1.1 Triangulated structure and comparison with E-theory

Let us start by fixing some standard conventions. For a $C^*$ -algebra A, we have a suspension functor $\Sigma A$ defined as $\Sigma A =C_0(\mathbb {R})\otimes A$ . For an equivariant $*$ -homomorphism of G-C $^*$ -algebras $f \colon A \to B$ , we define its associated mapping cone by

$$\begin{align*}\operatorname{\mathrm{Cone}}(f) = \{(a, b_*) \in A \oplus C_0((0, 1], B) \mid f(a) = b_1 \}. \end{align*}$$

This inherits a structure of G-C $^*$ -algebra from A and B.

An exact triangle in $\operatorname {\mathrm {KK}}^G$ is the data of a diagram of the form

$$\begin{align*}A \to B \to C \to \Sigma A, \end{align*}$$

and a $*$ -homomorphism $f\colon A' \to B'$ of G-C $^*$ -algebras, together with a commutative diagram

where the vertical arrows are equivalences in $\operatorname {\mathrm {KK}}^G$ , and the rightmost downward arrow is equal to the leftmost downward arrow, up to applying $\Sigma $ and the Bott periodicity isomorphism $\Sigma ^2 B' \simeq B'$ in $\operatorname {\mathrm {KK}}^G$ .

As we see from above, the most natural triangulated structure lives on the opposite category $(\operatorname {\mathrm {KK}}^G)^{\text {op}}$ . The opposite category of a triangulated category inherits a canonical triangulated category structure, which has ‘the same’ exact triangles. The passage to opposite categories exchanges suspensions and desuspensions and modifies some sign conventions. Thus, the functor $\Sigma $ becomes in principle a desuspension functor in $\operatorname {\mathrm {KK}}^G$ , but due to Bott periodicity $\Sigma $ and $\Sigma ^{-1}$ agree so that we can safely overlook this fact. Moreover, depending on the definition of triangulated category, one may want the suspension to be an equivalence or an isomorphism of categories. In the latter case, $\operatorname {\mathrm {KK}}^G$ should be replaced by an equivalent category (see [Reference Meyer and Nest40, Section 2.1]). This is not terribly important and will be ignored in the sequel.

The triangulated category axioms are discussed in greater detail in [Reference Neeman45, Reference Verdier and Maltsiniotis62]. Most of them amount to formal properties of mapping cones and mapping cylinders, which can be shown in analogy with classical topology. The fundamental axiom requires that any morphism $A \to B$ should be part of an exact triangle. In our setting, this can be proved as a consequence of the generalization of [Reference Meyer37] to groupoid-equivariant $\operatorname {\mathrm {KK}}$ -theory (see also [Reference Lafforgue29, Lemma A.3.2]). Having done that, the rest of the proof follows the same outline of [Reference Meyer and Nest40, Appendix A], where the triangulated structure is established in the case of action groupoids.

There is an alternative, perhaps more conceptual path which consists in defining the Kasparov category as a certain localization of the Spanier–Whitehead category associated to the standard tensor category of G- $C^*$ -algebras and $*$ -homomorphisms [Reference Dell’Ambrogio20]. The triangulated structure of the Spanier–Whithead category is proved in [Reference Dell’Ambrogio20, Theorem A.5.3]. The argument given there can be directly used to show that $\operatorname {\mathrm {KK}}^G$ is triangulated, because it makes use of only two facts, which we prove below.

Proof. In order to show the lifting properties above we make use of ‘extension triangles’. Let $f\in \operatorname {\mathrm {KK}}_{0}^{G}(Q,K)$ be a morphism and denote by $\tilde {f}$ the corresponding element $\tilde {f}\in \operatorname {\mathrm {KK}}_{1}^{G}({\Sigma Q, K})$ . By applying [Reference Lafforgue29, Lemma A.3.4], we can represent $\tilde {f}$ by a Kasparov module where the operator T is G-equivariant. Then the proof of [Reference Lafforgue29, Lemma A.3.2] gives that $\tilde {f}$ is represented by an equivariant (semi-)split extension which fits a diagram as follows (see [Reference Meyer and Nest40, Section 2.3]):

where $\beta _Q$ is the Bott isomorphism and $\epsilon _K$ is an equivalence. Hence, we have that $F(\iota _f)\cong f$ . Notice how this argument automatically shows that f is contained in an exact triangle (up to equivalence).

Now, given $g\in \operatorname {\mathrm {KK}}_{0}^{G}(K,D)$ , set $h=g\circ \epsilon _K^{-1},C_f=\operatorname {\mathrm {Cone}}(p_f)$ and consider the diagram

This shows that the pair $(f,g)$ can be lifted to a composable pair $(\Sigma ^2\iota _f,\iota _h)$ .

It should be clear from the definition above that $E_G$ , viewed as functor from the category of separable G- $C^*$ -algebras is the universal half-exact, $C^*$ -stable, and homotopy invariant functor. In this sense, we can understand E-theory as the universal ‘correction’ of $\operatorname {\mathrm {KK}}$ -theory in terms of excision properties. The universal property implies in particular that any functor between ‘concrete’ categories of $C^*$ -algebras such as $f_*$ and $f^*$ extends to E-theory the same way it does for $\operatorname {\mathrm {KK}}$ -theory.

By the same token, for a separable G- $C^*$ -algebra B we can define a functor $\sigma _B$ which is given by $\sigma _B(A)=A\otimes _X B$ on objects and $\sigma _B(\phi )=\phi \otimes \mathrm {1}_B$ on morphisms. It is important to discuss whether or not $\sigma _B$ is a triangulated functor on our K-theory categories $\operatorname {\mathrm {KK}}^G$ and $E_G$ . By this, we mean whether or not $\sigma _A$ preserves exact triangles. Since we are adopting the convention of using the maximal tensor product, the preservation of exact triangles is a simple consequence of the fact that $-\otimes B$ is an exact functor, and clearly it preserves semisplit extensions.

When B is $C_0(X)$ -nuclear, that is, a continuous field over X with nuclear fibers [Reference Bauval4], we have an isomorphism $A\otimes _X B\cong (A\otimes ^{\text {min}} B)_{\Delta _X}$ [Reference Blanchard7]. Note that this applies in particular to the pullback functor $f^*$ associated to an open map $f\colon Y\to X$ , such as the range and source maps $r,s\colon G\to G^{0}=X$ . Thus, if B is exact or $C_0(X)$ -nuclear the functor $\sigma _B$ is triangulated, regardless of the specific choice of tensor product.

The property of being $C_0(X)$ -nuclear, or rather its K-theoretic counterpart called $\operatorname {\mathrm {KK}}^X$ -nuclearity, is important to establish a useful identification between $\operatorname {\mathrm {KK}}$ - and E-theory groups as follows. More information on $\operatorname {\mathrm {KK}}^X$ -nuclearity can be found in [Reference Bauval4]; here, we limit ourselves to record the following simple fact, which is proved in [Reference Tu58, Proposition 5.1 & Corollary 5.2] (see Definition 3.1 for proper groupoids).

Having this, the following is a simple consequence of the universal properties.

Proof. Denote by F the standard $\operatorname {\mathrm {KK}}$ -functor from the category of separable $C^*$ -algebras. The universal property of $\operatorname {\mathrm {KK}}$ -theory gives us a map $\Phi _{C,B}\colon \operatorname {\mathrm {KK}}^G(C,B)\to E_G(C,B)$ . Let $F^\prime $ be the functor (from separable $C^*$ -algebras) given by $F^\prime (B)=\operatorname {\mathrm {KK}}^G(A,B)$ and $F^\prime (f\colon C\to B)$ induced by Kasparov product with $F(f)$ . Since $\operatorname {\mathrm {KK}}^G(A,-)$ is half-exact, the universal property of E-theory yields a map $\Psi _{C,B}\colon \operatorname {\mathrm {KK}}^G(A,C)\times E_G(C,B)\to \operatorname {\mathrm {KK}}^G(A,B)$ . It is clear that $\Psi (-\,,\Phi \circ F)=F^\prime $ . In particular, for $f\colon A\to B$ , we have

$$\begin{align*}\Psi_{A,B}(1_A, \Phi_{A,B}F(f))=F^\prime(f)(1_A)=F(f), \end{align*}$$

which implies that $\Psi _{A,B}(1_A,-)$ is a left inverse for $\Phi _{A,B}$ . The argument for showing it is a right inverse is analogous.

1.2 Complementary subcategories and cellular approximation

In this subsection, we recall some facts about complementary subcategories, homotopy colimits in triangulated categories and the fundamental notion of cellular approximation. The material in this section is summarized from [Reference Meyer38, Reference Meyer39, Reference Meyer and Nest40, Reference Meyer and Nest41].

Let $F\colon \mathcal {T} \to \mathcal {S}$ be an exact functor between triangulated categories. This means that F intertwines suspensions and preserves exact triangles. The kernel of F (on morphisms), denoted $\mathcal {I}=\ker F$ , will be called a homological ideal (see [Reference Meyer and Nest41, Remark 19]). We say that $\mathcal {I}$ is compatible with direct sums if F commutes with countable direct sums (see [Reference Meyer39, Proposition 3.14]). Note that triangulated categories involving $\operatorname {\mathrm {KK}}$ -theory have no more than countable direct sums because separability assumptions are needed for certain analytical results in the background.

An object $P\in \mathcal {T}$ is called $\mathcal {I}$ -projective if $\mathcal {I}(P,A)=0$ for all objects $A\in \mathcal {T}$ . An object $N\in \mathcal {T}$ is called $\mathcal {I}$ -contractible if $\mathrm {id}_N$ belongs to $\mathcal {I}(N,N)$ . Reference to $\mathcal {I}$ is often omitted in the sequel. Let $P_{\mathcal {I}}, N_{\mathcal {I}} \subseteq \mathcal {T}$ be the full subcategories of projective and contractible objects, respectively.

We denote by $\langle {P_{\mathcal {I}}}\rangle $ the localizing subcategory generated by the projective objects, that is, the smallest triangulated subcategory that is closed under countable direct sums and contains $P_{\mathcal {I}}$ . In particular, $\langle P_{\mathcal {I}}\rangle $ is closed under isomorphisms, suspensions, and if

is an exact triangle in $\mathcal {T}$ where any two of the objects $A,B,C$ are in $\langle P_{\mathcal {I}}\rangle $ , so is the third. Note that $N_{\mathcal {I}}$ is localizing, and any localizing subcategory is thick, that is, closed under direct summands (see [Reference Neeman45]).

Definition 1.9. Given an object $A\in \mathcal {T}$ and a chain complex

(3)

we say that Equation (3) is a projective resolution of A if

• – all the $P_n$ ’s are projective;

• – the chain complex below is split exact

  

We say that $\mathcal {T}$ has enough projectives if any object admits a projective resolution.

Definition 1.11. We call two thick triangulated subcategories $\mathcal {P},\mathcal {N}$ of $\mathcal {T}$ complementary if $\mathcal {T}(P,N)=0$ for all $P\in \mathcal {P},N\in \mathcal {N}$ and, for any $A\in \mathcal {T}$ , there is an exact triangle

where $P\in \mathcal {P}$ and $N\in \mathcal {N}$ .

The following result will be very important for us.

A pair of complementary subcategories helps clarify the degree to which a projective resolution ‘computes’ a homological functor into the category of abelian groups. The object $P(A)$ resulting from Proposition 1.12 is called the $P_{\mathcal {I}}$ -cellular approximation of A (it is called simiplicial approximation in [Reference Meyer and Nest40]).

Definition 1.14. In general, the homotopy direct limit of a countable inductive system $(A_n,\alpha _m^n)$ is defined as the object $A^h_\infty $ fitting into the exact triangle below:

where $S|_{A_n}\colon A_n\to A_{n+1}$ is just the connecting map $\alpha _n^{n+1}$ . We write $\text {ho-lim}(A_n,\alpha _m^n)=A^h_\infty $ , or simply $\text {ho-lim}\, A_n$ when the connecting maps are clear from context.

We mention a few more properties of this limit that will be useful for our later arguments. First of all, the last map in the triangle above is equivalent to a sequence of maps $\alpha _n^\infty \colon A_n \to A^h_\infty $ with the compatibility relation $\alpha _n^\infty \circ \alpha _m^n=\alpha _m^\infty $ when $m\leq n$ .

Let us consider the ordinary inductive limit of $C^*$ -algebras $A_\infty $ associated to the system $(A_n,\alpha _m^n)$ , where the maps $\alpha _m^n$ are equivariant $\ast $ -homomorphisms. We keep using $\alpha _n^\infty $ for the canonical maps $A_n\to A_\infty $ . The relation between $A^h_\infty $ and $A_\infty $ , as discussed in [Reference Meyer and Nest40, Section 2.4], is based on the notion of an admissible system in $\operatorname {\mathrm {KK}}^G$ . We do not need this definition here, but we recall a sufficient condition: The system $(A_n,\alpha _m^n)$ is admissible if there exist equivariant completely positive contractions $\phi _n\colon A_\infty \to A_n$ such that $\alpha _n^\infty \circ \phi _n\colon A_\infty \to A_\infty $ converges to the identity in the point norm topology [Reference Meyer and Nest40, Lemma 2.7]. The situation is simpler in $E_G$ -theory: By Definition 1.6, since all extensions in $E_G$ -theory are admissible, all inductive systems are admissible too.

1.3 Crossed products of Hilbert modules and descent

In this section, we recall the notion of crossed product of Hilbert modules and define the Kasparov descent morphism in the context of groupoids. We will focus on reduced crossed products. To this end, we start by recasting $C_0(X)$ -algebras under the perspective of $C^*$ -bundles. If A is a $C_0(X)$ -algebra, there exists a topology on $\mathcal {A}=\bigsqcup _{x\in X} A_x$ making the natural map $\mathcal {A}\to X$ an upper-semicontinuous $C^*$ -bundle. The associated algebra of sections vanishing at infinity, denoted $\Gamma _0(X,\mathcal {A})$ , admits a $C_0(X)$ -linear isomorphism onto A. The correspondence $A\mapsto \mathcal {A}$ sends $C_0(X)$ -linear morphisms to $C^*$ -bundles morphisms.

If $f\colon Y \to X$ is a continuous map, the pullback $C^*$ -algebra $f^*A$ can also be defined by first constructing the pullback bundle $f^*\mathcal {A}$ , then setting $f^*A=\Gamma _0(Y,f^*\mathcal {A})$ . A G-action on A can be given by defining a functor from G (viewed as a category) to the category of $C^*$ -algebras, sending $x\in X$ to $A_x$ , then imposing continuity on the resulting G-action on the topological space $\mathcal {A}$ . The definition of $A\rtimes G$ can then be reframed by endowing the compactly supported sections $\Gamma _c(G,r^*\mathcal {A})$ with a $\ast $ -algebra structure and completing in the appropriate norm as explained previously.

Given a G-algebra $(A,\alpha )$ and a Hilbert A-module $\mathcal {E}$ , for each $x\in X$ one defines the Hilbert $A_{x}$ -module $\mathfrak {E}_{x}$ to be the balanced tensor product $\mathcal {E}\otimes _{A}A_{x}$ . The space $\mathfrak {E}:=\bigsqcup _{x\in X}\mathfrak {E}_{x}$ may be topologized to obtain an upper-semicontinuous Hilbert $\mathcal {A}$ -module bundle $p_{\mathfrak {E}}:\mathfrak {E}\longrightarrow X$ . The space of sections $\Gamma _{0}(X;\mathfrak {E})$ is equipped with pointwise operations to furnish a Hilbert $\Gamma _{0}(X;\mathcal {A})$ -module, to which $\mathcal {E}$ is canonically isomorphic as a Hilbert A-module. We will identify $\mathcal {E}$ with its associated section space $\Gamma _{0}(X;\mathfrak {E})$ . We have associated bundles of $C^{*}$ -algebras $\mathcal {K}(\mathfrak {E})$ and $\mathcal {L}(\mathfrak {E})$ , whose fibres over $x\in X$ are $\mathcal {K}(\mathfrak {E}_{x})$ and $\mathcal {L}(\mathfrak {E}_{x})$ , respectively (the former bundle is upper-semicontinuous). By the identification $\mathcal {E} = \Gamma _{0}(X;\mathfrak {E})$ , we then also have $\mathcal {K}(\mathcal {E}) = \Gamma _{0}(X;\mathcal {K}(\mathfrak {E}))$ and $\mathcal {L}(\mathcal {E}) = \Gamma _{b}(X;\mathcal {L}(\mathfrak {E}))$ (strictly continuous bounded sections).

A G-action $\mathcal {E} = \Gamma _{0}(X;\mathfrak {E})$ consists of a family $\{W_{\gamma }\}_{\gamma \in G}$ such that:

• – for each $\gamma \in G$ , $W_{\gamma }:\mathfrak {E}_{s(\gamma )}\longrightarrow \mathfrak {E}_{r(\gamma )}$ is an isometric isomorphism of Banach spaces such that $\langle W_{\gamma }e, W_{\gamma }f\rangle _{r(\gamma )} = \alpha _{\gamma }(\langle e,f\rangle _{s(\gamma )})$ for all $e,f\in \mathfrak {E}_{s(\gamma )}$ ;

• – the map $G{}_s{\times }{_{p_{\mathfrak {E}}}}\mathfrak {E}\longrightarrow \mathfrak {E}$ , $(\gamma ,e)\mapsto W_{\gamma }e$ defines a continuous action of G on $\mathfrak {E}$ .

Conjugation by W gives rise to a strictly continuous action $\varepsilon :G{}_s{\times }{_{p_{\mathfrak {E}}}}\mathcal {L}(\mathfrak {E})\longrightarrow \mathcal {L}(\mathfrak {E})$ of G on the upper-semicontinuous bundle $\mathcal {L}(\mathfrak {E})$ (the restriction of $\varepsilon $ to the compact operators is continuous in the usual sense).

If $(B, \beta ) $ is a G-algebra and $\pi \colon B\to \mathcal {L}(\mathcal {E})$ a $C_0(X)$ -linear representation, we define a G-representation by requiring equivariance, namely for all $\gamma \in G$ we have

$$\begin{align*}\varepsilon_{\gamma}\circ\pi_{s(\gamma)} = \pi_{r(\gamma)}\circ \beta_{\gamma}. \end{align*}$$

Given a Kasparov module $(\pi ,\mathcal {E},T)$ representing a class in $\operatorname {\mathrm {KK}}^G(B,A)$ , let us consider the $B\rtimes _r G$ - $A\rtimes _r G$ -module $(\tilde {\pi },\mathcal {E}\widehat {\otimes }_A (A\rtimes _r G), T\widehat {\otimes } 1)$ , where $\tilde {\pi }$ is a representation of $B\rtimes _r G$ induced by $\pi $ as follows. First of all, note that $\mathcal {E}\widehat {\otimes }_A (A\rtimes _r G)$ is isomorphic to the completion of $\Gamma _c(G,r^*\mathfrak {E})$ with respect to the $\Gamma _{c}(G;r^{*}\mathcal {A})$ -valued inner product

$$\begin{align*}\langle\xi,\xi'\rangle(\gamma):=\int_{G}\alpha_{\eta}\big(\langle\xi(\eta^{-1}),\xi'(\eta^{-1}\gamma)\rangle_{s(\eta)}\big)\,d\lambda^{r(\gamma)}(\eta), \end{align*}$$

for $\xi ,\xi '\in \Gamma _{c}(G;r^{*}\mathfrak {E})$ and $\gamma \in G$ . We denote this completion $\mathcal {E}\rtimes G$ . Consider the formula below, defined for $f\in \Gamma _{c}(G;r^{*}\mathcal {A})$ , $\xi \in \Gamma _{c}(G;r^{*}\mathfrak {E})$ , and $\gamma \in G$ ,

$$\begin{align*}(f\cdot\xi)(\gamma):=\int_{G}\pi_{r(\eta)}(f(\eta))W_{\eta}\big(\xi(\eta^{-1}\gamma)\big)\,d\lambda^{r(\gamma)}(\eta). \end{align*}$$

This determines a bounded representation $\tilde {\pi }=\pi \rtimes G :A\rtimes _rG\longrightarrow \mathcal {L}(\mathcal {E}\rtimes G)$ (see, for example, [Reference MacDonald34, Prop. 7.6]).

Definition 1.18. We define the Kasparov descent morphism to be the homomorphism of abelian groups

$$\begin{align*}\jmath^G\colon \operatorname{\mathrm{KK}}^G(B,A)\to \operatorname{\mathrm{KK}}(B\rtimes_r G, A\rtimes_r G) \end{align*}$$

which sends the class of $(\pi ,\mathcal {E}, T)$ to the class of $(\tilde {\pi },\mathcal {E}\widehat {\otimes }_A (A\rtimes _r G),T\widehat {\otimes } 1)$ .

It can be checked that $\jmath ^G$ is compatible with the product in $\operatorname {\mathrm {KK}}^G$ , meaning that $\jmath ^G(x\,\widehat {\otimes }_D\, y)=\jmath ^G(x)\,\widehat {\otimes }_{D\rtimes _r G}\, \jmath ^G(y)$ , giving us a well-defined functor [Reference Le Gall31, Theorem 3.4].

2.1 The induction functor

Let $(B,\beta )\in \operatorname {\mathrm {KK}}^H$ with moment map $\rho \colon C_0(H^{0})\to Z(\mathcal {M}(B))$ . In this subsection, it is sufficient to assume H is locally closed in G. Recall $G_{H^{0}}$ is the subspace of G consisting of arrows with source in $H^{0}$ . We consider the restriction of the source map $\phi = s|_{ G_{H^0}}:G_{H^{0}}\rightarrow H^{0}$ and construct the pullback algebra

$$\begin{align*}\phi^*B= C_0(G_{H^{0}}) {\mathbin{\mathop{{}_{s}{\otimes}{_{\rho}}}{}_{{H^{0}}}}} B. \end{align*}$$

This balanced tensor product is then a $C_0(H^{0})$ -algebra in its own right and can be equipped with the diagonal action $\text {rt}\otimes \beta $ of H, where $\text {rt}$ denotes the action of H on $C_0(G_{H^{0}})$ induced by right translation. We define the induced algebra as the corresponding reduced crossed product

$$\begin{align*}\operatorname{\mathrm{Ind}}_H^G B:=(C_0(G_{H^{0}}) {\mathbin{\mathop{{}_{s}{\otimes}{_{\rho}}}{}_{{H^{0}}}}} B)\rtimes_{\text{rt}\otimes \beta} H. \end{align*}$$

To define a G-action on $\operatorname {\mathrm {Ind}}_H^G B$ , notice that G also acts on the balanced tensor product $C_0(G_{H^{0}}) \otimes _{H^{0}} B$ by $\mathrm {lt}\otimes \mathrm {id}_B$ , where $\mathrm {lt}$ denotes the action of G on $C_0(G_{H^{0}})$ induced by left translation. A straightforward computation reveals that the actions $\mathrm {rt}\otimes \beta $ and $\mathrm {lt}\otimes \mathrm {id}_B$ commute and therefore the left translation action of G descends to an action on the crossed product $(C_0(G_{H^{0}}) \otimes _{H^{0}}B)\rtimes _{\text {rt}\otimes \beta } H.$

Having defined $\operatorname {\mathrm {Ind}}_H^G$ on objects, let us consider the case of morphisms. Consider a right Hilbert B-module $\mathcal {E}$ . Considering the canonical action $B\longrightarrow \mathcal {M}(C_0(G_{H^{0}})\otimes _{H^{0}} B)$ given by multiplication in the second factor, we can form the $\phi ^*B$ -module

$$\begin{align*}\phi^*\mathcal{E}=\mathcal{E}\otimes_B \left(C_0(G_{H^{0}})\otimes_{H^{0}} B\right). \end{align*}$$

Note the module above corresponds to the space of section of the pullback bundle $\phi ^*\mathfrak {E}$ . Assume now that $\mathcal {E}$ carries an action of H (call it $\epsilon $ ) along with a nondegenerate equivariant representation $\pi \colon A\to \mathcal {L}(\mathcal {E})$ of an H-algebra A. First of all, we note that $\epsilon \otimes (\mathrm {rt}\otimes \beta )$ defines an H-action on $\phi ^*\mathcal {E}$ . Then we define a representation of $\phi ^*A$ on $\mathcal {\phi ^*\mathcal {E}}$ by considering elements $f\otimes a$ , with $f\in C_c(G_{H^{0}})$ and $a\in A$ , whose linear span is dense in $\Gamma _c(G_{H^{0}},\phi ^*\mathcal {A})\subseteq \phi ^*A$ and setting $\phi ^*\pi (f\otimes a) = \pi (a)\otimes (f\,\cdot )$ .

Now, if $(\pi ,\mathcal {E},T)$ is an A-B-Kasparov module, then $(\phi ^*{\pi },\phi ^*\mathcal {E},T\widehat {\otimes } 1)$ is a $\phi ^*A$ - $\phi ^*B$ -module equipped with an action of H, and we can define the induction functor by means of the descent morphism defined above, as follows:

$$\begin{align*}\operatorname{\mathrm{Ind}}_H^G({\pi},\mathcal{E},T)= \jmath_H(\phi^*{\pi},\phi^*\mathcal{E},T\widehat{\otimes} 1). \end{align*}$$

To complete the description of $\operatorname {\mathrm {Ind}}_H^G$ , we need two more observations. The $\phi ^*B$ -module $\mathcal {E}\otimes _B (C_0(G_{H^{0}})\otimes _{H^{0}} B)$ admits a G-action induced by left translation on $C_0(G_{H^{0}})$ . Notice this action is defined by fibreing over the range map. Clearly $T\widehat {\otimes } 1$ is equivariant with respect to this translation. To check the equivariance of $\phi ^*\pi $ , by definition it is sufficient to consider $\gamma \in G$ and $f\in C_c(G_{H^{0}})$ , and write

$$\begin{align*}[\gamma\cdot(f\cdot(\gamma^{-1}\cdot g))](\eta) = f(\gamma\eta)g(\gamma\gamma^{-1}\eta)= (\mathrm{lt}_\gamma(f)\cdot g)(\eta) \end{align*}$$

with $g\in C_0(G_{H^{0}})$ , $\eta \in G_{H^{0}}$ with $r(\eta )=s(\gamma )$ . This ensures the G-action commutes with the H-action on $(\phi ^*{\pi },\phi ^*\mathcal {E},T\widehat {\otimes } 1)$ , hence $\jmath _H(\phi ^*{\pi },\phi ^*\mathcal {E},T\widehat {\otimes } 1)\in \operatorname {\mathrm {KK}}^G(A,B)$ . Finally, as $\operatorname {\mathrm {Ind}}_H^G$ is defined as a composition of the pullback functor $\phi ^*$ with the descent functor $\jmath ^G$ , it is indeed a functor $\operatorname {\mathrm {Ind}}_H^G\colon \operatorname {\mathrm {KK}}^H\to \operatorname {\mathrm {KK}}^G$ .

2.2 Proof of the adjunction

Recall that if G acts freely and properly on a second countable, locally compact, Hausdorff space Y, then $G\ltimes Y$ is Morita equivalent as a groupoid to $Y/G$ and hence the groupoid $C^*$ -algebra $C_0(Y)\rtimes G\cong C^*(G\ltimes Y)$ is strongly Morita equivalent to $C_0(Y/G)$ [Reference Brown13]. Note that $G\ltimes Y$ is an amenable groupoid, so the reduced and full crossed products are isomorphic; see, for example, [Reference Anantharaman-Delaroche and Renault1, Corollary 2.1.17 & Proposition 6.1.10]).

In particular, when Y equals G itself and the action is given by right translation, the associated imprimitivity bimodule $X^G$ gives a $*$ -isomorphism $C_0(G)\rtimes _{\text {rt}} G\cong \mathcal {K}(L^2(G))$ , where $L^2(G)$ is the standard continuous field of Hilbert spaces associated to G. The $\operatorname {\mathrm {KK}}$ -class induced by $X^G$ will be important in a moment.

If $(A,G,\alpha )$ is a groupoid dynamical system, then the pushforward along the source map $s_*\alpha $ is an isomorphism of $C^*$ -dynamical systems:

$$\begin{align*}s_*\alpha: (s_*(C_0(G){\mathbin{\mathop{{}_{s}{\otimes}{_{\rho}}}{}_{{G^{0}}}}} A), G,\text{rt}\otimes\alpha)\to (s_*(C_0(G){\mathbin{\mathop{{}_{r}{\otimes}{_{\rho}}}{}_{{G^{0}}}}} A),G,\text{rt}\otimes \mathrm{id}_A), \end{align*}$$

where the intertwining map is given precisely by $\alpha $ [Reference Le Gall30]. As a consequence, we have the following.

Lemma 2.2. If $H\subseteq G$ is a locally closed subgroupoid and A is a G-algebra, then we have a canonical isomorphism

$$\begin{align*}\Phi \colon \operatorname{\mathrm{Ind}}_H^G\operatorname{\mathrm{Res}}_G^H A \cong (C_0(G_{H^{0}})\rtimes_{\mathrm{rt}} H) \otimes_{G^{0}} A. \end{align*}$$

After $\Phi $ , the G-action on the right-hand side is given by $\text {lt}\otimes \alpha $ , that is, left translation on $C_0(G_{H^{0}})\rtimes _{\mathrm {rt}} H$ , tensorized with the original action $\alpha $ on A.

Proof. Let $\alpha :s^*A\longrightarrow r^*A$ denote the $C_0(G)$ -linear isomorphism implementing the action of G on A. Now, we can consider the pushforward along the source maps to obtain a $C_0(G^{0})$ -linear isomorphism $\alpha =s_*\alpha :s_*s^*A\longrightarrow s_*r^*A$ . Now, $s_*s^*A$ is just the balanced tensor product $C_0(G)\otimes _{G^{0}}A$ with the canonical $C_0(G^{0})$ -algebra structure, while $s_*r^*A= \Gamma _0(G,r^*\mathcal {A})$ is equipped with the $C_0(G^{0})$ -algebra structure obtained by the formula $(\varphi \cdot f)(g)=\varphi (s(g))f(g)$ for $\varphi \in C_0(G^{0})$ and $f\in \Gamma _0(G,r^*\mathcal {A})$ . Note that this differs from the canonical structure it obtains as a balanced tensor product! With the structure defined above we can identify the fibre over a point $x\in G^{0}$ as $\Gamma _0(G,r^*\mathcal {A})_x=\Gamma _0(G_x,r^*\mathcal {A})$ and it makes sense to consider the action $\mathrm {rt}\otimes \mathrm {id}_A$ defined by

$$ \begin{align*}(\mathrm{rt}\otimes\mathrm{id}_A)_g(f)(h)=f(hg).\end{align*} $$

Summing up the discussion, we see that $\alpha $ implements an isomorphism of groupoid dynamical systems

$$\begin{align*}(C_0(G){\mathbin{\mathop{{}_{s}{\otimes}{_{\rho}}}{}_{{G^{0}}}}} A,G,\text{rt}\otimes\alpha)\to (C_0(G){\mathbin{\mathop{{}_{r}{\otimes}{_{\rho}}}{}_{{G^{0}}}}} A,G,\text{rt}\otimes \mathrm{id}_A). \end{align*}$$

Now, if we restrict these systems to the subgroupoid H we obtain an isomorphism

$$\begin{align*}(C_0(G_{H^{0}}){\mathbin{\mathop{{}_{s}{\otimes}{_{\rho}}}{}_{{H^{0}}}}} \operatorname{\mathrm{Res}}^H_G A,H,\text{rt}\otimes\alpha)\to (C_0(G_{H^{0}}){\mathbin{\mathop{{}_{r}{\otimes}{_{\rho}}}{}_{{G^{0}}}}} A,H,\text{rt}\otimes \mathrm{id}_A). \end{align*}$$

In particular, we obtain an isomorphism between the crossed products and hence conclude

$$ \begin{align*} \operatorname{\mathrm{Ind}}_H^G \operatorname{\mathrm{Res}}_G^H A & = (C_0(G_{H^{0}}) {\mathbin{\mathop{{}_{s}{\otimes}{_{\rho}}}{}_{{H^{0}}}}} \operatorname{\mathrm{Res}}_G^H A)\rtimes_{r,\text{rt}\otimes \alpha} H\\ &\cong (C_0(G_{H^{0}}) {\mathbin{\mathop{{}_{r}{\otimes}{_{\rho}}}{}_{{G^{0}}}}} A)\rtimes_{r,\text{rt}\otimes \mathrm{id}_A} H\\ &\cong (C_0(G_{H^{0}})\rtimes_{\mathrm{rt}}H)\otimes_{G^{0}} A.\\[-37pt] \end{align*} $$

Choosing $H=G$ in the result above yields an isomorphism

$$\begin{align*}\operatorname{\mathrm{Ind}}_G^G\operatorname{\mathrm{Res}}_G^G(B)\cong (C_0(G)\rtimes_{\text{rt}} G) \otimes_{G^{0}} B\cong \mathcal{K}(L^2(G))\otimes_{G^{0}} B. \end{align*}$$

We now prepare to prove the adjunction by defining some auxiliary maps. From now on, we assume $H\subseteq G$ to be an open subgroupoid. We get an induced embedding

$$\begin{align*}C_0(G_{H^{0}})\rtimes_{\mathrm{rt}}H\hookrightarrow C_0(G)\rtimes_{\mathrm{rt}}G\end{align*}$$

and hence, using the previous lemma, an embedding

$$\begin{align*}\kappa\colon \operatorname{\mathrm{Ind}}_H^G\operatorname{\mathrm{Res}}_G^H(B)\longrightarrow \mathcal{K}(L^2(G))\otimes_{G^{0}} B. \end{align*}$$

We can promote $X^G$ to a $\operatorname {\mathrm {KK}}^G$ -equivalence

$$\begin{align*}X^G_A\in \operatorname{\mathrm{KK}}^G(\operatorname{\mathrm{Ind}}_G^G\operatorname{\mathrm{Res}}_G^G(A),A) \end{align*}$$

given by the right A-module $L^2(G)_r{\otimes }{_\rho } A$ , where A acts pointwise as ‘constant functions’. The representation of the crossed product $r^*A\rtimes G\cong \operatorname {\mathrm {Ind}}_G^G\operatorname {\mathrm {Res}}_G^G(A)$ is the integrated form of the covariant pair given by the right regular representation of G, and pointwise multiplication of functions in $r^*A$ . We will denote this by $M_A\rtimes R_G$ .

Now, let $B\in \operatorname {\mathrm {KK}}^H$ and recall that

$$\begin{align*}\operatorname{\mathrm{Res}}_G^H\operatorname{\mathrm{Ind}}_H^G(B)=(C_0(G|_{ H^{0}})\otimes_{H^{0}}B)\rtimes H .\end{align*}$$

Then the inclusion $C_0(H)\subseteq C_0(G|_{ H^{0}})$ induces a map

$$\begin{align*}\iota\colon \operatorname{\mathrm{Ind}}_H^H B\cong (C_0(H)\otimes_{H^{0}}B)\rtimes H \to (C_0(G|_{ H^{0}})\otimes_{H^{0}}B)\rtimes H=\operatorname{\mathrm{Res}}_G^H\operatorname{\mathrm{Ind}}_H^G(B). \end{align*}$$

Theorem 2.3. Let G be a locally compact Hausdorff groupoid with Haar system. For every open subgroupoid $H\subseteq G$ , there is an adjunction

$$\begin{align*}(\epsilon,\eta)\colon \operatorname{\mathrm{Ind}}_H^G \dashv \operatorname{\mathrm{Res}}_G^H \end{align*}$$

with counit and unit

$$ \begin{align*} \epsilon\colon \operatorname{\mathrm{Ind}}_H^G\operatorname{\mathrm{Res}}_G^H&\to 1_{\operatorname{\mathrm{KK}}^G}\\ \eta\colon 1_{\operatorname{\mathrm{KK}}^H} &\to \operatorname{\mathrm{Res}}_G^H\operatorname{\mathrm{Ind}}_H^G \end{align*} $$

described as follows:

$$ \begin{align*} \epsilon_A &= X^G_A\circ \kappa \\ \eta_B &=\iota \circ \bigl(X^{H}_B\bigr)^{\text{op}}. \end{align*} $$

Here below we isolate a couple of technical lemmas which will be useful in the proof of the adjunction. The first lemma is just an observation on the compatibility of the canonical element $X_A^G$ with restriction and induction.

Let $L^2(G,B)$ denote the completion of $\Gamma _c(G,r^*\mathcal {B})$ with respect to the B-valued inner product $\langle \xi _1,\xi _2\rangle (x)=\int _{G^x} \xi _1(g)^*\xi _2(g)\,d\lambda ^x(g)$ . Note that $L^2(G,B)$ is canonically isomorphic to the B-module $L^2(G)\otimes _{G^{0}} B$ introduced above.

Let us make a point on notation before continuing the proof. So far, we have used A and $\mathcal {A}$ to denote a $C_0(X)$ - $C^*$ -algebra and its corresponding $C^*$ -bundle. However, this difference in font is not very convenient when A is replaced by a more complicated algebra, for example, $A=C_0(G)\rtimes H$ . In the sequel, we suppress this notational distinction, as the context suffices to disambiguate the usage.

Lemma 2.5. Let $H\subseteq G$ be an open subgroupoid and $B\in \operatorname {\mathrm {KK}}^H$ . Then there is an isometric G-equivariant homomorphism

$$\begin{align*}\Phi\colon\operatorname{\mathrm{Ind}}_H^G L^2(H,B) \longrightarrow L^2(G,\operatorname{\mathrm{Ind}}_H^G B) \end{align*}$$

of Hilbert $\operatorname {\mathrm {Ind}}_H^G B$ -modules.

Proof. Let us first describe the module $\operatorname {\mathrm {Ind}}_H^G L^2(H,B)$ more concretely. We have a canonical isomorphism $L^2(H,B)\otimes _B (C_0(G_{H^{0}})\otimes _{H^{0}} B)\cong L^2(H,C_0(G_{H^{0}})\otimes _{H^{0}} B)$ given by $\xi \otimes f\mapsto [h\mapsto \xi (h)f]$ . Hence, we can write $\operatorname {\mathrm {Ind}}_H^G L^2(H,B)$ as $L^2(H,C_0(G_{H^{0}})\otimes _{H^{0}} B)\rtimes H$ . So for a function $\xi \in \Gamma _c(H,r^*L^2(H,C_0(G_{H^{0}})\otimes _{H^{0}} B))$ , we define $\Phi (\xi )\in L^2(G,\operatorname {\mathrm {Ind}}_H^G B)$ as

$$ \begin{align*}\Phi(\xi)(g,h,x)=\left\{\begin{array}{ll} \beta_{x^{-1}g}(\xi(g^{-1}xh,g^{-1}x,g)), & g^{-1}x\in H \\ 0 ,& \text{otherwise} \end{array}\right\},\end{align*} $$

where $g\in G$ , $h\in H$ and $x\in G_{r(h)}^{r(g)}$ .

Given $\xi _1,\xi _2 \in \Gamma _c(H,r^*L^2(H,C_0(G_{H^{0}})\otimes _{H^{0}} B))$ , we compute (for $h\in H$ and $x\in G_{r(h)}$ ) that $\langle \Phi (\xi _1),\Phi (\xi _2)\rangle (h,x)$ equals

$$ \begin{align*} &\int\limits_{G}\left[\Phi(\xi_1)(g)^\ast\Phi(\xi_2)(g)\right](h,x)\,d\lambda^{r(x)}(g)\\ = &\int\limits_{G} \int\limits_{H}(\mathrm{rt}\otimes\beta)_{\tilde{h}}(\Phi(\xi_1)(g,\tilde{h}^{-1})^*\Phi(\xi_2)(g,\tilde{h}^{-1}h))(x)\,d\lambda^{r(h)}(\tilde{h})\,d\lambda^{r(x)}(g)\\ = &\int\limits_{G} \int\limits_{H}\beta_{\tilde{h}}(\Phi(\xi_1)(g,\tilde{h}^{-1},x\tilde{h})^*\Phi(\xi_2)(g,\tilde{h}^{-1}h,x\tilde{h}))\,d\lambda^{r(h)}(\tilde{h})\,d\lambda^{r(x)}(g)\\ =& \int\limits_{xH} \int\limits_{H}\beta_{x^{-1}g}(\xi_1(g^{-1}x,g^{-1}x\tilde{h},g)^*\xi_2(g^{-1}xh,g^{-1}x\tilde{h},g))\,d\lambda^{r(h)}(\tilde{h})\,d\lambda^{r(x)}(g).\end{align*} $$

At this point, we perform two change of variables and keep computing:

$$ \begin{align*} &\stackrel{g\mapsto xg}{=}\int\limits_{H}\int\limits_{H}\beta_g(\xi_1(g^{-1},g^{-1}\tilde{h},xg)^*\xi_2(g^{-1}h,g^{-1}\tilde{h},xg))\,d\lambda^{r(h)}(\tilde{h})\,d\lambda^{s(x)}(g)\\ & \stackrel{\tilde{h}\mapsto g\tilde{h}}{=}\int\limits_{H}\int\limits_{H}\beta_g(\xi_1(g^{-1},\tilde{h},xg)^*\xi_2(g^{-1}h,\tilde{h},xg))\,d\lambda^{s(g)}(\tilde{h})\,d\lambda^{s(x)}(g)\\ & = \int\limits_{H}\int\limits_{H}\beta_{g^{-1}}(\xi_1(g,\tilde{h},xg^{-1})^*\xi_2(gh,\tilde{h},xg^{-1}))\,d\lambda^{r(g)}(\tilde{h})\,d\lambda_{s(x)}(g)\\ & = \int\limits_{H} (\mathrm{rt}\otimes \beta)_{g^{-1}}(\langle \xi_1(g),\xi_2(gh)\rangle(x)\,d\lambda_{r(h)}(g)\\ & = \langle \xi_1,\xi_2\rangle (h,x). \end{align*} $$

This verifies that $\Phi $ extends to an isometry. Now, we proceed to checking that $\Phi $ is a right module map. Below, we have $\xi \in \Gamma _c(H,r^*L^2(H,C_0(G_{H^{0}})\otimes _{H^{0}} B))$ as before, and the element f belongs to $\Gamma _c(H,r^*(C_0(G_{H^{0}}){}_s{\otimes }{_\rho }B))$ .

$$ \begin{align*} (\Phi(\xi)f)(g,h,x) &= \Phi(\xi)(g,h,x)f(h,x)\\ &=\int_{H^{r(g)}} \Phi(\xi)(g,\tilde{h},x)\beta_{\tilde{h}}(f(\tilde{h}^{-1}h,x\tilde{h}))\,d\lambda^{r(h)}(\tilde{h})\\ & = \int_{H^{r(g)}} \beta_{x^{-1}g}(\xi(g^{-1}x\tilde{h},g^{-1}x,g))\beta_{\tilde{h}}(f(\tilde{h}^{-1}h,x\tilde{h}))\,d\lambda^{r(h)}(\tilde{h})\\ & \stackrel{\tilde{h}\mapsto x^{-1}g\tilde{h}}{=} \int_{H^{s(g)}} \beta_{x^{-1}g}(\xi(\tilde{h},g^{-1}x,g)\beta_{\tilde{h}}(f(\tilde{h}^{-1}g^{-1}xh,g\tilde{h})))\,d\lambda^{s(g)}(\tilde{h})\\ & = \beta_{x^{-1}g}((\xi f)(g^{-1}xh,g^{-1}x,g))\\ & = \Phi(\xi f)(g,h,x). \end{align*} $$

To complete the argument, we show that the left action of G commutes with $\Phi $ . Let us take $g'\in G$ with $r(g')=r(g)$ , and compute

$$ \begin{align*} (g'\Phi(\xi))(g,h,x)&=\Phi(\xi)(g^{\prime-1}g,h,g^{\prime-1}x)\\ &=\beta_{x^{-1}g} (\xi(g^{-1}xh,g^{-1}x,g^{\prime-1}g))\\ &=\beta_{x^{-1}g}((g' \xi)(g^{-1}xh,g^{-1}x,g))\\ &=\Phi(g'\xi)(g,h,x). \end{align*} $$

The proof is complete.

Proof of Theorem 2.3.

We need to verify the counit-unit equations. We start by proving that for every $A\in \operatorname {\mathrm {KK}}^G$ the composition

equals the identity in $\operatorname {\mathrm {KK}}^H(\operatorname {\mathrm {Res}}_G^HA,\operatorname {\mathrm {Res}}_G^HA)$ : Expanding the definitions of counit and unit in this case, we have $\operatorname {\mathrm {Res}}_G^H(\epsilon _A)\circ \eta _{\operatorname {\mathrm {Res}}_G^H A} = \operatorname {\mathrm {Res}} (X_A^G)\circ \operatorname {\mathrm {Res}}(\kappa )\circ \iota \circ (X_{\operatorname {\mathrm {Res}}_G^H A}^H)^{\text {op}}$ . Following the definitions, it is then easily seen that after identifying

$$ \begin{gather*} \operatorname{\mathrm{Ind}}_H^H(\operatorname{\mathrm{Res}}_G^H A)=(C_0(H)\rtimes_{\mathrm{rt}}H)\otimes \operatorname{\mathrm{Res}}_G^HA\\ \operatorname{\mathrm{Res}}_G^H(\operatorname{\mathrm{Ind}}_G^G A)\cong \operatorname{\mathrm{Res}}_G^H((C_0(G)\rtimes_{\mathrm{rt}}G)\otimes_{G^{0}} A)\cong (C_0(G^{H^{0}})\rtimes_{\mathrm{rt}}G)\otimes_{H^{0}}\operatorname{\mathrm{Res}}_G^H A, \end{gather*} $$

the composition $\operatorname {\mathrm {Res}}(\kappa )\circ \iota $ is just given by

(4)00E9 $$ \begin{align} (C_0(H)\rtimes H)\otimes_{H^{0}} \operatorname{\mathrm{Res}}_G^H A\stackrel{j\otimes \mathrm{id}}{\longrightarrow}(C_0(G^{H^{0}})\rtimes G)\otimes_{H^{0}} \operatorname{\mathrm{Res}}_G^H A, \end{align} $$

where $j:C_0(H)\rtimes H\longrightarrow C_0(G^{H^{0}})\rtimes G$ is induced by the inclusion of H as an open subgroupoid. Using Lemma 2.4, we have

$$ \begin{align*} \operatorname{\mathrm{Res}}_G^H(\epsilon_A)\circ\eta_{\operatorname{\mathrm{Res}}_G^H A} &= \operatorname{\mathrm{Res}} (X_A^G)\circ \operatorname{\mathrm{Res}}(\kappa)\circ \iota\circ (X_{\operatorname{\mathrm{Res}}_G^H A}^H)^{\text{op}}\\ &=\sigma_{\operatorname{\mathrm{Res}}_G^H A}(\operatorname{\mathrm{Res}}_G^H(X_{C_0(G^{0})}^G) \circ \sigma_{\operatorname{\mathrm{Res}}_G^H A}(j)\circ \sigma_{\operatorname{\mathrm{Res}}_H^G A}((X_{C_0(H^{0})}^H)^{\text{op}})\\ &= \sigma_{\operatorname{\mathrm{Res}}_G^H A}\left(\operatorname{\mathrm{Res}}_G^H(X_{C_0(G^{0})}^G)\circ j\circ (X_{C_0(H^{0})}^H)^{\text{op}}\right). \end{align*} $$

Hence, it is enough to show that the conclusion holds for $A=C_0(G^{0})$ . In this case, we can further use the isomorphisms $C_0(H)\rtimes _{\mathrm {rt}} H\cong \mathcal {K}(L^2(H))$ and $C_0(G^{H^{0}})\rtimes _{\mathrm {rt}} G\cong \mathcal {K}(L^2(G^{H^{0}}))$ to replace the map in Equation (4) by the canonical map

$$\begin{align*}i\colon \mathcal{K}(L^2(H))\to \mathcal{K}(L^2(G^{H^{0}})) \end{align*}$$

and the required verification is easily seen to be reduced to showing that the (interior) Kasparov product

$$\begin{align*}[(X^H_{C_0(H^{0})})^{\text{op}}]\,\widehat{\otimes}\,_{\mathcal{K}(L^2(H))} i^*[\operatorname{\mathrm{Res}}_G^H(X^G_{C_0(G^{0})})] \end{align*}$$

equals the class of identity $id_{C_0(H^0)}$ in $\operatorname {\mathrm {KK}}^H(C_0(H^{0}),C_0(H^{0}))$ .

The element $\operatorname {\mathrm {Res}}_G^H(X_{C_0(G^{0})}^G)\in \operatorname {\mathrm {KK}}^H(\mathcal {K}(L^2(G^{H^{0}})),C_0(H^{0}))$ can be represented by the triple $(L^2(G^{H^{0}}),\Phi ,0)$ , where $\Phi $ is the canonical action. Consequently, $i^*[\operatorname {\mathrm {Res}}_G^H(X^G_{C_0(G^{0})})$ is represented by $(L^2(G^{H^{0}}),\Phi \circ i,0)$ . The representation $\Phi \circ i$ fails to be nondegenerate, but we can replace $L^2(G^{H^{0}})$ by its ‘nondegenerate closure’ $\overline {\Phi \circ i(\mathcal {K}(L_s^2(H)))L^2(G^{H^{0}})}$ without changing its $\operatorname {\mathrm {KK}}^H$ -class (see [Reference Blackadar6, Proposition 18.3.6]). This module is easily seen to be (isomorphic to) $L^2(H)$ . Therefore, $i^*[\operatorname {\mathrm {Res}}_G^H(X^G_{C_0(G^{0})})]=[X^H_{C_0(H^{0})}]$ and the desired equality follows from

$$\begin{align*}[(X^H_{C_0(H^{0}))})^{\text{op}}]\,\widehat{\otimes}\,_{\mathcal{K}(L^2(H))} [X^H_{C_0(H^{0}))}]=1\in \operatorname{\mathrm{KK}}^H(C_0(H^{0}),C_0(H^{0})). \end{align*}$$

The next verification in order regards the composition

(5)

The map $\kappa \circ \operatorname {\mathrm {Ind}}_H^G(\iota )$ gives an inclusion

By using the isomorphisms introduced in Lemma 2.2 above, we can replace the previous inclusion into the more convenient map

Above, the Greek letters indicate our choice of notation for the variable on the given groupoid. These will be useful in a moment.

Recall the action on A is denoted by $\alpha $ . Suppressing notation for the inclusions $H\subseteq G$ and $C_0(H)\subseteq C_0(G)$ , the map i can be understood by

(6) $$ \begin{align} i(f)(\eta,\gamma,\mu,\nu)=\alpha_{\nu^{-1}\gamma}(f(\eta,\gamma,\mu,\gamma^{-1}\nu)), \end{align} $$

where f is in $\Gamma _c(H,r^*(C_0(G_{H^{0}}){}_s{\otimes }{_{r\otimes \rho }}(C_0(H)_r{\otimes }{_\rho }A)\rtimes _{\text {rt}\otimes \text {id}} H))$ . Note that the right-hand side is zero unless $\gamma ^{-1}\nu \in H$ and $\eta \in H$ (note $\gamma \in G_{H^{0}}$ follows). The composition in Equation (5) can be computed via the Kasparov product (over the domain of i)

$$\begin{align*}[\operatorname{\mathrm{Ind}}_H^G\bigl(\bigl(X^H_A\bigr)^{\text{op}}\bigr)]\,\widehat{\otimes}\, i^*[X^G_{\operatorname{\mathrm{Ind}}_H^G A}]. \end{align*}$$

We claim that

$$ \begin{align*}i^*[X^G_{\operatorname{\mathrm{Ind}}_H^G A}]=\operatorname{\mathrm{Ind}}_H^G(X_A^H)\end{align*} $$

The class $i^*[X^G_{\operatorname {\mathrm {Ind}}_H^G A}]$ is represented by the Kasparov triple

$$\begin{align*}\left(L^2(G,\operatorname{\mathrm{Ind}}_H^G A),(M_{\operatorname{\mathrm{Ind}}_H^G A}\rtimes R_G)\circ i,0\right) \end{align*}$$

while the class $\operatorname {\mathrm {Ind}}_H^G(X_A^H)$ is represented by

$$\begin{align*}\left(\operatorname{\mathrm{Ind}}_H^G L^2(H,A),\operatorname{\mathrm{Ind}}_H^G(M_A\rtimes R_H),0\right).\end{align*}$$

Consider the isometric embedding

$$\begin{align*}\Phi\colon\operatorname{\mathrm{Ind}}_H^G L^2(H,A) \longrightarrow L^2(G,\operatorname{\mathrm{Ind}}_H^G A) \end{align*}$$

from Lemma 2.5. We first verify that $\Phi $ intertwines the left actions of $\operatorname {\mathrm {Ind}}_H^G \operatorname {\mathrm {Ind}}_H^H A$ . To this end, recall that for $f\in \Gamma _c(H,r^*(C_0(G_{H^{0}})\otimes _{G^{0}}\operatorname {\mathrm {Ind}}_H^H A))$ , we have that $(\operatorname {\mathrm {Ind}}_H^G(M_A\rtimes R_H)(f)\xi )(g,h,x)$ equals

$$\begin{align*}\int\limits_H\int\limits_H f(h_1,x,h_2,h)\alpha_{h_1}(\xi(h_1^{-1}g,h_1^{-1}hh_2,xh_1))\,d\lambda^{s(h)}(h_2)\,d\lambda^{s(x)}(h_1). \end{align*}$$

Hence, considering elements $\xi \in \Gamma _c(H,r^*L^2(H,C_0(G_{H^{0}})\otimes _{H^{0}} B))$ and $f\in \Gamma _c(H,r^* (C_0(G){}_s{\otimes }{_{r}}(C_0(H){}_s{\otimes }{_\rho }A)\rtimes H))$ , we compute

$$ \begin{align*} &\Phi(\operatorname{\mathrm{Ind}}_H^G(M_A\rtimes R_H)(f)\xi)(g,h,x) =\alpha_{x^{-1}g}((\operatorname{\mathrm{Ind}}_H^G(M_A\rtimes R_H)(f)\xi)(g^{-1}xh,g^{-1}x,g))\\& \quad = \int\limits_H\int\limits_H\alpha_{x^{-1}g}(f(h_1,g,h_2,g^{-1}x)\alpha_{h_1}(\xi(h_1^{-1}g^{-1}xh,h_1^{-1}g^{-1}xh_2,gh_1)))\,d\lambda^{s(x)}(h_2)\,d\lambda^{s(g)}(h_1)\\& \quad = \int\limits_H \int\limits_H i(f)(h_1,g,h_2,x)\alpha_{h_2}(\Phi(\xi)(gh_1,h_2^{-1}h,xh_2))\,d\lambda^{s(x)}(h_2)\,d\lambda^{s(g)}(h_1)\\& \quad = \left((M_{\operatorname{\mathrm{Ind}}_H^G A}\rtimes R_G)(i(f))\Phi(\xi)\right)(g,h,x). \end{align*} $$

Since the representation $\operatorname {\mathrm {Ind}}_H^G(M_A\rtimes R_H)$ is nondegenerate, it follows immediately that $\text {Img}(\Phi )\subseteq \overline {((M_{\operatorname {\mathrm {Ind}}_H^G A}\rtimes R_G)\circ i)L^2(G,\operatorname {\mathrm {Ind}}_H^G A)}$ . In fact, since $\text {Img}(\Phi )$ is closed, in order to have equality it suffices to show the image is dense. From the definition of i in Equation (6), we see that

$$\begin{align*}\overline{((M_{\operatorname{\mathrm{Ind}}_H^G A}\rtimes R_G)\circ i)L^2(G,\operatorname{\mathrm{Ind}}_H^G A)}\subseteq \overline{L^2(G_{H^{0}},\operatorname{\mathrm{Ind}}_H^G A)\cap F}, \end{align*}$$

where F is spanned by those $L^2$ -functions such that $f(g,h,x)=0$ unless $g^{-1}x\in H$ (notation from Lemma 2.5). With this, the surjectivity is clear from the formula for $\Phi $ in Lemma 2.5. Since the element $i^*[X_{\operatorname {\mathrm {Ind}}_H^G A}^G]$ can equally well be represented by the submodule $\overline {((M_{\operatorname {\mathrm {Ind}}_H^G A}\rtimes R_G)\circ i)L^2(G,\operatorname {\mathrm {Ind}}_H^G A)}$ (see [Reference Blackadar6, Proposition 18.3.6]), we conclude that $i^*[X_{\operatorname {\mathrm {Ind}}_H^G B}^G]=\operatorname {\mathrm {Ind}}_H^G X_A^H$ and hence

$$\begin{align*}[\operatorname{\mathrm{Ind}}_H^G\bigl(\bigl(X^H_A\bigr)^{\text{op}}\bigr)]\,\widehat{\otimes}\, i^*[X^G_{\operatorname{\mathrm{Ind}}_H^G A}]=1_{\operatorname{\mathrm{Ind}}_H^G A}, \end{align*}$$

as desired.

2.3 Compatibility with other functors

Let $f\colon Y\to X$ be a continuous map, and A and B be $C_0(X)$ -algebras. There is a natural isomorphism $f^*(A\otimes _X B)=f^*(A)\otimes _Y f^*(B)$ because both algebras are naturally isomorphic to restrictions of $C_0(Y\times Y)\otimes A \otimes B $ to the same copy of $Y\times X$ in the topological space $Y\times Y\times X\times X$ (cf. [Reference Bönicke and Dell’Aiera12, Lemma 6.4])

Lemma 2.6. There is a natural isomorphism

$$\begin{align*}\operatorname{\mathrm{Ind}}_H^G(A)\otimes_{G^0} B\cong \operatorname{\mathrm{Ind}}_H^G(A\otimes_{H^0} \operatorname{\mathrm{Res}}_G^H(B)). \end{align*}$$

In particular, $\operatorname {\mathrm {Ind}}_H^G\circ f^* \cong f^*\circ \operatorname {\mathrm {Ind}}_H^G $ .

Proof. Let $\phi $ be the restriction of the source map to $G_{H^{0}}$ . We have

$$\begin{align*}\phi^*(A\otimes_{H^0} \operatorname{\mathrm{Res}}_G^H(B))\cong \phi^*A\otimes_{G_{H^0}} \phi^*\operatorname{\mathrm{Res}}_G^H(B) \end{align*}$$

by the observation above. Now, pushing forward along $\phi $ again we obtain an isomorphism of H- $C^*$ -algebras

$$ \begin{align*}(\phi_*(\phi^*(A\otimes_{H^0} \operatorname{\mathrm{Res}}_G^H(B)))\cong \phi_*(\phi^*A\otimes_{G_{H^0}} \phi^*\operatorname{\mathrm{Res}}_G^H(B))\cong \phi_*\phi^*A\otimes_{H^0} \operatorname{\mathrm{Res}}_G^H B.\end{align*} $$

Now, when we take crossed products by H for the leftmost system, we get $\operatorname {\mathrm {Ind}}_H^G (A\otimes _{H^0} \operatorname {\mathrm {Res}}_G^H(B))$ by definition. The rightmost system is just $C_0(G_{H^0})\otimes _{H^0}A\otimes _{H^0} \operatorname {\mathrm {Res}}_G^H B$ with the diagonal H-action $\mathrm {rt}\otimes \alpha \otimes \operatorname {\mathrm {Res}}_G^H(\beta )$ . So upon using commutativity of the tensor product and applying Lemma 2.2, we may replace it by the action $\mathrm {rt}\otimes \alpha \otimes \mathrm {id}_{B}$ .

Summing up, after taking crossed products by H we arrive at the desired conclusion:

$$\begin{align*}\phi_*(\phi^*(A\otimes_X \operatorname{\mathrm{Res}}_G^H(B)))\rtimes H \cong \phi_*\phi^*A\rtimes H\otimes_{G^0} B, \end{align*}$$

where $\phi _*\phi ^*A\rtimes H=\operatorname {\mathrm {Ind}}_H^G(A)$ by definition.

We conclude this section by listing other compatibility relations, which are straightforward as each of them involves a forgetful functor.

$$ \begin{gather*} \operatorname{\mathrm{Res}}_G^H(A\otimes_X B)\cong \operatorname{\mathrm{Res}}_G^H(A)\otimes_X \operatorname{\mathrm{Res}}_G^H(B)\\ \operatorname{\mathrm{Ind}}_H^G\circ f_* \cong f_*\circ \operatorname{\mathrm{Ind}}_H^G\qquad \operatorname{\mathrm{Res}}_G^H\circ f_* \cong f_*\circ \operatorname{\mathrm{Res}}_G^H\qquad \operatorname{\mathrm{Res}}_G^H\circ f^* \cong f^*\circ \operatorname{\mathrm{Res}}_G^H. \end{gather*} $$

4.1 The UCT

The article [Reference Bönicke and Dell’Aiera12] established a connection between the Baum–Connes conjecture for groupoids and the Künneth formula for groupoid crossed products. Now, the UCT introduced in [Reference Rosenberg and Schochet55] is formally stronger than the Künneth formula, so philosophically speaking it may not come as a surprise that a similar relation exists between the strong Baum–Connes conjecture and the UCT.

Proof. If A is a type I $\mathrm {C}^*$ -algebra and H is a proper groupoid, the crossed product $A\rtimes H$ is type I by [Reference Tu58, Proposition 10.3]. Given A as in the claim, and $H\subseteq G$ a proper open subgroupoid acting on A, then $C_0(G_{H^0})\otimes A$ is type I, $C_0(G_{H^0})\otimes _{H^0} A$ is type I (because it is a quotient) and $L_H(A):= \operatorname {\mathrm {Ind}}_H^G\operatorname {\mathrm {Res}}_G^H(A)$ is type I as well. Hence, $L_H(A)$ belongs to the bootstrap class. Since $L_H(A)\rtimes _r G$ is Morita equivalent to $A\rtimes _r H$ and $P(A)\rtimes _r G$ belongs to the localising subcategory of $\operatorname {\mathrm {KK}}$ generated by

$$\begin{align*}\{ L_{H_1}\cdots L_{H_n}(A)\rtimes G \mid n\in \mathbb{N}, H_i\subseteq G \text{ proper and open}\}, \end{align*}$$

it follows that $P(A)\rtimes _r G$ belongs to the bootstrap class as well.

Since the bootstrap class is closed under $\operatorname {\mathrm {KK}}$ -equivalence, the strong Baum–Connes conjecture yields the result.

We do in particular obtain the following corollary, generalising [Reference Barlak and Li3, Reference Kwaśniewski, Li and Skalski28]. To state it, recall that a twist over G is a central extension

$$ \begin{align*}G^0\times\mathbb{T}\rightarrow \Sigma\stackrel{j}{\rightarrow} G,\end{align*} $$

and that one can associate the twisted groupoid $C^*$ -algebra $C_r^*(G,\Sigma )$ to these data (see [Reference Renault54] for the details of this construction).

4.2 The going-down principle

We generalize some results obtained by the first author for ample groupoids [Reference Bönicke10] to the general étale case.

Theorem 4.3. Suppose there is an element $f\in \operatorname {\mathrm {KK}}^G(A,B)$ such that

$$\begin{align*}\operatorname{\mathrm{KK}}^H(D,\operatorname{\mathrm{Res}}_G^H(A))\xrightarrow{-\,\widehat{\otimes} \operatorname{\mathrm{Res}}_G^H(f)\,}\operatorname{\mathrm{KK}}^H(D, \operatorname{\mathrm{Res}}_G^H(B)) \end{align*}$$

is an isomorphism for all $H\in \mathcal {F}$ and separable H- $C^*$ -algebras D. Then f is a weak equivalence, and in particular the Kasparov product induces an isomorphism

(12) $$ \begin{align} - \,\widehat{\otimes}_A\, f:K_*^{\mathrm{top}}(G;A)\rightarrow K_*^{\mathrm{top}}(G;B). \end{align} $$

Proof. Using the induction-restriction adjunction the hypothesis is equivalent to the following map being an isomorphism for any $\tilde {D}\in \mathcal {C}\mathcal {I}$ ,

$$ \begin{align*} \operatorname{\mathrm{KK}}^G(\tilde{D}, A)\xrightarrow{-\,\widehat{\otimes} f\,}\operatorname{\mathrm{KK}}^G(\tilde{D}, B). \end{align*} $$

Applying the functor $\operatorname {\mathrm {KK}}^G(\tilde {D}, -)$ to a mapping cone triangle for f and using the five lemma we deduce that $\operatorname {\mathrm {KK}}^G(\tilde {D}, \operatorname {\mathrm {Cone}}(f))\cong 0$ for all $\tilde {D}$ in $\langle \mathcal {C}\mathcal {I}\rangle $ . Now, by Theorem 1.12 we get $\operatorname {\mathrm {Cone}}(f)\in N_{\mathcal {I}}$ . The rest follows from Theorems 3.12 and 3.14.

If we are only interested in studying the assembly map, then we might want to prove Equation (12) without necessarily proving that A and B have isomorphic cellular approximations. The following result is a version of the previous one ‘after $K_*(-\rtimes G)$ ’, and it can be proved with slightly weaker assumptions.

Theorem 4.4 (cf. [Reference Bönicke10, Theorem 7.10]).

Let $f\in \mathrm {KK}^G(A_1,A_2)$ be an element such that the induced map

$$ \begin{align*}K_*(\jmath_H(\mathrm{Res}^H_G(f))):K_*(\mathrm{Res}^H_G(A_1)\rtimes H)\rightarrow K_*(\mathrm{Res}^H_G(A_2)\rtimes H)\end{align*} $$

is an isomorphism for all proper open subgroupoids $H\subseteq Q$ for all $Q\in \mathcal {F}$ . Then

$$ \begin{align*}K_*(\jmath_G(P(f))):K_*(P(A_1)\rtimes_r G)\rightarrow K_*(P(A_2)\rtimes_r G)\end{align*} $$

is an isomorphism.

The proof requires some preparation. For a subgroupoid $H\subseteq G$ let $L_H:= \mathrm {Ind}_H^G\circ \mathrm {Res}_G^H$ . Consider the class $\mathcal {P}_0$ of G-algebras of the form $(L_{H_n}\circ \cdots \circ L_{H_1})(C_0(G^0))$ for $n\in \mathbb N$ and $H_i\in \mathcal {F}$ .

Proof. By [Reference Meyer39, Proposition 3.18], the $\mathcal {CI}$ -cellular approximation $P(C_0(G^0))$ can be computed as the homotopy limit of a phantom castle over $C_0(G^0)$ . Hence, it is enough to show that such a phantom castle can be found inside $\langle \mathcal {P}_0\rangle $ . Using the fact that $\langle \mathcal {P}_0\rangle $ is localising, an inspection of the construction of such a phantom castle in [Reference Meyer39] shows that it suffices to show that $C_0(G^0)$ admits a projective resolution by objects in $\langle \mathcal {P}_0\rangle $ . The standard way to construct such a projective resolution is by considering the algebras $(F^\dagger \circ F)^n(C_0(G^0))$ for $n\geq 1$ .

We will prove that this resolution is contained in $\langle \mathcal {P}_0\rangle $ by induction. First, we have $(F^\dagger \circ F)(C_0(G^0))=\bigoplus _{H\in \mathcal {F}} \mathrm {Ind}_H^G \mathrm {Res}_G^H C_0(G^0)\in \langle \mathcal {P}_0\rangle $ . Assuming now that the claim holds for $n-1$ , we compute

$$ \begin{align*}(F^\dagger\circ F)^n(C_0(G^0))=\bigoplus_{H\in \mathcal{F}} \mathrm{Ind}_H \mathrm{Res}_H((F^\dagger\circ F)^{n-1}(C_0(G^0))),\end{align*} $$

and the latter is contained in $\langle \mathcal {P}_0\rangle $ since $L_H(\langle \mathcal {P}_0\rangle )\subseteq \langle \mathcal {P}_0\rangle $ (we have $L_H(\mathcal {P}_0)\subseteq \mathcal {P}_0$ by definition of $\mathcal {P}_0$ and hence the general statement follows from the fact that $L_H$ is triangulated and compatible with direct sums).

Proof of Theorem 4.4.

We will show that

(13) $$ \begin{align} K_*((\jmath_G(\mathrm{id}_B\otimes_{G^0} f))):K_*((B\otimes_{G^0} A_1)\rtimes G)\rightarrow K_*((B\otimes_{G^0} A_2)\rtimes G) \end{align} $$

is an isomorphism for all $B\in \mathcal {P}_0$ . Once this is proven, we can complete the proof as follows: since K-theory is a homological functor (compatible with direct sums), these isomorphisms imply that Equation (13) is also an isomorphism for $B\in \langle \mathcal {P}_0\rangle $ by a routine argument involving the five lemma.

In particular, we can take $B=P(C_0(G^0))$ by the previous lemma. Noting further that $P(A)\rtimes G\cong P(A)\rtimes _r G$ in $\mathrm {KK}$ , the proof will be complete. Thus, in what follows we show that Equation (13) is an isomorphism for all $B\in \mathcal {P}_0$ .

Step 1: We will first prove that Equation (13) is an isomorphism for $B=L_H(C_0(G^0))=C_0(G/H)$ whenever $H\subseteq Q$ for some $Q\in \mathcal {F}$ .. In this case, we have natural G-equivariant isomorphisms

$$ \begin{align*}B\otimes_{G^0}A_i\cong \mathrm{Ind}_H^G(C_0(H^0))\otimes_{G^0} A_i\cong \mathrm{Ind}_H^G(\mathrm{Res}_G^H(A_i))\end{align*} $$

and hence $(B\otimes _{G^0} A_i)\rtimes G$ is Morita equivalent to $\mathrm {Res}_G^H(A_i)\rtimes H$ . Thus, this case follows directly from the assumption.

Step 2: Suppose $B=L_K(C_0(X))=\mathrm {Ind}_K^G C_0(X|_{K^0})$ , where X is any second countable proper étale G-space with anchor map $p:X\rightarrow G^0$ , and $K\in \mathcal {F}$ . We claim that Equation (13) is an isomorphism for this choice of B. Let $\mathcal {B}$ be a countable basis for the topology of $X|_{K^0}$ consisting of open subsets of $X|_{K^0}$ on which p restricts to a homeomorphism. Then we can write

$$ \begin{align*}X|_{K^0}=\bigcup_{S\in \mathcal{B}} KS.\end{align*} $$

Since $\mathcal {B}$ is countable, we may enumerate its elements writing $\mathcal {B}=\{S_n\mid n\in \mathbb N\}$ . Let $X_n:=\bigcup _{i=1}^n KS_n$ . Then $X_n$ is an open K-invariant subset of X. Moreover, $C_0(X|_{K^0})=\varinjlim _n C_0(X_n)$ where the connecting maps are just given by the canonical inclusions. Since the induction functor, tensor products and the maximal crossed product as well as K-theory are all compatible with inductive limits, it suffices to show that Equation (13) is an isomorphism for $B=\mathrm {Ind}_K^G C_0(X_n)$ . We will do this by induction on n.

For $n=1$ , observe that for every $S\in \mathcal {B}$ there are identifications $KS\cong K\times _{\mathrm {Stab}(S)} S$ , where $\mathrm {Stab}(S)$ is the proper open subgroupoid of K defined as $\mathrm {Stab}(S)=\{g\in K\mid gS\subseteq S\}$ . Note that the restriction of the anchor map induces a homeomorphism $S\cong \mathrm {Stab}(S)^0$ . It follows that

$$ \begin{align*}C_0(KS)\cong C_0(K\times_{\mathrm{Stab}(S)} S)\cong \mathrm{Ind}_{\mathrm{Stab}(S)}^K(C_0(\mathrm{Stab}(S)^0)),\end{align*} $$

and using induction in stages we conclude that

$$ \begin{align*}\mathrm{Ind}_K^G C_0(KS)=\mathrm{Ind}_{\mathrm{Stab}(S)}^G C_0(\mathrm{Stab}(S)^0)=C_0(G/\mathrm{Stab}(S)).\end{align*} $$

Since $\mathrm {Stab}(S)$ is a proper open subgroupoid of $K\in \mathcal {F}$ , it follows that Equation (13) is an isomorphism for $B=\mathrm {Ind}_K^G C_0(KS)$ by Step 1 above.

Next, consider a union $KS\cup KT$ for $S,T\in \mathcal {B}$ . Then we have two short exact sequences of K-algebras

$$ \begin{align*}0\rightarrow C_0(KS\cap KT)\rightarrow C_0(KS)\rightarrow C_0(KS\setminus KT)\rightarrow 0\end{align*} $$

and

$$ \begin{align*}0\rightarrow C_0(KT)\rightarrow C_0(KS\cup KT)\rightarrow C_0(KS\setminus KT)\rightarrow 0.\end{align*} $$

Using that the functors $\mathrm {Ind}_K^G -$ , $(-\, \otimes _{G^0} A_i)$ , and $(-\,\rtimes G)$ are all exact, we can apply them (in this order) to the above sequences and the result remains exact. Hence, we obtain induced six-term exact sequences in K-theory, which can be compared using the maps induced by f. Thus, using the case $n=1$ above, to prove the claim for the union $KS\cup KT$ for $S,T\in \mathcal {B}$ , it suffices to prove it for $KS\cap KT$ . To this end, note that

$$ \begin{align*}KS\cap KT=K(S\cap KT).\end{align*} $$

Considering the subgroupoid $\mathrm {Stab}(S\cap KT)$ of K defined as above, we can employ the same arguments as in the case $n=1$ to conclude that $C_0(KS\cap KT))\cong \mathrm {Ind}_{\mathrm {Stab}(S\cap KT)}^K(C_0(\mathrm {Stab}(S\cap KT)^0))$ , and hence using induction in stages again, we conclude that Equation (13) is an isomorphism for

$$ \begin{align*}B=\mathrm{Ind}_K^G C_0(KS\cap KT)=\mathrm{Ind}_{\mathrm{Stab}(S\cap KT)}^G C_0(\mathrm{Stab}(S\cap KT)^0)\cong C_0(G/\mathrm{Stab}(S\cap KT)).\end{align*} $$

Inductively, we can continue in this way to prove the isomorphism in line (13) for all $B=\mathrm {Ind}_K^G C_0(X_n)$ and hence complete step 2 by passing to the inductive limit.

Step 3: We can now prove that Equation (13) is an isomorphism for all $B\in \mathcal {P}_0$ by induction. The base case is contained in Step 1 above. For the induction step, note that $L_{H_n}\cdots L_{H_1}(C_0(G^0))\cong C_0(G/H_n\times _{G^0}\ldots \times _{G^0}G/H_1)$ and observe that the space $X:=G/H_n\times _{G^0}\ldots \times _{G^0}G/H_1$ is an étale proper G-space. Thus, we can just apply Step 2 to complete the proof.

This result directly allows to generalize several results obtained by the first author for ample groupoids to the general étale case.

4.2.1 Homotopies of twists

Let G be an étale groupoid. A homotopy of twists is a twist over $G\times [0,1]$ , that is, a central extension of the form

$$ \begin{align*}G^0\times [0,1]\times\mathbb{T}\rightarrow \Sigma\stackrel{j}{\rightarrow} G\times [0,1].\end{align*} $$

Theorem 4.6. Let G be a second countable étale groupoid satisfying the Baum–Connes conjecture with coefficients. If $\Sigma $ is a homotopy of twists over G, then for each $t\in [0,1]$ the canonical map $q_t:C_r^*(G\times [0,1],\Sigma )\rightarrow C_r^*(G,\Sigma _t)$ induces an isomorphism in K-theory.

Proof . The idea of the proof is the same as for the main result in [Reference Bönicke11]: Using a groupoid version of the Packer–Raeburn stabilisation trick and the going-down principle (Theorem 4.4), one only has to prove the result for all proper open subgroupoids of all elements $H\in \mathcal {F}$ in place of G. Recall that all the groupoids $H\in \mathcal {F}$ are (isomorphic to) transformation groupoids of finite groups. Hence, if the original homotopy of twists over G is topologically trivial in the sense that the map j has a continuous section (this means that the twist is equivalent to a continuous $2$ -cocycle), one can apply an earlier result of Gillaspy [Reference Gillaspy22] to finish the proof. In the setting of ample groupoids treated in [Reference Bönicke11], the requirement that the twist is topologically trivial is not actually a restriction by [Reference Bönicke11, Proposition 4.2].

In the étale setting twists are no longer automatically topologically trivial, so instead we use a refinement of the going-down principle. Observe that the constructions and results from the previous section allow some flexibility in choosing the family $\mathcal {F}$ of subgroupoids of G. Indeed, if $\mathcal {F}'$ is another family of subgroupoids of G with the property that every proper action of G is locally induced by members of $\mathcal {F}'$ , we can replace $\mathcal {F}$ by $\mathcal {F}'$ in all the results of Section 3 and hence also in Theorem 4.3.

Now, given a homotopy of twists with quotient map $j: \Sigma \rightarrow G\times [0,1]$ we claim that there exists a family $\mathcal {F}'$ of compact actions for G as above with the additional property that the restricted twist $j^{-1}(H\times [0,1])\rightarrow H\times [0,1]$ (this is now a homotopy of twists over H) admits a continuous cross section.

Let us explain how this works: By the proof of [Reference Bönicke11, Proposition 4.2] every $g\in G$ admits an open neighbourhood V such that there exists a local section $V\times [0,1]\rightarrow \Sigma $ of j. Now, given a proper action of G we will proceed as in the proof of Proposition 3.2, but (in the notation of that proof) we additionally choose the bisections $W_g$ to be the domains of local sections of j as above. Since the $W_g$ can be assumed to be pairwise disjoint and the remaining construction in the proof of Proposition 3.2 just shrinks them further, we can patch the resulting finitely many local sections $W_g\times [0,1]\rightarrow \Sigma $ together to obtain the desired continuous section $H\times [0,1]\rightarrow \Sigma $ . Since H is of the form $\Gamma \ltimes U$ for a finite group $\Gamma $ and an open subset $U\subseteq G^0$ we are again the position to apply Gillaspy’s result to conclude that $q_t$ induces an isomorphism for all $H\in \mathcal {F}'$ . To lift the result from this to all of G, one can follow the arguments in [Reference Bönicke11] again.

4.3 Amenability at infinity

Recall that a locally compact Hausdorff groupoid G is called amenable at infinity, if there exists a G-space Y with proper momentum map $p:Y\rightarrow G^{0}$ and such that $G\ltimes Y$ is (topologically) amenable.

It is called strongly amenable at infinity if, in addition, the momentum map p admits a continuous cross section. Since p is a proper map, it induces an equivariant $\ast $ -homomorphism $C_0(G^0)\rightarrow C_0(Y)$ and can hence be viewed as a morphism

$$ \begin{align*}\mathbf{p}\in \operatorname{\mathrm{KK}}^G(C_0(G^{0}),C_0(Y)).\end{align*} $$

It was shown in [Reference Anantharaman-Delaroche2, Lemma 4.9] that if G is strongly amenable at infinity, then the space Y witnessing this can be chosen second countable. Replacing this space further by the space of probability measures on Y supported in fibres we may also assume that each fibre (with respect to p) is a convex space and that G acts by affine transformations. The following result is [Reference Bönicke10, Proposition 8.2]:

We obtain the following consequence:

An element $\eta $ as in the theorem above is often called a dual Dirac morphism for G (see [Reference Meyer and Nest41, Definition 8.1]) and is unique (if it exists).

4.4 Permanence properties

In this section, we will often need to compare the subcategories $\langle \mathcal {C}\mathcal {I}\rangle $ and $\mathcal {N}$ for different groupoids. To highlight this, we will slightly adjust our notation and write $\mathcal {N}_G$ for the weakly contractible objects in $\operatorname {\mathrm {KK}}^G$ and $\mathcal {CI}_G$ for the compactly induced objects.

Sometimes we write ‘BC’ as a shorthand for ‘Baum–Connes conjecture’.

4.4.1 Subgroupoids

Given a second countable étale groupoid G and a subgroupoid $H\subseteq G$ , we may ask how the (strong) Baum–Connes conjectures for G and H are related. We need

Lemma 4.9. Suppose $H\subseteq G$ is a subgroupoid. Then the following hold:

1. 1. If $H\subseteq G$ is open, then $\operatorname {\mathrm {Res}}^H_G(\mathcal {N}_G)\subseteq \mathcal {N}_H$ .

2. 2. If H is closed in $G|_{H^0}$ , then $\operatorname {\mathrm {Res}}^H_G(\langle \mathcal {CI}_G\rangle )\subseteq \langle \mathcal {CI}_H\rangle $ .

3. 3. If H is open in G and closed in $G|_{H^0}$ , then $\operatorname {\mathrm {Res}}_G^H$ maps a Dirac triangle for G to a Dirac triangle for H.

Proof. To show the first item suppose H is an open subgroupoid of G and let $N\in \mathcal {N}_G\subseteq \operatorname {\mathrm {KK}}^G$ . Suppose that Q is a proper open subgroupoid of H. Then Q is also a proper open subgroupoid of G and hence $\operatorname {\mathrm {Res}}_H^Q(\operatorname {\mathrm {Res}}_G^H(\mathrm {id}_N))=\operatorname {\mathrm {Res}}_G^Q(\mathrm {id}_N)\stackrel {3.16}{=}0$ . Another application of Lemma 3.16 yields the result.

Next, suppose H is closed in $G|_{H^0}$ . Whenever G acts properly on a space Z with anchor map $p:Z\rightarrow G^0$ , then the action restricts to a proper action of H on $p^{-1}(H^0)$ . In particular, it follows that $\operatorname {\mathrm {Res}}^H_G(\mathcal {C}\mathcal {I}_G)\subseteq \Pr _H$ and hence $\operatorname {\mathrm {Res}}_G^H(\langle \mathcal {C}\mathcal {I}_G\rangle )\subseteq \langle \mathcal {C}\mathcal {I}_H\rangle $ by Theorem 3.10.

The final assertion is a direct consequence of the first two statements.

Lemma 4.10. Suppose $H\subseteq G$ is a subgroupoid such that H is closed in $G|_{H^0}$ . Then the following hold:

$$ \begin{align*}\operatorname{\mathrm{Ind}}_H^G:\operatorname{\mathrm{KK}}^H\rightarrow \operatorname{\mathrm{KK}}^G\end{align*} $$

is triangulated, $\operatorname {\mathrm {Ind}}_H^G(\mathcal {N}_H)\subseteq \mathcal {N}_G$ , and $\operatorname {\mathrm {Ind}}_H^G\langle \mathcal {C}\mathcal {I}_H\rangle \subseteq \langle \mathcal {C}\mathcal {I}_G\rangle $ . In particular, it maps Dirac triangles to Dirac triangles.

Proof. Induction in stages gives that a compactly induced object in $\operatorname {\mathrm {KK}}^H$ is mapped to a proper object in $\operatorname {\mathrm {KK}}^G$ . Indeed, if $Q\subseteq H$ is a compact action, then $\operatorname {\mathrm {Ind}}_H^G(\operatorname {\mathrm {Ind}}_Q^H A)=\operatorname {\mathrm {Ind}}_Q^G A$ . It follows from our assumption that Q is closed in $G|_{Q^0}$ , and hence the action of G on $G_{Q^0}/Q$ is proper. It follows immediately that $\operatorname {\mathrm {Ind}}_Q^G A$ is a proper G-algebra (see also the induction picture in [Reference Bönicke10]). Whence, $\operatorname {\mathrm {Ind}}_H^G\langle \mathcal {C}\mathcal {I}_H\rangle \subseteq \langle \mathcal {C}\mathcal {I}_G\rangle $ by Theorem 3.10.

Finally, let $A\in \mathcal {N}_H\subseteq \operatorname {\mathrm {KK}}^H$ . Then by Lemma 4.9.(2) we have

$$ \begin{align*}\operatorname{\mathrm{Res}}_G^H(P_G(C_0(G^0)))\otimes^{\text{max}}_{H^0}A\cong P_H(C_0(H^0))\otimes^{\text{max}}_{H^0}A\cong 0.\end{align*} $$

Using Lemma 2.6, we conclude that

$$ \begin{align*}P_G(C_0(G^0))\otimes_{G^0}\operatorname{\mathrm{Ind}}_H^G A\cong\operatorname{\mathrm{Ind}}_H^G(\operatorname{\mathrm{Res}}_G^H(P_G(C_0(G^0)))\otimes_{H^0}A)\cong 0\end{align*} $$

as well.

The following result was already observed by Tu [Reference Tu57] for the classical Baum–Connes conjecture. Unfortunately, his proof relies on [Reference Tu57, Lemma 3.9], which seems to be erroneous. A counterexample where G is the compact space $[0,1]$ (viewed as a trivial groupoid just consisting of units) is exhibited in [Reference Dadarlat and Meyer17, Example 5.6] and [Reference Bauval4, p.36].

Theorem 4.11. Let G be a second countable groupoid, $H\subseteq G$ be an étale subgroupoid that is closed in $G|_{H^0}$ , and $A\in \operatorname {\mathrm {KK}}^H$ . Then there is a natural $\operatorname {\mathrm {KK}}$ -equivalence between $P_G(\operatorname {\mathrm {Ind}}_H^G A)\rtimes _r G$ and $P_H(A)\rtimes _r H$ . Hence, the (strong) Baum–Connes conjecture with coefficients passes to closed subgroupoids and restrictions to open subsets.

Proof. From the previous lemma, we conclude that $P_G(\operatorname {\mathrm {Ind}}_H^G A)\rtimes _r G\cong \operatorname {\mathrm {Ind}}_H^G (P_H(A))\rtimes _r G$ . The latter, however, is canonically Morita-equivalent (and hence in particular $\operatorname {\mathrm {KK}}$ -equivalent) to $P_H(A)\rtimes _r H$ . The result about the (strong) Baum–Connes conjecture follows readily.