# Quasi-invariant lifts of completely positive maps for groupoid actions

Suvrajit Bhattacharjee  Matematisk institutt, Universitetet i Oslo, P.O. Box
1053, Blindern, 0316 Oslo, Norway<EMAIL_ADDRESS>and  Marzieh Forough
Department of Applied Mathematics, Faculty of Information Technology, Czech
Technical University in Prague, Thákurova 9, 160 00, Prague 6, Czech Republic
<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a locally compact, $\sigma$-compact, Hausdorff groupoid and $A$ be
a separable, $C_{0}(G^{(0)})$-nuclear, $G$-$C^{*}$-algebra. We prove the
existence of quasi-invariant, completely positive and contractive lifts for
equivariant, completely positive and contractive maps from $A$ into a
separable, quotient $C^{*}$-algebra. Along the way, we construct the Busby
invariant for $G$-actions.

###### Key words and phrases:

Groupoids; $C_{0}(X)$-algebras; completely positive maps; Busby invariant.

###### 1991 Mathematics Subject Classification:

22A22; 46L07; 46L35

##  1\. Introduction

In a series of papers [szabo:self-absorbing-1, szabo:self-absorbing-2,
szabo:self-absorbing-3], Szabó introduced and in the process, systematized,
strongly self-absorption of dynamical systems for actions of locally compact,
Hausdorff groups. Motivated by the work of [hirshberg-rordam-winter:self-
absorption] on strongly self-absorption of upper semi-continuous fields of
$C^{*}$-algebras, the second author and Gardella were naturally led in
[forough-gardella:absorption] to extend Szabó’s framework to Rieffel’s upper
semi-continuous fields of actions, see [rieffel:continuous-fields]*Definition
3.1. The second author and Gardella had the need for and proved, together with
Thomsen, the following (special case of the) lifting theorem in [forough-
gardella-thomsen:lifting], which is also the main technical tool in [forough-
gardella:absorption].

######  Theorem 1.1 ([forough-gardella-thomsen:lifting]*Theorem 3.4).

Let $G$ be a locally compact, $\sigma$-compact, Hausdorff group and
$(A,\alpha)$ be a separable, nuclear, $G$-$C^{*}$-algebra. Let

$0\rightarrow(I,\gamma)\rightarrow(B,\beta)\xrightarrow{p}(D,\delta)\rightarrow
0$

be a $G$-$C^{*}$-extension, with $I$ $\sigma$-unital. Let $\psi:A\rightarrow
D$ be a $G$-equivariant, completely positive and contractive map. Then there
exists a sequence of completely positive and contractive maps
$\varphi_{n}:A\rightarrow B$, $n\in\mathbb{N}$, such that $p\circ\varphi=\psi$
and

$\lim_{n\rightarrow\infty}\left\lVert\beta_{g}(\varphi_{n}(a))-\varphi_{n}(\alpha_{g}(a))\right\rVert=0,$

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

It is well-known by now that Arveson, in his work [arveson:essentially-
normal], used previous lifting theorems of Vesterstrøm, Anderson and Ando, to
prove that the Brown-Douglas-Fillmore group $\mathrm{Ext}(X)$ for a compact
metric space $X$ ([brown-douglas-fillmore:k-homology]), is indeed a group.
Soon afterwards, Choi-Effros in their remarkable paper [choi-effros:lifting]
generalized all the previously known lifting theorems for completely positive
(and contractive) maps. Within a year, Arveson, in his seminal paper
[arveson:extensions], introduced the ubiquitous quasi-central approximate
units and used it firstly, to simplify Choi and Effros’ proof of their lifting
theorem and secondly, to show that the Brown-Douglas-Fillmore group
$\mathrm{Ext}(A)$ for a separable, nuclear $C^{*}$-algebra $A$ is again indeed
a group, which was previously shown to be a semigroup by Voiculescu. Using
similar ideas, Kasparov in [kasparov:operator-k] showed that
$\mathrm{Ext}(A,B)$ for a separable, nuclear $C^{*}$-algebra $A$ and a stable,
$\sigma$-unital $C^{*}$-algebra $B$ is a group too. By now, the Choi-Effros
lifting theorem is a standard, important technical tool, which also plays a
crucial role in the recent breakthrough [tikuisis-white-
winter:quasidiagonality]. It may also be perceived as a noncommutative
analogue of the Michael’s selection theorem [michael:selection]; see, for
instance [kirchberg:classification, gabe:lifting].

Although, the framework in [forough-gardella:absorption] is general enough to
include most of the cases of interest, it is, however, in many ways natural to
let the group act on the base of the field of $C^{*}$-algebras, i.e., to
consider $G$-$C_{0}(X)$-algebras in Kasparov’s sense, [kasparov:equivariant-
kk]*Definition 1.5. As a first step towards this direction, one would need to
establish Theorem 1.1 for such algebras which does not follow automatically.
The difficulty maybe perceived from the observation that Theorem 1.1 enables
one to obtain only fiber-wise, quasi-invariant lifts; one then has to glue
them in order to construct a global lift which is nearly impossible except for
a few pathological cases. Another scenario, which is again natural, is to
consider, roughly speaking, dynamical systems $(A(x),G(x),\alpha(x))$ varying
continuously on some parameter space $X$. The obstruction is similar; one only
has fiber-wise, quasi-invariant lifts.

Our approach is to take the more general vantage point of groupoids and along
the way, include foliations into the framework. Introduced in operator
algebras more than four decades ago by Connes, by now, there is a substantial
literature on groupoids and their dynamical systems. The dynamical flavour of
groupoids was investigated in [renault:crossed-products] probably for the
first time and studied from the perspective of Kasparov’s bivariant theory in
[le_gall:equivariant-kk]. The latter was then successfully employed in
[tu:baum-connes-amenable] in the context of the Baum-Connes conjecture. We
take the very first steps in this article of a systematic exploration of
strongly self-absorption for groupoid dynamical systems, aiming towards
generalizations of [szabo:equivariant-kirchberg-phillips, suzuki:equivariant-
absorption] and the recent breakthrough papers [gabe-szabo:dynamical-
kirchberg-phillips, gabe-szabo:stable-uniqueness]. Let us now state (a special
case of) our main theorem, which is a vast generalization of Theorem 1.1.

######  Theorem 1.2.

Let $G$ be a locally compact, $\sigma$-compact, Hausdorff groupoid. Let
$(A,\alpha)$ be a separable, $C_{0}(G^{(0)})$-nuclear, $G$-$C^{*}$-algebra and
let

$0\rightarrow(I,\gamma)\rightarrow(B,\beta)\xrightarrow{p}(D,\delta)\rightarrow
0$

be a $G$-$C^{*}$-extension, with $I$ $\sigma$-unital. Let $\psi:A\rightarrow
D$ be a $G$-equivariant, $C_{0}(G^{(0)})$-linear, completely positive and
contractive map. 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 $p\circ\varphi=\psi$ and

$\lim_{n\rightarrow\infty}\left\lVert\beta_{g}((\varphi_{n})_{s(g)}(a(s(g))-(\varphi_{n})_{r(g)}(\alpha_{g}(a(s(g))))\right\rVert=0,$

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

We refer the reader to the next section for the undefined terminology and end
this introduction with a few words on the proof. Our approach is standard and
is already taken in [forough-gardella-thomsen:lifting] to prove Theorem 1.1.
Roughly speaking, $C_{0}(G^{(0)})$-nuclearity ensures the existence of a
$C_{0}(G^{(0)})$-linear, completely positive and contractive lift. We then
“average” this lift with respect to a quasi-invariant, quasi-central,
approximate units. This is achieved by first considering the case when the
codomain is a Calkin algebra and then using the Busby invariant to pass to the
general case. However, there are few a technical points that are worth
mentioning. Firstly, the existence of such an approximate units goes back to
[le_gall:equivariant-kk] and we modify his construction to suit our needs.
Secondly, while considering the case when the codomain is a Calkin algebra,
framing the right statement becomes cumbersome due to the fact that neither
the multiplier nor the Calkin algebra is a $C_{0}(G^{(0)})$-algebra. After
fleshing out the necessary technical details, one faces the problem of
constructing the Busby invariant for groupoid actions. We overcome this but in
a slightly less degree of generality than one might hope for. This is mainly
due to the fact that the corresponding theory in the case of group actions
relies heavily on the automatic continuity theorem of [thomsen:equivariant-kk]
and [brown:continuity] which we lack. We conjecture, however, that this should
continue to hold in our case as well. We note that Theorem 1.2 is similar to
[higson-kasparov:e-theory]*Theorem 8.1; however, the precise connection is yet
to be examined. While preparing a first draft of this article, we came to know
about the paper [gabe:lifting] which provides a generalization of the Choi-
Effros lifting theorem. We talk about this paper in relation to our results in
the somewhat longer appendix.

We end this introduction by briefly describing the organization of this
article. In Section 2, we gather the necessary background material on
groupoids and their actions in a self-contained fashion, as much as possible.
Section 3 fleshes out the technicalities of the Busby invariant for groupoid
actions. In Section 4, we construct the averaging tool, namely, the quasi-
invariant, quasi-central, approximate units. Section 5 contains the main
results. As mentioned above, the somewhat longer Appendix A is dedicated to
the clarification of one of our assumptions and on describing the precise
relation of our results to [gabe:lifting].

## Acknowledgments

The first author was supported by the NFR project 300837 “Quantum Symmetry”
and the Charles University PRIMUS grant Spectral Noncommutative Geometry of
Quantum Flag Manifolds PRIMUS/21/SCI/026. He expresses his sincere gratitude
to Sergey Neshveyev for several uplifting discussions on the subject and his
support throughout the preparation of this article. He also thanks Valerio
Proietti for pointing out the similarity of Theorem 1.2 to [higson-
kasparov:e-theory]*Theorem 8.1 and several fruitful discussions. The second
author thanks the University of Oslo for support during a visit in summer of
2023, when part of this work was carried out.

## Notations and Conventions

The norm-closed linear span of a subset $S$ of a Banach space is denoted by
$[S]$. For a right Hilbert $A$-module $E$, $\mathcal{L}(E)$ denotes the
$C^{*}$-algebra of adjointable operators on $E$. A $*$-homomorphism
$\pi:A\rightarrow\mathcal{L}(E)$ is called nondegenerate if $[\pi(A)E]=E$.

##  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 [le_gall:equivariant-
kk]. 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 [renault:book-groupoid-approach]*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
$G\tensor[^{s}]{\times}{{}_{G^{(0)}}}X$ (borrowed from [proietti-
yamashita:homology-1]) of the diagram

${X}$${G}$${G^{(0)},}$$\scriptstyle{p}$$\scriptstyle{s}$

i.e.,

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

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

######  Definition 2.1.

Let $G$ be a groupoid. A (left) $G$-space is a triple $(X,p,\alpha)$ where $X$
is a space, $p$ is a continuous map from $X$ to $G^{(0)}$, called the _moment
map_ , and $\alpha$ is a continuous map from $s^{*}X\rightarrow X$,
$\alpha(g,x):=g\cdot x$, satisfying the following conditions.

  * •

For all $x\in X$, $u_{p(x)}\cdot x=x$.

  * •

If $(g,h)\in G^{(2)}$ and $(h,x)\in s^{*}X$ then $(g,h\cdot x)\in s^{*}X$ and
$(gh)\cdot x=g\cdot(h\cdot x)$.

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.

######  Definition 2.2 ([kasparov:equivariant-kk]*Definition 1.5).

A $C_{0}(X)$-_algebra_ is a pair $(A,\theta_{A})$ consisting of a
$C^{*}$-algebra $A$ and a nondegenerate $*$-homomorphism
$\theta_{A}:C_{0}(X)\rightarrow\mathcal{M}(A)$ such that
$\theta_{A}(C_{0}(X))\subset\mathcal{Z}\mathcal{M}(A)$, the center of the
multiplier algebra $\mathcal{M}(A)$.

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.

######  Definition 2.3.

Let $A$ and $B$ be two $C_{0}(X)$-algebras. A $*$-homomorphism (or a
completely positive map) $\varphi:A\to B$ is $C_{0}(X)$-_linear_ if
$\varphi(fa)=f\varphi(a)$, for all $f\in C_{0}(X)$ and $a\in A$.

If $\varphi:A\to B$ is a $C_{0}(X)$-linear, $*$-homomorphism (or a completely
positive map), then it induces a $*$-homomorphism (respectively, a completely
positive map and conversely, by [borys:furstenberg]*Lemma 3.4)
$\varphi_{x}:A(x)\to B(x)$ for every $x\in X$. By [le_gall:equivariant-
kk]*Proposition 3.1, $\varphi$ is injective (respectively, surjective) if and
only if $\varphi_{x}$ is injective (respectively, surjective), for all $x\in
X$.

######  Definition 2.4.

Let $A$ be a $C_{0}(X)$-algebra.

  * •

Given an open subset $U$ of $X$, the _restriction_ of $A$ to $U$, denoted by
$A_{U}$, is the $C_{0}(U)$-algebra $C_{0}(U)A$.

  * •

Similarly, given a closed subset $F$ of $X$, the _restriction_ of $A$ to $F$,
denoted by $A_{F}$, is the $C_{0}(F)$-algebra $A/[m_{F}A]$, where

$m_{F}=\\{f\in C_{0}(X)\mid f(x)=0\text{ for all }x\in F\\}.$

Let $A$ be a $C_{0}(X)$-algebra, $B$ a $C_{0}(Y)$-algebra and consider the
$*$-homomorphism

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

By [blanchard:deformations]*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,
[blanchard:deformations]*Corollary 3.16 holds only if $X$ is compact. However,
as noted in [echterhoff-williams:c_0-actions], the same proof holds when the
space is $X$ is merely locally compact; see the discussion before Definition
2.3 of [echterhoff-williams:c_0-actions].

######  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

$\Delta(X)=\\{(x,x)\in X\times X\mid x\in X\\}.$

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
[blanchard:deformations]*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 [kasparov:equivariant-kk]*Definition
1.6 and [echterhoff-williams:c_0-actions]*Definition 2.5. The latter considers
the (closed, two-sided) ideal

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

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.

######  Definition 2.6.

Let $A$ be a $C_{0}(X)$-algebra, $Y$ be a locally compact, Hausdorff space,
and $p:Y\rightarrow X$ be a continuous map. The _pullback_ , denoted by
$p^{*}A$, is the $C_{0}(Y)$-algebra $A\otimes_{C_{0}(X)}C_{0}(Y)$.

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 [popescu:equivariant-e-theory]*Corollary 1.26, the
pullback functor is _exact_ , i.e., if

$0\rightarrow I\rightarrow A\rightarrow B\rightarrow 0$

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

$0\rightarrow p^{*}I\rightarrow p^{*}A\rightarrow p^{*}B\rightarrow 0,$

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 topologize $\mathcal{A}$ by taking the subsets

$W(a,U,\epsilon)=\\{b\in\mathcal{A}\mid\pi(b)\in U\text{ and }\left\lVert
a(\pi(b))-b\right\rVert<\epsilon\\},$ (2.1)

where $a\in A$, $U\subset X$ is open and $\epsilon>0$, 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, [kirchberg-
wassermann:continuous-bundles]*Introduction, [rieffel:continuous-
fields]*Section 1, or [williams:book-crossed-products]*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 [raeburn-
williams:pullbacks]*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 $C_{0}(G^{(0)})$-linear,
$*$-isomorphism $\alpha:s^{*}A\rightarrow r^{*}A$. Then $\alpha$ induces
$*$-isomorphisms $\alpha_{g}:A(s(g))\rightarrow A(r(g))$.

######  Definition 2.7 ([le_gall:equivariant-kk]*Definition 3.5).

Let $G$ be a groupoid. A $G$-$C^{*}$-_algebra_ is a triple
$(A,\theta_{A},\alpha)$ where $(A,\theta_{A})$ is a $C_{0}(G^{(0)})$-algebra
and $\alpha$ is a $C_{0}(G^{(0)})$-linear, $*$-isomorphism $\alpha:s^{*}A\to
r^{*}A$ such that $\alpha_{g}\circ\alpha_{h}=\alpha_{gh}$, whenever $(g,h)\in
G^{(2)}$.

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.

######  Lemma 2.8 ([muhly-williams:renault-theorem]*Lemma 4.3).

Let $G$ be a groupoid and $(A,\theta_{A})$ be a $C_{0}(G^{(0)})$-algebra.
Suppose that $(\alpha_{g})_{g\in G}$ is a family of $*$-isomorphisms
$\alpha_{g}:A(s(g))\to A(r(g))$ such that
$\alpha_{g}\circ\alpha_{h}=\alpha_{gh}$, whenever $(g,h)\in G^{(2)}$. Then the
following are equivalent.

  * •

There exists a $C_{0}(G^{(0)})$-linear, $*$-isomorphism $\alpha:s^{*}A\to
r^{*}A$ such that $\alpha(s^{*}(a))(g)=\alpha_{g}(a(s(g)))$, making
$(A,\theta_{A},\alpha)$ into a $G$-$C^{*}$-algebra.

  * •

For $(g,a)\in s^{*}\mathcal{A}$, setting $\tilde{\alpha}(g,a)=\alpha_{g}(a)$
makes $(\mathcal{A},\pi,\tilde{\alpha})$ into a $G$-space.

  * •

For any $a\in A$, the map $g\mapsto\alpha_{g}(a(s(g)))$ is a bounded
continuous section of $G\to r^{*}\mathcal{A}$.

We end this section with one more definition.

######  Definition 2.9.

Let $G$ be a groupoid, and $(A,\alpha)$ and $(B,\beta)$ be two $G$-algebras. A
$C_{0}(G^{(0)})$-linear map $\varphi\colon A\to B$ is said to be
$G$-_equivariant_ if $(r^{*}\varphi)\circ\alpha=\beta\circ(s^{*}\varphi)$,
i.e., for all $g\in G$,
$\varphi_{r(g)}\circ\alpha_{g}=\beta_{g}\circ\varphi_{s(g)}$.

##  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 Bubsy
invariants. This extends well-known results from [thomsen:equivariant-
kk]*Section 2 and [kasparov:operator-k]*Section 7 in the case when $G$ is a
group and from [blanchard:subtriviality]*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

$\xi_{x}\colon\mathcal{M}(A)\to\mathcal{M}(A(x)),\quad\xi_{x}(m)(e(x))=(me)(x),$
(3.1)

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, [akemann-
pedersen-tomiyama:multipliers]*Theorem 4.2. In turn, $\xi_{x}$ induces

$\zeta_{x}:\mathcal{Q}(A)\to\mathcal{Q}(A(x))$ (3.2)

such that the following diagram

${\mathcal{M}(A)}$${\mathcal{Q}(A)}$${\mathcal{M}(A(x))}$${\mathcal{Q}(A(x))}$$\scriptstyle{\xi_{x}}$$\scriptstyle{q_{A}}$$\scriptstyle{\zeta_{x}}$$\scriptstyle{q_{A(x)}}$
(3.3)

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

$\xi_{x}(\theta_{A}(f)m)=\zeta_{x}(q_{A}(\theta_{A}(f)m))=0.$ (3.4)

Given a $C_{0}(X)$-extension of $A$ by another $C_{0}(X)$-algebra $I$, i.e., a
short exact sequence

$0\rightarrow I\xrightarrow{\iota}B\xrightarrow{p}A\rightarrow 0,$ ($\eta$)

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

$0\rightarrow
I(x)\xrightarrow{\iota_{x}}B(x)\xrightarrow{p_{x}}A(x)\rightarrow 0,$
($\eta(x)$)

is exact and vice-versa. 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 [popescu:equivariant-e-theory]*Lemma 1.25. For such an
extension, we define

$\bar{\mu}(\eta):B\to\mathcal{M}(I),\quad\bar{\mu}(\eta)(b)(e)=b\iota(e)$
(3.5)

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

$\bar{\mu}(\eta(x)):B(x)\rightarrow\mathcal{M}(I(x))$ (3.6)

for the extension ($\eta(x)$). We observe that although $\mathcal{M}(I)$ is
not a $C_{0}(X)$-algebra (see, for instance, [williams:book-crossed-
products]*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

$\varphi(\theta_{A}(f)a)=\theta_{I}(f)\varphi(a),$

for all $f\in C_{0}(X)$ and $a\in A$.

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 (3.4), it follows that

$\xi_{x}(\varphi(\theta_{B}(f)b))=\xi_{x}(\theta_{I}(f)\varphi(b))=0,$

so that

$\varphi_{x}:B(x)\to\mathcal{M}(I(x)),\quad\varphi_{x}(b(x))=\xi_{x}(\varphi(b))$
(3.7)

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

${B}$${\mathcal{M}(I)}$${B(x)}$${\mathcal{M}(I(x))}$$\scriptstyle{\pi_{x}}$$\scriptstyle{\varphi}$$\scriptstyle{\xi_{x}}$$\scriptstyle{\varphi_{x}}$
(3.8)

commutes.

######  Lemma 3.2.

The $*$-homomorphism

$\bar{\mu}(\eta(x)):B(x)\rightarrow\mathcal{M}(I(x))$

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

$\bar{\mu}(\eta)_{x}:B(x)\rightarrow\mathcal{M}(I(x))$

as defined in (3.7) 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

$\displaystyle\bar{\mu}(\eta)_{x}(\pi_{x}(b))(e(x))$
$\displaystyle{}=\xi_{x}(\mu(\eta)(b))(e(x))$
$\displaystyle{}=((\bar{\mu}(\eta)(b))(e))(x)$
$\displaystyle{}=(b\iota(e))(x)$ $\displaystyle{}=b(x)\iota_{x}(e(x))$
$\displaystyle{}=\bar{\mu}(\eta(x))(b(x))e(x);$

where the first equality is by commutativity of (3.8); 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

${B}$${\mathcal{M}(I)}$${B(x)}$${\mathcal{M}(I(x))}$$\scriptstyle{\pi_{x}}$$\scriptstyle{\bar{\mu}(\eta)}$$\scriptstyle{\xi_{x}}$$\scriptstyle{\bar{\mu}(\eta(x))}$
(3.9)

commutes. As in the ordinary $C^{*}$-algebraic case, we define the _Busby
invariant_

$\mu(\eta):A\to\mathcal{Q}(I),\quad\mu(\eta)(p(b))=q_{I}(\bar{\mu}(\eta)(b)).$
(3.10)

Note that we also have the Busby invariant

$\mu(\eta(x)):A(x)\rightarrow\mathcal{Q}(I(x))$ (3.11)

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

$\psi(\theta_{A}(f)a)=q_{I}\theta_{I}(f)\psi(a),$

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 (3.4), it follows that

$\zeta_{x}(\psi(\theta_{A}(f)a))=\zeta_{x}(q_{I}(\theta_{I}(f)))\zeta_{x}(\psi(a))=0,$

so that

$\psi_{x}:A(x)\to\mathcal{Q}(I(x)),\quad\psi_{x}(a(x))=\zeta_{x}(\psi(a))$
(3.12)

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

${A}$${\mathcal{Q}(I)}$${A(x)}$${\mathcal{Q}(I(x))}$$\scriptstyle{\pi_{x}}$$\scriptstyle{\psi}$$\scriptstyle{\zeta_{x}}$$\scriptstyle{\psi_{x}}$
(3.13)

commutes.

######  Lemma 3.4.

The Busby invariant

$\mu(\eta(x)):A(x)\rightarrow\mathcal{Q}(I(x))$

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

$\mu(\eta)_{x}:A(x)\rightarrow\mathcal{Q}(I(x))$

as defined in (3.12) 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

$\displaystyle\mu(\eta)_{x}(\pi_{x}(p(b)))$
$\displaystyle{}=\zeta_{x}(\mu(\eta)(p(b)))$
$\displaystyle{}=\zeta_{x}(q_{I}(\bar{\mu}(\eta)(b)))$
$\displaystyle{}=q_{I(x)}\xi_{x}(\bar{\mu}(\eta)(b))$
$\displaystyle{}=q_{I(x)}\bar{\mu}(\eta(x))(\pi_{x}(b))$
$\displaystyle{}=\mu(\eta(x))(p_{x}\pi_{x}(b))$
$\displaystyle{}=\mu(\eta(x))(\pi_{x}p(b));$

where the first equality is by commutativity of (3.13); the second equality is
by definition of $\mu(\eta)$; the third equality is by commutativity of (3.3);
the fourth equality is by commutativity of (3.9); 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

${A}$${\mathcal{Q}(I)}$${A(x)}$${\mathcal{Q}(I(x))}$$\scriptstyle{\pi_{x}}$$\scriptstyle{\mu(\eta)}$$\scriptstyle{\zeta_{x}}$$\scriptstyle{\mu(\eta(x))}$
(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

$B=\\{(m,a)\in\mathcal{M}(I)\oplus A\mid q_{I}(m)=\mu(a)\\},$
$\iota:I\rightarrow B,e\mapsto(e,0),\quad p:B\rightarrow A,(m,a)\mapsto a.$

Then the following statements hold.

  1. (1)

$B$ can be 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

$B_{x}:=\\{(m^{\prime},a^{\prime})\in\mathcal{M}(I(x))\oplus A(x)\mid
q_{I(x)}(m^{\prime})=\mu_{x}(a^{\prime})\\}.$

  3. (3)

The extension

$0\rightarrow I\xrightarrow{\iota}B\xrightarrow{p}A\rightarrow 0$ ($\eta$)

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. To conclude (1), we need to show that
$\theta_{B}$ is nondegenerate. For that, we set $B_{X}=[C_{0}(X)B]$. By
$C_{0}(X)$-linearity of $\iota$ and $p$, we obtain that

$0\rightarrow I\xrightarrow{\iota}B_{X}\xrightarrow{p}A\rightarrow 0,$

is also exact. Thus we have the following commutative diagram

${0}$${I}$${B_{X}}$${A}$${0}$${0}$${I}$${B}$${A}$${0,}$$\scriptstyle{\iota}$$\scriptstyle{=}$$\scriptstyle{p}$$\scriptstyle{\subseteq}$$\scriptstyle{=}$$\scriptstyle{\iota}$$\scriptstyle{p}$

with exact rows. Five lemma then implies that $B=B_{X}$, i.e., $B$ is a
$C_{0}(X)$-algebra which concludes (1). Moreover, (3) follows as well since
($\eta$) is in particular an ordinary $C^{*}$-extension.

For (2), we fix $x\in X$. We observe that if $(m,a)$ is in $B$, then by
commutativity of (3.3) and (3.13), $(\xi_{x}(m),a(x))$ is in $B_{x}$. We set
$\iota^{\prime}_{x}:I(x)\rightarrow B_{x}$, $e(x)\mapsto(e(x),0)$,
$p^{\prime}_{x}:B_{x}\rightarrow A(x)$, $(m^{\prime},a^{\prime})\mapsto
a^{\prime}$ and $\varphi:B\rightarrow B_{x}$, $(m,a)\mapsto(\xi_{x}(m),a(x))$.
Then, by definition, we have the following commutative diagram

${0}$${I}$${B}$${A}$${0}$${0}$${I(x)}$${B_{x}}$${A(x)}$${0,}$$\scriptstyle{\iota}$$\scriptstyle{\pi_{x}}$$\scriptstyle{p}$$\scriptstyle{\varphi}$$\scriptstyle{\pi_{x}}$$\scriptstyle{\iota^{\prime}_{x}}$$\scriptstyle{p^{\prime}_{x}}$

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

$0\rightarrow[\mathfrak{m}_{x}I]\rightarrow\ker(\varphi)\rightarrow[\mathfrak{m}_{x}A]\rightarrow
0$

is exact. By (3.4), we observe that
$[\mathfrak{m}_{x}B]\subseteq\ker(\varphi)$. Moreover, we have the following
commutative diagram

${0}$${{}[\mathfrak{m}_{x}I]}$${{}[\mathfrak{m}_{x}B]}$${{}[\mathfrak{m}_{x}A]}$${0}$${0}$${{}[\mathfrak{m}_{x}I]}$${\ker(\varphi)}$${{}[\mathfrak{m}_{x}A]}$${0,}$$\scriptstyle{=}$$\scriptstyle{\subseteq}$$\scriptstyle{=}$

with exact rows. Again by the five lemma, we conclude that
$\ker(\varphi)=[\mathfrak{m}_{x}B]$, i.e., $\varphi$ descends to an
isomorphism from $B(x)$ to $B_{x}$. This concludes (2) 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 groupoid and $(A,\alpha)$ be a
separable, $G$-$C^{*}$-algebra. We will write

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

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

${\mathcal{M}(s^{*}A)}$${\mathcal{M}(r^{*}A)}$${\mathcal{Q}(s^{*}A)}$${\mathcal{Q}(r^{*}A)}$$\scriptstyle{q_{s^{*}A}}$$\scriptstyle{\bar{\alpha}}$$\scriptstyle{q_{r^{*}A}}$$\scriptstyle{\tilde{\alpha}}$
(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

${\mathcal{M}(s^{*}A)}$${\mathcal{M}(r^{*}A)}$${\mathcal{M}(A(s(g)))}$${\mathcal{M}(A(r(g))),}$$\scriptstyle{\xi_{g}}$$\scriptstyle{\bar{\alpha}}$$\scriptstyle{\xi_{g}}$$\scriptstyle{\bar{\alpha}_{g}}$${\mathcal{Q}(s^{*}A)}$${\mathcal{Q}(r^{*}A)}$${\mathcal{Q}(A(s(g)))}$${\mathcal{Q}(A(r(g))),}$$\scriptstyle{\zeta_{g}}$$\scriptstyle{\tilde{\alpha}}$$\scriptstyle{\zeta_{g}}$$\scriptstyle{\tilde{\alpha}_{g}}$
(3.16)

and

${\mathcal{M}(A(s(g)))}$${\mathcal{M}(A(r(g)))}$${\mathcal{Q}(A(s(g)))}$${\mathcal{Q}(A(r(g)))}$$\scriptstyle{q_{A(s(g))}}$$\scriptstyle{\bar{\alpha}_{g}}$$\scriptstyle{q_{A(r(g))}}$$\scriptstyle{\tilde{\alpha}_{g}}$
(3.17)

commute. An extension

$0\rightarrow I\xrightarrow{\iota}B\xrightarrow{p}A\rightarrow 0$ ($\eta$)

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.

######  Definition 3.6.

Let $(B,\beta)$ and $(I,\gamma)$ be $G$-$C^{*}$-algebras. A
$C_{0}(G^{(0)})$-linear $*$-homomorphism (or completely positive map)
$\varphi:B\rightarrow\mathcal{M}(I)$ is said to be $G$-equivariant if
$\varphi_{r(g)}\beta_{g}=\bar{\gamma}_{g}\varphi_{s(g)}$, i.e., the diagram

${B_{s(g)}}$${B_{r(g)}}$${\mathcal{M}(I(s(g)))}$${\mathcal{M}(I(r(g)))}$$\scriptstyle{\varphi_{s(g)}}$$\scriptstyle{\beta_{g}}$$\scriptstyle{\varphi_{r(g)}}$$\scriptstyle{\bar{\gamma}_{g}}$
(3.18)

commutes for all $g\in G$.

######  Lemma 3.7.

The $*$-homomorphism

$\bar{\mu}(\eta):B\rightarrow\mathcal{M}(I)$

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 $g\in G$,

$\displaystyle\bar{\mu}_{r(g)}(\beta_{g}(b(s(g))))(e(r(g)))=$
$\displaystyle{}\beta_{g}(b(s(g)))\iota_{r(g)}(e(r(g)))$ $\displaystyle=$
$\displaystyle{}\beta_{g}(b(s(g)))\iota_{r(g)}(\gamma_{g}(\gamma_{g}^{-1}(e(r(g)))))$
$\displaystyle=$
$\displaystyle{}\beta_{g}(b(s(g)))\beta_{g}(\iota_{s(g)}(\gamma_{g}^{-1}(e(r(g)))))$
$\displaystyle=$
$\displaystyle{}\gamma_{g}(b(s(g))\iota_{s(g)}(\gamma_{g}^{-1}(e(r(g)))))$
$\displaystyle=$
$\displaystyle{}\gamma_{g}(\bar{\mu}_{s(g)}(b(s(g)))(\gamma_{g}^{-1}(e(r(g)))))$
$\displaystyle=$
$\displaystyle{}\bar{\gamma}_{g}(\bar{\mu}_{s(g)}(b(s(g))))(e(r(g))),$

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.

######  Definition 3.8.

Let $(A,\alpha)$ and $(I,\gamma)$ be $G$-$C^{*}$-algebras. A
$C_{0}(G^{(0)})$-linear $*$-homomorphism (or completely positive map)
$\psi:A\rightarrow\mathcal{Q}(I)$ is said to be $G$-equivariant if
$\psi_{r(g)}\alpha_{g}=\tilde{\gamma}_{g}\psi_{s(g)}$, i.e., the diagram

${A_{s(g)}}$${A_{r(g)}}$${\mathcal{Q}(I(s(g)))}$${\mathcal{Q}(I(r(g)))}$$\scriptstyle{\psi_{s(g)}}$$\scriptstyle{\alpha_{g}}$$\scriptstyle{\psi_{r(g)}}$$\scriptstyle{\tilde{\gamma}_{g}}$
(3.19)

commutes for all $g\in G$.

######  Lemma 3.9.

The Busby invariant

$\mu:A\rightarrow\mathcal{Q}(I)$

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

###### Proof.

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

$\displaystyle\mu_{r(g)}(\alpha_{g}(p_{s(g)}(b_{s(g)})))=$
$\displaystyle{}\mu_{r(g)}(p_{r(g)}(\beta_{g}(b_{s(g)})))$ $\displaystyle=$
$\displaystyle{}q_{I(r(g))}(\bar{\mu}_{r(g)}(\beta_{g}(b(s(g)))))$
$\displaystyle=$
$\displaystyle{}q_{I(r(g))}(\bar{\gamma}_{g}(\bar{\mu}_{s(g)}(b(s(g)))))$
$\displaystyle=$
$\displaystyle{}\tilde{\gamma}_{g}(q_{I(s(g))}(\bar{\mu}_{s(g)}(b(s(g)))))$
$\displaystyle=$
$\displaystyle{}\tilde{\gamma}_{g}(\mu_{s(g)}(p_{s(g)}(b(s(g))))),$

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 (3.17) 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 [thomsen:equivariant-kk]*Theorem 2.2. However, we note
that [thomsen:equivariant-kk]*Theorem 2.2 is a corollary of the automatic
continuity result [thomsen:equivariant-kk]*Theorem 2.1 which was more
generally proved in [brown:continuity]*Theorem 2. Lacking such a result, we
follow [kasparov:operator-k]*Section 7 and make the following definition.

######  Definition 3.10.

Let $G$ be a groupoid, $(I,\gamma)$ and $(A,\alpha)$ be two separable,
$G$-$C^{*}$-algebras and $\mu:A\rightarrow\mathcal{Q}(I)$ be a
$C_{0}(G^{(0)})$-linear, $*$-homomorphism. We say $\mu$ is $G$-admissible if
given $a\in A$ and $m\in\mathcal{M}(I)$ with $q_{I}(m)=\mu(a)$, the section
$g\mapsto\gamma_{g}\xi_{s(g)}(m)\gamma_{g}^{-1}$ is continuous.

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

$\mu(\eta):A\rightarrow\mathcal{Q}(I)$

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

######  Remark 3.12.

We remark that to show $\mu:A\rightarrow\mathcal{Q}(I)$ is $G$-admissible, one
does not need a “global lift” $\bar{\mu}$ as in the previous corollary; it
suffices to have a “local” lift, i.e., for each $a\in A$ and
$m\in\mathcal{M}(I)$ with $q_{I}(m)=\mu(a)$, there must exist a
$G$-$C^{*}$-algebra $(D,\delta)$ and a $G$-equivariant, $*$-homomorphism
$\varphi:D\rightarrow\mathcal{M}(I)$ such that $m\in\varphi(D)$. It is,
however, not so conspicuous whether the existence of a “local” lift is
equivalent to $G$-admissibility.

The following corollary to Proposition 3.5 sets up the bijection between
$G$-$C^{*}$-extensions and $G$-admissible, $*$-homomorphisms as defined above.

######  Corollary 3.13.

Let $G$ be a groupoid, $(I,\gamma)$ and $(A,\alpha)$ be two separable,
$G$-$C^{*}$-algebras and let $\mu:A\to\mathcal{Q}(I)$ be a $G$-admissible,
$*$-homomorphism. Then the following statements hold.

  1. (1)

The $C_{0}(G^{(0)})$-algebra $B$ from Proposition 3.5 can be made into a
$G$-$C^{*}$-algebra such that $\iota$ and $p$ are $G$-equivariant.

  2. (2)

The extension

$0\rightarrow I\xrightarrow{\iota}B\xrightarrow{p}A\rightarrow 0$ ($\eta$)

is a $G$-$C^{*}$-extension such that $\mu(\eta)=\mu$.

######  Corollary 3.14.

Let $G$ be a locally compact, $\sigma$-compact, Hausdorff group, $(A,\alpha)$
and $(I,\gamma)$ be two separable, $G$-$C^{*}$-algebras, and let $\mu\colon
A\to\mathcal{Q}(I)$ be a $*$-homomorphism. Then $\mu$ is $G$-admissible if and
only if $\mu$ is $G$-equivariant.

###### Proof.

First, we assume that $\mu$ is $G$-admissible. Then by Corollary 3.13 and
Lemma 3.9, $\mu$ is $G$-equivariant. The converse follows from
[thomsen:equivariant-kk]*Theorem 2.1 and Corollary 3.11. ∎

We remark that by Corollary 3.13 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.

######  Conjecture 3.15.

Let $G$ be a $\sigma$-compact groupoid, $(A,\alpha)$ and $(I,\gamma)$ be two
separable, $G$-$C^{*}$-algebras, and let $\mu\colon A\to\mathcal{Q}(I)$ be a
$*$-homomorphism. If $\mu$ is $G$-equivariant then $\mu$ is $G$-admissible.

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

######  Definition 3.16.

Let $G$ be a groupoid, $(A,\alpha)$ be a $G$-$C^{*}$-algebra and $I$ be an
ideal in $A$. We say that $I$ is a $G$-invariant ideal if
$\alpha(s^{*}I)=r^{*}I$.

######  Lemma 3.17.

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

$0\rightarrow I\rightarrow A\rightarrow A/I\rightarrow 0$

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

$0\rightarrow I\rightarrow A\rightarrow A/I\rightarrow 0$

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
([kasparov:equivariant-kk]*Lemma 1.4) as presented in [forough-gardella-
thomsen:lifting].

######  Lemma 4.1.

Let $J$ be a $\sigma$-unital $C^{*}$-algebra, $0\leq d\leq 1$ be a strictly
positive element in $J$ and $M_{0}\subseteq\mathcal{M}(J)$ be a separable
$C^{*}$-subalgebra. Then there exists a countable approximate unit
$(x_{n})_{n\in\mathbb{N}}$ for $J$ contained in $C^{*}(d)$ with the following
properties.

  * •

For each $n\in\mathbb{N}$, $0\leq x_{n}\leq 1$ and $x_{n+1}x_{n}=x_{n}$.

  * •

For each $b\in M_{0}$, $\left\lVert x_{n}b-bx_{n}\right\rVert\rightarrow 0$ as
$n\rightarrow\infty$.

The following lemma is the groupoid analogue of [kasparov:equivariant-
kk]*Lemma 1.4 proved in [le_gall:equivariant-kk]*Corollary 6.1 and [tu:baum-
connes-amenable]*Corollary 4.4.

######  Lemma 4.2.

Let $G$ be a $\sigma$-compact groupoid, $(J,\gamma)$ be a $\sigma$-unital,
$G$-$C^{*}$-algebra, $0\leq d\leq 1$ be a strictly positive element in $J$ and
$M_{0}\subseteq\mathcal{M}(J)$ be a separable $C^{*}$-subalgebra. Then there
exists a countable approximate unit $(u_{n})_{n\in\mathbb{N}}$ of $J$
contained in $C^{*}(d)$ with the following properties.

  1. (1)

For each $n\in\mathbb{N}$, $0\leq u_{n}\leq 1$ and $u_{n+1}u_{n}=u_{n}$.

  2. (2)

For each $b\in M_{0}$, $\left\lVert u_{n}b-bu_{n}\right\rVert\rightarrow 0$ as
$n\rightarrow\infty$.

  3. (3)

For any compact subset $K\subseteq G$,

$\max_{g\in
K}\left\lVert\gamma_{g}(u_{n}(s(g))-u_{n}(r(g))\right\rVert\rightarrow 0\text{
as }n\rightarrow\infty.$

###### Proof.

We apply Lemma 4.1 to find a countable approximate unit
$(x_{n})_{n\in\mathbb{N}}$ contained in $C^{*}(d)$ satisfying (1)–(3). By the
proof of [tu:baum-connes-amenable]*Lemma 4.3, there exists a countable
approximate unit $(u_{n})_{n\in\mathbb{N}}$ which satisfies (4) and such that
for all $n\in\mathbb{N}$, $u_{n}\in\mathrm{Conv}(x_{m})_{m\geq n}$. We observe
that since each $u_{n}\in\mathrm{Conv}(x_{m})_{m\geq n}$,
$(u_{n})_{n\in\mathbb{N}}$ satisfies (1)–(3) as well. ∎

We recall the following lemma.

######  Lemma 4.3 ([arveson:extensions]*Lemma 1.1).

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

$\left\lVert ae-ea\right\rVert\leq\delta\implies\left\lVert
f(a)e-ef(a)\right\rVert\leq\epsilon.$

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

######  Lemma 4.4.

Let $G$ be a $\sigma$-compact groupoid, $(J,\gamma)$ be a $\sigma$-unital,
$G$-$C^{*}$-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

$\Delta_{0}=\sqrt{u_{0}},\quad\Delta_{n}=\sqrt{u_{n}-u_{n-1}},$

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

  1. (1)

For each $n\in\mathbb{N}$, $0\leq u_{n}\leq 1$ and $u_{n}=u_{n+1}u_{n}$.

  2. (2)

For each $n\in\mathbb{N}$, $\Delta_{n+1}u_{n}=0$.

  3. (3)

For any compact subset $K\subseteq G$,

$\max_{g\in
K}\left\lVert\gamma_{g}(u_{n}(s(g))-u_{n}(r(g))\right\rVert\rightarrow 0\text{
as }n\rightarrow\infty.$

  4. (4)

For each $n\in\mathbb{N}$ and for any $m\in W_{n}$,

$\left\lVert\Delta_{n}m-m\Delta_{n}\right\rVert\leq 1/n^{2}.$

  5. (5)

For each $n\in\mathbb{N}$ and for any $g\in K_{n}$,

$\left\lVert\gamma_{g}(\Delta_{n}(s(g)))-\Delta_{n}(r(g))\right\rVert\leq
1/n^{2}.$

  6. (6)

For each $n\in\mathbb{N}$, for any $m\in W_{n}$,

$\left\lVert m(1-u_{n})\right\rVert\leq\left\lVert
q_{I}(m)\right\rVert+1/n^{2}.$

###### Proof.

The proof is similar to the proof of [forough-gardella-thomsen:lifting]*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 (1)–(4) of Lemma 4.2. By approximating the square root
function on $[0,1]$ by polynomials, we see that given $\epsilon>0$, there is a
$\delta>0$ such that

$\left\lVert\gamma_{g}(a(s(g)))-a(r(g))\right\rVert\leq\delta\implies\left\lVert\gamma_{g}\left(\sqrt{a(s(g))}\right)-\sqrt{a(r(g))}\right\rVert\leq\epsilon,$

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 $\delta_{n}>0$ such
that given a positive contraction $a\in J$,

$\left\lVert\gamma_{g}(a(s(g)))-a(r(g))\right\rVert\leq\delta_{n}\implies\left\lVert\gamma_{g}\left(\sqrt{a(s(g))}\right)-\sqrt{a(r(g))}\right\rVert\leq\frac{1}{n^{2}},$

for any $g\in K_{n}$ and

$\left\lVert am-
ma\right\rVert\leq\delta_{n}\implies\left\lVert\sqrt{a}m-m\sqrt{a}\right\rVert\leq\frac{1}{n^{2}},$

for any $m\in W_{n}$. Now using (3) 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
$n\in\mathbb{N}$,

$\left\lVert\gamma_{g}((u_{n}-u_{n-1})(s(g)))-(u_{n}-u_{n-1})(r(g))\right\rVert\leq\delta_{n},$

for any $g\in K_{n}$ and

$\left\lVert(u_{n}-u_{n-1})m-m(u_{n}-u_{n-1})\right\rVert\leq\delta_{n},$

for any $m\in W_{n}$. Using the fact that

$\left\lVert q_{J}(m)\right\rVert=\lim_{n\to\infty}\left\lVert
m(1-y_{n})\right\rVert$

for all $m\in\mathcal{M}(J)$ and passing to a further subsequence, we obtain
$(u_{n})_{n\in\mathbb{N}}$ satisfying (1)–(6). ∎

##  5\. The main results

We now come, as the title suggests, to the main results of this article. The
approach is standard, as is already taken in [forough-gardella-
thomsen:lifting] and [gabe:lifting]; 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

$\left\lVert(\varphi,\alpha,\beta)\right\rVert_{F,K}=\max_{a\in F}\sup_{g\in
K}\left\lVert\beta_{g}(\varphi_{s(g)}(a(s(g))-\varphi_{r(g)}(\alpha_{g}(a(s(g))))\right\rVert.$
(5.1)

We remark that the above definition still makes sense when $B$ 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 $\sigma$-compact groupoid, $(A,\alpha)$ and $(I,\gamma)$ be two
$G$-$C^{*}$-algebras with $A$ separable and $I$ $\sigma$-unital. 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 $\epsilon>0$, there exists a $C_{0}(G^{(0)})$-linear,
completely positive and contractive map $\varphi^{\prime}:A\to\mathcal{M}(I)$
such that $q_{I}\circ\varphi^{\prime}=\psi$ and

$\left\lVert(\varphi^{\prime},\alpha,\bar{\gamma})\right\rVert_{F,K}\leq\left\lVert(\psi,\alpha,\tilde{\gamma})\right\rVert_{F,K}+\epsilon.$
(5.2)

###### 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
(1)–(6) as in Lemma 4.4 for the sequences $(W_{n})_{n\in\mathbb{N}}$ and
$(K_{n})_{n\in\mathbb{N}}$. By [manuilov-thomsen:e-theory-special-case]*Lemma
3.1, we see that for every $n\in\mathbb{N}$, the sequence

$\sum_{l=n}^{k}\Delta_{l}\varphi(a)\Delta_{l}$ (5.3)

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

$\left\lVert\sum_{l=n}^{\infty}\Delta_{l}\varphi(a)\Delta_{l}\right\rVert\leq
1.$ (5.4)

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

$\varphi_{n}\colon A\to\mathcal{M}(I),$ (5.5)

by setting for each $a\in A$,

$\varphi_{n}(a)=\sum_{l=n}^{\infty}\Delta_{l}\varphi(a)\Delta_{l}.$ (5.6)

We note that, since $\varphi$ is $C_{0}(G^{(0)})$-linear, so is $\varphi_{n}$,
for each $n\in\mathbb{N}$. Furthermore, property (4) in Lemma 4.4 implies that

$\sum_{l=n}^{k}\left\lVert\varphi(a)\Delta_{l}^{2}-\Delta_{l}\varphi(a)\Delta_{l}\right\rVert\leq\sum_{l=n}^{k}\left\lVert\varphi(a)\Delta_{l}-\Delta_{l}\varphi(a)\right\rVert\leq\sum_{l=n}^{k}\frac{1}{l^{2}},$
(5.7)

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

$\sum_{l=0}^{\infty}\left(\varphi(a)\Delta_{l}^{2}-\Delta_{l}\varphi(a)\Delta_{l}\right),$
(5.8)

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

$\varphi(a)-\varphi_{n}(a)=\sum_{l=0}^{\infty}(\varphi(a)\Delta_{l}^{2}-\Delta_{l}\varphi(a)\Delta_{l})+\sum_{l=0}^{n-1}\Delta_{l}\varphi(a)\Delta_{l},$
(5.9)

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
$x\in G^{(0)}$,

$\left\lVert q_{I(x)}(m)\right\rVert=\lim_{n\rightarrow\infty}\left\lVert
m((1-u_{n})(x))\right\rVert$ (5.10)

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 $n\geq n_{0}$,

$\sum_{l=n}^{\infty}\frac{1}{l^{2}}\leq\frac{\epsilon}{4\left\lVert
a\right\rVert},$ (5.11a) and
$\displaystyle{}\left\lVert(\bar{\gamma}_{g}(\varphi_{s(g)}(a(s(g))))-\varphi_{r(g)}(\alpha_{g}(a(s(g))))(1-u_{n})(r(g))\right\rVert$
(5.11b) $\displaystyle\leq$ $\displaystyle{}\left\lVert
q_{I(r(g))}(\bar{\gamma}_{g}(\varphi_{s(g)}(a(s(g))))-\varphi_{r(g)}(\alpha_{g}(a(s(g))))\right\rVert+\frac{\epsilon}{2}$
$\displaystyle=$
$\displaystyle{}\left\lVert\tilde{\gamma}_{g}(\psi_{s(g)}(a(s(g)))-\psi_{r(g)}(\alpha_{g}(a(s(g)))))\right\rVert+\frac{\epsilon}{2},$

where in the last step we use the commutativity of the square (3.17). By (5)
of Lemma 4.4, we obtain for all $l\in\mathbb{N}$ and for all $a\in F$,

$\displaystyle{}\left\lVert\gamma_{g}(\Delta_{l}(s(g)))\bar{\gamma}_{g}(\varphi_{s(g)}(a(s(g)))\gamma_{g}(\Delta_{l}(s(g)))-\Delta_{l}(r(g))\bar{\gamma}_{g}(\varphi_{s(g)}(a(s(g)))\Delta_{l}(r(g))\right\rVert$
(5.12) $\displaystyle{}\leq\frac{2\left\lVert a\right\rVert}{l^{2}}.$

Therefore, for all $n\geq n_{0}$, $a\in F$ and $g\in K$, we have

$\displaystyle{}\left\lVert\bar{\gamma}_{g}((\varphi_{n})_{s(g)}(a(s(g)))-\sum_{l=n}^{\infty}\Delta_{l}(r(g))\bar{\gamma}_{g}(\varphi_{s(g)}(a(s(g)))\Delta_{l}(r(g))\right\rVert$
(5.13) $\displaystyle=$
$\displaystyle{}\left\lVert\sum_{l=n}^{\infty}\gamma_{g}(\Delta_{l}(s(g)))\bar{\gamma}_{g}(\varphi_{s(g)}(a(s(g)))\gamma_{g}(\Delta_{l}(s(g)))-\sum_{l=n}^{\infty}\Delta_{l}(r(g))\bar{\gamma}_{g}(\varphi_{s(g)}(a(s(g)))\Delta_{l}(r(g))\right\rVert$
$\displaystyle\leq$ $\displaystyle{}\sum_{l=n}^{\infty}\frac{2\left\lVert
a\right\rVert}{l^{2}}\leq\frac{\epsilon}{2},$

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 ($a\in A$),

$(\varphi_{n})_{x}(a(x))=\sum_{l=n}^{\infty}\Delta_{l}(x)\varphi_{x}(a(x))\Delta_{l}(x)$

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 $n>n_{0}$. So
we fix $N>n_{0}$ and observe that for $n\geq N$,
$\Delta_{n}=(1-u_{n_{0}})\Delta_{n}$, since $u_{n_{0}}\Delta_{n}=0$.
Therefore,

$\displaystyle{}\left\lVert\sum_{n=k}^{l}\Delta_{n}(r(g))(\bar{\gamma}_{g}\varphi_{s(g)}(a(s(g)))-\varphi_{r(g)}(\alpha_{g}(a(s(g))))\Delta_{n}(r(g))\right\rVert$
(5.14) $\displaystyle=$
$\displaystyle{}\left\lVert\sum_{n=k}^{l}\Delta_{n}(r(g))(\bar{\gamma}_{g}\varphi_{s(g)}(a(s(g)))-\varphi_{r(g)}(\alpha_{g}(a(s(g))))(1-u_{n_{0}})(r(g))\Delta_{n}(r(g))\right\rVert$

for $l>k\geq N$. 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

$\displaystyle{}\left\lVert\sum_{n=k}^{l}\Delta_{n}(r(g))(\bar{\gamma}_{g}(\varphi_{s(g)}(a(s(g))))-\varphi_{r(g)}(\alpha_{g}(a(s(g))))(1-u_{n_{0}})(r(g))\Delta_{n}(r(g))e(r(g))\right\rVert$
(5.15) $\displaystyle\leq$
$\displaystyle{}\left\lVert(\bar{\gamma}_{g}(\varphi_{s(g)}(a(s(g))))-\varphi_{r(g)}(\alpha_{g}(a(s(g))))(1-u_{n_{0}})(r(g))\right\rVert\left\lVert\sum_{n=k}^{l}e(r(g))^{*}\Delta_{n}^{2}(r(g))e(r(g))\right\rVert$
$\displaystyle\leq$
$\displaystyle{}\left(\left\lVert\tilde{\gamma}_{g}(\psi_{s(g)}(a(s(g)))-\psi_{r(g)}(\alpha_{g}(a(s(g))))\right\rVert+\frac{\epsilon}{2}\right)\left\lVert(e^{*}(u_{l}-u_{k})e)(r(g))\right\rVert$

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,

$\sum_{n=N}^{\infty}\Delta_{n}(r(g))(\bar{\gamma}_{g}\varphi_{s(g)}(a(s(g)))-\varphi_{r(g)}(\alpha_{g}(a(s(g))))(1-u_{n_{0}})(r(g))\Delta_{n}(r(g))$
(5.16)

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

$\displaystyle{}\left\lVert\sum_{n=N}^{\infty}\Delta_{n}(r(g))(\bar{\gamma}_{g}\varphi_{s(g)}(a(s(g)))-\varphi_{r(g)}(\alpha_{g}(a(s(g))))(1-u_{n_{0}})(r(g))\Delta_{n}(r(g))\right\rVert$
(5.17) $\displaystyle\leq$
$\displaystyle{}\left\lVert\tilde{\gamma}_{g}(\psi_{s(g)}(a(s(g)))-\psi_{r(g)}(\alpha_{g}(a(s(g))))\right\rVert+\frac{\epsilon}{2},$

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

$\displaystyle{}\left\lVert(\bar{\gamma}_{g}((\varphi_{N})_{s(g)}(a(s(g))))-(\varphi_{N})_{r(g)}\alpha_{g}(a(s(g))))\right\rVert$
(5.18) $\displaystyle\leq$
$\displaystyle{}\left\lVert(\bar{\gamma}_{g}((\varphi_{N})_{s(g)}(a(s(g))))-\sum_{n=N}^{\infty}\Delta_{n}(r(g))\bar{\gamma}_{g}(\varphi_{s(g)}(a(s(g))))\Delta_{n}(r(g)))\right\rVert$
$\displaystyle+$
$\displaystyle{}\left\lVert(\sum_{n=N}^{\infty}\Delta_{n}(r(g))\bar{\gamma}_{g}(\varphi_{s(g)}(a(s(g))))\Delta_{n}(r(g))-(\varphi_{N})_{r(g)}\alpha_{g}(a(s(g))))\right\rVert$
$\displaystyle\leq$
$\displaystyle{}\frac{\epsilon}{2}+\left\lVert(\sum_{n=N}^{\infty}\Delta_{n}(r(g))(\bar{\gamma}_{g}\varphi_{s(g)}(a(s(g)))-\varphi_{r(g)}(\alpha_{g}(a(s(g))))\Delta_{n}(r(g)))\right\rVert$
$\displaystyle=$
$\displaystyle{}\frac{\epsilon}{2}+\left\lVert(\sum_{n=N}^{\infty}\Delta_{n}(r(g))(\bar{\gamma}_{g}\varphi_{s(g)}(a(s(g)))-\varphi_{r(g)}(\alpha_{g}(a(s(g))))(1-u_{n_{0}})(r(g))\Delta_{n}(r(g)))\right\rVert$
$\displaystyle\leq$
$\displaystyle{}\epsilon+\left\lVert\tilde{\gamma}_{g}(\psi_{s(g)}(a(s(g)))-\psi_{r(g)}(\alpha_{g}(a(s(g))))\right\rVert,$

where, the second inequality is by (5.13) and the definition of $\varphi_{N}$;
the third equality is by (5.14); and the final inequality is by (5.17). This
completes the proof. ∎

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

######  Theorem 5.3.

Let $G$ be a $\sigma$-compact groupoid and $(A,\alpha)$ be a separable,
$G$-$C^{*}$-algebra. Let

$0\rightarrow(I,\gamma)\rightarrow(B,\beta)\xrightarrow{p}(D,\delta)\rightarrow
0$

be a $G$-$C^{*}$-extension, with $I$ $\sigma$-unital. Let
$\varphi:A\rightarrow B$ and $\psi:A\rightarrow B/I$ be two
$C_{0}(G^{(0)})$-linear, completely positive and contractive maps such that
$p\circ\varphi=\psi$. Then given $\epsilon>0$, 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
$\varphi^{\prime}:A\rightarrow B$ such that $p\circ\varphi^{\prime}=\psi$ and

$\left\lVert(\varphi^{\prime},\alpha,\beta)\right\rVert_{F,K}\leq\left\lVert(\psi,\alpha,\delta)\right\rVert_{F,K}+\epsilon.$
(5.19)

###### Proof.

We begin by considering the $G$-$C^{*}$-extension

$0\rightarrow I\rightarrow B\rightarrow D\rightarrow 0,$ ($\eta$)

and the associated $*$-homomorphims

$\bar{\mu}(\eta):B\rightarrow\mathcal{M}(I),\quad\mu(\eta):D\rightarrow\mathcal{Q}(I).$
(5.20)

Setting

$E=\\{(m,c)\in\mathcal{M}(I)\oplus D:q_{I}(m)=\mu(\eta)(c)\\},$ (5.21)

and

$\Theta:B\to E,\quad\Theta(b)=(\bar{\mu}(\eta)(b),p(b))\text{ for all }b\in
B,$ (5.22)

by Corollary 3.13, we see that $E$ can be made into a $G$-$C^{*}$-algebra and
$\Theta$ is a $G$-equivariant isomorphism. Let

$\dot{\varphi}=\bar{\mu}(\eta)\circ\varphi:A\rightarrow\mathcal{M}(I),\text{
and }\dot{\psi}=\mu(\eta)\circ\psi:A\rightarrow\mathcal{Q}(I),$ (5.23)

so that

$q_{I}\circ\dot{\varphi}=q_{I}\circ\bar{\mu}(\eta)\circ\varphi=\mu(\eta)\circ
p\circ\varphi=\mu(\eta)\circ\psi=\dot{\psi},$ (5.24)

i.e., we are in the setting of Lemma 5.2. Therefore we find

$\dot{\varphi}^{\prime}:A\rightarrow\mathcal{M}(I)\text{ such that
}q_{I}\circ\dot{\varphi}^{\prime}=\dot{\psi}$ (5.25)

and

$\left\lVert(\dot{\varphi}^{\prime},\alpha,\bar{\gamma})\right\rVert_{F,K}\leq\left\lVert(\dot{\psi},\alpha,\tilde{\gamma})\right\rVert_{F,K}+\epsilon.$
(5.26)

This implies that for all $a\in A$,
$q_{I}(\dot{\varphi}^{\prime}(a))=\dot{\psi}(a)=\mu(\eta)(\psi(a))$, i.e.,
$(\dot{\varphi}^{\prime}(a),\psi(a))\in E$. We can therefore define

$\varphi^{\prime}:A\to
B\quad\varphi^{\prime}(a)=\Theta^{-1}(\dot{\varphi}^{\prime}(a),\psi(a)),\quad
a\in A,$ (5.27)

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

$\displaystyle\tilde{\gamma}_{g}(\dot{\psi}_{s(g)}(a(s(g))))=$
$\displaystyle{}\tilde{\gamma}_{g}(\mu(\eta)_{s(g)}(\psi_{s(g)}(a(s(g)))))$
(5.28) $\displaystyle=$
$\displaystyle{}\tilde{\gamma}_{g}(q_{I_{s(g)}}(\bar{\mu}(\eta)_{s(g)}(\varphi_{s(g)}(a(s(g))))))$
$\displaystyle=$
$\displaystyle{}q_{I_{r(g)}}(\bar{\gamma}_{g}(\bar{\mu}(\eta)_{s(g)}(\varphi_{s(g)}(a(s(g))))))$
$\displaystyle=$
$\displaystyle{}q_{I_{r(g)}}(\bar{\mu}(\eta)_{r(g)}(\beta_{g}(\varphi_{s(g)}(a(s(g))))))$
$\displaystyle=$
$\displaystyle{}\mu(\eta)_{r(g)}(p_{r(g)}(\beta_{g}(\varphi_{s(g)}(a(s(g))))))$
$\displaystyle=$
$\displaystyle{}\mu(\eta)_{r(g)}(\delta_{g}((p_{s(g)}(\varphi_{s(g)}(a(s(g)))))))$
$\displaystyle=$
$\displaystyle{}\mu(\eta)_{r(g)}(\delta_{g}(\psi_{s(g)}(a(s(g))))),$

where the first equality is by definition of $\dot{\psi}$, (5.23); the second
equality is by (5.24); the third equality is by (3.17); 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 (5.26) imply

$\displaystyle\left\lVert(\dot{\varphi}^{\prime},\alpha,\bar{\gamma})\right\rVert_{F,K}\leq\left\lVert(\psi,\alpha,\delta)\right\rVert_{F,K}+\epsilon.$
(5.29)

which, combined with the fact that $\Theta$ is $G$-equivariant, in turn yield
(5.19), as desired. This completes the proof. ∎

######  Remark 5.4.

We remark that the existence of $\varphi$ in the theorem is a nontrivial
question. We expound on this issue in the Appendix A below.

Since $A$ is separable and $G$ is $\sigma$-compact, we obtain the following
theorem as a corollary to Theorem 5.3.

######  Theorem 5.5.

Let $G$ be a $\sigma$-compact groupoid and $(A,\alpha)$ be a separable,
$G$-$C^{*}$-algebra. Let

$0\rightarrow(I,\gamma)\rightarrow(B,\beta)\xrightarrow{p}(D,\delta)\rightarrow
0$

be a $G$-$C^{*}$-extension, with $I$ $\sigma$-unital. 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
$\varphi^{\prime}_{n}:A\rightarrow B$, $n\in\mathbb{N}$, such that
$p\circ\varphi^{\prime}=\psi$ and

$\lim_{n\rightarrow\infty}\left\lVert\beta_{g}((\varphi^{\prime}_{n})_{s(g)}(a(s(g))-(\varphi^{\prime}_{n})_{r(g)}(\alpha_{g}(a(s(g))))\right\rVert=0,$
(5.30)

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

We thank Valerio Proietti for pointing out that the above theorem is similar
to [higson-kasparov:e-theory]*Theorem 8.1; however, the precise connection is
yet to be established.

## Appendix A

The purpose of this appendix is to clarify some of the lifting results related
to this article from [kasparov-skandalis:group-actions-buildings,
bauval:nuclear-rkk, gabe:lifting] and to address the issue mentioned in Remark
5.4 above. We begin with the latter.

######  Definition A.1 ([kasparov-skandalis:group-actions-buildings]*Section
6.2, [bauval:nuclear-rkk]*Definition 2.1).

Let $A$ and $B$ be two $C_{0}(X)$-algebras.

  * •

A completely positive, $C_{0}(X)$-linear map $\varphi:A\to B$ is said to be
$C_{0}(X)$-_factorable_ if it admits a factorization
$\varphi=\sigma\circ\tau$, where $\tau:A\to M_{n}(\mathbb{C})\otimes C_{0}(X)$
and $\sigma:M_{n}(\mathbb{C})\otimes C_{0}(X)\to B$ are completely positive,
$C_{0}(X)$-linear maps.

  * •

$\varphi$ is said to be strongly $C_{0}(X)$-factorable if moreover $\sigma$
and $\tau$ can be chosen to be contractive.

Let $\mathrm{fact}^{1}(X;A,B)$ denote the set of all $C_{0}(X)$-factorable and
contractive maps; further, let $\mathrm{fact}^{s}(X;A,B)$ denote the subset of
all strongly $C_{0}(X)$-factorable, completely positive maps. By
[bauval:nuclear-rkk]*Proposition 2.3, $\mathrm{fact}^{s}(X;A,B)$ is dense in
$\mathrm{fact}^{1}(X;A,B)$ in the point-norm topology. When $X=\\{pt\\}$, the
two coincide. Let us also write $\mathrm{cp}^{1}(X;A,B)$ for the set of all
$C_{0}(X)$-linear, completely positive and contractive maps.

######  Definition A.2.

Let $A$ and $B$ be two $C_{0}(X)$-algebras.

  * •

$\varphi\in\mathrm{cp}^{1}(X;A,B)$ is said to be $C_{0}(X)$-_nuclear_ if it is
in the closure of $\mathrm{fact}^{1}(X;A,B)$ in the point-norm topology.

  * •

$A$ is said to be $C_{0}(X)$-_nuclear_ if the identity map
$\mathrm{id}_{A}:A\rightarrow A$ is $C_{0}(X)$-nuclear.

We note that $A$ being $C_{0}(X)$-nuclear in particular implies that $A$ is
nuclear in the ordinary sense. Thus for each $x\in X$, $A(x)$ is nuclear as
well. The following (restatement of a) striking theorem of Bauval provides a
converse.

######  Theorem A.3 ([bauval:nuclear-rkk]*Theorem 7.2).

Let $A$ be a $C_{0}(X)$-algebra. Then the following are equivalent.

  * •

$A$ is $C_{0}(X)$-nuclear.

  * •

$A$ is nuclear and is a continuous field of $C^{*}$-algebras.

In [kasparov-skandalis:group-actions-buildings], the authors obtains the
following analogue of the Choi-Effros lifting theorem, more generally for
$C_{0}(X)$-nuclear maps.

######  Proposition A.4 ([kasparov-skandalis:group-actions-buildings]*Section
6.2).

Let $A$ be a separable, $C_{0}(X)$-nuclear $C^{*}$-algebra; let

$0\rightarrow I\rightarrow B\xrightarrow{p}D\rightarrow 0$

be a $C_{0}(X)$-extension, with $B$ separable and let
$\psi\in\mathrm{cp}^{1}(X;A,D)$. Then there is a
$\varphi\in\mathrm{cp}^{1}(X;A,B)$ such that $p\circ\varphi=\psi$.

We note that this proposition ensures the existence of $\varphi$ as in Remark
5.4. Now, although the proof is a slight adaptation of Arveson’s proof in
[arveson:extensions], we sketch it nevertheless, for the sake of completeness.
To that end, we fix a separable, $C_{0}(X)$-nuclear $C^{*}$-algebra $A$ and a
$C_{0}(X)$-extension

$0\rightarrow I\rightarrow B\xrightarrow{p}D\rightarrow 0,$

with $B$ separable. We write

$\mathrm{lift}(X;A,D)=\\{\psi\in\mathrm{cp}^{1}(X;A,D):\text{there is a
}\varphi\in\mathrm{cp}^{1}(X;A,B)\text{ such that }p\circ\varphi=\psi\\}.$

Then the following lemma is the $C_{0}(X)$-version of
[arveson:extensions]*Theorem 6 whose proof is identical and hence omitted.

######  Lemma A.5.

$\mathrm{lift}(X;A,D)$ is closed in $\mathrm{cp}^{1}(X;A,D)$ in the point-norm
topology.

The next step is to show that “factorable maps are liftable”.

######  Lemma A.6.

$\mathrm{fact}^{1}(X;A,D)\subseteq\mathrm{lift}(X;A,D)$.

The above lemma follows immediately from the following one.

######  Lemma A.7.

$\mathrm{cp}^{1}(X;M_{n}(\mathbb{C})\otimes
C_{0}(X),D)\subseteq\mathrm{lift}(X;M_{n}(\mathbb{C})\otimes C_{0}(X),D)$.

###### Proof.

To begin with, let us write $B_{b}=\\{m\in\mathcal{M}(B):fm\in B\text{ for all
}f\in C_{0}(X)\\}$ as in [le_gall:equivariant-kk] (which is denoted
$\mathcal{M}^{\\#}(B)$ in [fieux:pi-algebra]). Let
$\bar{p}:\mathcal{M}(B)\to\mathcal{M}(D)$ be the canonical, surjective
extension of $p:B\rightarrow D$, see for instance [akemann-pedersen-
tomiyama:multipliers]*Theorem 4.2. Since $p$ is a $C_{0}(X)$-linear, $\bar{p}$
maps $B_{b}$ onto $D_{b}$. Now let
$\psi\in\mathrm{cp}^{1}(X;M_{n}(\mathbb{C})\otimes C_{0}(X),D)$ and define
$\bar{\psi}:M_{n}(\mathbb{C})\to D_{b}$ by

$\bar{\psi}(a)=\lim_{\lambda}\psi(a\otimes f_{\lambda}),$

where $(f_{\lambda})$ is an approximate unit for $C_{0}(X)$ with $0\leq
f_{\lambda}\leq 1$, see the proof of [fieux:pi-algebra]*Proposition 3.6. It is
easy to see that $\bar{\psi}$ is completely positive and contractive and hence
there exists a completely positive and contractive
$\bar{\varphi}:M_{n}(\mathbb{C})\to B_{b}$ such that
$\bar{p}\circ\bar{\varphi}=\bar{\psi}$. Now set
$\varphi:M_{n}(\mathbb{C})\otimes C_{0}(X)\to B$ to be

$\varphi(a\otimes f)=f\bar{\varphi}(a).$

It is clear that $\varphi\in\mathrm{cp}^{1}(X;M_{n}(\mathbb{C})\otimes
C_{0}(X),B)$ such that $p\circ\varphi=\psi$. ∎

###### Proof of Proposition A.4.

The result follows from Lemmas A.5, A.6 and A.7. ∎

Let us remark that in [bauval:nuclear-rkk]*Section 3, the author obtains
similar results which coincide with the ones presented above in the case when
$B$ is unital. This completes the second purpose of this appendix and we now
move onto the first. In [gabe:lifting], the author proves a sweeping
generalization of Proposition A.4 in the context of $C^{*}$-algebras over
topological spaces. One should note, however, that [gabe:lifting] does not
refer to [kasparov-skandalis:group-actions-buildings] or [bauval:nuclear-rkk]
and it is therefore reasonable to relate the results of the former with the
latter ones. Let us begin with a few notations. Let $\mathfrak{X}$ be a
topological space and $A$ be a $C^{*}$-algebra. We will write
$\mathbb{O}(\mathfrak{X})$ for the complete lattice of open sets of
$\mathfrak{X}$ and $\mathbb{I}(A)$ for the complete lattice of closed, two-
sided ideals of $A$.

######  Definition A.8.

Let $\mathfrak{X}$ be a topological space. An $\mathfrak{X}$-$C^{*}$-algebra
is a pair $(A,\Theta_{A})$, where $A$ is a $C^{*}$-algebra and
$\Theta_{A}:\mathbb{O}(\mathfrak{X})\rightarrow\mathbb{I}(A)$ is an order
preserving map, i.e., a map such that if $U\subseteq V$ in
$\mathbb{O}(\mathfrak{X})$ then $\Theta_{A}(U)\subseteq\Theta_{A}(V)$.

It is customary to write only $A$ for an $\mathfrak{X}$-$C^{*}$-algebra
$(A,\Theta_{A})$, suppressing the structure map $\Theta_{A}$. It is also
customary to write $A(U)$ for $\Theta_{A}(U)$, whenever
$U\in\mathbb{O}(\mathfrak{X})$.

A remarkable feature (at least to the authors!) of the general nature of
$\mathfrak{X}$-$C^{*}$-algebras is the following. Let $A$ be a
$\mathfrak{X}$-$C^{*}$-algebra. Then $\mathcal{M}(A)$ and $\mathcal{Q}(A)$ can
both be made into $\mathfrak{X}$-$C^{*}$-algebras as follows. Given
$U\in\mathbb{O}(\mathfrak{X})$, one sets

$\mathcal{M}(A)(U)=\\{m\in\mathcal{M}(A):m(a)\in A(U)\text{ for all }a\in
A\\},$ (A.1a) $\mathcal{Q}(A)(U)=q_{A}(\mathcal{M}(A)(U)).$ (A.1b)

######  Definition A.9.

Let $A$ and $B$ be two $\mathfrak{X}$-$C^{*}$-algebras. A $*$-homomorphism (or
a completely positive map) $\varphi:A\rightarrow B$ is said to be
$\mathfrak{X}$-equivariant if $\varphi(A(U))\subseteq B(U)$ for all
$U\in\mathbb{O}(\mathfrak{X})$.

Because of the general nature of $\mathfrak{X}$-$C^{*}$-algebras, one
oftentimes needs some continuity conditions, made precise in the following
definition.

######  Definition A.10.

Let $A$ be an $\mathfrak{X}$-$C^{*}$-algebra. $A$ is said to be

  * •

_finitely lower semi-continuous_ if $A(\mathfrak{X})=A$ and given
$U,V\in\mathbb{O}(\mathfrak{X})$, one has

$A(U)\cap A(V)=A(U\cap V);$

  * •

_lower semi-continuous_ if $A(\mathfrak{X})=A$ and given a family
$(U_{\lambda})\subseteq\mathbb{O}(\mathfrak{X})$, one has

$\bigcap_{\lambda}A(U_{\lambda})=A(U),$

where $U$ is the interior of $\bigcap_{\lambda}U_{\lambda}$;

  * •

_finitely upper semi-continuous_ if $A(\emptyset)=0$ and given
$U,V\in\mathbb{O}(\mathfrak{X})$, one has

$A(U)+A(V)=A(U\cup V);$

  * •

_monotone upper semi-continuous_ if given an increasing net
$(U_{\lambda})\subseteq\mathbb{O}(\mathfrak{X})$, one has

$\overline{\bigcup_{\lambda}A(U_{\lambda})}=A\left(\bigcup_{\lambda}U_{\lambda}\right);$

  * •

_upper semi-continuous_ if it is finitely and monotone upper semi-continuous.

Let $\mathfrak{X}$ be a locally compact, Hausdorff space $X$ (note the
difference in notation as we exclusively used $X$ for a _space_). By [meyer-
nest:bootstrap-class]*Section 2, there is a bijective correspondence between
_finitely lower semi-continuous_ and _upper semi-continuous_
$\mathfrak{X}$-$C^{*}$-algebras and $C_{0}(X)$-algebras. Furthermore, if $A$
and $B$ are two $C_{0}(X)$-algebras and $\varphi:A\rightarrow B$ is
$*$-homomorphism or a completely positive map, then the following are
equivalent.

  * •

$\varphi$ is $C_{0}(X)$-linear.

  * •

$\varphi$ is $\mathfrak{X}$-equivariant.

  * •

$\varphi([\mathfrak{m}_{x}A])\subseteq[\mathfrak{m}_{x}B]$ for all $x\in X$,
i.e., $\varphi$ descends to $\varphi_{x}:A(x)\rightarrow B(x)$.

Actually, more is true.

######  Lemma A.11.

Let $\mathfrak{X}$ be locally compact, Hausdorff space $X$, $A$ and $B$ be two
$C_{0}(X)$-algebras, and $\varphi:A\to\mathcal{M}(B)$ be a $*$-homomorphism
(or a completely positive map). Then the following are equivalent.

  1. (1)

$\varphi$ is $C_{0}(X)$-linear.

  2. (2)

$\varphi$ is $\mathfrak{X}$-equivariant.

  3. (3)

For each $x\in X$, $\varphi$ induces a $*$-homomorphism (or a completely
positive map) $\varphi_{x}:A(x)\to\mathcal{M}(B(x))$ such that
$\varphi_{x}(a(x))=\xi_{x}(\varphi(a))$.

###### Proof.

(1)$\implies$(2). We recall that for
$U\in\mathbb{O}(\mathfrak{X})=\mathbb{O}(X)$, $A(U)=C_{0}(U)A$. Then

$\varphi(A(U))(B)=\varphi(C_{0}(U)A)(B)=C_{0}(U)\varphi(A)(B)\subseteq
C_{0}(U)B=B(U),$

i.e., $\varphi(A(U))\subseteq\mathcal{M}(B)(U)$ for all
$U\in\mathbb{O}(\mathfrak{X})=\mathbb{O}(X)$.

(2)$\implies$(3). It is enough to observe that for every $x\in X$,
$\varphi(a)\in\mathcal{M}(B)(X\setminus\\{x\\})$, whenever
$a\in\mathfrak{m}_{x}A=A(X\setminus\\{x\\})$.

(3)$\implies$(1). We fix $f\in C_{0}(X)$ and $a\in A$. Then for every $x\in
X$,

$\xi_{x}(\varphi(fa)-\theta_{B}(f)\varphi(a))=\varphi_{x}(f(x)a(x))-f(x)\varphi_{x}(a(x))=0,$

i.e., $((\varphi(fa)-\theta_{B}(f)\varphi(a))(b))(x)=0$ for all $x\in X$ and
all $b\in B$, i.e., $\varphi(fa)-\theta_{B}(f)\varphi(a)=0$, i.e., $\varphi$
is $C_{0}(X)$-linear. ∎

######  Lemma A.12.

Let $\mathfrak{X}$ be a locally compact, Hausdorff space $X$, $A$ and $B$ be
two $C_{0}(X)$-algebras, and $\psi:A\to\mathcal{Q}(B)$ be a $C_{0}(X)$-linear
$*$-homomorphism (or a completely positive map) such that there is a
$C_{0}(X)$-linear, completely positive lift
$\varphi:A\rightarrow\mathcal{M}(B)$ with $q_{B}\circ\varphi=\psi$. Then
$\psi$ is $\mathfrak{X}$-equivariant.

###### Proof.

We fix $U\in\mathbb{O}(\mathfrak{X})=\mathbb{O}(X)$. Then

$\psi(A(U))=q_{B}(\varphi(A(U)))\subseteq
q_{B}(\mathcal{M}(B)(U))=\mathcal{Q}(B)(U),$

which completes the proof. ∎

After these preliminaries, we can state (some special cases of) the results
from [gabe:lifting].

######  Proposition A.13 ([gabe:lifting]*Proposition 5.4).

Let $\mathfrak{X}$ be a topological space, $A$ be a separable, nuclear, lower
semi-continuous $\mathfrak{X}$-$C^{*}$-algebra, $B$ be a separable,
$\mathfrak{X}$-$C^{*}$-algebra with property (UBS) and
$\psi:A\rightarrow\mathcal{Q}(B)$ be an $\mathfrak{X}$-equivariant, completely
positive map. Then there is an $\mathfrak{X}$-equivariant, completely positive
map $\varphi:A\rightarrow\mathcal{M}(B)$ such that $q_{B}\circ\varphi=\psi$.

Several remarks are in order. Firstly, property (UBS) is another regularity
condition and is defined in [gabe:lifting]*Definition 4.12. Secondly, let
$\mathfrak{X}$ be a locally compact, Hausdorff space $X$ in the above
proposition. Then by the discussion before Lemma A.11, $A$ is a separable,
nuclear, continuous field of $C^{*}$-algebras, i.e., $A$ is
$C_{0}(X)$-nuclear. By [gabe:lifting]*Example 4.14, $B$ automatically has
property (UBS) in this case. The lift $\varphi$ is $C_{0}(X)$-linear by Lemma
A.11 and therefore $\psi$ itself is $C_{0}(X)$-linear, by Lemma A.12. Thirdly,
the proof of [gabe:lifting]*Proposition 5.4, shows that if
$\psi\in\mathrm{cp}^{1}(X;A,\mathcal{Q}(B))$ then
$\varphi\in\mathrm{cp}^{1}(X;A,\mathcal{M}(B))$, allowing a bit abuse of
notation, of course, because none of the codomain algebras are
$C_{0}(X)$-algebras. Let us summarize our discussion.

######  Proposition A.14.

Let $A$ and $B$ be two $C_{0}(X)$-algebras, $A$ be separable and
$C_{0}(X)$-nuclear, $B$ be separable and
$\psi\in\mathrm{cp}^{1}(X;A,\mathcal{Q}(B))$. Then there is a
$\varphi\in\mathrm{cp}^{1}(X;A,\mathcal{M}(B))$ such that
$q_{B}\circ\varphi=\psi$.

We note, however, in this form, we are able to prove the above proposition
differently than the proof in [gabe:lifting]. More precisely, although
$\mathcal{Q}(B)$ is not a $C_{0}(X)$-algebra and one cannot therefore apply
Proposition A.4, one can, however, still run the proof as described above,
especially, Lemma A.7. Thus we obtain another justification of the assumption
of Lemma 5.2. Finally, we have the following (special case of the)
$\mathfrak{X}$-equivariant Choi-Effros lifting theorem. Before stating it, let
us recall that an $\mathfrak{X}$-$C^{*}$-extension is a short exact sequence

$0\rightarrow I\rightarrow B\rightarrow D\rightarrow 0,$

of $\mathfrak{X}$-$C^{*}$-algebras and $\mathfrak{X}$-equivariant,
$*$-homomorphims such that for each $U\in\mathbb{O}(\mathfrak{X})$, the
sequence

$0\rightarrow I(U)\rightarrow B(U)\rightarrow D(U)\rightarrow 0$

is exact.

######  Theorem A.15 ([gabe:lifting]*Theorem 5.6).

Let $\mathfrak{X}$ be a topological space and $A$ be a separable, nuclear,
lower semi-continuous $\mathfrak{X}$-$C^{*}$-algebra. Let

$0\rightarrow I\rightarrow B\xrightarrow{p}D\rightarrow 0$

be an $\mathfrak{X}$-$C^{*}$-extension, such that $I$ is separable and has
property (UBS). Let $\psi:A\rightarrow D$ be an $\mathfrak{X}$-equivariant,
completely positive map. Then there is an $\mathfrak{X}$-equivariant,
completely positive map $\varphi:A\rightarrow B$ such that
$p\circ\varphi=\psi$.

Again several remarks are in order. Firstly, let $\mathfrak{X}$ be a locally
compact, Hausdorff space $X$ in the above theorem. Then $A$ is a separable,
nuclear, continuous field of $C^{*}$-algebras, i.e., $A$ is
$C_{0}(X)$-nuclear. Secondly,

$0\rightarrow I\rightarrow B\rightarrow D\rightarrow 0$

is a $C_{0}(X)$-extension with $I$ separable. Finally, by the remarks just
before Lemma A.11, $\psi$ is a $C_{0}(X)$-linear, completely positive map.
Thus Theorem A.15, in this very special case, is a slightly weaker version of
Proposition A.4, in that both $\varphi$ and $\psi$ being contractions is
omitted, which is, however, easy to deduce from the proof.

To conclude, let us note that [gabe:lifting]*Proposition 5.5 is similarly
related to our Proposition 3.5; to show equivalence of the latter with the
former, one needs to show that the pullback in the former proposition is
finitely lower semi-continuous and upper semi-continuous. In this regard, our
Proposition 3.5 seems to be slightly more efficient.

## References