAKA
Abstract.
Let A ⊂ C and B ⊂ D be inclusions of C ∗ -algebras with AC = C , BD = D . Let A B A ( C, A ) (resp. B B B ( D, B )) be the space of all bounded A -bimodule (resp. B -bimodule) linear maps from C (resp. D ) to A (resp. B ).We suppose that A ⊂ C and B ⊂ D are strongly Morita equivalent. In thispaper, we shall show that there is an isometric isomorphism f of B B B ( D, B )onto A B A ( C, A ) and we shall study on basic properties about f . And, wedefine the Picard group for a bimodule linear map and discuss on the Picardgroup of a bimodule linear map. Introduction
Let A ⊂ C and B ⊂ D be inclusions of C ∗ -algebras with AC = C , BD = D .Let A B A ( C, A ), B B B ( D, B ) be the spaces of all bounded A -bimodule linear mapsand all bounded B -bimodule linear maps from C and D to A and B , respectively.We suppose that they are strongly Morita equivalent with respect to a C − D -equivalence bimodule Y and its closed subspace X . In this paper, we shall definean isometric isomorphism f of B B B ( D, B ) onto A B A ( C, A ) induced by Y and X in the same way as in [8]. We shall study on the basic properties about f . And, wedeifne the Picard group for a bimodule linear map and discuss on the Picard groupof a bimodule linear map.For a C ∗ -algebra A , we denote by 1 A and id A the unit element in A and theidentity map on A , respectively. If no confusion arises, we denote them by 1 andid, respectively. For each n ∈ N , we denote by M n ( C ) the n × n -matrix algebraover C and I n denotes the unit element in M n ( C ). Also, we denote by M n ( A ) the n × n -matrix algebra over A and we identify M n ( A ) with A ⊗ M n ( C ) for any n ∈ N .For a C ∗ -algebra A , let M ( A ) be the multiplier C ∗ -algebra of A .Let K be the C ∗ -algebra of all compact operators on a countably infinite dimen-sional Hilbert space.Let A and B be C ∗ -algebras. Let X be an A − B -equivalence bimodule. For any a ∈ A , b ∈ B , x ∈ X , we denote by a · x the left A -action on X and by x · b theright B -action on X , respectively. Let A K ( X ) be the C ∗ -algebra of all “ compact”adjointable left A -linear operators on X and we identify A K ( X ) with B . Similarlywe define K B ( X ) and we identify K B ( X ) with A .2. Construction
Let A ⊂ C and B ⊂ D be inclusions of C ∗ -algebras with AC = C and CD = D .Let A B A ( C, A ), B B B ( D, B ) be the spaces of all bounded A -bimodule linear mapsand all bounded B -bimodule linear maps from C and D to A and B , respectively.We suppose that A ⊂ C and B ⊂ D are strongly Morita equivalent with respectto a C − D -equivalence bimodule Y and its closed subspace X . We construct an Mathematics Subject Classification.
Key words and phrases. bimodule maps, inclusions of C ∗ -algebras, conditional expectations,strong Morita equivalence. sometric isomorphism of B B B ( D, B ) onto A B A ( C, A ). For any φ ∈ B B B ( D, B ),we define the linear map τ from Y to X by h x , τ ( y ) i B = φ ( h x, y i D )for any x ∈ X , y ∈ Y . Lemma 2.1.
With the above notation, τ satisfies the following conditions: (1) τ ( x · d ) = x · φ ( d ) , (2) τ ( y · b ) = τ ( y ) · b , (3) h x , τ ( y ) i B = φ ( h x, y i D ) for any b ∈ B , d ∈ D , x ∈ X , y ∈ Y . Also, τ is bounded and || τ || ≤ || φ || .Furthermore, τ is the unique linear map from Y to X satisfying Condition (3) .Proof. We can prove this lemma in the same way as in the proof of [8, Lemma2.1]. (cid:3)
Lemma 2.2.
With the above notation, τ ( a · y ) = a · τ ( y ) for any a ∈ A , y ∈ Y .Proof. This can be proved in the same way as in the proof of [8, Lemma 2.2].Indeed, for any x, z ∈ X , y ∈ Y , τ ( A h x, z i · y ) = τ ( x · h z, y i D ) = x · φ ( h z, y i D ) = x · h z, τ ( y ) i B = A h x, z i · τ ( y ) . Since A h X , X i = A and τ is bounded, we obtain the conclusion. (cid:3) Let ψ be the linear map from C to A defined by ψ ( c ) · x = τ ( c · x )for any c ∈ C , x ∈ X , where we identify K B ( X ) with A as C ∗ -algebras by the map a ∈ A T a ∈ K B ( X ), which is defined by T a ( x ) = a · x for any x ∈ X . Lemma 2.3.
With the above notation, ψ is a linear map from C to A satisfyingthe following conditions: (1) τ ( c · x ) = ψ ( c ) · x , (2) ψ ( C h y, x i ) = A h τ ( y ) , x i for any c ∈ C , x ∈ X , y ∈ Y . Also, ψ is a bounded A -bimodule linear map from C to A with || ψ || ≤ || τ || . Furthermore, ψ is the unique linear map from C to D satisfying Condition (1) .Proof. We can prove this lemma in the same way as in the proof of [8, Lemma2.3]. (cid:3)
Proposition 2.4.
Let A ⊂ C and B ⊂ D be inclusions of C ∗ -algebras with AC = C and BD = D . We suppose that A ⊂ C and B ⊂ D are strongly Morita equivalentwith respect to a C − D -equivalence bimodule Y and its closed subspace X . Let φ be any element in B B B ( D, B ) . Then there are the unique linear map τ from Y to X and the unique element ψ in A B A ( C, A ) satisfying the following conditions: (1) τ ( c · x ) = ψ ( c ) · x , (2) τ ( a · y ) = a · τ ( y ) , (3) A h τ ( y ) , x i = ψ ( C h y, x i ) , (4) τ ( x · d ) = x · φ ( d ) , (5) τ ( y · b ) = τ ( y ) · b , (6) φ ( h x, y i D ) = h x , τ ( y ) i B for any a ∈ A , b ∈ B , c ∈ C , d ∈ D , x ∈ X , y ∈ Y . Furthermore, || ψ || ≤ || τ || ≤|| φ || . Also, for any element ψ ∈ A B A ( C, A ) , we have the same results as above.Proof. This is immediate by Lemmas 2.1, 2.2 and 2.3. (cid:3) e denote by f ( X,Y ) the map from φ ∈ B B B ( D, B ) to the above ψ ∈ A B A ( C, A ).By the definition of f ( X,Y ) and Proposition 2.4, we can see that f ( X,Y ) is an isometricisomorphism of B B B ( D, B ) onto A B A ( C, A ). Lemma 2.5.
With the above notation, let φ be any element in B B B ( D, B ) . Then f ( X,Y ) ( φ ) is the unique linear map from C to A satisfying that h x , f ( X,Y ) ( φ )( c ) · z i B = φ ( h x , c · z i D ) for any c ∈ C , x, z ∈ X .Proof. We can prove this lemma in the same way as in the proof of [8, Lemma2.6]. (cid:3)
Let Equi(
A, C, B, D ) be the set of all pairs (
X, Y ) such that Y is a C − D -equivalence bimodule and X is its closed subspace satisfying Conditions (1), (2) in[9, Definition 2.1]. We define an equivalence relation “ ∼ ” in Equi( A, C, B, D ) asfollows: For any (
X, Y ) , ( Z, W ) ∈ Equi(
A, C, B, D ), we say that (
X, Y ) ∼ ( Z, W )in Equi(
A, C, B, D ) if there is a C − D - equivalence bimodule Φ of Y onto W suchthat Φ | X is a bijection of X onto Z . Then Φ | X is an A − B -equivalence bimoduleof X onto Z by [6, Lemma 3.2]. We denote by [ X, Y ] the equivalence class of(
X, Y ) ∈ Equi(
A, C, B, D ). Lemma 2.6.
With the above notation, let ( X, Y ) , ( Z, W ) ∈ Equi(
A, C, B, D ) with ( X, Y ) ∼ ( Z, W ) in Equi(
A, C, B, D ) . Then f ( X,Y ) = f ( Z,W ) .Proof. This can be proved in the same way as in the proof [8, Lemma 6.1]. (cid:3)
We denote by f [ X,Y ] the isometric isomorphism of B B B ( D, B ) into A B A ( C, A )induced by the equivalence class [
X, Y ] of (
X, Y ) ∈ Equi(
A, C, B, D ).Let L ⊂ M be an inclusion of C ∗ -algebras with LM = M , which is strongyMorita equivalent to the inclusion B ⊂ D with respect to a D − M -equivalencebomodule W and its closed subspace Z . Then the inclusion A ⊂ C is stronglyMorita equivalent to the inclusion L ⊂ M with respect to the C − M -equivalencebimodule Y ⊗ D W and its closed subspace X ⊗ B Z . Lemma 2.7.
With the above notation, f [ X ⊗ B Z , Y ⊗ D W ] = f ] X,Y ] ◦ f [ Z,W ] . Proof.
This can be proved in the same way as in the proof of [8, Theorem 6.2]. (cid:3) Strong Morita equivalence
Let A ⊂ C and B ⊂ D be inclusions of C ∗ -algebras with AC = C and BD = D .Let ψ ∈ A B A ( C, A ) and φ ∈ B B B ( D, B ). Definition . We say that ψ and φ are strongly Morita equivalent if there is anelement ( X, Y ) ∈ Equi(
A, C, B, D ) such that f [ X,Y ] ( φ ) = ψ . Also, we say that φ and ψ are strongly Morita equivalent with respect to ( X, Y ) in Equi(
A, C, B, D ). Remark . By Lemma 2.7, strong Morita equivalence for bimodule linear mapsare equivlence relation.Let ψ ∈ A B A ( C, A ) and φ ∈ B B B ( D, B ). We suppose that φ and ψ are stronglyMorita equivalent with respect to ( X, Y ) in Equi(
A, C, B, D ). Let L X and L Y bethe linking C ∗ -algebras for X and Y , respectively. Then in the same way as in [6,Section 3] or Brown, Green and Rieffel [2, Theorem 1.1], L X is a C ∗ -subalgebra of Y and by easy computations, L X L Y = L Y . Furthermore, there are full projections p, q ∈ M ( L X ) with p + q = 1 M ( L X ) satisfying the following conditions: pL X p ∼ = A, pL Y p ∼ = C,qL X q ∼ = B, qL Y q ∼ = D as C ∗ -algebras. We note that M ( L X ) ⊂ M ( L Y ) by Pedersen [10, Section 3.12.12]since L X L Y = L Y .Let φ , ψ be as above. We suppose that φ and ψ are selfadjoint. Let τ be theunique bounded linear map from Y to X satisfying Conditions (1)-(6) in Proposition2.4. Let ρ be the map from L Y to L X defined by ρ ( (cid:20) c y e z d (cid:21) ) = (cid:20) ψ ( c ) τ ( y ) g τ ( z ) φ ( d ) (cid:21) for any c ∈ C , d ∈ D , y, z ∈ Y . By routine computations ρ is a selfadjoint element in L X B L X ( L Y , L X ), where L X B L X ( L Y , L X ) is the space of all bounded L X -bimodulelinear maps from L Y to L X . Furthermore, ρ | pL Y p = ψ and ρ | qL Y q = φ , wherewe identify A, C and
B, D with pL X p , pL Y p and qL X q , qL Y q in the usual way,respectively. Thus we obtain the following lemma: Lemma 3.2.
With the above notation, let ψ ∈ A B A ( C, A ) and φ ∈ B B B ( D, B ) .We suppose that ψ and φ are selfadjoint and strongly Morita equivalent with re-spect to ( X, Y ) ∈ Equi(
A, C, B, D ) . Then there is a selfadjoint element ρ ∈ L X B L X ( L Y , L X ) such that ρ | pL Y p = ψ, ρ | qL Y q = φ. Also, we have the inverse direction:
Lemma 3.3.
Let A ⊂ C and B ⊂ D be as above and let ψ ∈ A B A ( C, A ) and φ ∈ B B B ( D, B ) be selfadjoint elements. We suppose that there are an inclusion K ⊂ L of C ∗ -algebras with KL = L and full projections p, q ∈ M ( K ) with p + q = 1 M ( K ) such that A ∼ = pKp, C ∼ = pLp, B ∼ = qKq, D ∼ = qLq, as C ∗ -algebras. Also, we suppose that there is a selfadjoint element ρ in K B K ( L, K ) such that ρ | pLp = ψ, ρ | qLq = φ. Then φ and ψ are strongly Morita equivalent, where we identify pKp , pLp and qKq , qLq with A, C and
B, D , respectively.Proof.
We note that (
Kp, Lp ) ∈ Equi(
K, L, A, C ), where we identify A and C with pKp and pLp , respectively. By routine computations, we can see that h kp , ρ ( l ) · k p i A = ψ ( h kp , l · k p i C )for any k, k ∈ K , l ∈ L . Thus by Lemma 2.5, f [ Kp , Lp ] ( ψ ) = ρ . Similarly, f [ Kq , Lq ] ( φ ) = ρ . Since f − Kq , Lq ] ( ρ ) = φ ,( f − Kq , Lq ] ◦ f [ Kp , Lp ] )( ψ ) = φ. Since f − Kq , Lq ] = f [ qK , qL ] , by Lemma 2.7 φ = f [ qK , qL ][ Kp , Lp ] ( ρ ) = f [ qKp , qLp ] ( ψ ) . Therefore, we obtain the conclusion. (cid:3) roposition 3.4. Let A ⊂ C and B ⊂ D be inclusions of C ∗ -algebras with AC = C and BD = D . Let ψ and φ be selfadjoint elements in A B A ( C, A ) and B B B ( D, B ) ,respectively. Then the following conditions are equivalent: (1) ψ and φ are strongly Morita equivalent, (2) There are an inclusion K ⊂ L of C ∗ -algebras with KL = L , full projections p, q ∈ M ( K ) with p + q = 1 M ( K ) and a selfadjoint element ρ ∈ K B K ( L, K ) satis-fying that A ∼ = pKp, C ∼ = pLp, B ∼ = qKq, D ∼ = qLq, as C ∗ -algebras and that ρ | pLp = ψ, ρ | qLq = φ, where we identify pKp , pLp and qKq , qLq with A , C and B , D , respectively.Proof. This is immediate by Lemmas 3.2 and 3.3. (cid:3) Stable C ∗ -algebras Let A ⊂ C be an inclusion of C ∗ -algebras with AC = C . Let A s = A ⊗ K and C s = C ⊗ K . Let { e ij } ∞ i,j =1 be a system of matrix units of K . Clearly A s ⊂ C s and A ⊂ C are strongly Morita equivalent with respect to the C s − C -equivalencebimodule C s (1 M ( A ) ⊗ e ) and its closed subspace A s (1 M ( A ) ⊗ e ), where we identify A and C with (1 ⊗ e ) A s (1 ⊗ e ) and (1 ⊗ e ) C s (1 ⊗ e ), respectively. Lemma 4.1.
With the above notation, for any φ ∈ A B A ( C, A ) , f [ A s (1 ⊗ e ) , C s (1 ⊗ e )] ( φ ) = φ ⊗ id K . Proof.
It suffices to show that h a (1 ⊗ e ) , ( φ ⊗ id K )( c ) · b (1 ⊗ e ) i A = φ ( h a (1 ⊗ e ) , c · b (1 ⊗ e ) i C )for any a, b ∈ A s , c ∈ C s by Lemma 2.5. Indeed, for any a, b ∈ A s , c ∈ C s , h a (1 ⊗ e ) , ( φ ⊗ id K )( c ) · b (1 ⊗ e ) i A = (1 ⊗ e ) a ∗ ( φ ⊗ id K )( c ) b (1 ⊗ e )= ( φ ⊗ id K )((1 ⊗ e ) a ∗ cb (1 ⊗ e )) . On the other hand, φ ( h a (1 ⊗ e ) , c · b (1 ⊗ e ) i C ) = φ ((1 ⊗ e ) a ∗ cb (1 ⊗ e )) . Since we identify C with (1 ⊗ e ) C s (1 ⊗ e ), h a (1 ⊗ e ) , ( φ ⊗ id K )( c ) · b (1 ⊗ e ) i A = φ ( h a (1 ⊗ e ) , c · b (1 ⊗ e ) i C )for any a, b ∈ A s , c ∈ C s . Therefore, we obtain the conclusion. (cid:3) Let ψ ∈ A B A ( C, A ). Let { u λ } λ ∈ Λ be an approximate units of A s with || u λ || ≤ λ ∈ Λ. Since AC = C , { u λ } λ ∈ Λ is an approximate units of C s . Let c beany element in C . For any a ∈ A , { aψ ( cu λ ) } λ ∈ Λ and { ψ ( cu λ ) a } λ ∈ Λ are Cauchynets in A . Hence there is an element x ∈ M ( A ) such that { ψ ( cu λ ) } λ ∈ Λ is strictlyconvergent to x ∈ M ( A ). Let ψ be the map from M ( C ) to M ( A ) defined by ψ ( c ) = x for any c ∈ C . By routine computations ψ is a bounded M ( A )-bimodulelinear map from M ( C ) to M ( A ) and ψ = ψ | C .Let q be a full projection in M ( A ), that is, AqA = A . Since AC = C , M ( A ) ⊂ M ( C ) by [10, Section 3.12.12]. Thus CqC = CAqAC = CAC = C. We regard qC and qA as a qCq − C -equivalence bimodule and a qAq − A -equivalencebimodule, respectively. Then ( qA, qC ) ∈ Equi( qAq, qCq, A, C ). Lemma 4.2.
With the above notation, for any ψ ∈ A B A ( C, A ) f [ qA,qC ] ( ψ ) = ψ | qCq . roof. By easy computations, we see that h qx , ψ | qCq ( c ) · qz i A = ψ ( h qx , c · qz i C )for any x, z ∈ A , c ∈ C since ψ ( q ) = q . Thus we obtain the conclusion by Lemma2.5. (cid:3) Let A ⊂ C and B ⊂ D be inclusions of C ∗ -algebras such that A and B are σ -unital and AC = C and BD = D . Let B s = B ⊗ K and D s = D ⊗ K . Wesuppose that A ⊂ C and B ⊂ D are strongly Morita equivalent with respect to( X, Y ) ∈ Equi(
A, C, B, D ). Let X s = X ⊗ K and Y s = Y ⊗ K , an A s − B s -equivalence bimodule and a C s − D s -equivalence bimodule, respectively. We notethat ( X s , Y s ) ∈ Equi( A s , C s , B s , D s ). Let L X s and L Y s be the linking C ∗ -algebrasfor X s and Y s , respectively. Let p = (cid:20) M ( A s )
00 0 (cid:21) , p = (cid:20) M ( B s ) (cid:21) . Then p and p are full projections in M ( L X s ). By easy computations, we can seethat L X s L Y s = L Y s . Hence by [10, Section 3.12.12], M ( L X s ) ⊂ M ( L Y s ). Since p and p are full projections in M ( L X ), by Brown [1, Lemma 2.5], there is a partialisometry w ∈ M ( L X s ) such that w ∗ w = p , ww ∗ = p . We note that w ∈ M ( L Y s ).Let Ψ be the map from p L Y s p to p L Y s p defined byΨ( (cid:20) d (cid:21) ) = w ∗ (cid:20) d (cid:21) w for any d ∈ D s . In the same way as in the discussions of [2], Ψ is an isomorphism of p L Y s p onto p L Y s p and Ψ | p L Xs p is an isomorphism of p L X s p onto p L X s p .Also, we note the following: p L Y s p ∼ = C s , p L X s p ∼ = A s p L Y s p ∼ = D s , p L X s p ∼ = B s as C ∗ -algebras. We identify A s , C s and B s , D s with p L X s p , p L Y s p and p L X s p , p L Y s p , respectively. Also, we identify X s , Y s with p L X s p , p L Y s p .Let A s Ψ be the A s − B s -equivalence bimodule induced by Ψ | B s , that is, A s Ψ = A s as C -vector spaces. The left A s -action and the A s -valued inner product on A s Ψ aredefined in the usual way. The right B s -action and B s -valued inner product on A s Ψ are defined as follows: For any x, y ∈ A s Ψ , b ∈ B s , x · b = x Ψ( b ) , h x, y i B s = Ψ − ( x ∗ y ) . Similarly, we define the C s − D s -equivalence bimodule C s Ψ induced by Ψ. We notethat A s Ψ is a closed subspace of C s Ψ and ( A s Ψ , C s Ψ ) ∈ Equi( A s , C s , B s , D s ). Lemma 4.3.
With the above notation, ( A s Ψ , C s Ψ ) is equivalent to ( X s , Y s ) in Equi( A s , C s , B s , D s ) .Proof. We can prove this lemma in the same way as in the proof of [2, Lemma 3.3].Indeed, let π be the map from Y s to C s Ψ defined by π ( y ) = (cid:20) y (cid:21) w for any y ∈ Y s . By routine computations, π is a C s − D s -equivalence bimoduleisomorphism of Y s onto C s Ψ and π | X s is a bijection from X s onto A s . Hence by [6,Lemma 3.2], we obtain the conclusion. (cid:3) Lemma 4.4.
With the above notation, for any φ ∈ B s B B s ( D s , B s ) , f [ X s ,Y s ] ( φ ) = Ψ ◦ φ ◦ Ψ − . roof. We claim that h x , (Ψ ◦ φ ◦ Ψ − )( d ) · z i B s = φ ( h x, d · z i D s )for any φ ∈ B s B B s ( D s , B s ), x, z ∈ A s Ψ , d ∈ D s . Indeed, h x , (Ψ ◦ φ ◦ Ψ − )( d ) · z i B s = Ψ − ( x ∗ (Ψ ◦ φ ◦ Ψ − )( d ) z )= Ψ − ( x ∗ )( φ ◦ Ψ − )( d )Ψ − ( z ) . On the other hand, φ ( h x, d · z i D s ) = φ (Ψ − ( x ∗ dz )) = φ (Ψ − ( x ∗ )Ψ − ( d )Ψ − ( z ))= Ψ − ( x ∗ )( φ ◦ Ψ − )( d )Ψ − ( z )since Ψ − ( x ∗ ), Ψ − ( z ) ∈ B s . Thus h x , (Ψ ◦ φ ◦ Ψ − )( d ) · z i B s = φ ( h x, d · z i D s )for any φ ∈ B s B B s ( D s , B s ), x, z ∈ A s Ψ , d ∈ D s . Hence by Lemma 2.5, f [ A s Ψ ,C s Ψ ] ( φ ) =Ψ ◦ φ ◦ Ψ − for any φ ∈ B s B B s ( D s , B s ). Therefore, f [ X s ,Y s ] ( φ ) = Ψ ◦ φ ◦ Ψ − byLemmas 2.6 and 4.3. (cid:3) Let Ψ be the strictly continuous isomorphism of M ( D s ) onto M ( C s ) extendingΨ to M ( D s ), which is defined in Jensen and Thomsen [4, Corollary 1.1.15]. ThenΨ | M ( B s ) is an isomorphism of M ( B s ) onto M ( A s ). Let q = Ψ(1 ⊗ e ). Then q is a full projection in M ( A s ) with C s qC s = C s and qA s q ∼ = A , qC s q ∼ = C as C ∗ -algebras. We identify with qA s q and qC s q with A and C , respectively. Thenwe obtain the following proposition: Proposition 4.5.
Let A ⊂ C and B ⊂ D be inclusions of C ∗ -algebras such that A and B are σ -unital and AC = C and BD = D . Let Ψ be the isomorphismof D s onto C s defined before Lemma 4.3 and let q = Ψ(1 ⊗ e ) . Let ( X, Y ) ∈ Equi(
A, C, B, D ) . For any φ ∈ B B B ( D, B ) , f [ X,Y ] ( φ ) = (Ψ ◦ ( φ ⊗ id K ) ◦ Ψ − ) | qC s q , where we identify qA s q and qC s q with A and C , respectively.Proof. We note that (1 ⊗ e ) B s (1 ⊗ e ) and (1 ⊗ e ) D s (1 ⊗ e ) are identified with B and D , respectively. Also, we identify qA s q and qC s q with A and C , respectively.Thus we see that[ qA s ⊗ A s X s ⊗ B s B s (1 ⊗ e ) , qC s ⊗ C s Y s ⊗ D s D s (1 ⊗ e )] = [ X, Y ]in Equi(
A, C, B, D ). Hence by Lemma 2.7, f [ X,Y ] ( φ ) = ( f [ qA s , qC s ] ◦ f [ X s , Y s ] ◦ f [ B s (1 ⊗ e ) , D s (1 ⊗ e )] )( φ ) . Therefore, by Lemmas 4.1, 4.2 and 4.4, f [ X,Y ] ( φ ) = (Ψ ◦ ( φ ⊗ id K ) ◦ Ψ − ) | qC s q . (cid:3) Basic properties
Let A ⊂ C and B ⊂ D be inclusions of C ∗ -algebras with AC = C and BD = D . We suppose that they are strongly Morita equivalent with respect to( X, Y ) ∈ Equi(
A, C, B, D ). Let A B A ( C, A ) and B B B ( D, B ) be as above and let f [ X,Y ] be the isometric isomorphism of B B B ( D, B ) onto A B A ( C, A ) induced by(
X, Y ) ∈ Equi(
A, C, B, D ) which is defined in Section 2. In this section, we givebasic properties about f [ X,Y ] . emma 5.1. With the above notation, we have the following: (1)
For any selfadjoint linear map φ ∈ B B B ( D, B ) , f [ X,Y ] ( φ ) is selfadjoint. (2) For any positive linear map φ ∈ B B B ( D, B ) , f [ X,Y ] ( φ ) is positive.Proof. (1) Let φ be any selfadjoint linear map in B B B ( D, B ) and let c ∈ C , x, z ∈ X . By lemma 2.5, h x , f [ X,Y ] ( φ )( c ∗ ) · z i B = φ ( h x , c ∗ · z i D ) = φ ( h c · x , z i D )= φ ( h z , c · x i D ) ∗ = h z , f [ X,Y ] ( φ )( c ) · x i ∗ B = h f [ X,Y ] ( φ )( c ) · x , z i B = h x , f [ X,Y ] ( φ )( c ) ∗ · z i B . Hence f [ X,Y ] ( φ )( c ∗ ) = f [ X,Y ] ( φ )( c ) ∗ for any c ∈ C .(2) Let φ be any positive linear map in B B B ( D, B ) and let c be any positive elementin C . Then h x, c · x i D ≥ x ∈ X by Raeburn and Williams [11, Lemma2.28]. Hence φ ( h x, c · x i D ) ≥ x ∈ X . That is, h x , f [ X,Y ] ( φ )( c ) · x i B ≥ x ∈ X . Thus f [ X,Y ] ( φ )( c ) ≥ (cid:3) Proposition 5.2.
Let A ⊂ C an B ⊂ D be as in Lemma 5.1. If φ is a conditionalexpectation from D onto B , then f [ X,Y ] ( φ ) is a conditional expectation from C onto A .Proof. Since φ ( b ) = b for any b ∈ B , for any a ∈ A , x, z ∈ X , h x , f [ X,Y ] ( φ )( a ) · z i B = φ ( h x , a · z i B ) = h x , a · z i B by Lemma 2.5. Thus f [ X,Y ] ( φ )( a ) = a for any a ∈ A . By Proposition 2.4 andLemma 5.1, we obtain the conclusion. (cid:3) Since A ⊂ C and B ⊂ D are strongly Morita equivalent with respect to ( X, Y ) ∈ Equi(
A, C, B, D ), A s ⊂ C s and B s ⊂ D s are strongly Morita equivalent withrespect to ( X s , Y s ) ∈ Equi( A s , C s , B s , D s ). Let φ be any element in B B B ( D, B ).Then φ ⊗ id K ∈ B s B B s ( D s , B s ) . Lemma 5.3.
With the above notation, for any φ ∈ B B B ( D, B ) f [ X s ,Y s ] ( φ ⊗ id K ) = f [ X,Y ] ( φ ) ⊗ id K . Proof.
This can be proved by routine computations. Indeed, for any c ∈ C , x, z ∈ X , k , k , k ∈ K , h x ⊗ k , f [ X s ,Y s ] ( φ ⊗ id)( c ⊗ k ) · z ⊗ k i B s = ( φ ⊗ id)( h x ⊗ k , c ⊗ k · z ⊗ k i B s )= ( φ ⊗ id)( h x ⊗ k , c · z ⊗ k k i D s )= ( φ ⊗ id)( h x , c · z i D ⊗ k ∗ k k )= h x , f [ X,Y ] ( φ )( c ) · z i B ⊗ k ∗ k k = h x ⊗ k , f [ X,Y ] ( φ )( c ) ⊗ k · z ⊗ k ] i B s by Lemma 2.5. Therefore we obtain the conclusion by Lemma 2.5. (cid:3) Corollary 5.4.
With the above notation, let n ∈ N . Then for any φ ∈ B B B ( D, B ) , f [ X ⊗ M n ( C ) , Y ⊗ M n ( C )] ( φ ⊗ id) = f [ X,Y ] ( φ ) ⊗ id M n ( C ) . Proposition 5.5.
With the above notation, let φ ∈ B B B ( D, B ) . If φ is n -positive,then f [ X,Y ] ( φ ) is n -positive for any n ∈ N .Proof. This is immediate by Lemma 5.1 and Corollary 5.4. (cid:3) . The Picard groups
Let A ⊂ C be an inclusion of C ∗ -algebras with AC = C . Let A B A ( C, A ) be asabove. Let Pic(
A, C ) be the Picard group of the inclusion A ⊂ C . Definition . Let φ ∈ A B A ( C, A ). We define Pic( φ ) byPic( φ ) = { [ X, Y ] ∈ Pic(
A, C ) | f [ X,Y ] ( φ ) = φ } . We call Pic( φ ) the Picard group of φ .Let B ⊂ D be an inclusion of C ∗ -algebras with BD = D . Let φ ∈ B B B ( D, B )and ψ ∈ A B A ( C, A ). Lemma 6.1.
With the above notation, if φ and ψ are strongly Morita equivalentwith respect to ( Z, W ) ∈ Equi(
A, C, B, D ) , then Pic( φ ) ∼ = Pic( ψ ) as groups.Proof. Let g be the map from Pic( φ ) to Pic( A, C ) defined by g ([ X, Y ]) = [ Z ⊗ B X ⊗ B e Z , W ⊗ D Y ⊗ D f W ]for any [ X, Y ] ∈ Pic( φ ). Then since f [ Z,W ] ( φ ) = ψ , by Lemma 2.7 f [ Z ⊗ B X ⊗ B e Z , W ⊗ D Y ⊗ D f W ] ( ψ ) = ( f [ Z,W ] ◦ f [ X,Y ] ◦ f [ e Z, f W ] )( ψ )= ( f [ Z,W ] ◦ f [ X,Y ] ◦ f − Z,W ] )( ψ ) = ψ. Hence [ Z ⊗ B X ⊗ B e Z , W ⊗ D Y ⊗ D f W ] ∈ Pic( ψ ) and by easy computations, we cansee that g is an isomorphism of Pic( φ ) onto Pic( ψ ). (cid:3) Let φ ∈ A B A ( C, A ). Let α be an automorphism of C such that the restrictionof α to A , α | A is an automorphism of A . Let Aut( A, C ) be the group of all suchautomorphisms and letAut(
A, C, φ ) = { α ∈ Aut(
A, C ) | α ◦ φ ◦ α − = φ } . Then Aut(
A, C, φ ) is a subgroup of Aut(
A, C ). Let π be the homomorphism ofAut( A, C ) to Pic(
A, C ) defined by π ( α ) = [ X α , Y α ]for any α ∈ Aut(
A, C ), where ( X α , Y α ) is an element in Equi( A, C ) induced by α ,which is defined in [6, Section 3], where Equi( A, C ) = Equi(
A, C, A, C ). Let u be aunitary element in M ( A ). Then u ∈ M ( C ) and Ad( u ) ∈ Aut(
A, C ) since AC = C .Let Int( A, C ) be the group of all such automorphisms in Aut(
A, C ). We note thatInt(
A, C ) = Int( A ), the subgroup of Aut( A ) of all generalized inner automorphismsof A . Let ı be the inclusion map of Int( A, C ) to Aut(
A, C ). Lemma 6.2.
With the above notation, let φ ∈ A B A ( C, A ) . Then the followinghold: (1) For any α ∈ Aut(
A, C ) , f [ X α ,Y α ] ( φ ) = α ◦ φ ◦ α − . (2) The map π | Aut(
A,C,φ ) is a homomorphism of Aut(
A, C, φ ) to Pic( φ ) , where π | Aut(
A,C,φ ) is the restriction of π to Aut(
A, C, φ ) . (3) Int( A, C ) ⊂ Aut(
A, C, φ ) and the following sequence −→ Int(
A, C ) ı −→ Aut(
A, C, φ ) π −→ Pic( φ ) is exact.Proof. (1) Let α ∈ Aut(
A, C ). Then for any c ∈ C , x, z ∈ X α , h x , ( α ◦ φ ◦ α − )( c ) · z i A = h x , ( α ◦ φ ◦ α − )( c ) z i A = α − ( x ∗ ( α ◦ φ ◦ α − )( c ) z )= α − ( x ∗ )( φ ◦ α − )( c ) α − ( z ) . n the other hand, φ ( h x , c · z i C ) = φ ( α − ( x ∗ cz )) = φ ( α − ( x ∗ ) α − ( c ) α − ( z ))= α − ( x ∗ )( φ ◦ α − )( c ) α − ( z ) . Thus by Lemma 2.5, f [ X α ,Y α ] ( φ ) = α ◦ φ ◦ α − .(2) Let α be any element in Aut( A, C, φ ). Then by (1), f [ X α ,Y α ] ( φ ) = α ◦ φ ◦ α − = φ .Hence [ X α , Y α ] ∈ Pic( φ ).(3) Let Ad( u ) ∈ Int(
A, C ). Then u ∈ M ( A ) ⊂ M ( C ). For any c ∈ C ,(Ad( u ) ◦ φ ◦ Ad( u ∗ ))( c ) = uφ ( u ∗ cu ) u ∗ = uu ∗ φ ( c ) uu ∗ = φ ( c )since φ ( u ) = u . Thus Int( A, C ) ⊂ Aut(
A, C, φ ). It is clear by [6, Lemma 3.4] thatthe sequence 1 −→ Int(
A, C ) ı −→ Aut(
A, C, φ ) π −→ Pic( φ )is exact. (cid:3) Proposition 6.3.
Let A ⊂ C be an inclusion of C ∗ -algebras with AC = C and wesuppose that A is σ -unital. Let φ ∈ A s B A s ( C s , A s ) . Then the sequence −→ Int( A s , C s ) ı −→ Aut( A s , C s , φ ) π −→ Pic( φ ) −→ is exact.Proof. It suffices to show that π is surjective by Lemma 6.2 (3). Let [ X, Y ] be anyelement in Pic( φ ). Then by [6, Proposition 3.5], there is an element α ∈ Aut( A s , C s )such that π ( α ) = [ X, Y ]in Pic(
A, C ). Since [
X, Y ] ∈ Pic( φ ), f [ X,Y ] ( φ ) = φ . Also, by Lemma 2.6, f [ X,Y ] = f [ X α ,Y α ] , where [ X α , Y α ] is the element in Pic( A, C ) induced by α . Hence f [ X α ,Y α ] ( φ ) = f [ X,Y ] ( φ ) = φ. Since f [ X α ,Y α ] ( φ ) = α ◦ φ ◦ α − by Lemma 6.2(1), φ = α ◦ φ ◦ α − . Hence α ∈ Aut( A s , C s , φ ). (cid:3) The C ∗ -basic construction Let A ⊂ C be a unital inclusion of unital C ∗ -algebras and let E A be a conditionalexpectation of Watatani index-finite type from C onto A . Let e A be the Jones’projection for E A and C the C ∗ -basic construction for E A . Let E C be its dualconditional expectation from C onto C . Let e C be the Jones’ projection for E C and C the C ∗ -basic construction for E C Let E C be the dual conditional expectationof E C from C onto C . Since E A and E C are of Watatani index-finite type, C and C can be regarded as a C − A -equivalence bimodule and a C − C -equivalencebimodule induced by E A and E C , respectively. We suppose that the Watatani indexof E A , Ind W ( E A ) ∈ A . Then by [9, Examples], inclusions A ⊂ C and C ⊂ C arestrongly Morita equivalent with respect to the C − C equivalence bimodule C andits closed subspace C , where we regard C as a closed subspace of C by the map θ C ( x ) = Ind W ( E A ) xe A for any x ∈ C (See [9, Examples]). Lemma 7.1.
With the above notation, we suppose that
Ind W ( E A ) ∈ A . Then E A and E C are strongly Morita equivalent with respect to ( C, C ) ∈ Equi( C , C , A, C ) . roof. By [9, Lemma 4.2], A ⊂ C and C ⊂ C are strongly Morita equivalent withrespect to ( C, C ) ∈ Equi( C , C , A, C ). Since we regard C as a closed subspace of C by the linear map θ C , we have only to show that h x , E C ( c e A c e C d e A d ) · z i A = E A ( h θ C ( x ) , c e A c e C d e A d · θ C ( z ) i C )for any c , c , d , d ∈ C , x, z ∈ C . Indeed, h x , E C ( c e A c e C d e A d ) · z i A = h x , Ind W ( E A ) − c e A c d e A d · z i A = Ind W ( E A ) − h x , c E A ( c d ) E A ( d z ) i A = Ind W ( E A ) − E A ( x ∗ c ) E A ( c d ) E A ( d z ) . for any c , c , d , d ∈ C , x, z ∈ C . On the other hand, E A ( h θ C ( x ) , c e A c e C d e A d · θ C ( z ) i C )= Ind W ( E A ) E A ( h xe A , c e A c E C ( d e A d ze A ) i C )= E A ( h xe A , c e A c d E A ( d z ) i C )= E A ( E C ( e A x ∗ c e A c d )) E A ( d z )= E A ( x ∗ c ) E A ( E C ( e A c d )) E A ( d z )= Ind W ( E A ) − E A ( x ∗ c ) E A ( c d ) E A ( d z ) . Hence h x , E C ( c e A c e C d e A d ) · z i A = E A ( h θ C ( x ) , c e A c e C d e A d · θ C ( z ) i C )for any c , c , d , d ∈ C , x, z ∈ C . Thus by Lemma 2.5, f [ C,C ] ( E A ) = E C .Therefore, we obtain the conclusion. (cid:3) Let B ⊂ D be another unital inclusion of unital C ∗ -algebras and let E B be a con-ditional expectation of Watatani index-finite type from D onto B . Let e B , D , E D , e D , D , E D be as above. Lemma 7.2.
With the above notation, we suppose that E A and E B are stronglyMorita equivalent with respect to ( X, Y ) ∈ Equi(
A, C, B, D ) . Then E C and E D arestrongly Morita equivalent.Proof. Since E A and E B are strongly Morita equivalent with respect to ( X, Y ) ∈ Equi(
A, C, B, D ), there is the unique linear map E X from Y to X , which is called aconditional expectation from Y onto X satisfying Conditions (1)-(6) in [9, Definition2.4]. Let Y be the upward basic construction of Y for E X defined in [9, Definition6.5]. Then by [9, Corollary 6.3 and Lemma 6.4], f [ Y,Y ] ( E D ) = E C , that is, E C and E D are strongly Morita equivalent with respect to ( Y, Y ) ∈ Equi(
C, C , D, D ). (cid:3) Lemma 7.3.
With the above notation, we suppose that
Ind W ( E A ) ∈ A . If E C and E D are strongly Morita equivalent with respect to ( Y, Z ) ∈ Equi(
C, C , D, D ) ,then E A and E B are strongly Morita equivalent.Proof. By Lemma 7.2, there is an element (
Z, Z ) ∈ Equi( C , C , D , D ) such that f [ Z,Z ] ( E D ) = E C . Since Ind W ( E B ) ∈ B by [9, Lemma 6.7], f [ C,C ] ( E A ) = E C , f [ D,D ] ( E B ) = E D by Lemma 7.1. Thus[ e C ⊗ C Z ⊗ D D , f C ⊗ C Z ⊗ D D ] ∈ Equi(
A, C, B, D )and f [ e C ⊗ C Z ⊗ D D , f C ⊗ C Z ⊗ D D ] ( E B ) = ( f − C,C ] ◦ f [ Z,Z ] ◦ f [ D,D ] )( E B ) = E A by Lemma 2.7. Therefore, we obtain the conclusion. (cid:3) roposition 7.4. Let A ⊂ C and B ⊂ D be unital inclusions of unital C ∗ -algebras.Let E A and E B be conditional expectations from C and D onto A and B , which areof Watatani index-finite type, respectively. Let E C and E D be the dual conditionalexpectations of E A and E B , respectively. We suppose that Ind W ( E A ) ∈ A . Thenthe following conditions are equivalent: (1) E A and E B are strongly Morita equivalent, (2) E C and E D are strongly Morita equivalent.Proof. This is immediate by Lemmas 7.1 and 7.3. (cid:3)
Let A ⊂ C and C , C be as above. Let E A , E C and E C be also as above.We suppose that Ind W ( E A ) ∈ A . We consider the Picard groups Pic( E A ) andPic( E C ) of E A and E C , respectively. For any [ X, Y ] ∈ Pic( E A ), there is theunique conditional expectation E X from Y onto X satisfying Conditions (1)-(6)in [9, Definition 2.4] since f [ X,Y ] ( E A ) = E A . Let F be the map from Pic( E A ) toPic( E C ) defined by F (([ X, Y ]) = [
Y, Y ]for any [ X, Y ] ∈ Pic( E A ), where Y is the upward basic construction for E X and byProposition 7.4, [ Y, Y ] ∈ Pic( E C ). Since E X is the unique conditional expectationfrom Y onto X satisfying Conditions (1)-(6) in [9, Definition 2.4] we can see thatthe same results as [6, Lemmas 4.3-4.5] hold. Hence in the same way as in the proofof [6, Lemma 5.1], we obtain that F is a homomorphism of Pic( E A ) to Pic( E C ).Let G be the map from Pic( E A ) to Pic( E C ) defined by for any [ X, Y ] ∈ Pic( E A ) G ([ X, Y ]) = [ C ⊗ A X ⊗ A e C , C ⊗ C Y ⊗ C f C ] , where ( C, C ) is regarded as an element in Equi( C , C , A, C ). By the proof ofLemma 6.1, G is an isomorphism of Pic( E A ) onto Pic( E C ). Let F be the homo-morphism of Pic( E C ) to Pic( E C ) defined as above. Then in the same way as inthe proof of [6, Lemma 5.2], F ◦ F = G on Pic( E A ). Furthermore, in the sameway as in the proofs of [6, Lemmas 5.3 and 5.4], we obtain that F ◦ G − ◦ F = idon Pic( E C ). Therefore, we obtain the same result as [6, Theorem 5.5]. Theorem 7.5.
Let A ⊂ C be a unital inclusion of unital C ∗ -algebras. We supposethat there is a conditional expectation E A of Watatani index-finite type from C onto A and that Ind W ( E A ) ∈ A . Then Pic( E A ) ∼ = Pic( E C ) , where E C is the dualconditional expectation of E A from C onto C and C is the C ∗ -basic constructionfor E A . Relative commutants
Let A ⊂ C and B ⊂ D be unital inclusions of C ∗ -algebras and let E A and E B beconditional expectations of Watatani index-finite type from C and D onto A and B , respectively. We suppose that there is an element ( X, Y ) ∈ Equi(
A, C, B, D )such that E A is strongly Morita equivalent to E B , that is, f [ X,Y ] ( E B ) = E A . For any element h ∈ A ′ ∩ C , let h E A be defined by h E A ( c ) = E A ( ch )for any c ∈ C . We also define k E B in the same way as above for any k ∈ B ′ ∩ D . Lemma 8.1.
With the above notation, for any h ∈ A ′ ∩ C , there is the uniqueelement k ∈ B ′ ∩ D such that f [ X,Y ] ( k E B ) = h E A . roof. Since A ⊂ C and B ⊂ D are strongly Morita equivalent with respect to( X, Y ) ∈ Equi(
A, C, B, D ), there are a positive integer n ∈ N and a projection p ∈ M n ( A ) with M n ( A ) pM n ( A ) = M n ( A ) and M n ( C ) pM n ( C ) = M n ( C ) such thatthe inclusion B ⊂ D is regarded as the inclusion pM n ( A ) p ⊂ pM n ( C ) p and suchthat X and Y are identified with (1 ⊗ f ) M n ( A ) p and (1 ⊗ f ) M n ( C ) p (See [9, Section2]), where M n ( A ) and M n ( C ) are identified with A ⊗ M n ( C ) and C ⊗ M n ( C ),respectively, f is a minimal projection in M n ( C ) and we identified A and C with(1 ⊗ f )( A ⊗ M n ( C ))(1 ⊗ f ) and (1 ⊗ f )( C ⊗ M n ( C ))(1 ⊗ f ), respectively. Then wecan see that for any h ∈ A ′ ∩ C , there is the unique element k ∈ B ′ ∩ D such that h · x = x · k for any x ∈ X . Indeed, by the above discussions, we may assume that B = pM n ( A ) p , D = pM n ( C ) p , X = (1 ⊗ f ) M n ( A ) p . Let h be any element in A ′ ∩ C .Then for any x ∈ M n ( A ) h · (1 ⊗ f ) xp = (1 ⊗ f )( h ⊗ I n ) xp = (1 ⊗ f ) x ( h ⊗ I n ) p = (1 ⊗ f ) xp ( h ⊗ I n ) p = (1 ⊗ f ) xp · ( h ⊗ I n ) p. By the proof of [9, Lemma 10.3], ( h ⊗ I n ) p ∈ ( pM n ( A ) p ) ′ ∩ pM n ( C ) p . Thus, forany h ∈ A ′ ∩ C , there is an element k ∈ B ′ ∩ D such that h · x = x · k for any x ∈ X . Next, we suppose that there is another element k ∈ B ′ ∩ D suchthat h · x = x · k for any x ∈ X . Then ( c · x ) · k = ( c · x ) · k for any c ∈ C , x ∈ X .Since C · X = Y by [9, Lemma 10.1], k = k . Hence k is unique. Furthermore, forany x, z ∈ X , c ∈ C , h x , h E A ( c ) · z i B = h x , E A ( ch ) · z i B = E B ( h x , ch · z i D ) = E B ( h x , c · z · k i D )= E B ( h x , c · z i D k ) = k E B ( h x , c · z i D ) . Therefore, we obtain the conclusion by Lemma 2.5. (cid:3)
Remark . Let π be the map from A ′ ∩ C to ( pM n ( A ) p ) ′ ∩ pM n ( C ) p definedby π ( h ) = ( h ⊗ I n ) p for any h ∈ A ′ ∩ C . Then π is an isomorphism of A ′ ∩ C onto ( pM n ( A ) p ) ′ ∩ pM n ( C ) p by the proof of [9, Lemma 10.3]. We regard π as anisomorphism of A ′ ∩ C onto B ′ ∩ D . By the above proof, we can see that k = π ( h ).Thus we obtain that f [ X,Y ] ( π ( h ) E B ) = h E A for any h ∈ A ′ ∩ C , that is, for any h ∈ A ′ ∩ C , h E A and π ( h ) E B are strongly Morita equivalent. Proposition 8.3.
With the above notation,
Pic( h E A ) ∼ = Pic( π ( h ) E B ) for any h ∈ A ′ ∩ C .Proof. This is immediate by Lemma 6.1 (cid:3)
Corollary 8.4.
Let A ⊂ C be a unital inclusion of unital C ∗ -algebras. Let E A be a conditional expectation of Watatani index-finite type from C onto A . Let [ X, Y ] ∈ Pic( E A ) . Then there is an automorphism α of A ′ ∩ C such that f [ X,Y ] ( α ( h ) E A ) = h E A for any h ∈ A ′ ∩ C .Proof. This is immediate by Lemma 8.1 and Remark 8.2. (cid:3)
Let ρ A and ρ B be the (not ∗ -) anti-isomorphism of A ′ ∩ C and B ′ ∩ D onto C ′ ∩ C and D ′ ∩ D , which are defined in [14, pp.79], respectively. By the discussions as bove or the discussions in [9, Section 2], there are a positive integer n and aprojection p in M n ( A ) satisfying M n ( A ) pM n ( A ) = M n ( A ) , M n ( C ) pM n ( C ) = M n ( C ) ,M n ( C ) pM n ( C ) = M n ( C ) ,B ∼ = pM n ( A ) , D ∼ = pM n ( C ) p, D ∼ = pM n ( C ) p as C ∗ -algebras. Then by the proof of [9, Lemma 10. 3],( pM n ( A ) p ) ′ ∩ pM n ( C ) p = { ( h ⊗ I n ) p | h ∈ A ′ ∩ C } , ( pM n ( C ) p ) ′ ∩ pM n ( C ) p = { ( h ⊗ I n ) p | h ∈ C ′ ∩ C } . And by easy computations, the anti-isomorphism ρ of ( pM n ( A ) p ) ′ ∩ pM n ( C ) p onto( pM n ( C ) p ) ′ ∩ pM n ( C ) p defined in the same way as in [14, pp.79] is following: ρ (( h ⊗ I n ) p ) = ( ρ A ( h ) ⊗ I n ) p for any h ∈ A ′ ∩ C . This proves that π ◦ ρ A = ρ B ◦ π , where π and π are theisomorphisms of A ′ ∩ C and C ′ ∩ C onto ( pM n ( A ) p ) ′ ∩ pM n ( C ) p and ( pM n ( C ) p ) ′ ∩ pM n ( C ) p defined in [9, Lemma 10.3], respectively and we regard π and π asisomorphisms of A ′ ∩ C and C ′ ∩ C onto B ′ ∩ D and D ′ ∩ D , respectively. Thenwe have the following: Remark . (1) If f [ X,Y ] ( E B ) = E A , then f [ Y,Y ] ( ρ B ( π ( h )) E D ) = ρ A ( h ) E C for any h ∈ A ′ ∩ C . Indeed, by Lemma 7.2 f [ Y,Y ] ( E D ) = E C . Thus by Remark 8.2, forany c ∈ C ′ ∩ C , f [ Y,Y ] ( π ( c ) E D ) = c E C . Hence for any h ∈ A ′ ∩ C , f [ Y,Y ] ( ρ B ( π ( h )) E D ) = f [ Y,Y ] ( π ( ρ A ( h )) E D ) = ρ A ( h ) E C since π ◦ ρ A = ρ B ◦ π .(2) We suppose that Ind W ( E A ) ∈ A and f [ Y,Y ] ( E D ) = E C . Then we can obtainthat f [ X,Y ] ( ( ρ − B ( π (( c )) E B ) = ρ − A ( c ) E A for any c ∈ C ′ ∩ C . In the same way asabove, this is immediate by Lemma 7.2 and by Remark 8.2.9. Examples
In this section, we shall give some easy examples of the Picard groups of bimodulemaps.
Example 9.1.
Let A ⊂ C be a unital inclusion of unital C ∗ -algebras and E A aconditional expectation of Watatani index-finite type from C onto A . We supposethat A ′ ∩ C = C . Then Pic( E A ) = Pic( A, C ) .Proof. Since E A is the unique conditional expectation by [14, Proposition 1.4.1],for any [ X, Y ] ∈ Pic(
A, C ), f [ X,Y ] ( E A ) = E A . Thus Pic( E A ) = Pic( A, C ). (cid:3) Let ( α, w ) be a twisted action of a countable discrete group G on a unital C ∗ -algebra A and let A ⋊ α,w,r G be the reduced twisted crossed product of A by G .Let E A be the canonical conditional expectation from A ⋊ α,w,r G onto A defined by E A ( x ) = x ( e ) for any x ∈ K ( G, A ), where K ( G, A ) is the ∗ -algebra of all complexvalued functions on G with a finite support and e is the unit element in G . Example 9.2.
If the twisted action ( α, w ) is free, then E A is the unique conditionalexpectation from A ⋊ α,w,r G onto A by [7, Proposition4.1] . By the same reason asabove, Pic( E A ) = Pic( A, A ⋊ α,w,r G ) . Let A be a unital C ∗ -algebra such that the sequence1 −→ Int( A ) −→ Aut( A ) −→ Pic( A ) −→ s exact, where Int( A ) is the subgroup of Aut( A ) of all inner automorphisms of A . We consider the unital inclusion of unital C ∗ -algebras C ⊂ A . Let φ be abounded linear functional on A . We regard φ as a C -bimodule map from A to C .Let Aut φ ( A ) be the subgroup of Aut( A ) defined byAut φ ( A ) = { α ∈ Aut( A ) | φ = φ ◦ α } . Also, let U φ ( A ) be the subgroup of U ( A ) defined by U φ ( A ) = { u ∈ U ( A ) | φ ◦ Ad( u ) = φ } . By [6, Lemma 7.2 and Example 7.3],Pic( C , A ) ∼ = U ( A ) /U ( A ′ ∩ A ) ⋊ s Pic( A ) , that is, Pic( C , A ) is isomorphic to a semidirect product group of U ( A ) /U ( A ′ ∩ A )by Pic( A ) and generated by { [ C u, A ] ∈ Pic( C , A ) | u ∈ U ( A ) } and { [ C , X α ] ∈ Pic( C , A ) | α ∈ Aut( A ) } , where U ( A ) is the group of all unitary elements in A and X α is the A − A -equivalencebimodule induced by α ∈ Aut( A ) (See [6, Example 7.3]). Example 9.3.
Let A be a unital C ∗ -algebra such that the sequence −→ Int( A ) −→ Aut( A ) −→ Pic( A ) −→ is exact. Let φ be a bounded linear functional on A . Let Pic φ ( A ) be the subgroupof Pic( A ) defined by Pic φ ( A ) = { [ X α ] | α ∈ Aut φ ( A ) } . Then
Pic( φ ) ∼ = U ( A ) /U ( A ′ ∩ A ) ⋊ s Pic φ ( A ) .Proof. Let α ∈ Aut( A ). Then by Lemma 6.2(1), f [ C ,X α ] ( φ ) = α ◦ φ ◦ α − = φ ◦ α − . Hence α ∈ Pic φ ( A ) if and only if f [ C ,X α ] ( φ ) = φ . Also, by Lemma 2.5, for any a ∈ A , h u , f [ C u,A ] ( φ )( a ) · u i C = φ ( h u , a · u i A ) = φ ( u ∗ au ) , that is, f [ C u,A ] ( φ )( a ) = φ (Ad( u ∗ )( a )). Hence by [6, Example 7.3],Pic( φ ) ∼ = U φ ( A ) /U ( A ′ ∩ A ) ⋊ s Pic φ ( A ) . (cid:3) Remark . If τ is the unique tracial state on A , Pic τ ( A ) = Pic( A ). HencePic( τ ) ∼ = Pic( C , A ) ∼ = U ( A ) /U ( A ′ ∩ A ) ⋊ s Pic( A ) . Let A be a unital C ∗ -algebra such that the sequence1 −→ Int( A ) −→ Aut( A ) −→ Pic( A ) −→ n be any positive integer with n ≥
2. We consider the unital inclusionof unital C ∗ -algebras a ∈ A a ⊗ I n ∈ M n ( A ), where I n is the unit element in M n ( A ). We regard A as a C ∗ -subalgebra of M n ( A ) by the above unital inclusionmap. Let E A be the conditional expectation from M n ( A ) onto A defined by E A ([ a ij ] ni,j =1 ) = 1 n n X i =1 a ii or any [ a ij ] ni,j =1 ∈ M n ( A ). Let Aut ( A, M n ( A )) be the group of all automorphisms β of M n ( A ) with β | A = id on A . By [6, Example 7.6],Pic( A, M n ( A )) ∼ = Aut ( A, M n ( A )) ⋊ s Pic( A )and the sequence1 −→ Aut ( A, M n ( A )) ı −→ Pic(
A, M n ( A )) f A −→ Pic( A ) −→ ı is the inclusion map of Aut ( A, M n ( A )) defined by ı ( β ) = [ A, Y β ]for any β ∈ Aut ( A, M n ( A )) and f A is defined by f A ([ X, Y ]) = [ X ] for any [ X, Y ] ∈ Pic(
A, M n ( A )). Also, let be the homomorphism of Pic( A ) to Pic( A, M n ( A ))defined by ([ X α ]) = [ X α , X α ⊗ id ] for any α ∈ Aut( A ). Example 9.5.
Let A be a unital C ∗ -algebra such that the sequence −→ Int( A ) −→ Aut( A ) −→ Pic( A ) −→ is exact. Let n be any positive integer with n ≥ . Let E A be as above. Let Aut E A ( A, M n ( A )) be the subgroup of Aut ( A, M n ( A )) defined by Aut E A ( A, M n ( A )) = { β ∈ Aut ( A, M n ( A )) | E A = E A ◦ β } . Then
Pic( E A ) ∼ = Aut E A ( A, M n ( A )) ⋊ s Pic( A ) .Proof. Let β ∈ Aut ( A, M n ( A )). Then by Lemma 6.2(1), f [ X β ,Y β ] ( E A ) = β ◦ E A ◦ β − = E A ◦ β − . Hence β ∈ Aut E A ( A, M n ( A )) if and only if f [ X β ,Y β ] ( E A ) = E A . Also, by Lemma6.2(1) for any α ∈ Aut( A ), f [ X α ,X α ⊗ id ] ( E A ) = α ◦ E A ◦ ( α − ⊗ id) = E A since we identify A with A ⊗ I n . Thus by [6, Example 7.6],Pic( E A ) ∼ = Aut E A ( A, M n ( A )) ⋊ s Pic( A ) . (cid:3) References [1] L. G. Brown,
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Index for C ∗ -subalgebras , Mem. Amer. Math. Soc., , Amer. Math. Soc.,1990. Department of Mathematical Sciences, Faculty of Science, Ryukyu University,Nishihara-cho, Okinawa, 903-0213, Japan
E-mail address: [email protected]@math.u-ryukyu.ac.jp