An identification of the Baum-Connes and Davis-Lück assembly maps
AAn identification of theBaum-Connes and Davis-Lückassembly maps
Julian KranzSeptember 28, 2020
The Baum-Connes conjecture predicts that a certain assembly map is anisomorphism. We identify the homotopy theoretical construction of the as-sembly map by Davis and Lück [DL98] with the category theoretical con-struction by Meyer and Nest [MN04]. This extends the result of Hamble-ton and Pedersen [HP04] to arbitrary coefficients. Our approach uses ab-stract properties rather than explicit constructions and is formally similarto Meyer’s and Nest’s identification of their assembly map with the originalconstruction of the assembly map by Baum, Connes and Higson [BCH94].
Contents
Or( G ) -spectrum K GA
135 Identification of the assembly maps 246 Exotic crossed products 29 a r X i v : . [ m a t h . K T ] S e p Introduction
Let G be a countable discrete group and A a separable G - C ∗ -algebra. TheBaum-Connes conjecure predicts that the Baum-Connes assembly map µ : K G ∗ ( E Fin
G, A ) → K ∗ ( A (cid:111) r G ) is an isomorphism. The map was defined by Baum, Connes and Higson[BCH94] using the equivariant KK -theory of Kasparov [Kas88]. Later, ahomotopy theoretical definition of the assembly map was given by Davis andLück [DL98]. They developed an abstract machinery to study isomorphismconjectures like the Baum-Connes conjecture or the Farrell-Jones conjecturein a common framework. Their machinery takes as input a family F ofsubgroups of G and an Or( G ) -spectrum E , i.e. a functor from the categoryof all homogeneous G -spaces G/H to the category of spectra. Every
Or( G ) -spectrum E has a natural extension to the category of G - CW -complexesand defines a G -equivariant homology theory H G ∗ ( − , E ) by taking homotopygroups. In this setting, the ( E , F , G ) -assembly map is the map H G ∗ ( E F G, E ) → H G ∗ (pt , E ) (1)induced by the projection E F G → pt where E F G denotes a classifying spacefor the family F .To obtain the Baum-Connes assembly map in (1), one takes F to be thefamily of finite subgroups and K GA to be an Or( G ) -spectrum satisfying π ∗ ( K GA ( G/H )) ∼ = K ∗ ( A (cid:111) r H ) (2)for all subgroups H ⊆ G . We call the resulting assembly map H G ∗ ( E Fin G, K GA ) → H G ∗ (pt , K GA ) the Davis-Lück assembly map . The construction of K GA has been done byDavis and Lück in the case A = C and by Mitchener [Mit04] in the gen-eral case. We will give a variant of Mitchener’s construction using MichaelJoachims K -theory spectrum for C ∗ -categories [Joa03].It is not at all obvious that this construction gives rise to the same assem-bly map as in [BCH94]. Identifications have been made in [HP04] for thecase A = C and in [Mit04] for the general case. However, both works relyon heavy machinery and omit a lot of detail. Furthermore, the constructionof the assembly map in [Mit04] contains some inconsistencies. For example,it is not clear to the author of this paper whether the K -theory class [ E K ] in[Mit04, Definition 6.2] is well-defined for a non-compact G -space K .2he main ingredient for our identification is yet another construction ofthe assembly map by Meyer and Nest [MN04]. Recall that the equivariant KK -groups KK G ( A, B ) are the morphism sets of a triangulated category KK G with separable G - C ∗ -algebras as objects. Let CI ⊆ KK G be the fullsubcategory of G - C ∗ -algebras Ind GH B induced from finite subgroups H ⊆ G .Let (cid:104)CI(cid:105) be the localizing subcategory generated by CI , i.e. the small-est full subcategory containing CI which is closed under KK G -equivalence,suspension, mapping cone extensions and countable direct sums. Every G - C ∗ -algebra can be approximated by a G - C ∗ -algebra in (cid:104)C I (cid:105) in the followingsense: Theorem 1.1 ([MN04, Proposition 4.6]) . Let A be a separable G - C ∗ -algebra.Then there is a G - C ∗ -algebra ˜ A ∈ (cid:104)CI(cid:105) and an element D ∈ KK G ( ˜ A, A ) ,which restricts to a KK H -equivalence for every finite subgroup H ⊆ G . Meyer and Nest identify the Baum-Connes assembly map with the map D ∗ : K ∗ ( ˜ A (cid:111) r G ) → K ∗ ( A (cid:111) r G ) , (3)which we call the Meyer-Nest assembly map . In fact, they achieve the iden-tification as follows:
Theorem 1.2 ([MN04, Theorem 5.2]) . The indicated maps in the followingdiagram are isomorphisms. K G ∗ ( E Fin G, ˜ A ) K ∗ ( ˜ A (cid:111) r G ) K G ∗ ( E Fin
G, A ) K ∗ ( A (cid:111) r G ) ∼ = µ ∼ = D ∗ D ∗ µ . We use the same strategy, to identify the Davis-Lück assembly map to theMeyer-Nest assembly map:
Theorem 1.3 (Theorem 5.3) . The indicated maps in the following diagramare isomorphisms: H G ∗ ( E Fin G, K G ˜ A ) H G ∗ (pt , K G ˜ A ) H G ∗ ( E Fin G, K GA ) H G ∗ (pt , K GA ) ∼ =pr ∗ ∼ = D ∗ D ∗ pr ∗ . (4)Here the lower hand map is the Davis-Lück assembly map and the righthand map is identical to the Meyer-Nest assembly map by (2).3et us outline the proof of the above theorem. First we prove that themap H G ∗ ( G/H, K G ˜ A ) → H G ∗ ( G/H, K GA ) is an isomorphism for any finite subgroup H ⊆ G . Indeed, by (2) this mapcan be identified with the map K ∗ ( ˜ A (cid:111) r H ) → K ∗ ( A (cid:111) r H ) . It is an isomorphism since D ∈ KK G ( ˜ A, A ) is a KK H -equivalence. Usingexcision we conclude that the map H G ∗ ( E Fin G, K G ˜ A ) → H G ∗ ( E Fin G, K GA ) is an isomorphism as well.To prove that the upper hand map in (4) is an isomorphism, we pro-ceed in two steps: First we show that the class of all ˜ A ∈ KK G , for whichit is an isomorphism, is localizing. This boils down to translating KK G -equivalences, suspensions, mapping cone extensions and direct sums in KK G to stable equivalences, loops, fiber sequences and wedge sums in spectra. Thenext step is to show that the upper hand map in (4) is an isomorphism forall generators ˜ A = Ind GH B ∈ CI . To see this, we use Green’s imprimitivitytheorem to construct a natural induction isomorphism H H ∗ ( X | H , K HB ) ∼ = H G ∗ ( X, K G Ind GH B ) , (5)for any G - CW -complex X . We can then identify the map in question withthe map H H ∗ ( E Fin G | H , K HB ) → H H ∗ (pt , K HB ) . This map is an isomorphism since H is finite. Outline of the paper
The paper is organized as follows: In section 2 we describe the category KK G and recall the construction of the Meyer-Nest assembly map. Section 3 con-tains the construction of equivariant homology theories and assembly mapsfrom Or( G ) -spectra as well as some basic homotopy theory for Or( G ) -spectra.The results are well known and can be found either explicitly or implicitlyin [DL98] and [LRV03]. But we hope that including them keeps the exposi-tion reasonably self-contained. In section 4 we construct the Or( G ) -spectrum K GA . We begin by discussing groupoid C ∗ -algebras and their reduced crossedproducts. For better functoriality properties, we consider the reduced crossedproduct of a groupoid C ∗ -algebra as a C ∗ - category rather than a C ∗ -algebra.Our construction is similar to the construction in [Mit04]. We then recall the4onstruction of Michael Joachims K -theory spectrum K for C ∗ -categories(see [Joa03]). Finally we define K GA by the formula K GA ( G/H ) := K ( A (cid:111) r G/H ) , where G/H denotes the transformation groupoid associated to the G -space G/H . We end the section by discussing some homotopy theoretical proper-ties of the functor A (cid:55)→ K GA . In section 5 we use all the technology developedso far to construct the induction isomorphism (5) and to prove Theorem 1.3.We include a discussion on variants of our results for other crossed productfunctors in section 6. Acknowledgements
This work arose from the authors master thesis at WWU Münster. Theauthor would like to thank Siegfried Echterhoff, Michael Joachim, ThomasNikolaus, Markus Schmetkamp and Felix Janssen for valuable discussions,comments, motivation and inspiration.
Notation If C is a category, we denote its homomorphism sets by C ( x, y ) , its collectionof objects by Ob( C ) and its opposite category by C op . All C ∗ -algebras arecomplex. If A is a C ∗ -algebra, we denote by M ( A ) its multiplier algebra andby Z ( A ) its center. If X is a set, we denote by (cid:96) ( X, A ) the right Hilbert- A -module ⊕ x ∈ X A and by L A ( (cid:96) ( X, A )) its adjointable operators. In this section, we recall the basic properties of equivariant KK -theory andthe definition of the Meyer-Nest assembly map. Throughout this section, G isa countable discrete group and all C ∗ -algebras are assumed to be separable.By a G -Hilbert space we mean a Hilbert space H together with a unitaryrepresentation u : G → U ( H ) . We denote the algebra of compact operatorson H by K ( H ) and equip it with the G -action given by conjugation with u . We denote by C ∗ G the category of all separable G - C ∗ -algebras with G -equivariant ∗ -homomorphisms. For two G - C ∗ -algebras A and B , the tensorproduct A ⊗ B denotes the minimal tensor product with the natural G -action. We denote the reduced crossed product of A and G by A (cid:111) r G . The suspension of A is the G - C ∗ -algebra SA := C ((0 , ⊗ A ∼ = C ((0 , , A ) with the trivial G -action on the first factor. The mapping cone of a morphism5 : A → B is given by Cone( π ) := { ( a, b ) ∈ A ⊕ C ((0 , , B ) : π ( a ) = b (1) } . The mapping cone triangle associated to π is the sequence SB → Cone( π ) → A π −→ B where the first map is given by inclusion and the second map is given byevaluation at . We also call Cone( π ) → A → B a mapping cone extension .A short exact sequence → I → A π −→ B → of G - C ∗ -algebras is called split exact , if there is a G -equivariant ∗ -homomorphism σ : B → A satisfying πσ = id B . For a subgroup H ⊆ G , we denoteby Res HG : C ∗ G → C ∗ H the obvious restriction functor. Let B be an H - C ∗ -algebra with H -action β . The induced algebra Ind GH B is the C ∗ -algebra ofall bounded functions f : G → B satisfying f ( gh ) = β h − ( f ( g )) for all g ∈ G and h ∈ H , such that the function gH (cid:55)→ (cid:107) f ( gH ) (cid:107) belongs to C ( G/H ) . Weequip Ind GH B with the G -action given by left translation.The following theorem is a collection of well-known results on equivariant KK -theory. For more details we refer to [Mey08] and the references therein. Theorem 2.1.
There is an additive category KK G with the same objects as C ∗ G and a functor KK G : C ∗ G → KK G with the following properties:i) KK G is G -homotopy invariant.ii) For any two separable G -Hilbert spaces H , H (cid:48) and any G - C ∗ -algebra A ,the stabilization morphism A ⊗ K ( H ) → A ⊗ K ( H ⊕ H (cid:48) ) is mapped to an isomorphism in KK G .iii) Any split exact sequence → I → A → B → of G - C ∗ -algebras ismapped to a split exact sequence in KK G .iv) KK G : C ∗ G → KK G is universal with the above properties in the sensethat any other functor from C ∗ G into an additive category with the aboveproperties uniquely factors through KK G .v) The category KK G is triangulated with respect to the suspension functor S and the mapping cone triangles SB → Cone( π ) → A π −→ B. e write KK Gn ( A, B ) := KK G ( A, S n B ) := KK G ( A, S n B ) .vi) We have Bott periodicity: KK Gn ( A, B ) ∼ = KK Gn +2 ( A, B ) .vii) Topological K -theory is given by K ∗ ( A ) ∼ = KK ∗ ( C , A ) := KK { e }∗ ( C , A ) .viii) Let H ⊆ G be a subgroup and A a G - C ∗ -algebra. Then the functors Ind GH : C ∗ H → C ∗ G Res HG : C ∗ G → C ∗ H (cid:111) r G : C ∗ G → C ∗ ⊗ A : C ∗ G → C ∗ G uniquely extend to functors Ind GH : KK H → KK G Res HG : KK G → KK H (cid:111) r G : KK G → KK ⊗ A : KK G → KK G . Furthermore,
Ind GH : KK H → KK G is left adjoint to Res HG : KK G → KK H . The isomorphisms in KK G are also called KK G -equivalences. A G - C ∗ -algebra A is called KK G -contractible , if it is isomorphic to in KK G . Definition 2.2.
A full subcategory
C ⊆ KK G is called localizing , if it isclosed under KK G -equivalence, suspension, mapping cone extensions andcountable direct sums. Being closed under mapping cone extensions meansthat if Cone( π ) → A π −→ B is a mapping cone extension, then Cone( π ) , A and B belong to C if at least two of them belong to C .The restriction to countable direct sums in the above definition is neces-sary in order to stay in the realm of separable C ∗ -algebras. For any fullsubcategory C ⊆ KK G , there is a smallest localizing subcategory (cid:104)C(cid:105) ⊆ KK G containing C . Definition 2.3 ([MN04, Definition 4.1]) . Let
CI ⊆ KK G denote the fullsubcategory of G - C ∗ -algebras of the form Ind GH B , where H ⊆ G is a finitesubgroup and B is an H - C ∗ -algebra. Let CC ⊆ KK G denote the full subcat-egory of G - C ∗ -algebras N , such that N is KK H -contractible for any finitesubgroup H ⊆ G . Theorem 2.4 ([MN04, Theorem 4.7]) . The localizing subcategories (cid:104)CI(cid:105) ⊆ KK G and CC ⊆ KK G are complementary in the sense that for any G - C ∗ -algebra A , there is an exact triangle SN → ˜ A D −→ A → N ith N ∈ CC and ˜ A ∈ (cid:104)CI(cid:105) . The above triangle is unique up to isomorphism. Remark 2.5.
The morphism D : ˜ A → A is called the Dirac morphism .Note that it follows from the adjunction of
Ind GH and Res HG that D is a KK H -equivalence for any finite subgroup H ⊆ G . Theorem 2.6 ([MN04, Theorem 5.2]) . The indicated maps in the followingdiagram are isomorphisms. K G ∗ ( E Fin G, ˜ A ) K ∗ ( ˜ A (cid:111) r G ) K G ∗ ( E Fin
G, A ) K ∗ ( A (cid:111) r G ) ∼ = µ ∼ = D ∗ D ∗ µ In particular, the Baum-Connes assembly map can canonically be identifiedwith the map D ∗ : K ∗ ( ˜ A (cid:111) r G ) → K ∗ ( A (cid:111) r G ) . We call the above map the
Meyer-Nest assembly map.
In this section, we recall the basic machinery of [DL98] in order to writedown the Davis-Lück assembly map. We also state some homotopy theore-tical results which will allow us to prove that the class of G - C ∗ -algebras, forwhich the Davis-Lück assembly map is an isomorphism, is localizing.Throughout this section, we work in the category of compactly generatedweak Hausdorff spaces with continuous maps (see [Str09]) and denote thiscategory by Top . Similarly, we denote the category of pointed compactlygenerated weak Hausdorff spaces with pointed continuous maps by
Top ∗ .These categories are closed symmetric monoidal with respect to the product X × Y respectively the smash product X ∧ Y . We denote the mapping spacesby Top ( X, Y ) respectively Top ∗ ( X, Y ) . We write X + := X (cid:96) { + } to equip aspace X with a disjoint basepoint + and reserve the notation Y + for the one-point compactification of a locally compact space Y . We use the notation Ω X := Top ∗ ( S , X ) and Σ X := S ∧ X to denote the loop space and thesuspension of a pointed space X . Recall that there is a natural adjunctionhomeomorphism Top ∗ (Σ X, Y ) ∼ = Top ∗ ( X, Ω Y ) . (6)We denote by π n ( X ) := π (Ω n X ) , n ≥ the n -th homotopy group of apointed space X . A pointed map is called a weak equivalence , if it induces anisomorphism on all homotopy groups. For a discrete group G , we denote by Top G the category of (compactly generated weak Hausdorff) G -spaces and8 -equivariant maps. We equip the mapping spaces Top G ( X, Y ) with thetopology inherited from the inclusion Top G ( X, Y ) ⊆ Top ( X, Y ) . Spaces and spectra over the orbit category
Definition 3.1.
A spectrum E is a sequence of pointed spaces E n , n ≥ together with pointed maps E n → Ω E n +1 called structure maps . A map f : E → F of spectra is a sequence of pointed maps f n : E n → F n whichcommute with the structure maps. We denote the category of spectra by Sp . Definition 3.2.
Let E be a spectrum. For n ∈ Z , the n -th homotopy group of E is the group π n ( E ) := colim k →∞ π n + k ( E k ) . Here the colimit is taken with respect to the maps π n + k ( E k ) → π n + k (Ω E k +1 ) ∼ = π n + k +1 ( E k +1 ) . A map of spectra is called a stable equivalence , if it induces an isomorphismon all homotopy groups.
Definition 3.3.
Let G be a discrete group. The orbit category Or( G ) is thecategory of all homogeneous G -sets G/H together with G -equivariant maps. Definition 3.4 ([DL98, Definition 1.2]) . A pointed
Or( G ) -space is a functor X : Or( G ) → Top ∗ . A map of pointed Or( G ) -spaces is a natural transforma-tion of the underlying functors. Analogously, we define (pointed) Or( G ) op -spaces and Or( G ) -spectra. Example 3.5.
Let X be a G -space. We can define a pointed Or( G ) op -space G/H (cid:55)→
Top G ( G/H, X ) + ∼ = X H + where X H ⊆ X denotes the space of H -fixed-points. Definition 3.6 ([DL98, Definition 1.4]) . Let X be a pointed Or( G ) op -spaceand Y a pointed Or( G ) -space. The balanced smash product of X and Y isthe pointed space X ∧ Or( G ) Y := (cid:95) G/H ∈ Or( G ) X ( G/H ) ∧ Y ( G/H ) / ∼ where the equivalence relation ∼ is generated by the relations f ∗ x ∧ y ∼ x ∧ f ∗ y, x ∈ X ( G/H ) , y ∈ Y ( G/K ) , f ∈ Or( G )( G/K, G/H ) . E is an Or( G ) -spectrum, we define the balanced smash product X ∧ Or( G ) E of X and E as the spectrum given by the sequence of pointed spaces X ∧ Or( G ) E n , n ∈ N with structure maps given by the adjoints of the natural maps ( X ∧ Or( G ) E n ) ∧ S ∼ = X ∧ Or( G ) ( E n ∧ S ) → X ∧ E n +1 under the adjunction (6). Definition 3.7 (c.f. [DL98, Definition 4.3]) . Let X be a G - CW -complex and E an Or( G ) -spectrum. The G -equivariant homology of X with coefficientsin E is given by H G ∗ ( X, E ) := π ∗ ( Top G ( − , X ) + ∧ Or( G ) E ) . Remark 3.8.
Note that there is a natural homeomorphism
Top G ( − , G/H ) + ∧ Or( G ) E → E ( G/H ) , f ∧ x (cid:55)→ f ∗ ( x ) for any subgroup H ⊆ G . In particular, we have a natural isomorphism H G ∗ ( G/H, E ) ∼ = π ∗ E ( G/H ) . Proposition 3.9 ([DL98, Lemma 4.4]) . The functor H G ∗ ( − , E ) defines ageneralized homology theory for G - CW -complexes. Definition 3.10.
A collection F of subgroups of G is called a family of sub-groups , if it is closed under conjugation and taking subgroups. A classifyingspace for F is a G - CW -complex E F G such that the fixed points ( E F G ) H with respect to a subgroup H ⊆ G are contractible for H ∈ F and emptyfor H / ∈ F . Lemma 3.11 ([DL98, Section 7]) . For any family F of subgroups of G , thereis a classifying space E F G . Furthermore, E F G is unique up to G -homotopyequivalence. Definition 3.12 ([DL98, Section 5.1]) . Let G be a discrete group, F a familyof subgroups and E an Or( G ) -spectrum. The ( E , F , G ) -assembly map is themap H G ∗ ( E F G, E ) → H G ∗ (pt , E ) induced by the projection E F G → pt .The following Lemma is a special case of [DL98, Lemma 1.9]. Lemma 3.13.
Let H ⊆ G be a subgroup. Consider the induction functor I : Or( H ) → Or( G ) , H/K (cid:55)→ G × H H/K ∼ = G/K. et E be an Or( H ) -spectrum and denote by I ∗ E the Or( G ) -spectrum givenby I ∗ E ( G/K ) :=
Top G ( I ( − ) , G/K ) + ∧ Or( H ) E ∼ = Top H ( − , G/K ) + ∧ Or( H ) E . Let X be a G - CW -complex and denote by X | H the same space with the actionrestricted to H . Then there is a natural isomorphism H H ∗ ( X | H , E ) ∼ = H G ∗ ( X, I ∗ E ) . Homotopy theory for
Or( G ) -spectra The last part of this section deals with those homotopy theoretical state-ments which guarantee that the class of G - C ∗ -algebras A , for which the ( G, Fin , K GA ) -assembly map is an isomorphism, is localizing. We recall somebasic homotopy theoretical terminology and refer to [tD08] for more details.Let f : X → Y be a pointed map and let x ∈ X, y ∈ Y be the basepoints.We denote by C f the cone of f , i.e. the pointed space obtained from ( X × [0 , ∪ Y by gluing X × { } to Y along f and by collapsing X × { } ∪ { x } × [0 , to a point. The homotopy fiber of f is the pointed space F f := { ( x, γ ) ∈ X × Top ([0 , , Y ) : f ( x ) = γ (1) , γ (0) = y } whose basepoint is given by ( x , y ) . Both the cone and the homotopy fiberdefine functors on a category with pointed maps as objects and commutativesquares as morphisms. There are natural pointed homeomorphisms C Σ f ∼ = Σ C f , F Ω f ∼ = Ω F f . (7)We define the cone C f of a map f : E → F of spectra as the sequence ofspaces C f n with structure maps given by the adjoints of the maps Σ C f n ∼ = C Σ f n → C f n +1 . The homotopy fiber of f is defined analogously, using the second homeomor-phism of (7). Let X f −→ Y g −→ Z be a sequence of pointed maps together witha homotopy h : [0 , × X → Z of pointed maps such that h is equal to gf and h is the constant map. We call X f −→ Y g −→ Z a cofiber sequence , if thecanonical map C f → Z, (cid:40) ( x, t ) (cid:55)→ h t ( x ) y (cid:55)→ g ( y ) is a weak equivalence. Dually, we call X f −→ Y g −→ Z a fiber sequence, if thecanonical map X → F g , x (cid:55)→ ( f ( x ) , t (cid:55)→ h t ( x ))
11s a weak equivalence. We drop the homotopy from our notation wheneverit is clear from context. By replacing (homotopies of) pointed maps by(homotopies of) maps of (
Or( G ) -)spectra and by replacing weak equivalencesby stable equivalences, we obtain analogous notions of (co-)fiber sequencesof ( Or( G ) -)spectra. Lemma 3.14 ([LRV03, Lemma 2.6]) . A sequence E → F → G of maps ofspectra is a fiber sequence if and only if it is a cofiber sequence. In this casethere is a natural long exact sequence · · · → π n +1 ( G ) → π n ( E ) → π n ( F ) → π n ( G ) → π n − ( F ) → · · · of homotopy groups. The following well known Lemma is an easy consequence of Lemma 3.14.
Lemma 3.15 (c.f. [Sch12, Proposition 6.12(i)]) . Let E i , i ∈ I be a collectionof spectra. Then the natural map π ∗ (cid:32)(cid:95) i ∈ I E i (cid:33) → (cid:77) i ∈ I π ∗ ( E i ) is an isomorphism. Lemma 3.16 ([DL98, Lemma 4.6]) . Let E → F be a stable equivalence of Or( G ) -spectra and X a G - CW -complex. Then the induced map H G ∗ ( X, E ) → H G ∗ ( X, F ) is an isomorphism. The following lemma is inspired by [DL98, Definition 3.13].
Lemma 3.17.
Let X be a G - CW -complex. Then the functor Top G ( − , X ) + ∧ Or( G ) − maps cofiber sequences of Or( G ) -spectra to cofiber sequences of spectra.Proof. The preceding lemma shows that the functor
Top G ( − , X ) + ∧ Or( G ) − commutes with stable equivalences. It therefore suffices to show that italso commutes with taking cones. To see that this is indeed the case, wereformulate the definition of the cone. Consider the category C representedby the following diagram: c c c (8)12here is a natural C op -space E C given by E C ( c ) = [0 , , E C ( c ) = { } , E C ( c ) = { } on objects and by the obvious inclusions on morphisms. A morphism f : E → F of spectra gives rise to a C -spectrum D f by mapping the diagram (8) to thediagram E F pt f . Similarly, a morphism f : E → F of Or( G ) -spectra gives rise to a C ×
Or( G ) -spectrum D f . Now the cone of f can be rewritten as C f = E C + ∧ C D f . Using associativity of balanced smash products and observing that the con-struction f (cid:55)→ D f commutes with our functor Top G ( − , X ) + ∧ Or( G ) − , weobtain the desired formula C Top G ( − ,X ) + ∧ Or( G ) f = E C + ∧ C D Top G ( − ,X ) + ∧ Or( G ) f = E C + ∧ C ( Top G ( − , X ) + ∧ Or( G ) D f )= Top G ( − , X ) + ∧ Or( G ) ( E C + ∧ C D f )= Top G ( − , X ) + ∧ Or( G ) C f Or( G ) -spectrum K GA In this section, we associate an
Or( G ) -spectrum K GA to every G - C ∗ -algebra A , closely following [Mit04] and [Joa03]. We call the resulting assembly map H G ∗ ( E Fin G, K GA ) → H G ∗ (pt , K GA ) the Davis-Lück assembly map , where
Fin denotes the family of finite sub-groups of G . Let us motivate the construction of K GA . In order for the right13and sides of the Baum-Connes and Davis-Lück assembly maps to match,we need an isomorphism H G ∗ (pt , K GA ) = π ∗ ( K GA ( G/G )) ! ∼ = K ∗ ( A (cid:111) r G ) . In order for the left hand sides to match, we expect an isomorphism H G ∗ ( X, K GA ) ! ∼ = KK G ∗ ( C ( X ) , A ) for all cocompact proper G -spaces X . For a finite subgroup H ⊆ G and X = G/H , this boils down to the isomorphism H G ∗ ( G/H, K GA ) ! ∼ = KK G ∗ ( C ( G/H ) , A ) ∼ = KK H ∗ ( C , A ) ∼ = K ∗ ( A (cid:111) r H ) . Now we are tempted to define K GA ( G/H ) := K ( A (cid:111) r H ) , where K : C ∗ → Sp is a functor representing K -theory for C ∗ -algebras. Unfortunately, theassignment G/H (cid:55)→ A (cid:111) r H does not define a functor on the orbit category.To solve this, we replace A (cid:111) r H by a Morita-equivalent C ∗ - category A (cid:111) r G/H . We then define K GA ( G/H ) to be the K -theory spectrum in the sense of[Joa03] of that C ∗ -category. The construction of the C ∗ -category A (cid:111) r G/H given here is a minor modification of the construction in [Mit04].
Groupoid actions and crossed products
Definition 4.1. A unital C ∗ -category is a small category A , whose morphismsets A ( x, y ) are complex Banach spaces equipped with conjugate linear in-volution maps ∗ : A ( x, y ) → A ( y, x ) satisfying the axiomsi) ( a ∗ ) ∗ = a ii) (cid:107) ab (cid:107) ≤ (cid:107) a (cid:107)(cid:107) b (cid:107) iii) (cid:107) a ∗ a (cid:107) = (cid:107) a (cid:107) iv) ( ab ) ∗ = b ∗ a ∗ v) a ∗ a ≥ for all morphisms a ∈ A ( y, z ) , b ∈ A ( x, y ) . A unital C ∗ -functor is a functorbetween C ∗ -categories which is linear on morphism sets and preserves theinvolution. A (non-unital) C ∗ -category is defined in the same way as a uni-tal C ∗ -category except that the morphism sets are not required to containidentity morphisms. A (non-unital) C ∗ -functor is defined in the same wayas a C ∗ -functor except that it does not need to preserve identity morphisms.By dropping the norm from the definition, we obtain analogous notions of ∗ -categories and ∗ -functors. 14 efinition 4.2. A groupoid G is a small category with all morphisms invert-ible. We do not equip groupoids with any topology. A groupoid morphism F : G → H is a functor between the underlying categories. A G - C ∗ -algebra A is a functor x (cid:55)→ A x from G to the category of C ∗ -algebras. A G -equivariantmorphism A → B is a natural transformation of the underlying functors.Sticking to the notation for G - C ∗ -algebras, we denote the action of anelement g ∈ G ( x, y ) by α g : A x → A y and say that the G -action is denotedby α . Remark 4.3.
Our definition of G - C ∗ -algebras is adapted from [Mit04] andformally differs from the classical definition in [LG99]. Usually, a G - C ∗ -algebra A is defined as a single C ∗ -algebra A together with a non-degenerate ∗ -homomorphism ϕ : C (Ob( G )) → ZM ( A ) and an additional datum imple-menting the action. Our definition can be obtained from the classical one bytaking the fibers A x := A/ϕ ( C (Ob( G ) \ { x } ) A ) . Example 4.4.
Let G be a discrete group acting on a set X . The trans-formation groupoid X has the points of X as objects and morphisms givenby X ( x, y ) := { g ∈ G : gx = y } . Every G -equivariant map X → Y gives rise to a faithful (i.e. injective onmorphism sets) groupoid morphism X → Y . In particular, there is a naturalmorphism X → G = pt . By precomposition with this morphism, every G - C ∗ -algebra can be regarded as an X - C ∗ -algebra as well. Definition 4.5.
Let G be a groupoid and A a G - C ∗ -algebra with G -actiondenoted by α . The convolution category A G is the category with the sameobjects as G and morphism sets given by formal sums A G ( x, y ) := (cid:40) n (cid:88) i =1 a i u g i , n ∈ N , g i ∈ G ( x, y ) , a i ∈ A y (cid:41) . We define composition and involution on A G by linear extension of the for-mulas au g · bu h := aα g ( b ) u gh , ( au g ) ∗ := α g − ( a ) ∗ u g − for a ∈ A z , b ∈ A y , h ∈ G ( x, y ) , g ∈ G ( y, z ) . In this way, A G becomes a ∗ -category. Definition 4.6.
Let A be a G - C ∗ -algebra with G -action α . Let x, y ∈ Ob( G ) and choose z ∈ Ob( G ) such that G ( z, x ) is nonempty. To each f ∈ A G ( x, y ) ,we associate an adjointable operator Λ A, G ,z ( f ) : (cid:96) ( G ( z, x ) , A z ) → (cid:96) ( G ( z, y ) , A z )
15f Hilbert- A z -modules, defined by linear extension of the formula Λ A, G ,z ( au g ) ξ ( h ) := α h − ( a ) ξ ( g − h ) for a ∈ A y , ξ ∈ (cid:96) ( G ( z, x ) , A z ) , g ∈ G ( x, y ) and h ∈ G ( z, y ) . The reducednorm of f is given by (cid:107) f (cid:107) r := (cid:107) Λ A, G ,z ( f ) (cid:107) . (9)The reduced crossed product A (cid:111) r G is the C ∗ -category obtained from A G bycompleting all the morphism sets with respect to the reduced norm. Remark 4.7.
The norm in (9) does not depend on the choice of z . Indeed,if z (cid:48) ∈ Ob( G ) is another object such that G ( x, z (cid:48) ) is nonempty, we may pick amorphism g ∈ G ( z, z (cid:48) ) . A calculation then shows that for every f ∈ A G ( x, y ) ,the diagram (cid:96) ( G ( z, x ) , A z ) (cid:96) ( G ( z (cid:48) , x ) , A z (cid:48) ) (cid:96) ( G ( z, y ) , A z ) (cid:96) ( G ( z (cid:48) , y ) , A z (cid:48) ) Λ A, G ,z ( f ) ρ g ⊗ α g ∼ = Λ A, G ,z (cid:48) ( f ) ρ g ⊗ α g ∼ = (10)commutes where ρ g ⊗ α g is defined by the formula ( ρ g ⊗ α g ) ξ ( h ) = α g ( ξ ( hg )) . Thus, we have (cid:107) Λ A, G ,z ( − ) (cid:107) = (cid:107) Λ A, G ,z (cid:48) ( − ) (cid:107) .It is sometimes convenient to have a fixed representation of the reducedcrossed product. We call the representation Λ A, G := (cid:89) z ∈ Ob( G ) Λ A, G ,z : A G → (cid:89) z ∈ Ob( G ) L A z (cid:77) x ∈ Ob( G ) (cid:96) ( G ( z, x ) , A z ) (11)the regular representation of A G . Lemma 4.8. i) Let
A, B be G - C ∗ -algebras and ϕ : A → B a G -equivariantmorphism. Then the canonical ∗ -functor ϕ G : A G → B G , defined as the identity on objects and as au g (cid:55)→ ϕ ( a ) u g on morphisms,extends to a C ∗ -functor ϕ (cid:111) r G : A (cid:111) r G → B (cid:111) r G . ii) Let A be a G - C ∗ -algebra and ϕ : H → G a faithful groupoid morphism.Denote the H - C ∗ -algebra obtained by precomposition with ϕ also by A . hen the natural ∗ -functor id A ϕ : A H → A G , defined by x (cid:55)→ ϕ ( x ) on objects and au g (cid:55)→ au ϕ ( g ) on morphisms extendsto an isometric C ∗ -functor id A (cid:111) r ϕ : A (cid:111) r H → A (cid:111) r G . Proof.
For the first statement, fix x, y ∈ Ob( G ) and fix z ∈ Ob( G ) such that G ( z, x ) is non-empty. Consider the following commutative diagram. A G ( x, y ) L A z (cid:16)(cid:76) w ∈ Ob( G ) (cid:96) ( G ( z, w ) , A z ) (cid:17) M ( A z ⊗ K ( (cid:76) w ∈ Ob( G ) (cid:96) G ( z, w ))) B G ( x, y ) L B z (cid:16)(cid:76) w ∈ Ob( G ) (cid:96) ( G ( z, w ) , B z ) (cid:17) M ( B z ⊗ K ( (cid:76) w ∈ Ob( G ) (cid:96) G ( z, w ))) ϕ G Λ A, G ,z ∼ = ϕ z ⊗ idΛ B, G ,z ∼ = . The horizontal isomorphisms are the standard identifications (c.f. [Lan95,Theorem 2.4 and p. 37]). In general, ϕ z ⊗ id does not extend to the wholemultiplier algebra. However, it extends to a C ∗ -subalgebra which containsthe image of Λ A, G ,z by [EKQR06, Definition A.3, Proposition A.6 (i)]. Inany case, the extension of ϕ z ⊗ id is norm-decreasing. Since the horizontalarrows in the above diagram are isometric by definition, ϕ G must be norm-decreasing as well.For the second statement of the Lemma, fix x, y ∈ Ob( H ) and pick z ∈ Ob( H ) such that H ( z, x ) is nonempty. We have to prove the follow-ing equation. (cid:107) Λ A, H ,z ( f ) (cid:107) = (cid:107) Λ A, G ,ϕ ( z ) ◦ (id A ϕ )( f ) (cid:107) , ∀ f ∈ A H ( x, y ) . (12)Let S ⊆ G ( ϕ ( z ) , ϕ ( z )) be a system of coset representatives for H ( z, z ) \G ( ϕ ( z ) , ϕ ( z )) (this expression makes sense since ϕ is faithful). We get a direct sum decom-position (cid:96) ( G ( ϕ ( z ) , ϕ ( x )) , A ϕ ( z ) ) = (cid:77) g ∈ S (cid:96) ( H ( z, x ) g, A ϕ ( z ) ) and a similar decomposition for y instead of x . As in (10), we have a com-mutative diagram (cid:96) ( H ( z, x ) g, A ϕ ( z ) ) (cid:96) ( H ( z, x ) , A ϕ ( z ) ) (cid:96) ( H ( z, y ) g, A ϕ ( z ) ) (cid:96) ( H ( z, y ) , A ϕ ( z ) ) Λ A, H ,z ( f ) ρ g ⊗ α g ∼ = Λ A, H ,z ( f ) ρ g ⊗ α g ∼ = f ∈ A H ( x, y ) and g ∈ S . Thus, the representation Λ A, G ,ϕ ( z ) ◦ (id A ϕ ) of A H ( x, y ) is equivalent to a direct sum of | S | -many copies of therepresentation Λ A, H ,z . This proves (12). C ∗ -algebras associated to C ∗ -categories We now recall the construction from [Joa03] of a K -theory spectrum for C ∗ -categories. The idea is to first associate a C ∗ -algebra to a C ∗ -category andthen associate a K -theory spectrum to this C ∗ -algebra. There are two KK -equivalent constructions of the associated C ∗ -algebra. The first constructionis easier to compute for our examples while the second construction has betterfunctoriality properties. Definition 4.9 ([Joa03]) . Let A be a C ∗ -category. We equip C ∗ A := (cid:77) x,y A ( x, y ) with the structure of a ∗ -algebra by inheriting the involution from A and bydefining the product of two elements f ∈ A ( x, y ) , g ∈ A ( z, w ) to be g · f := (cid:40) gf, y = z , y (cid:54) = z . Suppose that B is a C ∗ -algebra and that F : A → B is a C ∗ -functor satisfying F ( f ) F ( g ) = 0 whenever f and g are non-composable morphisms in A . Thenwe get a well-defined ∗ -homomorphism C ∗ F : C ∗ A → B, A ( x, y ) (cid:51) f (cid:55)→ F ( f ) which induces a seminorm on C ∗ A . We denote by C ∗ A the completion of C ∗ A by the supremum of all such seminorms (which has been shown to be anorm in [Joa03]).In [Joa03], the above C ∗ -algebra is denoted by A A rather than C ∗ A . Remark 4.10.
By construction, the C ∗ -category C ∗ A has the following uni-versal property: Given any C ∗ -algebra B and any C ∗ -functor F : A → B satisfying F ( f ) F ( g ) = 0 for all non-composable morphisms f and g , there isa unique ∗ -homomorphism C ∗ F : C ∗ A → B such that the following diagramcommutes: A B C ∗ A F C ∗ F A (cid:55)→ C ∗ A is not functorial with respect toarbitrary C ∗ -functors. Before giving a functorial construction, we list someuseful properties of C ∗ A . Lemma 4.11.
Let A be a C ∗ -category, B a C ∗ -algebra and F : A → B a C ∗ -functor satisfying F ( f ) F ( g ) = 0 whenever f and g are non-composablemorphisms in A . Suppose that F is isometric on morphism sets and that C ∗ F : C ∗ A → B is injective. Then C ∗ F : C ∗ A → B is isometric.Proof. The proof is a variant of the proof of [Joa03, Lemma 3.6]. We haveto show that C ∗ F is isometric. By construction, we have C ∗ A = (cid:91) A (cid:48) C ∗ A (cid:48) , where the union is taken over all full subcategories A (cid:48) ⊆ A with only finitelymany objects. It suffices to show, that C ∗ F is isometric on each C ∗ A (cid:48) . Since F is isometric and C ∗ F is injective, it suffices to show that there is only one C ∗ -norm (cid:107) · (cid:107) on C ∗ A (cid:48) which restricts to the given norm on the morphism sets(note that the inclusions A (cid:48) ( x, y ) (cid:44) → C ∗ A (cid:48) are isometric since F is isometricand C ∗ F norm-decreasing). Write a ∈ C ∗ A (cid:48) as a finite sum a = (cid:88) a xy , a xy ∈ A (cid:48) ( x, y ) and denote by N the number of objects of A (cid:48) . Then the estimate max x,y (cid:107) a xy (cid:107) ≤ (cid:107) a (cid:107) ≤ N max x,y (cid:107) a xy (cid:107) (13)shows that (cid:107) · (cid:107) is already complete on C ∗ A (cid:48) and therefore the unique C ∗ -norm with this property. The first inequality in (13) can be verified by writing a xy = lim λ u λ av λ for approximate units u λ ∈ A (cid:48) ( y, y ) and v λ ∈ A (cid:48) ( x, x ) . Corollary 4.12.
Let A be a G - C ∗ -algebra. Then C ∗ ( A (cid:111) r G ) is naturallyisomorphic to the classical reduced crossed product C ∗ -algebra of A as definedin [AD16, Section 1.4].Proof. Denote the classical reduced crossed product of A by (cid:94) A (cid:111) r G . Al-though using different notation, it is precisely defined as the closed imageof the regular representation Λ A, G from (11). Since Λ A, G is by definition19sometric on morphism sets, Lemma 4.11 provides us with an isomorphism C ∗ Λ A, G : C ∗ ( A (cid:111) r G ) → (cid:94) A (cid:111) r G . In particular, we obtain the following special case:
Corollary 4.13.
Let G be a discrete group acting on set X . Let A be a G - C ∗ -algebra, considered as a X - C ∗ -algebra. Then there is a natural isomorphism C ∗ ( A (cid:111) r X ) ∼ = C ( X, A ) (cid:111) r G. Corollary 4.14.
Let A be a G - C ∗ -algebra and B a C ∗ -algebra (endowed withtrivial G -action). Then there is a canonical ∗ -isomorphism C ∗ (( A ⊗ B ) (cid:111) r G ) ∼ = C ∗ ( A (cid:111) r G ) ⊗ B. Proof.
By Lemma 4.11, the representation C ∗ Λ A ⊗ B, G : C ∗ (( A ⊗ B ) (cid:111) r G ) → (cid:89) z L A z ⊗ B ( ⊕ x (cid:96) ( G ( z, x ) , A z ⊗ B )) is faithful. Its image coincides with the image of the faithful representation C ∗ ( A (cid:111) r G ) ⊗ B → (cid:89) z L A z ( ⊕ x (cid:96) ( G ( z, x ) , A z )) ⊗ B → (cid:89) z L A z ⊗ B ( ⊕ x (cid:96) ( G ( z, x ) , A z ⊗ B )) . Definition 4.15 ([Joa03, Definition 3.7]) . Let A be a C ∗ -category. Wedenote by C ∗ f A the universal C ∗ -algebra generated by symbols ( f ) for mor-phisms f ∈ A ( x, y ) subject to the relations ( λf + g ) = λ ( f ) + ( g ) , ( f ∗ ) = ( f ) ∗ , ( hg ) = ( h )( g ) for f, g ∈ A ( x, y ) , h ∈ A ( y, z ) and λ ∈ C . By construction, A (cid:55)→ C ∗ f A is theleft adjoint functor of the inclusion functor from the category of C ∗ -categoriesto the category of C ∗ -algebras.In [Joa03], the above algebra is denoted by A f A rather than C ∗ f A . Proposition 4.16 ([Joa03, Proposition 3.8]) . Let A be a C ∗ -category withcountably many objects and separable morphism sets. Then the canonical ∗ -homomorphism C ∗ f A → C ∗ A is a stable homotopy equivalence and thereforea KK -equivalence. The reader should not be concerned about the unitality assumptions in[Joa03] since they are not used in the proof of the above proposition.20 K -theory spectrum We now recall very briefly the construction of the K -theory spectrum K from[Joa03]. We use this particular model because it is quite easy to show that K maps mapping cone extensions to fiber sequences, KK -equivalences to stableequivalences, suspensions to loops and direct sums to wedge sums. The defi-nition of K involves graded C ∗ -algebras. We only recall the basic definitionsand refer to [Bla98] for a more detailed account on graded C ∗ -algebras. Agraded C ∗ -algebra is a Z - C ∗ -algebra, i.e. a C ∗ -algebra A together with aself-inverse grading automorphism α . We will need the following examples:i) We denote by ˆ K the graded C ∗ -algebra of compact operators on (cid:96) N ⊕ (cid:96) N with grading automorphism given by conjugation with the unitary (cid:18) (cid:19) .ii) We denote by ˆ S the C ∗ -algebra C ( R ) with grading automorphism givenby reflecting functions at the origin ∈ R .iii) The Clifford algebra C n on n generators is the universal C ∗ -algebragenerated by self-adjoint unitaries e , . . . , e n satisfying e i e j = − e j e i , i (cid:54) = j The grading automorphism of C n is given by e i (cid:55)→ − e i .iv) If not specified otherwise, we equip any C ∗ -algebra with a trivial grad-ing.For any two graded C ∗ -algebras A and B , there is a spatial graded tensorproduct A ˆ ⊗ B . It is a completion of the algebraic tensor product A (cid:12) B equipped with a non-standard multiplication and involution depending onthe grading [Bla98, Definition 14.4.1]. If one of the factors is trivially graded,then A ˆ ⊗ B is naturally isomorphic to the usual spatial tensor product. Wedenote the space of Z -equivariant ∗ -homomorphisms A → B by C ∗ Z ( A, B ) and endow it with the compact-open topology and the zero morphism as abasepoint. Proposition 4.17 ([Joa03, Proposition 4.1]) . Let
A, B be graded C ∗ -algebrasand X a locally compact space. Then there is a natural homeomorphism C ∗ Z ( A, B ˆ ⊗ C ( X )) → Top ∗ ( X + , C ∗ Z ( A, B )) , f (cid:55)→ ( x (cid:55)→ (id B ˆ ⊗ ev x ) ◦ f ) , where X + denotes the one-point compactification and ev x : C ( X ) → C theevaluation at x . Definition 4.18.
Let A be a separable C ∗ -algebra. The spectrum K ( A ) isgiven by the sequence of pointed spaces K ( A ) n := C ∗ Z ( ˆ S , A ˆ ⊗ C n ˆ ⊗ ˆ K ) K ( A ) n → Ω K ( A ) n +1 given by C ∗ Z ( ˆ S , A ˆ ⊗ C n ˆ ⊗ ˆ K ) β −→ ∼ C ∗ Z ( ˆ S , A ˆ ⊗ C n +1 ˆ ⊗ C ((0 , ⊗ ˆ K ) . ∼ = Ω C ∗ Z ( ˆ S , A ˆ ⊗ C n +1 ˆ ⊗ ˆ K ) , where β denotes the Bott map from [HG04, Lecture 1].Let A be a C ∗ -category with countably many objects and separable mor-phism sets. The K -theory spectrum of A is given by K ( A ) := K ( C ∗ f A ) (cid:39) K ( C ∗ A ) . Proposition 4.19 ([Joa03, Theorem A.2]) . Let A be a trivially graded sep-arable C ∗ -algebra. Then there is a natural isomorphism π ∗ K ( A ) ∼ = K ∗ ( A ) . Definition 4.20.
Let G be a countable discrete group and A a separable G - C ∗ -algebra. We define an Or( G ) -spectrum K GA by K GA ( G/H ) := K ( A (cid:111) r G/H ) . The
Davis-Lück assembly map for G with coefficients in A is the map H G ∗ ( E Fin G, K GA ) → H G ∗ (pt , K GA ) where Fin denotes the family of subgroups.Note that the functoriality of C ∗ f and Lemma 4.8 guarantee functoriality of K GA ( G/H ) both in G/H for G -equivariant maps and in A for G -equivariant ∗ -homomorphisms. Lemma 4.21.
The functor A (cid:55)→ K GA from the category of separable G - C ∗ -algebras to the category of Or( G ) -spectra has the following properties:i) It maps KK G -equivalences to stable equivalences.ii) It maps mapping cone extensions to fiber sequences.iii) For any separable G - C ∗ -algebra A , we have K GSA ∼ = Ω K GA .iv) Let A i , i ∈ I be a countable family of separable G - C ∗ -algebras. Thenthere is a natural stable equivalence (cid:95) i ∈ I K GA i (cid:39) K G ⊕ i ∈ I A i . Proof.
We fix
G/H ∈ Or( G ) throughout the proof.i) Every KK G -equivalence A → B induces a KK -equivalence C ( G/H, A ) (cid:111) r → C ( G/H, B ) (cid:111) r G . Therefore, the induced map π ∗ ( K GA ( G/H )) . ∼ = K ∗ ( C ( G/H, A ) (cid:111) r G ) → K ∗ ( C ( G/H, B ) (cid:111) r G ) . ∼ = π ∗ ( K GB ( G/H )) is an isomorphism.ii) We claim that the functor A (cid:55)→ C ∗ ( A (cid:111) r G/H ) preserves mapping coneextensions and that the functor A (cid:55)→ K ( A ) maps mapping cone exten-sions to fiber sequences. Let Cone( π ) → A π −→ B be a mapping coneextension. For the first claim, use Corollaries 4.13 and 4.14 to identify C ∗ (Cone( π ) (cid:111) r G/H ) with the cone of the map C ∗ ( A (cid:111) r G/H ) → C ∗ ( B (cid:111) r G/H ) . For the second claim, use Proposition 4.17 to identify K (Cone( π )) withthe homotopy fiber of the map K ( A ) → K ( B ) . iii) By Corollary 4.14, the functor A (cid:55)→ C ∗ ( A (cid:111) r G/H ) commutes withsuspensions. Now the claim follows from Proposition 4.17.iv) There is a natural map (cid:95) i ∈ I K GA i ( G/H ) → K G ⊕ i ∈ I A i ( G/H ) which we claim to be a stable equivalence. On homotopy groups, theabove map can be written as the composition π ∗ (cid:32)(cid:95) i ∈ I K ( C ∗ ( A (cid:111) r G/H )) (cid:33) ∼ = −→ (cid:77) i ∈ I π ∗ K ( C ∗ ( A i (cid:111) r G/H )) ∼ = −→ π ∗ K (cid:32)(cid:77) i ∈ I C ∗ ( A i (cid:111) r G/H ) (cid:33) ∼ = −→ π ∗ K (cid:32) C ∗ (cid:32)(cid:32)(cid:77) i ∈ I A i (cid:33) (cid:111) r G/H (cid:33)(cid:33) . The first map is an isomorphism by Lemma 3.15, the second map is anisomorphism since K -theory commutes with countable direct sums (c.f.[WO93, Proposition 6.2.9]) and the third isomorphism arises from anisomorphism of the underlying C ∗ -algebras.23 Identification of the assembly maps
In this section we finally identify the Davis-Lück assembly map H G ∗ ( E Fin G, K GA ) → H G ∗ (pt , K GA ) with the Meyer-Nest assembly map K ∗ ( ˜ A (cid:111) r G ) → K ∗ ( A (cid:111) r G ) . The strategy is to use the Dirac morphism D ∈ KK G ( ˜ A, A ) from Theorem2.4 to compare the Davis-Lück map with coefficients in A to the Davis-Lückmap with coefficients in ˜ A . To do so, we need the functor A (cid:55)→ K GA toextend from the category of separable G - C ∗ -algebras to the category KK G .Due to the choice of our specific model of the K -theory spectrum K , it isnot obvious how to construct such an extension. One solution could be tochoose a KK -functorial model for K which also satisfies Lemma 4.21 andshow that the functor A (cid:55)→ C ∗ ( A (cid:111) r G/H ) extends to a functor KK G → KK .But we can also give an elementary solution to this problem using zig-zags: Definition 5.1. A zig-zag of G -equivariant ∗ -homomorphisms is a diagram A ϕ −→ B ψ ←− A ϕ −→ · · · ϕ n −→ B n ψ n ←− A n +1 . of G -equivariant ∗ -homomorphisms, such that each ψ k is a KK G -equivalence.Such a zig-zag naturally defines a KK G -class [ ψ n ] − ◦ [ ϕ n ] ◦ · · · ◦ [ ψ ] − ◦ [ ϕ ] ∈ KK G ( A , A n +1 ) . Similarly, a zig-zag of (
Or( G ) -)spectra is a diagram E f −→ F g ←− E f −→ · · · f n −→ F n g n ←− E n +1 of ( Or( G ) -)spectra, such that each g k is a stable equivalence. By Lemma3.16, every such zig-zag gives rise to a well-defined natural transformation (id X ⊗ g n ) − ∗ ◦ · · · ◦ (id X ⊗ f ) ∗ : H G ∗ ( X, E ) → H G ∗ ( X, E n +1 ) . on homology. Thus, any zig-zag of G -equivariant ∗ -homomorphisms givesrise to a natural transformation on homology. The following lemma shows,that we can always restrict to the case of zig-zags: Lemma 5.2.
Every morhism in KK G can be represented by a zig-zag of G -equivariant ∗ -homomorphisms.Proof. This follows from the proofs of [Mey00, Proposition 6.1, Theorem24.5].From now on, we pretend that all KK G -classes are represented by G -equivariant ∗ -homomorphisms. We leave it to the reader to replace the rel-evant maps of spectra by zig-zags. Recall from 3.8 that there is a naturalisomorphism H G ∗ (pt , K GA ) ∼ = K ∗ ( A (cid:111) r G ) . Thus, the Meyer-Nest assemblymap can be identified with the map H G ∗ (pt , K G ˜ A ) → H G ∗ (pt , K GA ) . We are now ready to state our main theorem:
Theorem 5.3.
The indicated maps in the following diagram are isomor-phisms. H G ∗ ( E Fin G, K G ˜ A ) H G ∗ (pt , K G ˜ A ) H G ∗ ( E Fin G, K GA ) H G ∗ (pt , K GA ) pr ∗ ∼ = D ∗ ∼ = D ∗ pr ∗ In particular, the Meyer-Nest assembly map can be identified with the Davis-Lück assembly map.
We reduce the proof to the trivial case of finite groups by a series oflemmas. A key ingredient is the following classical theorem. A simple proofof it for discrete groups can be found in [Ech17, Proposition 6.8].
Theorem 5.4 (Green’s imprimitivity theorem) . Let G be a countable dis-crete group, H a subgroup and B a separable H - C ∗ -algebra with H -action β .Then the H -equivariant ∗ -homomorphism ψ B : B → Ind GH B, b (cid:55)→ (cid:32) g (cid:55)→ (cid:40) β g − ( b ) , g ∈ H , g / ∈ H (cid:33) (14) gives rise to an inclusion B (cid:111) r H ψ B (cid:111) r H −−−−→ (Ind GH B ) (cid:111) r H (cid:44) −→ (Ind GH B ) (cid:111) r G whose image is a full corner. In particular, the inclusion B (cid:111) r H (cid:44) → (Ind GH B ) (cid:111) r G is a KK -equivalence. Lemma 5.5.
The map D ∗ : H G ∗ ( E Fin G, K G ˜ A ) → H G ∗ ( E Fin G, K GA ) s an isomorphism.Proof. Let H ⊆ G be a finite subgroup and consider the following commu-tative diagram: H G ∗ ( G/H, K G ˜ A ) H G ∗ ( G/H, K GA ) K ∗ ( C ( G/H, ˜ A ) (cid:111) r G ) K ∗ ( C ( G/H, A ) (cid:111) r G ) K ∗ ( ˜ A (cid:111) r H ) K ∗ ( A (cid:111) r H ) ∼ = ∼ = ∼ = ∼ = ∼ = Here the horizontal maps are induced by D . The vertical isomorphisms areobtained from Corollary 4.13 and Theorem 5.4. The lower horizontal map isan isomorphism since D is a KK H -equivalence. Thus, the upper horizontalmap is an isomorphism. Since H ⊆ G was an arbitrary finite subgroup, itfollows from a Mayer-Vietoris argument that the map D ∗ : H G ∗ ( E Fin G, K G ˜ A ) → H G ∗ ( E Fin G, K GA ) is an isomorphism too. Lemma 5.6.
Let
D ⊆ KK G be the full subcategory of all G - C ∗ -algebras, forwhich the Davis-Lück assembly map is an isomorphism. Then D is localizingin the sense of Definition 2.2. In particular, we can reduce the proof of Theorem 5.3 to the case ˜ A =Ind GH B for a finite subgroup H ⊆ G and a separable H - C ∗ -algebra B . Proof.
By Lemma 4.21 i) and Lemma 3.16, D is closed under KK G -equivalence.By Lemma 4.21 iii), the Davis-Lück map for a suspension can be identi-fied with the Davis-Lück map for the original algebra with homology groupsshifted by one. Thus, D is closed under suspension. By Lemma 3.15, Lemma4.21 iv) and compatibility of the balanced smash product ∧ Or( G ) with wedgesums, the Davis-Lück map for a countable direct sum (cid:76) i ∈ I A i can be identi-fied with the direct sum of the Davis-Lück maps for the individual summands A i , i ∈ I . Thus, D is closed under countable direct sums. It remains to ver-ify stability under mapping cone extensions. Let Cone( π ) → A π −→ B be amapping cone extension. By Lemma 4.21 ii), the sequence K G Cone( π ) → K GA → K GB is a fiber sequence. Now if X is any G - CW -complex, the sequence Top G ( − , X ) + ∧ Or( G ) K G Cone( π ) → Top G ( − , X ) + ∧ Or( G ) K GA → Top G ( − , X ) + ∧ Or( G ) K GB
26s still a fiber sequence by Lemmas 3.14 and 3.17. In particular, the rows inthe following diagram are exact: · · · H G ∗ ( E Fin G, K G Cone( π ) ) H G ∗ ( E Fin G, K GA ) H G ∗ ( E Fin G, K GB ) · · ·· · · H G ∗ (pt , K G Cone( π ) ) H G ∗ (pt , K GA ) H G ∗ (pt , K GB ) · · · It follows from the five-lemma that
Cone( π ) , A and B belong to D if at leasttwo of them do. Theorem 5.7.
Let H ⊆ G be a finite subgroup, B an H - C ∗ -algebra and X a G - CW -complex. Then there is a natural induction isomorphism H H ∗ ( X | H , K HB ) ∼ = −→ H G ∗ ( X, K G Ind GH B ) , where X | H denotes the restriction of X to H .Proof. Consider the induction functor I : Or( H ) → Or( G ) , H/K (cid:55)→ G × H H/K ∼ = G/K.
By Lemma 3.13, there is a natural isomorphism H H ∗ ( X | H , K HB ) ∼ = H G ∗ ( X, I ∗ K HB ) . It thus suffices to construct a natural stable equivalence I ∗ K HB (cid:39) K G Ind GH B of Or( G ) -spectra. We prove this in two steps. Our first claim is, that thenatural map I ∗ K HB ( G/K ) =
Top H ( − , G/K ) + ∧ Or( H ) K HB → K ( B (cid:111) r ( G/K ) | H ) (15)given by f ∧ x (cid:55)→ f ∗ ( x ) is a stable equivalence for each G/K ∈ Or( G ) . Tosee this, decompose G/K ∼ = (cid:96) i H/L i into H -orbits. We get a commutativediagram Top H ( − , G/K ) + ∧ Or( H ) K HB K ( B (cid:111) r ( G/K ) | H ) (cid:87) i Top H ( − , H/L i ) + ∧ Or( H ) K HB (cid:87) i K ( B (cid:111) r H/L i ) ∼ = ∼ = (cid:39) . Here the left vertical map is an isomorphism by compatibility of balanced27mash products with wedge sums. The lower horizontal map is an isomor-phism by Remark 3.8. To see that the right hand map is an equivalence, useCorollary 4.13 to identify C ∗ ( B (cid:111) r ( G/K ) | H ) ∼ = (cid:77) i C ∗ ( B (cid:111) r H/L i ) and apply Lemma 4.21 iv). This proves the first claim.Our second claim is that there is a natural C ∗ -functor F : B (cid:111) r ( G/K ) | H → (Ind GH B ) (cid:111) r G/K which induces a stable equivalence of K -theory spectra. To construct F , de-note the H -action on B by β and consider the H -equivariant ∗ -homomorphism ψ B : B → Ind GH B, b (cid:55)→ (cid:32) g (cid:55)→ (cid:40) β g − ( b ) , g ∈ H , g / ∈ H (cid:33) . Then ψ B is automatically ( G/K ) | H -equivariant and induces a C ∗ -functor F : B (cid:111) r ( G/K ) | H ψ B (cid:111) r ( G/K ) | H −−−−−−−−→ (Ind GH B ) (cid:111) r ( G/K ) | H (cid:44) −→ (Ind GH B ) (cid:111) r G/K.
To see that F induces a stable equivalence of K -theory spectra, we claimthat C ∗ F can be identified with the ∗ -homomorphism C ( G/K, B ) (cid:111) r H ψ C G/K,B ) (cid:111) r H −−−−−−−−−→ Ind GH ( C ( G/K, B )) (cid:111) r H (cid:44) −→ Ind GH ( C ( G/K, B )) (cid:111) r G from Theorem 5.4. Indeed this identification can easily be made by usingCorollary 4.13 and checking commutativity of the diagram C ( G/K, B ) C ( G/K,
Ind GH B )Ind GH ( C ( G/K, B )) id C G/K ) ⊗ ψ B ψ C G/K,B ) α ∼ = , where α is defined by α ( f )( g )( hK ) := f ( ghK )( g ) , f ∈ C ( G/K,
Ind GH B ) , g ∈ G, hK ∈ G/K as in [GHT00, Lemma 12.6]. This proves the second claim and provides uswith a natural equivalence I ∗ K HB ( G/K ) (cid:39) K ( B (cid:111) r ( G/K ) | H ) (cid:39) K G Ind GH B ( G/K ) , G/K ∈ Or( G ) . roof of Theorem 5.3. By Lemma 5.5, the map H G ∗ ( E Fin G, K G ˜ A ) → H G ∗ ( E Fin G, K GA ) is an isomorphism. By Lemma 5.6, it suffices to prove that H G ∗ ( E Fin G, K G Ind GH B ) → H G ∗ (pt , K G Ind GH B ) is an isomorphism for every finite subgroup H ⊆ G and every H - C ∗ -algebra B . Theorem 5.7 provides us with a commutative diagram H G ∗ ( E Fin G, K G Ind GH B ) H G ∗ (pt , K G Ind GH B ) H H ∗ ( E Fin G | H , K HB ) H H ∗ (pt , K HB ) ∼ = ∼ = ∼ = . Since H is finite, E Fin G is H -contractible. Thus, the lower map in the diagramis an isomorphism and so is the upper one. As shown in [HLS02], the Baum-Connes conjecture (with coefficients) turnsout to be false in general. The problem is that for certain discrete groups G ,the functor A (cid:55)→ K ∗ ( A (cid:111) r G ) is not exact in the middle. Motivated by thesecounterexamples, Baum, Guentner and Willett gave a new formulation of theBaum-Connes conjecture in [BGW16] which fixes these counterexamples andis equivalent to the old conjecture for all exact groups. The idea is to replacethe reduced crossed product by a better behaved crossed product functor.Such exotic crossed product functors were also studied extensively by Buss,Echterhoff and Willett, see [BEW16] for a survey.The fact that we used the reduced crossed product in this paper was nottoo important. In fact we only made use of the following facts:i) The reduced crossed product (cid:111) r G : C ∗ G → C ∗ extends to a functor KK G → KK .ii) Green’s imprimitivity theorem: Let H ⊆ G be a subgroup and B an H - C ∗ -algebra. Then there is a full corner embedding B (cid:111) r H (cid:44) → (Ind GH B ) (cid:111) r G. iii) Let A be a G - C ∗ -algebra and X a G -set. There is a C ∗ -category A (cid:111) r X ,which is sufficiently functorial in the sense of Lemma 4.8 and satisfies29orollary 4.13, i.e. there is a natural isomorphism C ∗ ( A (cid:111) r X ) ∼ = C ( X, A ) (cid:111) r G. In this section we reformulate our main result for more general
Morita-compatible crossed product functors and indicate how to adapt the proof tothis situation.
Definition 6.1 ([BGW16, Definition 2.1]) . Let G be a countable discretegroup. A crossed product functor µ for G is a functor A (cid:55)→ A (cid:111) µ G from thecategory of G - C ∗ -algebras to the category of C ∗ -algebras, such that every A (cid:111) µ G contains the convolution algebra AG as a dense subalgebra, togetherwith natural transformations A (cid:111) max G → A (cid:111) µ G → A (cid:111) r G which extend the identity on AG . Definition 6.2 ( [BGW16, Definition 3.3]) . Equip H G := (cid:96) ( G ) ⊗ (cid:96) ( N ) witha G -action U : G → U ( H G ) given by left translation on the first factor andthe trivial G -action on the second factor. A crossed product functor µ for G is called Morita-compatible if the algebraic ∗ -isomorphism ( A (cid:12) K ( H G )) G ∼ = −→ ( AG ) (cid:12) K ( H G ) , ( a ⊗ k ) u g (cid:55)→ ( au g ) ⊗ ( kU g ) extends to a ∗ -isomorphism ( A ⊗ K ( H G )) (cid:111) µ G ∼ = ( A (cid:111) µ G ) ⊗ K ( H G ) . Remark 6.3.
Note that for separable G - C ∗ -algebras, the definition of Moritacompatible crossed product functors coincides with the notion of correspon-dence functors in [BEW18, Theorem 4.9] by [BEW18, Proposition 8.10].The following proposition and lemma subsume those properties of Morita-compatible crossed product functors which are most important for us. Proposition 6.4 ([BEW18, Proposition 6.1]) . Let µ be a Morita-compatiblecrossed product functor for G . Then µ naturally extends to a functor (cid:111) µ G : KK G → KK . Lemma 6.5 ([BEW18, Lemma 3.6, Lemma 4.11]) . Let µ be a Morita-compatible crossed product functor for G and A a separable G - C ∗ -algebra.i) For any (second countable) locally compact space X , there is a canonical ∗ -isomorphism C ( X ) ⊗ ( A (cid:111) µ G ) → ( C ( X ) ⊗ A ) (cid:111) µ G, f ⊗ ( au g ) (cid:55)→ ( f ⊗ a ) u g here X is equipped with the trivial G -action.ii) Let I ⊆ A be a G -invariant ideal. Then the map I (cid:111) µ G → A (cid:111) µ G isinjective. Using the functor (cid:111) µ G : KK G → KK , the µ -Baum-Connes assembly map K G ∗ ( E Fin
G, A ) → K ∗ ( A (cid:111) µ G ) is constructed in [BEW18, Section 6] analogously to the classical constructionin [BCH94].Similarly, we can define the µ -Meyer-Nest assembly map K ∗ ( ˜ A (cid:111) µ G ) → K ∗ ( A (cid:111) µ G ) by composition with the image of the Dirac morphism D ∈ KK G ( ˜ A, A ) under the functor (cid:111) µ G : KK G → KK . We obtain Theorem 6.6 (c.f. [MN04, Theorem 5.2]) . Let G be a countable discretegroup, A a separable G - C ∗ -algebra and µ a Morita-compatible crossed prod-uct functor for G . Then the indicated maps in the following diagram areisomorphisms. K G ∗ ( E Fin G, ˜ A ) K ∗ ( ˜ A (cid:111) µ G ) K G ∗ ( E Fin
G, A ) K ∗ ( A (cid:111) µ G ) ∼ = ∼ = In particular, the µ -Baum-Connes assembly map can be identified with the µ -Meyer-Nest assembly map.Proof. The proof is analogous to the proof of [MN04, Theorem 5.2]: Theleft hand map is an isomorphism by [CEOO04, Theorem 1.5]. By [BEW16,Proposition 6.6], the upper hand map is an isomorphism for ˜ A = Ind GH B whenever H ⊆ G is a finite subgroup and B a separable H - C ∗ -algebra. Aneasy application of Lemma 6.5 shows that the functor A (cid:55)→ A (cid:111) µ G preserves KK G -equivalences, suspensions, mapping cone extensions and countable di-rect sums. Thus, the same arguments as in [MN04] show that the class ofall ˜ A ∈ KK G , for which the upper hand map is an isomorphism, is localizing.This concludes the proof.In order to write down a Davis-Lück assembly map for a group G anda Morita-compatible crossed product µ , we need to define crossed product C ∗ -categories A (cid:111) µ G for certain groupoids G and G - C ∗ -algebras A . Forgeneral groupoid actions this is hopeless unless µ is defined by some universalproperty which is independent from G . But for our applications it sufficesto consider the following cases: 31) G = G ii) G = G/H for a finite subgroup H ⊆ G iii) G = X where X is an H -set for a finite subgroup H ⊆ G We already have a definition for A (cid:111) µ G . For the second and third case,we simply define A (cid:111) µ G := A (cid:111) r G . The following lemma shows that this isa reasonable definition. Lemma 6.7.
Let G be a groupoid whose morphism sets G ( x, y ) are all finiteand let A be a G - C ∗ -algebra. Then the reduced norm (cid:107) · (cid:107) r is the unique C ∗ -norm on the convolution category A G .Proof. By the C ∗ -identity (cid:107) f ∗ f (cid:107) = (cid:107) f (cid:107) , the norm on a C ∗ -category isuniquely determined by its value on endomorphism sets. Therefore it sufficesto show that there is only one C ∗ -norm on the endomorphism sets A G ( x, x ) .But it is a well-known fact that the convolution algebra associated to a finitegroup is already complete with respect to the reduced norm. Definition 6.8. i) Denote by ˜Or( G ) ⊆ Or( G ) the full subcategory of all G/H ∈ Or( G ) such that H is either finite or H = G .ii) Let X be a G - CW -complex such that the fixed points X H are emptywhenever H (cid:40) G is an infinite proper subgroup. Let E : ˜Or( G ) → Sp be a functor. We define the G -equivariant homology of X withcoefficients in E by H G ∗ ( X, E ) := π ∗ ( Top G ( − , X ) + ∧ ˜Or( G ) E ) where the balanced smash product ∧ ˜Or( G ) is defined as in Definition 3.6.iii) Let µ be a Morita-compatible crossed product functor for G and A aseparable G - C ∗ -algebra. We define a functor K GA,µ : ˜Or( G ) → Sp , K GA,µ ( G/H ) := K ( A (cid:111) µ G/H ) . iv) The µ -Davis-Lück assembly map for G with coefficients in A is the map H G ∗ ( E Fin G, K GA,µ ) → H G ∗ (pt , K GA,µ ) induced by the projection E Fin G → pt .It is now easy to check that the analogue of Lemma 4.8 holds as well forthe µ -crossed product A (cid:111) µ G as long as G is one of the groupoids consideredabove. Using Lemma 6.5 one can also prove the analogue of Lemma 4.21 for K GA,µ . Moreover, the analogue of Green’s imprimitivity theorem 5.4 for the µ -crossed product and a finite subgroup H ⊆ G holds for trivial reasons. In factfor any H - C ∗ -algebra B , the crossed products (Ind GH B ) (cid:111) r G = (Ind GH B ) (cid:111) µ G coincide since Ind GH B is proper. Using the same proofs as in Section 5, weobtain the following analogue of our main result:32 heorem 6.9 (c.f. Theorem 5.3) . Let G be a countable discrete group, µ a Morita-compatible crossed product functor for G and A a separable G - C ∗ -algebra. Then the indicated maps in the following diagram are isomorphisms. H G ∗ ( E Fin G, K G ˜ A,µ ) H G ∗ (pt , K G ˜ A,µ ) H G ∗ ( E Fin G, K GA,µ ) H G ∗ (pt , K GA,µ ) pr ∗ ∼ = D ∗ ∼ = D ∗ pr ∗ In particular, the µ -Davis-Lück assembly map and the µ -Meyer-Nest assem-bly map can be identified. Remark 6.10.
Admittedly, the introduction of ˜Or( G ) is only an ad hocsolution. The main problem with extending the definition of A (cid:111) µ X to arbitrary G -sets X is the analogue of Lemma 4.8 ii), i.e. functoriality in X . Say we define A (cid:111) µ X by the condition C ∗ ( A (cid:111) µ X ) ∼ = C ( X, A ) (cid:111) µ G as in Corollary 4.13. Then functoriality in X essentially boils down to thequestion whether we have a canonical map A (cid:111) µ | H H → A (cid:111) µ G for anysubgroup H ⊆ G where µ | H is defined as in [BEW16, Definition 6.1]. Itis unclear to the author whether this question has an affirmative answerfor arbitrary Morita-compatible crossed product functors µ . However, thecanonical map A (cid:111) µ | H H → A (cid:111) µ G does exist if µ is the smallest exactMorita-compatible crossed product [BEW16, Corollary 7.6]. References [AD16] Claire Anantharaman-Delaroche. Some remarks about the weakcontainment property for groupoids and semigroups. arXivpreprint arXiv:1604.01724 , 2016.[BCH94] Paul Baum, Alain Connes, and Nigel Higson. Classifying space forproper actions and K-theory of group C*-algebras.
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