A Morita characterisation for algebras and spaces of operators on Hilbert spaces
aa r X i v : . [ m a t h . OA ] S e p A MORITA CHARACTERISATION FOR ALGEBRAS ANDSPACES OF OPERATORS ON HILBERT SPACES
G.K. ELEFTHERAKIS AND E. PAPAPETROS
Abstract.
We introduce the notion of ∆ and σ ∆ − pairs for operator algebrasand characterise ∆ − pairs through their categories of left operator modules overthese algebras. Furthermore, we introduce the notion of ∆-Morita equivalentoperator spaces and prove a similar theorem about their algebraic extensions.We prove that σ ∆-Morita equivalent operator spaces are stably isomorphic andvice versa. Finally, we study unital operator spaces, emphasising their left (resp.right) multiplier algebras, and prove theorems that refer to ∆-Morita equivalenceof their algebraic extensions. Introduction
In what follows, if X is a subset of B ( H , H ) and Y is a subset of B ( H , H ),then we denote by [ YX ] the norm-closure of the linear span of the set { y x ∈ B ( H , H ) , y ∈ Y , x ∈ X } . Similarly, if Z is a subset of B ( H , H ), we define the space [ ZYX ] . If H , K are Hilbert spaces, then a linear subspace M ⊆ B ( H, K ) is called aternary ring of operators (TRO) if
M M ⋆ M ⊆ M. It then follows that M is an A − B equivalence bimodule in the sense of Rieffel for the C ⋆ -algebras A = [ M M ⋆ ]and B = [ M ⋆ M ].We call a norm closed ternary ring of operators M, σ -TRO if there exist sequences { m i ∈ M , i ∈ N } and { n j ∈ M , j ∈ N } such thatlim t t X i =1 m i m ⋆i m = m , lim t t X j =1 m n ⋆j n j = m , ∀ m ∈ M and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) t X i =1 m i m ⋆i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) t X j =1 n ⋆j n j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ , ∀ t ∈ N . Equivalently, a TRO M is a σ -TRO if and only if the C ⋆ -algebras [ M ⋆ M ] , [ M M ⋆ ]have a σ -unit.At the beginning of the 1970s, M. A. Rieffel introduced the idea of Morita equiv-alence of C ⋆ -algebras. In particular, he gave the following definitions: Mathematics Subject Classification.
Key words and phrases.
Operator algebras, C ∗ -algebras, TRO, Stable isomorphism, Moritaequivalence. i ) Two C ⋆ -algebras, A and B , are said to be Morita equivalent if they have equiv-alent categories of ⋆ -representations via ⋆ -functors. ii ) The same algebras are said to be strongly Morita equivalent if there exists an A − B module of equivalence or if there exists a TRO M such that the C ⋆ algebras[ M ⋆ M ] and A (resp. [ M M ⋆ ] and B ) are ⋆ -isomorphic. We write A ∼ R B . If A ∼ R B , then A and B have equivalent categories of representations. The conversedoes not hold. For further details, see [17, 18, 19].Brown, Green and Rieffel proved the following fundamental theorem for C ⋆ -algebras ([7, 8]). Theorem 1.1. If A , B are C ⋆ -algebras with σ -units, then A ∼ R B if and onlyif they are stably isomorphic, which means that the algebras A ⊗ K , B ⊗ K are ⋆ -isomorphic. Here, K is the algebra of compact operators acting on ℓ ( N ) , and ⊗ isthe minimal tensor product. The next step in this theory came from Blecher, Muhly and Paulsen. Theydefined the notion of strong Morita equivalence ∼ BMP for operator algebras, self-adjoint or not, and they proved that if
A ∼
BMP B , their categories of left operatormodules are equivalent ([6]). Later, Blecher proved that the converse is also true([4]). Therefore, he proved that two C ⋆ -algebras A , B have equivalent categories ofleft operator modules if and only if A ∼ R B .A third notion of Morita equivalence was introduced by the first author of thisarticle. According to this theory, two operator algebras, A , B , are said to be ∆-equivalent and we write A ∼ ∆ B if they have completely isometric representations α : A → α ( A ) ⊆ B ( H ) , β : B → β ( B ) ⊆ B ( K ) and there exists a TRO M ⊆ B ( H, K ) such that α ( A ) = [ M ⋆ β ( B ) M ] , β ( B ) = [ M α ( A ) M ⋆ ]([11]). If M is a σ -TRO, we write A ∼ σ ∆ B . G .K. Eleftherakis proved that
A ∼ σ ∆ B if and only if A , B are stably isomorphic([12]). If we define C = [ M ⋆ M ] , D = [ M M ⋆ ], then the spaces A = α ( A ) + C , B = β ( B ) + D are operator algebras with contractive approximate identities, even if A , B do nothave, and they are also ∆-equivalent since A = [ M ⋆ B M ] , B = [ M A M ⋆ ] . Also observe that(1.1) A = A C = CA , B = B D = DB and that α ( A ) (resp. β ( B )) is an ideal of A (resp. B ).Generally, if A is an operator algebra and C ⊆ A is a C ⋆ -algebra satisfyingrelation (1.1), we call ( A , C ) a ∆-pair. Furthermore, if C has a σ -unit, we call( A , C ) a σ ∆-pair.In section 2, we characterise the ∆-equivalence and stable isomorphism of ∆-pairs under the notion of equivalence of categories of their left operator modules.In section 3, using the above theory, we characterise the ∆-equivalence and stable MORITA CHARACTERISATION FOR ALGEBRAS AND SPACES OF OPERATORS ON HILBERT SPACES3 isomorphism of the operator spaces X and Y through the equivalence of the cat-egories of left operator modules of operator algebras A X , A Y , on which X and Y naturally embed completely isometrically. If X and Y are unital operator spaces, weget stronger results using the algebras Ω X , Ω Y generated by X , Y and the diagonalsof their multiplier algebras (see section 4).If X is an operator space, then K ⊗ X is completely isometrically isomorphicwith the space K ∞ ( X ), which is the norm closure of the finitely supported matricesin M ∞ ( X ) . Here, M ∞ ( X ) is the space of ∞ × ∞ matrices, which define boundedoperators. Also, by X ⊗ h Y , we denote the Haagerup tensor product of the operatorspaces X and Y . If A is an operator algebra, X is a right A -module and Y is a left A -module, we denote by X ⊗ h A Y the balanced Haagerup tensor product of X and Y over A ([6]).For further details about operator spaces, operator algebras, Morita theory andcategory theory, we refer the reader to [1, 5, 9, 15, 16, 20].2. ∆ -Morita equivalence of operator algebras Definition 2.1.
Let
A ⊆ B ( H ) , B ⊆ B ( K ) be operator algebras. We call themTRO-equivalent (resp. σ -TRO equivalent) if there exists a TRO (resp. σ -TRO) M ⊆ B ( H, K ) such that A = [ M ⋆ B M ] , B = [ M A M ⋆ ] . We write
A ∼
T RO B , resp. A ∼ σT RO B . Definition 2.2.
Let A , B be operator algebras. We call them ∆ -equivalent (resp. σ ∆ - equivalent) if there exist completely isometric homomorphisms a : A → B ( H ) and β : B → B ( K ) such that a ( A ) ∼ T RO β ( B ) (resp. a ( A ) ∼ σT RO β ( B ) . ) We write A ∼ ∆ B (resp. A ∼ σ ∆ B ) Definition 2.3.
Let A be an operator algebra and C be a C ⋆ -algebra such that C ⊆ A . If A = [ A C ] = [ C A ] , we call the pair ( A , C ) a ∆ -pair. If C has a σ -unit,we call ( A , C ) a σ ∆ -pair. If A is an operator algebra, then A OM OD is the category with objects theessential left A -operator modules, namely operator spaces U such that there existsa completely contractive bilinear map θ : A × U → U such that U = [ A U ], where A U = { θ ( a, x ) ∈ U , a ∈ A , x ∈ U } . For our convenience, we write a x instead of θ ( a, x ). If U , U ∈ A OM OD is the space of homomorphisms between U and U is the space of completely bounded maps, which are left operator maps over A , andwe denote this space by A CB ( U , U ). Observe that if ( A , C ) is a ∆-pair, then A OM OD is a subcategory of C OM OD .A functor F : A OM OD → B OM OD is called completely contractive if for all U , U ∈ A OM OD the map F : A CB ( U , U ) → B CB ( F ( U ) , F ( U ))is completely contractive. G.K. ELEFTHERAKIS AND E.PAPAPETROS
Definition 2.4.
Let ( A , C ) , ( B , D ) be ∆ -pairs. We call them ∆ -Morita equivalentif there exist completely contractive functors F : C OM OD → D OM OD and G : D OM OD → C OM OD such that G ◦ F ∼ = Id C OMOD , F ◦ G ∼ = Id D OMOD and G | B OMOD ◦ F | A OMOD ∼ = Id A OMOD , F | A OMOD ◦ G | B OMOD ∼ = Id B OMOD . Here, ∼ = is the natural equivalence. If A , B , C , D are operator algebras such that C ⊆ A , D ⊆ B and
A ∼ ∆ B , C ∼ ∆ D , we say that ∆-equivalence is implemented in both cases by the same TRO ifthere exist completely isometric homomorphisms α : A → α ( A ) ⊆ B ( H ) , β : B → β ( B ) ⊆ B ( K ) and a TRO M ⊆ B ( H, K ) such that α ( A ) = [ M ⋆ β ( B ) M ] , β ( B ) = [ M α ( A ) M ⋆ ]and α ( C ) = [ M ⋆ β ( D ) M ] , β ( D ) = [ M α ( C ) M ⋆ ] . We now prove our main theorem for operator algebras.
Theorem 2.1.
Let ( A , C ) , ( B , D ) be ∆ -pairs. The following are equivalent: i ) A ∼ ∆ B , C ∼ ∆ D , where ∆ -equivalence is implemented in both cases by thesame TRO. ii ) The pairs ( A , C ) , ( B , D ) are ∆ -Morita equivalent.Proof. We start with the proof of i ) = ⇒ ii ) . Assume that A = [ M ⋆ B M ] and B = [ M A M ⋆ ] and also C = [ M ⋆ D M ] , D =[ M C M ⋆ ] for the same TRO M ⊆ B ( H, K ) . Let U ∈ A OM OD and E = [ M ⋆ M ] . We notice that[
E U ] = [ M ⋆ M U ] = [ M ⋆ M A U ] = [ M ⋆ M M ⋆ B M U ] ⊆ [ M ⋆ B M U ] = [ A U ] = U (so U is a left E -operator module).We set F ( U ) = M ⊗ hE U. We fix v = r X i =1 m i a i n ⋆i ∈ [ M A M ⋆ ] . We define the bilinear map f v : M × U → M ⊗ hE U , f v ( ℓ, x ) = r X i =1 m i ⊗ E a i n ⋆i ℓ x and then there exists a linear map denoted again by f v : M ⊗ U → M ⊗ hE U suchthat f v ( ℓ ⊗ x ) = r X i =1 m i ⊗ E a i n ⋆i ℓ x , ℓ ∈ M , x ∈ U MORITA CHARACTERISATION FOR ALGEBRAS AND SPACES OF OPERATORS ON HILBERT SPACES5
Let u = k X j =1 ℓ j ⊗ x j ∈ M ⊗ U. Since M is a TRO, there exists a net m λ = ( m ⋆ λ , ..., m ⋆n λ ) t ∈ M n , ( M ⋆ ) such that || m ⋆λ || ≤ , ∀ λ ∈ Λ and also m λ m ⋆λ ℓ → ℓ , ∀ ℓ ∈ M, (see [5]).Let ǫ > . We choose λ ∈ Λ such that for every λ ≥ λ holds || f v ( u ) || − ǫ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X i =1 k X j =1 m i ⊗ E a i n ⋆i ℓ j x j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − ǫ ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X i =1 k X j =1 m λ m ⋆λ m i ⊗ E a i n ⋆i ℓ j x j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Using now the fact that || y ⊗ C b || ≤ || y || || b || , y ∈ M p,q ( M ⋆ ) , b ∈ M q,s ( U ) , p, q, s ∈ N , we get || f v ( u ) || − ǫ ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X i =1 k X j =1 m λ ⊗ E m ⋆λ m i a i n ⋆i ℓ j x j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m λ ⊗ E r X i =1 m i a i n ⋆i k X j =1 m ⋆λ ℓ j x j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ || m λ || (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r X i =1 m i a i n ⋆i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k X j =1 m ⋆λ ℓ j x j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ || v || (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k X j =1 m ⋆λ ℓ j ⊗ x j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h ≤ || v || || ( m ⋆λ ℓ , ..., m ⋆λ ℓ k ) || (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) x ...x k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ || v || || ( ℓ , ..., ℓ k ) || (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) x ...x k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) We have shown that the above procedure is independent of λ , so if ǫ → + ,and by taking infimum over all representations of u , we get || f v ( u ) || h ≤ || v || h || u || h . Therefore, f v is continuous and contractive, since || f v || ≤ || v || h . Let n ∈ N and thecorresponding map ( f v ) n : M n ( M ⊗ U ) → M n ( M ⊗ hE U ) . We have to prove that ( f v ) n is contractive, that is, f v is completely contractivewith respect to the Haagerup norm. This statement is true since M n ( M ) ⊗ h M n ( E ) M n ( U ) ∼ = M n ( M ⊗ hE U ) , n ∈ N . For more details check [5].
G.K. ELEFTHERAKIS AND E.PAPAPETROS
Furthermore, for every ℓ ∈ M , z ⋆ w ∈ M ⋆ M , x ∈ U holds f v ( ℓ z ⋆ w ⊗ x ) = r X i =1 m i ⊗ E a i n ⋆i ℓ z ⋆ w x = r X i =1 m i ⊗ E a i n ⋆i ℓ ( z ⋆ w x )= f v ( ℓ ⊗ z ⋆ w x )Since f v is continuous and linear and E = [ M ⋆ M ], we get f v ( ℓ e ⊗ x ) = f v ( ℓ ⊗ e x )for every ℓ ∈ M , e ∈ E , x ∈ U. Therefore, f v extends to a linear and completelycontractive map ˆ f v : M ⊗ hE U → M ⊗ hE U with the property ˆ f v ( ℓ ⊗ E x ) = r X i =1 m i ⊗ E a i n ⋆i ℓ x , ℓ ∈ M , x ∈ U So, we have the map ˆ f : [ M A M ⋆ ] → CB ( F ( U )) , v ˆ f v , which is completelycontractive and therefore extends to a completely contractive map denoted againby ˆ f : B → CB ( F ( U )), where CB ( F ( U )) is the space of all linear and completelybounded maps of F ( U ) to itself. The algebra B acts to F ( U ) via the mapˆ θ : B × F ( U ) → F ( U ) , ˆ θ ( b, y ) = ˆ f ( b )( y ) , such that [ B F ( U )] = F ( U ) and thus F ( U ) = M ⊗ hE U ∈ B OM OD.
Therefore, we have a correspondence between the objects F : A OM OD → B OM OD, U
7→ F ( U ) = M ⊗ hE U. Let U , U ∈ A OM OD.
We fix f ∈ A CB ( U , U ) and we define the map F ( f ) : M × U → M ⊗ hE U = F ( U ) , F ( f )( ℓ, x ) := ℓ ⊗ E f ( x )The map F ( f ) is linear, completely contractive and E -balanced, so we denoteagain by F ( f ) the linear and completely contractive map F ( f ) : M ⊗ hE U = F ( U ) → M ⊗ hE U = F ( U )with the property F ( f )( ℓ ⊗ E x ) = ℓ ⊗ E f ( x ) , ℓ ∈ M , x ∈ U Furthermore, F ( f )( m a n ⋆ · ℓ ⊗ E x ) = F ( f )( m ⊗ E a n ⋆ ℓ x ) = m ⊗ E f ( an ∗ lx ) = m ⊗ E a n ⋆ ℓ f ( x ) = m a n ⋆ · F ( f )( ℓ ⊗ E x ) , m , n , ℓ ∈ M , x ∈ U , a ∈ A and since B = [ M A M ⋆ ], we have F ( f )( b · y ) = b · F ( f )( y ) , b ∈ B , y ∈ M ⊗ hE U . We proved that F ( f ) ∈ B CB ( F ( U ) , F ( U )) MORITA CHARACTERISATION FOR ALGEBRAS AND SPACES OF OPERATORS ON HILBERT SPACES7
Therefore, we have a completely contractive map F : A CB ( U , U ) → B CB ( F ( U ) , F ( U )) , f
7→ F ( f )Similarly, we have a functor G : B OM OD → A OM OD defined as G ( V ) = M ⋆ ⊗ hE ′ V , V ∈ B OM OD, where E ′ = [ M M ⋆ ] and the corresponding functor G : B CB ( V , V ) → A CB ( G ( V ) , G ( V ))for every V , V ∈ B OM OD .We are going to prove that G is the natural inverse of F . If U ∈ A OM OD , wehave that( G F )( U ) = G ( M ⊗ hE U )= M ⋆ ⊗ hE ′ ( M ⊗ hE U ) ∼ = ( M ⋆ ⊗ hE ′ M ) ⊗ hE U ∼ = E ⊗ hE U ∼ = U = Id C OMOD ( U )(Similarly, ( F G )( V ) ∼ = V , ∀ V ∈ B OM OD ).We note that if U ∈ A OM OD , then there exists an isometry f U : ( G F )( U ) = M ⋆ ⊗ hE ′ ( M ⊗ hE U ) → U = Id A OMOD ( U )such that f U ( m ⋆ ⊗ E ′ ( ℓ ⊗ E x )) = m ⋆ ℓ x , m , ℓ ∈ M , x ∈ U. We have to prove that for every U , U ∈ A OM OD , f ∈ A CB ( U , U ), thefollowing diagram ( G F )( U ) U ( G F )( U ) U G F )( f ) f U ff U is commutative, or equivalently, the following diagram is commutative M ⋆ ⊗ hE ′ ( M ⊗ hE U ) U M ⋆ ⊗ hE ′ ( M ⊗ hE U ) U G ( F ( f )) f U ff U So, we have to prove that f ◦ f U = f U ◦ G ( F ( f )).Indeed, G.K. ELEFTHERAKIS AND E.PAPAPETROS ( f U ◦ G ( F ( f ))( m ⋆ ⊗ E ′ ( ℓ ⊗ E x )) = f U ( G ( F ( f ))( m ⋆ ⊗ E ′ ( ℓ ⊗ E x )) = f U ( m ⋆ ⊗ E ′ F ( f )( ℓ ⊗ E x )) = f U ( m ⋆ ⊗ E ′ ℓ ⊗ E f ( x )) = m ⋆ ℓ f ( x ) = f ( m ⋆ ℓ x )and on the other hand( f ◦ f U )( m ⋆ ⊗ E ′ ( ℓ ⊗ E x )) = f ( f U ( m ⋆ ⊗ E ′ ( ℓ ⊗ E x )) = f ( m ⋆ ℓ x )for every m , ℓ ∈ M , x ∈ U . The functor F extends to a functor F δ to the category C OM OD in the samesense that is F δ ( U ) = M ⋆ ⊗ hC U , U ∈ OMOD C and F δ | A OMOD = F (similarly for G δ ). In conclusion, we have proved that the pairs ( A , C ) , ( B , D ) are ∆-Moritaequivalent.We are now going to complete the remaining proof of ii ) = ⇒ i ). Suppose thatthe pairs ( A , C ) , ( B , D ) are ∆-Morita equivalent. We fix an equivalence functor F : C OM OD → D OM OD with inverse G : D OM OD → C OM OD such that F ( A OM OD ) = B OM OD , G ( B OM OD ) = A OM OD
Let F ( C ) = Y , G ( D ) = X . By [2] we have that Y is a TRO and X ∼ = Y ⋆ .Also, C ∼ = X ⊗ h D Y , D ∼ = Y ⊗ hC X . We also assume that F ( A ) = Y , G ( B ) = X ,and by [4] we get A ∼ = X ⊗ h B Y , B ∼ = Y ⊗ h A X . From both the above papers, we have that F ( U ) ∼ = Y ⊗ h A U , ∀ U ∈ A OM OD , F ( U ) ∼ = Y ⊗ hC U , ∀ U ∈ C OM OD and then we get
Y ⊗ h A U ∼ = Y ⊗ hC U , ∀ U ∈ A OM OD.
Similarly,
X ⊗ h B V ∼ = X ⊗ hD V , ∀ V ∈ B OM OD.
Now, we have that X ⊗ hD B ⊗ hD X ⋆ ∼ = X ⊗ hD ( Y ⊗ h A X ) ⊗ hD Y ∼ = ( X ⊗ hD Y ) ⊗ h A ( X ⊗ hD Y ) ∼ = ( X ⊗ h B Y ) ⊗ h A ( X ⊗ h B Y ) ∼ = A ⊗ h A A∼ = A Similarly, X ⋆ ⊗ hC A ⊗ hC X ∼ = B . The following lemma implies that A ∼ ∆ B and C ∼ ∆ D , where ∆-equivalence is implemented in both cases by the same TRO. Theproof of Theorem 1.1 is complete. (cid:3) Lemma 2.2.
Suppose that A , B are operator algebras and D ⊆ B be a C ⋆ - algebrasuch that [ D B ] = [ B D ] = B . Let M ⊆ B ( K, H ) be a TRO such that [ M ⋆ M ] ∼ = D (as C ⋆ algebras) and assume that A ∼ = M ⊗ hD B ⊗ hD M ⋆ . Then
A ∼ ∆ B . MORITA CHARACTERISATION FOR ALGEBRAS AND SPACES OF OPERATORS ON HILBERT SPACES9
Proof.
We fix a completely isometric homomorphism β : B → B ( K ), and we havethat β | D is also a ⋆ -homomorphism. We can consider that the operator space M ⊗ hD K is a Hilbert space with the inner product given by h n ⊗ D x, ℓ ⊗ D y i := h β ( φ ( ℓ ⋆ n ))( x ) , y i , n , ℓ ∈ M , x , y ∈ K, where φ : [ M ⋆ M ] → D is an isometric ⋆ -isomorphism ([6]). Instead of φ ( ℓ ⋆ n ), wemay write h n ⊗ D x, ℓ ⊗ D y i := h β ( ℓ ⋆ n )( x ) , y i , n , ℓ ∈ M , x , y ∈ K, and for the action of D on K , we denote ℓ ⋆ n x instead of φ ( ℓ ⋆ n ) x where ℓ , n ∈ M , x ∈ K. For each m ∈ M , we define r m : K → M ⊗ hD K by r m ( x ) = m ⊗ D x . Obviously, r m is a linear map and r m ∈ B ( K, M ⊗ hD K ).We can easily see that r m r ⋆m r m = r m m ⋆ m ∈ r ( M ) , therefore, r ( M ) is aTRO. Also, with similar arguments, we have that β ( D ) = [ r ( M ) ⋆ r ( M )] (since r ( M ) ⋆ r ( M ) = β ( φ ( M ⋆ M ))). We also claim that r is completely isometric. ByLemma 8.3.2 (Harris-Kaup) of [5], it is sufficient to prove that r is one-to-one.Indeed, for every m ∈ M holds( r ⋆m r m )( x ) = r ⋆m ( m ⊗ D x ) = β ( m ⋆ m )( x ) , ∀ x ∈ K so || r m || = || r ⋆m r m || = || β ( m ⋆ m ) || = || m ⋆ m || = || m || , which means that r isisometric and also one-to-one. Therefore, M ∼ = r ( M ), and using Lemma 5.4 in[13], we get M ⊗ hD B ⊗ hD M ⋆ ∼ = M ⊗ hD [ β ( B ) r ( M ) ⋆ ] ∼ = [ r ( M ) β ( B ) r ( M ) ⋆ ] . Therefore, there exists a completely isometric map such that a ( A ) = [ r ( M ) β ( B ) r ( M ) ⋆ ] (4),so [ r ( M ) ⋆ a ( A ) r ( M )] = [ r ( M ) ⋆ r ( M ) β ( B ) r ( M ) ⋆ r ( M )]= [ β ( D ) β ( B ) β ( D )]= [ β ( D B D )]= β ( B ) (5)By (4) , (5), we get a ( A ) ∼ T RO β ( B ) = ⇒ A ∼ ∆ B . (cid:3) Corollary 2.3.
The relation ∼ ∆ is an equivalence relation for ∆ -pairs. Remark 2.4.
We consider that the ∆-pairs ( A , C ) , ( B , D ) are equivalent in thesense of Theorem 2.1, and F is the functor defined in its proof. For every U , U ∈ A OM OD , the map F : A CB ( U , U ) → B CB ( F ( U ) , F ( U )) is a complete isom-etry. Proof.
For every g ∈ B CB ( F ( U ) , F ( U )), we define θ = f U ◦ g ◦ f − U ∈ A CB ( U , U ) . So, for every f ∈ A CB ( U , U ), we have that F ( f ) ∈ B CB ( F ( U ) , F ( U )) and ( θ ◦ G )( F ( f )) = θ ( G ( F ( f ))= f U ◦ G F ( f ) ◦ f − U = f ◦ f U ◦ f − U = f Since θ ◦ G is completely contractive, we get || f || cb = || ( θ ◦ G )( F ( f ) || cb ≤ ||F ( f ) || cb . (cid:3) Theorem 2.5.
Let ( A , C ) , ( B , D ) be σ ∆ -pairs. The following are equivalent: i ) A ∼ σ ∆ B , C ∼ σ ∆ D , where σ ∆ -equivalence is implemented in both cases by thesame σ -TRO. ii ) The pairs ( A , C ) , ( B , D ) are ∆ -Morita equivalent. iii ) There exists a completely isometric isomorphism φ : A ⊗ K → B ⊗ K such that φ ( C ⊗ K ) = D ⊗ K , where K is the algebra of compact operators of ℓ ( N ) . Proof. i ) ⇐⇒ ii ) It is obvious according to the previous Theorem 2.1 i ) = ⇒ iii ). We may consider A = [ M ⋆ B M ] , B = [ M A M ⋆ ] and also C = [ M ⋆ D M ] , D = [ M C M ⋆ ] . Since
C , D have a σ -unit by Lemma 3.4 of [11], M is a σ -TRO. By Theorem 3.2 in the same article, there exists a completely iso-metric onto map φ : A ⊗ K → B ⊗ K such that φ ( C ⊗ K ) = D ⊗ K .iii ) = ⇒ i ) We have that ( A , C ) ∼ ∆ ( A⊗K , C ⊗K ), so ( A , C ) ∼ ∆ ( B⊗K , D ⊗K ),but also, ( B , D ) ∼ ∆ ( B ⊗K , D ⊗K ). Since ∼ ∆ is an equivalence relation for ∆-pairs,we get ( A , C ) ∼ ∆ ( B , D ) . (cid:3) In the rest of this section, we consider that the ∆-pairs ( A , C ) , ( B , D ) are equiv-alent in the sense of Theorem 2.1, and F is the functor defined in its proof.We consider the subcategory of representations of A denoted by A HM OD . If H ′ ∈ A HM OD , there exists a completely contractive morphism π : A → B ( H ′ )such that π ( A )( H ′ ) = [ π ( a )( h ) ∈ H ′ : a ∈ A , h ∈ H ′ ]. The space F ( H ′ ) = M ⊗ hE H ′ is also a Hilbert space. Its inner product is given by h m ⊗ E ξ, ℓ ⊗ E w i := h π ( ℓ ⋆ m )( ξ ) , w i H ′ , m , ℓ ∈ M , ξ , w ∈ H ′ (for more details check [17, 19, 6]).Also, the map F ( π ) : A → B ( F ( H ′ )) given by F ( π )( m b n ⋆ )( ℓ ⊗ E ξ ) = m ⊗ E π ( b n ℓ ⋆ )( ξ )is completely contractive.We are going to prove that the functor F maintains the complete isometricrepresentations. So, let π : A → B ( H ′ ) be a homomorphism and also a complete MORITA CHARACTERISATION FOR ALGEBRAS AND SPACES OF OPERATORS ON HILBERT SPACES11 isometry. We set ρ = F ( π ) : B → B ( F ( H ′ )), where F ( H ′ ) = M ⊗ hE H ′ and ρ ( m a n ⋆ )( ℓ ⊗ E h ) = m ⊗ E π ( a n ℓ ⋆ )( h ) , m , n , ℓ ∈ M , a ∈ A , h ∈ H ′ We define the unitary operator U : G F ( H ′ ) → H ′ , U ( k ⋆ ⊗ E ′ ( n ⊗ E h )) := π ( k ⋆ n )( h ) , and we consider φ = G ( ρ ) : A → B ( G F ( H ′ )) given by φ ( m ⋆ b n )( ℓ ⋆ ⊗ E ′ x ) = m ⋆ ⊗ E ′ ρ ( b n ℓ ⋆ )( x ) , m , n , ℓ ∈ M , b ∈ B , x ∈ F ( H ′ ) Lemma 2.6.
It holds that
U φ ( a ) U ⋆ = π ( a ) , ∀ a ∈ A .Proof. For every m , k , s , n , t , ℓ ∈ M , a ∈ A , h ∈ H ′ , we have that φ ( m ⋆ k a s ⋆ n )( ℓ ⋆ ⊗ E ′ ( t ⊗ E h )) = m ⋆ ⊗ E ′ ρ ( k a s ⋆ n ℓ ⋆ )( t ⊗ E h )= m ⋆ ⊗ E ′ k ⊗ E π ( a s ⋆ n ℓ ⋆ t )( h )Therefore, U ( φ ( m ⋆ k a s ⋆ n )( ℓ ⋆ ⊗ E ′ ( t ⊗ E h ))) = U ( m ⋆ ⊗ E ′ k ⊗ E π ( a s ⋆ n ℓ ⋆ t )( h ))= π ( m ⋆ k )( π ( a s ⋆ n ℓ ⋆ t )( h ))= π ( m ⋆ k a s ⋆ n ) π ( ℓ ⋆ t )( h )= π ( m ⋆ k a s ⋆ n ) U ( ℓ ⋆ ⊗ E ′ ( t ⊗ E h ))So, U ( φ ( m ⋆ k a s ⋆ n )) = π ( m ⋆ k a s ⋆ n ) U , but since A = [ M ⋆ M A M ⋆ M ], we getthat U φ ( a ) U ⋆ = π ( a ) . (cid:3) We conclude that since π is an isometry and U φ ( a ) U ⋆ = π ( a ) , a ∈ A , where U is unitary, φ is also an isometry. Observe now that if ( m i ) i ∈ I is a net of M n i , ( M ⋆ )such that || m i || ≤ ∀ i ∈ I and also m i m ⋆i m → m , ∀ m ∈ M , then we have that φ ( m ⋆i b m i )( ℓ ⋆ ⊗ E ′ x ) = m ⋆i ⊗ E ′ ρ ( b m i ℓ ⋆ )( x )= m ⋆i ⊗ E ′ ρ ( b ) V ( m i ⊗ E ( ℓ ⋆ ⊗ E ′ x ))for every b ∈ B , x ∈ F ( H ′ ) , ℓ ∈ M where V is the unitary operator V : M ⊗ hE ( M ⋆ ⊗ hE ′ K ′ ) → K ′ , K ′ = F ( H ′ ) . Therefore, for all w ∈ M ⋆ ⊗ K ′ holds φ ( m ⋆i b m i )( w ) = m i ⊗ E ρ ( b ) V ( m i ⊗ E w ) . So, || φ ( m ⋆i b m i )( w ) || ≤ || m i || || ρ ( b ) || || m ⋆i || || w || ≤ || ρ ( b ) || || w || , which means that || φ ( m ⋆i b m i ) || ≤ || ρ ( b ) || , but φ is an isometry, and we concludethat || m ⋆i b m i || ≤ || ρ ( b ) || , ∀ b ∈ B (3) . Since lim i m i m ⋆i b m i m ⋆i = b , we have thatsup i ∈ I || m i m ⋆i b m i m ⋆i || = || b || . On the other hand, || m i m ⋆i b m i m ⋆i || ≤ || m ⋆i b m i || ≤ || b || , therefore,sup i ∈ I || m ⋆i b m i || = || b || , ∀ b ∈ B . We conclude from (3) that || b || ≤ || ρ ( b ) || , ∀ b ∈ B , but also that ρ is completelycontractive and therefore || ρ ( b ) || = || b || , ∀ b ∈ B , so ρ is an isometry. Similarly, ρ is a complete isometry. Using the above facts, we can prove that F restricts to anequivalence functor from A HM OD to B HM OD.
This functor maps completelyisometric representations to completely isometric representations.3. ∆ -Morita equivalence of operator spaces
Definition 3.1.
Let
X ⊆ B ( H , H ) , Y ⊆ B ( K , K ) be operator spaces. Wecall them TRO-equivalent (resp. σ -TRO equivalent) if there exist TROs (resp. σ -TROs) M i ⊆ B ( H i , K i ) , i = 1 , such that X = [ M ⋆ Y M ] , Y = [ M X M ⋆ ] We write
X ∼
T RO Y , resp. X ∼ σT RO Y . Definition 3.2.
Let X , Y be operator spaces. We call them ∆ -equivalent (resp. σ Delta -equivalent) if there exist completely isometric maps φ : X → B ( H , H ) , ψ : Y → B ( K , K ) such that φ ( X ) ∼ T RO ψ ( Y ) (resp. φ ( X ) ∼ σT RO ψ ( Y ) . We write
X ∼ ∆ Y , resp. X ∼ σ ∆ Y . Definition 3.3.
Let X be an operator space and D , D be C ⋆ -algebras (resp. σ unital C ⋆ -algebras) such that X = [ D X ] = [ D X ] Then, the space A X = (cid:18) D X D (cid:19) is an operator algebra, which we call an algebraic ∆ -extension of X (resp. σ ∆ -extension of X ). Definition 3.4.
Let X , Y be operator spaces. We call them ∆ -Morita equiva-lent (resp. σ ∆ -Morita equivalent) if they have algebraic ∆ -extensions (resp. σ ∆ -extensions) A X , A Y such that the ∆ -pairs ( A X , ∆( A X )) , ( A Y , ∆( A Y )) to be ∆ -Morita equivalent. Lemma 3.1.
The TRO-equivalence (resp. σ -TRO) of operator spaces is an equiv-alence relation.Proof. The fact TRO-equivalence is an equivalence relation has been proved in [13].The proof that σ -TRO-equivalence is an equivalence relation is similar. (cid:3) MORITA CHARACTERISATION FOR ALGEBRAS AND SPACES OF OPERATORS ON HILBERT SPACES13
Theorem 3.2.
Let X , Y be operator spaces. The following are equivalent: i ) X ∼ ∆ Y ii ) X and Y are ∆ -Morita equivalent.Proof. i ) = ⇒ ii ) We may assume that X = [ M ⋆ Y M ] , Y = [ M X M ⋆ ] for TROs M ⊆ B ( H , K ) and M ⊆ B ( H , K ) . If we consider the C ⋆ -algebras D = [ M ⋆ M ] , D = [ M ⋆ M ] , E = [ M M ⋆ ] , E = [ M M ⋆ ] , we get X = [ D X ] = [ X D ] , Y = [ E Y ] = [ Y E ] . So, the operator algebras A X = (cid:18) D X D (cid:19) ⊆ B ( H ⊕ H ) , A Y = (cid:18) E Y E (cid:19) ⊆ B ( K ⊕ K )are ∆-algebraic extensions of X , Y , respectively, such that∆( A X ) = (cid:18) D D (cid:19) , ∆( A Y ) = (cid:18) E E (cid:19) Clearly M = (cid:18) M M (cid:19) ⊆ B ( H ⊕ H , K ⊕ K )is a TRO.Furthermore, [ M ⋆ A Y M ] = A X and A Y = [ M A X M ⋆ ] , ∆( A X ) = [ M ⋆ ∆( A Y ) M ] , ∆( A Y ) = [ M ∆( A X ) M ⋆ ] , so the pairs ( A X , ∆( A X )) , ( A Y , ∆( A Y )) are ∆-Morita equivalent. That is, X and Y are ∆-Morita equivalent. ii ) = ⇒ i ) Suppose that X and Y are ∆-Morita equivalent. There exist C ⋆ -algebras D i , E i , i = 1 , X = [ D X ] = [ X D ] , Y = [ E Y ] = [ Y E ] andthe pairs ( A X , ∆( A X )) , ( A Y , ∆( A Y )) are ∆-Morita equivalent, where A X = (cid:18) D X D (cid:19) , A Y = (cid:18) E Y E (cid:19) so ∆( A X ) = (cid:18) D D (cid:19) , ∆( A Y ) = (cid:18) E E (cid:19) Let N be a TRO such that A X = [ N ⋆ A Y N ] , A Y = [ N A X N ⋆ ] , ∆( A X ) = [ N ⋆ ∆( A Y ) N ] , ∆( A Y ) = [ N ∆( A X ) N ⋆ ] . We define M = [∆( A Y ) N ] = [ N ∆( A X )], then M is a TRO since M M ⋆ M = [∆( A Y ) N N ⋆ ∆( A Y ) ∆( A Y ) N ]= [∆( A Y ) N N ⋆ ∆( A Y ) N ] = [∆( A Y ) N ∆( A X )] = M Using the fact that [∆( A Y ) A Y ∆( A Y )] = A Y , we get[ M ⋆ A Y M ] = [ N ⋆ ∆( A Y ) A Y ∆( A Y ) N ] = [ N ⋆ A Y N ] = A X , and with similar arguments we have that A Y = [ M A X M ⋆ ] , ∆( A X ) = [ M ⋆ M ] , ∆( A Y ) = [ M M ⋆ ]We define the TROs M = (cid:18) E
00 0 (cid:19) M (cid:18) D
00 0 (cid:19) and M = (cid:18) E (cid:19) M (cid:18) D (cid:19) Since Y = [ E Y E ], we have that (cid:18) Y (cid:19) = (cid:18) E
00 0 (cid:19) A Y (cid:18) E (cid:19) = (cid:18) E
00 0 (cid:19) M A X M ⋆ (cid:18) E (cid:19) = (cid:18) E
00 0 (cid:19) (cid:20) M ∆( A X ) M ⋆ + M (cid:18) X (cid:19) M ⋆ (cid:21) (cid:18) E (cid:19) But M ∆( A X ) M ⋆ = ∆( A Y ) = (cid:18) E E (cid:19) , so it holds that (cid:18) E
00 0 (cid:19) M ∆( A X ) M ⋆ (cid:18) E (cid:19) = 0 , and thus (cid:18) Y (cid:19) = (cid:18) E
00 0 (cid:19) M (cid:18) X (cid:19) M ⋆ (cid:18) E (cid:19) = (cid:18) E
00 0 (cid:19) M (cid:18) D
00 0 (cid:19) (cid:18) X (cid:19) (cid:18) D (cid:19) M ⋆ (cid:18) E (cid:19) = M (cid:18) X (cid:19) M ⋆ Similarly, (cid:18) X (cid:19) = M ⋆ (cid:18) Y (cid:19) M ,and therefore (cid:18) X (cid:19) ∼ T RO (cid:18) Y (cid:19) . Since ( C , (cid:18) X (cid:19) (cid:18) C (cid:19) = X and (cid:18) C (cid:19) X (0 , C ) = (cid:18) X (cid:19) .we have that X ∼ T RO (cid:18) X (cid:19) .Similarly, Y ∼
T RO (cid:18) Y (cid:19) . Therefore, according to Lemma 3.1, we get X ∼ ∆ Y . (cid:3) Theorem 3.2 and Theorem 3.11 in [13] imply the following corollary:
Corollary 3.3. ∆ -Morita equivalence of operator spaces is an equivalence relation. Theorem 3.4.
Let X , Y be operator spaces. The following are equivalent: i ) X ∼ σ ∆ Y MORITA CHARACTERISATION FOR ALGEBRAS AND SPACES OF OPERATORS ON HILBERT SPACES15 ii ) X and Y are σ ∆ -Morita equivalent iii ) X and Y are stably isomorphic.Proof. i ) ⇐⇒ ii ) Check the proof of the Theorem 3.2. i ) = ⇒ iii ) See [13, Theorem 4.6]. iii ) = ⇒ i ) Since X ∼ st Y , we have that K ∞ ( X ) ∼ = K ∞ ( Y ), but since X ∼ σ T RO K ∞ ( X ), we get X ∼ σ ∆ K ∞ ( Y ) . Also,
Y ∼ σT RO K ∞ ( Y ), so Y ∼ σ ∆ K ∞ ( Y ), anddue to the fact that ∼ σ ∆ is an equivalence relation, we have X ∼ σ ∆ Y . (cid:3)
4. ∆ -Morita equivalence of unital operator spaces
Definition 4.1.
We call an operator space X unital if there exists a completelyisometric map φ : X → B ( H ) such that I H ∈ φ ( X ) . If Y is an operator space that is bimodule over the C ⋆ algebra A , we say thatthe map ( π, ψ, π ) : A Y A → B ( H )is a completely contractive bimodule map if ψ : Y → B ( H ) is a completely con-tractive map and π : A → B ( H ) is a ∗− homomorphism such that φ ( asb ) = π ( a ) φ ( s ) π ( b ) , ∀ a, b ∈ A , s ∈ Y . Lemma 4.1.
Let X , Y be operator spaces and M be a TRO such that X = [ M Y M ⋆ ] , Y = [ M X M ⋆ ] . We denote A = [ M ⋆ M ] , B = [ M M ⋆ ] . For every completely isometric bimodule map ( π, ψ, π ) : A Y A → B ( H ) there exists a completely isometric bimodule map ( σ, φ, σ ) : B X B → B ( K ) and a TRO N ⊆ B ( H, K ) such that ψ ( Y ) = [ N ⋆ φ ( X ) N ] , φ ( X ) = [ N ψ ( Y ) N ⋆ ] , π ( A ) = [ N ⋆ N ] , σ ( B ) = [ N N ⋆ ] . Proof.
Suppose that K = M ⊗ hA H is the Hilbert space with the inner productgiven by h m ⊗ ξ, n ⊗ ω i = h π ( n ⋆ m ) ξ, ω i m, n ∈ M, ξ, ω ∈ H. By the usual arguments, we can define a completely isometric map φ : X → B ( K )given by φ ( msn ⋆ )( l ⊗ ξ ) = m ⊗ ψ ( sn ⋆ l )( ξ ) , m, n, l ∈ M, s ∈ S , ξ ∈ H and the ∗− homomorphism σ : B → B ( K ) given by σ ( mn ⋆ )( l ⊗ ξ ) = m ⊗ π ( n ⋆ l )( ξ ) . We also define the map µ : M → B ( L, K ) given by µ ( m )( π ( n ⋆ l )( ξ )) = ( mn ⋆ l ) ⊗ ξ and the map ν : M ⋆ → B ( K, L ) given by ν ( m ⋆ )( l ⊗ ξ ) = π ( m ⋆ l )( ξ ) . We can easily see that ν ( m ⋆ ) = µ ( m ) ⋆ for all m , and N = µ ( M ) is a TROsatisfying ψ ( Y ) = [ N ⋆ φ ( X ) N ] , φ ( X ) = [ N ψ ( Y ) N ⋆ ] , π ( A ) = [ N ⋆ N ] , σ ( B ) = [ N N ⋆ ] . (cid:3) Lemma 4.2.
Let
X ⊆ B ( K ′ ) , Y ⊆ B ( H ′ ) be unital operator spaces and M ⊆ B ( H ′ , K ′ ) be a TRO such that X = [ M Y M ⋆ ] , Y = [ M ⋆ X M ] . We denote A = [ M ⋆ M ] , B = [ M M ⋆ ] . For every completely isometric bimodule map ( π, ψ, π ) : A Y A → B ( H ) there exists a unital completely isometric bimodule map ( σ, φ, σ ) : B X B → B ( K ) and a TRO N ⊆ B ( H, K ) such that ψ ( Y ) = [ N ⋆ φ ( X ) N ] , φ ( X ) = [ N ψ ( Y ) N ⋆ ] , π ( A ) = [ N ⋆ N ] , σ ( B ) = [ N N ⋆ ] . Proof.
Suppose that
K, ψ, µ, σ, N are as in the proof of Lemma 4.1. We can seethat φ ( msn ⋆ ) = µ ( m ) ψ ( s ) µ ( n ) ⋆ , ∀ m, n ∈ M, s ∈ X . Since Y is unital, we have that φ ( mn ⋆ ) = µ ( m ) ψ ( I H ′ ) µ ( n ) ⋆ , ∀ m, n ∈ M. Assume that I K ′ = lim λ k λ X i =1 m λi ( m λi ) ⋆ . If l ∈ M, ξ ∈ H , we have φ ( I K ′ )( l ⊗ ξ ) = lim λ k λ X i =1 µ ( m λi ) ψ ( I H ′ ) µ (( m λi )) ⋆ ( l ⊗ ξ ) =lim λ k λ X i =1 µ ( m λi ) ψ ( I H ′ )( π (( m λi ) ⋆ l )( ξ ) = lim λ k λ X i =1 µ ( m λi ) ψ (( m λi ) ⋆ l )( ξ ) . We can easily see that ψ | M ⋆ M = π, thus φ ( I K ′ )( l ⊗ ξ ) = lim λ k λ X i =1 µ ( m λi ) π (( m λi ) ⋆ l )( ξ ) = lim λ ( k λ X i =1 m λi ( m λi ) ⋆ l ) ⊗ ξ = l ⊗ ξ. Therefore, φ ( I K ′ ) = I K . MORITA CHARACTERISATION FOR ALGEBRAS AND SPACES OF OPERATORS ON HILBERT SPACES17 (cid:3)
Lemma 4.3.
Let X , Y be unital operator spaces such that X ∼ ∆ Y . Then, thereexist completely isometric maps φ : X → B ( H ) , ψ : Y → B ( K ) such that I H ∈ φ ( X ) , I K ∈ ψ ( Y ) and a σ -TRO L ⊆ B ( K, H ) such that ψ ( Y ) = [ L ⋆ φ ( X ) L ] , φ ( X ) = [ L ψ ( Y ) L ⋆ ] . Proof.
We have that
Y ∼
T RO K ∞ ( Y ), and the TRO equivalence is implemented byone TRO. Since X and Y are unital, by [13] K ∞ ( Y ) ∼ = K ∞ ( X ) as K ∞ − operatormodules. Lemma 4.1 implies that there exists a completely isometric map ζ : Y → ζ ( Y ) such that ζ ( Y ) ∼ T RO K ∞ ( X ), and this TRO equivalence is implemented byone TRO. Since X ∼
T RO K ∞ ( X ) with one TRO as in the proof of Theorem 2.1in [11], we have that ζ ( Y ) ∼ T RO X with one TRO. From Lemma 4.1, given thecomplete isometry ζ − : ζ ( Y ) → Y , there exists a complete isometry φ : X → φ ( X )and a TRO M such that Y = [ M φ ( X ) M ⋆ ] , φ ( X ) = [ M ⋆ Y M ] . By Lemma 4.9 in [13], the algebra [ M ⋆ M ] is unital, thus φ ( X ) is a unital operatorspace. The map φ − : φ ( X ) → X is a complete isometry, thus by Lemma 4.2 thereexists a unital complete isometry ψ : Y → ψ ( Y ) and a TRO L such that ψ ( Y ) = [ L ⋆ X L ] , X = [ Lψ ( S ) L ⋆ ] . (cid:3) If X is an operator space, we denote by M ℓ ( X ) (resp. M r ( X )) the left (resp.right) multiplier algebra of X . We also denote A ℓ ( X ) = ∆( M ℓ ( X )) , A r ( X ) = ∆( M r ( X )). Remark 4.4.
If we consider X as unital subspace of its C ⋆ -envelope, C ⋆env ( X ) , then by Proposition 4.3 in [3], we have M ℓ ( X ) = { a ∈ C ⋆env ( X ) : a X ⊆ X } and M r ( X ) = { a ∈ C ⋆env ( X ) : X a ⊆ X } Lemma 4.5. If X , Y are ∆ -equivalent unital operator spaces, we can considerthat X ⊆ C ⋆env ( X ) ⊆ B ( H ) , Y ⊆ C ⋆env ( Y ) ⊆ B ( K ) and there exists a TRO M ⊆ B ( H, K ) such that X = [ M ⋆ Y M ] , Y = [ M X M ⋆ ] and also C ⋆env ( X ) = [ M ⋆ C ⋆env ( Y ) M ] , C ⋆env ( Y ) = [ M C ⋆env ( X ) M ] M l ( X ) = [ M ⋆ M l ( Y ) M ] , M l ( Y ) = [ M M l ( X ) M ⋆ ] M r ( X ) = [ M ⋆ M r ( Y ) M ] , M r ( Y ) = [ M M r ( X ) M ⋆ ] Proof.
From Lemma 4.3, we may assume that X and Y have TRO equivalent com-pletely isometric representations whose images are TRO equivalent by one TRO.Using this fact and the proof of Theorem 5.10 in [13], we may consider that thereexists a TRO M such that C ⋆env ( X ) = [ M ⋆ C ⋆env ( Y ) M ] , C ⋆env ( Y ) = [ M C ⋆env ( X ) M ] . Let us prove that M l ( X ) = [ M ⋆ M l ( Y ) M ] . Let a ∈ M l ( Y ), that is a ∈ C ⋆env ( Y )and a Y ⊆ Y . For all m , n ∈ M , we have that m ⋆ a n ∈ C ⋆env ( Y ) and m ⋆ a n X = m ⋆ a n M ⋆ Y M ⊆ m ⋆ a Y M ⊆ M ⋆ Y M = X , so m ⋆ a n ∈ M l ( X ), that is M ⋆ M l ( Y ) M ⊆ M l ( X ) . Similarly,
M M l ( X ) M ⋆ ⊆ M l ( Y ), so M ⋆ M M l ( X ) M ⋆ M ⊆ M ⋆ M l ( Y ) M ⊆ M l ( X ) , but M ⋆ M C ⋆env ( X ) = C ⋆env ( X ). (cid:3) The proof of the previous Lemma implies the following corollary:
Corollary 4.6. If X , Y are ∆ -equivalent unital operator spaces, then M l ( X ) ∼ ∆ M l ( Y ) , and thus M l ( X ) and M l ( Y ) are stably isomorphic. The same assertionholds for M r ( X ) and M r ( Y ) . Definition 4.2. If X is an operator space, then we define the operator algebra Ω X = (cid:18) A l ( X ) X A r ( X ) (cid:19) Theorem 4.7. If X , Y are unital operator spaces, the following are equivalent: i ) X and Y are stably isomorphic. ii ) X ∼ σ ∆ Y .iii ) X ∼ ∆ Y iv ) Ω X and Ω Y are stably isomorphic. v ) Ω X ∼ σ ∆ Ω Y vi ) Ω X ∼ ∆ Ω Y . Proof.
We have proved the equivalence i ) ⇐⇒ ii ) at the Theorem 3.4. Also, ii ) = ⇒ iii ) is obvious. iii ) = ⇒ ii ) Suppose that φ ( X ) = [ M ⋆ ψ ( Y ) M ] , ψ ( Y ) = [ M φ ( X ) M ⋆ ] for someTRO M. By Lemma 4.9 in [13], the C ⋆ -algebras [ M ⋆ M ] , [ M M ⋆ ] are unital, so itfollows that X ∼ σ ∆ Y . Similarly, we have the equivalence iv ) ⇐⇒ v ) ⇐⇒ vi ) . It remains to provethat iii ) ⇐⇒ vi ) .iii ) = ⇒ vi ). If X ∼ ∆ Y , then by Lemma 4.5, there exists a TRO M such that X = [ M ⋆ Y M ] , Y = [ M X M ⋆ ] MORITA CHARACTERISATION FOR ALGEBRAS AND SPACES OF OPERATORS ON HILBERT SPACES19 M l ( X ) = [ M ⋆ M l ( Y ) M ] , M l ( Y ) = [ M M l ( X ) M ⋆ ] M r ( X ) = [ M ⋆ M r ( Y ) M ] , M r ( Y ) = [ M M r ( X ) M ⋆ ]Since A l ( X ) = ∆( M l ( X )) , A l ( Y ) = ∆( M l ( Y )) , A r ( X ) = ∆( M r ( X )) , A r ( Y ) =∆( M r ( Y )), we get A l ( X ) = [ M ⋆ A l ( Y ) M ] , A l ( Y ) = [ M A l ( X ) M ⋆ ] A r ( X ) = [ M ⋆ A r ( Y ) M ] , A r ( Y ) = [ M A r ( X ) M ⋆ ] , so Ω X = (cid:18) A l ( X ) X A r ( X ) (cid:19) = (cid:18) M ⋆ A l ( Y ) M M ⋆ Y M M ⋆ A r ( Y ) M (cid:19) = (cid:18) M ⋆ M ⋆ (cid:19) (cid:18) A l ( Y ) Y A r ( Y ) (cid:19) (cid:18) M M (cid:19) = (cid:18) M M (cid:19) ⋆ (cid:18) A l ( Y ) Y A r ( Y ) (cid:19) (cid:18) M M (cid:19) where (cid:18) M M (cid:19) is TRO. Similarly, Ω Y = (cid:18) M M (cid:19) (cid:18) A l ( Y ) Y A r ( Y ) (cid:19) (cid:18) M M (cid:19) ⋆ ,and we conclude that Ω X ∼ ∆ Ω Y (Theorem 2.1). vi ) = ⇒ iii ) Let Ω X ∼ ∆ Ω Y . The operator algebras Ω X , Ω Y are ∆-algebraicextensions of X , Y , respectively, so X , Y are ∆-Morita equivalent. According toTheorem 3.2, we conclude that X ∼ ∆ Y . (cid:3) Corollary 4.8. If X , Y are unital operator spaces, the following are equivalent: i ) X ∼ ∆ Y ii ) X and Y are ∆ -Morita equivalent. iii ) The ∆ -pairs (Ω X , ∆(Ω X )) , (Ω Y , ∆(Ω Y )) are ∆ -Morita equivalent.Proof. i ) ⇐⇒ ii ) It has been proven previously at Theorem 3.2. iii ) = ⇒ ii ) It is obvious since Ω X , Ω Y are algebraic ∆-extensions of X , Y ,respectively. i ) = ⇒ iii ) We may consider, using again the Lemma 4.5, that there exists aTRO M such that X = [ M ⋆ Y M ] , Y = [ M X M ⋆ ] A l ( X ) = [ M ⋆ A l ( Y ) M ] , A l ( Y ) = [ M A l ( X ) M ⋆ ] A r ( X ) = [ M ⋆ A r ( Y ) M ] , A r ( Y ) = [ M A r ( X ) M ⋆ ] Using the TRO N = (cid:18) M M (cid:19) , we have that Ω X = [ N ⋆ Ω Y N ] , Ω Y = [ N Ω X N ⋆ ].Also, ∆(Ω X ) = (cid:18) A l ( X ) 00 A r ( X ) (cid:19) , ∆(Ω Y ) = (cid:18) A l ( Y ) 00 A r ( Y ) (cid:19) , and it is obvious that ∆(Ω X ) = [ N ⋆ ∆(Ω Y ) N ] , ∆(Ω Y ) = [ N ∆(Ω X ) N ⋆ ], so Ω X ∼ ∆ Ω Y and ∆(Ω X ) ∼ ∆ ∆(Ω Y ) with the same TRO, which means that the ∆-pairs(Ω X , ∆(Ω X )) , (Ω Y , ∆(Ω Y )) are ∆-Morita equivalent. (cid:3) References [1] H. Bass,
The Morita Theorems , Lecture Notes, University of Oregon, Eugene, 1962.[2] D. P. Blecher,
On Morita’s fundemental theorem for C* algebras , Mathematica Scandinavica (2001), 137–153.[3] D. P. Blecher,
The Shilov boundary of an operator space and the characterization theorems ,J. Funct. Anal. (2001), 280–343.[4] D. P. Blecher,
A Morita theorem for algebras of operators on Hilbert space , J. Pure Appl.Algebra (2001), 153–169.[5] D. P. Blecher and C. Le Merdy,
Operator algebras and their modules–an operator space ap-proach.
London Mathematical Society Monographs. New Series, . Oxford Science Publica-tions. The Clarendon Press, Oxford University Press, Oxford, 2004.[6] D. P. Blecher, P. S. Muhly and V. I. Paulsen, Categories of operator modules (Morita equiv-alence and projective modules) , Mem. Amer. Math. Soc. (2000), no. 681.[7] L. G. Brown,
Stable isomorphism of hereditary subalgebras of C*-algebras , Pacific J. Math. (1977), 335–348 .[8] L. G. Brown, P. Green and M. A. Rieffel,
Stable isomorphism and strong Morita equivalenceof C*-algebras , Pacific J. Math. (1977), 349–363.[9] E. Effros and Z.-J. Ruan,
Operator Spaces , London Mathematical Society Monographs. NewSeries, 23. The Clarendon Press, Oxford University Press, New York, 2000.[10] G. K. Eleftherakis,
TRO equivalent algebras , Houston J. of Mathematics (2012), 153–175.[11] G. K. Eleftherakis,
Stable isomorphism and strong Morita equivalence of operator algebras ,Houston J. of Mathematics, (2016), 1245-1266.[12] G. K. Eleftherakis,
On stable maps of operator algebras , J. of Math. Analysis and Applications (2019), 1951-1975.[13] G. K. Eleftherakis and E.T.A. Kakariadis,
Strong Morita equivalence of operator spaces , J. ofMath. Analysis and Applications (2017), 1632-1653.[14] M. R. Hestenes,
A ternary algebra with applications to matrices and linear transformations ,J. of Archive for Rational Mechanics and Analysis (1962), 138-194.[15] V. I. Paulsen,
Completely bounded maps and operator algebras , Cambridge Studies in Ad-vanced Mathematics, vol. . Cambridge University Press, Cambridge, 2002.[16] G. Pisier, Introduction to operator space theory , London Mathematical Society Lecture NoteSeries, 294. Cambridge University Press, Cambridge, 2003.[17] M. A. Rieffel,
Morita equivalence for C*-algebras and W*-algebras , J. Pure Appl. Algebra (1974), 51–96.[18] M. A. Rieffel, Induced representations of C*-algebras , Adv. Math. (1974), 176–257.[19] M. A. Rieffel, Morita equivalence for operator algebras , Operator Algebras and Applications,Part I (Kingston, ON, Canada, 1980), pp. 285–298. Proc. Sympos. Pure Math. , Amer.Math. Soc., Providence, R.I., 1982. MORITA CHARACTERISATION FOR ALGEBRAS AND SPACES OF OPERATORS ON HILBERT SPACES21 [20] J.Rotman,
An introduction to homomlogical algebra , Springer Science and Business Media,2008.
G. K. Eleftherakis, University of Patras, Faculty of Sciences, Department ofMathematics, 265 00 Patras Greece
E-mail address : [email protected] E. Papapetros, University of Patras, Faculty of Sciences, Department of Mathe-matics, 265 00 Patras Greece
E-mail address ::