aa r X i v : . [ m a t h . OA ] A ug E -theory for C ∗ -categories Sarah L. Browne and Paul D. MitchenerAugust 31, 2020
Abstract E -theory was originally defined concretely by Connes and Higson [CH]and further work followed this construction. We generalise the definitionto C ∗ -categories. C ∗ -categories were formulated to give a theory of opera-tor algebras in a categorical picture and play important role in the study ofmathematical physics. In this context, they are analogous to C ∗ -algebrasand so have invariants defined coming from C ∗ -algebra theory but theydo not yet have a definition of E -theory. Here we define E -theory for bothcomplex and real graded C ∗ -categories and prove it has similar propertiesto E -theory for C ∗ -algebras. Throughout this article, we use F to denote either the field of real numbers R orthe field of complex numbers C . C ∗ -categories are analogous to C ∗ -algebras butgive a much more general framework which do not require a choice of Hilbertspace. In particular, we can consider all Hilbert spaces and the collection of allbounded linear operators on them simultaneously, and we can view a C ∗ -algebraas a one object C ∗ -category. E -theory is an invariant for C ∗ -algebras. K -theory for C ∗ -algebras wasgeneralised to C ∗ -categories in [Joa03, Mit01] and we take a similar approachto generalising E -theory here. We build on the work of Guentner-Higson [HG04]and Browne [Bro17] on constructing E -theory for graded complex and real C ∗ -categories.The plan in future papers is to build spectra related to E -theory, and eventu-ally equivariant E -theory. With these defined, we will be ready to develop appli-cations in the field of analytic assembly. Ultimately this machinery should givea smooth proof that the Baum-Connes conjecture implies the stable Gromov-Lawson-Rosenberg conjecture, as in [Sto02], and similar results that requirelooking at analytic assembly from a homotopy-theoretic point of view.This paper details the generalisation of a structure of asymptotic morphismsto asymptotic functors for C ∗ -categories, and then the notion of homotopy for AMS subject classification (2020): 46L80 ( K -theory and operator algebras).SB was partially supported by NSF grant E -theory forcomplex and real graded C ∗ -categories. We then construct a product and provethat E -theory is a bivariant functor going from the category where objects are C ∗ -categories and morphisms are ∗ -functors to the category where objects areabelian groups and morphisms are group homomorphisms.We check properties of standard E -theory also hold in this context. Namelywe check that the functor E is half exact, homotopy invariant and has Bottperiodicity. We give details and show that we have long exact sequences inducedfrom short exact sequences. C ∗ -categories and asymptotic functors In this section we give a review on the notions of C ∗ -category, representationsand ideals in this setting before giving the definition of an asymptotic functorand the notion of homotopy of these which will form part of the definition of E -theory for C ∗ -categories.Recall (see [GLR85, Mit02]) that a unital C ∗ -category over the field F is acategory A where: • Each morphism set Hom( a, b ) A is a Banach space over the field F . Com-position of morphismsHom( b, c ) A × Hom( a, b ) A → Hom( a, c ) A is bilinear and satisfies the inequality k xy k ≤ k x k · k y k for all x ∈ Hom( b, c ) A and y ∈ Hom( a, b ) A . • We have an involution , that is to say conjugate-linear maps Hom( a, b ) A → Hom( b, a ) A , written x x ∗ such that ( x ∗ ) ∗ = x for all x ∈ Hom( a, b ) A ,and ( xy ) ∗ = y ∗ x ∗ for all x ∈ Hom( b, c ) A and y ∈ Hom( a, b ) A . • The C ∗ -identity k x ∗ x k = k x k holds for all x ∈ Hom( a, b ) A . Further, theelement x ∗ x is a positive element of the unital C ∗ -algebra Hom( a, a ) A .Similarly we can define a non-unital C ∗ -category A by dropping the insis-tence of unit elements 1 a ∈ Hom( a, a ) A . Hence a non-unital C ∗ -category is notactually a category, but there are objects and morphisms with an associativecomposition rule that satisfy the above axioms. The key example of a C ∗ -category is L , the category of all Hilbert spaces and bounded linear operators.The norm on each morphism set is the operator norm, and the involution isdefined by taking adjoints.The following comes from [Mit02]. Definition 2.1
Let A be a C ∗ -category, and let J be a collection of objects andmorphisms which are closed under composition (effectively, J is a“non-unitalsubcategory”). We call J a C ∗ -ideal if: • Ob( J ) = Ob( A ); 2 Each space Hom( a, b ) J is a vector subspace of Hom( a, b ) A ; • If x ∈ Hom( a, b ) J , then x ∗ ∈ Hom( b, a ) J ; • For all x ∈ Hom( a, b ) J , u ∈ Hom ( a ′ , a ) A , and v ∈ Hom ( b, b ′ ) B the com-posites vx and xu are morphisms in the category J .It is shown in [Mit02] that the morphism sets of a C ∗ -ideal are automaticallyclosed under the norm, so any C ∗ -ideal is a C ∗ -category. Further, we can formthe quotient A / J ; the objects are the same as those of the categories A and J ,and the morphism set Hom( a, b ) in the quotient A / J is the quotient Banachspace Hom( a, b ) A / Hom( a, b ) J .If A and B are C ∗ -categories, a ∗ -functor α : A → B consists of a map α : Ob( A ) → Ob( B ) and linear maps α : Hom( a, b ) A → Hom( α ( a ) , α ( b )) A suchthat: • α ( xy ) = α ( x ) α ( y ) for all x ∈ Hom( b, c ) A and y ∈ Hom( a, b ) A . • α ( x ∗ ) = α ( x ) ∗ for all x ∈ Hom( a, b ) A .For example, if A is a C ∗ -category and J is a C ∗ -ideal, we have a quotient ∗ -functor π : A → A / J defined by taking π to be the identity map on the setof objects, and the quotient map π : Hom( a, b ) A → Hom( a, b ) A / Hom( a, b ) J oneach morphism set.It is shown in [Mit02] that any ∗ -functor is norm-decreasing, that is to say k α ( x ) k ≤ k x k for all x ∈ Hom( a, b ) A . In particular, each map α : Hom( a, b ) A → Hom( α ( a ) , α ( b )) B is continuous. If the ∗ -functor α is faithful , that is to sayinjective on each morphism set, then it is an isometry. It is shown in [GLR85,Mit02] that for any C ∗ -category A there is a faithful ∗ -functor ρ : A → L ; wecall such a ∗ -functor a representation .A ∗ -functor is a logical generalisation of a ∗ -homomorphism between C ∗ -categories. We can also generalise asymptotic morphisms to the notion of anasymptotic functor for C ∗ -categories as below. Definition 2.2
Let A and B be C ∗ -categories. An asymptotic functor ϕ = ϕ t : A B consists of: • ϕ : Ob( A ) → Ob( B ), • For each t ∈ [1 , ∞ ) a map ϕ t : Hom( a, b ) A → Hom( ϕ ( a ) , ϕ ( b )) B ,with the following properties: • For each x ∈ Hom( a, b ) A the map [1 , ∞ ) → Hom( ϕ ( a ) , ϕ ( b )) B defined bywriting t ϕ t ( x ) is continuous and bounded. • For all x, y ∈ Hom( a, b ) A and λ, µ ∈ F we havelim t →∞ k ϕ t ( λx + µy ) − λϕ t ( x ) − µϕ t ( y ) k = 0 . For all x ∈ Hom( b, c ) A and y ∈ Hom( a, b ) A we havelim t →∞ k ϕ t ( xy ) − ϕ t ( x ) ϕ t ( y ) k = 0 . • For all x ∈ Hom( a, b ) A we havelim t →∞ k ϕ t ( x ∗ ) − ϕ t ( x ) ∗ k = 0 . A ∗ -functor α : A → B can also be considered as an asymptotic functor bywriting α t ( x ) = α ( x ) for all t ∈ [1 , ∞ ) and x ∈ Hom( a, b ) A .As noted in [Mit02] we can form the spatial tensor product of C ∗ -categories.We first form the algebraic tensor product. If a ∈ Ob( A ) and b ∈ Ob( B ) letus write the pair ( a, b ) ∈ Ob( A ) × Ob( B ) as a ⊗ b . Then the algebraic tensorproduct is an algebroid (see [Mit85]) A ⊙ B , with objects Ob( A ) × Ob( B ). Themorphism sets are algebraic tensor products of vector spacesHom( a ⊗ b, a ′ ⊗ b ′ ) A⊙B = Hom( a, a ′ ) A ⊙ Hom( b, b ′ ) B . Composition of morphisms and the involution are defined by the formulae • ( u ⊗ v )( x ⊗ y ) = ux ⊗ vy where x ∈ Hom( a, a ′ ) A , u ∈ Hom( a ′ , a ′′ ) A , and y ∈ Hom( b, b ′ ) B , v ∈ Hom( b ′ , b ′′ ) B , • ( x ⊗ y ) ∗ = x ∗ ⊗ y ∗ where x ∈ Hom( a, a ′ ) A and y ∈ Hom( b, b ′ ) B ,and extending by linearity.Let H and H ′ be Hilbert spaces. Consider the algebraic tensor product H ⊙ H ′ with bilinear form defined by h v ⊗ v ′ , w ⊗ w ′ i = h v, w i · h v ′ , w ′ i v, w ∈ H, v ′ , w ′ ∈ H ′ . We then take the quotient by elements x ∈ H ⊙ H ′ such that h x, x i = 0and complete with respect to the resulting norm to obtain the tensor product H ⊗ H ′ .Pick faithful representations ρ A : A → L and ρ B : B → L . As noted above,such always exist, and are isometries on each morphism set. We can define afaithful ∗ -functor ρ : A⊙B → L by mapping the object a ⊗ b to the Hilbert space ρ ( a ⊗ b ) = ρ A ( a ) ⊗ ρ B ( b ), and the morphism x ⊗ y where x ∈ Hom( a, a ′ ) A and y ∈ Hom( b, b ′ ) B to the bounded linear map ρ ( x ⊗ y ) : ρ ( a ⊗ b ) → ρ ( a ′ ⊗ b ′ ) definedby the formula ρ ( x ⊗ y )( v ⊗ w ) = ρ A ( x )( v ) ⊗ ρ B ( y )( w ) where v ⊗ w ∈ ρ ( a ⊗ b ).We extend where necessary by linearity.It follows that we can define a norm on the morphism sets of the algebraictensor product by writing k z k = k ρ ( z ) k for each morphism z in A ⊙ B . Wedefine the spatial tensor product
A ⊗ B by completion of each morphism set inthe algebraic tensor product under this norm. As shown in [Mit02], the spatialtensor product
A ⊗ B is a C ∗ -category, and the norm does not depend on ourchoices of faithful representations. 4 xample 2.3 Let X be a compact Hausdorff space. Then the C ∗ -algebra C ( X ) can be viewed as a C ∗ -category with one object. Let A be another C ∗ -category. Then the tensor product C ( X ) ⊗ A is a C ∗ -category with thesame objects as A and morphism setsHom( a, b ) C ( X ) ⊗A = { f : X → Hom( a, b ) A | f continuous , lim x →∞ k f ( x ) k → } Proposition 2.4
Let ϕ : A B be an asymptotic functor. Let C be a C ∗ -category. Then there is an asymptotic functor ϕ ⊗ id : A ⊗ C
B ⊗ C such that ( ϕ ⊗ id)( a ⊗ c ) = ϕ ( a ) ⊗ c for all a ∈ Ob( A ) and c ∈ Ob( C ) , and ( ϕ ⊗ id) t ( x ⊗ y ) = ϕ t ( x ) ⊗ y if x ∈ Hom( a, a ′ ) A , y ∈ Hom( c, c ′ ) C and t ∈ [1 , ∞ ) . Proof:
We need to check the asymptotic properties of the maps ( ϕ ⊗ id) t : Hom( a ⊗ c, b ⊗ c ) A⊗C → Hom( ϕ ( a ) , ϕ ( b )) B⊗C .Firstly, let x ⊗ y ∈ Hom( a, b ) A ⊗ Hom( c, c ′ ) C . We know the map [0 , → Hom( ϕ ( a ) , ϕ ( b )) A given by t ϕ t ( x ) is continuous and bounded. The map t ϕ t ( x ) ⊗ y is certainly continuous. Suppose k ϕ t ( x ) k ≤ M for all t ∈ [1 , ∞ ).Then k ϕ t ( x ) ⊗ y k ≤ M k y k by definition of the spatial tensor product, and themap t ϕ t ( x ) ⊗ y is certainly bounded.Let x, y ∈ Hom( a, b ) A , λ, µ ∈ F . Let z ∈ Hom( c, c ′ ) C . Then k ϕ t ( λx + µy ) ⊗ z − λϕ t ( x ) ⊗ z − µϕ t ( y ) ⊗ z k ≤ k z kk ϕ t ( λx + µy ) − λϕ t ( x ) − µϕ t ( y ) k which converges to 0 as t → ∞ .The other required limit properties follow similarly. ✷ We can similarly define asymptotic functors of the form id ⊗ ϕ : C⊗A → C⊗B . Definition 2.5
Let α : A → B be a ∗ -functor. Then the mapping cylinder C α is the C ∗ -category with objectsOb( C α ) = { ( a, b ) ∈ Ob( A ) × Ob( B ) | α ( a ) = b } and morphism setsHom(( a, b ) , ( a ′ , b ′ )) = ( ( x, f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ∈ Hom( a,a ′ ) A , f (0)= α ( x ) ,f : [0 , −→ Hom( b,b ′ ) B continuous ) . The mapping cylinder fits into a pullback diagram C α (cid:15) (cid:15) / / A α (cid:15) (cid:15) C [0 , ⊗ B E / / B
5n the category of C ∗ -categories and ∗ -functors.In the special case where the categories A and B have the same set of objects,and the ∗ -functor α is the identity on the set of objects, the mapping cylinder C α has a simpler description. The set of objects of C α can be the same asthe set of the objects of the categories A and B . The morphism set Hom( a, b )in the mapping cylinder C α is the same as the mapping cylinder of the map α : Hom( a, b ) A → Hom( a, b ) B , that is to say the set { ( x, f ) | x ∈ Hom( a, b ) A , f : [0 , → Hom( a, b ) B continuous , f (0) = α ( x ) } . Using tensor products, we can form a notion of homotopy of asymptoticfunctors. Specifically, if A is a C ∗ -category, define the C ∗ -category I A = A ⊗ C [0 , ∗ -homomorphisms e , e : C [0 , → F defined by writing e ( f ) = f (0) and e ( f ) = f (1) respectively. Hence, by the above, there are ∗ -homomorphisms E , E : I A → A given by writing E = e ⊗ id and E = e ⊗ id. Definition 2.6
Let ϕ, ψ : A B be asymptotic functors. Then a homotopy between ϕ and ψ is an asymptotic functor θ : A I B such that E ◦ θ = ϕ and E ◦ θ = ψ . Definition 2.7
Two asymptotic functors ϕ, ψ : A B are called equivalent if ϕ ( a ) = ψ ( a ) for each object a ∈ Ob( A ), and for each morphism x ∈ Hom( a, b ) A lim t →∞ || ϕ t ( x ) − ψ t ( x ) || = 0 . Proposition 2.8
Equivalent asymptotic functors are homotopic.
Proof:
Let ϕ, ψ : A B be equivalent asymptotic functors. Define a ho-motopy θ : A IB by θ t ( a ) = ϕ ( a ) = ψ ( a ) for all a ∈ Ob( A ) and for all x ∈ Hom( a, b ) A , s ∈ [0 ,
1] by θ t ( x )( s ) = (1 − s ) ϕ t ( x ) + sψ t ( x ) = ϕ t ( x ) + s ( ψ t ( x ) − ϕ t ( x )) . It is easy to verify that this is an asymptotic functor, and gives the requiredhomotopy between ϕ and ψ .Then this is an asymptotic functor which defines a homotopy as required. ✷ Definition 2.9
Let B be a C ∗ -category, and let J be a C ∗ -ideal. A quasicentralset of approximate units for the pair ( B , J ) consists of a norm-continuous family { u at | t ∈ [1 , ∞ ) } of elements of each space Hom( a, a ) J such that: • For each a ∈ Ob( B ) and t ∈ [1 , ∞ ), the elements u at and 1 − u at are bothpositive. • lim t →∞ k u bt x − x k = 0 for all x ∈ Hom( a, b ) J .6 lim t →∞ k u bt x − xu at k = 0 for all x ∈ Hom( a, b ) B . Definition 2.10
We call a C ∗ -category A separable if for each a, b ∈ Ob( A ) wehave compact sets k n ( a, b ) such that: • The union ∪ n ∈ N k n ( a, b ) is dense in Hom( a, b ) A ; • For each n , and a, b ∈ Ob( A ) we have k n ( a, b ) + k n ( a, b ) ⊆ k n +1 ( a, b ), k ∗ n ( a, b ) = k n ( a, b ), and λk n ( a, b ) ⊆ k n +1 ( a, b ) whenever | λ | ≤ n ; • For each n , and a, b, c ∈ Ob( A ), we have k n ( b, c ) k n ( a, b ) ⊆ k n +1 ( a, c ).To decode the above notation, we mean: • k n ( a, b ) + k n ( a, b ) = { x + y | x, y ∈ k n ( a, b ) } ; • k ∗ n ( a, b ) = { x ∗ | x ∈ k n ( a, b ); • λk n ( a, b ) = { λx | x ∈ k n ( a, b ); • k n ( b, c ) k n ( a, b ) = { xy | x ∈ K n ( b, c ) , y ∈ k n ( a, b ) } .If a C ∗ -category A is separable then each Banach space Hom( a, b ) A is sep-arable. Conversely, if A is a separable C ∗ -algebra, then A is separable in thesense of the above definition. Lemma 2.11
Let B be a separable C ∗ -category, and let J be a C ∗ -ideal. Thenthe pair ( B , J ) has a quasi-central set of approximate units. Proof:
It is well-known from the theory of C ∗ -algebras (see for example[GHT00]) that each pair of C ∗ -algebras (Hom( a, a ) B , Hom( a, a ) J ) has a quasi-central approximate unit { u at | t ∈ [1 , ∞ ) } .Let x ∈ Hom( a, b ) J . Then xx ∗ ∈ Hom( b, b ) J , and so for all x ∈ Hom( a, b ) J ,lim t →∞ k u bt xx ∗ − xx ∗ k = 0.By functional calculus, define v bt = ( u bt ) . Then again by functional calculus, u bt xx ∗ = ( v bt x )( v bt x ) ∗ and so, by the C ∗ -identitylim t →∞ k v bt x − x k = 0 . Similarly lim t →∞ k xv at x − x k = 0 . By the triangle inequality, we can combine the above two limits to seelim t →∞ k v bt x − xv at k = 0so by the C ∗ -identity lim t →∞ k u bt x − xu at k = 0and we are done. ✷ Note that if f : [0 , → F is a continuous function, we can define by functionalcalculus f ( u at ) ∈ Hom( a, a ) B for each u at ∈ Hom( a, a ) B .7 emma 2.12 Let B be a separable C ∗ -category, and let J be a C ∗ -ideal. Let ( u at ) be a quasicentral set of approximate units. Let f : [0 , → F be a continuousfunction such that f (0) = 0 . • Let b ∈ Hom( a, b ) B . Then lim t →∞ bf ( u at ) − f ( u bt ) b = 0 . • If f (1) = 0 then lim t →∞ k f ( u bt ) b k = 0 for all b ∈ Hom( a, b ) J . Proof:
The set of polynomials with constant coefficient 0 is dense in the set { f ∈ C [0 , | f (0) = 0 } by the Stone-Weierstrass theorem. Hence it suffices toprove the result for such polynomials. To do this, it suffices to prove the resultfor the polynomial f ( x ) = x . The first part of the result therefore follows fromthe definition of quasicentral set of approximate units.As for the second part of the result, observe that the set of functions f ∈ C [0 ,
1] such that lim t →∞ k f ( u bt ) b k = 0 for all b ∈ Hom( a, b ) J is an ideal inthe C ∗ -algebra of functions { f ∈ C [0 , | f (0) = f (1) = 1 } . By the Stone-Weierstrass theorem, as above, this algebra is generated by the function f ( x ) = x (1 − x ), so it suffices to prove the result for this function. By the definition ofa quasicentral set of approximate unitslim t →∞ k f ( u bt ) b k = lim t →∞ k u bt (1 − ut b ) b k ≤ lim t →∞ k (1 − ut b ) b k = 0for all b ∈ Hom( a, b ) J . ✷ Definition 2.13
Let A be a C ∗ -category. Then we define the additive comple-tion , A ⊕ , to be the category in which the objects are formal sequences of theform a ⊕ · · · ⊕ a n a i ∈ Ob( A )Repetitions are allowed in such formal sequences. The empty sequence is alsoallowed, and labelled 0. The morphism set Hom( a ⊕ · · · ⊕ a m , b ⊕ · · · ⊕ b n ) isdefined to be the set of matrices of the form x , · · · x ,m ... . . . ... x n, · · · x n,m x i,j ∈ Hom( a j , b i )and composition of morphisms is defined by matrix multiplication. The involu-tion is defined by the formula x , · · · x ,m ... . . . ... x n, · · · x n,m ∗ = x ∗ , · · · x ∗ n, ... . . . ... x ∗ ,m · · · x ∗ n,m ∗ x i,j ∈ Hom( a j , b i ) . ∗ - functor α : A → B , there is an induced ∗ -functor α ⊕ : A ⊕ → B ⊕ defined by writing α ⊕ ( a ⊕ · · · a n ) = α ( a ) ⊕ · · · ⊕ α ( a n ) a i ∈ Ob( A )and α ⊕ x , · · · x ,m ... . . . ... x n, · · · x n,m = α ( x , ) · · · α ( x ,m )... . . . ... α ( x n, ) · · · α ( x n,m ) x i,j ∈ Hom( a j , b i )If we have a faithful representation ρ : A → L , this therefore extends to afaithful representation ρ ⊕ : A ⊕ → L . We now use the same trick as we didin the spatial tensor product to define the norm on the morphism sets of theadditive completion A ⊕ . Definition 2.14
Let A and B be C ∗ -categories. We call a ∗ -functor α : A ⊕ →B ⊕ additive if: • α ( a ⊕ a ) = α ( a ) ⊕ α ( a ) for all objects a , a ∈ Ob( A ). • α ⊕ x , · · · x ,m ... . . . ... x n, · · · x n,m = α ( x , ) · · · α ( x ,m )... . . . ... α ( x n, ) · · · α ( x n,m ) for all x i,j ∈ Hom( a j , b i ).Given a ∗ -functor α : A → B , the induced ∗ -functor α ⊕ : A ⊕ → B ⊕ is clearlyadditive. If we have a ∗ -functor α : A → B ⊕ , we can also define an additive ∗ -functor α ⊕ : A ⊕ → B ⊕ by the same construction as above.It is easy to check that with the above induced additive ∗ -functors, thethe assignment A 7→ A ⊕ is a functor from the category of C ∗ -categories and ∗ -functors to the category of C ∗ -categories and additive ∗ -functors.We can proceed in much the same way with asymptotic functors. To beprecise, let ϕ : A B ⊕ be an asymptotic functor. Then we have an asympoticfunctor ϕ ⊕ : A ⊕ B ⊕ by writing • ϕ ( a ⊕ a ) = ϕ ( a ) ⊕ ϕ ( a ) for all objects a , a ∈ Ob( A ). • ( ϕ ⊕ ) t x , · · · x ,m ... . . . ... x n, · · · x n,m = ϕ t ( x , ) · · · ϕ t ( x ,m )... . . . ... ϕ t ( x n, ) · · · ϕ t ( x n,m ) for all x i,j ∈ Hom( a j , b i ) and t ∈ [1 , ∞ ).The induced asymptotic morphism ϕ ⊕ is additive in the same sense as theabove. 9 emma 2.15 Let → J i → B j → A → be a short exact sequence of graded C ∗ -categories. Then the induced sequence → J ⊕ i ⊕ → B ⊕ j ⊕ → A ⊕ → is also short exact. Proof:
Let a i and b j be objects. Since the ∗ -functor i is injective at the levelof objects, and i ⊕ x , · · · x ,m ... . . . ... x n, · · · x n,m = i ( x , ) · · · i ( x ,m )... . . . ... i ( x n, ) · · · i ( x n,m ) x i,j ∈ Hom( a j , b i )it follows that i ⊕ is also injective. Similarly, the ∗ -functor j ⊕ is surjective.So it suffices to check that Ker j ⊕ = Im i ⊕ . Since Ker j = Im i , we haveagreement in each term of the matrices. and the result follows. ✷ Let A , B , and C be C ∗ -categories, and let ϕ : A B and ψ : B C beasymptotic functors. We would like to be able to define the composition ψ ◦ ϕ : A C . Unfortunately, we cannot simply define ( ϕ ◦ ψ ) t = ϕ t ◦ ψ t for eachvalue t and expect an asymptotic morphism; this fails even in the C ∗ -algebracase.Fortunately, there are cases where the obvious construction works, and areparametrisation procedure which works more generally. Proposition 3.1
Let α : A → B be a ∗ -functor, and let ϕ : B → C be an asymp-totic functor. Then we have an asymptotic functor ϕ ◦ α : A → C defined bywriting ( ϕ ◦ α ) t = ϕ t ◦ α where t ∈ [1 , ∞ ) . Proof:
As noted in [Mit02], the ∗ -functor α is norm-decreasing on each mor-phism set, as well as being linear, and compatible with the involution and com-position in B and C . The required properties of an asymptotic morphism arenow easy to check. ✷ The following is similar.
Proposition 3.2
Let ϕ : A B be an asymptotic functor, and let α : B → C be a ∗ -functor. Then we have an asymptotic functor α ◦ ϕ : A C defined bywriting ( α ◦ ϕ ) t = α ◦ ϕ t where t ∈ [1 , ∞ ) . We have set things up with definition 2.10 so that the proof of the followingresult is the same as the proof of theorem 25.3.1 of [Bla98].10 heorem 3.3
Let A , B , and C be separable C ∗ -categories. Let ϕ : A B and ψ : B C be asymptotic functors. Then there is a continuous increasing func-tion r : [1 , ∞ ) → [1 , ∞ ) such that we have an asymptotic functor ψ ◦ r ϕ : B C defined by the formula ( ψ ◦ r ϕ ) t = ψ r ( t ) ◦ ϕ t .Further: • If we have continuous functions r, s : [1 , ∞ ) → [1 , ∞ ) such that ψ ◦ r ϕ is anasymptotic functor, and s ( t ) ≥ r ( t ) for all t , then ψ ◦ s ϕ is an asymptoticfunctor. • The homotopy class of the asymptotic functor ψ ◦ r ϕ is independent of r ,and depends only on the homotopy classes of the asymptotic functors φ and ϕ . E -theory Here we give the definition of gradings and generalise the framework already setout to C ∗ -categories with gradings. Then we prove we have an abelian groupstructure on the homotopy class of asymptotic functors from S b ⊗A ⊕ to B ⊕ b ⊗K . Definition 4.1
Let A be a C ∗ -category. A grading on A is a ∗ -functor δ A : A →A such that for each object a ∈ A , δ A ( a ) = a , and δ A = id.For each morphism set Hom( a, b ) A in a graded C ∗ -category we have a de-composition Hom( a, b ) A = Hom( a, b ) even A ⊕ Hom( a, b ) odd A . where x ∈ Hom( a, b ) even A if δ A ( x ) = x , and x ∈ Hom( a, b ) odd A if δ A ( x ) = − x . Wedefine the degree , deg( x ), of a morphism x to be 0 if x ∈ Hom( a, b ) even A and 1 if x ∈ Hom( a, b ) odd A .A C ∗ -category equipped with a grading is called a graded C ∗ -category. Wecan also consider graded ∗ -functors and graded asymptotic functors.A graded C ∗ -algebra can be considered the same thing as a graded C ∗ -category with one object. In our constructions, we make particular use of thefollowing graded C ∗ -algebra. Definition 4.2
We define S to be the C ∗ -algebra C ( R ) equipped with thegrading defined by saying an element f ∈ C ( X ) has degree 0 if the function f is even, and degree 1 if the function f is odd.We can realise the grading on S by the ∗ -homomorphism δ S : S → S definedby the formula δ ( f )( x ) = f ( − x ). We distinguish the graded C ∗ -algebra S from C ( R ), which as a graded C ∗ -algebra has the trivial grading, where ever elementhas degree 0. 11 efinition 4.3 Let A and B be graded C ∗ -categories with gradings δ A and δ B respectively. A graded ∗ -functor α : A → B is a ∗ -functor such that for all x ∈ Hom( a, b ) A we have δ B ( α ( x )) = α ( δ A ( x )).A graded asymptotic functor ϕ = ϕ t : A B is an asymptotic functor withthe additional property that for all x ∈ Hom( a, b ) A we havelim t →∞ k δ B ( ϕ t ( x )) − ϕ t ( δ A ( x )) k = 0 . For a morphism x ∈ Hom( a, b ) A let us define the degree of x by writingdeg( x ) = 0 if x ∈ Hom( a, b ) even A , and deg( x ) = 1 if x ∈ Hom( a, b ) odd A . We callmorphisms of degree 0 even, and morphisms of degree 1 odd. Definition 4.4
Let A and B be graded C ∗ -categories, with gradings δ A and δ B respectively. Then we define the graded tensor product A ˆ ⊗ B to be thesame collection of objects and Banach spaces of morphisms as the spatial tensorproduct A⊗B , but with involution and composition law defined by the formulae • ( x ⊗ y ) ∗ = ( − deg( x ) deg( y ) x ∗ ⊗ y ∗ ; • ( u ⊗ v )( x ⊗ y ) = ( − deg( v ) deg( x ) ux ⊗ vy ,and extending by linearity.With the above alternative composition law, the graded tensor product A ˆ ⊗ B is a C ∗ -category. We have a grading, δ , defined by the formula δ ( x ⊗ y ) = δ A ( x ) ⊗ δ B ( y )on elementary tensors; we again extend by linearity.Let us equip C [0 ,
1] with the trivial grading, where every morphism is even.Then if A is a graded C ∗ -category, we can form the tensor product I A = A ˆ ⊗ C [0 , ∗ -functors E , E : I A → A , and we can form a notion of homotopy.
Definition 4.5
Let ϕ, ψ : A B be graded asymptotic functors. Then a homotopy between ϕ and ψ is a graded asymptotic functor θ : A I B suchthat E ◦ θ = ϕ and E ◦ θ = ψ .Looking at objects, observe that • θ t ( a ) = ϕ t ( a ) = ψ t ( a ) for each object a ∈ Ob( A ). • Given x ∈ Hom( a, b ) A . we have θ t ( x )(0) = ϕ t ( x ) and θ t ( x )(1) = ψ t ( x ) forall t ∈ [1 , ∞ ).Being homotopic is an equivalence relation on the set of all graded asymp-totic functors between graded C ∗ -categories A and B . We denote the set ofequivalence classes by J A , B K . We also write J A ⊕ , B ⊕ K ⊕ to denote the set of ho-motopy equivalence classes of additive graded asymptotic functors. The obviousmap J A , B ⊕ K → J A ⊕ , B ⊕ K ⊕
12s a one to one correspondence.We call a Hilbert space H graded if we have a decomposition H = H ⊕ H .If H and H ′ are graded Hilbert spaces, and T : H → H ′ is a bounded linearmap, we call T even if T [ H ] ⊆ H ′ and T [ H ] ⊆ H ′ , and odd if T [ H ] ⊆ H ′ and T [ H ] ⊆ H ′ . The odd and even bounded linear maps give us a decompositionHom( H, H ′ ) L = Hom( H, H ′ ) even L ⊕ Hom(
H, H ′ ) odd L on each morphism set on the C ∗ -category L made up of all graded separa-ble Hilbert spaces and bounded linear maps. Hence we view L as a graded C ∗ -category. We have a non-unital graded C ∗ -subcategory K of all compactoperators between graded separable Hilbert spaces.Giiven a graded Hilbert space H = H ⊕ H , let H opp be the Hilbert spacewe obtain by reversing the grading, that is to say H = H ⊕ H .Observe that we have natural graded ∗ -isomorphisms K ⊕ K → K K ˆ ⊗ K → K Observe that K ⊕ = K , and so for any C ∗ -category A we have( A ˆ ⊗ K ) ⊕ = A ⊕ ˆ ⊗ K = A ˆ ⊗ K Now, observe that the algebra S is generated by two functions u, v ∈ S described by the formulae u ( x ) = e − x and v ( x ) = xe − x respectively. Wedefine a graded ∗ -homomorphism ∆ : S → S ˆ ⊗ S by writing ∆( u ) = u ⊗ u and∆( v ) = u ⊗ v + v ⊗ u . As shown in section 1.3 of [HG04], ∆ : S → S ˆ ⊗ S is theunique ∗ -homomorphism with the property∆( f ) = f ⊗ ⊗ f whenever f has compact support. Lemma 4.6
We have an abelian semigroup operation on the set J A ⊕ , B ⊕ ˆ ⊗ K K ⊕ defined by the formula [( ϕ t )] + [( ψ t )] = [( ϕ t ) ⊕ ( ψ t )] . Further, the abelian semi-group we define has an identity defined by taking the class of the zero asumptoticfunctor. Proof:
We can take the direct sum of two additive graded asymptotic functors A ⊕ B ⊕ ˆ ⊗ K in the obvious way. The resulting operation, defined above, iscertainly well-defined and associative, and the the class of the zero asymptoticfunctor, mapping every object of A ⊕ to the zero object of B ⊕ is the identity.So we must verify commutativity. In other words, given additive gradedasymptotic morphisms ϕ t , ψ t : A ⊕ B ⊕ ˆ ⊗ K , we want to define an additivegraded asymptotic homotopy between ϕ t ⊕ ψ t and ψ t ⊕ ϕ t .Let R θ = (cid:18) cos( πθ/ − sin( πθ/ πθ/
2) cos( πθ/ (cid:19) θ ∈ [0 , R θ is a morphism in the category K , so for elements x, y ∈ Hom( a, b ) B ) ⊕ ⊗ K we can form the element R θ ( x ⊕ y ) R ∗ θ .Now R ( x ⊕ y ) R ∗ = x ⊕ y and R ( x ⊕ y ) R ∗ = (cid:18) −
11 0 (cid:19) (cid:18) x y (cid:19) (cid:18) − (cid:19) = y ⊕ x The Hence, more generally we have a homotopy between asymptotic functors( ϕ t ) ⊕ ( ψ t ) and ( ψ t ) ⊕ ( ϕ t ) which makes the semigroup J A ⊕ , B ⊕ ˆ ⊗ K K ⊕ commu-tative ✷ Lemma 4.7
The set J S b ⊗A ⊕ , B ⊕ b ⊗K K ⊕ is an abelian group under the groupoperation defined above. Proof:
By the above it suffices to check that we have inverses.Let ϕ t : S ˆ ⊗ A ⊕ B ⊕ ˆ ⊗ K be a graded asymptotic additive functor, andlet δ : S ˆ ⊗ A ⊕ → S ˆ ⊗ A ⊕ be the grading, defined by taking the tensor productof the gradings on S and on A ⊕ . Define ϕ opp t = ϕ t ◦ δ .Let s ≥ st : S ˆ ⊗ S ˆ ⊗ A ⊕ ˆ ⊗ H B ⊕ ˆ ⊗ K by:Φ st ( a ) = ϕ t ( a ) a ∈ Ob( A ⊕ )and Φ st ( f ⊗ x ) = f ( x )( ϕ t ( x ) ⊕ ϕ opp t ( x )) f ∈ S, x ∈ Hom( a, b ) S ˆ ⊗ A ⊕ Then (Φ st ) t ∈ [1 , ∞ ) is a graded asymptotic functor. Moreover, as s → ∞ wehave that Φ st →
0. ObserveΦ st ( u ⊗ x ) = ϕ t ( x ) ⊕ ϕ opp t ( x ) Φ st ( v ⊗ x ) = 0for the functions u ( x ) = e − x and v ( x ) = xe − x which generate S .The function s s/ (1 − s ) is a monotone increasing bijection [0 , → [0 , ∞ ).We have a graded ∗ -homomorphism ∆ : S → S ˆ ⊗ S defined by writing ∆( u ) = u ⊗ u and ∆( v ) = u ⊗ v + v ⊗ u . So we can define θ : S ˆ ⊗ A ⊕ I B ⊕ ˆ ⊗ K bythe formula. θ ( x ) = Φ s/ (1 − s ) t ∆ ˆ ⊗ id A b ⊗K ( H ) ( x )Observe Φ s/ (1 − s ) t (∆ b ⊗ id A b ⊗K ( H ) )( u b ⊗ x ) = Φ s/ (1 − s ) t (∆( u ) b ⊗ x )= Φ s/ (1 − s ) t ( u b ⊗ u b ⊗ x )= u ( s )( ϕ t ⊕ ϕ opp t )( u b ⊗ x ) , and when s = 0, u ( s ) = e − = 1 and so we obtain ϕ t ⊕ ϕ opp t and when s → v ,Φ s/ (1 − s ) t (∆ b ⊗ id A b ⊗K ( H ) )( v b ⊗ x ) = Φ s/ (1 − s ) t (∆( v ) b ⊗ x )= Φ s/ (1 − s ) t (( u b ⊗ v + v b ⊗ u ) b ⊗ x ) ∼ Φ s/ (1 − s ) t (( u b ⊗ v ) b ⊗ x ) + Φ st (( v b ⊗ u ) b ⊗ x )= u ( s )( ϕ t ⊕ ϕ opp t )( v b ⊗ x ) + v ( s )( ϕ t ⊕ ϕ opp t )( u b ⊗ x ) , where the equivalence is valid, since equivalent graded asymptotic morphismsare homotopic by Proposition 2.8. Now at s = 0, v ( s ) = 0 and as s → v ( s ) →
0, so the second term is equal to 0 at the end points, and hence weobtain the same endpoints as above and we are done. ✷ Definition 4.8
For C ∗ -categories A and B , we define the graded E -theorygroups by E n ( A , B ) = J S b ⊗A ⊕ b ⊗K , Σ n B ⊕ b ⊗K K ⊕ As a special case we write E ( A , B ) = E ( A , B ) = J S b ⊗A ⊕ b ⊗K , B ⊕ b ⊗K K ⊕ . We can define a graded ∗ -functor c : S → F by evaluating a function atzero. Given an asympotic functor ϕ : A B we have an equivalence class[ α ] = [ c ⊗ α ⊕ ⊗ id] ∈ E ( A , B ). To explore the other properties of E -theory, we first set up an associativeproduct. Let ϕ : S ˆ ⊗ A ⊕ ˆ ⊗ K B ⊕ ˆ ⊗ K and ψ : S ˆ ⊗ B ⊕ ˆ ⊗ K C ⊕ ˆ ⊗ K begraded asymptotic functors. We have a canonical graded ∗ -homomorphism∆ : S → S ˆ ⊗ S . So by theorem 3.3 we have an increasing map r : [1 , ∞ ) → [1 , ∞ )giving us an asymptotic functor ψ ◦ r ϕ : S ˆ ⊗ A ⊕ ˆ ⊗ K → C ⊕ ˆ ⊗ K S ˆ ⊗ A ⊕ ˆ ⊗ K ∆ ⊗ id −→ S ˆ ⊗ S ˆ ⊗ A ⊕ ˆ ⊗ K id ⊗ ϕ t −→ S ˆ ⊗ B ⊕ ˆ ⊗ K ψ r ( t ) −→ C ⊕ ˆ ⊗ K Further, by theorem 3.3 the homotopy class of the asymptotic morphism ψ ◦ r ϕ depends only on the homotopy classes ψ and ϕ and not on the map r ,so we have a well-defined map of E -theory groups E ( A , B ) × E ( B , C ) → E ( A , C )given by the formula ([ ϕ ] , [ ψ ]) [ ψ ◦ r ϕ ]15ote also that composition of maps is associative, so the above product isassociative. To be precise, this means that the products( E ( A , B ) × E ( B , C )) × E ( C , D ) → E ( A , C ) × E ( C , D ) → E ( A , D )and E ( A , B ) × ( E ( B , C ) × E ( C , D )) → E ( A , B ) × E ( B , D ) → E ( A , D )are equal.If A and B are categories, a bivariant functor , F , from A to B is a functor F : A op × A → B . In other words, for objects a, b ∈ Ob( A ) we obtain an object F ( a, b ) ∈ Ob( B ), which is a contravariant functor in the first variable and acovariant functor in the second variable. Lemma 5.1 (Functoriality) E is a bivariant functor from the category whereobjects are graded C ∗ -categories and morphisms are ∗ -functors to the categorywhere objects are abelian groups and morphisms are group homomorphisms. Proof:
The identity property required for functors is clearly satisfied.Let A , B , C , D be graded C ∗ -categories. Let α : A → B be a graded ∗ -functor,then we have an abelian group E ( A , D ) = J S b ⊗A ⊕ b ⊗K , D ⊕ b ⊗K K ⊕ for all A anda group homomorphism α ∗ : E ( B , D ) → E ( A , D ) defined by α ∗ ( J x K ) = J x.α K ,where ( x.α ) t = x t ◦ (id S b ⊗ α ⊕ b ⊗ id K ) for all J x K ∈ E ( B , D ).Now consider the composition of graded ∗ -functors A α −→ B β −→ C on a repre-sentative x of J x K ,( β ◦ α ) ∗ ( x t ) = x t ◦ (id S b ⊗ ( β ⊕ ◦ α ⊕ ) b ⊗ id K )= x t ◦ (id S b ⊗ β ⊕ b ⊗ id K ) ◦ (id S b ⊗ α ⊕ b ⊗ id K )= β ∗ ( x t ) ◦ (id S b ⊗ α ⊕ b ⊗ id K )= α ∗ β ∗ ( x t )= ( α ∗ ◦ β ∗ )( x t ) . Similarly, we have for each graded C ∗ -category A , an abelian group E ( D , A ) = J S b ⊗D ⊕ b ⊗K , A ⊕ b ⊗K K ⊕ and for a graded ∗ -functor α , a group ho-momorphism α ∗ : E ( D , A ) → E ( D , B ) defined by α ∗ ( J y K ) = J α.y K , where( α.y ) t = ( α ⊕ b ⊗ id K ) ◦ y t for all J y K ∈ E ( D , A ). Considering the composition16f morphisms above and taking a representative y of J y K ∈ E ( D , A ) we see that( β ◦ α ) ∗ ( y t ) = (( β ⊕ ◦ α ⊕ ) b ⊗ id K ) ◦ y t = (( β ⊕ ◦ α ⊕ ) b ⊗ id K ) ◦ y t = ( β ⊕ b ⊗ id K ) ◦ ( α ⊕ b ⊗ id K ) ◦ y t = ( β ⊕ b ⊗ id K ) ◦ α ∗ ( y t )= β ∗ α ∗ ( y t )= ( β ∗ ◦ α ∗ )( y t ) . ✷ Note that the above functorially induced maps are a special case of theproduct. It follows that the product is a natural map, that is to say it commuteswith the homomorphisms α ∗ and α ∗ arising from a ∗ -functor α : A → B . Let A and B be unital graded C ∗ -categories, and let ϕ, ψ : A → B be graded ∗ -functors. As in [Mit01], we say ϕ and ψ are unitarily equivalent if for eachobject a ∈ Ob( A ) we have an even unitary element U a ∈ Hom( ϕ ( a ) , ψ ( a )) B such that for each element x ∈ Hom( a, b ) A we have that U b ϕ ( x ) = ψ ( x ) U a . Definition 6.1
We call a graded ∗ -functor ϕ : A → B an even unitary equiv-alence if there is a ∗ -functor ψ : B → A such that the compositions ϕ ◦ ψ and ψ ◦ ϕ are the unitarily equivalent to the identity ∗ -functor. Theorem 6.2
Let A , B , and C be unital graded C ∗ -categories, and let ϕ, ψ : B → C be graded ∗ -homomorphisms. Suppose that ϕ and ψ are unitarilyequivalent. Then the functors ψ ∗ , ϕ ∗ : E ( A , B ) → E ( A , C ) are equal. Proof:
Let U ∈ Hom( c, d ) C be a unitary element. Then as noted in [Mit01]we have a path of unitaries between the two elements (cid:18) U ∗ U (cid:19) and 1in the C ∗ -algebra Hom( c ⊕ d, c ⊕ d ) C .Let x ∈ Hom( a, b ) B . Then (cid:18) U ∗ b U b (cid:19) ϕ ( x ) ⊕ (cid:18) U ∗ a U a (cid:19) = 1 ⊕ ψ ( x )Hence the maps ϕ ⊕ ⊕ ψ are homotopic on each morphism set via theabove paths of unitaries. It follows that the additive ∗ -functors ϕ ⊕ , φ ⊕ : B ⊕ →C ⊕ are homotopic. The result now follows. ✷ Corollary 6.3
Let A , B , and C be unital graded C ∗ -categories, and let ϕ : B →C be a unitary equivalence. Then the map ϕ ∗ : E ( A , B ) → E ( A , C ) is an iso-morphism. We have a similar result in the other variable.
Proposition 6.4
Let A , B , and C be graded C ∗ -categories, and let ϕ, ψ : A → B be graded ∗ -homomorphisms. Suppose that ϕ and ψ are unitarily equivalent.Then the functors ψ ∗ , ϕ ∗ : E ( B , C ) → E ( A , C ) are equal. Corollary 6.5
Let A , B , and C be graded C ∗ -categories, and let ϕ : B → C be aunitary equivalence. Then the map ϕ ∗ : E ( B , C ) → E ( A , C ) is an isomorphism. Definition 6.6
Let A be a C ∗ -category. We call A additive if: • We have a zero object ∈ Ob( A ) such that for any object a ∈ Ob( A ) wehave unique morphisms 0 ∈ Hom(0 , a ) and 0 ∈ Hom( a, • For any two objects a, b ∈ Ob( A ) we have an object a ⊕ b equipped withmorphisms i a ∈ Hom( a, a ⊕ b ), i b ∈ Hom( b, a ⊕ b ), p a ∈ Hom( a ⊕ b, a ), and p b ∈ Hom( a ⊕ b, b ) such that p a i a = 1 a , p b i b = 1 b , and i a p a + i b p b = 1 a ⊕ b .It is clear from the definition that for any C ∗ -category A , the additive com-pletion A ⊕ is additive. Proposition 6.7
Let A be a unital additive C ∗ -category. Then A ⊕ is unitarilyequivalent to A . In particular, the category ( A ⊕ ) ⊕ is unitarily equivalent to A . Moreover,the C ∗ -category K is additive, so for any C ∗ -category B the categories B ˆ ⊗ K , B ⊕ ˆ ⊗ K , and ( B ˆ ⊗ K ) ⊕ are all equivalent. This yields the slightly simpler for-mulation of the E -theory groups E ( A , B ) = J S b ⊗A ˆ ⊗ K , B ˆ ⊗ K K . We shall use this formulation without further comment when convenientfrom now on.
Theorem 6.8 (Homotopy invariance)
Suppose α, β : A → B are homotopicgraded ∗ -functors and C is a C ∗ -category. Then the induced maps1. α ∗ , β ∗ : E ( C , A ) → E ( C , B ) , and2. α ∗ , β ∗ : E ( B , C ) → E ( A , C ) are equal. roof:
1. Let f : S b ⊗C b ⊗K A b ⊗K be a graded asymptotic morphism representing J f K ∈ E ( C , A ), then we can define α ∗ and β ∗ by α ∗ ( f t ) = ( α b ⊗ id K ) ◦ f t ,β ∗ ( f t ) = ( β b ⊗ id K ) ◦ f t . Since α and β are homotopic, and hence α ∗ and β ∗ are equal since we arehomotopic on representatives of our class.2. Similarly to (1) we can define α ∗ and β ∗ , for an representative g of J g K ∈ E ( B , C ) by α ∗ ( g t ) = g t ◦ (id S b ⊗ α b ⊗ id K ) ,β ∗ ( g t ) = g t ◦ (id S b ⊗ β b ⊗ id K ) . Then again since α and β are homotopic, we can clearly see that α ∗ and β ∗ are equal at the level of homotopy classes. ✷ The following immediately follow from the fact that we have an isomorphism K ˆ ⊗ K ∼ = K . Proposition 6.9
Let A and B be graded C ∗ -categories. Then we have naturalisomorphisms E ( A ⊗ K , B ) ∼ = E ( A , B ) and E ( A , B ⊗ K ) ∼ = E ( A , B )Finally, just as in the case of ordinary E -theory and KK -theory, we havethe following abstract form of duality. Theorem 6.10
Let D and E be graded C ∗ -categories with E -theory elements α ∈ E ( D , E ) and β ∈ E ( E , D ) such that the product of α and β is the identityon E ( D , D ) and the product of β and α is the identity on E ( E , E ) .Then for graded C ∗ -categories A and B we have natural isomorphisms E ( A ˆ ⊗ D , B ) ∼ = E ( A , B ˆ ⊗ E ) and E ( A ˆ ⊗ E , B ) ∼ = E ( A , B ˆ ⊗ D )19 Bott Periodicity
Now in E -theory we have a Bott periodicity result that can be found in [HG04]for the complex graded case and in [Bro19] for the real case. Before we state theresult we note that F p,q is the Clifford algebra on generators e , e , . . . e n suchthat e i = 1, f j = − e i e j = − e j e i , f i f j = − f j f i , and e i f j = − f j e i for all i, j .We define a grading by deeming the generators e i and f j to be odd. Then wehave the following: Theorem 7.1
There is a ∗ -homomorphism b : S → Σ b ⊗ F , inducing an iso-morphism which induces an invertible E -theory element [ b ] ∈ E ( F , Σ b ⊗ F , ) . In particular, the ∗ -homomorphism b : S → Σ b ⊗ F , induces an invertible E -theory element [ b ] ∈ E ( F , Σ b ⊗ F , ) . Let A and B be C ∗ -categories. Then we have a ∗ -functor b B : S B → Σ b ⊗B b ⊗ F , = Σ B b ⊗ F , defined by taking the tensor product with the identiy ∗ -functor on B . This ∗ -functor also, by construction of the E -theory product, yields an isomorphism[ b B ] ∈ E ( B , Σ B b ⊗ F , ) . Taking the E -theory product of an element of E ( A , B ) with [ b B ] gives us anatural isomorphism of E -theory groups E ( A , B ) → E ( A , Σ B b ⊗ F , ) . There is similarly an isomorphism E ( A , B ) → E (Σ A ˆ ⊗ F , , B )By theorem 6.10 we also have natural isomorphisms E ( A ˆ ⊗ F , , B ) ∼ = E ( A , Σ B ) E (Σ A , B ˆ ⊗ F , )To use this result we need a few facts on Clifford algebras. Observe F p,q ˆ ⊗ F r,s ∼ = F p + r,q + s Clifford algebras have the following form of periodicity [LM90] in the complexand real cases respectively C , ∼ = C , ∼ = C , R , ∼ = R , ∼ = R , To apply these results to E -theory, note the following:20 roposition 7.2 We have an isomorphism K ˆ ⊗ F , ≡ K . Proof:
Need to say something here. ✷ It follows that given C ∗ -categories A and B we have natural isomorphisms E ( A , B ) ∼ = E ( A ˆ ⊗ F , , B ) ∼ = E ( A , B ˆ ⊗ F , ) ∼ = E ( A ˆ ⊗ F , , B ˆ ⊗ F , )Hence we have natural isomorphisms E ( A ˆ ⊗ F , , B ) ∼ = E (Σ A ˆ ⊗ F , , B ) ∼ = E (Σ A , B )and E ( A , B ˆ ⊗ F , ) ∼ = E ( A , Σ B )By Bott periodicity and periodicity of Clifford algebras, in the complex caseit follows that we have a natural isomorphism E ( A , B ) ∼ = E ( A , B ˆ ⊗ C , ) ∼ = E ( A , Σ B )Similarly, in the real case we have a natural isomorphism E ( A , B ) ∼ = E ( A , B ˆ ⊗ R , ) ∼ = E ( A , Σ B ) Proposition 7.3
Let A and B be C ∗ -categories. We a natural isomorphism E ( A ˆ ⊗ F p,q , B ) ∼ = E ( A , B ˆ ⊗ F p,q ) Proof:
By Bott periodicity we have E ( A ˆ ⊗ F , , B ) ∼ = E ( A , Σ B ) ∼ = E ( A , B ˆ ⊗ F , )By periodicity of Clifford algebras, whether real or complex, we have E ( A ˆ ⊗ F , , B ) ∼ = E ( A ˆ ⊗ F , , B ) ∼ = E ( A , B )and E ( A , B ˆ ⊗ F , ) ∼ = E ( A , B )Hence, by properties of tensor products of Clifford algebras E ( A ˆ ⊗ F , , B ) ∼ = E ( A ˆ ⊗ F , , B ) ∼ = E ( A ˆ ⊗ F , , B ) . Similarly E ( A , B ˆ ⊗ F , ) ∼ = E ( A , B ˆ ⊗ F , ). So by the above E ( A ˆ ⊗ F , , B ) ∼ = E ( A , B ˆ ⊗ F , )The result now follows. ✷ Combining the above proposition with Bott periodicty, we immediately havethe following
Corollary 7.4
We have a natural isomorphism E ( A , B ) → E (Σ A , Σ B )21 Long Exact Sequences
Definition 8.1
Let J , B , and A be C ∗ -categories with the same set of objects.Let i : J → B and j : B → A be ∗ -functors such that i ( a ) = a and j ( a ) = a foreach object a ∈ Ob( A ). Then we call the sequence of ∗ -functors0 → J i → B j → A → short exact sequence if for all a, b ∈ Ob( J ) the sequence of morphism sets0 → Hom( a, b ) J i → Hom( a, b ) B j → Hom( a, b ) A → → J i → B j → A → C ∗ -categories, the C ∗ -category J is isomorphicunder the ∗ -functor i to a C ∗ -ideal in B , and the C ∗ -category A is isomorphicto the quotient B / J , with the map j the quotient map. The proof is similar tothe corresponding result for abelian groups. Proposition 8.2
Let → J → B π → A → be a short exact sequence of C ∗ -categories, where the C ∗ -category J is a C ∗ -ideal in the category B . Let { u at | t ∈ [1 , ∞ ) , a ∈ Ob( B ) } be a quasi-central setof approximate units for the pair ( B , J ) . Then there is an asymptotic functor σ : Σ A J such that if s : Hom( a, b ) A → Hom( a, b ) B is a set of continuoussections, then lim t →∞ k σ t ( f ⊗ x ) − f ( u t ) s ( x ) k = 0 for all f ∈ Σ , x ∈ Hom( a, b ) A . Proof:
Let Σ = Σ F . Observe that Σ A = Σ ⊗ A . Then by Lemma 2.12, wecan define a ∗ -functor σ : Σ A J by writing σ t ( f ⊗ x ) = f ( u t ) s ( x ) . Certainly lim t →∞ k σ t ( f ⊗ x ) − f ( u t ) s ( x ) k = 0 . Let s ′ : Hom( a, b ) B → Hom( a, b ) A be a different set of continuous sections.Define σ ′ : Σ A J by writing σ ′ t ( f ⊗ x ) = f ( u t ) s ′ ( x ). Then again by lemma2.12, lim t →∞ k f ( u t )( s ( x ) − s ′ ( x )) k = 0In particular, it follows thatlim t →∞ k σ t ( f ⊗ x ) − f ( u t ) s ′ ( x ) k = 022nd we are done. ✷ In order to show that E -theory is half exact in this framework, we first showthat we have exactness when we consider the mapping cylinder C α of the map α : A → A / J gives an exact sequence and then use some theory to show thatfor an ideal we have that J and C α are inverses in the E -theory category. Lemma 8.3
For a graded C ∗ -category B , the following are exact in the middle.1. E ( B , C α ) β ∗ −→ E ( B , A ) α ∗ −−→ E ( B , A / J ) is exact,2. E ( A / J , B ) α ∗ −−→ E ( A , B ) β ∗ −→ E ( C α , B ) is exact. Proof:
1. Let ϕ t : S b ⊗B b ⊗K A b ⊗K , and [ ϕ t ] ∈ E g ( B , A ) be such that α ∗ [ ϕ t ] = 0,i.e [ ϕ t ] ∈ Ker α ∗ . Then α ◦ ϕ t ∼ h
0. Now let θ : S b ⊗B b ⊗K A b ⊗K b ⊗ C [0 , α ◦ θ t is a homotopy between α ◦ ϕ t and 0. Then write θ : S b ⊗B b ⊗K A b ⊗ C [0 , b ⊗K ∼ = C A b ⊗K , and define ψ : S b ⊗B b ⊗K C α b ⊗K on objects as the identity and on mor-phisms such that ψ t = ϕ t ⊕ θ t . Then β ◦ ψ t = ϕ t , and β ∗ [ ψ t ] = [ ϕ t ] and so [ ϕ t ] ∈ Im β ∗ . So Ker α ∗ ⊆ Im α ∗ .Conversely, from homological algebra we know that Im β ∗ ⊆ Ker α ∗ isequivalent to α ∗ β ∗ = 0. Now for x t ∈ E ( B , C α ) α ∗ β ∗ ( x t ) = α ∗ ( β b ⊗ id K ◦ x t )= α b ⊗ id K ◦ β b ⊗ id K ◦ x t = ( α ◦ β ) b ⊗ id K ◦ x t = 0since α ◦ β is equal to the identity.2. Firstly β ∗ ◦ α ∗ = 0 so Im α ∗ ⊆ Ker β ∗ by a similar method to part(1). Nowwe show that Ker β ∗ ⊆ Im α ∗ . Let ϕ t : S b ⊗A b ⊗K B b ⊗K and q : C α b ⊗K →A b ⊗K be the identity on objects and a projection on morphisms. Then wewant θ t : S b ⊗ Σ A / J b ⊗K Σ B b ⊗K such that[ θ t ◦ (id S ◦ Σ α )] = Σ[ ϕ ] ∈ E (Σ A , Σ B ) , where α : S b ⊗A b ⊗K → S b ⊗A / J b ⊗K . Now let η : S b ⊗C α b ⊗K → I B b ⊗K (where I = [0 , ϕ t ◦ (id b ⊗ q ) and 0. Then by symmetryof homotopy we can obtain e η t : S b ⊗C α b ⊗K → I B b ⊗K , I = [0 , i : Σ S b ⊗A / J b ⊗K ∼ = S b ⊗ Σ A / J b ⊗K → S b ⊗C α b ⊗K , defined on objects as the identity and on morphisms by i ( g b ⊗ f b ⊗ k ) = g b ⊗ (0 ⊕ f ) b ⊗ k, where g ∈ S , f : [ − , → A / J and k ∈ K . Then (id S b ⊗ q ) ◦ i = 0. Define θ t : S b ⊗ Σ A / J b ⊗K Σ B b ⊗K by θ t = e η t ◦ i . Then we need to show that θ t ◦ (id S ◦ Σ α ) is homotopic to Σ ϕ . Now Σ ϕ t = e η t ◦ i ◦ (id S ◦ Σ α ), and so || θ t ◦ (id S ◦ Σ α ) − (Σ ϕ t ) || → , as t → ∞ and asymptotic equivalence implies homotopic equivalence byProposition 2.8, half exactness follows. ✷ Proposition 8.4
Let / / J / / (cid:15) (cid:15) B / / (cid:15) (cid:15) A (cid:15) (cid:15) / / / / J / / B / / A / / be a commuting diagram of short exact sequences of separable C ∗ -categories.Then this gives a commuting diagram Σ A / / (cid:15) (cid:15) J (cid:15) (cid:15) Σ A / / J in the E -theory category. The proof follows precisely from that of Proposition 5.8 in [GHT00], since all C ∗ -categories have the same objects, the morphisms sets have approximate unitsindexed the same and the morphisms satisfy the properties that the short exactsequence has for C ∗ -algebras. Likewise the Proposition below is proves just asin the case of Proposition 5.9 in [GHT00]. Proposition 8.5
Let σ : Σ A J be the asymptotic functor associated to theshort exact sequence → J → B → A → . hen if D is a separable C ∗ -category and if σ D : Σ A b ⊗D J b ⊗D is the morphism associated to the short exact sequence → J b ⊗D → B b ⊗D → A b ⊗D → , then σ D is equal to σ ⊗ id D . The following is just taken from [GHT00]; for a proof see Proposition 5.11. For C as a C ∗ -category, so viewed as a one object C ∗ -category, we obtain a naturalasymptotic functor. Proposition 8.6
The asymptotic morphism σ : Σ C C C coming from theshort exact sequence → Σ C → C C → C → , is just the identity morphism in the E -theory category. Let α : B → A be a ∗ -functor, and C α be the associated mapping cylinder.Let J ⊂ B be a C ∗ -ideal with quasi-central approximate unit. Then considerthe inclusion ∗ -functor, τ : J → C α . Consider the short exact sequence:0 → Σ J → C B → C α → . By proposition 8.2, we have an associated asymptotic functor σ : Σ C α Σ J .Then the following result follows from proposition 8.4; for exact details seeProposition 5.14 in [GHT00]. Proposition 8.7
The ∗ -functor Σ τ : Σ J → Σ C α and the asymptotic functor Σ σ : Σ C α Σ J are mutually inverse morphisms in the E -theory category. Lemma 8.8
Let → J β −→ B α −→ A → be a short exact sequence of graded C ∗ -categories and let D be a graded C ∗ -category. Then1. E ( D , J ) β ∗ −→ E ( D , B ) α ∗ −−→ E ( D , A ) is exact,2. E ( A , D ) α ∗ −−→ E ( B , D ) β ∗ −→ E ( J , D ) is exact. Proof:
1. First we consider the connection between J and C α . From Lemma 8.3,with α : B → A , it follows that the sequence E ( D , C α ) β ∗ −→ E ( D , B ) α ∗ −−→ E ( D , A )is exact. Then by Proposition 8.7, we have that E (Σ C α , Σ J ) ∼ = E (Σ J , Σ C α ) and hence by Corollary 7.4, J and C α are isomorphic in25 -theory. Then half exactness of the required sequence follows from thecommutative diagram:0 / / E ( D , C α ) / / ∼ = (cid:15) (cid:15) E ( D , B ) / / E ( D , A ) / / / / E ( D , J ) β ∗ / / E ( D , B ) α ∗ / / E ( D , J ) / /
02. Similarly, the proof of 1 works in the contravariant case. ✷ Now to extend to long exact sequences, we require the following Proposition.
Proposition 8.9
Let → J i −→ B α −→ A → , be a short exact sequence of graded C ∗ -categories. Then if A is contractible,then i induces an isomorphism between E ( D , J ) and E ( D , B ) . Proof:
Since A is contractible, it follows that for any separable C ∗ -category E ( D , A ) = 0 , since E is a homotopy invariant functor, we have that this isequivalent to having A as the zero category. Then applying half exactness toour short exact sequence, and using the above, we have E ( D , J ) → E ( D , B ) → and so i ∗ is surjective. Now we check that i ∗ is injective. Consider the shortexact sequence, → Σ A → C α → B → and once again apply half exactness: E ( D , Σ A ) → E ( D , C α ) → E ( D , B ) . Then since A is contractible, Σ A is also contracible and as J ∼ = C α in the E -theory category, we obtain the exact sequence → E ( D , J ) → E ( D , B ) , and i ∗ is injective, and hence E ( D , J ) ∼ = E ( D , B ) as required. ✷ Theorem 8.10
For the functor E from the category of graded C ∗ -categoriesand ∗ -functors to the category of abelian groups and group homomorphisms toevery short exact sequence of separable graded C ∗ -categories → J → B → A → , e obtain a long exact sequence of abelian groups · · · → E ( D , Σ B ) → E ( D , Σ A ) ∂ ∗ −→ E ( D , J ) → E ( D , B ) → E ( D , A ) , where the connecting map ∂ ∗ fits in to the commutative diagram E ( D , Σ A ) β ∗ & & ▼▼▼▼▼▼▼▼▼▼ δ ∗ / / E ( D , J ) E ( D , C α ) τ ∗ rrrrrrrrrr Proof:
Since E is half exact by Lemma 8.8, it suffices to show that we haveexactness at E ( D , Σ A ) and E ( D , J ). For exactness at E ( D , J ) consider theshort exact sequence 0 → Σ A → C α → B → . Then by half exactness, we have that E ( D , Σ A ) → E ( D , C α ) → E ( D , B )is exact, and then by equivalence of J and C α in the E -theory category, it follwsthat E ( D , Σ A ) ∂ ∗ −→ E ( D , J ) → E ( D , B )is exact as required. Now for exactness at E ( D , Σ A ), consider the C ∗ -category T = C (( − , , B ) with Ob( T ) = { Ob( A ) } Hom(( a, b ) , ( a ′ , b ′ )) = ( ( x, f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ∈ Hom( a,a ′ ) A , f ( − f (1)= α ( x ) ,f : [ − , −→ Hom( b,b ′ ) B continuous ) . Then we have a canonical embedding κ : Σ A → T , which induces a map from κ ∗ : E ( D , Σ A ) → E ( D , T ), which is an isomorphism since the quotient of T by Σ A is contractible and by Proposition 8.9, κ ∗ is an isomorphism. NowΣ B ∼ = C (( − , , B ) and we have a canonical embedding C (( − , , B ) → T ,which is homotopic to the composition Σ B Σ α −−→ Σ A κ −→ T . Then applying halfexactness to the following short exact sequence0 → C (( − , , B ) → T → C α → E ( D , C (( − , , B )) / / ∼ = (cid:15) (cid:15) E ( D , T ) / / (cid:15) (cid:15) E ( D , C α ) ∼ = (cid:15) (cid:15) E ( D , Σ B ) / / E ( D , Σ A ) / / E ( D , J )and hence exactness at E ( D , Σ A ) follows. ✷ The contravariant case is proved similarly.By Bott periodicity and Theorem 8.10, we obtain the following result.27 heorem 8.11
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