2. Groupoids and their actions

In this section, we collect a few basic definitions concerning (locally compact, Hausdorff) groupoids and their actions. We will give precise references as we proceed but we will generally follow [ Reference Le Gall25 ]. To fix notation, we recall that a groupoid is a small category in which each arrow is invertible. If G is a groupoid, we write $G^{(0)}$ for the set of objects; $G^{(2)}$ is the set of composable arrows; r and s are the range and source maps, respectively; u denotes the unit map from $G^{(0)}$ to G. For more details, we refer to [ Reference Renault33 , Chapter 1]. In what follows, by a space we mean a locally compact, Hausdorff space and by a groupoid we mean a locally compact, Hausdorff groupoid.

Given a groupoid G, a space X and a continuous map $p\,:\, X \rightarrow G^{(0)}$ , we write, by a slight abuse of notation, $s^{*}X$ for the pullback $G^{\,\,s}{\times}{_{G^{(0)}}} X$ (borrowed from [ Reference Proietti and Yamashita31 ]) of the diagram

i.e.,

\begin{align*}s^{*}X =\{(g,x) \in G \times X \mid s(g)=p(x)\},\end{align*}

and equip it with the relative topology of $G \times X$ . Similarly, we shall write $r^{*}X$ for $G ^{\,\,r}{\times}{_{G^{(0)}}} X$ .

It follows that $p(g \cdot x)=r(g)$ , i.e., $(g,g \cdot x) \in r^{*}X$ . Thus one may view $\alpha$ as a continuous map from $s^{*}X \rightarrow r^{*}X$ . To define G-actions on $C^{*}$ -algebras, we need some preparation. Let X be a space.

By slightly abusing notation, we will only write A for a $C_{0}(X)$ -algebra $(A,\theta_{A})$ , suppressing the structural map $\theta_{A}$ . We will also write fa for $\theta_{A}(f)(a)$ , where $f \in C_{0}(X)$ and $a \in A$ .

Let A be a $C_{0}(X)$ -algebra. For $x \in X$ , we denote by $\mathfrak{m}_{x}$ the maximal ideal in $C_{0}(X)$ consisting of continuous functions on X that vanish at x. The norm-closed linear span $[\mathfrak{m}_{x}A]$ is a (closed, two-sided) ideal in A. The fiber of A at x, denoted by A(x), is the quotient $C^{*}$ -algebra $A/[\mathfrak{m}_{x}A]$ . We denote the quotient map by $\pi_{x}\,:\, A \rightarrow A(x)$ and write $\pi_{x}(a)=a(x)$ , for all $a \in A$ . For $f \in C_{0}(X)$ and $a \in A$ , one has $(fa)(x)=f(x)a(x)$ . For $a \in A$ , the map $x \mapsto \left\lVert a(x) \right\rVert$ is upper semi-continuous and $\left\lVert a \right\rVert=\sup_{x \in X}\left\lVert a(x) \right\rVert$ . If the map $x \mapsto \left\lVert a(x) \right\rVert$ is continuous for all $a \in A$ , we say that A is a continuous field of $C^{*}$ -algebras.

If $\varphi\,:\, A \to B$ is a $C_0(X)$ -linear, $*$ -homomorphism, then it induces a $*$ -homomorphism $\varphi_x\,:\, A(x) \to B(x)$ for every $x \in X$ . By [ Reference Le Gall25 , Proposition 3·1], $\varphi$ is injective (respectively, surjective) if and only if $\varphi_x$ is injective (respectively, surjective), for all $x \in X$ . Similarly, if $\varphi\,:\, A \to B$ is a $C_0(X)$ -linear, completely positive map, then by [ Reference Borys7 , Lemma 3·4], then it induces a completely positive map $\varphi_x\,:\, A(x) \to B(x)$ for every $x \in X$ .

Let A be a $C_0(X)$ -algebra, B a $C_0(Y)$ -algebra and consider the $*$ -homomorphism

\begin{align*}C_{0}(X) \otimes C_{0}(Y) \to \mathcal{Z}\mathcal{M}(A) \otimes\mathcal{Z}\mathcal{M}(B) \rightarrow \mathcal{Z}\mathcal{M}(A\otimes^{\mathrm{max}} B). \end{align*}

By [ Reference Blanchard5 , Corollary 3·16], $A \otimes^{\mathrm{max}} B$ together with the above $*$ -homomorphism becomes a $C_{0}(X \times Y)$ -algebra. Moreover, for $(x,y) \in X \times Y$ , the fiber $(A \otimes^{\mathrm{max}} B)(x,y)$ can naturally be identified with $A(x) \otimes^{\mathrm{max}} B(y)$ . Strictly speaking, [ Reference Blanchard5 , Corollary 3·16] holds only if X is compact. However, as noted in [ Reference Echterhoff and Williams11 ], the same proof holds when the space is X is merely locally compact; see the discussion before Definition 2·3 of [ Reference Echterhoff and Williams11 ].

Definition 2·5. Let A and B be two $C_{0}(X)$ -algebras. The balanced or relative tensor product, denoted by $A \otimes^{\mathrm{max}}_{C_{0}(X)} B$ , is the $C_{0}(X)$ -algebra $(A \otimes^{\mathrm{max}} B)_{\Delta(X)}$ , obtained from restricting the $C_{0}(X \times X)$ -algebra $A \otimes^{\mathrm{max}} B$ to the diagonal

\begin{align*}\Delta(X)=\{(x,x) \in X \times X \mid x \in X\}.\end{align*}

Instead of the maximal tensor product, we could have as well chosen the minimal tensor product $\otimes^{\mathrm{min}}$ in the above definition to define $\otimes^{\mathrm{min}}_{C_{0}(X)}$ . However, $\otimes^{\mathrm{min}}_{C_{0}(X)}$ is not associative; see [ Reference Blanchard5 , Section 3·3]. There are at least two more different ways of defining $\otimes^{\mathrm{max}}_{C_{0}(X)}$ (and of course, $\otimes^{\mathrm{min}}_{C_{0}(X)}$ ); see [ Reference Kasparov21 , Definition 1·6] and [ Reference Echterhoff and Williams11 , Definition 2.5]. The latter considers the (closed, two-sided) ideal

\begin{align*}I\,:\!=\,\{ fa \otimes^{\mathrm{max}} b - a \otimes^{\mathrm{max}}fb \mid f \in C_{0}(X), a \in A, b \in B\}\end{align*}

of $A \otimes^{\mathrm{max}} B$ and defines $A \otimes^{\mathrm{max}}_{C_{0}(X)} B$ to be the quotient $A \otimes^{\mathrm{max}} B/I$ . Using the relative tensor product, we can now define pullbacks.

Note that, $p^*A$ is, a priori, only a $C_0(X)$ -algebra; the $C_0(Y)$ -algebra structure comes from the canonical inclusion $C_0(Y) \rightarrow \mathcal{M}(A \otimes_{C_0(X)} C_0(Y))$ . Note further, that it is not necessary to specify whether we use the minimal tensor product $\otimes^{\mathrm{min}}$ or the maximal tensor product $\otimes^{\mathrm{max}}$ because $C_{0}(Y)$ is nuclear. Alternatively, one can define $p^{*}A$ as the restriction of the $C_{0}(X \times Y)$ -algebra $A \otimes C_{0}(Y)$ to the closed set $\{(p(y),y) \mid y \in Y\}$ of $X \times Y$ . For $y \in Y$ , $(p^{*}A)(y)$ is naturally identified with A(p(y)). Given two $C_{0}(X)$ -algebras A and B, one can identify $p^{*}A \otimes^{\mathrm{max}}_{C_{0}(Y)} p^{*}B$ with $p^{*}(A \otimes^{\mathrm{max}}_{C_{0}(X)} B)$ ; similar identification holds if we replace $\otimes^{\mathrm{max}}$ with $\otimes^{\mathrm{min}}$ . Given a $C_{0}(X)$ -linear, $*$ -homomorphism (or a completely positive map) $\varphi\,:\, A \rightarrow B$ , there is a $C_{0}(Y)$ -linear, $*$ -homomorphism (respectively, a completely positive map) $p^{*}\varphi\,:\, p^{*}A \rightarrow p^{*}B$ . Finally, by [ Reference Popescu30 , Corollary 1·26], the pullback functor is exact, i.e., if

\begin{align*}0 \rightarrow I \rightarrow A \rightarrow B \rightarrow 0\end{align*}

is a short exact sequence of $C_{0}(X)$ -algebras and $C_{0}(X)$ -linear, $*$ -homomorphisms, then

\begin{align*}0 \rightarrow p^{*}I \rightarrow p^{*}A \rightarrow p^{*}B \rightarrow 0,\end{align*}

too is a short exact sequence of $C_{0}(Y)$ -algebras and $C_{0}(Y)$ -linear $*$ -homomorphisms, for every continuous map $p\,:\, Y \rightarrow X$ . This is will be important for us. In particular, this implies that the “fiber” functor, i.e., that sends A to A(x), being the pullback functor via ${x} \rightarrow X$ , is exact.

Before going to the definition of a groupoid action on a $C^{*}$ -algebra, let us mention that oftentimes, it is helpful to view a $C_{0}(X)$ -algebra A as an (upper semi-continuous) bundle of $C^{*}$ -algebras. For that purpose, we set $\mathcal{A}=\bigsqcup_{x \in X} A(x)$ and let $\pi\,:\, \mathcal{A} \to X$ to be the projection map. We topologise $\mathcal{A}$ by taking the subsets

(2·1) \begin{equation}W(a, U, \epsilon)=\{b \in \mathcal{A} \mid \pi(b) \in U \text{ and } \left\lVert a(\pi(b))-b \right\rVert \lt \epsilon \},\end{equation}

where $a \in A$ , $U \subset X$ is open and $\epsilon \gt0$ , as a basis. Let $\Gamma_0(X;\,\mathcal{A})$ denote the algebra of continuous sections of the upper semi-continuous $C^{*}$ -bundle $\mathcal{A}$ vanishing at infinity, then A and $\Gamma_0(X;\,\mathcal{A})$ are isomorphic as $C_{0}(X)$ -algebras. Moreover, this process can be reversed; see, for example, [ Reference Kirchberg and Wassermann24 , Introduction], [ Reference Rieffel35 , Section 1], or [ Reference Williams44 , Appendix C] for more details. As an example, let A be a $C_0(X)$ -algebra and let $\pi\,:\, \mathcal{A} \to X$ be the upper semi-continuous $C^*$ -bundle associated to A. If $p\,:\, Y \to X$ is a continuous map, one defines the pullback $p^*\mathcal{A}=\{(c, y) \in \mathcal{A} \times Y \mid \pi(c)=p(y)\}$ , which is again an upper semi-continuous $C^*$ -bundle. It follows from [ Reference Raeburn and Williams32 , Proposition 1·3] that $p^*A$ and $\Gamma_{0}(X;\,p^{*} \mathcal{A})$ are isomorphic as $C_{0}(Y)$ -algebras.

Returning back to defining groupoid actions on $C^{*}$ -algebras, suppose now that we have a $C_{0}(G^{(0)})$ -algebra A and a $C_{0}(G)$ -linear, $*$ -isomorphism $\alpha\,:\, s^{*}A \rightarrow r^{*}A$ . Then $\alpha$ induces $*$ -isomorphisms $\alpha_{g}\,:\, A(s(g)) \rightarrow A(r(g))$ for each $g \in G$ .

Again by slightly abusing notation, we will only write $(A,\alpha)$ for a G- $C^*$ -algebra $(A,\theta_{A},\alpha)$ , suppressing the structural map $\theta_{A}$ . The following lemma puts the above definition in terms of the bundle picture.

We end this section with one more definition.

3. Equivariant extensions and the Busby invariant

This section describes how for a groupoid G, extensions of G- $C^*$ -algebras are in one-one correspondence with their Busby invariants. This extends well-known results from [ Reference Thomsen41 , Section 2] and [ Reference Kasparov20 , Section 7] in the case when G is a group and from [ Reference Blanchard6 , Section 1] for $C_0(X)$ -algebras with X compact. Most of the results are straightforward; however, for the readers’ convenience, we provide full details. To begin with, we will throughout this article write $q_A\,:\, \mathcal{M}(A) \rightarrow \mathcal{Q}(A)$ for the canonical projection. Let X be a space and let A be a separable, $C_0(X)$ -algebra. For any $x \in X$ , we define

(3·1) \begin{equation}\xi_{x} \colon \mathcal{M}(A) \to \mathcal{M}(A(x)), \quad \xi_x(m)(e(x))=(me)(x),\end{equation}

for any $m \in \mathcal{M}(A)$ and $e \in A$ , i.e., $\xi_x$ is the canonical surjective extension of $\pi_x\,:\, A \to A(x)$ , see, for instance, [ Reference Akemann, Pedersen and Tomiyama1 , Theorem 4·2]. In turn, $\xi_x$ induces

(3·2) \begin{equation}\zeta_x\,:\, \mathcal{Q}(A) \to \mathcal{Q}(A(x))\end{equation}

such that the following diagram

(3·3)

commutes. It is an easy check that if $f \in \mathfrak{m}_x$ and $m \in \mathcal{M}(A)$ , then

(3·4) \begin{equation}\xi_x(\theta_A(f)m)=\zeta_x(q_A(\theta_A(f)m))=0.\end{equation}

Given a $C_0(X)$ -extension of A by another $C_0(X)$ -algebra I, i.e., a short exact sequence

(η) \begin{equation}\qquad\qquad\qquad\qquad\qquad\qquad\qquad0 \rightarrow I \xrightarrow{\iota} B \xrightarrow{p} A \rightarrow 0,\end{equation}

of $C_0(X)$ -algebras and $C_0(X)$ -linear, $*$ -homomorphisms $\iota\,:\, I \rightarrow B$ and $p\,:\, B \rightarrow A$ , for any $x \in X$ , the sequence

(η(x)) \begin{equation}\qquad\qquad\qquad\qquad\qquad\qquad\qquad 0 \rightarrow I(x) \xrightarrow{\iota_x} B(x) \xrightarrow{p_x} A(x) \rightarrow 0,\end{equation}

is exact; conversely, given ( $\eta$ ) such that for each $x \in X$ , ( $\eta$ ( $\chi$ )) is exact then ( $\eta$ ) is exact. This follows from the fact that the “fiber” functor, i.e., that sends A to A(x) is the pullback functor via ${x} \rightarrow X$ , which is exact and [ Reference Popescu30 , Lemma 1·25]. For such an extension, we define

(3·5) \begin{equation}\bar{\mu}(\eta)\,:\, B \to \mathcal{M}(I), \quad \bar{\mu}(\eta)(b)(e)=b \iota(e)\end{equation}

for all $b \in B$ and $e \in I$ . Note that we also have the $*$ -homomorphism

(3·6) \begin{equation}\bar\mu(\eta(x))\,:\, B(x) \rightarrow \mathcal{M}(I(x))\end{equation}

for the extension ( $\eta(x)$ ). We observe that although $\mathcal{M}(I)$ is not a $C_0(X)$ -algebra (see, for instance, [ Reference Williams44 , Remark C·14]), the $*$ -homomorphism $\bar \mu$ satisfies a type of “ $C_0(X)$ -linearity” in the sense of the following definition.

Definition 3·1. Let B and I be $C_0(X)$ -algebras. A $*$ -homomorphism (or a completely positive map) $\varphi\,:\, B \to \mathcal{M}(I)$ is said to be $C_0(X)$ -linear if

\begin{align*}\varphi(\theta_A(f)b)=\theta_I(f)\varphi(b),\end{align*}

for all $f \in C_0(X)$ and $b \in B$ .

If I is separable, then given a $C_0(X)$ -linear $\varphi\,:\, B \rightarrow \mathcal{M}(I)$ , $b \in B$ , $x \in X$ and $f \in \mathfrak{m}_x$ , by (3·5), it follows that

\begin{align*}\xi_x(\varphi(\theta_B(f)b))=\xi_x(\theta_I(f)\varphi(b))=0,\end{align*}

so that

(3·7) \begin{equation}\varphi_x\,:\, B(x) \to \mathcal{M}(I(x)), \quad \varphi_x(b(x))=\xi_x(\varphi(b))\end{equation}

is a well-defined $*$ -homomorphism such that the following diagram

(3·8)

commutes.

Lemma 3·2. The $*$ -homomorphism

\begin{align*}\bar\mu(\eta(x))\,:\, B(x) \rightarrow \mathcal{M}(I(x))\end{align*}

for the extension ( $\eta(x)$ ) coincides with the $*$ -homomorphism

\begin{align*}\bar\mu(\eta)_x\,:\, B(x) \rightarrow \mathcal{M}(I(x))\end{align*}

as defined in (3·8) above, whenever I is separable.

Proof. The proof is a plain computation which goes as follows. We fix $x \in X$ , $b \in B$ and $e \in I$ . We then have

\begin{align*}\begin{aligned}\bar\mu(\eta)_x(\pi_x(b))(e(x)) &{}=\xi_x(\mu(\eta)(b))(e(x))\\&{}=((\bar\mu(\eta)(b))(e))(x)\\&{}=(b\iota(e))(x)\\&{}=b(x)\iota_x(e(x))\\&{}=\bar\mu(\eta(x))(b(x))e(x);\end{aligned}\end{align*}

where the first equality is by commutativity of (3·9); the second equality is by definition of $\xi_x$ ; the third equality is by definition of $\bar\mu(\eta)$ ; and the fifth equality is by definition of $\bar\mu(\eta(x))$ . Since $\pi_x$ is surjective, the proof is complete.

In particular, the following square

(3·9)

commutes. As in the ordinary $C^*$ -algebraic case, we define the Busby invariant

(3·10) \begin{equation}\mu(\eta)\,:\, A \to \mathcal{Q}(I), \quad \mu(\eta)(p(b))=q_I(\bar{\mu}(\eta)(b)).\end{equation}

Note that we also have the Busby invariant

(3·11) \begin{equation}\mu(\eta(x))\,:\, A(x) \rightarrow \mathcal{Q}(I(x))\end{equation}

for the extension ( $\eta(x)$ ). Again although $\mathcal{Q}(I)$ is not a $C_0(X)$ -algebra, the Busby invariant $\mu$ satisfies a type of “ $C_0(X)$ -linearity” in the sense of the following definition.

Definition 3·3. Let A and I be $C_0(X)$ -algebras. A $*$ -homomorphism (or a completely positive map) $\psi\,:\, A \to \mathcal{Q}(I)$ is said to be $C_0(X)$ -linear if

\begin{equation*}\psi(\theta_A(f)a)=q_I\theta_I(f)\psi(a),\end{equation*}

for all $f \in C_0(X)$ and $a \in A$ .

Once again, if I is separable, given such a $C_0(X)$ -linear $\psi\,:\, A \rightarrow \mathcal{Q}(I)$ , $a \in A$ , $x \in X$ and $f \in \mathfrak{m}_x$ , by (3·5), it follows that

\begin{align*}\zeta_x(\psi(\theta_A(f)a))=\zeta_x(q_I(\theta_I(f)))\zeta_x(\psi(a))=0,\end{align*}

so that

(3·12) \begin{equation}\psi_x\,:\, A(x) \to \mathcal{Q}(I(x)), \quad \psi_x(a(x))=\zeta_x(\psi(a))\end{equation}

is a well-defined $*$ -homomorphism such that the following diagram

(3·13)

commutes.

Lemma 3·4. The Busby invariant

\begin{align*}\mu(\eta(x))\,:\, A(x) \rightarrow \mathcal{Q}(I(x))\end{align*}

for the extension ( $\eta(x)$ ) coincides with the $*$ -homomorphism

\begin{align*}\mu(\eta)_x\,:\, A(x) \rightarrow \mathcal{Q}(I(x))\end{align*}

as defined in (3·13) above, whenever I is separable.

Proof. The proof is a plain computation which goes as follows. We fix $x \in X$ and $b \in B$ . We then have

\begin{align*}\begin{aligned}\mu(\eta)_x(\pi_x(p(b))) &{}=\zeta_x(\mu(\eta)(p(b)))\\[4pt] &{}=\zeta_x(q_I(\bar \mu(\eta)(b)))\\[4pt] &{}=q_{I(x)}\xi_x(\bar \mu(\eta)(b))\\[4pt] &{}=q_{I(x)}\bar \mu(\eta(x))(\pi_x(b))\\[4pt] &{}=\mu(\eta(x))(p_x\pi_x(b))\\[4pt] &{}=\mu(\eta(x))(\pi_xp(b));\end{aligned}\end{align*}

where the first equality is by commutativity of (3·14); the second equality is by definition of $\mu(\eta)$ ; the third equality is by commutativity of (3·4); the fourth equality is by commutativity of (3·10); the fifth equality is by definition of $\mu(\eta(x))$ and the sixth equality is because p is $C_0(X)$ -linear. Since p and $\pi_x$ are surjective, the proof is complete.

In particular, the following square

(3·14)

commutes.

Proposition 3·5. Let I and A be two separable, $C_0(X)$ -algebras and let $\mu\,:\, A \to \mathcal{Q}(I)$ be a $C_0(X)$ -linear, $*$ -homomorphism. Set

\begin{align*}B=\{(m,a) \in \mathcal{M}(I) \oplus A \mid q_I(m)=\mu(a) \},\end{align*}

\begin{align*}\iota\,:\, I \rightarrow B, e \mapsto (e,0), \quad p\,:\, B \rightarrow A, (m,a) \mapsto a.\end{align*}

Then the following statements hold:

1. (1) B can be uniquely made into a $C_0(X)$ -algebra such that $\iota$ and p are $C_0(X)$ -linear;

2. (2) for any $x \in X$ , the fiber B(x) is isomorphic to

   \begin{align*}B_x\,:\!=\,\{(m', a') \in \mathcal{M}(I(x)) \oplus A(x) \mid q_{I(x)}(m')=\mu_x(a') \} ; \end{align*}

3. (3) the extension

   (η) \begin{equation} \qquad\qquad\qquad\qquad\qquad\qquad 0 \rightarrow I \xrightarrow{\iota} B \xrightarrow{p} A \rightarrow 0\end{equation}
   is a $C_0(X)$ -extension such that $\mu(\eta)=\mu$ .

Proof. We begin by observing that since $\mu$ is $C_0(X)$ -linear, for any $f \in C_0(X)$ , and $(m,a) \in B$ , $(\theta_I(f)m,\theta_A(f)a)$ is in B too. Let then $\theta_B\,:\, C_0(X) \rightarrow \mathcal{M}(B)$ to be the $*$ -homomorphism defined as $\theta_B(f)(m,a)=(\theta_I(f)m,\theta_A(f)a)$ , for $f \in C_0(X)$ , and $(m,a) \in B$ . It is easy to see that $\theta_B(C_0(X)) \subset \mathcal{ZM}(B)$ . It is equally easy to see that $\iota$ and p are $C_0(X)$ -linear. Next, we show that $\theta_B$ is nondegenerate. For that, we set $B_X=[C_0(X)B]$ . By $C_0(X)$ -linearity of $\iota$ and p together with exactness of ( $\eta$ ), we obtain that

\begin{align*}0 \rightarrow I \xrightarrow{\iota} B_X \xrightarrow{p} A \rightarrow 0,\end{align*}

is also exact. Thus we have the following commutative diagram

with exact rows. The Five lemma then implies that $B=B_X$ , i.e., B is a $C_0(X)$ -algebra. To conclude (i), we now show uniqueness of this $C_0(X)$ -algebra structure. For that, let $\theta'_B\,:\, C_0(X) \rightarrow \mathcal{ZM}(B)$ be another nondegenerate, $*$ -homomorphism which makes B into a $C_0(X)$ -algebra such that $\iota$ and p are $C_0(X)$ -linear. We fix $f \in C_0(X)$ , $(m,a) \in B$ , and write $\theta'_B(f)(m,a)=(m',a')$ . Let $(u_n)_{n \in \mathbb{N}}$ be an approximate unit of I. Then we note that $(m'u_n,0)=\theta'_B(f)(m,a)(u_n,0)=(m,a)\theta'_B(f)(u_n,0)=(m,a)(\theta_I(f)u_n,0)=(\theta_I(f)mu_n,0)$ . Letting $n \rightarrow \infty$ , we see that $m'=\theta_I(f)m$ ; here we have used the centrality of $\theta_B$ and $\theta'_B$ together with the $C_0(X)$ -linearity of $\iota$ . Now since p is also $C_0(X)$ -linear, we obtain that $a'=\theta_A(f)a$ , by applying p to the both sides of $\theta'_B(f)(m,a)=(m',a')$ . Thus $\theta_B(f)(m,a)=(\theta_I(f)m,\theta_A(f)a)$ and this concludes (i). Moreover, (iii) follows as well since ( $\eta$ ) is in particular an ordinary $C^*$ -extension.

For (ii), we fix $x \in X$ . We observe that if (m, a) is in B, then by commutativity of (3·4) and (3·14), $(\xi_x(m),a(x))$ is in $B_x$ . We set $\iota'_x\,:\, I(x) \rightarrow B_x$ , $e(x) \mapsto (e(x),0)$ , $p'_x\,:\, B_x \rightarrow A(x)$ , $(m',a') \mapsto a'$ and $\varphi\,:\, B \rightarrow B_x$ , $(m,a) \mapsto (\xi_x(m),a(x))$ . Then, by definition, we have the following commutative diagram

with exact rows. The Snake lemma then implies that the following sequence

\begin{align*}0 \rightarrow [\mathfrak{m}_xI] \rightarrow \ker(\varphi) \rightarrow [\mathfrak{m}_xA] \rightarrow 0\end{align*}

is exact. By (3.5), we observe that $[\mathfrak{m}_xB] \subseteq \ker(\varphi)$ . Moreover, we have the following commutative diagram

with exact rows. Again by the five lemma, we conclude that $\ker(\varphi)=[\mathfrak{m}_xB]$ , i.e., $\varphi$ descends to an isomorphism from B(x) to $B_x$ . This concludes (ii) and the proof.

We now proceed towards a G-equivariant version of Proposition 3·5 and we begin by fixing some notations. Let G be a second countable groupoid and $(A,\alpha)$ be a separable, G- $C^*$ -algebra. We will write

\begin{align*}\bar \alpha\,:\, \mathcal{M}(s^*A) \rightarrow \mathcal{M}(r^*A), \quad \tilde \alpha\,:\, \mathcal{Q}(s^*A) \rightarrow \mathcal{Q}(r^*A),\end{align*}

for the induced $*$ -isomorphisms, so that the diagram

(3·15)

commutes. Since $\alpha$ is $C_0(G)$ -linear, so are $\bar \alpha$ and $\tilde \alpha$ . Thus for each $g \in G$ , the following diagrams

(3·16)

and

(3·17)

commute. An extension

(η) \begin{equation}\qquad\qquad\qquad\qquad\qquad\qquad0 \rightarrow I \xrightarrow{\iota} B \xrightarrow{p} A \rightarrow 0\end{equation}

of G- $C^*$ -algebras and G-equivariant, $*$ -homomorphisms, will be called a G- $C^*$ -extension. In particular, ( $\eta$ ) is a $C_0(G^{(0)})$ -extension and therefore $\bar \mu(\eta)$ is $C_0(G^{(0)})$ -linear. As one might expect, $\bar \mu(\eta)$ is G-equivariant in a sense, which we make precise now.

(3·18)

commutes for all $g \in G$ .

Lemma 3·7. The $*$ -homomorphism

\begin{align*}\bar \mu(\eta)\,:\, B \rightarrow \mathcal{M}(I)\end{align*}

for the extension ( $\eta$ ) is G-equivariant, whenever I is separable.

Proof. We fix $b \in B$ , $e \in I$ . We then have for all $g \in G$ ,

\begin{align*}\begin{aligned}\bar \mu_{r(g)}(\beta_g(b(s(g))))(e(r(g)))=&{}\beta_g(b(s(g)))\iota_{r(g)}(e(r(g)))\\=&{}\beta_g(b(s(g)))\iota_{r(g)}(\gamma_g(\gamma_g^{-1}(e(r(g)))))\\=&{}\beta_g(b(s(g)))\beta_g(\iota_{s(g)}(\gamma_g^{-1}(e(r(g)))))\\=&{}\gamma_g(b(s(g))\iota_{s(g)}(\gamma_g^{-1}(e(r(g)))))\\=&{}\gamma_g(\bar \mu_{s(g)}(b(s(g)))(\gamma_g^{-1}(e(r(g)))))\\=&{}\bar \gamma_g (\bar \mu_{s(g)}(b(s(g))))(e(r(g))),\end{aligned}\end{align*}

where the first equality is by definition of $\bar \mu_{r(g)}$ ; the third equality is by G-equivariance of $\iota$ ; the fifth equality is again by definition of $\bar \mu_{s(g)}$ and the sixth equality is by definition of $\bar \gamma_g$ . This completes the proof.

Similarly, the Busby invariant $\mu\,:\, A \rightarrow \mathcal{Q}(I)$ is $C_0(G^{(0)})$ -linear. Moreover, it is G-equivariant in the following sense.

(3·19)

commutes for all $g \in G$ .

Lemma 3·9. The Busby invariant

\begin{align*}\mu\,:\, A \rightarrow \mathcal{Q}(I)\end{align*}

for the extension ( $\eta$ ) is G-equivariant, whenever I is separable.

Proof. We fix $b \in B$ . We then have for all $g \in G$ ,

\begin{align*}\begin{aligned}\mu_{r(g)}(\alpha_g(p_{s(g)}(b_{s(g)})))=&{}\mu_{r(g)}(p_{r(g)}(\beta_g(b_{s(g)})))\\=&{}q_{I(r(g))}(\bar \mu_{r(g)}(\beta_g(b(s(g)))))\\=&{}q_{I(r(g))}(\bar \gamma_g(\bar \mu_{s(g)}(b(s(g)))))\\=&{}\tilde \gamma_g(q_{I(s(g))}(\bar \mu_{s(g)}(b(s(g)))))\\=&{}\tilde \gamma_g(\mu_{s(g)}(p_{s(g)}(b(s(g))))),\end{aligned}\end{align*}

where the first equality is by equivariance of p; the second equality is by definition of $\mu_{r(g)}$ ; the third equality is by Lemma 3·7; the fourth equality is by the commutativity of the square (3·18) and the fifth equality is again by definition of $\mu_{s(g)}$ . This completes the proof.

One might expect that the converse to the lemma above holds too, i.e., for a G-equivariant, $*$ -homomorphism $\mu\,:\, A \rightarrow \mathcal{Q}(I)$ , the pullback B in Proposition 3·5 is a G- $C^*$ -algebra and the extension ( $\eta$ ) is a G- $C^*$ -extension such that $\mu(\eta)=\mu$ . Indeed, when G is a group, this is [ Reference Thomsen41 , Theorem 2·2]. However, we note that [ Reference Thomsen41 , Theorem 2·2] is a corollary of the automatic continuity result [ Reference Thomsen41 , Theorem 2·1] which was more generally proved in [ Reference Brown8 , Theorem 2]. Lacking such a result, we follow [ Reference Kasparov20 , Section 7] and make the following definition.

Some remarks are in order. Firstly, we topologise $\bigsqcup_{g \in G} \mathcal{M}(I(r(g)))$ as in [ Reference Akemann, Pedersen and Tomiyama1 , Section 3] and note that although the reference deals with continuous fields, the topology nevertheless makes sense with upper semi-continuity instead of continuity; see also [ Reference Williams44 , Lemma C·11]. Secondly, with this topology, the section $g \mapsto \gamma_g\xi_{s(g)}(m)\gamma_g^{-1}$ in the above definition is required to be norm-continuous instead of strictly-continuous. Finally, that the class of G-admissible, $*$ -homomorphisms is not empty follows from the following corollary to Lemma 3·7.

Corollary 3·11. The Busby invariant

\begin{align*}\mu(\eta)\,:\, A \rightarrow \mathcal{Q}(I)\end{align*}

for the extension ( $\eta$ ) is G-admissible, whenever B is separable.

The following corollary to Proposition 3·5 sets up the bijection between G- $C^*$ -extensions and G-admissible, $*$ -homomorphisms as defined above.

We remark that by Corollary 3·12 and Lemma 3·9, it also follows that for a groupoid G, G-admissibility implies G-equivariance. Therefore it is natural to conjecture the following.

We end this section with a definition and an easy lemma.

Lemma 3·16. Let $(A,\alpha)$ be a G- $C^*$ -algebra and I be a G-invariant ideal in A. Then the quotient $A/I$ can be made into a G- $C^*$ -algebra such that the extension

\begin{align*}0 \rightarrow I \rightarrow A \rightarrow A/I \rightarrow 0\end{align*}

is a G- $C^*$ -extension.

Proof. It is easy to see that $A/I$ can be made into a $C_0(G^{(0)})$ -algebra via $\theta_A$ such that the extension

\begin{align*}0 \rightarrow I \rightarrow A \rightarrow A/I \rightarrow 0\end{align*}

is a $C_0(G^{(0)})$ -extension. The result now is immediate from the definition of a G-invariant ideal and the fact that the pullback functor is exact.

4. Some technical results on quasi-central approximate units

As the title suggests, in this section, we gather some technical results on quasi-central approximate units that form one of the main technical tools for our main theorem. More precisely, we present variations of the Kasparov lemma ([ Reference Kasparov21 , Lemma 1·4]) as presented in [ Reference Forough, Gardella and Thomsen14 ].

The following lemma is the groupoid analogue of [ Reference Kasparov21 , Lemma 1·4] proved in [ Reference Le Gall25 , Corollary 6·1] and [ Reference Tu43 , Corollary 4·4].

We recall the following lemma.

Lemma 4·3 ([ Reference Arveson3 , Lemma 1·1]). Let J be a $C^*$ -algebra, $\epsilon \gt0$ and f be a continuous function on [0, 1] satisfying $f(0)=0$ . Then there is a $\delta \gt0$ such that for each pair (a,e) of contractions in J with $a \geq 0$ , one has

\begin{align*}\left\lVert ae-ea \right\rVert \leq \delta \implies \left\lVert f(a)e-ef(a) \right\rVert \leq \epsilon.\end{align*}

The following lemma is one of the main technical tools in the proof of our main theorem.

Lemma 4·4. Let G be a second countable groupoid, $(J,\gamma)$ be a separable, G- $C^*$ -algebra, $0 \leq d \leq 1$ be a strictly positive element in J, $(W_n)_{n \in \mathbb{N}}$ be an increasing sequence of compact subsets in $\mathcal{M}(J)$ and $(K_n)_{n \in \mathbb{N}}$ be an increasing sequence of compact subsets in G. Then there exists a countable approximate unit $(u_n)_{n \in \mathbb{N}}$ of J contained in $C^*(d)$ such that if we set

\begin{align*}\Delta_0=\sqrt{u_0}, \quad \Delta_n=\sqrt{u_n-u_{n-1}},\end{align*}

for $n \geq 1$ , the following properties are satisfied:

1. (i) for each $n \in \mathbb{N}$ , $0 \leq u_n \leq 1$ and $u_n=u_{n+1}u_n$ ;

2. (ii) for each $n \in \mathbb{N}$ , $\Delta_{n+1}u_n=0$ ;

3. (iii) for any compact subset $K \subseteq G$ ,

   \begin{align*}\max_{g \in K} \left\lVert \gamma_g(u_n(s(g))-u_n(r(g)) \right\rVert \rightarrow 0 \text{ as } n \rightarrow \infty ;\end{align*}

4. (iv) for each $n \in \mathbb{N}$ and for any $m \in W_n$ ,

   \begin{align*} \left\lVert \Delta_n m-m\Delta_n \right\rVert \leq 1/n^2 ;\end{align*}

5. (v) for each $n \in \mathbb{N}$ and for any $g \in K_n$ ,

   \begin{align*} \left\lVert \gamma_g(\Delta_n(s(g)))-\Delta_n(r(g)) \right\rVert \leq 1/n^2;\end{align*}

6. (vi) for each $n \in \mathbb{N}$ , for any $m \in W_n$ ,

   \begin{align*} \left\lVert m(1-u_n) \right\rVert \leq \left\lVert q_J(m) \right\rVert + 1/n^2.\end{align*}

Proof. The proof is similar to the proof of [ Reference Forough, Gardella and Thomsen14 , Lemma 2·3]. Let $M_0 \subseteq \mathcal{M}(J)$ be the separable $C^*$ -subalgebra generated by $\bigcup_{n \in \mathbb{N}} W_n$ and let $(y_n)_{n \in \mathbb{N}}$ be a countable approximate unit of J contained in $C^*(d)$ satisfying (i)–(iv) of Lemma 4·2. By approximating the square root function on [0, 1] by polynomials, we see that given $\epsilon \gt 0$ , there is a $\delta \gt0$ such that

\begin{align*}\left\lVert \gamma_g(a(s(g)))-a(r(g)) \right\rVert \leq \delta \implies \left\lVert \gamma_g\left(\sqrt{a(s(g))}\right)-\sqrt{a(r(g))} \right\rVert \leq \epsilon,\end{align*}

for any positive contraction $a \in J$ and $g \in G$ . Together with Lemma 4·3, this implies that for each $n \in \mathbb{N}$ , there is a $\delta_n \gt 0$ such that given a positive contraction $a \in J$ ,

\begin{align*}\left\lVert \gamma_g(a(s(g)))-a(r(g)) \right\rVert \leq \delta_n \implies \left\lVert \gamma_g\left(\sqrt{a(s(g))}\right)-\sqrt{a(r(g))} \right\rVert \leq \frac{1}{n^2},\end{align*}

for any $g \in K_n$ and

\begin{align*}\left\lVert am-ma \right\rVert \leq \delta_n \implies \left\lVert \sqrt{a}m-m\sqrt{a} \right\rVert \leq \frac{1}{n^2},\end{align*}

for any $m \in W_n$ . Now using (iii) of Lemma 4·2, we find a subsequence $(u_n)_{n \in \mathbb{N}}$ of $(y_n)_{n \in \mathbb{N}}$ such that given any $n \in \mathbb{N}$ ,

\begin{align*}\left\lVert \gamma_g((u_{n}-u_{n-1})(s(g)))-(u_{n}-u_{n-1})(r(g)) \right\rVert \leq \delta_n,\end{align*}

for any $g \in K_n$ and

\begin{align*}\left\lVert (u_{n}-u_{n-1})m-m(u_{n}-u_{n-1}) \right\rVert \leq \delta_n,\end{align*}

for any $m \in W_n$ . Using the fact that

\begin{align*}\left\lVert q_J(m) \right\rVert=\lim_{n \to \infty}\left\lVert m(1-y_n) \right\rVert\end{align*}

for all $m \in \mathcal{M}(J)$ and passing to a further subsequence, we obtain $(u_n)_{n \in \mathbb{N}}$ satisfying (i)–(vi).

5. The main results

We now come, as the title suggests, to the main results of this paper. The approach is standard, as is already taken in [ Reference Forough, Gardella and Thomsen14 ] and [ Reference Gabe15 ]; namely, we first consider the case when the codomain is a Calkin algebra and then use the Busby invariant to pass to the general case. We begin with the following definition.

Definition 5·1. Let G be a groupoid, $(A,\alpha)$ and $(B,\beta)$ be two G- $C^*$ -algebras, $\varphi\,:\, A \rightarrow B$ be a completely positive and contractive map, $F \subset A$ be a finite subset of A and $K \subset G$ be a compact subset of G. The equivariance modulus of $\varphi$ with respect to $\alpha$ , $\beta$ , F and K, denoted $\left\lVert (\varphi,\alpha,\beta) \right\rVert_{F,K}$ is defined to be the number

(5·1) \begin{equation}\left\lVert (\varphi,\alpha,\beta) \right\rVert_{F,K}=\max_{a \in F}\sup_{g \in K}\left\lVert \beta_g (\varphi_{s(g)}(a(s(g))-\varphi_{r(g)}(\alpha_g(a(s(g)))) \right\rVert.\end{equation}

We remark that the above definition still makes sense when G is second countable and B is $\mathcal{M}(I)$ or $\mathcal{Q}(I)$ for a separable, G- $C^*$ -algebra $(I,\gamma)$ , with $\bar \gamma$ , respectively $\tilde \gamma$ , playing the role of $\beta$ and we will continue to write $\left\lVert (\varphi,\alpha,\bar \gamma) \right\rVert_{F,K}$ , respectively $\left\lVert (\varphi,\alpha,\tilde \gamma) \right\rVert_{F,K}$ , for the equivariance modulus. Now we can state the technical lemma for our main result.

Lemma 5·2. Let G be a second countable groupoid, $(A,\alpha)$ and $(I,\gamma)$ be two G- $C^*$ -algebras with A and I separable. Let $\varphi\,:\, A \to \mathcal{M}(I)$ and $\psi\,:\, A \to \mathcal{Q}(I)$ be two $C_0(G^{(0)})$ -linear, completely positive and contractive maps such that $q_I \circ \varphi=\psi$ . Then given a finite subset $F \subset A$ , a compact set $K \subset G$ and $\epsilon \gt0$ , there exists a $C_0(G^{(0)})$ -linear, completely positive and contractive map $\varphi'\,:\, A \to \mathcal{M}(I)$ such that $q_I \circ \varphi'=\psi$ and

(5·2) \begin{equation}\left\lVert (\varphi',\alpha,\bar \gamma) \right\rVert_{F,K} \leq \left\lVert (\psi,\alpha,\tilde \gamma) \right\rVert_{F,K} + \epsilon.\end{equation}

Proof. We begin the proof by fixing an increasing sequence of finite subsets $(F_n)_{n \in \mathbb{N}}$ of A with dense union in A and $F \subseteq F_0$ . Similarly, we fix an increasing sequence of compact subsets $(K_n)_{n \in \mathbb{N}}$ of G whose union equals G itself and $K \subseteq K_0$ . Now we set $W_n=\varphi(F_n)$ , $n \in \mathbb{N}$ , and by Lemma 4·4 we obtain an approximate unit $(u_n)_{n \in \mathbb{N}}$ of I, hence the sequence $(\Delta_n)_{n \in \mathbb{N}}$ satisfying the conditions (i)–(vi) as in Lemma 4·4 for the sequences $(W_n)_{n \in \mathbb{N}}$ and $(K_n)_{n \in \mathbb{N}}$ . By [ Reference Manuilov and Thomsen26 , Lemma 3·1], we see that for every $n \in \mathbb{N}$ , the sequence

(5·3) \begin{equation}\sum_{l=n}^k \Delta_l\varphi(a)\Delta_l\end{equation}

converges in the strict topology of $\mathcal{M}(I)$ as $k \to \infty$ and

(5·4) \begin{equation}\left\lVert \sum_{l=n}^\infty \Delta_l\varphi(a)\Delta_l \right\rVert \leq 1.\end{equation}

Therefore, for each $n \in \mathbb{N}$ , we can define a completely positive and contractive map

(5·5) \begin{equation}\varphi_n \colon A \to \mathcal{M}(I),\end{equation}

by setting for each $a \in A$ ,

(5·6) \begin{equation}\varphi_n(a)=\sum_{l=n}^\infty \Delta_l \varphi(a) \Delta_l.\end{equation}

We note that, since $\varphi$ is $C_0(G^{(0)})$ -linear, so is $\varphi_n$ , for each $n \in \mathbb{N}$ . Furthermore, property (iv) in Lemma 4·4 implies that

(5·7) \begin{equation}\sum_{l=n}^{k} \left\lVert \varphi(a) \Delta_l^2 - \Delta_l \varphi(a) \Delta_l \right\rVert \leq \sum_{l=n}^{k} \left\lVert \varphi(a) \Delta_l - \Delta_l \varphi(a) \right\rVert \leq \sum_{l=n}^{k} \frac{1}{l^{2}},\end{equation}

for all $n \in \mathbb{N}$ , $a \in F_n$ and $k \gt n$ . Therefore, the series

(5·8) \begin{equation}\sum_{l=0}^{\infty} \left(\varphi(a) \Delta_l^2 -\Delta_l \varphi(a) \Delta_l \right),\end{equation}

converges in norm, for all $a \in \bigcup_{l \in \mathbb{N}}F_l$ , hence for all $a \in A$ , to an element in I as each summand is an element of I. Since $\sum_{l=0}^{\infty}\Delta_l^2$ converges strictly to 1, we have

(5·9) \begin{equation}\varphi(a) - \varphi_n(a) = \sum_{l=0}^{\infty} (\varphi(a) \Delta_l^2- \Delta_l \varphi(a) \Delta_l) + \sum_{l=0}^{n-1} \Delta_l \varphi(a) \Delta_l,\end{equation}

for all $a\in A$ and $n \in \mathbb{N}$ . Since each summand of the above sum is in I, $\varphi(a) - \varphi_n(a)$ is in I as well, i.e., $q_I \circ \varphi_n=q_I \circ \varphi=\psi$ , for all $n \in \mathbb{N}$ . Now since $((u_n)(x))_{n \in \mathbb{N}}$ is an approximate unit of I(x) for all $x \in G^{(0)}$ ,

(5·10) \begin{equation}\left\lVert q_{I(x)}(m) \right\rVert=\lim_{n \rightarrow \infty}\left\lVert m((1-u_n)(x)) \right\rVert\end{equation}

for $m \in \mathcal{M}(I(x))$ . We can then find $n_0 \in \mathbb{N}$ such that whenever $a \in F$ , $g \in K$ and $n \geq n_0$ ,

(5·11a) \begin{equation}\sum_{l=n}^\infty \frac{1}{l^2} \leq \frac{\epsilon}{4\left\lVert a \right\rVert},\end{equation}

and

(5·11b) \begin{equation}\begin{aligned}&{}\left\lVert (\bar \gamma_g (\varphi_{s(g)}(a(s(g))))-\varphi_{r(g)}(\alpha_g(a(s(g))))(1-u_n)(r(g)) \right\rVert\\&\quad\leq {} \left\lVert q_{I(r(g))}(\bar \gamma_g (\varphi_{s(g)}(a(s(g))))-\varphi_{r(g)}(\alpha_g(a(s(g)))) \right\rVert + \frac{\epsilon}{2}\\&\quad={}\left\lVert \tilde \gamma_g(\psi_{s(g)}(a(s(g)))-\psi_{r(g)}(\alpha_g(a(s(g))))) \right\rVert + \frac{\epsilon}{2},\end{aligned}\end{equation}

where in the last step we use the commutativity of the square (3·18). By (v) of Lemma 4·4, we obtain for all $l \in \mathbb{N}$ and for all $a \in F$ ,

(5·12) \begin{align}&{} \left\lVert \gamma_g(\Delta_l(s(g)))\bar \gamma_g(\varphi_{s(g)}(a(s(g)))\gamma_g(\Delta_l(s(g)))-\Delta_l(r(g))\bar \gamma_g(\varphi_{s(g)}(a(s(g)))\Delta_l(r(g)) \right\rVert \\&\quad{} \leq \frac{2\left\lVert a \right\rVert}{l^2}.\end{align}

Therefore, for all $n \geq n_0$ , $a \in F$ and $g \in K$ , we have

(5·13) \begin{align}&{} \left\lVert \bar \gamma_g((\varphi_n)_{s(g)}(a(s(g)))-\sum_{l=n}^\infty \Delta_l(r(g))\bar \gamma_g(\varphi_{s(g)}(a(s(g)))\Delta_l(r(g)) \right\rVert \\&\quad={} \left\lVert \sum_{l=n}^\infty \gamma_g(\Delta_l(s(g))) \bar \gamma_g(\varphi_{s(g)}(a(s(g)))\gamma_g(\Delta_l(s(g)))\right.\\&\quad\quad-{}\left.\sum_{l=n}^\infty \Delta_l(r(g))\bar \gamma_g(\varphi_{s(g)}(a(s(g)))\Delta_l(r(g)) \right\rVert \\&\quad\leq {} \sum_{l=n}^\infty \frac{2\left\lVert a \right\rVert}{l^2}\\ &\quad\leq {} \frac{\epsilon}{2},\end{align}

where the first inequality is by (5·12) and the second inequality is by (5·11a). We also note that we have used the identity ( $a \in A$ ),

\begin{align*}(\varphi_n)_x(a(x))=\sum_{l=n}^\infty \Delta_l(x) \varphi_x(a(x))\Delta_l(x)\end{align*}

which holds since $\xi_x$ is continuous with respect to the strict topology, for all $x \in G^{(0)}$ ; and the fact that $\bar \gamma_g$ is continuous with respect to the strict topology, for all $g \in G$ . We now show that $\varphi_n$ satisfies the inequality (5·2) in the statement for $n \gt n_0$ . So we fix $N \gt n_0$ and observe that for $n \geq N$ , $\Delta_n=(1-u_{n_0})\Delta_n$ , since $u_{n_0}\Delta_n=0$ . Therefore, for $l \gt k \geq N$ , using Cauchy-Schwarz inequality for the Hilbert module $\mathcal{M}(I(r(g)))^{l-k+1}$ at the first step, and (5·11b) at the second, we obtain

(5·14) \begin{align}&{}\left\lVert \sum_{n=k}^l\Delta_n(r(g))(\bar \gamma_g\varphi_{s(g)}(a(s(g)))-\varphi_{r(g)}(\alpha_g(a(s(g))))\Delta_n(r(g)) e(r(g))\right\rVert \\&\quad={} \left\lVert \sum_{n=k}^l\Delta_n(r(g))(\bar \gamma_g(\varphi_{s(g)}(a(s(g))))\right.\\&\quad\quad-{}\left.\vphantom{\sum_{n=k}^l}\varphi_{r(g)}(\alpha_g(a(s(g))))(1-u_{n_0})(r(g))\Delta_n(r(g))e(r(g)) \right\rVert \\&\quad\leq {} \left\lVert (\bar \gamma_g(\varphi_{s(g)}(a(s(g))))- \varphi_{r(g)}(\alpha_g(a(s(g))))(1-u_{n_0})(r(g)) \right\rVert \\&\quad\quad{} \left\lVert \sum_{n=k}^l e(r(g))^*\Delta_n^2(r(g))e(r(g)) \right\rVert \\&\quad\leq {} \left(\left\lVert \tilde \gamma_g(\psi_{s(g)}(a(s(g)))-\psi_{r(g)}(\alpha_g(a(s(g)))) \right\rVert + \frac{\epsilon}{2} \right) \left\lVert (e^*(u_l-u_k)e)(r(g)) \right\rVert\end{align}

for all $a \in F$ , $g \in K$ and $e \in I$ . We note that $e^*(u_l-u_k)e$ converges to zero in norm as $l,k \to \infty$ , since $(u_l)_{l \in \mathbb{N}}$ is an approximate unit of I. Hence,

(5·15) \begin{equation}\sum_{n=N}^\infty \Delta_n(r(g))(\bar \gamma_g\varphi_{s(g)}(a(s(g)))-\varphi_{r(g)}(\alpha_g(a(s(g))))(r(g))\Delta_n(r(g))\end{equation}

converges in the strict topology of $\mathcal{M}(I(r(g)))$ such that

(5·16) \begin{align}&{} \left\lVert \sum_{n=N}^\infty \Delta_n(r(g)) (\bar \gamma_g\varphi_{s(g)}(a(s(g)))-\varphi_{r(g)}(\alpha_g(a(s(g))))(r(g))\Delta_n(r(g)) \right\rVert \\&\quad\leq {} \left\lVert \tilde \gamma_g(\psi_{s(g)}(a(s(g)))-\psi_{r(g)}(\alpha_g(a(s(g)))) \right\rVert + \frac{\epsilon}{2},\end{align}

for all $a \in F$ and $g \in K$ . Now we have all the estimates to end the proof, so to that end, fix $a \in F$ and $g \in K$ . Then

(5·17) \begin{equation}\begin{aligned}&{} \left\lVert (\bar \gamma_g((\varphi_N)_{s(g)}(a(s(g))))-(\varphi_N)_{r(g)}\alpha_g(a(s(g)))) \right\rVert \\&\quad\leq \left\lVert (\bar \gamma_g((\varphi_N)_{s(g)}(a(s(g))))-\sum_{n=N}^\infty \Delta_n(r(g))\bar \gamma_g(\varphi_{s(g)}(a(s(g))))\Delta_n(r(g))) \right\rVert \\\end{aligned}\end{equation}

(5·17) \begin{equation}\begin{aligned} &\quad\quad+ \left\lVert (\sum_{n=N}^\infty \Delta_n(r(g))\bar \gamma_g(\varphi_{s(g)}(a(s(g))))\Delta_n(r(g))-(\varphi_N)_{r(g)}\alpha_g(a(s(g)))) \right\rVert \\&\quad\leq \frac{\epsilon}{2} + \left\lVert (\sum_{n=N}^\infty \Delta_n(r(g))(\bar \gamma_g\varphi_{s(g)}(a(s(g)))-\varphi_{r(g)}(\alpha_g(a(s(g))))\Delta_n(r(g))) \right\rVert \\&\quad\leq \epsilon + \left\lVert \tilde \gamma_g(\psi_{s(g)}(a(s(g)))-\psi_{r(g)}(\alpha_g(a(s(g)))) \right\rVert,\end{aligned} \\[6pt] \end{equation}

where, the second inequality is by (5·13) and the definition of $\varphi_N$ ; and the final inequality is by (5·16). This completes the proof.

Now we are in the position to prove our main result.

Theorem 5·3. Let G be a second countable groupoid and $(A,\alpha)$ be a separable, G- $C^*$ -algebra. Let

\begin{align*}0 \rightarrow (I,\gamma) \rightarrow (B,\beta) \xrightarrow{p} (D,\delta) \rightarrow 0\end{align*}

be a G- $C^*$ -extension with B separable. Let $\varphi\,:\, A \rightarrow B$ and $\psi\,:\, A \rightarrow D$ be two $C_0(G^{(0)})$ -linear, completely positive and contractive maps such that $p \circ \varphi=\psi$ . Then given $\epsilon \gt 0$ , a finite subset $F \subset A$ of A, a compact subset $K \subset G$ of G, there exists a $C_0(G^{(0)})$ -linear, completely positive and contractive map $\varphi'\,:\, A \rightarrow B$ such that $p \circ \varphi'=\psi$ and

(5·18) \begin{equation}\left\lVert (\varphi',\alpha,\beta) \right\rVert_{F,K} \leq \left\lVert (\psi,\alpha,\delta) \right\rVert_{F,K} + \epsilon.\end{equation}

Proof. We begin by considering the G- $C^*$ -extension

(η) \begin{equation} \qquad\qquad\qquad\qquad\qquad\qquad 0 \rightarrow I \rightarrow B \rightarrow D \rightarrow 0,\end{equation}

and the associated $*$ -homomorphims

(5·19) \begin{equation}\bar \mu(\eta)\,:\, B \rightarrow \mathcal{M}(I), \quad \mu(\eta)\,:\, D \rightarrow \mathcal{Q}(I).\end{equation}

Setting

(5·20) \begin{equation}E=\{(m,c) \in \mathcal{M}(I) \oplus D\,:\, q_I(m)=\mu(\eta)(c)\},\end{equation}

and

(5·21) \begin{equation}\Theta\,:\, B \to E, \quad \Theta(b)=(\bar \mu(\eta)(b),p(b)) \text{ for all } b \in B,\end{equation}

by Corollary 3·12, we see that E can be made into a G- $C^*$ -algebra and $\Theta$ is a G-equivariant isomorphism. Let

(5·22) \begin{equation}\dot{\varphi}=\bar \mu(\eta) \circ \varphi\,:\, A \rightarrow \mathcal{M}(I), \text{ and } \dot{\psi}=\mu(\eta) \circ \psi\,:\, A \rightarrow \mathcal{Q}(I),\end{equation}

so that

(5·23) \begin{equation}q_I \circ \dot{\varphi}=q_I \circ \bar \mu(\eta) \circ \varphi=\mu(\eta) \circ p \circ \varphi=\mu(\eta) \circ \psi=\dot{\psi},\end{equation}

i.e., we are in the setting of Lemma 5·2. Therefore we find

(5·24) \begin{equation}\dot{\varphi}'\,:\, A \rightarrow \mathcal{M}(I) \text{ such that } q_I \circ \dot{\varphi}'=\dot{\psi}\end{equation}

and

(5·25) \begin{equation}\left\lVert (\dot \varphi',\alpha,\bar \gamma) \right\rVert_{F,K} \leq \left\lVert (\dot \psi,\alpha,\tilde \gamma) \right\rVert_{F,K} + \epsilon.\end{equation}

This implies that for all $a \in A$ , $q_I(\dot \varphi'(a))=\dot \psi(a)=\mu(\eta)(\psi(a))$ , i.e., $(\dot \varphi'(a), \psi(a)) \in E$ . We can therefore define

(5·26) \begin{equation}\varphi'\,:\, A \to B \quad \varphi'(a)=\Theta^{-1}(\dot \varphi'(a),\psi(a)), \quad a \in A,\end{equation}

which is then clearly a $C_0(G^{(0)})$ -linear, completely positive and contractive map such that $p \circ \varphi'=\psi$ . To end the proof, we need the control on the equivariance modulus of $\varphi'$ . To that end, we fix $g \in G$ and $a \in A$ and observe that

(5·27) \begin{equation}\begin{aligned}\tilde \gamma_g(\dot \psi_{s(g)}(a(s(g)))) & = \tilde \gamma_g(\mu(\eta)_{s(g)}(\psi_{s(g)}(a(s(g)))))\\ & = \tilde \gamma_g(q_{I_{s(g)}}(\bar \mu(\eta)_{s(g)}(\varphi_{s(g)}(a(s(g))))))\\ & = q_{I_{r(g)}}(\bar \gamma_g(\bar \mu(\eta)_{s(g)}(\varphi_{s(g)}(a(s(g))))))\\ & = q_{I_{r(g)}}(\bar \mu(\eta)_{r(g)}(\beta_g(\varphi_{s(g)}(a(s(g))))))\\ & = \mu(\eta)_{r(g)}(p_{r(g)}(\beta_g(\varphi_{s(g)}(a(s(g))))))\\ & = \mu(\eta)_{r(g)}(\delta_g((p_{s(g)}(\varphi_{s(g)}(a(s(g)))))))\\ & = \mu(\eta)_{r(g)}(\delta_g(\psi_{s(g)}(a(s(g))))),\end{aligned}\end{equation}

where the first equality is by definition of $\dot \psi$ , (5·22); the second equality is by (5·23); the third equality is by (3·18); the fourth equality is by Lemma 3·7; the fifth equality is by definition of $\mu(\eta)$ ; the sixth equality is because p is G-equivariant and the final equality is by hypothesis. This, together with (5·25), implies

(5·28) \begin{equation}\begin{aligned}\left\lVert (\dot \varphi',\alpha,\bar \gamma) \right\rVert_{F,K} \leq \left\lVert (\psi,\alpha,\delta) \right\rVert_{F,K} + \epsilon.\end{aligned}\end{equation}

which, combined with the fact that $\Theta$ is G-equivariant, in turn yields (5·18), as desired. This completes the proof.

Since A is separable and G is second countable, we obtain the following theorem as a corollary to Theorem 5·3.

Theorem 5·5. Let G be a second countable groupoid and $(A,\alpha)$ be a separable, G- $C^*$ -algebra. Let

\begin{align*}0 \rightarrow (I,\gamma) \rightarrow (B,\beta) \xrightarrow{p} (D,\delta) \rightarrow 0\end{align*}

be a G- $C^*$ -extension with B separable. Let $\varphi\,:\, A \rightarrow B$ and $\psi\,:\, A \rightarrow D$ be two $C_0(G^{(0)})$ -linear, completely positive and contractive maps with $\psi$ equivariant and such that $p \circ \varphi=\psi$ . Then there exists a sequence of $C_0(G^{(0)})$ -linear, completely positive and contractive maps $\varphi'_n\,:\, A \rightarrow B$ , $n \in \mathbb{N}$ , such that $p \circ \varphi'_n=\psi$ and

(5·29) \begin{equation}\lim_{n \rightarrow \infty} \sup_{g \in K}\left\lVert \beta_g ((\varphi'_n)_{s(g)}(a(s(g))-(\varphi'_n)_{r(g)}(\alpha_g(a(s(g)))) \right\rVert=0,\end{equation}

for all $a \in A$ and for all compact subsets $K \subseteq G$ .

We thank Valerio Proietti for pointing out that the above theorem is similar to [ Reference Higson and Kasparov18 , Theorem 8·1]; however, the precise connection is yet to be established.