The relative tensor product and a minimal fiber product in the setting of C ∗ -algebras
aa r X i v : . [ m a t h . OA ] N ov The relative tensor product and a minimal fiber product inthe setting of C ∗ -algebras Thomas Timmermann ∗ University of Muenster,Einsteinstr. 62, 48149 Muenster, Germany,[email protected]: ++49 251 8332724, Fax: ++49 251 8332708November 12, 2018
Abstract
We introduce a relative tensor product of C ∗ -bimodules and a spatial fiber product of C ∗ -algebras that are analogues of Connes’ fusion of correspondences and the fiber productof von Neumann algebras introduced by Sauvageot, respectively. These new constructionsform the basis for our approach to quantum groupoids in the setting of C ∗ -algebras that ispublished separately. Keywords:
Hilbert module, relative tensor product, fiber product, quantum groupoid
MSC:
Contents C ∗ -algebras 43 The spatial fiber product of C ∗ -algebras 114 Relation to the setting of von Neumann algebras 235 The special case of a commutative base 31 ∗ This work was supported by the SFB 478 “Geometrische Strukturen in der Mathematik” funded by the DeutscheForschungsgemeinschaft (DFG) and initiated during a stay at the “Special Programme on Operator Algebras” at theFields Institute in Toronto, Canada, 2007. Introduction
Background
The relative tensor product of Hilbert modules over von Neumann algebras wasintroduced by Connes in an unpublished manuscript [4, 10, 20] and later used by Sauvageot todefine a fiber product of von Neumann algebras relative to a common (commutative) von Neu-mann subalgebra [21]. These constructions and Haagerups theory of operator-valued weights onvon Neumann algebras [12, 13] form the basis for the theory of measured quantum groupoidsdeveloped by Enock, Lesieur and Vallin [8, 9, 18, 30, 31].In this article, we introduce a new notion of a bimodule in the setting of C ∗ -algebras, constructrelative tensor products of such bimodules, and define a fiber product of C ∗ -algebras repre-sented on such bimodules. These constructions form the basis for a series of articles on quantumgroupoids in the setting of C ∗ -algebras, individually addressing fundamental unitaries [29], ax-iomatics of the compact case [25], and coactions of quantum groupoids on C ∗ -algebras [28].Moreover, our previous approach to quantum groupoids in the setting of C ∗ -algebras [27] em-beds functorially into this new framework [26], and the latter overcomes the serious restrictionsof the former one.Already in the definition of a quantum groupoid, the relative tensor product and a fiber productappear as follows. Roughly, such an object consists of the following ingredients: an algebra B , thought of as the functions on the unit space, an algebra A , thought of as functions on thetotal space, a homomorphism r : B → A and an antihomomorphism s : B → A correspondingto the range and the source map, and a comultiplication D : B → A ∗ B A corresponding to themultiplication of the quantum groupoid. Here, A ∗ B A is a fiber product whose precise definitiondepends on the class of the algebras involved. In the setting of operator algebras, A acts naturallyon some bimodule H and product A ∗ B A is a certain subalgebra of operators acting on a relativetensor product H ⊗ B H . This relative tensor product is important also because it forms the domainor range of the fundamental unitary of the quantum groupoid. Overview
Let us now sketch the problems and constructions studied in this article.The first problem is the construction of a tensor product H ⊗ B K of modules H , K over somealgebra B . In the algebraic setting, H ⊗ B K is simply a quotient of the full tensor product H ⊗ K .In the setting of von Neumann algebras, H and K are Hilbert spaces, and Connes explained thatthe right tensor product is not a completion of the algebraic one but something more complicated.If B is commutative and of the form B = L ¥ ( X , µ ) , then the modules H , K can be disintegratedinto two measurable fields of Hilbert spaces in the form H = R ⊕ X H x dµ ( x ) and K = R ⊕ X K x dµ ( x ) ,and H ⊗ B K is obtained by taking tensor products of the fibers and integrating again: H ⊗ B K = R ⊕ X H x ⊗ K x dµ ( x ) . For the situation where B is a C ∗ -algebra, we propose an approach that is basedon the internal tensor product of Hilbert C ∗ -modules and essentially consists of an algebraicreformulation of Connes’ fusion. Central to this approach is a new notion of a bimodule in thesetting of C ∗ -algebras.The second problem is the construction of a fiber product A ∗ B C of two algebras A , C relative toa subalgebra B . If B is central in A and the opposite B op is central in C , this fiber product is2ust a relative tensor product. In the algebraic setting, it coincides with the tensor product ofmodules; in the setting of operator algebras, it can be obtained via disintegration and a fiberwisetensor product again. This approach was studied by Sauvageot for Neumann algebras [21], andby Blanchard [1] for C ∗ -algebras.The case where the subalgebra B ( op ) is no longer central in A or C is more difficult. In the al-gebraic setting, the fiber product was introduced by Takeuchi [24] and is, roughly, the largestsubalgebra of the relative tensor product A ⊗ B C where componentwise multiplication is still welldefined. In the setting of von Neumann algebras, Sauvageot’s definition of the fiber productcarries over to the general case and takes the form A ∗ B C = ( A ′ ⊗ B C ′ ) ′ , where A and C are rep-resented on Hilbert spaces H and K , respectively, and A ′ ⊗ B C ′ acts on Connes’ relative tensorproduct H ⊗ B K . Here, it is important to note that A ′ ⊗ B C ′ is a completion of an algebraic tensorproduct spanned by elementary tensors, but in general, A ∗ B C is not. Similarly, in the setting of C ∗ -algebras, one can not start from some algebraic tensor product and define the fiber product tobe some completion; rather, a new idea is needed. We propose such a new fiber product for C ∗ -algebras represented on the new class of modules mentioned above. Unfortunately, several im-portant questions concerning this construction remain open, but the applications in [25, 28, 29]already prove its usefulness. Plan
This article is organized as follows.The introduction ends with a short summary on terminology and some background on Hilbert C ∗ -modules.Section 2 is devoted to the relative tensor product in the setting of C ∗ -algebras. It starts with somemotivation, then presents a new notion of modules and bimodules in the setting of C ∗ -algebras,and finally gives the construction and its formal properties like functoriality, associativity andunitality.Section 3 introduces a fiber product of C ∗ -algebras. It starts with an overview and then proceedsto C ∗ -algebras represented on the class of modules and bimodules introduced in Section 2. Thefiber product is first defined and studied for such represented C ∗ -algebras, including a discussionof functoriality, slice maps, lack of associativity, and unitality. A natural extension to non-represented C ∗ -algebras is indicated at the end.Section 4 relates our constructions for the setting of C ∗ -algebras to the corresponding construc-tions for the setting of von Neumann algebras. Adapting our constructions to von Neumannalgebras, one recovers Connes fusion and Sauvageot’s fiber product; moreover, the construc-tions are related by functors going from the C ∗ -level to the W ∗ -level. The section ends with acategorical interpretation of Sauvageot’s fiber product.Section 5 shows that for a commutative base B = C ( X ) , the relative tensor product of the newclass of modules corresponds to the fiberwise tensor product of continuous Hilbert bundles over X , and the fiber product of represented C ∗ -algebras is related to the relative tensor product ofcontinuous C ( X ) -algebras studied by Blanchard.3 reliminaries and notation Given a category C , we write A , B ∈ C to indicate that A , B areobjects of C , and denote by C ( A , B ) the associated set of morphisms.Given a subset Y of a normed space X , we denote by [ Y ] ⊂ X the closed linear span of Y .All sesquilinear maps like inner products on Hilbert spaces are assumed to be conjugate-linearin the first component and linear in the second one.Given a Hilbert space H and an element x ∈ H , we define ket-bra operators | x i : C → H , l lx ,and h x | = | x i ∗ : H → C , x ′
7→ h x | x ′ i .We shall make extensive use of (right) Hilbert C ∗ -modules; a standard reference is [16].Let A and B be C ∗ -algebras. Given Hilbert C ∗ -modules E and F over B , we denote by L ( E , F ) the space of all adjointable operators from E to F . Let E and F be Hilbert C ∗ -modules over A and B , respectively, and let p : A → L ( F ) be a ∗ -homomorphism. Then the internal tensor product E ⊗ p F is a Hilbert C ∗ -module over B [16, §4] and the closed linear span of elements h ⊗ p x ,where h ∈ E and x ∈ F are arbitrary, and h h ⊗ p x | h ′ ⊗ p x ′ i = h x | p ( h h | h ′ i ) x ′ i and ( h ⊗ p x ) b = h ⊗ p x b for all h , h ′ ∈ E , x , x ′ ∈ F , b ∈ B . We denote the internal tensor product by “ = ” and dropthe index p if the representation is understood; thus, E = F = E = p F = E ⊗ p F .We define a flipped internal tensor product F p < E as follows. We equip the algebraic tensorproduct F ⊙ E with the structure maps h x ⊙ h | x ′ ⊙ h ′ i : = h x | p ( h h | h ′ i ) x ′ i , ( x ⊙ h ) b : = x b ⊙ h ,form the separated completion, and obtain a Hilbert C ∗ - B -module F p < E which is the closedlinear span of elements x p < h , where h ∈ E and x ∈ F are arbitrary, and h x p < h | x ′ p < h ′ i = h x | p ( h h | h ′ i ) x ′ i and ( x p < h ) b = x b p < h for all h , h ′ ∈ E , x , x ′ ∈ F , b ∈ B . As above, we usuallydrop the index p and simply write “ < ” instead of “ p < ”. Evidently, there exists a unitary S : F = E ∼ = −→ E < F , h = x x < h .Let E , E be Hilbert C ∗ -modules over A , let F , F be Hilbert C ∗ -modules over B with ∗ -homomorphisms p i : A → L ( F i ) for i = ,
2, and let S ∈ L ( E , E ) , T ∈ L ( F , F ) such that T p ( a ) = p ( a ) T for all a ∈ A . Then there exists a unique operator S = T ∈ L ( E = F , E = F ) such that ( S = T )( h = x ) = S h = T x for all h ∈ E , x ∈ F , and ( S = T ) ∗ = S ∗ = T ∗ [7, Proposition1.34]. C ∗ -algebras The aim of this section is to construct a relative tensor product of suitably defined left and rightmodules over a general C ∗ -algebra B such that i) the construction shares the main properties ofthe ordinary tensor product of bimodules over rings like functoriality and associativity and ii) themodules admit representations of C ∗ -algebras that do not commute with the module structures.The latter condition will be needed to construct fiber products of C ∗ -algebras; see Section 3.The internal tensor product of Hilbert C ∗ -modules meets condition i) but not ii) because C ∗ -algebras represented on such modules necessarily commute with the right module structure. Anapproach to quantum groupoids based on the internal tensor product was developed in [27] butremained restricted to very special cases.What we are looking for is an analogue of Connes’ fusion of correspondences. Here, B is a vonNeumann algebra, and left and right modules are Hilbert spaces equipped with suitable repre-4entation or antirepresentation of B , respectively. The relative tensor product of a right module H and a left module K is then constructed as follows. Choose a normal, semi-finite, faithful(n.s.f.) weight µ on B , construct a B -valued inner product h · | · i µ on the dense subspace H ⊆ H of all bounded vectors, and define H ⊗ µ K to be the separated completion of the algebraic tensorproduct H ⊙ K with respect to the sesquilinear form given by h x ⊙ h | x ′ ⊙ h ′ i = h h |h x | x ′ i µ h ′ i .The definition of bounded vectors involves the GNS-space H : = H µ for µ which — by Tomita-Takesaki theory — is bimodule over B , and each bounded vector x ∈ H gives rise to a map L ( x ) ∈ L ( H B , H B ) of right B -modules such that h x | x ′ i µ = L ( x ) ∗ L ( x ′ ) ∈ B ⊆ L ( H ) . Example.
Assume that B = L ¥ ( X , µ ) for some nice measure space ( X , µ ) , and denote the weighton B given by integration by µ as well. Then H = L ( X , µ ) , and we can disintegrate H and K into measurable fields ( H x ) x and ( K x ) x of Hilbert spaces over X such that H ∼ = R ⊕ X H x dµ ( x ) and K ∼ = R ⊕ X K x dµ ( x ) . Each vector x of H or K corresponds to a measurable section x x ( x ) withsquare-integrable norm function | x | : x
7→ k x x k , and is bounded with respect to µ if and only ifthis norm function is essentially bounded. Then for all x , x ′ ∈ H , x ∈ X , h , h ′ ∈ K , h x | x ′ i µ ( x ) = h x ( x ) | x ′ ( x ) i H x , h x ⊙ h | x ′ ⊙ h ′ i = Z X h x ( x ) | x ′ ( x ) ih h ( x ) | h ′ ( x ) i dµ ( x ) , and H ⊗ µ K ∼ = R ⊕ X H x ⊗ K x dµ ( x ) . Note that the sesquilinear form above need not extend to H ⊙ K because the integrand need not be in L ( X , µ ) for arbitrary x , x ′ ∈ H and h , h ′ ∈ K .For our purpose, the following algebraic description of H ⊗ µ K is useful. This relative tensorproduct can be identified with the separated completion of algebraic tensor product L ( H B , H B ) ⊙ H ⊙ L ( B H , B K ) (1)with respect to the sesquilinear form h S ⊙ z ⊙ T | S ′ ⊙ z ′ ⊙ T ′ i = h z | S ∗ S ′ T ∗ T ′ z ′ i = h z | T ∗ T ′ S ∗ S ′ z ′ i ,where L ( H B , H B ) and L ( B H , B K ) are all bounded maps of right or left B -modules, respectively.We adapt this definition to the setting of C ∗ -algebras, making the following modifications:(A) The construction above depends on the choice of some n.s.f. weight µ or, more precisely,the triple ( H µ , p µ ( B ) , p µ ( B ) ′ ) , but any other µ yields a triple which is unitarily equivalent.In the setting of C ∗ -algebras, such a canonical triple does not exist but has to be chosen.(B) The module structure of H and K can equivalently be described in terms of (anti)repre-sentations of B or in terms of the spaces L ( H B , H B ) and L ( B H , B K ) . In the setting of C ∗ -algebras, this equivalence breaks down, and we shall make suitable closed subspacesof intertwiners the primary object. In the commutative case, a representation correspondsto a measurable field of Hilbert spaces, and the subspaces fix a continuous structure.(C) If H and K are bimodules, then so is H ⊗ µ K . Here, a bimodule structure on H is given bythe additional choice of a representation of some von Neumann algebra A that commuteswith the antirepresentation of B or, equivalently, satisfies A L ( H B , H B ) = L ( H B , H B ) . Ifwe pass to C ∗ -algebras, then commutation is too weak, and we shall adopt the secondcondition, where L ( H B , H B ) is replaced by the subspace of intertwiners mentioned above.5 .2 Modules and bimodules over C ∗ -bases Observation (A) leads us to adopt the following terminology.
Definition 2.1.
A C ∗ -base b = ( K , B , B † ) consists of a Hilbert space H and commuting non-degenerate C ∗ -algebras B , B † ⊆ L ( K ) , respectively. The opposite of b is the C ∗ -base b † : =( K , B † , B ) . A C ∗ -base ( H , A , A † ) is equivalent to b if Ad V ( A ) = B and Ad V ( A † ) = B † forsome unitary V ∈ L ( H , K ) . Clearly, the Hilbert space C and twice the algebra C ≡ L ( C ) form a trivial C ∗ -base t = ( C , C , C ) . Example 2.2.
Let µ be a proper, faithful KMS-weight on a C ∗ -algebra A [15] with GNS-space H µ , GNS-representation p µ : A → L ( H µ ) , modular conjugation J µ : H µ → H µ , and opposite GNS-representation p µ op : A op → L ( H µ ) , a J µ p µ ( a ∗ ) J µ . Then ( H µ , p µ ( A ) , p µ op ( A op )) is a C ∗ -base.Its opposite is equivalent to the C ∗ -base associated to the opposite weight µ op on A op . Indeed, H µ can be considered as the GNS-space for µ op via the opposite GNS-map L µ op : N µ op → H µ , a op J µ L µ ( a ∗ ) , and then J µ op p µ op ( A op ) J µ op = p µ ( A ) .Let b = ( K , B , B † ) be a C ∗ -base. We define C ∗ -modules over b as indicated in comment (B). Definition 2.3.
A C ∗ - b -module H a = ( H , a ) is a Hilbert space H with a closed subspace a ⊆ L ( K , H ) satisfying [ a K ] = H, [ a B ] = a , [ a ∗ a ] = B . A semi-morphism between C ∗ - b -modulesH a and K b is an operator T ∈ L ( H , K ) satisfying T a ⊆ b . If additionally T ∗ b ⊆ a , we call T a morphism . We denote the set of all (semi-)morphisms by L ( s ) ( H a , K b ) . Evidently, the class of all C ∗ - a -modules forms a category with respect to all semi-morphisms,and a C ∗ -category in the sense of [11] with respect to all morphisms. Lemma 2.4. i) Let H , K be Hilbert spaces and I ⊆ L ( H , K ) such that [ IH ] = K. Then thereexists a unique normal, unital ∗ -homomorphism r I : ( I ∗ I ) ′ → ( II ∗ ) ′ such that r I ( x ) S = Sxfor all x ∈ ( I ∗ I ) ′ , S ∈ I.ii) Let H , K , L be Hilbert spaces and I ⊆ L ( H , K ) , J ⊆ L ( K , L ) such that [ IH ] = K, [ JK ] = L,and J ∗ JI ⊆ I. Then r I (( I ∗ I ) ′ ) ⊆ ( J ∗ J ) ′ and r J ◦ r I = r JI .Proof. i) Uniqueness is evident. Let x ∈ ( I ∗ I ) ′ and S , . . . , S n ∈ I , x , . . . , x n ∈ H . Since x ∗ x commutes with each S ∗ i S j , the matrix ( S ∗ i S j x ∗ x ) i , j ∈ M n ( L ( H )) is dominated by k x ∗ x k ( S ∗ i S j ) i , j ,and k (cid:229) i S i x x i k = (cid:229) i , j h x i | S ∗ i S j x ∗ x x j i ≤ k x k (cid:229) i , j h x i | S ∗ i S j x j i = k x k k (cid:229) i S i x i k . Hence, there exists an operator r I ( x ) ∈ L ( K ) as claimed. One easily verifies that the assignment x r I ( x ) is a ∗ -homomorphism. It is normal because [ IH ] = K and for all S , T ∈ I , x , h ∈ K ,the functional x
7→ h S x | r I ( x ) T h i = h x | xS ∗ T h i is normal.ii) Let x ∈ ( I ∗ I ) ′ . Then r I ( x ) ∈ J ∗ J because S ∗ T r I ( x ) R = S ∗ T Rx = r I ( x ) S ∗ T R for all S , T ∈ J , R ∈ I , and r JI ( x ) = r J ( r I ( x )) because r JI ( x ) T R = T Rx = r J ( r I ( x )) T R for all T ∈ J , R ∈ I . Lemma 2.5.
Let H a be a C ∗ - b -module. ) a is a Hilbert C ∗ - B -module with inner product ( x , x ′ ) x ∗ x ′ .ii) There exist isomorphisms a = K → H, x = z xz , and K < a → H, z < x xz .iii) There exists a unique normal, unital and faithful representation r a : B ′ → L ( H ) such that r a ( x )( xz ) = x x z for all x ∈ B ′ , x ∈ a , z ∈ K .iv) Let K b be a C ∗ - b -module and T ∈ L s ( H a , K b ) . Then T r a ( x ) = r b ( x ) T for all x ∈ B ′ . Ifadditionally T ∈ L ( H a , K b ) , then left multiplication by T defines an operator in L B ( a , b ) ,again denoted by T .Proof. Assertions i) and ii) are obvious, and iii) follows from the preceding lemma. To proveiv), let x ∈ B ′ , x ∈ a , z ∈ K . Then T x ∈ b and T r a ( x ) xz = T x x z = r b ( x ) T xz . Example 2.6.
Let Z be a locally compact Hausdorff space, µ a Radon measure on Z of fullsupport, and H = ( H z ) z a continuous bundle of Hilbert spaces on Z with full support. Then theHilbert space K = L ( Z , µ ) together with the C ∗ -algebras B = B † = C ( Z ) ⊆ L ( K ) forms a C ∗ -base. Let H = R ⊕ Z H z dµ ( z ) and a = m ( G ( H )) , where for each section x ∈ G ( H ) , the operator m ( x ) ∈ L ( K , H ) is given by pointwise multiplication, m ( x ) f = ( x ( z ) f ( z )) z ∈ Z . Then H a is a C ∗ - b -module and r a : B ′ = L ¥ ( Z , µ ) → L ( H ) is given by pointwise multiplication of sections byfunctions. Every C ∗ - b -module arises in this way from a continuous bundle; see Section 5.Let also a = ( H , A , A † ) be a C ∗ -base. We define C ∗ - ( a † , b ) -bimodules as indicated in (C). Definition 2.7.
A C ∗ - ( a † , b ) -module is a triple a H b = ( H , a , b ) , where H is a Hilbert space, ( H , a ) a C ∗ - a † -module, ( H , b ) a C ∗ - b -module, and [ r a ( A ) b ] = b , [ r b ( B † ) a ] = a . The set of (semi-)morphisms between C ∗ - ( a † , b ) -modules a H b and g K d is L ( s ) ( a H b , g K d ) : = L ( s ) ( H a , K g ) ∩ L ( s ) ( H b , K d ) . Remark 2.8.
By Lemma 2.5, [ r a ( A ) , r b ( B † )] = C ∗ - ( a † , b ) -module a H b .Again, the class of all C ∗ - ( a † , b ) -modules forms a category with respect to all semi-morphisms,and a C ∗ -category with respect to all morphisms. Examples 2.9. i) H A is a C ∗ - a -module, r A ( x ) = x for all x ∈ A ′ , and A † H A is a C ∗ - ( a † , a ) -module because [ r A † ( A ) A ] = [ AA ] = A and [ r A ( A † ) A † ] = A † .ii) Let H b be a C ∗ - b -module, let t = ( C , C , C ) be the trivial C ∗ -base, and let a = L ( C , H ) .Then a H b is a C ∗ - ( t , b ) -module.iii) Let ( H i ) i be a family of C ∗ - ( a † , b ) -modules, where H i = ( H i , a i , b i ) for each i . Denote by ⊞ i a i ⊆ L (cid:0) H , ⊕ i H i (cid:1) the norm-closed linear span of all operators of the form z ( x i z ) i ,where ( x i ) i is in the algebraic direct sum L alg i a i , and similarly define ⊞ i b i ⊆ L (cid:0) K , ⊕ i H i (cid:1) .Then the triple ⊞ i H i : = (cid:0) ⊕ i H i , ⊞ i a i , ⊞ i b i (cid:1) is a C ∗ - ( a † , b ) -module, for each j , the canon-ical inclusions i j : H j → ⊕ i H i and projection p j : ⊕ i H i → H j are morphisms H j → ⊞ i H i and ⊞ i H i → H j , and with respect to these maps, ⊞ i H i is the direct sum of the family ( H i ) i .The following example shows how bimodules arise from conditional expectations.7 xample 2.10. Let B be a C ∗ -algebra with a KMS-state µ and associated C ∗ -base b (Example2.2), let A be a unital C ∗ -algebra containing B such that 1 A ∈ B , and let f : A → B be a faithfulconditional expectation such that n : = µ ◦ f is a KMS-state and f ◦ s n t = s µt ◦ f for all t ∈ R .Fix a GNS-construction p n : A → L ( H n ) for n with modular conjugation J n : H n → H n , anddefine p op n : A op → L ( H n ) by a J n p n ( a ∗ ) J n . Then the inclusion B ֒ → A extends to an isometry z : K = H µ ֒ → H n = H , and we obtain a C ∗ - ( b † , b ) -module a H b , where H = H n , a = [ J n p n ( A ) z ] , b = [ p n ( A ) z ] , and r a ◦ p µ op = p op n , r b ◦ p µ = p n . Moreover, p n ( A ) + p op n (( A ∩ B ′ ) op ) ⊆ L ( H a ) , p n op ( A op ) + p n ( A ∩ B ′ ) ⊆ L ( H b ) . For details, see [25, §2–3]. The concepts introduced above allow us to adapt the algebraic formulation of Connes’ fusion tothe setting of C ∗ -algebras as follows. Let b = ( K , B , B † ) be a C ∗ -base, H b a C ∗ - b -module, and K g a C ∗ - b † -module. Then the relative tensor product of H b and K g is the Hilbert space H b ⊗ b g K : = b = K < g , which is spanned by elements x = z < h , where x ∈ b , z ∈ K , h ∈ g , the inner product being givenby h x = z < h | x ′ = z ′ < h ′ i = h z | x ∗ x ′ h ∗ h ′ z ′ i = h z | h ∗ h ′ x ∗ x ′ z ′ i for all x , x ′ ∈ b , z , z ′ ∈ K , h , h ′ ∈ g . Examples 2.11. i) If b is the trivial C ∗ -base t = ( C , C , C ) , then b = L ( C , H ) , g = L ( C , K ) ,and H b ⊗ b g K ∼ = H ⊗ K via x = z < h xz ⊗ h = x ⊗ hz .ii) Let Z be a locally compact Hausdorff space, µ a Radon measure on Z of full support, H = ( H z ) z and K = ( K z ) z continuous bundles of Hilbert spaces on Z with full support,and H a , K b the associated C ∗ - b -modules as defined in Example 2.6. One easily checksthat then we have an isomorphism H b ⊗ b g K → Z ⊕ Z H z ⊗ K z dµ ( z ) , m ( x ) = z < m ( h ) ( x ( z ) z ( z ) ⊗ h ( z )) z ∈ Z . Let us list some easy observations and a few definitions.i) The isomorphisms in Lemma 2.5 ii), applied to H b and K g , respectively, yield the followingidentifications which we shall use without further notice: b = r g K ∼ = H b ⊗ b g K ∼ = H r b < g , x = hz ≡ x = z < h ≡ xz < h . ii) For each x ∈ b and h ∈ g , there exist bounded linear operators | x i : K → b = r g K = H b ⊗ b g K , w x = w , | h i : H → H r b < g = H b ⊗ b g K , w w < h , whose adjoints h x | : = | x i ∗ and h h | : = | h i ∗ are given by h x | : x ′ = w r g ( x ∗ x ′ ) w , h h | : w < h ′ r b ( h ∗ h ′ ) w . We put | b i : = {| x i | x ∈ b } ⊆ L ( K , H b ⊗ b g K ) and similarly define h b | , | g i , h g | .8ii) For all S ∈ r b ( B † ) ′ and T ∈ r g ( B ) ′ , we have operators S < id ∈ L ( H r b < g ) = L ( H b ⊗ b g K ) , id = T ∈ L ( b = r g K ) = L ( H b ⊗ b g K ) . If these operators commute, we let S ⊗ b T : = ( S < id )( id = T ) = ( id = T )( S < id ) . Thecommutativity condition holds in each of the following cases:(a) S ∈ L s ( H b ) ; then ( S ⊗ b T )( x = w ) = S x = T w for each x ∈ b , w ∈ K ;(b) T ∈ L s ( K g ) ; then ( S ⊗ b T )( w < h ) = S w < T h for each w ∈ H , h ∈ g ;(c) ( B † ) ′ = B ′′ ; then for all x , x ′ ∈ b and h , h ′ ∈ g , the elements h ∗ T h ′ ∈ B ′ and x ∗ S x ′ ∈ ( B † ) ′ commute, and if z , z ′ ∈ K and w = x = z < h , w ′ = x ′ = z ′ < h ′ , then h w | ( id = T )( S < id ) w ′ i = h z | ( h ∗ T h ′ )( x ∗ S x ′ ) z ′ i = h z | ( x ∗ S x ′ )( h ∗ T h ′ ) z ′ i = h w | ( S < id )( id = T ) w ′ i .Let a = ( H , A , A † ) and c = ( L , C , C † ) be further C ∗ -bases. Then the relative tensor product ofbimodules over ( a † , b ) and ( b † , c ) is a bimodule over ( a † , c ) : Proposition 2.12.
Let H = a H b be a C ∗ - ( a † , b ) -module, K = g K d a C ∗ - ( b † , c ) -module, and a ⊳ g : = [ | g i a ] ⊆ L ( H , H b ⊗ b g K ) , b ⊲ d : = [ | b i d ] ⊆ L ( L , H b ⊗ b g K ) . (2) Then H ⊗ b K : = ( a ⊳ g ) ( H b ⊗ b g K ) ( b ⊲ d ) is a C ∗ - ( a † , c ) -module and r ( a ⊳ g ) ( x ) = r a ( x ) < id for all x ∈ ( A † ) ′ , r ( b ⊲ d ) ( y ) = id = r d ( y ) for all y ∈ C ′ . (3) Proof. ( H b ⊗ b g K ) ( a ⊳ g ) is a C ∗ - a † -module because [ a ∗ h g | | g i a ] = [ a ∗ r b ( B † ) a ] = A † , [ | g i a A † ] =[ | g i a ] , and [ | g i a H ] = [ | g i H ] = H b ⊗ b g K . Likewise, ( H b ⊗ b g K ) ( b ⊲ d ) is a C ∗ - c -module.For all x ∈ ( A † ) ′ , z ∈ H , q ∈ a , h ∈ g , we have | h i q ∈ a ⊳ g and hence r ( a ⊳ g ) ( x )( qz < h ) = r ( a ⊳ g ) ( x ) | h i qz = | h i q x z = r a ( x ) qz < h = ( r a ( x ) < id )( qz < h ) . The first equation in (3) follows, and a similar agument proves the second one.Finally, ( a ⊳ g ) ( H b ⊗ b g K ) ( b ⊲ d ) is a C ∗ - ( a † , c ) -module because [ r ( a ⊳ g ) ( A ) | b i d ] = [ | r a ( A ) b i d ] =[ | b i d ] and [ r ( b ⊲ d ) ( C † ) | g i a ] = [ | g i a ] .In the situation above, we call H ⊗ b K the relative tensor product of H and K . Note the follow-ing commutative diagram of Hilbert spaces and closed spaces of operators between them: H a ) ) TTTTTT a ⊳ g . . K b s s gggggggg g + + WWWWWWWW L d u u jjjjjj b ⊲ d p p H | g i * * TTTTTT K | b i t t jjjjjj H b ⊗ b g K C ∗ - b -module H = H b and a C ∗ - ( b † , c ) -module K = g K d , we abbreviate H b ⊗ b g K d : =( H b ⊗ b g K ) b ⊲ d . Likewise, we write a H b ⊗ b g K for ( H b ⊗ b g K ) a ⊳ g and a H b ⊗ b g K d for a ⊳ g ( H b ⊗ b g K ) b ⊲ d .The relative tensor product is functorial, associative, unital, and compatible with direct sums inthe following sense: Proposition 2.13.
Let H = a H b , H = a H b , H = a H b be C ∗ - ( a † , b ) -modules, K = g K d , K = g K d , K = g K d C ∗ - ( b † , c ) -modules, and L = e L f a C ∗ - ( c † , d ) -module.i) S ⊗ b T ∈ L (cid:0) H ⊗ b K , H ⊗ b K (cid:1) for all S ∈ L ( H , H ) , T ∈ L ( K , K ) .ii) The composition of the isomorphisms ( H b ⊗ b g K d ) ⊗ c e L ∼ = ( H b ⊗ b g K ) r ( b ⊲ d ) < e ∼ = b = r g K r d < e and b = r g K r d < e ∼ = b = r ( g ⊳ e ) ( K d ⊗ c e L ) ∼ = H b ⊗ b ( g K d ⊗ c e L ) is an isomorphism of C ∗ - ( a † , c ) -modules a a , b , c , d ( L , K , H ) : ( H ⊗ b K ) ⊗ c L → H ⊗ b ( K ⊗ c L ) .iii) Put U : = B † K B . Then there exist isomorphismsr a , b ( H ) : H ⊗ b U → H , x = z < b † x b † z = r b ( b † ) xz , l b , c ( K ) : U ⊗ b K → K , b = z < h h b z = r g ( b ) hz . iv) Let ( H i ) i be a family of C ∗ - ( a † , b ) -modules and ( K j ) j a family of C ∗ - ( b † , c ) -modules.For each i , j, denote by i i H : H i → ⊞ i ′ H i ′ , i j K : K j → ⊞ j ′ K j ′ and p i H : ⊞ i ′ H i ′ → H i , p j K : ⊞ j ′ K j ′ → K j the canonical inclusions and projections, respectively. Then thereexist inverse isomorphisms ⊞ i , j ( H i ⊗ b K j ) ⇆ ( ⊞ i H i ) ⊗ b ( ⊞ j K j ) , given by ( w i , j ) i , j (cid:229) i , j ( i i H ⊗ b i j K )( w i , j ) and (cid:0) ( p i H ⊗ b p j K )( w ) (cid:1) i , j ← [ w , respectively.Proof. i) If S , T are as above and H i = a i H i b i , K j = g j K j d j for i , j = ,
2, then ( S ⊗ b T ) | g i a = | T g i S a ⊆ | g i a and similarly ( S ⊗ b T ) | b i d ⊆ | b i d , ( S ⊗ b T ) ∗ | g i a ⊆ | g i a , ( S ⊗ b T ) ∗ | b i d ⊆ | b i d .ii) Straightforward.iii) r a , b ( H ) · ( a ⊳ B † ) = [ r b ( B † ) a ] = a and r a , b ( H ) · ( b ⊲ B ) = [ b B ] = b . For l b , c ( K ) , thearguments are similar.iv) Straightforward. Remark 2.14.
The relative tensor product of modules and morphisms can be considered asthe composition in a bicategory as follows. Recall that a bicategory B consists of a class ofobjects ob B , a category B ( A , B ) for each A , B ∈ ob B whose objects and morphisms are called and , respectively, a functor c A , B , C : B ( B , C ) × B ( A , B ) → B ( A , C ) (“composition”)for each A , B , C ∈ ob B , an object 1 A ∈ B ( A , A ) (“identity”) for each A ∈ ob B , an isomorphism a A , B , C , D ( f , g , h ) : c A , B , D ( c B , C , D ( h , g ) , f ) → c A , C , D ( h , c A , B , C ( g , f )) in B ( A , D ) (“associativity”) foreach triple of 1-cells A f −→ B g −→ C h −→ D in B , and isomorphisms l A ( f ) : c A , A , B ( f , A ) → f and10 B ( f ) : c A , B , B ( B , f ) → f in B ( A , B ) for each 1-cell A f −→ B in B , subject to several axioms [17].Tedious but straightforward calculations show that there exists a bicategory C ∗ - bimod such thati) the objects are all C ∗ -bases and C ∗ - bimod ( a , b ) is the category of all C ∗ - ( a † , b ) -moduleswith morphisms (not semi-morphisms) for all C ∗ -bases a , b ;ii) the functor c a , b , c is given by ( g K d , a H b ) a H b ⊗ b g K d and ( T , S ) S ⊗ b T , respectively, andthe identity 1 a is A † H A for all C ∗ -bases a , b , c , d ;iii) a , r , l are as in Proposition 2.13. C ∗ -algebras We now use the relative tensor product to construct a fiber product of C ∗ -algebras that are rep-resented on C ∗ -modules over C ∗ -bases. To motivate our approach, let us first review severalrelated constructions. In each case, the task is to construct a relative tensor product or “fiberproduct” of two algebras A and C with respect to a common subalgebra B .First, assume that we are working in the category of unital commutative rings. Then the fiberproduct is just the push-out of the diagram formed by A , B , C . Explicitly, it is the algebraictensor product A ⊙ B C , where A and C are considered as modules over B , and the multiplicationis defined componentwise. In the category of commutative C ∗ -algebras, the push-out is themaximal completion of the algebraic tensor product A ⊙ B C and, as usual in the setting of C ∗ -algebras, also other interesting completions exist [1]. For example, if B = C ( X ) for some locallycompact Hausdorff space and if A and C are represented on Hilbert spaces H and K , respectively,then H and K can be disintegrated over X with respect to some measure µ (see Subsection 2.1),and the algebra A ⊙ B C has a natural representation p on the relative tensor product H ⊗ µ K = R ⊕ X H x ⊗ K x dµ ( x ) , leading to a minimal completion p ( A ⊙ B C ) . In the setting of von Neumannalgebras, H and K are intrinsic, and the desired fiber product is p ( A ⊙ B C ) ′′ ⊆ L ( H ⊗ µ K ) . Notethat all of these constructions do not depend on commutativity of A and C and make sense aslong as B is central in A and in C .Next, consider the case where A , B , C are non-commutative, B is a subalgebra of A , and theopposite B op is a subalgebra of C . Then one can consider A and C as modules over B via rightmultiplication, and form the algebraic tensor product A ⊙ B C , but componentwise multiplicationis well defined only on the subspace A × B C ⊆ A ⊙ B C which consists of all elements (cid:229) i a i ⊙ c i satisfying (cid:229) i ba i ⊙ c i = (cid:229) i a i ⊙ b op c i for all b ∈ B . This subspace was first considered by Takeuchiand provides the right notion of a fiber product for the algebraic theory of quantum groupoids[2, 32]. In the setting of C ∗ -algebras, the Takeuchi product A × B C may be 0 even when we expecta nontrivial fiber product on the level of C ∗ -algebras; therefore, the latter can not be obtainedas the completion of the former. In the setting of von Neumann algebras, a fiber product can11e constructed as follows [21]. If A and C act on Hilbert spaces H and K , respectively, one canform the Connes fusion H ⊗ µ K with respect to some weight µ on B and the actions of B on H and B op on K which — by functoriality — carries a representation p : A ′ ⊙ C ′ → L ( H ⊗ µ K ) , andthe desired fiber product is A ∗ µ C = p ( A ′ ⊙ C ′ ) ′ . A categorical interpretation of this constructionis given in 4.3.We now modify the last construction to define a fiber product for C ∗ -algebras A and C as follows.(A) We assume that A and C are represented on a C ∗ - b -module H b and a C ∗ - b † -module K g ,respectively, where b = ( K , B , B † ) is a C ∗ -base, such that r b ( B ) and r g ( B † ) take theplaces of B and B op , respectively.(B) On the relative tensor product H b ⊗ b g K , we define C ∗ -algebras Ind | g i ( A ) and Ind | b i ( C ) which, roughly, take the places of p ( A ′ ⊙ id K ) ′ and p ( id H ⊙ C ′ ) ′ .(C) The fiber product is then the intersection A b ∗ b g B = Ind | g i ( A ) ∩ Ind | b i ( C ) ⊆ L ( H b ⊗ b g K ) . C ∗ -algebras represented on C ∗ -modules Let b = ( K , B , B † ) be a C ∗ -base. As indicated in step (A), we adopt the following terminology. Definition 3.1.
A C ∗ - B † -algebra ( A , r ) , briefly written A r , is a C ∗ -algebra A with a ∗ -homo-morphism r : B † → M ( A ) . A morphism of C ∗ - B † -algebras A r and B s is a ∗ -homomorphism p : A → B satisfying s ( x ) p ( a ) = p ( r ( x ) a ) for all x ∈ B † , a ∈ A. We denote the category of allC ∗ - B † -algebras by C ∗ B † .A (nondegenerate) C ∗ - b -algebra is a pair A a H = ( H a , A ) , where H a is a C ∗ - b -module, A ⊆ L ( H ) a (nondegenerate) C ∗ -algebra, and r a ( B † ) A ⊆ A. A (semi-)morphism between C ∗ - b -algebrasA a H , B b K is a ∗ -homomorphism p : A → B satisfying b = [ L p ( s ) ( H a , K b ) a ] , where L p ( s ) ( H a , K b ) : = { T ∈ L ( s ) ( H a , K b ) | ∀ a ∈ A : Ta = p ( a ) T } . We denote the category of all C ∗ - b -algebras togetherwith all (semi-)morphisms by C ∗ b ( s ) . We first give some examples of C ∗ - b -algebras and then study the relation between C ∗ B † and C ∗ b . Examples 3.2. i) If H is a Hilbert space and A ⊆ L ( H ) a C ∗ -algebra, then A a H is a C ∗ - t -algebra, where t = ( C , C , C ) denotes the trivial C ∗ -base and a = L ( C , H ) .ii) Let A a H be a nondegenerate C ∗ - b -algebra. If we identify M ( A ) with a C ∗ -subalgebra of L ( H ) in the canonical way, M ( A ) a H becomes a C ∗ - b -algebra.iii) Let ( A i ) i be a family of C ∗ - b -algebras, where A i = ( H i , A i ) for each i . Then the c -sum L i A i and the l ¥ -product (cid:213) i A i are naturally represented on the underlying Hilbert spaceof ⊞ i H i , and we obtain C ∗ - b -algebras ⊞ i A i : = (cid:0) ⊞ i H i , L i A i (cid:1) and (cid:213) i A i : = (cid:0) ⊞ i H i , (cid:213) i A i (cid:1) .For each j , the canonical maps A j → L i A i → (cid:213) i A i → A j are evidently morphisms of C ∗ - b -algebras A j → ⊞ i A i → (cid:213) i A i → A j .The following example is a continuation of Example 2.10.12 xample 3.3. Let B be a C ∗ -algebra with a KMS-state µ and associated C ∗ -base b , and let A bea C ∗ -algebra containing B with a conditional expectation f : A → B as in Example 2.10. Withthe notation introduced before, p n ( A ) b H is a nondegenerate C ∗ - b -algebra because r b ( B ) p n ( A ) = p n ( B ) p n ( A ) ⊆ p n ( A ) , and similarly, ( p op n ( A op )) a H is a nondegenerate C ∗ - b † -algebra [25, §2–3].The categories C ∗ s b and C ∗ B † are related by a pair of adjoint functors, as we shall see now. Lemma 3.4.
Let p be a semi-morphism of C ∗ - b -algebras A a H and B b K . Then p is normal and p ( a r a ( x )) = p ( a ) r b ( x ) for all x ∈ B † , a ∈ A.Proof.
Let T , T ′ ∈ L p s ( H a , K b ) , x , x ′ ∈ a , z , z ′ ∈ K , a ∈ A , x ∈ B † . Then h T xz | p ( a ) T ′ x ′ z ′ i = h xz | aT ∗ T ′ x ′ z ′ i and p ( a r a ( x )) T xz = Ta r a ( x ) xz = p ( a ) T x x z = p ( a ) r b ( x ) T xz because T x ∈ b .Now, the assertions follow since K = [ L p s ( H a , K b ) a K ] .The preceding lemma shows that there exists a forgetful functor U b : C ∗ s b → C ∗ B † , ( A a H A r a for each object A a H , p p for each morphism p . We shall see that this functor has a partial adjoint that associates to a C ∗ - B † -algebra a universalrepresentation on a C ∗ - b -module. For the discussion, we fix a C ∗ - B † -algebra C s . Definition 3.5. A representation of C s in C ∗ s b is a pair ( A , f ) , where A = A a H ∈ C ∗ s b and f ∈ C ∗ B † ( C s , U A ) . Denote by Rep b ( C s ) the category of all such representations, where the mor-phisms between objects ( A , f ) and ( B , y ) are all p ∈ C ∗ s b ( A , B ) satisfying y = U p ◦ f . Note that
Rep b ( C s ) is just the comma category ( C s ↓ U b ) [19]. Unfortunately, we have no gen-eral method like the GNS-construction to produce representations of C s in in C ∗ s b . In particular,we have no good criteria to decide whether there are any and, if so, whether there exists a faithfulone. However, we now show that if there are any representations, then there also is a universalone. The proof involves the following direct product construction. Example 3.6.
Let ( A i , f i ) ∈ Rep b ( C s ) for all i , where A i = ( H i , A i ) , and define f : C → (cid:213) i A i by c ( f i ( c )) i . Then (cid:213) i ( A i , f i ) : = ( (cid:213) i A i , f ) ∈ Rep b ( C s ) , and the canonical maps A j → (cid:213) i A i → A j are morphisms between ( A j , f j ) and ( (cid:213) i A i , f ) for each j . Proposition 3.7.
If the category
Rep b ( C s ) is non-empty, then it has an initial object.Proof. Assume that
Rep b ( C s ) is non-empty. We first use a cardinality argument to show that Rep b ( C s ) has an initial set of objects, and then apply the direct product construction to this setto obtain an initial object.Given a topological vector space X and a cardinal number c , let us call X c-separable if X hasa linearly dense subset of cardinality c . Choose a cardinal number d such that B and C × K are d -separable, and let e : = | N | (cid:229) n d n . Then the isomorphism classes of e -separable Hilbert C ∗ - B -modules form a set, and hence there exists a set R of objects in Rep b ( C s ) such that each ( A a H , f ) ∈ Rep b ( C s ) with e -separable a is isomorphic to some element of R . Let ( A a H , f ) = ⊞ R ∈ R R . We show that ( f ( C ) a H , f ) is initial in Rep b ( C s ) .13et ( B b K , y ) ∈ Rep b ( C s ) . We show that there exists a morphism p ∈ C ∗ s b ( f ( C ) a H , B b K ) such that y = p ◦ f , and uniqueness of such a p is evident. Let x ∈ b be given. Since B and C × K are d -separable, we can inductively choose subspaces b ⊆ b ⊆ · · · ⊆ b and cardinal numbers d , d , . . . such that x ∈ b , [ b ∗ b ] = B , d ≤ d + b is d -separable and for all n ≥ b n B ⊆ b n + , y ( C ) b n K ⊆ [ b n + K ] , d n + ≤ | N | dd n , b n + is d n + -separable . Let ˜ b : = [ S n b n ] ⊆ b and ˜ K : = [ ˜ b K ] ⊆ K . By construction, [ ˜ b ∗ ˜ b ] = B , ˜ b B ⊆ ˜ b , y ( C ) ˜ K ⊆ ˜ K ,so that ( y ( C ) | ˜ K ) ˜ b ˜ K is in C ∗ b . Define ˜ y : C → y ( C ) | ˜ K by c y ( c ) | ˜ K . Then ( ˜ y ( C ) ˜ b ˜ K , ˜ y ) is in Rep b ( C s ) . Since ˜ b is e -separable, ( ˜ y ( C ) ˜ b ˜ K , ˜ y ) is isomorphic to some element of R . Hence, thereexists a surjection ˜ T : H → ˜ K such that ˜ T a = ˜ b , and the composition with the inclusion ˜ K → K gives an operator T ∈ L s ( H a , K b ) such that y ( c ) T = T f ( c ) for all c ∈ C . Since x ∈ ˜ b = T a and x ∈ b was arbitrary, we can conclude the existence of p as desired.Evidently, every morphism F between C ∗ - B † -algebras C s and D t yields a functor F ∗ : Rep b ( D t ) → Rep b ( C s ) , ( ( A a H , f ) ( A a H , f ◦ F ) for each object ( A a H , f ) , p p for each morphism p . Denote by C ∗ r B † the full subcategory of C ∗ B † consisting of all objects C s for which Rep ( C s ) isnon-empty. Theorem 3.8.
There exist a functor R b : C ∗ r B † → C ∗ s b and natural transformations h : id C ∗ r B † → U b R b and e : R b U b → id C ∗ s b such that for every C s , D t ∈ C ∗ r B † , F ∈ C ∗ r B † ( C s , D t ) , A a H ∈ C ∗ s b , • R b ( C s ) ∈ Rep b ( C s ) is an initial object and R b ( F ) is the unique morphism from R b ( C s ) to F ∗ ( R b ( D t )) , • h C s = f if R b ( C s ) = ( B b K , f ) , and e A a H is the unique morphism from R b U b ( A a H ) to ( A a H , id A ) .Moreover, R b is left adjoint to U b and h , e are the unit and counit of the adjunction, respectively.Proof. This follows from Proposition 3.7 and [19, §IV Theorem 2].We next consider C ∗ -algebras represented on C ∗ -bimodules. Let a = ( H , A , A † ) be a C ∗ -base. Definition 3.9.
A C ∗ - ( A , B † ) -algebra is a triple ( A , r , s ) , briefly written A r , s , where A r is a C ∗ - A -algebra, A s a C ∗ - B † -algebra, and [ r ( A ) , s ( B † )] = . A morphism of C ∗ - ( A , B † ) -algebrasis a morphism of the underlying C ∗ - A -algebras and C ∗ - B † -algebras. We denote the category ofall C ∗ - ( A , B † ) -algebras by C ∗ ( A , B † ) .A (nondegenerate) C ∗ - ( a † , b ) -algebra is a pair A a , b H = ( a H b , A ) , where a H b is a C ∗ - ( a † , b ) -module, A a H a (nondegenerate) C ∗ - a † -algebra, and A b H a C ∗ - b -algebra. A (semi-)morphism ofC ∗ - ( a † , b ) -algebras A a , b H and B g , d K is a ∗ -homomorphism p : A → B satisfying g = [ L p ( s ) ( a H b , g K d ) a ] and d = [ L p ( s ) ( a H b , g K d ) b ] , where L p ( s ) ( a H b , g K d ) : = { T ∈ L ( s ) ( a H b , g K d ) | ∀ a ∈ A : Ta = p ( a ) T } .We denote the category of all C ∗ - ( a † , b ) -algebras together with all (semi-)morphisms by C ∗ ( s )( a † , b ) . emark 3.10. Note that the condition on a (semi-)morphism between C ∗ - ( a † , b ) -algebras aboveis stronger than just being a (semi-)morphism of the underlying C ∗ - a † -algebras and C ∗ - b -algebras.Examples 3.2 ii) and iii) naturally extend to C ∗ - ( a † , b ) -algebras, and the categories C ∗ ( A , B † ) and C ∗ s ( a † , b ) are again related by a pair of adjoint functors. Theorem 3.11.
There exists a functor U ( a † , b ) : C ∗ s ( a † , b ) → C ∗ ( A , B † ) , given by A a , b H A r a , r b onobjects and p p on morphisms. Denote by C ∗ r ( A , B † ) the full subcategory of C ∗ ( A , B † ) consist-ing of all objects C s , r for which the comma category ( C s , r ↓ U ( a † , b ) ) is non-empty. Then thecorestriction of U ( a † , b ) to C ∗ r ( A , B † ) has a left adjoint R ( a † , b ) : C ∗ r ( A , B † ) → C ∗ s ( a † , b ) .Proof. The proof proceeds as in the case of C ∗ - b -algebras with straightforward modifications, sowe only indicate the necessary changes for the second half of the proof of Proposition 3.7. Givena C ∗ - ( A , B † ) -algebra C s , t and a C ∗ - ( a † , b ) -algebra B g , d K with a morphism y : C s , t → B r g , r d , oneconstructs ˜ g ⊆ g and ˜ d ⊆ d for given x ∈ g , h ∈ d as follows. One first fixes a cardinal number d such that A , A † , H , B , B † , H are d -separable, and then inductively chooses cardinal numbers d , d , . . . and closed subspaces g ⊆ g ⊆ · · · ⊆ g and d ⊆ d ⊆ · · · ⊆ d such that x ∈ g , h ∈ d , [ g ∗ g ] = A † , [ d ∗ d ] = B , d ≤ d + , g , d are d -separable , r d ( B † ) g n + g n A † ⊆ g n + , r g ( A ) g n + d n B ⊆ d n + , y ( C ) g n H + y ( C ) d n K ⊆ [ g n + H ] ∩ [ d n + K ] , d n + ≤ | N | d d n , g n + , d n + are d n + -separablefor all n ≥
0, and finally lets ˜ g : = [ S n g n ] , ˜ d : = [ S n d n ] , ˜ K : = [ ˜ g H ] = [ ˜ d K ] . Remark 3.12.
Let C r , s be a C ∗ - ( A , B † ) -algebra, A a , b H = R ( a † , b ) ( C r , s ) , and f = h C r , s : C r , s → A r a , r b the morphism given by the unit of the adjunction above. Then ( A a , f ) ∈ Rep a † ( C r ) and ( A b , f ) ∈ Rep b ( C s ) , whence we have semi-morphisms R a † ( C s ) → A a H and R b ( C r ) → A b H . C ∗ -algebras represented on C ∗ -modules Our definition of the fiber product of C ∗ -algebras represented on C ∗ -modules — more precisely,step (B) in the introduction — involves the following construction.Let H and K be Hilbert spaces, I ⊆ L ( H , K ) a subspace and A ⊆ L ( H ) a C ∗ -algebra such that [ IH ] = K , [ I ∗ K ] = H , [ II ∗ I ] = I , I ∗ IA ⊆ A . We define a new C ∗ -algebraInd I ( A ) : = { T ∈ L ( K ) | T I + T ∗ I ⊆ [ IA ] } ⊆ L ( K ) . Definition 3.13.
The I -strong- ∗ , I -strong, and I -weak topology on L ( K ) are the topologiesinduced by the families of semi-norms T
7→ k T x k + k T ∗ x k ( x ∈ I), T
7→ k T x k ( x ∈ I ) , andT
7→ k x ∗ T x ′ k ( x , x ′ ∈ I ) , respectively. Given a subset X ⊆ L ( K ) , denote by [ X ] I the closure of span X with respect to the I-strong- ∗ topology. Evidently, the multiplication in L ( K ) is separately continuous with respect to the topologiesintroduced above, and the involution T T ∗ is continuous with respect to the I -strong- ∗ andthe I -weak topology. Define r I : ( I ∗ I ) ′ → L ( K ) as in Lemma 2.4.15 emma 3.14. i) [ I ∗ Ind I ( A ) I ] ⊆ A and
Ind I ( A ) = [ IAI ∗ ] I .ii) Ind I ( M ( A )) ⊆ M ( Ind I ( A )) .iii) Ind I ( A ) ⊆ L ( K ) is nondegenerate if and only if A ⊆ L ( H ) is nondegenerate.iv) If A ⊆ L ( H ) is nondegenerate, then A ′ ⊆ ( I ∗ I ) ′ and Ind I ( A ) ⊆ r I ( A ′ ) ′ .Proof. i) We have [ I ∗ Ind I ( A ) I ] ⊆ [ I ∗ IA ] ⊆ A by definition and [ IAI ∗ ] I ⊆ Ind I ( A ) because [ IAI ∗ ] I I ⊆ [ IAI ∗ I ] ⊆ [ IA ] . To see that [ IAI ∗ ] I ⊇ Ind I ( A ) , choose a bounded approximate unit ( u n ) n forthe C ∗ -algebra [ II ∗ ] and observe that for each T ∈ Ind I ( A ) , the net ( u n Tu n ) n lies in the space [ II ∗ Ind I ( A ) II ∗ ] ⊆ [ IAI ∗ ] and converges to T in the I -strong- ∗ topology because lim n T ( ∗ ) u n x = T ( ∗ ) x ∈ [ IA ] for all x ∈ I and lim n u n w = w for all w ∈ [ IA ] .ii) If S ∈ Ind I ( M ( A )) , T ∈ Ind I ( A ) , then ST ∈ Ind I ( A ) because ST I ⊆ [ SIA ] ⊆ [ IM ( A ) A ] = [ IA ] and T ∗ S ∗ I ⊆ [ T IM ( A )] ⊆ [ IAM ( A )] = [ IA ] .iii) If Ind I ( A ) ⊆ L ( K ) is nondegenerate, then [ AH ] ⊇ [ I ∗ Ind I ( A ) IH ] = [ I ∗ Ind I ( A ) K ] = [ I ∗ K ] = H . Conversely, if A is nondegenerate, then [ IAI ∗ ] and hence also Ind I ( A ) is nondegenerate.iv) Assume that A is nondegenerate. Then I ∗ I ⊆ M ( A ) ⊆ L ( H ) and hence A ′ ⊆ ( I ∗ I ) ′ . Forall x ∈ Ind I ( A ) , y ∈ A ′ , S , T ∈ I , we have S ∗ x r I ( y ) T = S ∗ xTy = yS ∗ xT = S ∗ r I ( y ) xT because S ∗ xT ∈ A , and since [ IH ] = K , we can conclude that x r I ( y ) = r I ( y ) x .Let b = ( K , B , B † ) be a C ∗ -base, A b H a C ∗ - b -algebra, and B g K a C ∗ - b † -algebra. We apply theconstruction above to A , B and | g i ⊆ L ( H , H b ⊗ b g K ) , | b i ⊆ L ( K , H b ⊗ b g K ) , respectively, anddefine the fiber product of A b H and B g K to be the C ∗ -algebra A b ∗ b g B : = Ind | g i ( A ) ∩ Ind | b i ( B )= { T ∈ L ( H b ⊗ b g K ) | T | g i + T ∗ | g i ⊆ [ | g i A ] , T | b i + T ∗ | b i ⊆ [ | b i B ] } . The spaces of operators involved are visualized as arrows in the following diagram: H A (cid:15) (cid:15) | g i / / H b ⊗ b g K A b ∗ b g B (cid:15) (cid:15) K | b i o o B (cid:15) (cid:15) H | g i / / H b ⊗ b g K K | b i o o Even in very special situations, it seems to be difficult to give a more explicit description of thefiber product. The main drawback of the definition above is that apart from special situations,we do not know how to produce elements of the fiber product.Let a = ( H , A , A † ) and c = ( L , C , C † ) be further C ∗ -bases. Proposition 3.15.
Let A = A a , b H be a C ∗ - ( a † , b ) -algebra and B = B g , d K a C ∗ - ( b † , c ) -algebra. Then A ∗ b B : = ( a H b ⊗ b g K d , A b ∗ b g B ) is a C ∗ - ( a † , c ) -algebra. roof. The product X : = r ( a ⊳ g ) ( A † )( A b ∗ b g B ) is contained in A b ∗ b g B because X | b i ⊆ [ | r a ( A ) b i B ] = [ | b i B ] , X ∗ | b i = ( A b ∗ b g B ) | r a ( A ) b i ⊆ [ | b i B ] , X | g i ⊆ [ | g i r a ( A ) A ] ⊆ [ | g i A ] , X ∗ | g i = ( A b ∗ b g B ) | g i r a ( A ) ⊆ [ | g i A ] by equation (3). A similar argument shows that r ( b ⊲ d ) ( C † )( A b ∗ b g B ) ⊆ A b ∗ b g B .In the situation above, we call A ∗ b B the fiber product of A and B . Forgetting a or d , we obtain a C ∗ - c -algebra A b ∗ b g B d : = A b H ∗ b B g , d H : = ( H b ⊗ b g K d , A b ∗ b g B ) and a C ∗ - a † -algebra a A b ∗ b g B = A a , b H ∗ b B g K .Denote by A ′ ⊆ L ( H ) and B ′ ⊆ L ( K ) the commutants of A and B , respectively, and let A ( b ) : = A ∩ L ( H b ) , B ( g ) : = B ∩ L ( K g ) , X : = ( A ( b ) ⊗ b id ) + ( id ⊗ b B ( g ) ) , M s ( A ( b ) ⊗ b B ( g ) ) : = { T ∈ L ( H b ⊗ b g K ) | T X , X T ⊆ A ( b ) ⊗ b B ( g ) } . Lemma 3.16. i) h b | ( A b ∗ b g B ) | b i ⊆ B, h g | ( A b ∗ b g B ) | g i ⊆ A, and M ( A ) b ∗ b g M ( B ) ⊆ M ( A b ∗ b g B ) .ii) A ( b ) ⊗ b B ( g ) ⊆ A b ∗ b g B.iii) If [ A ( b ) b ] = b and [ B ( g ) g ] = g , then A b ∗ b g B is nondegenerate and M s ( A ( b ) ⊗ b B ( g ) ) ⊆ A b ∗ b g B.iv) If r b ( B † ) ⊆ A, then id H ⊗ b B ( g ) ⊆ A b ∗ b g B. If r g ( B ) ⊆ B, then A ( b ) ⊗ b id K ⊆ A b ∗ b g B.v) id ( H b ⊗ b g K ) ∈ A b ∗ b g B if and only if r b ( B † ) ⊆ A and r g ( B ) ⊆ B.vi) If A a , b H is a C ∗ - ( a † , b ) -algebra and B g , d K a C ∗ - ( b † , c ) -algebra such that r a ( A ) + r b ( B † ) ⊆ Aand r g ( B ) + r d ( C † ) ⊆ B, then r ( a ⊳ g ) ( A ) + r ( b ⊲ d ) ( C † ) ⊆ A b ∗ b g B.vii) If A b ∗ b g B is nondegenerate, then the C ∗ -algebra [ b ∗ A b ] ∩ [ g ∗ B g ] ⊆ L ( K ) is nondegenerate.viii) If A and B are nondegenerate, then A ′ ⊆ r b ( B † ) ′ , B ′ ⊆ r g ( B ) ′ , and A b ∗ b g B ⊆ r | g i ( A ′ ) ∩ r | b i ( B ′ ) = ( A ′ ⊗ b id K ) ′ ∩ ( id H ⊗ b B ′ ) ′ .Proof. i) Immediate from Lemma 3.14.ii) Use ( A ( b ) ⊗ b B ( g ) ) | b i ⊆ [ | A ( b ) b i B ( g ) ] ⊆ [ | b i B ] , ( A ( b ) ⊗ b B ( g ) ) | g i ⊆ [ | B ( g ) g i A ( b ) ] ⊆ [ | g i A ] .iii) Assume [ A ( b ) b ] = b and [ B ( g ) g ] = g . Then A ( b ) ⊗ b B ( g ) ⊆ A b ∗ b g B is nondegenerate and foreach T ∈ M s ( A ( b ) ⊗ b B ( g ) ) , we have T | b i ⊆ [ T ( A ( b ) ⊗ b id ) | b i ] ⊆ [( A ( b ) ⊗ b B ( g ) ) | b i ] ⊆ [ | b i B ] and similarly T ∗ | b i ⊆ [ | b i B ] , T | g i + T ∗ | g i ⊆ [ | g i A ] .iv) If r g ( B ) ⊆ B , then ( A ( b ) ⊗ b id K ) | g i = | g i A ( b ) and [( A ( b ) ⊗ b id K ) | b i ] ⊆ | b i = [ | b B i ] =[ | b i r g ( B )] ⊆ [ | b i B ] . The second assertion follows similarly.17) If id ( H b ⊗ b g K ) ∈ A b ∗ b g B , then r b ( B † ) = [ h g | | g i ] ⊆ A , r g ( B ) = [ h b | | b i ] ⊆ B by i). Conversely,if the last two inclusions hold, then | g i = [ | g B † i ] = [ | g i r b ( B † )] ⊆ [ | g i A ] and similarly | b i ⊆ [ | b i B ] , whence id ( H b ⊗ b g K ) ∈ A b ∗ b g B .vi) Immediate from iv).vii) The C ∗ -algebra C : = [ b ∗ A b ] ∩ [ g ∗ B g ] contains b ∗ h g | ( A b ∗ b g B ) | g i b = g ∗ h b | ( A b ∗ b g B ) | b i g . If A b ∗ b g B is nondegenerate, we therefore must have [ C K ] ⊇ [ b ∗ h g | ( A b ∗ b g B )( H b ⊗ b g K )] = K .viii) Immediate from Lemma 3.14.Even in the case of a trivial C ∗ -base, we have no explicit description of the fiber product. Examples 3.17.
Let H and K be Hilbert spaces, b = L ( C , H ) , g = L ( C , K ) , b = t the trivial C ∗ -base ( C , C , C ) , and identify H b ⊗ b g K with H ⊗ K as in Example 2.11.i) Let A ⊆ L ( H ) and B ⊆ L ( K ) be nondegenerate C ∗ -algebras. Then A ( b ) = A , B ( g ) = B ,and by Lemma 3.16, A b ∗ b g B contains the minimal tensor product A ⊗ B ⊆ L ( H ⊗ K ) and M s ( A ⊗ B ) = { T ∈ L ( H ⊗ K ) | T ( ∗ ) ( ⊗ B ) , T ( ∗ ) ( A ⊗ ) ⊆ A ⊗ B } . If A or B is non-unital,then id H ⊗ K A b ∗ b g B by Lemma 3.16 and so M ( A ⊗ B ) A b ∗ b g B . In Example 5.3 iii), weshall see that also A b ∗ b g B * M ( A ⊗ B ) is possible.ii) Assume that H = K = l ( N ) and identify b = g = L ( C , H ) with H . Then the flip S : H ⊗ H → H ⊗ H , x ⊗ h h ⊗ x , is not contained in L ( H ) b ∗ b g L ( H ) . Indeed, let ( x n ) n be anorthonormal basis for H and let h ∈ H be non-zero. Then h x n | S | h i = | h ih x n | for each n and hence (cid:229) n h x n | S | h i does not converge in norm. On the other hand, one easily verifiesthat (cid:229) n h x n | S converges in norm for each S ∈ [ | H i L ( H )] . Hence, S | h i [ | H i L ( H )] . We first show that the fiber product constructed above is functorial, and then consider variousslice maps. The results concerning functoriality were stated in slightly different form in [25,28, 29] with proofs referring to unpublished material. We use the opportunity to rectify thissituation. As before, let a = ( H , A , A † ) , b = ( K , B , B † ) , c = ( L , C , C † ) be C ∗ -bases. Lemma 3.18.
Let p be a (semi-)morphism of C ∗ - b -algebras A b H and C l L , let g K d be a C ∗ - ( b † , c ) -module, and let I : = L p ( s ) ( H b , L l ) ⊗ b id ⊆ L ( H b ⊗ b g K , L l ⊗ b g K ) .i) X : = ( H b ⊗ b g K d , ( I ∗ I ) ′ ) and Y : = ( L l ⊗ b g K d , ( II ∗ ) ′ ) are nondegenerate C ∗ - c -algebras.ii) There exists a unique r I ∈ Mor ( s ) ( X , Y ) such that r I ( x ) S = Sx for all x ∈ ( I ∗ I ) ′ , S ∈ I.iii) There exists a unique linear contraction j p : [ | g i A ] → [ | g i C ] given by | h i a
7→ | h i p ( a ) .iv) Ind | g i ( A ) ⊆ ( I ∗ I ) ′ and r I ( x ) | h i = j p ( x | h i ) for all x ∈ Ind | g i ( A ) , h ∈ g . ) Let B g K be a C ∗ - b † -algebra. Then A b ∗ b g B ⊆ ( I ∗ I ) ′ and r I ( A b ∗ b g B ) ⊆ C l ∗ b g B.Proof. i) Clearly, ( I ∗ I ) ′ and ( II ∗ ) ′ are nondegenerate C ∗ -algebras, and X and Y are C ∗ - c -algebras because r ( b ⊲ d ) ( C † ) = id b ⊗ b g r d ( C † ) ⊆ ( I ∗ I ) ′ and r ( l ⊲ d ) ( C † ) = id l ⊗ b g r d ( C † ) ⊆ ( II ∗ ) ′ .ii) There exists a unique ∗ -homomorphism r I : ( I ∗ I ) ′ → ( II ∗ ) ′ satisfying the formula above byLemma 2.4, and this is a (semi-)morphism because [ I ( b ⊲ d )] = [ l ⊲ d ] by assumption on p .iii) Let h , . . . , h n ∈ g and a , . . . , a n ∈ A . Then k (cid:229) j | h j i p ( a j ) k = k (cid:229) i , j p ( a ∗ i ) r l ( h ∗ i h j ) p ( a j ) k ≤k (cid:229) i , j a ∗ i r b ( h ∗ i h j ) a j k = k (cid:229) j | h j i a j k by Lemma 3.4. The claim follows.iv) The first assertion follows from Lemma 3.14 and the relation I ∗ I ⊆ A ′ ⊗ b id = r | g i ( A ′ ) , and thesecond one from the fact that for all x ∈ Ind | g i ( A ) , h ∈ g , S ∈ L p ( s ) ( H b , L l ) , we have r I ( x ) | h i S = r I ( x )( S ⊗ b id ) | h i = ( S ⊗ b id ) x | h i = j p ( x | h i ) S .v) First, A b ∗ b g B ⊆ ( I ∗ I ) ′ by Lemma 3.16. The second assertion follows from the relations r I ( A b ∗ b g B ) | g i ⊆ r I ( Ind | g i ( A )) | g i ⊆ j p ([ | g i A ]) = [ | g i C ] , r I ( A b ∗ b g B ) | l i = r I ( A b ∗ b g B )[ I | b i ] ⊆ [ I ( A b ∗ b g B ) | b i ] ⊆ [ I | b i B ] = [ | l i B ] . Theorem 3.19.
Let f be a (semi-)morphism of C ∗ - ( a , b ) -algebras A = A a , b H and C = C k , l L , and y a (semi-)morphism of C ∗ - ( b † , c ) -algebras B = B g , d K and D = D µ , n M . Then there exists a unique(semi-)morphism of C ∗ - ( a , c ) -algebras f ∗ y from A ∗ b B to C ∗ b D such that ( f ∗ y )( x ) R = Rx for all x ∈ A b ∗ b g B and R ∈ I M J H + J L I K , where I X = L f ( s ) ( H b , L l ) ⊗ b id X and J Y = id Y ⊗ b L y ( s ) ( K g , M µ ) for X ∈ { K , M } , Y ∈ { H , L } .Proof. By Lemma 3.18, we can define f ∗ y to be the restriction of r I M ◦ r J H or of r J L ◦ r I K to A b ∗ b g B . Uniqueness follows from the fact that [ I M J H ( H b ⊗ b g K )] = [ J L I K ( H b ⊗ b g K )] = L l ⊗ b µ M . Remark 3.20.
Let A b H , C l L be C ∗ - b -algebras, B g K , D µM C ∗ - b † -algebras, and f ∈ Mor ( A b H , M ( C ) l L ) , y ∈ Mor ( B g K , M ( D ) µM ) such that [ f ( A ) C ] = C , [ y ( B ) D ] = D . Then there exists a ∗ -homomorphism f ∗ b y : A b ∗ b g B → M ( C ) l ∗ b µ M ( D ) ֒ → M ( C l ∗ b µ D ) , but in general, we do not know whether this isnondegenerate.Next, we briefly discuss two kinds of slice maps on fiber products. For applications and furtherdetails, see [29]. The first class of slice maps arises from a completely positive map on onefactor and takes values in operators on a certain KSGNS-construction, that is, an internal tensorproduct with respect to a completely positive linear map [16, §4–§5]. Proposition 3.21.
Let A b H be a C ∗ - b -algebra, K g a C ∗ - b † -module, L a Hilbert space, f : [ A + r b ( B † )] → L ( L ) a c.p. map, and q = f ◦ r b : B † → L ( L ) . Then there exists a unique c.p. map f ∗ id : Ind | g i ( A ) → L ( L q < g ) such that for all z , z ′ ∈ L , h , h ′ ∈ g , x ∈ Ind | g i ( A ) , h z < h | ( f ∗ id )( x )( z ′ < h ′ ) i = h z | f ( h h | x | h ′ i ) z ′ i . (4) If B g K is a C ∗ - b † -algebra, then ( f ∗ id )( A b ∗ b g B ) ⊆ ( f ( A ) ′ q < ( B ′ ∩ L ( K g )) ′ ⊆ L ( L q < g ) . roof. Let x = ( x i j ) i , j ∈ M n ( Ind | g i ( A )) be positive, z , . . . , z n ∈ L , h , . . . , h n ∈ g , where n ∈ N ,and d = diag ( | h i , . . . , | h n i ) . Then 0 ≤ ( h h i | x i j | h j i ) i , j = d ∗ xd ≤ k x k d ∗ d and hence 0 ≤ ( f ( h h i | x i j | h j i )) i , j ≤ k x k f ( d ∗ d ) and0 ≤ (cid:229) i , j h z i | f ( h h i | x i j | h j i ) z j i ≤ k x k (cid:229) i , j h z i < h i | z j < h j i . Hence, there exists a map f ∗ id as claimed. The verification of the assertion concerning B g K isstraightforward. Remark 3.22. If C l L is a C ∗ - b † -algebra and f | A is a semi-morphism of C ∗ - b † -algebras, then themap f ∗ id extends the fiber product f ∗ id defined in Theorem 3.19.Second, we show that the fiber product is functorial with respect to the following class of maps.A spatially implemented map of C ∗ - b -algebras A b H and C l L is a map f : A → C admitting se-quences ( S n ) n and ( T n ) n in L ( L l , H b ) such thati) (cid:229) n S ∗ n S n and (cid:229) n T ∗ n T n converge in norm , ii) f ( a ) = (cid:229) n S ∗ n aT n for all a ∈ A . (5)Note that condition i) implies norm-convergence of the sum in ii). Evidently, such a map islinear, extends to a normal map ¯ f : A ′ → C ′ , its norm is bounded by k (cid:229) n S ∗ n S n k / k (cid:229) n T ∗ n T n k / ,and the composition of spatially implemented maps is spatially implemented again. Proposition 3.23.
Let f be a spatially implemented map of C ∗ - b -algebras A b H and C l L , and letB g , d K be a C ∗ - ( b † , c ) -algebra. Then there exists a spatially implemented map from A b H ∗ b B g , d K toC l H ∗ b B g , d K such that h h | ( f ∗ id )( x ) | h ′ i = f ( h h | x | h ′ i ) for all x ∈ A b ∗ b g B, h , h ′ ∈ g .Proof. Uniqueness is clear. Fix sequences ( S n ) n , ( T n ) n as in (5) and let ˜ S n : = S n ⊗ b id K , ˜ T n : = T n ⊗ b id K for all n . Then ˜ S n , ˜ T n ∈ L ( L l ⊗ b g K d , H b ⊗ b g K d ) for all n , we have k (cid:229) n ˜ S ∗ n ˜ S n k = k (cid:229) n S ∗ n S n k , k (cid:229) n ˜ T ∗ n ˜ T n k = k (cid:229) n T ∗ n T n k , and the map f ∗ id : A b ∗ b g B → L ( L l ⊗ b g K ) given by x (cid:229) n ˜ T ∗ n x ˜ S n hasthe desired properties. Indeed, let x ∈ A b ∗ b g B , h , h ′ ∈ g . Then ˜ S n | h i = | h i S n and ˜ T n | h ′ i = | h ′ i T n for all n , and hence h h | ( f ∗ id )( x ) | h ′ i = f ( h h | x | h ′ i ) . It remains to show that ( f ∗ id )( x ) ∈ C l ∗ b g B . Consider the expression ( f ∗ id )( x ) | h ′ i = (cid:229) n ˜ S ∗ n x | h ′ i T n . This sum converges innorm and each summand lies in [ | g i L ( H )] because x | h ′ i ∈ [ | g i A ] and [ ˜ S ∗ n | g i ] = [ | g i S ∗ n ] . Since h h ′′ | ( f ∗ id )( x ) | h ′ i ∈ C for each h ′′ ∈ g , we can conclude that the sum lies in [ | g i C ] . Finally,consider the expression ( f ∗ id )( x ) | x i = (cid:229) n ˜ S n x ˜ T n | x i , where x ∈ l . Again, the sum convergesin norm and each summand lies in [ | l i B ] because ˜ S ∗ n x ˜ T n | x i = ˜ S ∗ n x | T n x i ∈ ˜ S ∗ n ( A b ∗ b g B ) | b i ⊆ [ ˜ S ∗ n | b i B ] ⊆ [ | l i B ] . Remarks 3.24. i) The map f ∗ id constructed above is a “slice map” in the case where C l L = L ( K ) BK and S n , T n ∈ b ⊆ L ( K B , H b ) for all n . Then, we can identify C l ∗ b g B with a C ∗ -subalgebra of L ( K ) , and f ∗ id is just the map A b ∗ b g B → B given by x (cid:229) n h S n | X | T n i .20i) Assume that the extension ˜ f : [ A + r b ( B † )] → C given by x (cid:229) n S ∗ n xT n is completelypositive. Here, we use the notation of the proof above. Then the map ˜ f ∗ id constructedin Proposition 3.21 extends the map f ∗ id of Proposition 3.23 because then q = r l andhence h h | ( ˜ f ∗ id )( x ) | h ′ i = ˜ f ( h h | x | h ′ i ) for all x ∈ A b ∗ b g B and h , h ′ ∈ g .Of course, slice maps of the form id ∗ f can be constructed in a similar way. The fiber product of C ∗ -algebras is neither associative, unital, nor compatible with infinite sums. Non-associativity
Let A = A a , b H be a C ∗ - ( a † , b ) -algebra, B = B g , d K a C ∗ - ( b † , c ) -algebra, and C = C e , f L a C ∗ - ( c † , d ) -algebra. Then we can form the fiber products ( A ∗ b B ) ∗ c C and A ∗ b ( B ∗ c C ) .The following example shows that these C ∗ -algebras need not be identified by the canonicalisomorphism a a , b , c , d ( e L f , g K d , a H b ) of Proposition 2.13. A similar phenomenon occurs in thepurely algebraic setting with the Takeuchi × R -product [24]. Example 3.25.
Let a = b = c = d be the trivial C ∗ -base, H = l ( N ) , a = L ( C , H ) , A = B = C = L ( H ) a , a H . Identify H a ⊗ b a K a ⊗ c a L ∼ = a ⊗ H ⊗ a with H ⊗ H ⊗ H via | x i = z < | h i ≡ x ⊗ z ⊗ h , fixan orthonormal basis ( e n ) n ∈ N of H , and define T ∈ L ( H ⊗ ) by T ( e k ⊗ e l ⊗ e m ) = ( e k ⊗ e l ⊗ e m for all k , l , m ∈ N s.t. m ≤ k + l , e l ⊗ e k ⊗ e m for all k , l , m ∈ N s.t. m > k + l . We show that T belongs to the underlying C ∗ -algebra of ( A ∗ b B ) ∗ c C , but not of A ∗ b ( B ∗ c C ) .For each x ∈ H and w ∈ H ⊗ , define | x i , | x i ∈ L ( H ⊗ , H ⊗ ) and | w i ∈ L ( H , H ⊗ ) by u x ⊗ u , u u ⊗ x , and z w ⊗ z , respectively. Then for all k , l , m ∈ N , T | e k ⊗ e l i = | e k ⊗ e l i P l + k + | e l ⊗ e k i ( id − P l + k ) , where P l + k : = (cid:229) m ≤ k + l | e m ih e m | , T | e m i = | e m i ( id + S m ) , where S m : = (cid:229) k , lk + l < m | e l ⊗ e k − e k ⊗ e l ih e k ⊗ e l | , and therefore, T | H ⊗ i ∈ [ | H ⊗ i L ( H )] , T | a i ∈ [ | a i ( id + K ( H ) ⊗ K ( H ))] ⊆ [ | a i ( L ( H ) a ∗ b a L ( H ))] . Since T = T ∗ , we can conclude that T belongs to ( L ( H ) a ∗ b a L ( H ) a ) ∗ b a L ( H ) . However, T | e i = | e i Q + (cid:229) l | e l i Q l , where Q = (cid:229) m ≤ l | e l ⊗ e m ih e l ⊗ e m | and Q l = (cid:229) m > l | e ⊗ e m ih e l ⊗ e m | , | e i Q ∈ [ | a i L ( H ⊗ H )] , but (cid:229) l | e l i Q l [ | a i L ( H ⊗ H )] because the sum (cid:229) l Q ∗ l Q l = (cid:229) l (cid:229) m > l | e l ⊗ e m ih e l ⊗ e m | does not converge in norm. Hence, T | e i [ | a i L ( H ⊗ H )] and T L ( H ) a ∗ b ( a L ( H ) a ∗ b a L ( H )) . Unitality
A unit for the fiber product relative to b would be a C ∗ - ( b † , b ) -algebra U = U B † , BK such that for all C ∗ - ( a † , b ) -algebras A = A a , b H and all C ∗ - ( b † , c ) -algebras B = B g , d K , we have A = Ad r ( A ∗ b U ) and B = Ad l ( U ∗ b B ) , where r = r a , b ( a H b ) and l = l b , c ( g K d ) (see Proposition2.13). The relations r | b i = b , r | B † i = r b ( B † ) , l | g i = g , l | B i = r g ( B ) implyAd r ( A b ∗ b B † U ) = Ind b ( U ) ∩ Ind r b ( B † ) ( A ) , Ad l ( U B ∗ b g B ) = Ind r g ( B ) ( B ) ∩ Ind g ( U ) . (6)If B † and B are unital, then Ind r b ( B † ) ( A ) = A and Ind r g ( B ) ( B ) = B , and then the C ∗ - ( b † , b ) -algebra L ( K ) B † , BK is a unit for the fiber product on the full subcategories of all A a , b H and B g , d K satisfying A ⊆ Ind b ( L ( K )) and B ⊆ Ind g ( L ( K )) . Remarks 3.26. i) If A ⊆ Ind a ( L ( H )) and B ⊆ Ind g ( L ( L )) , then A b ∗ b g B ⊆ Ind ( a ⊳ g ) ( L ( H )) ∩ Ind ( b ⊲ d ) ( L ( K )) .ii) Ind b ( B † ) = L ( H b ) , and if B † is unital, then Ad r ( A b ∗ b B † B † ) = A ∩ L ( H b ) = A ( b ) .iii) Ad r ( B B ∗ bB † B † ) = L ( K B ) ∩ L ( K B † ) = M ( B ) ∩ M ( B † ) . Compatibility with sums and products
The fiber product is compatible with finite sumsin the following sense. Let ( A i ) i be a finite family of C ∗ - ( a † , b ) -algebras and ( B j ) j a finitefamily of C ∗ - ( b † , c ) -algebras. For each i , j , denote by i i A : A i → ⊞ i ′ A i ′ , i j B : B j → ⊞ j ′ B j ′ and p i A : ⊞ i ′ A i ′ → A i , p j B : ⊞ j ′ B j ′ → B j the canonical inclusions and projections, respectively. Oneeasily verifies that there exist inverse isomorphisms ⊞ i , j A i ∗ b B j ⇆ ( ⊞ i A i ) ∗ b ( ⊞ j B j ) , given by ( x i , j ) i , j (cid:229) i , j ( i i A ∗ b i j B )( x i , j ) and (cid:0) ( p i A ∗ b p j B )( y ) (cid:1) i , j ← [ y , respectively. However, the fiber productis neither compatible with infinite sums nor infinite products: Examples 3.27.
Let t = ( C , C , C ) be the trivial C ∗ -base.i) For each i , j ∈ N , let A i and B j be the C ∗ - t -algebra C CC . Identify the Hilbert space L i , j C C ⊗ t C C with l ( N × N ) in the canonical way. Then L i , j A i ∗ t B j corresponds to C ( N × N ) , represented on l ( N × N ) by multiplication operators, but ( L i A i ) ∗ t ( L j B j ) ∼ = C ( N ) ∗ t C ( N ) is strictly larger and contains, for example, the characteristic function of thediagonal { ( x , x ) | x ∈ N } (see Example 5.3).22i) Let H = l ( N ) , a = L ( C , H ) , and let A and B j be the C ∗ - t -algebra K ( H ) a H for all j .Identify H a ⊗ t a H with H ⊗ H as in Example 2.11 i), choose an orthonormal basis ( e k ) k ∈ N of H , and put y j : = | e j ⊗ e ih e ⊗ e | ∈ K ( H ⊗ H ) for each j ∈ N . Then y : = ( y j ) j ∈ (cid:213) j A ∗ t B j because y j ∈ K ( H ) ⊗ K ( H ) ⊂ A ∗ t B j for all j ∈ N , but with respect to thecanonical identification L j H ⊗ H ∼ = H ⊗ (cid:0) L j ⊗ H (cid:1) , we have y A ∗ t ( (cid:213) j B j ) because y | e i corresponds to the family ( | e j i | e ih e | ) j ∈ (cid:213) j L ( H , H ⊗ H ) ⊆ L ( L j H , L j H ⊗ H ) which is not contained in the space [ | a i L ( L j H )] . C ∗ -algebras The spatial fiber product of C ∗ -algebras represented on C ∗ -modules yields a fiber product ofnon-represented C ∗ -algebras as follows.Let b = ( K , B , B † ) be a C ∗ -base. In Subsection 3.2, we constructed a functor R b : C ∗ r B † → C ∗ s b that associates to each C ∗ - B † -algebra a universal representation in form of a C ∗ - b -algebra.Replacing b by b † , we obtain a functor R b † : C ∗ r B → C ∗ s b , and composition of these with thespatial fiber product gives a fiber product of non-represented C ∗ -algebras in form of a functor C ∗ r B † × C ∗ r B R b × R b † −−−−−→ C ∗ s b × C ∗ s b † → C ∗ , ( C s , D t ) R b ( C s ) ∗ b R b † ( D t ) , where C ∗ denotes the category of C ∗ -algebras and ∗ -homomorphisms. In categorical terms, thisis the right Kan extension of the spatial fiber product on C ∗ s b × C ∗ s b † along the product of theforgetful functors U b × U b † : C ∗ s b × C ∗ s b † → C ∗ r B † × C ∗ r B [19, §X].Given further C ∗ -bases a = ( H , A , A † ) and c = ( L , C , C † ) , we similarly obtain a functor C ∗ r ( A , B † ) × C ∗ r ( B , C † ) R ( a † , b ) × R ( b † , c ) −−−−−−−−→ C ∗ s ( a † , b ) × C ∗ s ( b † , c ) → C ∗ s ( a † , c ) U ( a † , c ) −−−→ C ∗ r ( A , C † ) , and, using Remark 3.12, a natural transformation between the compositions in the square C ∗ r ( A , B † ) × C ∗ r ( B , C † ) / / (cid:15) (cid:15) C ∗ r ( A , C † ) (cid:15) (cid:15) q y llllllllllllllllll C ∗ r B † × C ∗ r B / / C ∗ , , where the vertical maps are the forgetful functors. Throughout this section, let N be a von Neumann algebra with a n.s.f. weight µ , denote by N µ , H µ , p µ , J µ the usual objects of Tomita-Takesaki theory [23], and define the antirepresentation p opµ : N → L ( H µ ) by x J µ p µ ( x ∗ ) J µ . 23 .1 Adaptation to von Neumann algebras The definitions and constructions presented in Sections 2 and 3 can be adapted to a varietyof other settings. We now briefly explain what happens when we pass to the setting of vonNeumann algebras. Instead of a C ∗ -base, we start with the triple b = ( K , B , B † ) , where K = H µ , B = p µ ( N ) , and B † = J µ p µ ( N ) J µ . Next, we define W ∗ - b -modules, W ∗ - ( b † , b ) -modules, theirrelative tensor product, W ∗ - b -algebras, and the fiber product by just replacing the norm closure [ · ] by the closure with respect to the weak operator topology [ · ] w everywhere in Sections 2 and3. We then recover Connes’ fusion of Hilbert bimodules over N and Sauvageot’s fiber product: Modules
Let H be some Hilbert space. If ( H , r ) is a right N -module, then the space a = L (( K , p opµ ) , ( H , r )) : = { T ∈ L ( K , H ) : T p opµ ( x ) = r ( x ) T for all x ∈ N } satisfies [ a K ] = H , [ a ∗ a ] w = B , a B ⊆ a , and r a ◦ p opµ (see Lemma 2.4) coincides with r . Conversely, if a ⊆ L ( K , H ) is a weakly closed subspace satisfying the three precedingequations, then ( H , r a ◦ p opµ ) is a right N -module and a = L (( K , p opµ ) , ( H , r a ◦ p opµ )) [22].We thus obtain a bijective correspondence between right N -modules and W ∗ - b -modules.This correspondence is an isomorphism of categories since for every other right N -module ( K , s ) , an operator T ∈ L ( H , K ) intertwines r and s if and only if T a is contained in b : = L (( K , p opµ ) , ( K , s )) . For W ∗ - b -modules, the notions of morphisms and semi-morphismscoincide. Algebras
Let H , r , a be as above and let A ⊆ L ( H ) be a von Neumann algebra. Then r ( N ) ⊆ A if and only if r a ( B ) A ⊆ A . Thus, W ∗ - b -algebras correspond with von Neumann alge-bras equipped with a normal unital embedding of N . Moreover, let K , s , b be as above,let B ⊆ L ( K ) be a von Neumann algebra, assume r ( N ) ⊆ A and s ( N ) ⊆ B , and let p : A → B be a ∗ -homomorphism satisfying p ◦ r = s . Then p is normal if and only if [ L p ( H a , K b ) a ] w = b . Indeed, the “if” part is straightforward (see Lemma 3.4), and the“only if” part follows easily from the fact that every normal ∗ -homomorphism is the com-position of an amplification, reduction, and unitary transformation [5, §4.4]. Bimodules
Let ( H , r ) be a left N -module, ( H , s ) a right N -module, a = L (( K , p µ ) , ( H , r )) and b = L (( K , p opµ ) , ( H , s )) . Then ( H , r , s ) is an N -bimodule if and only if r ( N ) b = b and s ( N ) a = a , and thus we obtain an isomorphism between the category of N -bimodulesand the category of W ∗ - ( b † , b ) -modules. Fusion
The preceding considerations and formula (1) show that the relative tensor product of W ∗ - ( b † , b ) -modules corresponds to Connes’ fusion of N -bimodules. Fiber product
Let ( H , r ) be a right N -module, ( K , s ) a left N -module, a = L (( K , p opµ ) , ( H , r )) , b = L (( K , p µ ) , ( K , s )) , and let A ⊆ L ( H ) and B ⊆ L ( K ) be von Neumann algebras satis-fying r ( N ) ⊆ H and s ( N ) ⊆ K . One easily verifies the equivalence of the following con-ditions for each x ∈ L ( H b ⊗ b g K ) : i) x | a i ⊆ [ | a i B ] w , ii) h a | x | a i ⊆ B , iii) x ∈ ( id H ⊗ b B ′ ) ′ .Consequently, the fiber product of A and B , considered as a W ∗ - b -algebra and a W ∗ - b † -algebra, coincides with the fiber product ( id H ⊗ b B ′ ) ′ ∩ ( A ′ ⊗ b id K ) ′ = ( A ′ ⊗ b B ′ ) ′ of Sauvageot.24 .2 Relation to Connes’ fusion and Sauvageot’s fiber product Let b = ( K , B , B † ) be a C ∗ -base such that K = H µ , B ′′ = p µ ( N ) , ( B † ) ′′ = p opµ ( N ) = B ′ .Denote by C ∗ - mod ( b † , b ) the category of all C ∗ - ( b † , b ) -modules with all semi-morphisms, and by W ∗ - bimod ( N , N op ) the category of all N -bimodules, respectively. Lemmas 2.4 and 2.5 imply: Proposition 4.1.
There exists a faithful functor F : C ∗ - mod ( b † , b ) → W ∗ - bimod ( N , N op ) , given by a H b ( H , r a ◦ p µ , r b ◦ p opµ ) on objects and T T on morphisms.
The categories C ∗ - mod ( b † , b ) and W ∗ - bimod ( N , N op ) carry the structure of a monoidal category[19], and we now show that the functor F above is monoidal. Let H b be a C ∗ - b -module, K g a C ∗ - b † -module, and let r = r b ◦ p opµ , X = L (( K , p opµ ) , ( H , r )) , s = r g ◦ p µ , Y = L (( K , p µ ) , ( K , s )) . Given subspaces X ⊆ X and Y ⊆ Y , we define a sesquilinear form h · | · i on the algebraic tensorproduct X ⊙ K ⊙ Y such that for all x , x ′ ∈ X , z , z ′ ∈ K , h , h ′ ∈ Y , h x ⊙ z ⊙ h | x ′ ⊙ z ′ ⊙ h ′ i = h z | ( x ∗ x ′ )( h ∗ h ′ ) h ′ i = h z | ( h ∗ h ′ )( x ∗ x ′ ) h ′ i Denote by X = K < Y the Hilbert space obtained by forming the separated completion. Lemma 4.2.
Let X ⊆ X and Y ⊆ Y be subspaces satisfying [ X K ] = H and [ Y K ] = K. Thenthe natural map X = K < Y → X = K < Y is an isomorphism.Proof.
Injectivity is clear. The natural map X = K < Y → X = K < Y is surjective becauseboth spaces coincide with the separated completion of the algebraic tensor product H ⊙ Y withrespect to the sesquilinear inner form given by h w ⊙ h | w ′ ⊙ h ′ i = h w | r b ( h ∗ h ′ ) w ′ i , and a similarargument shows that the natural map X = K < Y → X = K < Y is surjective.We conclude that Connes’ original definition of the relative tensor product H r ⊗ µ s K via boundedvectors coincides with the algebraic one given in (1) and with the relative tensor product H b ⊗ b g K . Theorem 4.3.
There exists a natural isomorphism between the compositions in the square C ∗ - mod ( b † , b ) × C ∗ - mod ( b † , b ) −⊗ b − / / F × F (cid:15) (cid:15) C ∗ - mod ( b † , b ) n v ffffffffffffffffffffffffffffffffffff F (cid:15) (cid:15) W ∗ - bimod ( N , N op ) × W ∗ - bimod ( N , N op ) −⊗ µ − / / W ∗ - bimod ( N , N op ) , given for each object ( a H b , g K d ) ∈ C ∗ - mod ( b † , b ) × C ∗ - mod ( b † , b ) by the natural mapH b ⊗ b g K = b = K < g → X = K < Y = H r ⊗ µ s K . (7) With respect to this isomorphism, the functor F : C ∗ - mod ( b † , b ) → W ∗ - bimod ( N , N op ) is monoidal. roof. Lemma 4.2 implies that the map (7) is an isomorphism. Evidently, this map is naturalwith respect to a H b and g K d . The verification of the assertion concerning F is now tedious butstraightforward.Denote by C ∗ s , nd ( b † , b ) the category of all C ∗ - ( b † , b ) -algebras A a , b H satisfying r a ( B ) + r b ( B † ) ⊆ A together with all semi-morphisms, and by W ∗ ( N , N op ) the category of all von Neumann alge-bras A equipped with a normal, unital embedding and anti-embedding i ( op ) A : N → A such that [ i A ( N ) , i opA ( N )] =
0, together with all morphisms preserving these (anti-)embeddings. Lemma3.4 implies:
Proposition 4.4.
There exists a faithful functor G : C ∗ s , nd ( b † , b ) → W ∗ ( N , N op ) , given by ( a H b , A ) ( A ′′ , r a ◦ p µ , r b ◦ p opµ ) on objects and f f ′′ on morphisms, where f ′′ denotes the normal exten-sion of f . By Lemma 3.16, A ∗ b B ∈ C ∗ s , nd ( b † , b ) for all A , B ∈ C ∗ s , nd ( b † , b ) , but C ∗ s , nd ( b † , b ) is not a monoidal categorywith respect to the fiber product because the latter is not associative (see Subsection 3.5). Proposition 4.5.
There exists a natural transformation C ∗ s , nd ( b † , b ) × C ∗ s , nd ( b † , b ) −∗ b − / / G × G (cid:15) (cid:15) C ∗ s , nd ( b † , b ) G (cid:15) (cid:15) p x iiiiiiiiiiiiiiiiiiiiiiiiiiii W ∗ ( N , N op ) × W ∗ ( N , N op ) −∗ µ − / / W ∗ ( N , N op ) , given for each object A a , b H and B g , d K by conjugation with the isomorphism (7) .Proof. Immediate from Theorem 4.3 and Lemma 3.16.
We keep the notation introduced above, denote by
Hilb the category of Hilbert spaces andbounded linear operators, and call a subcategory of W ∗ - mod ( N , N op ) a ∗ -subcategory if it is closedwith respect to the involution T T ∗ of morphisms. Definition 4.6. A category over W ∗ - mod ( N , N op ) is a category C equipped with a functor U C : C → W ∗ - mod ( N , N op ) such that U C C is a ∗ -subcategory of W ∗ - mod ( N , N op ) . Let ( C , U C ) be such a cat-egory. We loosely refer to C as a category over W ∗ - mod ( N , N op ) without mentioning U C explicitly,and denote by H C the composition of U C with the forgetful functor W ∗ - mod ( N , N op ) → Hilb . Wecall an object G ∈ C separating if [ H C C ( G , X )( H C G )] = H C X for each X ∈ C .We denote by Cat ( N , N op ) the category of all categories over W ∗ - mod ( N , N op ) having a separatingobject, where the morphisms between objects ( C , U C ) and ( D , U D ) are all functors F : C → D satisfying U D F = U C . xample 4.7. For each A ∈ W ∗ ( N , N op ) , denote by W ∗ - mod A the category of all normal, unitalrepresentations p : A → L ( H ) for which p ◦ i A and p ◦ i opA are faithful, and all intertwiners. Thisis a category over W ∗ - mod ( N , N op ) , where U A : W ∗ - mod A → W ∗ - mod ( N , N op ) is given by ( L , p ) ( L , p ◦ i A , p ◦ i opA ) on objects and T T on morphisms. The only non-trivial thing to check isthat W ∗ - mod A has a separating object; by [3, Lemma 2.10] or [23, IX Theorem 1.2 iv)], one cantake the GNS-representation for a n.s.f. weight on A .For each morphism f : A → B in W ∗ ( N , N op ) , we obtain a functor f ∗ : W ∗ - mod B → W ∗ - mod A ,given by ( L , p ) ( L , p ◦ f ) on objects and T T on morphisms. Remark 4.8.
In the definition above,
Cat ( N , N op ) ( C , D ) need not be a set, and this may causeproblems. There are several possible solutions: we can fix a “universe” to work in, or replacethe category W ∗ - mod ( N , N op ) by a small subcategory and require categories over W ∗ - mod ( N , N op ) to be small, too. It is clear how to modify the preceding example in that case. Proposition 4.9.
There exists a contravariant functor
Mod : W ∗ ( N , N op ) → Cat ( N , N op ) given byA Mod ( A ) : = ( W ∗ - mod A , U A ) on objects and f Mod ( f ) : = f ∗ on morphisms. For each category C ∈ Cat ( N , N op ) , choose a separating object G C . Fix C ∈ Cat ( N , N op ) , let U = U C , H = H C G = G C , ( H , r , s ) = U G , and define End ( C ) : = H ( C ( G , G )) ′ ⊆ L ( H ) . Then r ( N ) + s ( N ) ⊆ End ( C ) because H ( C ( G , G )) ⊆ ( r ( N ) + s ( N )) ′ , and we can consider End ( C ) as an element of W ∗ ( N , N op ) with respect to r and s . Lemma 4.10.
There exists a morphism h C : C → Mod ( End ( C )) in Cat ( N , N op ) , given by X ( U X , r X ) on objects and T H T on morphisms, where r X = r HC ( G , X ) for each X ∈ C . Inparticular, r X ( End ( C )) ⊆ H ( C ( X , X )) ′ for each X ∈ C .Proof. Let X ∈ C and ( K , f , y ) = U X . Lemma 2.4, applied to I : = HC ( G , X ) ⊆ L ( H G , H X ) ,gives a normal representation r I : ( I ∗ I ) ′ → L ( K ) . Since I ∗ I ⊆ HC ( G , X ) by assumption on C ,we have End ( C ) ⊆ ( I ∗ I ) ′ and can define r X = r I | End ( C ) . Each element of I intertwines r with f and s with y , whence U X = ( K , r I ◦ r , r I ◦ s ) = U End ( C ) ( h C X ) .Next, let Y ∈ C , T ∈ C ( X , Y ) , J : = HC ( G , Y ) . Then H ( T ) r I ( S ) = r J ( S ) H ( T ) for all S ∈ End ( G ) because H ( T ) I ∈ J , and therefore H ( T ) is a morphism from ( H X , r X ) to ( H Y , r Y ) . By definition, H End ( C ) ( h C ( T )) = H T . Remark 4.11. If G ′ ∈ C is another separating object, then r G ′ : H ( C ( G , G )) ′ → H ( C ( G ′ , G ′ )) ′ is an isomorphism with inverse r HC ( G ′ , G ) .We eventually show that the assignment C → End ( C ) extends to a functor End : Cat ( N , N op ) → W ∗ ( N , N op ) that is adjoint to Mod . The key is a more careful analysis of functors from a cat-egory C ∈ Cat ( N , N op ) to categories of the form Mod ( A ) , where A ∈ W ∗ ( N , N op ) . Such functorsthemselves can be considered as objects of a category as follows.For all C , D ∈ Cat ( N , N op ) , the elements of Cat ( N , N op ) ( C , D ) are the objects of a category, wherethe morphisms are all natural transformations with the usual composition.Similarly, for all A , B ∈ Cat ( N , N op ) , the morphisms in W ∗ ( N , N op ) ( A , B ) can be considered as ob-jects of a category, where the morphisms between f , y are all b ∈ B satisfying b f ( a ) = y ( a ) b for all a ∈ A , and where composition is given by multiplication.27 roposition 4.12. Let A ∈ W ∗ ( N , N op ) and C ∈ Cat ( N , N op ) . Then there exists an isomorphism F C , A : Cat ( N , N op ) ( C , Mod ( A )) → W ∗ ( N , N op ) ( A , End ( C )) with inverse Y C , A : = F − C , A such thati) F C , A ( F ) is defined by F G C = ( H C G C , F C , A ( F )) for each functor F : C → Mod ( A ) and F C , A ( a ) = a G C for each natural transformation a in Cat ( N , N op ) ( C , Mod ( A )) ,ii) Y C , A ( p ) = Mod ( p ) ◦ h C : C → Mod ( End ( C )) → Mod ( A ) for each object p and Y C , A ( S ) =( r X ( S )) X ∈ C for each morphism S in W ∗ ( N , N op ) ( A , End ( C )) . Explicitly, Y C , A ( p ) is given by X ( H C X , r X ◦ p ) on objects and T H C T on morphisms.The proof of Proposition 4.12 involves the following result. Lemma 4.13.
Write U C G C = ( H C G C , r , s ) . Then the assignments a a G C and ( r X ( S )) X ∈ C ← [ S are inverse bijections between all natural transformations a of H C (or h C ) and all elementsS ∈ End ( G C ) (or S ∈ End ( G C ) ∩ ( r ( N ) + s ( N )) ′ , respectively).Proof. A family of morphisms ( a X : H C X → H C X ) X ∈ C is a natural transformation of H C if andonly if a X T = T a X for all X ∈ C and T ∈ H C ( G C , X ) , that is, if a X = r X ( a G C ) and a G C ∈ End ( C ) . Such a family is a natural transformation of h C if and only if additionally, a X = r X ( a G C ) is a morphism of U C X for each X ∈ C or, equivalently, if a G C ∈ ( r ( N ) + s ( N )) ′ . Proof of Proposition 4.12.
Lemma 4.13 implies that Y : = Y C , A is well defined by ii). Let usshow that F : = F C , A is well defined by i). For each F as above, the image H Mod ( A ) ( F ( C ( G C , G C ))) = H C ( C ( G C , G C )) consists of intertwiners for F ( F ) and hence ( F ( F ))( A ) ⊆ H C ( C ( G C , G C )) ′ = End ( C ) . Likewise, for each a as above, a G C intertwines H C ( C ( G C , G C )) and hence a G C ∈ End ( C ) . Finally, F ( a ◦ b ) = a G C ◦ b G C = F ( a ) F ( b ) for all composable a , b .Next, F ◦ Y = id because for each p as above, Y ( p )( G C ) = ( H C G C , r G C ◦ p ) so that F ( Y ( p )) = r G C ◦ p = p , and for each S as above, the component of ( r X ( S )) X ∈ C at X = G C is r G C ( S ) = S .Finally, we prove Y ◦ F = id. Let F be as above and define f X by F X = ( H C X , f X ) for each X ∈ C . Then F ( F ) = f G C , and for each a ∈ A , the family ( f X ( a )) X ∈ C is a natural transformationof H Mod ( A ) ◦ F = H C and coincides by Lemma 4.13 with ( r X ( f G C ( a ))) X ∈ C . Therefore, F X =( H C X , f X ) = ( H C X , r X ◦ F ( F )) = Y ( F ( F ))( X ) for each X ∈ C . On morphisms, Y ( F ( F )) and F coincide anyway. For each a as above, Y ( F ( a )) = ( r X ( a G C )) X ∈ C = a by Lemma 4.13. Corollary 4.14. i) Let A ∈ W ∗ ( N , N op ) and consider id A as an object of C : = Mod ( A ) . Then F C , A ( id C ) : A → End ( Mod ( A )) is an isomorphism in W ∗ ( N , N op ) with inverse e A : = r id A .ii) Let A , B ∈ W ∗ ( N , N op ) . The the isomorphism Mod ( A , B ) : = Y Mod ( B ) , A ◦ ( e − B ) ∗ : W ∗ ( N , N op ) ( A , B ) → W ∗ ( N , N op ) ( A , End ( Mod ( B ))) → Cat ( N , N op ) ( Mod ( B ) , Mod ( A )) is given by f Mod ( f ) on objects and b ( p ( b )) ( L , p ) on morphisms.iii) Let C , D ∈ Cat ( N , N op ) . Then the functor End ( C , D ) : = F C , End ( D ) ◦ ( h D ) ∗ : Cat ( N , N op ) ( C , D ) → Cat ( N , N op ) ( C , Mod ( End ( D ))) → W ∗ ( N , N op ) ( End ( D ) , End ( C )) is given by F r F G C onobjects and a H D ( a G C ) on morphisms. roof. Assertions i) and iii) follow immediately from the definitions and Proposition 4.12. Letus prove ii). For each object f , we have G Mod ( B ) = ( H Mod ( B ) , e − B ) and F Mod ( B ) , A ( Mod ( f )) = e − B ◦ f , whence Y Mod ( B ) , A ( e − B ◦ f ) = Mod ( f ) , and for each morphism b , the family a : =( p ( b )) ( L , p ) is a natural transformation and F Mod ( B ) , A ( a ) = a G Mod ( B ) = e − B ( b ) .The relative tensor product on W ∗ - mod ( N , N op ) induces a product on Cat ( N , N op ) as follows. Let C , D ∈ Cat ( N , N op ) . Then C × D and the functor U C × D = ( −⊗ µ − ) ◦ ( U C × U D ) : C × D → W ∗ - mod ( N , N op ) , form a category over W ∗ - mod ( N , N op ) with separating object ( G C , G D ) . Thus, we obtain a monoidalstructure on Cat ( N , N op ) , given by ( C , D ) C × D on objects and ( F , G ) F × G on morphisms. Corollary 4.15.
For all A , B , C ∈ W ∗ ( N , N op ) , there exists an isomorphism X : W ∗ ( N , N op ) ( A , B ∗ µ C ) → Cat ( N , N op ) ( Mod ( B ) × Mod ( C ) , Mod ( A )) such that for each object p , the functor X ( p ) is given by (( L , t ) , ( M , u )) ( L ⊗ µ M , ( t ∗ µ u ) ◦ p ) and ( S , T ) S ⊗ µ T , and for each morphism x : p → p , the transformation X ( b ) : X ( p ) → X ( p ) isgiven by X ( b ) (( L , t ) , ( M , u )) = ( t ∗ µ u )( x ) .Proof. Let B : = Mod ( B ) , C : = Mod ( C ) , D : = B × C . Then G : = ( G B , G C ) is separating and r G : End ( D ) → H D ( D ( G , G )) ′ = ( End ( B ) ′ ⊗ µ End ( C ) ′ ) ′ = End ( B ) ∗ µ End ( C ) ∼ = B ∗ µ C is an isomorphism by Remark 4.11. Moreover, if X = ( L , t ) ∈ B , Y = ( M , u ) ∈ C , then r ( X , Y ) =( t ∗ µ u ) ◦ r G by Lemma 2.4 because t ∗ µ u = r J , where J = H B ( B ( G B , X )) ⊗ µ H C ( C ( G C , Y )) , and J · H D ( D ( G D , G )) ⊆ H D ( D ( G D , ( X , Y ))) . Now, the assertion follows from Proposition 4.12.The categories W ∗ ( N , N op ) and Cat ( N , N op ) are enriched over the monoidal category Cat of smallcategories [14], or, equivalently, are 2-categories, meaning that the morphisms between fixedobjects are themselves objects of a small category, as explained before Proposition 4.12, andthat the composition of morphisms between fixed objects extends to a functor, where B y ( ( y ⇓ c C ◦ A f ( ( f ⇓ b B = A y ◦ f * * y ◦ f ⇓ y ( b ) c C in W ∗ ( N , N op ) , (8) C G ) ) G ⇓ b D ◦ B F ) ) F ⇓ a C = B G ◦ F + + G ◦ F ⇓ b F ◦ G a D in Cat ( N , N op ) . (9)Recall that a contravariant functor between enriched categories C , D consists of an assignment F : ob C → ob D and, for each pair of objects X , Y ∈ C , a functor F ( X , Y ) : C ( X , Y ) → D ( F Y , F X ) Mod , End defined above are functors in this sense and that the isomorphisms in Proposition4.12 form part of an adjunction between
Mod and
End . For background on enriched categories,see [14].
Theorem 4.16.
The assignments
Mod , End define contravariant functors
Mod : W ∗ ( N , N op ) → Cat ( N , N op ) and End : Cat ( N , N op ) → W ∗ ( N , N op ) of enriched categories, and the isomorphisms ( F C , A ) C , A define an adjunction whose unit is ( h C ) C ∈ Cat ( N , Nop ) and counit is ( e A ) A ∈ W ∗ ( N , Nop ) .Proof. We first show that
Mod and
End are functors of enriched categories. By Corollary4.14, it suffices to prove this for
End . Consider a diagram as in (9) and let a = End ( B , C ) ( a ) , b = End ( C , D ) ( b ) , c = End ( B , D ) ( b F ◦ G a ) . We have to show that then the cells End ( C ) End ( B , C ) ( F ) , , ⇓ a End ( B , C ) ( F ) End ( B ) ◦ End ( D ) End ( C , D ) ( G ) , , ⇓ b End ( C , D ) ( G ) End ( C ) and End ( D ) End ( B , D ) ( G F ) , , End ( B , D ) ( G F ) ⇓ c End ( B ) are equal. By definition, a = H C ( a G B ) , b = H D ( b G C ) , and by Lemma 4.13, c = H D ( b F G B · G ( a G B )) = r F G B ( H D ( b G C )) · H C ( a G B ) = End ( F )( b ) · a . It remains to show that for all morphisms f : A → B in W ∗ ( N , N op ) and F : C → D in Cat ( N , N op ) ,the diagram Cat ( N , N op ) ( D , Mod ( B )) F D , B / / (cid:15) (cid:15) W ∗ ( N , N op ) ( B , End ( D )) (cid:15) (cid:15) Cat ( N , N op ) ( C , Mod ( A )) F C , A / / W ∗ ( N , N op ) ( A , End ( C )) commutes, where the vertical maps are induced by F and Mod ( A , B ) ( f ) on the left and f and End ( C , D ) ( F ) on the right, respectively, or, more precisely, that for each object G and each mor-phism a in Cat ( N , N op ) ( D , Mod ( B )) , End ( C , D ) ( F ) ◦ F D , B ( G ) ◦ f = F C , A ( Mod ( A , B ) ( f ) ◦ G ◦ F ) , End ( C , D ) ( F )( a ) = Mod ( A , B ) ( f )( a F ) . The second equation holds because of Lemma 4.13 and the relation
End ( C , D ) ( F )( a G C ) = r F G C ( a G D ) = a F G C = Mod ( A , B ) ( f )( a F G C ) first one holds because by Corollary 4.14, End ( C , D ) ( F ) ◦ F D , B ( G ) ◦ f = r F G C ◦ F D , B ( G ) ◦ f , ( Mod ( A , B ) ( f ) ◦ G ◦ F )( G C ) = ( H C G C , r F G C ◦ F D , B ( G ) ◦ f ) . The special case of a commutative base
Let Z be a locally compact Hausdorff space with a Radon measure µ of full support, and iden-tify C ( Z ) with multiplication operators on L ( L ( Z , µ )) . Then the relative tensor product andthe fiber product over the C ∗ -base b = ( L ( Z , µ ) , C ( Z ) , C ( Z )) can be related to the fiberwiseproduct of bundles as follows. Modules and their relative tensor product
Denote by
Mod b , Mod C ( Z ) , Bdl Z the categoriesof all C ∗ - b -modules with all morphisms, of all Hilbert C ∗ -modules over C ( Z ) , and of all con-tinuous Hilbert bundles over Z ; for the precise definition of the latter, see [6]. Each of thesecategories carries a monoidal structure, where the product • of E , F ∈ Mod C ( Z ) is the separated completion of E ⊙ F with respect to the inner product h x ⊙ h | x ′ ⊙ h ′ i = h x | x ′ ih h | h ′ i , denoted by E ⊗ C ( Z ) F , • of E , F ∈ Bdl Z is the fibrewise tensor product of E and F , • of H b , K g ∈ Mod b is ( H b ⊗ b g K , b ⊲⊳ g ) , where b ⊲⊳ g : = [ | g i b ] = [ | b i g ] ; here, note that b H b , g K g are C ∗ - ( b , b ) -modules.There exist equivalences of monoidal categories Mod C ( Z ) B ⇄ G Bdl Z and Mod C ( Z ) F ⇄ U Mod b suchthat for each E ∈ Mod C ( Z ) , F ∈ Bdl Z , H b ∈ Mod b , • B E = F z ∈ Z E z is and G ( B E ) = { ( x z ) z | x ∈ E } , where E z is the completion of E withrespect to the inner product ( x , h )
7→ h x | h i ( z ) , and x x z denotes the quotient map E → E z , • the operations on the space of sections G ( F ) ∈ Mod C ( Z ) are defined fiberwise, • F E = ( E ⊗ C ( Z ) L ( Z , µ ) , l ( E )) , where l ( x ) h = x ⊗ C ( Z ) h for each x ∈ E , h ∈ L ( Z , µ ) , • U H b = b ∈ Mod C ( Z ) .The first equivalence is explained in [6], and the second one is easily verified. Compare alsoExamples 2.6 and 2.11 ii). Algebras
Denote by C ∗ C ( Z ) the category of all continuous C ( Z ) -algebras with full support [],where the morphisms between A , B ∈ C ∗ C ( Z ) are all C ( Z ) -linear nondegenerate ∗ -homomorphisms p : A → M ( B ) , and by ˜ C ∗ b the category of all C ∗ - b -algebras A b H satisfying [ r b ( C ( Z )) A ] = A and [ A b ] = b , where the morphisms between A b H , B g K ∈ ˜ C ∗ b are all p ∈ C ∗ b ( A b H , M ( B ) g K ) satisfying [ p ( A ) B ] = B . Then there exists a functor ˜ C ∗ b → C ∗ C ( Z ) , given by A b H ( A , r a ) and p p , andthis functor has a full and faithful left adjoint which embeds C ∗ C ( Z ) into ˜ C ∗ b [28, Theorem 6.6].31 he fiber product of commutative C ∗ - b -algebras We finally discuss the fiber product ofcommutative C ∗ - b -algebras and start with preliminaries. Let Z be a locally compact space, E a Hilbert C ∗ -module over C ( Z ) , and B E = F z ∈ Z E z the corresponding Hilbert bundle. Thetopology on B E is generated by all open sets of the form U V , h , e = { z | z ∈ V , z ∈ E z , k h z − z k E z < e } , where V ⊆ Z is open, h ∈ E , e >
0. Denote by q : F z ∈ Z L ( E z ) → Z the natural projectionand define for each h , h ′ ∈ E maps w h , h ′ : G z ∈ Z L ( E z ) → C , T
7→ h h q ( T ) | T h ′ q ( T ) i , u ( ∗ ) h : : G z ∈ Z L ( E z ) → G z ∈ Z E z , T T ( ∗ ) h q ( T ) . The weak topology (strong-*-topology) on F z ∈ Z L ( E z ) is the weakest one that makes q and allmaps of the form w h , h ′ (of the form u ( ∗ ) h ) continuous.Let A be a commutative C ∗ -algebra, p : C ( Z ) → M ( A ) a ∗ -homomorphism, and c ∈ b A . Thenwe identify E ⊗ f ∗ A ⊗ c C with E z , where z ∈ Z corresponds to c ◦ p ∈ \ C ( Z ) , via h ⊗ p a ⊗ c l lc ( a ) h z . A map T : b A → F z ∈ Z L ( E z ) is weakly vanishing (strong- ∗ -vanishing) at infinity if forall h , h ′ ∈ E , the map w h , h ′ ◦ T (the maps c
7→ k u ( ∗ ) h ( T ( c )) k ) vanish at infinity. Lemma 5.1.
Let A b H be a C ∗ - b -algebra, K g a C ∗ - b † -module, x ∈ L ( H b ⊗ b g K ) . Assume thatA is commutative, [ r b ( C ( Z )) A ] = A, and h g | x | g i ⊆ A. Define F x : b A → F z ∈ Z L ( g z ) by c ( c ∗ id )( x ) . Then:i) F x is weakly continuous, weakly vanishing at infinity.ii) x ∈ Ind | g i ( A ) if and only if F x is strong- ∗ continuous, strong- ∗ -vanishing at infinity.Proof. First, note that for all h , h ′ ∈ g and c ∈ ˆ A , c ( h h | x | h ′ i ) = h ( c ◦ r b ) < h | ( c ∗ id )( x )( ( c ◦ r b ) < h ′ ) i = h h ( c ◦ r b ) | F x ( c ) h ′ ( c ◦ r b ) i . i) For each h ′ , h ∈ g , the map c
7→ h h ( c ◦ r b ) | F x ( c ) h ′ ( c ◦ r b ) i equals h h | x | h ′ i ∈ A .ii) Assume that F x is strong- ∗ continuous vanishing at infinity and let h ∈ g . Then the map c F x ( c ) h ( c ◦ r b ) lies in G ( g = r b A ) . Hence, there exists an w ∈ g = r b A such that F x ( c ) h ( c ◦ r b ) = w c for all c ∈ b A . We identify g = r b A with [ | g i A ] ⊆ L ( H , H b ⊗ b g K ) in the canonical manner andfind that x | h i = w because c ( h h ′ | x | h i ) = h h ′ ( c ◦ r b ) | w ( c ◦ r b ) i = c ( h h ′ | w ) for all c ∈ ˆ A , h ′ ∈ g .Since h ∈ g was arbitrary, we can conclude x | g i ⊆ [ | g i A ] . A similar argument, applied to x ∗ instead of x , shows that x ∗ | g i ⊆ [ | g i A ] , and therefore x ∈ Ind | g i ( A ) . Reversing the arguments,we obtain the reverse implication.Let X be a locally compact Hausdorff space with a continuous surjection p : X → Z and a familyof Radon measures f = ( f z ) z ∈ Z such that (i) supp f z = X z : = p − ( z ) for each z ∈ Z and (ii)the map f ∗ ( f ) : z R X z f d f z is continuous for each f ∈ C c ( X ) . Define a Radon measure n X on X such that R X f d n X = R Z f ∗ ( f ) dµ for all f ∈ C c ( X ) . Then there exists a map j X : C c ( X ) → L ( L ( Z , µ ) , L ( X , n X )) such that j X ( f ) h = f p ∗ ( h ) and j X ( f ) ∗ g = f ∗ ( f g ) for all f , g ∈ C c ( X ) , h ∈ C c ( Z ) . Similarly, let Y be a locally compact Hausdorff space with a continuous map q : Y → Z y = ( y z ) z ∈ Z satisfying the same conditions as X , p , f , and define aRadon measure n Y on Y and an embedding j Y : C c ( Y ) → L ( L ( Z , µ ) , L ( Y , n Y )) as above. Let H : = L ( X , n X ) , b : = [ j X ( C c ( X ))] , A : = C ( X ) ⊆ L ( L ( X , n X ) = L ( H ) , K : = L ( Y , n Y ) , g : = [ j Y ( C c ( Y ))] , B : = C ( Y ) ⊆ L ( L ( Y , n Y )) = L ( K ) . Then H b , K g are C ∗ - b -modules and A b H , B g K are C ∗ - b -algebras, as one can easily check. Consid-ering b and g as Hilbert C ∗ -modules over C ( Z ) , we can canonically identify b z ∼ = L ( X z , f z ) and g z ∼ = L ( Y z , y z ) . Finally, define a Radon measure n on X p × Z q Y such that for all h ∈ C c ( X p × Z q Y ) , Z X p × Zq Y h d n = Z Z Z X z Z Y z h ( x , y ) d y z ( y ) d f z ( x ) d µ ( z ) . Proposition 5.2. i) There exists a unitary U : H b ⊗ b g K → L ( X p × Z q Y , n ) such that ( F ( j X ( f ) = h < j Y ( g )))( x , y ) = f ( x ) h ( p ( x )) g ( y ) for all f ∈ C c ( X ) , g ∈ C c ( Y ) , h ∈ C c ( Z ) , ( x , y ) ∈ X p × Z q
Y .ii) Ad U ( A b ∗ b g B ) is the C ∗ -algebra of all f ∈ L ¥ ( X p × Z q Y , n ) that have representatives f X , f Y such that the maps X → Tot L ( g ) and Y → Tot L ( b ) given by x f X ( x , · ) ∈ L ¥ ( Y p ( x ) , y p ( x ) ) and y f Y ( · , y ) ∈ L ¥ ( X q ( y ) , f q ( y ) ) respectively, are strong- ∗ continuous vanishing at in-finity.Proof. The proof of assertion i) is straightforward, and assertion ii) follows immediately fromProposition Lemma 3.16 viii) and Lemma 5.1 ii).
Examples 5.3. i) Let X , Y be discrete, Z = { } , and let f , y be the counting measures on X , Y , respectively. Then C ( X ) b ∗ b g C ( Y ) ∼ = { f ∈ C b ( X × Y ) | f ( x , · ) ∈ C ( Y ) for all x ∈ X , f ( · , y ) ∈ C ( X ) for all y ∈ Y } . This follows from Proposition 5.2 and the fact that for each f ∈ C b ( X × Y ) , the maps X → L ( l ( Y )) , x f ( x , · ) , and Y → L ( l ( X )) , y f ( · , y ) , are strong- ∗ continuousvanishing at infinity if and only if f ( · , y ) ∈ C ( X ) and f ( x , · ) ∈ C ( Y ) for each y ∈ Y and x ∈ X .ii) Let X = N , Z = { } , and let f be the counting measure. Then C ( N ) b ∗ b g C ( Y ) ∼ = { f ∈ C b ( N × Y ) | ( f ( x , · )) x is a sequence in C ( Y ) that converges strongly to 0 ∈ L ( L ( Y , y )) } because for each f ∈ L ¥ ( N × Y ) , the map Y → L ( l ( N )) , y f ( · , y ) , is strong- ∗ contin-uous vanishing at infinity if and only if f ( x , · ) ∈ C ( Y ) for all x ∈ N .33ii) Let X = Y = [ , ] , Z = { } , and let f = y be the Lebesgue measure. For each subset I ⊆ [ , ] , denote by c I its characteristic function. Then the function f ∈ L ¥ ([ , ] × [ , ]) given by f ( x , y ) = y ≤ x and f ( x , y ) = C ([ , ]) b ∗ b g C ([ , ]) because the functions [ , ] → L ¥ ([ , ]) ⊆ L ( L ([ , ])) given by x f ( x , · ) = c [ , x ] and y f ( · , y ) = c [ y , ] are strong- ∗ continuous. In particular, we see that C ([ , ]) b ∗ b g C ([ , ]) * C ([ , ] × [ , ]) = C ([ , ]) ⊗ C ([ , ]) . References [1] ´E. Blanchard. D´eformations de C ∗ -alg`ebres de Hopf. Bull. Soc. Math. France , 124(1):141–215, 1996.[2] G. B ¨ohm. Hopf algebroids. In
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