Strong Morita equivalence for completely positive linear maps on C^*-algebras
aa r X i v : . [ m a t h . OA ] F e b STRONG MORITA EQUIVALENCE FOR COMPLETELYPOSITIVE LINEAR MAPS ON C ∗ -ALGEBRAS KAZUNORI KODAKA
Abstract.
We will introduce the notion of strong Morita equivalence for com-pletely positive linear maps and study its basic properties. Also, we will discussthe relation between strong Morita equivalence for bounded C ∗ -bimodule lin-ear maps and strong Morita equivalence for completely positive linear maps.Furthermore, we will show that if two unital C ∗ -algebras are strongly Moritaequivalent, then there is a 1 − C ∗ -algebras and we will show that the corresponding two classes ofthe completely positive linear maps are also strongly Morita equivalent. Introduction
In the previous paper [7], we introduced the notions of strong Morita equivalencefor bounded C ∗ -bimodule linear maps and the Picard group of a bounded C ∗ -bimodule linear map.In this paper, we will introduce the notion of strong Morita equivalence forcompletely positive linear maps applying its minimal Stinespring representationand [4, Definition 2.1] introduced by Echterhoff and Raeburn.To do this, following [4, Definition 2.1], in Section 3, we will introduce the notionof strong Morita equivalence for non-degenerate representations of C ∗ -algebras andwe will discuss its basic properties.In Section 4, we will define strong Morita equivalence for completely positivelinear maps and discuss its basic properties.In Section 5, we will consider inclusions of C ∗ algebras A ⊂ C with AC = C and conditional expectations. Conditional expectations are regarded as bounded C ∗ -bimodule linear maps. Also, they are regarded as completely positive linearmaps. We will discuss the relation between strong Morita equivalence for bounded C ∗ -bimodule linear maps and strong Morita equivalence for completely positivelinear maps.In Section 6, we will consider two unital C ∗ -algebras A and B , which are stronglyMorita equivalent. We will construct a 1 − A to the C ∗ -algebra of all bounded linear operators on a Hilbert space and the set of allstrong Morita equivalence classes of completely positive linear maps from B to the C ∗ -algebra of all bounded linear operators on a Hilbert space.2. Preliminaries
Let A be a C ∗ -algebra. We denote by id A the identity map on A . When A is unital, we denote by 1 A the unit element in A . We simply them by id and 1,respectively if no confusion arises. Mathematics Subject Classification.
Key words and phrases. completely positive linear maps, inclusions of C ∗ -algebras, conditionalexpectations, strong Morita equivalence. or each n ∈ N , let M n ( A ) be the n × n -matrix algebra over A . We identify M n ( A ) with A ⊗ M n ( C ). Let I n be the unit element in M n ( C ). For each a ∈ M n ( A ),we denote by a ij , the i × j -entry of the matrix a .Let M ( A ) be the multiplier C ∗ -algebra of A and for any automorphism α of A , let α be the automorphism of M ( A ) extending α to M ( A ), which is defined inJensen and Thomsen [5, Corollary 1.1.15].Let A and B be C ∗ -algebras and X an A − B -equivalence bimodule. We denoteits left A -action and right B -action on X by a · x and x · b for any a ∈ A , b ∈ B , x ∈ X , respectively. Let e X be the dual B − A -equivalence bimodule of X and let e x denote the element in e X associated to an element x ∈ X .For Hilbert spaces H and K , let B ( K , H ) be the space of all bounded linearoperators from K to H and if H = K , we denote B ( H , H ) by B ( H ). For a Hilbertspace H , we denote by h· , ·i H the inner product of H .Let K be the C ∗ -algebra of all compact operators on a countably infinite dimen-sional Hilbert space H . Let { ǫ i } ∞ i =1 be an orthogonal basis of H and { e ij } ∞ i,j =1 the system of matrix units of K with respect to { ǫ i } ∞ i =1 .3. Definition and properties of Strong Morita equivalence fornon-degenerate representations of C ∗ -algebras Following [4, Definition 2.1], we give the definition of a representation of equiv-alence bimodule.
Definition . Let A and B be C ∗ -algebras. A representation of an A − B -equivalence bimodule X on the pair of Hilbert spaces ( H , K ) is a triple ( π A , π X , π B )consisting of non-degenerate representations π A : A → B ( H ), π B : B → B ( K ) anda linear map π X : X → B ( K , H ) satisfying the following: for any a ∈ A , b ∈ B , x, y ∈ X ,(1) π X ( x ) π X ( y ) ∗ = π A ( A h x, y i ),(2) π X ( x ) ∗ π X ( y ) = π B ( h x, y i B ),(3) π X ( a · x · b ) = π A ( a ) π X ( x ) π B ( b ).Let ( π A , H ) and ( π B , K ) be non-degenerate representations of C ∗ -algebras A and B , respectively. Definition . The non-degenerate representation ( π A , H ) of A is strongly Moritaequivalent to the non-degenerate representation ( π B , K ) of B if there are an A − B -equivalence bimodule X and a linear map π X : X → B ( K , H ) such that ( π A , π X , π B )is a representation of X on the pair of Hilbert spaces ( H , K ). Lemma 3.1.
With the above notation, strong Morita equivalence for non-degeneraterepresentations of C ∗ -algebras is equivalence relation.Proof. Let ( π A , H ) be a non-degenerate representation of a C ∗ -algebra A . Weregard A as the trivial A − A -equivalence bimodule in the usual way. We denoteit by X . Let π X be the linear map from X to B ( H ) defined by π X ( x ) = π A ( x )for any x ∈ X . Then we can see that π X satisfies Conditions (1)-(3) in Definition3.1. Hence ( π A , H ) is strongly Morita equivalent to itself.Let ( π B , K ) be a non-degenerate representation of a C ∗ -algebra B and we supposethat ( π A , H ) is strongly Morita equivalent to ( π B , K ). Then there are an A − B -equivalence bimodule X and a linear map π X from X to B ( K , H ) satisfyingConditions (1)-(3) in Definition 3.1. Let f π X be the linear map from e X to B ( H , K )defined by f π X ( e x ) = π X ( x ) ∗ for any x ∈ X . Then we see that f π X satifies Conditions(1)-(3) in Definition 3.1. Thus, ( π B , K ) is strongly Morita equivalent to ( π A , H ).Let ( π C , L ) be a non-degenerate representation of a C ∗ -algebra C . We sup-pose that ( π A , H ) is strongly Morita equivalent to ( π B , K ) with respect to an − B -equivalence bimodule X and a linear map π X : X → B ( K , H ) such that( π A , π X , π B ) is a representation of X on the pair of Hilbert spaces ( H , K ). Also,we suppose that ( π B , K ) is strongly Morita equivalent to ( π C , L ) with respect toa B − C -equivalence bimodule Y and a linear map π Y : Y → B ( L , K ) such that( π B , π Y , π C ) is a representation of Y on the pair of Hilbert spaces ( K , L ). Let π X ⊗ B Y be the linear map from X ⊗ B Y to B ( L , H ) defined by π X ⊗ Y ( x ⊗ y ) = π X ( x ) π Y ( y )for any x ∈ X , y ∈ Y . Then by routine computations, we can see that π X ⊗ B Y satisfies Conditions (1)-(3) in Definition 3.1. Thus ( π A , π X ⊗ B Y , π C ) is a represen-tation of X ⊗ Y on a pair of Hilbert spaces ( H , L ) and ( π A , H ) is strongly Moritaequivalent to ( π C , L ). Therefore, we obtain the conclusion. (cid:3) Lemma 3.2.
Let ( π , H ) and ( π , H ) be non-degenerate representations of a C ∗ -algebra A . If ( π , H ) and ( π , H ) are unitarily equivalent, they are strongly Moritaequivalent.Proof. Since ( π , H ) and ( π , H ) are unitarily equivalent, there is an isometry u from H onto H such that π = Ad( u ) ◦ π . Let X be the trivial A − A -equivalencebimodule defined in the proof of Lemma 3.1. Let π X be the linear map from X to B ( H , H ) defined by π X ( x ) = π ( x ) u ∗ . Then by easy computations, we can seethat ( π , π X , π ) is a representation of X on the pair of Hilbert spaces ( H , H ).Hence ( π , H ) and ( π , H ) are strongly Morita equivalent. (cid:3) Let ( π A , H ) and ( π B , K ) be non-degenerate representations of C ∗ -algebras A and B , respectively. We suppose that ( π A , H ) and ( π B , K ) are strongly Moritaequivalent, that is, there are an A − B -equivalence bimodule X and a linear map π X from X to B ( K , H ) such that ( π A , π X , π B ) is representation of X on the pairof Hilbert spaces ( H , K ).Let L X be the linking C ∗ -algebra for X , that is, L X = { (cid:20) a x e y b (cid:21) | a ∈ A, b ∈ B, x, y ∈ X } . Let ρ be the representation of L X on H ⊕ K defined by ρ ( (cid:20) a x e y b (cid:21) ) = (cid:20) π A ( a ) π X ( x ) π X ( y ) ∗ π B ( b ) (cid:21) for any a ∈ A , b ∈ B , x, y ∈ X . Since ( π A , H ) and ( π B , K ) are non-degenerate, sois ( ρ, H ⊕ K ) by [4, §
2, Remark (3)]. Let p = (cid:20) M ( A )
00 0 (cid:21) , q = (cid:20) M ( B ) (cid:21) . Then p and q are projections in M ( L X ) with L X pL X = L X , L X qL X = L X , pL X p ∼ = A, qL X q ∼ = B as C ∗ -algebras, respectively. We identify pL X p and qL X q with A and B , respec-tively. Lemma 3.3.
With the above notation, ( ρ, H ⊕ K ) is strongly Morita equivalent to ( π A , H ) and ( π B , K ) .Proof. We have only to show that ( ρ, H⊕K ) is strongly Morita equivalent to ( π A , H )by Lemma 3.1. Let Y = pL X . Since we identify pL X p with A , Y can be regardedas an A − L X -equivalence bimodule in the usual way. Let π Y be the linear mapfrom Y to B ( H ⊕ K , H ) defined by π Y ( (cid:20) a x (cid:21) ) = (cid:20) π A ( a ) π X ( x )0 0 (cid:21) or any a ∈ A , x ∈ X , where we identify pL X with the space { (cid:20) a x (cid:21) | a ∈ A, x ∈ X } . Then by routine computations, we can see that ( π A , π Y , ρ ) is a representation of Y on the pair of Hilbert spaces ( H , H ⊕ K ). Therefore, ( ρ, H ⊕ K ) is strongly Moritaequivalent to ( π A , H ). (cid:3) Since ( ρ, H ⊕ K ) is a non-degenerate representation of L X , by Pedersen [10,Theorem 3.7.7], there is a unique normal homomorphism ρ ′′ of L ′′ X onto ρ ( L X ) ′′ ,which extends ρ , where L ′′ X is the enveloping von Neumann algebra of L X . Also, by[10, 3.12.1] we can regard M ( L X ) as a C ∗ -subalgebra of L ′′ X . Furthermore, we seethat ρ is the restriction of ρ ′′ to M ( L X ). Hence ρ ( p ) and ρ ( q ) are the projections in B ( H ⊕ K ) with their ranges are
H ⊕ ⊕ K , respectively. That is, P H = ρ ( p )and P K = ρ ( q ), where P H and P K are projections from H ⊕ K onto
H ⊕ ⊕ K , respectively. Furthermore, we assume that A and B are σ -unital stable C ∗ -algebras. Then in the same way as in the proof of Brown, Green and Rieffel [2,Theorem 3.4], there is a partial isometry w ∈ M ( L X ) such that w ∗ w = p , ww ∗ = q .Let θ be the map from pL X p to qL X q defined by θ ( (cid:20) a
00 0 (cid:21) ) = w (cid:20) a
00 0 (cid:21) w ∗ for any a ∈ A . Then θ is an isomorphism of pL X p onto qL X q . Identifying A and B with pL X p and qL X q , respectively, we can regard θ as an isomorphism of A onto B . Let W = ρ ( w ). Then W ∈ B ( H ⊕ K ) and W ∗ W = ρ ( w ∗ w ) = ρ ( p ) = P H , W W ∗ = ρ ( ww ∗ ) = ρ ( q ) = P K . Lemma 3.4.
With the above notation, there is an isometry f W from H onto K such that π B ( θ ( a )) = f W π A ( a ) f W ∗ for any a ∈ A .Proof. Let f W = W | H = W P H . Then for any ξ ∈ H , f W ξ = W P H ξ = W ξ = W W ∗ W ξ = P K W ξ ∈ K . Hence f W ∈ B ( H , K ). For any η ∈ K , W ∗ η = W ∗ W W ∗ η = P H W ∗ η ∈ H . Then f W W ∗ η = W W ∗ η = P K η = η. Thus f W is surjective. Furthermore, for any ξ , ξ ∈ H , h f W ξ , f W ξ i K = h W ξ , W ξ i H⊕K = h ξ , W ∗ W ξ i H⊕K = h ξ , P H ξ i H⊕K = h ξ , ξ i H . Hence f W is an isometry from H onto K . Finally, for any a ∈ A , π B ( θ ( a )) = π B ( w (cid:20) a
00 0 (cid:21) w ∗ ) = π B ( ww ∗ w (cid:20) a
00 0 (cid:21) w ∗ ww ∗ )= π B ( qw (cid:20) a
00 0 (cid:21) w ∗ q ) = ρ ( qw (cid:20) a
00 0 (cid:21) w ∗ q )= ρ ( qw ) (cid:20) π A ( a ) 00 0 (cid:21) ρ ( w ∗ q ) = P K W (cid:20) π A ( a ) 00 0 (cid:21) W ∗ P K = W P H π A ( a ) P H W ∗ = f W π A ( a ) f W ∗ . herefore, we obtain the conclusion. (cid:3) Next, we will give an easy example of non-degenerate representations of C ∗ -algebras which are strongly Morita equivalent.Let ( π, H ) be a non-degenerate representation of a C ∗ -algebra A . Let A s = A ⊗ K and let π s = π ⊗ id K and H s = H ⊗ H . Then ( π s , H s ) is a non-degeneraterepresentation of A s on H s . Example 3.5.
With the above notation, ( π, H ) and ( π s , H s ) are strongly Moritaequivalent.Proof. Let X = A s (1 ⊗ e ). Then X is an A s − A -equivalence bimodule in theusual way, where we identify A with (1 M ( A ) ⊗ e )( A ⊗ K )(1 M ( A ) ⊗ e ). Let ı H be the linear map from H to H s defined by ı H ξ = ξ ⊗ ǫ for any ξ ∈ H . Then ı H is an isometry and ı ∗H ( ξ ⊗ ǫ i ) = ( ξ if i = 10 if i ≧ ξ ∈ H by routine computations. Thus ı H ı ∗H = 1 ⊗ e and ı ∗H ı H = id H on H . Let π X be the linear map from X to B ( H , H s ) defined by π X ( a (1 M ( A ) ⊗ e )) = π s ( a (1 M ( A ) ⊗ e )) ı H = π s ( a )(1 M ( A ) ⊗ e ) ı H for any a ∈ A s . Then for any a, b ∈ A s , π s ( A s h a (1 M ( A ) ⊗ e ) , b (1 M ( A ) ⊗ e ) i ) = π s ( a (1 M ( A ) ⊗ e ) b ∗ ) . On the other hand, π X ( a (1 M ( A ) ⊗ e )) π X ( b (1 M ( A ) ⊗ e )) ∗ = π s ( a )(1 ⊗ e ) ı H ı ∗H (1 ⊗ e ) π s ( b ) ∗ = π s ( a )(1 ⊗ e ) π s ( b ) ∗ since ı H i ∗H = 1 ⊗ e . Thus π s ( A s h a (1 M ( A ) ⊗ e ) , b (1 M ( A ) ⊗ e ) i ) = π X ( a (1 M ( A ) ⊗ e )) π X ( b (1 M ( A ) ⊗ e )) ∗ Also, for any a, b ∈ A s , π ( h a (1 ⊗ e ) , b (1 ⊗ e ) i A ) = π s ((1 ⊗ e ) a ∗ b (1 ⊗ e )) = (1 ⊗ e ) π s ( a ∗ b )(1 ⊗ e ) . Since we identify A with (1 ⊗ e ) A s (1 ⊗ e ), we regard H as the closed subspace H ⊗ ǫ of H s . Hence we can regard (1 ⊗ e ) π s ( a ∗ b )(1 ⊗ e ) as an element ı ∗H (1 ⊗ e ) π s ( a ∗ b )(1 ⊗ e ) ı H in B ( H ). On the other hand, π X ( a (1 ⊗ e )) ∗ π X ( b (1 ⊗ e )) = ( π s ( a )(1 ⊗ e ) ı H ) ∗ ( π s ( b )(1 ⊗ e ) ı H )= ı ∗H (1 ⊗ e ) π s ( a ∗ b )(1 ⊗ e ) ı H . Thus π ( h a (1 ⊗ e ) , b (1 ⊗ e ) i A ) = π X ( a (1 ⊗ e )) ∗ π X ( b (1 ⊗ e ))for any a, b ∈ A s . Furthermore, for any a ∈ A s , b ∈ A , x ∈ A s , π X ( a · x (1 ⊗ e ) · b ) = π X ( ax ( b ⊗ e )) = π s ( ax ( b ⊗ e )) ı H = π s ( a ) π s ( x )(1 ⊗ e )( π ( b ) ⊗ e ) ı H . For any ξ ∈ H ,( π ( b ) ⊗ e ) ı H ξ = ( π ( b ) ⊗ e )( ξ ⊗ ǫ ) = π ( b ) ξ ⊗ ǫ = ı H π ( b ) ξ. Hence ( π ( b ) ⊗ e ) ı H = ı H π ( b ). Thus π X ( a · x (1 ⊗ e ) · b ) = π s ( a ) π s ( x )(1 ⊗ e ) ı H π ( b ) = π s ( a ) π X ( x (1 ⊗ e )) π ( b ) . Therefore, ( π s , π X , π ) is a representation of X on the pair of spaces ( H s , H ), thatis, ( π s , H s ) and ( π, H ) are strongly Morita equivalent. (cid:3) . Definition and properties of strong Morita equivalence forcompletely positive linear maps
In this section, we define strong Morita equivalence for completely positive linearmaps from a C ∗ -algebra to a C ∗ -algebra of all bounded linear operators on a Hilbertspace.Let φ be a completely positive linear map from a C ∗ -algebra A to B ( H ), where H is a Hilbert space H . Then by a Stinespring dilation theorem, there are a Hilbertspace H φ , a representation π φ of A on H φ and V φ ∈ B ( H , H φ ) with || V φ || = || φ || such that φ ( a ) = V ∗ φ π φ ( a ) V φ for any a ∈ A and such that H φ = π φ ( A ) V φ H . For more information, see Blackadar[1, II. 6.9.7], Paulsen [9, Theorem 4.1] and Stinespring [11]. We call the above( π φ , V φ , H φ ) a minimal Stinespring representation for φ . Furthermore, we can seethat ( π φ , V φ , H φ ) is unique in the sense of [9, Proposition 4.2], that is, Proposition 4.1. ([9, Proposition 4.2])
With the above notation, if ( ρ, W, K ) isanother minimal Stinespring representation for φ , then there is an isometry U from H φ onto K satisfying that U V φ = W and that U π φ ( a ) U ∗ = ρ ( a ) for any a ∈ A .Remark . We note that a minimal Stinespring representation for φ is non-degenerate. Indeed, let ( π φ , V φ , H φ ) be a minimal Stinespring representation for φ . Then since V φ H ⊂ H φ , H φ = π φ ( A ) V φ H ⊂ π φ ( A ) H φ ⊂ H φ . Thus H φ = π φ ( A ) H φ , that is, ( π φ , H φ ) is non-degenerate.Let φ and ψ be completely positive linear maps from C ∗ -algebras A and B to B ( H ) and B ( K ), respectively, where H and K are Hilbert spaces. Definition . We say that φ is strongly Morita equivalent to ψ if a minimalStinespring representation for φ is strongly Morita equivalent to that for ψ . Proposition 4.3.
Strong Morita equivalence for completely positive linear mapson C ∗ -algebras is equivalence relation.Proof. This is immediate by Lemmas 3.1, 3.2 and Proposition 4.1. (cid:3)
Let φ and ψ be completely positive linear maps from C ∗ -algebra A and B to B ( H )and B ( K ), respectively, which are strongly Morita equivalent. Let ( π φ , V φ , H φ ) and( π ψ , V ψ , H ψ ) be minimal Stinespring representations for them. Then since ( π φ , H φ )and ( π ψ , H ψ ) are strongly Morita equivalent, there are an A − B -equivalence bi-module X and a linear map π X from X to B ( H ψ , H φ ) such that ( π φ , π X , π ψ ) is arepresentation of X on the pair of Hilbert spaces ( H φ , H ψ ). Let L X be the linking C ∗ -algebra for X and let ρ be the representation of L X on H φ ⊕ H ψ induced bythe representation ( π φ , π X , π ψ ) of X , which is defined in Section 3. Let τ be thecompletely positive linear map from L X to B ( H ⊕ K ) defined by τ ( (cid:20) a x e y b (cid:21) ) = (cid:20) V ∗ φ V ∗ ψ (cid:21) ρ ( (cid:20) a x e y b (cid:21) ) (cid:20) V φ V ψ (cid:21) for any (cid:20) a x e y b (cid:21) ∈ L X . Lemma 4.4.
With the above notation, ( ρ, V φ ⊕ V ψ , H φ ⊕ H ψ ) is a minimal Stine-spring representation for τ . roof. It suffices to show that ρ ( L X )( V φ ⊕ V ψ )( H ⊕ K ) is dense in H φ ⊕ H ψ . Indeed, π φ ( A ) V φ H ⊕ π ψ ( B ) V ψ K ⊂ ρ ( L X )( V φ ⊕ V ψ )( H ⊕ K ) . Since π φ ( A ) V φ H = H φ and π ψ ( B ) V ψ K = H ψ , ρ ( L X )( V φ ⊕ V ψ )( H ⊕ K ) = H φ ⊕ H ψ . Therefore, we obtain the conclusion. (cid:3)
Proposition 4.5.
Let φ , ψ and τ be as above. Then τ is strongly Morita equivalentto φ and ψ and φ ( a ) = P H τ ( (cid:20) a
00 0 (cid:21) ) , ψ ( b ) = P K τ ( (cid:20) b (cid:21) ) , for any a ∈ A and b ∈ B , respectively, where we identify H and K with H ⊕ and ⊕ K , respectively and P H and P K are projections from H ⊕ K onto H and K ,respectively.Proof. This is immediate by Lemmas 3.3, 4.4 and the definition of τ . (cid:3) Furthermore, we assume that A and B are σ -unital stable C ∗ -algebras. Then bythe discussions before Lemma 3.4 and Lemma 3.4, there are an isomorphism θ of A onto B and an isometry f W from H φ onto H ψ such that π ψ ( θ ( a )) = f W π φ ( a ) f W ∗ for any a ∈ A . By easy computations, we obtain that ψ ( θ ( a )) = V ∗ ψ π ψ ( θ ( a )) V ψ = V ∗ ψ f W π φ ( a ) f W ∗ V ψ for any a ∈ A . Proposition 4.6.
Let A and B be σ -unital stable C ∗ -algebras. Let φ and ψ becompletely positive linear maps from C ∗ -algebra A and B to B ( H ) and B ( K ) , re-spectively. Let ( π φ , V φ , H φ ) and ( π ψ , V ψ , H ψ ) be minimal Stinespring representa-tions for φ and ψ , respectively. Then there are an isomorphism θ of A onto B and an isometry f W from H φ onto H ψ satisfying that ( π φ , g W ∗ V ψ , H φ ) is a minimalStinespring representation for ψ ◦ θ .Proof. Let f W and θ be as before Lemma 3.4 and in the proof of Lemma 3.4. Thenit suffices to show that ( π φ , f W ∗ V ψ , H π ) is a minimal Stinespring representation for ψ ◦ θ . Since ( π ψ , V ψ , H ψ ) is a minimal Stinespring representation for ψ , f W π φ ( A ) f W ∗ V ψ K = π ψ ( θ ( A )) V ψ K = π ψ ( B ) V ψ K = H ψ . Since f W is an isometry of H φ onto H ψ , H φ = f W ∗ H ψ = f W ∗ ( f W π φ ( A ) f W ∗ V ψ K ) = π φ ( A ) f W ∗ V ψ K . Thus, ( π φ , f W ∗ V ψ , H φ ) is a minimal Stinespring representation for ψ ◦ θ . (cid:3) Let φ be a completely positive linear map from A to B ( H ). We will show that φ and φ ⊗ id K are strongly Morita equivalent. Example 4.7.
With the above notation, φ and φ ⊗ id K are strongly Morita equiv-alent.Proof. Let ( π φ , V φ , H φ ) be a minimal Stinespring representation for φ . Then ( π sφ , V φ ⊗ H , H sφ ) is a minimal Stinespring representation for φ ⊗ id K , where 1 H is the iden-tity operator on H . Indeed, for any a ∈ A , k ∈ K ,( V φ ⊗ H ) ∗ π sφ ( a ⊗ k )( V φ ⊗ H ) = V ∗ φ π φ ( a ) V φ ⊗ k = ( φ ⊗ id K )( a ⊗ k ) . lso, π sφ ( A s )( V φ ⊗ H )( H sφ ) = π φ ( A ) V φ H φ ⊗ K H . Since π φ ( A ) V φ H φ = H φ , we can see that π sφ ( A s )( V φ ⊗ H )( H sφ ) = H sφ . Since by Example 3.5, ( π φ , H φ ) and ( π sφ , H sφ ) are strongly Morita equivalent, weobtain the conclusion. (cid:3) Relation between strong Morita equivalence for bimodule linearmaps and strong Morita equivalence for completely positivelinear maps
Let A ⊂ C and B ⊂ D be inclusions of C ∗ -algebras with AC = C and BD = D . Let E A and E B be conditional expectations from C and D onto A and B ,respectively. We assume that E A and E B are strongly Morita equivalent withrespect to a C − D -equivalence bimodule Y and its closed subspace X as bimodulelinear maps (See [7, Definition 3.1]). We note that A ⊂ C and B ⊂ D are stronglyMorita equivalent with respect to Y and its closed subspace X (See [8, Definition2.1]) and that E A and E B are completely positive linear maps from C and D onto A and B , respectively. It is natural that we consider whether E A and E B arestrongly Morita equivalent as completely positive linear maps.In this section, first, we will give the following result: We consider E A and E B which are strongly Morita equivalent as bimodule linear maps. For any non-degenerate representation ( π B , K B ) of B , there is a non-degenerate representation( π A , H A ) of A such that π A ◦ E A and π B ◦ E B are strongly Morita equivalent ascompletely positive linear maps. Also, we will consider its inverse direction.We will use the same notation as above. We note that π B ◦ E B is a completelypositive linear map from D to B ( K B ). Hence by [1, II. 6.9.7] or [9, Theorem 4.1],there is a minimal Stinespring representation ( π D , V D , K D ) for π B ◦ E B such that( π B ◦ E B )( d ) = V ∗ D π D ( d ) V D for all d ∈ D , where V D ∈ B ( K B , K D ). Modifying the proof of [9, Theorem 4.1],we define K D and V D . Let D ⊙ K B be the algebraic tensor product of D and K B .We define a symmetric bilinear function h· , ·i on D ⊙ K B by setting h d ⊗ ξ , d ⊗ η i = h ξ , ( π B ◦ E B )( d ∗ d ) η i K B for any d, d ∈ D , ξ, η ∈ K B and extending linearly. Let N D be the subspace of D ⊙ K B defined by N D = { u ∈ D ⊙ K B | h u, u i = 0 } . The induced bilinear form on the quotient space ( D ⊙ K B ) / N D defined by h u + N D , v + N D i = h u , v i , is an inner product, where u, v ∈ D ⊙ K D . We denote by K D and h· , ·i K D theHilbert space, the completion of the innner product space ( D ⊙ K B ) / N D and itsinner product, respectively.We define V D : K B → K D by setting V D ( ξ ) = lim λ ( d λ ⊗ ξ + N D )for any ξ ∈ K B , where { d λ } is an approximate unit of D with 0 ≤ d λ and || d λ || ≤ K D . Also, we note thatlim λ ( d λ ⊗ ξ + N D ) is independent of the choice of an approximate unit of D .Furthermore, V ∗ D ( d ⊗ ξ + N D ) = ( π B ◦ E B )( d ) ξ or any d ∈ D , ξ ∈ K B . Indeed, for any d ∈ D , ξ, η ∈ K B , h V ∗ D ( d ⊗ ξ + N D ) , η i K B = h d ⊗ ξ + N D , lim λ ( d λ ⊗ η + N D ) i K D = lim λ h ( π B ◦ E B )( d λ d ) ξ , η i K B = h ( π B ◦ E B )( d ) ξ , η i K B . Next, following the proof of [4, Lemma 2.2], we define non-degenerate represen-tations ( π A , H A ) and ( π C , H C ) of A and C , which are strongly Morita equivalentto ( π B , K B ) and ( π D , K D ), respectively.We regard K B as a Hilbert B − C -bimodule using the representation ( π B , K B )and let H A = X ⊗ B K B , a Hilbert space, where its inner product h· , ·i H A is definedby h x ⊗ ξ , y ⊗ η i H A = h ξ , π B ( h x, y i B ) η i K B for any x, y ∈ X , ξ, η ∈ K B . We define π A by setting π A ( a )( x ⊗ ξ ) = ( a · x ) ⊗ ξ for all a ∈ A , x ∈ X , ξ ∈ K B . Furthermore, we define a linear map π X from X to B ( K B , H A ) by setting π X ( x ) = x ⊗ ξ for any x ∈ X , ξ ∈ K B . By [4, Lemma 2.2], ( π A , π X , π B ) is a representation of X on the pair of Hilbert spaces ( H A , K B ). Thus ( π A , H A ) and ( π B , K B ) are stronglyMorita equivalent. Similarly we define a representation of Y on the pair of Hilbertspaces ( H C , K D ) as follows: H C = Y ⊗ D K D , π Y ( y ) ξ = y ⊗ ξ, π C ( c )( y ⊗ ξ ) = ( c · y ) ⊗ ξ for any c ∈ C , y ∈ Y , ξ ∈ K D .We consider π A ◦ E A , which is a completely positive linear map from C to B ( H A ).By [9, Theorem 4.1] or [1, II. 6.9.7], there is a minimal Stinespring representation( π ′ C , V ′ C , H ′ C ). We will show that H C ∼ = H ′ C as Hilbert spaces. In order to dothis, we introduce the following Hilbert space E . We regard Y as a Hilbert C − B -bimodule in the following way: We define the left C -action, the right B -action andthe left C -valued inner product in the usual way. We define the right B -valuedinner product by setting h x, y i B = E B ( h x, y i D )for any x, y ∈ Y . We denote it by the symbol Y B . We define the Hilbert space E by E = Y B ⊗ B K B in the same way as the definition of Hilbert space H A = X ⊗ B K B .First, we show that H C is isomorphic to E as Hilbert spaces. Before doing it, weprepare the following lemma. Lemma 5.1.
With the above notation, let { d λ } λ ∈ Λ be an approximate unit of D with ≤ d λ and || d λ || ≤ for any λ ∈ Λ . For any y ∈ Y , || y − y · d λ || → λ → ∞ ) .Proof. Let y be any element in Y . For any ǫ >
0, there are y , y , . . . , y n ∈ Y , d , d , . . . , d n ∈ D such that || y − n X i =1 y i · d i || < ǫ, y [3, Proposition 1.7]. Then || y − y · d λ ||≤ || y − n X i =1 y i · d i || + || n X i =1 y i · d i − n X i =1 y i · d i d λ || + || n X i =1 y i · d i d λ − y · d λ || < ǫ + || n X i =1 y i · ( d i − d i d λ ) || . Since || d i − d i d λ || → λ → ∞ ) for i = 1 , , . . . , n , there is a λ ∈ Λ such that || n X i =1 y i · ( d i − d i d λ ) || < ǫ for any λ ≥ λ . Thus || y − y · d λ || → λ → ∞ ). (cid:3) Lemma 5.2.
With the above notation, H C ∼ = E as Hilbert spaces.Proof. Let Φ be the linear map from Y ⊙ K D to E defined byΦ( y ⊗ ( d ⊗ ξ )) = ( y · d ) ⊗ ξ for any y ∈ Y , d ∈ D , ξ ∈ K B , where Y ⊙ K D is the algebraic tensor product of Y and K D . Let y, y ∈ Y , d, d ∈ D , ξ, ξ ∈ K B . Then h y ⊗ ( d ⊗ ξ ) , y ⊗ ( d ⊗ ξ ) i H C = h d ⊗ ξ , π D ( h y , y i D ) d ⊗ ξ i K D = h d ⊗ ξ , h y , y i D d ⊗ ξ i K D = h ξ , ( π B ◦ E B )( d ∗ h y , y i D d ) ξ i K B . On the other hand, h Φ( y ⊗ ( d ⊗ ξ )) , Φ( y ⊗ ( d ⊗ ξ )) i E = h ( y · d ) ⊗ ξ , ( y · d ) ⊗ ξ i E = h ξ, π B ( h y · d , y · d i B ) ξ i K B = h ξ , ( π B ◦ E B )( h y · d , y · d i D ) ξ i K B = h ξ , ( π B ◦ E B )( d ∗ h y , y i D d ) ξ i K B . Thus Φ preserves the inner products on the algebraic tensor products. We canextend Φ to H C . We denote it by the same symbol Φ. Then Φ is an isometryfrom H C to E . Next, we show that Φ is surjective. Let y and ξ be elements in Y B and K B , respectively. Let { d λ } λ ∈ Λ be an approximate units of D with d λ ≥ || d λ || ≤
1. Then by Lemma 5.1, y = lim λ y · d λ . Also, y ⊗ ( d λ ⊗ ξ ) is an element in Y ⊗ D K D and Φ( y ⊗ ( d λ ⊗ ξ )) = y · d λ ⊗ ξ → y ⊗ ξ ( λ → ∞ ) . Since Φ is isometric, { y ⊗ ( d λ ⊗ ξ ) } λ ∈ Λ is a Cauchy net in Y ⊗ D K D . Hence thereexists an element z ∈ H C such that y ⊗ ( d λ ⊗ ξ ) → z ( λ → ∞ ). Thus Φ( z ) = y ⊗ ξ ,that is, Φ is surjective. Therefore Φ is an isometry from H C onto E . (cid:3) By the proof of Lemma 5.2Φ ∗ ( y ⊗ ξ ) = lim λ y ⊗ ( d λ ⊗ ξ )for any y ∈ Y , ξ ∈ K B , where { d λ } λ ∈ Λ is an approximate units of D with 0 ≤ d λ and || d λ || ≤ λ ∈ Λ.Next, we will show that H ′ C ∼ = E as Hilbert spaces. Lemma 5.3.
With the above notation H ′ C ∼ = E as Hilbert spaces. roof. Let c ∈ C , x ∈ X , ξ ∈ K B . We denote by the same notation c ⊗ ( x ⊗ ξ ), theequivalence class of c ⊗ ( x ⊗ ξ ) ∈ C ⊙ H A , where C ⊙ H A is the algebraic tensorproduct of C and H A . Let Ψ be the linear map from C ⊙ H A to E defined byΨ( c ⊗ ( x ⊗ ξ )) = ( c · x ) ⊗ ξ for any c ∈ C , x ∈ X , ξ ∈ K B , where we note that X is a closed subspace of Y .Let c, c ∈ C , x, x ∈ X , ξ, ξ ∈ K B . Then h c ⊗ ( x ⊗ ξ ) , c ⊗ ( x ⊗ ξ ) i H ′ C = h x ⊗ ξ , ( π A ◦ E A )( c ∗ c )( x ⊗ ξ ) i H A = h x ⊗ ξ , [ E A ( c ∗ c ) · x ] ⊗ ξ i H A = h ξ , π B ( h x , E A ( c ∗ c ) · x i B ) ξ i K B . Since E A and E B are strongly Morita equivalent with respect to Y and its closedsubspace X as bimodule linear maps, we have the equation h z , E A ( a ) · z i B = E B ( h z , a · z i D )for any z, z ∈ X , a ∈ C . Hence we obtain that h c ⊗ ( x ⊗ ξ ) , c ⊗ ( x ⊗ ξ ) i H ′ C = h ξ , ( π B ◦ E B )( h x , c ∗ c · x i D ) ξ i K B . On the other hand, h Ψ( c ⊗ ( x ⊗ ξ )) , Ψ( c ⊗ ( x ⊗ ξ )) i E = h ( c · x ) ⊗ ξ , ( c · x ) ⊗ ξ i E = h ξ , ( π B ◦ E B )( h c · x , c · x i D ) ξ i K B . Thus Ψ preserves the inner products on the algebraic tensor products. We canextend Ψ to H ′ C . We denote it by the same symbol Ψ. Then Ψ is an isometry from H ′ C to E . We show that Ψ is surjective. Let y and ξ be any elements in Y B and K B , respectively. Brown, Mingo and Shen [3, Proposition 1.7], Y = Y · D . Hence[8, Definition 2.1] Y = Y · D = Y · h Y, X i D = C h Y, Y i · X = C · X. For any m ∈ N , there are elements c , c , . . . , c n m ∈ C and x , x . . . , x n m ∈ X such that || y − n m X i =1 c i · x i || < m . Let z m = P n m i =1 c i ⊗ ( x i ⊗ ξ ). Then z m ∈ H ′ C for any m ∈ N and Ψ( z m ) = P n m i =1 ( c i · x i ) ⊗ ξ → y ⊗ ξ ( m → ∞ ). Since Ψ is isometric, { z m } is a Cauchysequence in H ′ C . Thus there exists an element z ∈ H ′ C such that z m → z ( m → ∞ ).Hence Ψ( z ) = y , that is, Ψ is surjective. Therefore, Ψ is an isometry from H ′ C onto E . (cid:3) Lemma 5.4.
With the above notation, the non-degenerate representations ( π C , H C ) and ( π ′ C , H ′ C ) are unitarily equivalent.Proof. Let c ∈ C , x ∈ X and ξ ∈ K B . Then(Φ ∗ Ψ)( c ⊗ ( x ⊗ ξ )) = Φ ∗ (( c · x ) ⊗ ξ ) = lim λ ( c · x ) ⊗ ( d λ ⊗ ξ ) , where { d λ } λ ∈ Λ is an approximate unit of D with d λ ≥ || d λ || ≤ λ ∈ Λ. Thus for any c ∈ C ,(Φ ∗ Ψ π ′ C ( c ))( c ⊗ ( x ⊗ ξ )) = lim λ ( c c · x ) ⊗ ( d λ ⊗ ξ ) . On the other hand,( π C ( c )Φ ∗ Ψ)( c ⊗ ( x ⊗ ξ )) = lim λ π C ( c )(( c · x ) ⊗ ( d λ ⊗ ξ )) = lim λ ( c c · x ) ⊗ ( d λ ⊗ ξ ) . ence π C ( c )Φ ∗ Ψ = Φ ∗ Ψ π ′ C ( c ) for any c ∈ C . Therefore, we obtain the conclu-sion. (cid:3) Let U = Φ ∗ Ψ. Then U is an isometry from H ′ C onto H C and π C ( c ) = U π ′ C ( c ) U ∗ for any c ∈ C . Lemma 5.5.
With the above notation, ( π C , U V ′ C , H C ) is a minimal Stinespringrepresentation for π A ◦ E A .Proof. For any c ∈ C ,( U V ′ C ) ∗ π C ( c ) U V ′ C = V ′ ∗ C U ∗ π C ( c ) U V ′ C = V ′ ∗ C π ′ C ( c ) V ′ C = ( π A ◦ E A )( c ) . Also, π C ( C ) U V ′ C H A = U π ′ C ( C ) V ′ C H A = U H ′ C = H C . Hence ( π C , U V ′ C , H C ) is a minimal Stinespring representation for π A ◦ E A . (cid:3) Lemma 5.6.
With the above notation, π A ◦ E A and π B ◦ E B are strongly Moritaequivalent as completely positive linear maps.Proof. By the definition of ( π C , H C ), ( π C , H C ) and ( π D , K D ) are strongly Moritaequivalent. Also, since ( π C , U V ′ C , H C ) is a minimal Stinespring representation for π A ◦ E A by Lemma 5.5, π A ◦ E A and π B ◦ E B are strongly Morita equivalent ascompletely positive linear maps. (cid:3) Combining the above discussions, we obtain the following theorem.
Theorem 5.7.
Let A ⊂ C and B ⊂ D be inclusions of C ∗ -algebras with AC = C and BD = D . Let E A and E B conditional expectations from C and D onto A and B , respectively. We assume that E A and E B are strongly Morita equivalentwith respect to a C − D -equivalence bimodule Y and its closed subspace X . Thenfor any non-degenerate representation ( π B , K B ) of B , there exists a non-degeneraterepresentation ( π A , H A ) of A such that π A ◦ E A and π B ◦ E B are strongly Moritaequivalent as completely positive linear maps.Proof. This is immediate by Lemma 5.6. (cid:3)
Next, we will consider the inverse direction. Let A ⊂ C and B ⊂ D be asabove. We suppose that A ⊂ C and B ⊂ D are strongly Morita equivalent withrespect to a C − D -equivalence bimodule Y and its closed subspace X . Let E A and E B be conditional expectations from C and D onto A and B , respectively. Let( π B , K B ) be a non-degenerate representation B and ( π A , H A ) be the non-degeneraterepresentation of A induced by X and ( π B , K B ), which is defined in [4, Lemma 2,2].Let ( π D , V D , K D ) be a minimal Stinespring representation for π B ◦ E B and ( π C , H C )the non-degenerate representation of C induced by Y and ( π D , K D ). First, we showthe following lemma. Lemma 5.8.
Let { d λ } λ ∈ Λ be an approximate unit of D with d λ ≥ and || d λ || ≤ for any λ ∈ Λ . Then { y ⊗ ( d λ ⊗ ξ ) } λ ∈ Λ is a Cauchy net in H C with respect to theweak topology of H C for any y ∈ Y , ξ ∈ K B .Proof. Since the linear span of the set { y ⊗ ( d ⊗ ξ ) ∈ H C | y ∈ Y, d ∈ D, ξ ∈ K B } s dense in H C , it suffices to show that for any y ∈ Y , ξ ∈ K B , d ∈ D , the net {h y ⊗ ( d λ ⊗ ξ ) , y ⊗ ( d ⊗ ξ ) i H C } λ is a Cauchy net. For any λ, µ ∈ Λ, h y ⊗ (( d λ − d µ ) ⊗ ξ ) , y ⊗ ( d ⊗ ξ ) i H C = h ( d λ − d µ ) ⊗ ξ , π D ( h y , y i D )( d ⊗ ξ ) i K D = h ( d λ − d µ ) ⊗ ξ , h y , y i D d ⊗ ξ i K D = h ξ , ( π B ◦ E B )(( d λ − d µ ) h y , y i D d ) ξ i K B → λ , µ → ∞ ) . Thus, we obtain the conclusion. (cid:3)
Let V C be the linear map from H A to H C defined by V C ( x ⊗ ξ ) = lim λ x ⊗ ( d λ ⊗ ξ )for any x ∈ X , ξ ∈ K B , where { d λ } λ ∈ Λ is an approximate unit of D with d λ ≥ || d λ || ≤ λ ∈ Λ and the limit is taken under the weak topology of H C .By Lemma 5.8, the above limit is convergent with respect to the weak topology of H C and by routine computations, V C is well-defined and independent of the choiceof an approximate unit of D . Lemma 5.9.
With the above notation, V C is an isometry from H A to H C .Proof. For any x, x ∈ X , ξ, ξ ∈ K B , h V C ( x ⊗ ξ ) , V C ( x ⊗ ξ ) i H C = lim λ, µ h x ⊗ ( d λ ⊗ ξ ) , x ⊗ ( d µ ⊗ ξ ) i H C = lim λ, µ h d λ ⊗ ξ , π D ( h x, x i B )( d µ ⊗ ξ ) i K D = lim λ, µ h d λ ⊗ ξ , h x, x i B d µ ⊗ ξ i K D = lim λ, µ h ξ , ( π B ◦ E B )( d λ h x, x i B d µ ) ξ i K B = h ξ , π B ( h x, x i B ) ξ i K B = h x ⊗ ξ , x ⊗ ξ i H A . Thus, we obtain the conclusion. (cid:3)
Proposition 5.10.
Let V C be the isometry from H A to H C defined by V C ( x ⊗ ξ ) = lim λ x ⊗ ( d λ ⊗ ξ ) for any x ∈ X , ξ ∈ K B , where { d λ } λ ∈ Λ is an approximate unit of D with d λ ≥ and || d λ || ≤ for any λ ∈ Λ and the limit is taken under the weak topology of H C .We suppose that ( π B , K B ) is faithful and that ( π A ◦ E A )( c ) = V ∗ C π C ( c ) V C for any c ∈ C . Then E A and E B are strongly Morita equivalent as bimodule linearmaps.Proof. For any c ∈ C , x, x ∈ X and ξ, ξ ∈ K B , h ( π A ◦ E A )( c )( x ⊗ ξ ) , x ⊗ ξ i H A = h ( E A ( c ) · x ) ⊗ ξ , x ⊗ ξ i H A = h ξ , π B ( h ( E A ( c ) · x ) , x i B ) ξ i K B . lso, h V ∗ C π C ( c ) V C ( x ⊗ ξ ) , x ⊗ ξ i H A = h π C ( c ) V C ( x ⊗ ξ ) , V C ( x ⊗ ξ ) i H C = lim λ, µ h π C ( c )( x ⊗ ( d λ ⊗ ξ )) , x ⊗ ( d µ ⊗ ξ ) i H C = lim λ, µ h ( c · x ) ⊗ ( d λ ⊗ ξ ) , x ⊗ ( d µ ⊗ ξ ) i H C = lim λ, µ h d λ ⊗ ξ , π D ( h c · x , x i D ) d µ ⊗ ξ i K D = lim λ h d λ ⊗ ξ , h c · x , x i D ⊗ ξ i K B = lim λ h ξ , ( π B ◦ E B )( d λ h c · x , x i D ) ξ i K B = h ξ , ( π B ◦ E B )( h c · x , x i D ) ξ i K B . Since ( π A ◦ E A )( c ) = V ∗ C π C ( c ) V C for any c ∈ C and π B is faithful, E B ( h c · x , x i D ) = h ( E A ( c ) · x ) , x i B . for any c ∈ C , x, x ∈ X . By [7, Lemma 2.5], E A and E B are strongly Moritaequivalent as bimodule linear maps. (cid:3) A correspondence of strong Morita equivalence classes ofcompletely positive linear maps
Let A and B be C ∗ -algebras, which are strongly Morita equivalent with respectto an A − B -equivalence bimodule X . In this section, we will construct a 1 − A and the set of all strong Morita equivalence classes ofcompletely positive linear maps on B and we will show that the correspondingpositive linear maps are strongly Morita equivalent.Let ψ be a completely positive linear map from B to B ( K ), where K is aHilbert space. Let ( π ψ , V ψ , K ψ ) be a minimal Stinespring representation for ψ . Let( π A , H A ) be the non-degenerate representation of A induced by X and ( π ψ , V ψ , K ψ ).Let { u i } ni =1 be a finite subset of X . Let φ be the linear map from A to B ( K ) ⊗ M n ( C )defined by [ φ ( a ) ij ] ni,j =1 = [ ψ ( h u i , a · u j i B )] ni,j =1 for any a ∈ A . Since ψ ( b ) = V ∗ ψ π ψ ( b ) V ψ for any b ∈ B ,[ φ ( a ) ij ] ni,j =1 = (cid:2) V ∗ ψ π ψ ( h u i , a · u j i B ) V ψ (cid:3) ni,j =1 = ( V ∗ ψ ⊗ I n ) [ π ψ ( h u i , a · u j i B )] ni,j =1 ( V ψ ⊗ I n ) . First, we will show that φ is a completely positive linear map from A to B ( K ) ⊗ M n ( C ).Let X n be the n -times direct sum of X and we regard X n as an A − M n ( B )-equivalence bimodule as follows: For any a ∈ A , [ b ij ] ni,j =1 ∈ M n ( B ), [ x , . . . , x n ],[ z , . . . , z n ] ∈ X n , we define the left A -action, right M n ( B )-action and the left A -valued inner product, the right M n ( B )-valued inner product on X n by setting a · [ x , . . . , x n ] = [ a · x , . . . , a · x n ] , [ x , . . . , x n ] · [ b ij ] ni,j =1 = " n X i =1 x i · b i , . . . , n X i =1 x i · b i n , A h [ x , . . . , x n ] , [ z , . . . , z n ] i = n X i =1 A h x i , z i i , h [ x , . . . , x n ] , [ z , . . . , z n ] i M n ( B ) = [ h x i , z j i B ] ni,j =1 . et Y = X n and D = M n ( B ). We regard Y as an A − D -equivalence bimodule inthe abovev way. For each m ∈ N , let M m ( Y ) be the C -linear space of all matricesover Y . We regard M m ( Y ) as an M m ( A ) − M m ( D )-equivalence bimodule as follows:For any [ a kl ] mk,l =1 ∈ M m ( A ), [ d kl ] mk,l =1 ∈ M m ( D ), [ y kl ] mk,l =1 , [ z kl ] mk,l =1 M m ( Y ), wedefine the left M m ( A )-action, the right M m ( D )-action and the left M m ( A )-valuedinner product, the right M m ( D )-valued inner product on M m ( Y ) by setting[ a kl ] mk,l =1 · [ y kl ] mk,l =1 = " m X t =1 a kt · b tl mk,l =1 , [ y kl ] mk,l =1 · [ d kl ] mk,l =1 = " m X t =1 y kt · d tl mk,l =1 , M m ( A ) h [ y kl ] mk,l =1 , [ z kl ] mk,l =1 i = " m X t =1 A h y kt , z lt i mk,l =1 , h [ y kl ] mk,l =1 , [ z kl ] mk,l =1 i M m ( D ) = " m X t =1 h y tk , z tl i D mk,l =1 . Lemma 6.1.
With the above notation, φ is a completely positive linear map from A to B ( K ) ⊗ M n ( C ) .Proof. Since φ is clearly linear, we have only to show that for any [ a kl ] mk,l =1 ∈ M m ( A ) with [ a kl ] mk,l =1 ≥
0, [ φ ( a kl )] mk,l =1 ≥ . Let [ a kl ] mk,l =1 be any positive element in M m ( A ). Then by the definition of φ , φ ( a kl ) ij = ψ ( h u i , a kl · u j i B ) . Thus ( φ ⊗ id M m ( C ) )([ a kl ] mk,l =1 )= [ ψ ( h u i , a · u j i B )] ni,j =1 . . . [ ψ ( h u i , a m · u j i B )] ni,j =1 ... . . . ...[ ψ ( h u i , a m · u j i B )] ni,j =1 . . . [ ψ ( h u i , a m m · u j i B )] ni,j =1 mk,l =1 = h [ ψ ( h u i , a kl · u j i B )] ni,j =1 i mk,l =1 . Since ψ is a completely positive linear map from B to B ( K ), we have only to showthat the element h [ h u i , a kl · u j i B ] ni,j =1 i mk,l =1 is positive in B ⊗ M n ( C ) ⊗ M m ( C ). Let y = [ u , . . . , u n ] ∈ Y . Let [ y kl ] mk.l =1 be anelement in M m ( Y ) defined by y kl = ( y if k = l k = l . Then h [ y kl ] mk,l =1 , [ a kl ] mk,l =1 · [ y kl ] mk,l =1 i M m ( D ) ≥ ince [ a kl ] mk,l =1 ≥
0. On the other hand, by the definition of [ y kl ] mk,l =1 and y , h [ y kl ] mk,l =1 , [ a kl ] mk,l =1 · [ y kl ] mk,l =1 i M m ( D ) = h [ y kl ] mk,l =1 , " m X t =1 a kt · y tl mk,l =1 i M m ( D ) = h [ y kl ] mk,l =1 , [ a kl · y ] mk,l =1 i M m ( D ) = " m X t =1 h y tk , a tl · y i D mk,l =1 = [ h y , a kl · y i D ] mk,l =1 = (cid:2) h [ u , . . . , u n ] , [ a kl · u , . . . , a kl · u n ] i M n ( B ) (cid:3) mk.l =1 = h [ h u i , a kl · u j i B ] ni,j =1 i mk,l =1 . Therefore, we obtain the conclusion. (cid:3)
Let ( π φ , V φ , H φ ) be a minimal Stinespring representation for φ . Let A ⊙ ( K⊗ C n )be the algebraic tensor product of A and K ⊗ C n . Let { b p } p ∈ P be an approximateunit of B with b p ≥ || b p || ≤ p ∈ P . We define a map U from A ⊙ ( K ⊗ C n ) to H A by setting U ( a ⊗ ξ ⊗ λ ) = lim p n X i =1 λ i ( a · u i ) ⊗ b p ⊗ ξ for any a ∈ A , ξ ∈ K , λ ∈ C n and λ = λ ... λ n , and extending linearly, wherewe identify B ( K ) ⊗ M n ( C ) with B ( K ⊗ C n ) and the limit is taken under the weaktopology of H A . Lemma 6.2.
With the above notation, U is an isometry from A ⊙ ( K ⊗ C n ) to H A . Hence we can extend U to an isometry from H φ to H A .Proof. Let a, b ∈ A , ξ, η ∈ K , λ = λ ... λ n , µ = µ ... µ n ∈ C n . Then h U ( a ⊗ ξ ⊗ λ ) , U ( b ⊗ ξ ⊗ µ ) i H A = h lim p n X i =1 λ i ( a · u i ) ⊗ b p ⊗ ξ , lim q n X j =1 µ j ( b · u j ) ⊗ b q ⊗ η i H A = lim p,q n X i,j =1 h b p ⊗ ξ , π ψ ( h λ i ( a · u i ) , µ j ( b · u j ) i B )( b q ⊗ η ) i K ψ = n X i,j =1 h V ψ ξ , π ψ ( h λ i ( a · u i ) , µ j ( b · u j ) i B ) V ψ η i K ψ = n X i,j =1 h ξ , V ∗ ψ π ψ ( h λ i ( a · u i ) , µ j ( b · u j ) i B ) V ψ η i K = n X i,j =1 h ξ , ψ ( h λ i ( a · u i ) , µ j ( b · u j ) i B ) η i K . n the other hand, h a ⊗ ξ ⊗ λ , b ⊗ η ⊗ µ i H φ = h λ ξ ... λ n ξ , [ φ ( a ∗ b ) ij ] ni,j =1 µ η ... µ n η i K⊗ C n = h λ ξ ... λ n ξ , P nj =1 φ ( a ∗ b ) j µ j η ... P nj =1 φ ( a ∗ b ) nj µ j η i K⊗ C n = n X i,j =1 h λ i ξ , φ ( a ∗ b ) ij µ j η i K = n X i,j =1 h λ i ξ , ψ ( h u i , a ∗ b · u j i B ) µ j η i K = n X i,j =1 h ξ , ψ ( h λ i ( a · u i ) , µ j ( b · u j ) i B ) η i K . Hence U preserves the inner products on the algebraic tensor products. We canextend U to an isometry from H φ to H A . (cid:3) We denote by the same symbol the above isometry from H φ to H A . From nowon, we assume that A and B are strongly Morita equivalent unital C ∗ -algebras.Since X is an A − B -equivalence bimodule, by Kajiwara and Watatani [6, Corollary1.19], there is a left A -basis in X , which is a finite subset of X . We will show that φ is strongly Morita equivalent to ψ if { u i } ni =1 is a left A -basis in X . Lemma 6.3.
With the above notation, we assume that { u i } ni =1 is a left A -basis in X . Then U is surjective.Proof. Since { u i } ni =1 is a left A -basis in X , for any x ∈ X , x = P ni =1 A h x, u i i · u i .Thus the set { P ni =1 a · u i | a ∈ A } is equal to X . Also, by [3, Proposition 1.7], X · B = X . Hence we can see that the set { n X i =1 ( a · u i ) ⊗ B ⊗ ξ | a ∈ A , ξ ∈ K} is dense in H A . Therefore, U is surjective. (cid:3) Lemma 6.4.
With the above notation, we assume that { u i } ni =1 is a left A -basis in X . Then π φ ( c ) = U ∗ π A ( c ) U for any c ∈ A . roof. Let a, b, c ∈ A , ξ, η ∈ K , λ = λ ... λ n , µ = µ ... µ n ∈ C n . Then h U ∗ π A ( c ) U ( a ⊗ ξ ⊗ λ ) , b ⊗ η ⊗ µ i H φ = h π A ( c ) n X i =1 λ i ( a · u i ) ⊗ B ⊗ ξ , n X j =1 µ j ( b · u j ) ⊗ B ⊗ η i H A = n X i,j =1 h λ i ( ca · u i ) ⊗ B ⊗ ξ , µ j ( b · u j ) ⊗ B ⊗ η i H A = n X i,j =1 h B ⊗ ξ , π ψ ( h λ i ( ca · u i ) , µ j ( b · u j ) i B )(1 B ⊗ η ) i K ψ = n X i,j =1 h V ψ ξ , π ψ ( h λ i ( ca · u i ) , µ j ( b · u j ) i B ) V ψ η i K ψ = n X i,j =1 h ξ , ψ ( h λ i ( ca · u i ) , µ j ( b · u j ) i B ) η i K . On the othere hand, h π φ ( c )( a ⊗ ξ ⊗ λ ) , b ⊗ η ⊗ µ i H φ = h ξ ⊗ λ , φ ( a ∗ c ∗ b )( η ⊗ µ ) i K⊗ C n = h λ ξ ... λ n ξ , [ ψ ( h u i , a ∗ c ∗ b · u j i B )] ni,j =1 µ η ... µ n η i K⊗ C n = h λ ξ ... λ n ξ , P nj =1 ψ ( h u , a ∗ c ∗ b · u j i B ) µ j η ... P nj =1 ψ ( h u n , a ∗ c ∗ b · u j i B ) µ j η i K⊗ C n = n X i,j =1 h ξ , ψ ( h λ i u i , a ∗ c ∗ b · u j i B ) µ j η i K . Thus we obtain the conclusion. (cid:3)
We give the main theorem in the paper.
Theorem 6.5.
Let A and B be unital C ∗ -algebras, which are strongly Moritaequivalent with respect to an A − B -equivalence bimodule X . Let { u i } ni =1 be a left A -basis in X . Let ψ be a completely positive linear map from B to B ( K ) , where K is a Hilbert space. Let φ be a map from A to B ( K ⊗ C n ) defined by [ φ ( a ) ij ] ni,j =1 = [ ψ ( h u i , a · u j i B )] ni,j =1 for any a ∈ A . Then φ is a completely positive linear map from A to B ( K ⊗ C n ) ,which is strongly Morita equivalent to ψ .Proof. By Lemma 6.1, we can see that φ is a completely positive linear map from A to B ( K ⊗ C n ). Also, by Lemmas 6.2, 6.3 and 6.4 and the definition of strongMorita equivalence for completely positive maps, we can see that φ is stronglyMorita equivalent to ψ . (cid:3) Corollary 6.6.
Let A and B be unital C ∗ -algebras, which are strongly Moritaequivalent. Then there is a − correspondence between the set of all strong orita equivalence classes of completely positive linear maps on A and the set ofall strong Morita equivalence classes of completely positive linear maps on B .Proof. This is immediate by Theorem 6.5. (cid:3)
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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