# The Haagerup property for twisted groupoid dynamical systems

Bartosz K. Kwaśniewski  Bartosz K. Kwaśniewski, Faculty of Mathematics,
University of Białystok, ul. K. Ciołkowskiego 1M, 15-245 Białystok, Poland
<EMAIL_ADDRESS>,  Kang Li  Kang Li, Institute of Mathematics of the
Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland
Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg
(FAU), Cauerstraße 11, 91058 Erlangen, Germany<EMAIL_ADDRESS>and  Adam
Skalski  Adam Skalski, Institute of Mathematics of the Polish Academy of
Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland<EMAIL_ADDRESS>
###### Abstract.

We introduce the Haagerup property for twisted groupoid $C^{*}$-dynamical
systems in terms of naturally defined positive-definite operator-valued
multipliers. By developing a version of ‘the Haagerup trick’ we prove that
this property is equivalent to the Haagerup property of the reduced crossed
product $C^{*}$-algebra with respect to the canonical conditional expectation
$E$. This extends a theorem of Dong and Ruan for discrete group actions, and
implies that a given Cartan inclusion of separable $C^{*}$-algebras has the
Haagerup property if and only if the associated Weyl groupoid has the Haagerup
property in the sense of Tu. We use the latter statement to prove that every
separable $C^{*}$-algebra which has the Haagerup property with respect to some
Cartan subalgebra satisfies the Universal Coefficient Theorem. This
generalises a recent result of Barlak and Li on the UCT for nuclear Cartan
pairs.

###### Key words and phrases:

étale groupoids; twisted groupoid crossed products; Haagerup property; UCT
problem; Cartan subalgebras

###### 2010 Mathematics Subject Classification:

Primary 46L05; Secondary 22A22, 43A22, 46L55

##  1\. Introduction

One of the most active areas of research at the intersection of geometric
group theory and the theory of operator algebras in recent years is the study
of interactions between the geometric properties of a group and analytic
properties of the associated $C^{*}$\- or von Neumann algebras. The
prototypical example is the equivalence of amenability of a discrete group
$\Gamma$ and the nuclearity of the group $C^{*}$-algebra $C^{*}_{r}(\Gamma)$
or injectivity of the group von Neumann algebra $VN(\Gamma)$, see e.g. [BrO].
Similarly, the Haagerup property for $\Gamma$ can be characteristed by
approximation properties of $C^{*}_{r}(\Gamma)$ or $VN(\Gamma)$, see [BrO,
CCJJV, Cho, Don]. The aforementioned correspondences, at a technical level,
are achieved in two stages: first one encodes some property of $\Gamma$ in
terms of the existence of certain functions on $\Gamma$, and then one shows
that an abstract approximation property of the associated operator algebra can
be achieved with the approximating maps being Herz-Schur multipliers, i.e.
maps determined by functions on $\Gamma$. This second step is sometimes called
‘averaging maps into Herz-Schur multipliers’, and is essentially based on a
simple but very efficient trick attributed to Haagerup (see [Haa2]).

With the increased interest in various constructions generalising the group
operator algebras there is a need to develop and apply appropriate versions of
the above procedure in other contexts. This has been done, for example, for
(unimodular) discrete quantum groups ([KrR], [DFSW]) and for $C^{*}$-algebraic
crossed products ([MSTT], [DoR]). The latter case shows an interesting
dichotomy: if one is interested directly in the Haagerup property of the
crossed product by an action on a $C^{*}$-algebra $A$, one has to allow a
broader class of analogues of Herz-Schur multipliers, sometimes called mapping
multipliers ([MTT], [BC2]), whereas if one wants to consider only the
$A$-valued multipliers, one arrives rather at a relative Haagerup property of
the crossed product with respect to $A$ viewed as a subalgebra, as defined in
[DoR].

Recent years have also brought a renewed interest in $C^{*}$-algebras
associated to twisted étale groupoids and their actions. This is partially due
to the fact that such structures lead to many new interesting examples of
$C^{*}$-algebras, but also due to a certain universality: the celebrated
result of Renault from [Ren5] says that any Cartan pair $(B,A)$, i.e. a
$C^{*}$-algebra $B$ and a regular maximal abelian subalgebra $A\subseteq B$
that admits a faithful conditional expectation, comes from a (unique) twisted-
groupoid model. Also Cartan pairs are ubiquitous. For example Li proves in
[Li] that any simple classifiable $C^{*}$-algebra admits a Cartan subalgebra,
so also a twisted groupoid model. On the other hand Barlak and Li show in
[BaL] that nuclear algebras admitting Cartan subalgebras satisfy the UCT. They
exploit results of Tu ([Tu]) that connect the UCT property for étale groupoid
$C^{*}$-algebras with what we call the Haagerup property of a groupoid. Our
first initial motivation was to generalise the result of [BaL] by replacing
nuclearity with a weaker property of a Cartan pair, equivalent to the Haagerup
property of the associated Weyl groupoid. A relevant variant of the weaker
property mentioned above was introduced by Dong and Ruan [DoR]. The authors of
[DoR] defined the relative Haagerup property for a unital $C^{*}$-inclusion
$A\subseteq B$ and characterised it when $B=A\rtimes_{r}\Gamma$ is the reduced
crossed product, of a discrete group action $\Gamma$ on a unital
$C^{*}$-algebra $A$, via $Z(A)$-valued positive-definite multipliers defined
on $\Gamma$. Thus our second key motivation was to extend this result to the
twisted actions of locally compact étale groupoids on not necessarily unital
$C^{*}$-algebras. The necessity to consider the twist comes from the fact that
Renault’s theorem involves twisted groupoids. However, it suffices to consider
only _Kumjian’s twists_ by a circle bundle [Kum1], and in the present paper we
restrict ourselves to such twists. Again a central role in the proofs is based
on a suitable version of the Haagerup Trick.

In particular we establish the following two results (see Theorems 5.17, 6.1)
that are far reaching generalisations of the main results of [DoR] and [BaL],
and apply to $C^{*}$-algebras admitting Cartan subalgebras (see Corollary
6.7). Pertinent definitions of the Haagerup property can be found in
Definitions 5.6 and 5.12 below.

######  Theorem A.

Let $B$ be the reduced crossed product of a twisted action
$(\mathcal{A},G,\Sigma,\alpha)$ of an étale locally compact Hausdorff groupoid
$G$ on the $C^{*}$-algebra $A:=C_{0}(\mathcal{A})$ where $\mathcal{A}$ has a
continuous unit section. Denote the canonical expectation from $B$ to $A$ by
$E$. Then $B$ has the $E$-Haagerup property, if and only if
$(\mathcal{A},G,\Sigma,\alpha)$ has the Haagerup property.

######  Theorem B.

For any continuous separable saturated Fell bundle $\mathcal{B}$ over a second
countable locally compact Hausdorff étale groupoid $G$ with the Haagerup
property such that $C^{*}(\mathcal{B}|_{{G^{(0)}}})$ is of type I, both the
reduced $C^{*}$-algebra $C^{*}_{r}(\mathcal{B})$ and the full $C^{*}$-algebra
$C^{*}(\mathcal{B})$ satisfy the UCT.

######  Corollary C.

Let $B$ be a separable $C^{*}$-algebra which admits a Cartan subalgebra $A$.
If the inclusion $A\subseteq B$ has the Haagerup property, then $B$ satisfies
the UCT.

An important fact which distinguishes our work is that we treat groupoids with
not neccessarily compact unit spaces. This forces us to work with non-unital
algebras and has significant consequences even at the level of the relevant
definitions. Since the proof of Theorem B relies heavily on the work of Tu (in
fact it is a combination of the arguments of [BaL] and the Stabilization
Theorem from [IKSW]), our model initial condition is existence of a _locally
proper function of negative type_ in the sense of [Tu]. This local properness,
when translated to an approximation property of positive-definite maps, means
that the maps in question are only _locally $C_{0}$_ (see Proposition 5.4).
Accordingly, we adapt the relative Haagerup property of [DoR] to not-
neccesarily unital $C^{*}$-inclusions $A\subseteq B$, mimicking the ‘locally
$C_{0}$ condition’ by using the Pedersen’s ideal of $A$ (see Definition 5.12).

As alluded to above, the key step in proving Theorem A is a version of _the
Haagerup Trick_ that gives a systematic way of passing from general maps
acting on the crossed product to certain functions on $G$, which in turn lead
to analogues of Herz-Schur multipliers. In the context of operator algebras
associated to groupoids, generalisations of Herz-Schur multipliers were
studied for example in [RaW], [Ren4], [Tak] and [BrO, Subsection 5.6]. We
assume in Theorem A that $\mathcal{A}$ has a continuous unit section only to
facilitate the Haagerup trick. This is enough for all of our applications and
allows us to consider very concrete and natural operator-valued multipliers.
The genuinely new element is that we are dealing with the multipliers of
_twisted_ groupoid $C^{*}$-dynamical systems, with twisting in the sense of
Green-Renault [Gre], [Ren2], [Ren3]. In the group case this brings us close to
some considerations in [BC1]. The authors of [BC1] employ twists by cocycles
in the spirit of [Z-M], [BuS], rather than that of Green [Gre], but as shown
in [PaR] cocycle twists cover those coming from the group extension by a
choice of a global continuous section. This can not be directly generalised to
groupoids, as groupoid extensions admit non-trivial continuous sections only
locally. Nevertheless, it seems that we need a variant of this result to make
the technicalities related to our framework tractable. We found a way out by
describing twisted groupoid dynamical systems using suitable pictures of
inverse semigroup actions, that exploit the ideas of [BuE2], [BuM], [BEM] and
[KwM1]. We establish the relevant equivalences, describing them in detail, as
we believe they are of independent interest and could be used in other
contexts. In particular, for étale groupoids twisted crossed products in the
sense of Green-Kumjian-Renault can be replaced by inverse semigroup crossed
products twisted by a cocycle, as introduced in [BuE2]. This is the content of
the following theorem (see Theorem 2.16).

######  Theorem D.

Let $(\mathcal{A},G,\Sigma,\alpha)$ be a twisted groupoid $C^{*}$-dynamical
system with $G$ étale locally compact and Hausdorff. There is a natural
twisted inverse semigroup action $(A,S,\beta,\omega)$ (described in Lemma 2.8)
where $A=C^{*}(\mathcal{A})$ and $S$ is an inverse semigroup consisting of
open bisections, and we have natural isomorphisms for the corresponding
crossed products: $C^{*}(\mathcal{A},G,\Sigma,\alpha)\cong
A\rtimes_{\beta}^{\omega}S$, $C^{*}_{r}(\mathcal{A},G,\Sigma,\alpha)\cong
A\rtimes_{\beta,r}^{\omega}S.$

We use the $S$-grading of the reduced crossed product from Theorem D to
establish our version of the Haagerup Trick and study its properties (see
Propositions 4.1, 4.4). We believe this tool will be useful in the study of
other approximation properties of groupoid crossed products.

The detailed plan of the paper is as follows: after discussing preliminary
facts and notations concerning groupoid Fell bundles in the beginning of
Section 2, we provide in the same section an extended discussion of various
equivalent pictures of twisted groupoid actions and associated crossed product
operator algebras, focusing on the interpretation in terms of inverse
semigroup actions on Hilbert bimodules. In Section 3 we introduce various
classes of Herz-Schur type multipliers for such twisted crossed products and
show that positive-definite $\mathcal{A}$-valued functions correspond to
completely positive multipliers. Section 4 contains a version of the Haagerup
Trick in our framework: every map $\Phi$ on the resulting crossed product
algebra which is a bimodule map (with respect to the subalgebra given by the
$C^{*}$-bundle $\mathcal{A}$ on which our groupoid acts) is shown to determine
in a canonical way a function $h^{\Phi}$ on the groupoid with values in the
associated bundle. There we also establish the interplay between properties of
$\Phi$ and $h^{\Phi}$, which are key for the following results. In Section 5
we discuss the Haagerup property for groupoids, as defined in [Tu], and its
variant (equivalent in the $\sigma$-compact setting) which provides the right
notion for twisted groupoid dynamical systems. Then we exploit the content of
the previous sections to show one of the main results of the paper, namely the
equivalence between the Haagerup property for such a system and a relative
Haagerup property of the pair of associated $C^{*}$-algebras. We also record
in this section certain immediate corollaries for Cartan pairs. Finally in
Section 6 we present some consequences of the theorem mentioned above for the
UCT property of Cartan pairs, based on the observation that the key arguments
of [BaL] require only the Haagerup property of the associated groupoid, and
not neccessarily amenability, as was used in that paper.

##  2\. Groupoid Fell bundles, twisted groupoid actions and their
$C^{*}$-algebras

In this section we establish the notation and conventions regarding groupoids
and their Fell bundles. We also discuss the concept of twisted groupoid
actions on $C^{*}$-algebras and relate them to certain twisted inverse
semigroup actions. Further we describe the relevant convolution operator
algebras and prove their isomorphism (Theorem D of the introduction).

###  2.1. Banach bundles

Throughout this paper $X$ is a locally compact Hausdorff space. We consider
Fell bundles that are upper semicontinuous, see for instance [DuG], [BuE2]. So
by a _Banach bundle over_ $X$ we mean a topological space
$\mathcal{B}=\bigsqcup_{x\in X}B_{x}$ such that the canonical projection
$p:\mathcal{B}\twoheadrightarrow X$ is continuous and open, each fiber $B_{x}$
is a complex Banach space, and the Banach space structure is consistent with
the topology of $\mathcal{B}$ in the sense that the maps
$\mathcal{B}\times\mathcal{B}\supseteq\bigsqcup_{x\in X}B_{x}\times
B_{x}\ni(v,w)\to v+w\in\mathcal{B}$ and
$\mathbb{C}\times\mathcal{B}\ni(\lambda,v)\to\lambda v\in\mathcal{B}$ are
continuous, but we require the map $\mathcal{B}\ni v\mapsto\|v\|\in\mathbb{R}$
only to be upper semicontinuous. One also requires another technical
condition: if $\\{v_{i}\\}_{i\in\mathcal{I}}$ is a net in $\mathcal{B}$ such
that $\|v_{i}\|\stackrel{{\scriptstyle i\in\mathcal{I}}}{{\longrightarrow}}0$
and $p(v_{i})\stackrel{{\scriptstyle i\in\mathcal{I}}}{{\longrightarrow}}x$ in
$X$, then $v_{i}$ converges to $0_{x}$, where $0_{x}$ is the zero element in
$B_{x}$. If the map $\mathcal{B}\ni v\mapsto\|v\|\in\mathbb{R}$ is continuous,
then we say the _bundle is continuous_. We will also use an analogous notion
of (upper semicontinuous) $C^{*}$-bundles. The bundle as above is said to be
_locally trivial_ if for any point $x\in X$ there is an open neighbourhood $U$
of $x$ in $X$ and a Banach space $\mathsf{Y}$ such that $\mathcal{B}|_{U}\cong
U\times\mathsf{Y}$, where the isomorphism respects the bundle structure.

In a number of sources, including the original work of Fell, Banach bundles
are continuous by definition, see, for instance, [FeD], [Kum2], [Tak].
However, nowadays it seems more standard to consider upper-semicontinuous
bundles. Firstly, this more general setting is often very useful. Secondly,
upper semicontinuity is usually sufficient to prove important theorems for
Banach bundles. For instance, the reconstruction theorem [FeD, Theorem
II.13.18] is generalised (to upper semicontinuous bundles) in [BuE2,
Proposition 2.4], and the Douady-dal Soglio-Herault theorem [FeD, Appendix C]
is generalised in [Laz, Corollary 2.10]. The reconstruction theorem says that
whenever we have a disjoint sum of Banach spaces $\mathcal{B}=\bigsqcup_{x\in
X}B_{x}$ and a set $\Gamma$ of sections of $\mathcal{B}$ such that (a) for
each $f\in\Gamma$ the function $X\ni x\mapsto\|f(x)\|\in\mathbb{R}$ is upper
semicontinuous, and (b) for each $x\in X$ the set $\\{f(x):f\in\Gamma\\}$ is
dense in $B_{x}$, then there is a unique topology on $\mathcal{B}$ making it
into a Banach bundle such that $\Gamma\subseteq C(\mathcal{B})$, i.e. all the
sections in $\Gamma$ become continuous. The Douady-dal Soglio-Herault theorem
states that for every Banach bundle $\mathcal{B}=\bigsqcup_{x\in X}B_{x}$ and
every $v\in B_{x}$ there is a continuous section $f\in C(\mathcal{B})$ such
that $f(x)=v$. In fact, one can choose $f$ to be vanishing at infinity in the
sense that for every $\varepsilon>0$ the set $\\{x\in
X:\|f(x)\|\geq\varepsilon\\}$ is compact. In particular, the space of
continuous sections vanishing at infinity, which we denote by
$C_{0}(\mathcal{B})$, is a Banach $C_{0}(X)$-module (with the norm
$\|f\|=\sup_{x\in X}\|f(x)\|$), and we can recover
$\mathcal{B}=\bigsqcup_{x\in X}B_{x}$ from this module.

A _line bundle_ is a Banach bundle $\mathcal{B}=\bigsqcup_{x\in X}B_{x}$ where
each space $B_{x}$, $x\in X$, is one-dimensional (we will usually use
$\mathcal{L}$ to denote line bundles). It is well known that every _continuous
line bundle is locally trivial_. More specifically, if $\mathcal{B}$ is a
continuous line bundle, then for every point $x_{0}\in X$ there is a section
$f\in C_{0}(\mathcal{B})$ with $|f(x_{0})|=1$ so that $U:=\\{x\in
X:|f(x)|>1/2\\}$ is an open neighborhood of $x_{0}$. Then
$C_{0}(\mathcal{B}|_{U})\cong C_{0}(U)$ because for every $g\in
C_{0}(\mathcal{B}|_{U})$ there is a unique function $h:U\to\mathbb{C}$ such
that $g(x)=h(x)f(x)$ for every $x\in U$. Continuity of $g$ and $f$ forces
continuity of $h$ and since $g\in C_{0}(\mathcal{B}|_{U})$ and $\|f|_{U}\|\geq
1/2$ we get $h\in C_{0}(U)$. Thus $\mathcal{B}|_{U}=\bigsqcup_{x\in U}B_{x}$
is isomorphic to the trivial bundle. We stress that general (upper
semicontinuous) Banach bundles need not be locally trivial. For instance, we
may consider a bundle $\mathcal{B}=\bigsqcup_{x\in X}\mathbb{C}$ with topology
in which all upper semicontinuous functions are continuous sections.

###  2.2. Fell bundles over groupoids and groupoid dynamical systems

In this paper $G$ will always denote an étale locally compact Hausdorff
groupoid. Equivalently, $G$ is a topological groupoid such that the unit space
$X:=G^{(0)}$ is locally compact and Hausdorff, the source and range maps
$s,r:G\to X$ are local homeomorphisms and $X$ is a clopen subset of $G$ (note
that it is in fact automatically open as a range of a local homeomorphism).
Yet another equivalent phrasing is that $G$ is a locally compact Hausdorff
groupoid with a topological basis consisting of bisections. A bisection for
$G$ is a subset (usually assumed to be open) $U\subseteq G$ such that the
restrictions $s:U\to s(U)$, $r:U\to r(U)$ are homeomorphisms. The set of all
open bisections of $G$ will be denoted $\operatorname{Bis}(G)$.

A _Fell bundle over the groupoid $G$_ is (an upper-semicontinuous) Banach
bundle $\mathcal{B}=\bigsqcup_{\gamma\in G}B_{\gamma}$ equipped with a
continuous involution ${}^{*}:\mathcal{B}\to\mathcal{B}$ and a continuous
multiplication ${\cdot}:\\{(a,b)\in\mathcal{B}\times\mathcal{B}:a\in
B_{\gamma_{1}},\ b\in B_{\gamma_{2}},\ (\gamma_{1},\gamma_{2})\in
G^{(2)}\\}\to\mathcal{B}$, that satisfy the standard set of axioms, for
details see, for instance, [BuE2, Section 2] or [Kum2], [Tak]. Accordingly,
this structure turns the fibres $B_{x}$ for $x\in X$ into $C^{*}$-algebras and
the fibres $B_{\gamma}$ for $\gamma\in G$ into Hilbert
$B_{r(\gamma)}$-$B_{s(\gamma)}$-bimodules, such that the multiplication map
yields isometric Hilbert bimodule maps
$B_{\gamma_{1}}\otimes_{B_{s(\gamma_{1})}}B_{\gamma_{2}}\to
B_{\gamma_{1}\gamma_{2}}$ for all $(\gamma_{1},\gamma_{2})\in G^{(2)}$. If
these maps are surjective, the Fell bundle $\mathcal{B}$ is called
_saturated_. This is equivalent to assuming that all Hilbert
$B_{s(\gamma)}$-modules $B_{\gamma}$ are _full_ , i.e. for each $\gamma\in G$
the linear span of $\langle
B_{\gamma},B_{\gamma}\rangle_{B_{s(\gamma)}}:=\\{\langle\xi,\eta\rangle_{B_{s(\gamma)}}:\xi,\eta\in
B_{\gamma}\\}$ is dense in $B_{s(\gamma)}$. Obviously, all Fell line bundles
are saturated.

For every Fell bundle $\mathcal{B}=\bigsqcup_{\gamma\in G}B_{\gamma}$ its
restriction $\mathcal{B}|_{X}:=\bigsqcup_{x\in X}B_{x}$ to the unit space is
an upper semicontinuous $C^{*}$-bundle and thus $A:=C_{0}(\mathcal{B}|_{X})$
is a _$C_{0}(X)$ -$C^{*}$-algebra_, i.e. $A$ is a $C^{*}$-algebra and a
$C_{0}(X)$-module with the module map taking values in $ZM(A)$ \- the center
of the multiplier algebra. Every $C_{0}(X)$-$C^{*}$-algebra $A$ arises as the
algebra of continuous sections for some $C^{*}$-bundle $\mathcal{B}|_{X}$, cf.
[Wil, Appendix C]; further $A$ is called a _continuous
$C_{0}(X)$-$C^{*}$-algebra_ if the corresponding bundle is continuous.

(Twisted) groupoid actions on (continuous) $C_{0}(X)$-$C^{*}$-algebras were
introduced in [Ren2].

######  Definition 2.1.

A _groupoid $C^{*}$-dynamical system_ is a triple $(\mathcal{A},G,\alpha)$
where $\mathcal{A}=\bigsqcup_{x\in X}A_{x}$ is an upper semicontinuous
$C^{*}$-bundle, $G$ is a topological groupoid with unit space $X$ and $\alpha$
is a _continuous groupoid homomorphism_ , i.e. a family
$\\{\alpha_{\gamma}\\}_{\gamma\in G}$ of $*$-isomorphisms
$\alpha_{\gamma}:A_{s(\gamma)}\to A_{r(\gamma)}$, $\gamma\in G$, such that
$\alpha_{\gamma\eta}=\alpha_{\gamma}\circ\alpha_{\eta}$ for all
$(\gamma,\eta)\in G^{(2)}$ and the map $(\gamma,a)\to\alpha_{\gamma}(a)$ from
$G*_{s}\mathcal{A}:=\\{(\gamma,a)\in G\times\mathcal{A}:a\in A_{s(\gamma)}\\}$
to $\mathcal{A}$ is continuous.

######  Remark 2.2.

Sometimes groupoid $C^{*}$-dynamical systems are written as $(A,G,\alpha)$
where $A=C_{0}(\mathcal{A})$ is the $C_{0}(X)$-$C^{*}$-algebra corresponding
to the $C^{*}$-bundle $\mathcal{A}$. If $A$ is (isomorphic to) $C_{0}(X)$,
that is $\mathcal{A}=X\times\mathbb{C}$ is a trivial line bundle, then there
is a unique action $\alpha$ of $G$ on $\mathcal{A}$ consisting of identities
on $\mathbb{C}$. Thus the triple $(C_{0}(X),G,\alpha)$ might be then
identified with $G$. Note also that for every groupoid $C^{*}$-dynamical
system the ‘unit’ automorphisms $\alpha_{x}\in\textup{Aut}(A_{x})$ for
$x\in{G^{(0)}}$ are all trivial.

######  Example 2.3 ([Kum2]).

Every groupoid $C^{*}$-dynamical system $(\mathcal{A},G,\alpha)$ yields a
saturated Fell bundle $\mathcal{A}\rtimes_{\alpha}G$ over $G$ that extends the
$C^{*}$-bundle $\mathcal{A}$. As a topological space
$\mathcal{A}\rtimes_{\alpha}G$ is the pullback bundle subspace
$\mathcal{A}*_{r}G:=\\{(a,\gamma)\in\mathcal{A}\times G:a\in
A_{r(\gamma)},\gamma\in G\\}$. We may also write
$\mathcal{A}\rtimes_{\alpha}G=\bigsqcup_{\gamma\in G}B_{\gamma}$ where
$B_{\gamma}:=A_{r(\gamma)}$ for $\gamma\in G$. The multiplication maps and
involutions are defined by $B_{\eta}\times B_{\gamma}\to B_{\eta\gamma}$,
$(a,b)\mapsto a\cdot\alpha_{\eta}(b)$, and $B_{\gamma}\to B_{\gamma^{-1}}$,
$a\mapsto\alpha_{\gamma}^{-1}(a^{*})$.

###  2.3. Twisted groupoid $C^{*}$-dynamical systems vs twisted inverse
semigroup actions

In most of this paper (with a brief exception in Section 6) we will only
consider Kumjian twists ([Kum1]), i.e. twists by the trivial circle bundle
$X\times\mathbb{T}$. Thus by a _twist of a groupoid_ $G$ we mean a topological
groupoid $\Sigma$ such that we have a central groupoid extension

$X\times\mathbb{T}\stackrel{{\scriptstyle
i}}{{\hookrightarrow}}\Sigma\stackrel{{\scriptstyle\pi}}{{\twoheadrightarrow}}G,$

where $X\times\mathbb{T}$ is the group bundle over $X$. More specifically, cf.
[Sim, Definition 5.1.1], $i$ and $\pi$ are groupoid homomorphisms that
restrict to the identity on the unit spaces $X=\Sigma^{(0)}\cong G^{(0)}$, $i$
is injective, $\pi$ is surjective and $\pi^{-1}(X)=i(X\times\mathbb{T})$. So
suppressing the map $i$ we may assume that $X\times\mathbb{T}$ is a
subgroupoid of $\Sigma$. This subgroupoid is assumed to be central in the
sense that for every $\sigma\in\Sigma$ and $z\in\mathbb{T}$ we have
$(r(\sigma),z)\sigma=\sigma(s(\sigma),z)$; we denote this common element by
$z\cdot\sigma$. Moreover, one also requires that $\Sigma$ is a locally trivial
$G$-bundle in the sense that every point $\gamma\in G$ has a bisection
neighbourhood $U$ on which there exists a continuous section $c:U\to\Sigma$
satisfying $\pi\circ c=id_{U}$, and such that the map
$U\times\mathbb{T}\ni(\gamma,z)\mapsto z\cdot c(\gamma)\in\pi^{-1}(U)$ is a
homeomorphism $U\times\mathbb{T}\cong\pi^{-1}(U)$. Thus $\Sigma$ is a
principal $\mathbb{T}$-space and $\Sigma/\mathbb{T}\cong G$. We will also call
the pair $(G,\Sigma)$ as above simply a _twisted groupoid_.

Renault [Ren2], [Ren3] generalised Green’s notion of twisted group actions
[Gre] to groupoid actions as follows. We denote by $UM(A)$ the group of
unitaries in the multiplier algebra of a $C^{*}$-algebra $A$.

######  Definition 2.4.

A _twisted groupoid $C^{*}$-dynamical system_, usually denoted by a quadruple
$(\mathcal{A},G,\Sigma,\alpha)$, consists of a twisted groupoid $(G,\Sigma)$
and a groupoid $C^{*}$-dynamical system $(\mathcal{A},\Sigma,\alpha)$ that are
combined by a “twisting” continuous groupoid homomorphism
$u:X\times\mathbb{T}\to\bigsqcup_{x\in X}UM(A_{x})$. This means that putting
$u_{x}(z):=u(x,z)$, $x\in X$, $z\in\mathbb{T}$, we have a family
$\\{u_{x}\\}_{x\in X}$ of group homomorphisms $u_{x}:\mathbb{T}\to UM(A_{x})$,
$x\in X$, such that

  1. (1)

the map $\mathcal{A}\times\mathbb{T}\to\mathcal{A}$, where
$A_{x}\times\mathbb{T}\ni(a_{x},z)\to a_{x}u_{x}(z)\in A_{x}$, is continuous;

  2. (2)

$\alpha_{(x,z)}:A_{x}\to A_{x}$ is given by $u_{x}(z)(\cdot)u_{x}(z)^{-1}$,
for every $(x,z)\in X\times\mathbb{T}$;

  3. (3)

$u_{r(\sigma)}(z)=\alpha_{\sigma}(u_{s(\sigma)}(z))$ for every
$\sigma\in\Sigma$ and $z\in\mathbb{T}$; here
$\alpha_{\sigma}:M(A_{s(\sigma)})\to M(A_{r(\sigma)})$ is the unique strictly
continuous extension of $\alpha_{\sigma}:A_{s(\sigma)}\to A_{r(\sigma)}$.

######  Example 2.5.

Every twisted groupoid $C^{*}$-dynamical system
$(\mathcal{A},G,\Sigma,\alpha)$ yields a saturated Fell bundle over $G$, cf.
[Lal, Section 3.1]. We define it as the quotient
$\mathcal{A}\rtimes_{\alpha}\Sigma/\mathbb{T}$ of the Fell bundle
$\mathcal{A}\rtimes_{\alpha}\Sigma$ associated to
$(\mathcal{A},\Sigma,\alpha)$ as in Example 2.3, by the $\mathbb{T}$-action
defined by

(2.1)
$z\cdot(a,\sigma):=(au_{r(\sigma)}(z)^{-1},z\cdot\sigma),\qquad(a,\sigma)\in\mathcal{A}\rtimes_{\alpha}\Sigma,z\in\mathbb{T}.$

We denote by $[a,\sigma]$ the corresponding class - the $\mathbb{T}$-orbit of
$(a,\sigma)$. We also write $\dot{\sigma}$ for the image of $\sigma\in\Sigma$
in $G=\Sigma/\mathbb{T}$. Then the fibers of the corresponding bundle are
$B_{\dot{\sigma}}=\\{[a,\sigma]:a\in A_{r(\sigma)}\\}$. The multiplication
maps and involutions on $\mathcal{A}\rtimes_{\alpha}\Sigma$ factor through to
well defined operations on the quotient
$\mathcal{A}\rtimes_{\alpha}\Sigma/\mathbb{T}$. So the formulas
$[a,\sigma]\cdot[b,\tau]:=[a\alpha_{\sigma}(b),\sigma\tau]$, and
$[a,\sigma]^{*}:=[\alpha_{\sigma}^{-1}(a^{*}),\sigma^{-1}]$, $a\in
A_{r(\sigma)}$, $b\in A_{r(\tau)}$, yield the structure of the Fell bundle on
$\mathcal{B}:=\bigsqcup_{\gamma\in G}B_{\gamma}$. We will refer to it as the
_Fell bundle associated to the twisted groupoid $C^{*}$-dynamical system_
$(\mathcal{A},G,\Sigma,\alpha)$.

######  Remark 2.6.

If $\mathcal{A}=X\times\mathbb{C}$ is a trivial line bundle (and $u_{x}$ are
identities), then the construction in Example 2.5 specializes to the standard
construction of a continuous line bundle $\mathcal{L}=\bigsqcup_{\gamma\in
G}L_{\gamma}$ from a twisted groupoid $(G,\Sigma)$, see [Kum2, 2.5.iv] or
[Ren5]. This gives a part of the well known equivalence between continuous
Fell line bundles over $G$ and twists of $G$. For the other part, recall that
if $\mathcal{L}=\bigsqcup_{\gamma\in G}L_{\gamma}$ is a continuous Fell line
bundle then the circle bundle $\Sigma:=\\{\xi\in\mathcal{L}:|\xi|=1\\}$ with
the topology and multiplication inherited from $\mathcal{L}$ is a twist of
$G$. Note that in this case both the line bundle $\mathcal{L}$ and the
extension $\mathbb{T}\times X\hookrightarrow\Sigma\twoheadrightarrow G$ are
locally trivial.

Busby and Smith [BuS] introduced twisted actions by groups that involve
cocycles rather than group extensions. As shown by Packer and Raeburn [PaR]
this formalism covers Green’s approach to twisted crossed products. Also the
original Renault’s twisted groupoid $C^{*}$-algebras [Ren1] were defined in
the spirit of Busby and Smith (dating back in fact already to Zeller-Meier
[Z-M]). Then Renault switched to Green’s approach in [Ren2], [Ren3], as it was
not clear whether the Packer-Raeburn result generalises to groupoids. This
question is raised, for instance, in [Sim, Remark 5.1.6], and we were informed
by Tristan Bice that the answer to it is negative. Bice’s example of a
groupoid twist that does admit a continuous global section comes from the
Pedersen-Petersen $C^{*}$-Algebras, see [BiF]. The relevant $C^{*}$-algebra
$\mathfrak{B}_{1}$ is a cross-sectional algebra of a continuous line bundle
over the principal groupoid $G=\mathbb{C}P^{1}\times\\{0,1\\}^{2}$ whose every
continuous section has to vanish by the Borsuk-Ulam Theorem.

Nevertheless, we will show that Kumjian-Renault twisted actions are covered by
a version of Busby-Smith twisted actions adapted to inverse semigroups by Buss
and Exel [BuE1].

######  Definition 2.7.

A _twisted action of an inverse semigroup_ $S$ on a $C^{*}$-algebra $A$
consists of partial automorphisms $\beta_{t}\colon D_{t^{*}}\to D_{t}$ of $A$
for $t\in S$ – that is, $D_{t}$ is an ideal in $A$ and $\beta_{t}$ is a
$*$-isomorphism – and unitary multipliers $\omega(t,u)\in UM(D_{tu})$ for
$t,u\in S$, such that $D_{1}=A$ and the following conditions hold for
$r,t,u\in S$ and $e,f\in E(S)$, where $E(S)$ is the set of units in $S$:

  1. (T1)

$D_{(rt)^{*}}=D_{t^{*}}\cap\beta_{t}^{-1}(D_{r^{*}})$ and
$\beta_{r}\circ\beta_{t}=Ad_{\omega(r,t)}\beta_{rt}$;

  2. (T2)

$\beta_{r}\big{(}a\omega(t,u)\big{)}\omega(r,tu)=\beta_{r}(a)\omega(r,t)\omega(rt,u)$
for $a\in D_{r^{*}}\cap D_{tu}$;

  3. (T3)

$\omega(e,f)=1_{ef}$ and $\omega(r,r^{*}r)=\omega(rr^{*},r)=1_{r}$, where
$1_{r}$ is the unit of $M(D_{r})$;

  4. (T4)

$\omega(t^{*},e)\omega(t^{*}e,t)a=\omega(t^{*},t)a$ for all $a\in
D_{t^{*}et}$.

Given an étale groupoid $G$, the set of its open bisections
$\operatorname{Bis}(G)$ forms naturally an inverse semigroup with operations

$U\cdot V:=\\{\gamma\eta:\gamma\in U,\eta\in V,s(\gamma)=r(\eta)\\},\qquad
U^{*}=\\{\gamma^{-1}:\gamma\in U\\},\qquad U,V\in\operatorname{Bis}(G).$

An inverse semigroup $S\subseteq\operatorname{Bis}(G)$ defines an inverse
semigroup action $\\{h_{U}\\}_{U\in S}$ on the unit space $X=G^{(0)}$, where
$h_{U}:=r\circ s|_{U}^{-1}:s(U)\to r(U)$. The transformation groupoid
$X\rtimes_{h}S$ for this action is naturally isomorphic to $G$ if and only if
the semigroup $S$ is _wide_ , that is $\bigcup S=G$ and $U\cap V$ is a union
of bisections in $S$ for all $U,V\in S$; the corresponding notions and facts
mentioned above are described for example in [KwM1, Section 2].

Note that the conclusion of the next lemma does not depend on the choice of
the section $c_{U}$; the resulting actions for different choices of sections
need not explicitly coincide, but the resulting crossed products are
isomorphic (see Theorem 2.16 below).

######  Lemma 2.8.

Let $(\mathcal{A},G,\Sigma,\alpha)$ be a twisted groupoid $C^{*}$-dynamical
system with $G$ étale. Let $S$ be the family of open bisections $U$ of $G$ on
which the $\mathbb{T}$-bundle $\pi:\Sigma\to G$ is trivial, and for each $U\in
S$ choose a continuous section $c_{U}:U\to\Sigma$ that induces the
homeomorphism $U\times\mathbb{T}\cong\pi^{-1}(U)$. If $U\subseteq X$ we choose
$c_{U}$ to be given by $c_{U}(x):=(x,1)\subseteq
X\times\mathbb{T}\subseteq\Sigma$, for $x\in U$. Then the following statements
hold.

  1. (1)

$S$ is a wide unital inverse subsemigroup of $\operatorname{Bis}(G)$.

  2. (2)

For each $U\in S$ we may treat $D_{U}:=C_{0}(\mathcal{A}|_{r(U)})$ as an ideal
in $A:=C_{0}(\mathcal{A})$. The formula

$\beta_{U}(a)(r(\gamma))=\alpha_{c_{U}(\gamma)}(a(s(\gamma)),\qquad a\in
D_{U^{*}}=C_{0}(\mathcal{A}|_{s(U)}),\,\,\gamma\in U,$

gives a well defined $*$-isomorphism $\beta_{U}:D_{U^{*}}\to D_{U}$.

  3. (3)

For every $U$, $V\in S$ the formula

$\omega(U,V)(r(\gamma\eta))=u(c_{U}(\gamma)c_{V}(\eta)c_{UV}(\gamma\eta)^{-1}),\qquad\gamma\in
U,\eta\in V$

gives a well defined element in $UM(D_{UV})$.

The quadruple $(A,S,\beta,\omega)$ described above yields a twisted action of
an inverse semigroup in the sense of Definition 2.7.

###### Proof.

(1) Let $U,V\in S$ and $c_{U}:U\to\Sigma$ and $c_{V}:V\to\Sigma$ be continuous
sections that trivialise $\pi^{-1}(U)$ and $\pi^{-1}(V)$. Then putting
$d_{U^{*}}(\gamma):=c_{U}(\gamma^{-1})^{-1}$, for $\gamma\in U^{*}$, and
$d_{UV}(\gamma\eta):=c_{U}(\gamma)c_{V}(\eta)$ for $\gamma\in U,\eta\in V$, we
get continuous sections that trivializes $\pi^{-1}(U^{*})$ and $\pi^{-1}(UV)$,
respectively. Thus $S$ is an inverse subsemigroup of $\operatorname{Bis}(G)$.
It is unital because $X\in S$ and it is wide as it is closed under inclusions
and every $\gamma\in G$ has a neighbourhood $U\in S$.

(2) Identifying sections of $\mathcal{A}|_{r(U)}$ with sections of
$\mathcal{A}$ that vanish outside the open set $r(U)$ we have that
$D_{U}:=C_{0}(\mathcal{A}|_{r(U)})\subseteq C_{0}(\mathcal{A})=A$ is an ideal.
Using continuity of the map
$\Sigma\,*_{s}\mathcal{A}\ni(\sigma,a)\to\alpha_{\sigma}(a)\in\mathcal{A}$ and
that $r:U\to r(U)$, $s:U\to s(U)$ and $c_{U}:U\to c_{U}(U)\subseteq\Sigma$ are
homeomorphisms, we see that the formula
$\beta_{U}(f)(x):=\alpha_{c_{U}(r|_{U}^{-1}(x))}(f(s(r|_{U}^{-1}(x)))$, for
$f\in C_{0}(\mathcal{A}|_{s(U)})$, $x\in r(U)$, gives a well defined element
$\beta_{U}(f)$ in $C_{0}(\mathcal{A}|_{r(U)})$. This is exactly the formula in
(2). The map $\beta_{U}:D_{U^{*}}\to D_{U}$ is a $*$-isomorphism with the
inverse given by
$\beta_{U}^{-1}(f)(s(\gamma))=\alpha_{c_{U}(\gamma)^{-1}}(f(r(\gamma)))$ for
$f\in D_{U}=C_{0}(\mathcal{A}|_{r(U)})$, $\gamma\in U$.

(3) Note first that we have $UV\ni\gamma\eta\mapsto
c_{U}(\gamma)c_{V}(\eta)c_{UV}(\gamma\eta)^{-1}\in
r(UV)\times\mathbb{T}\subseteq X\times\mathbb{T}$ because
$\pi(c_{U}(\gamma)c_{V}(\eta)c_{UV}(\gamma\eta)^{-1})=\pi(r(\gamma))$.
Moreover this function is continuous, as so is the natural map
$UV\ni\gamma\eta\to(\gamma,\eta)\in U\times V$. Since $r:UV\to r(UV)$ is a
homeomorphism and $u:X\times\mathbb{T}\to\bigsqcup_{x\in X}UM(A_{x})$ is
continuous, we get that for every $a\in D_{UV}=C_{0}(\mathcal{A}|_{r(UV)})$
the formula

$[\omega(U,V)a](r(\gamma\eta))=u(c_{U}(\gamma)c_{V}(\eta)c_{UV}(\gamma\eta)^{-1})a(r(\gamma\eta))$

defines an element of $C_{0}(\mathcal{A}|_{r(UV)})$. Now one readily infers
that $\omega(U,V)$ is a multiplier of $C_{0}(\mathcal{A}|_{r(UV)})$ with the
adjoint given by
$\omega(U,V)^{*}=u(c_{U}(\gamma)c_{V}(\eta)c_{UV}(\gamma\eta)^{-1})^{*}$.
Hence $\omega(U,V)\in UM(D_{UV})$. Note that in terms of the notation
introduced in Definition 2.4 we have ($\gamma\in U,\eta\in V,\gamma\eta\in
UV$)

$\omega(U,V)(r(\gamma\eta))=u_{r(\gamma)}(p(c_{U}(\gamma)c_{V}(\eta)c_{UV}(\gamma\eta)^{-1})),$

where $p:X\times\mathbb{T}\to\mathbb{T}$ is the projection onto the second
coordinate. Now we need to check conditions (T1)-(T4) in Definition 2.7. To
simplify the notation in what follows we will sometimes write $\omega_{U,V}$
instead of $\omega(U,V)$. Choose $\gamma\in U\in S,\eta\in V\in S$ so that
$\gamma\eta\in UV$. Then for $x:=r(\gamma\eta)=r(\gamma)$ and $a\in
D_{(UV)^{*}}$, using Definition 2.4 (2), we have

$\displaystyle\left((Ad_{\omega_{U,V}}\circ\beta_{UV})(a)\right)(x)$
$\displaystyle=\alpha_{c_{U}(\gamma)c_{V}(\eta)c_{UV}(\gamma\eta)^{-1}}(\alpha_{c_{UV}(\gamma\eta)}(a(x))$
$\displaystyle=\alpha_{c_{U}(\gamma)c_{V}(\eta)}(a(x))=\left((\beta_{U}\circ\beta_{V})(a)\right)(x).$

This proves (T1).

Now take $a\in D_{Z^{*}}\cap D_{UV}=C_{0}(\mathcal{A}|_{s(Z)\cap r(UV)})$
where $U,V,Z\in S$. Take any $x\in r(ZUV)$ and choose $\gamma\in U$, $\eta\in
V$ and $\sigma\in Z$ so that $x=r(\sigma\gamma\eta)$. Denoting by $p$ again
the projection onto the second coordinate and using Definition 2.4(3) we get

$\displaystyle\beta_{Z}\Big{(}a\cdot\omega_{U,V}\Big{)}(x)$
$\displaystyle=\alpha_{c_{Z}(\sigma)}\Big{(}a(s(\sigma))u(c_{U}(\gamma)c_{V}(\eta)c_{UV}(\gamma\eta)^{-1})\Big{)}$
$\displaystyle=\alpha_{c_{Z}(\sigma)}\left(a(s(\sigma))\cdot
u_{r(\gamma)}\Big{(}p(c_{U}(\gamma)c_{V}(\eta)c_{UV}(\gamma\eta)^{-1})\Big{)}\right)$
$\displaystyle=\beta_{Z}(a)(x)\cdot
u_{r(\sigma)}\Big{(}p(c_{U}(\gamma)c_{V}(\eta)c_{UV}(\gamma\eta)^{-1})\Big{)}.$

Using that the extension is central, for the pointwise muliplication on
$r(ZUV)$ we have

$\displaystyle p(c_{U}c_{V}c_{UV}^{-1})p(c_{Z}c_{UV}c_{ZUV}^{-1})$
$\displaystyle=p(c_{Z}p(c_{U}c_{V}c_{UV}^{-1})c_{UV}c_{ZUV}^{-1})=p(c_{Z}c_{U}c_{V}c_{UV}^{-1}c_{UV}c_{ZUV}^{-1})$
$\displaystyle=p(c_{Z}c_{U}c_{V}c_{ZUV}^{-1})=p(c_{Z}c_{U}(c_{ZU}^{-1}c_{ZU})c_{V}c_{ZUV}^{-1})$
$\displaystyle=p(c_{Z}c_{U}c_{ZU}^{-1})p(c_{ZU}c_{V}c_{ZUV}^{-1}).$

Combining the above and using that
$\omega_{Z,UV}(x)=u_{r(\sigma)}(p(c_{Z}(\sigma)c_{UV}(\gamma\eta)c_{ZUV}(\sigma\gamma\eta)^{-1})$,
we get

$\displaystyle\beta_{Z}\Big{(}a\cdot\omega_{U,V}\Big{)}\cdot\omega_{Z,UV}$
$\displaystyle=\beta_{Z}(a)\cdot
u\Big{(}p(c_{U}(\gamma)c_{V}(\eta)c_{UV}(\gamma\eta)^{-1})c_{Z}(\sigma)c_{UV}(\gamma\eta)c_{ZUV}(\sigma\gamma\eta)^{-1}\Big{)}$
$\displaystyle=\beta_{Z}(a)\cdot
u\big{(}c_{Z}(\sigma)c_{U}(\gamma)c_{ZU}(\sigma\gamma)^{-1}\big{)}u\big{(}c_{ZU}(\sigma\gamma)c_{V}(\eta)c_{ZUV}(\sigma\gamma\eta)^{-1}\big{)}$
$\displaystyle=\beta_{Z}(a)\omega_{Z,U}\omega_{ZU,V}$

on $r(ZUV)$. This proves (T2).

Idempotents in $S$ correspond to open subsets of $X$. Thus axiom (T3) follows
readily from our choice of $c_{U}$ for $U\subseteq X$. To check (T4) let
$U,V\in S$ where $V\subseteq X$, so that $c_{V}(V)=V\times\\{1\\}\subseteq
V\times\mathbb{T}\subseteq\Sigma$. Then on the set $r(U^{*}VU)$ we have
$\omega(U^{*},V)\omega(U^{*}V,U)=u(c_{U^{*}}c_{V}c_{U^{*}V}^{-1})u(c_{U^{*}V}c_{U}c_{U^{*}VU}^{-1})=u(c_{U^{*}}c_{U}c_{U^{*}VU}^{-1})=\omega(U^{*},U)$,
where we for example use the fact that $U^{*}VU$ is an idempotent, so that
$c_{U^{*}VU}=c_{U^{*}U}|_{U^{*}VU}$. ∎

_Saturated Fell bundles over inverse semigroups_ ([Exe], [BuE1], [BuE2]) are
equivalent to inverse semigroup actions by Hilbert bimodules ([BuM], [BEM]),
whose definition is relatively concise.

######  Definition 2.9 (Definition 4.7, [BuM]).

Let $S$ be a unital inverse semigroup and $A$ a $C^{*}$-algebra. _An action of
$S$ on $A$ by Hilbert bimodules_ is a semigroup $\mathcal{C}=\bigsqcup_{t\in
S}C_{t}$ where each fiber $C_{t}$ is a Hilbert $A$-bimodule, $C_{1}=A$ is a
trivial bimodule, and the semigroup product in $\mathcal{C}$ induces
isomorphisms $C_{t}\otimes C_{s}\cong C_{ts}$ of Hilbert $A$-bimodules, for
all $t,s\in S$.

######  Remark 2.10.

Any inverse semigroup action by Hilbert bimodules $\mathcal{C}=\bigsqcup_{t\in
S}C_{t}$ induces uniquely defined canonical involutions $C_{t}^{*}\to
C_{t^{*}}$, $x\mapsto x^{*}$, and inclusion maps $j_{s,t}\colon C_{t}\to
C_{s}$ for $t,s\in S$, $t\leq s$ that satisfy the conditions required for a
saturated Fell bundle, as studied in [Exe], [BuE1] and [BuE2], see [BuM,
Theorem 4.8]. Note further that the structure of $\mathcal{C}=\bigsqcup_{t\in
S}C_{t}$ is also uniquely determined by the multiplication and involution in
$\mathcal{C}$ (these operations determine the Hilbert $A$-module operations on
each fiber $C_{t}$).

######  Example 2.11.

Any twisted action $(A,S,\beta,\omega)$ of an inverse semigroup $S$ on a
$C^{*}$-algebra $A$ yields a natural action of $S$ on $A$ by Hilbert bimodules
given by a family of the standard bimodules associated to partial
automorphisms $\\{\beta_{t}\\}_{t\in S}$ of $A$. Namely, we put
$\mathcal{C}:=\bigsqcup_{t\in S}D_{t}$, where each $D_{t}$ is a Hilbert
$A$-bimodule with operations

$a\cdot\xi:=a\xi,\qquad\xi\cdot
a:=\beta_{t}(\beta^{-1}_{t}(\xi)a),\qquad\langle\xi,\psi\rangle_{A}:=\beta_{t}^{-1}(\xi^{*}\psi).$

where $a\in A$, $\xi,\psi\in D_{t}$. Multiplication between different fibers
is given by

$D_{s}\times
D_{t}\ni(\xi_{s},\xi_{t})\longmapsto\beta_{s}(\beta^{-1}_{s}(\xi_{s})\xi_{t})\omega(s,t)\in
D_{st}.$

The induced involution is given by
$D_{s}\ni\xi_{s}\longmapsto\beta_{s}^{-1}(\xi_{s}^{*})\omega(s^{*},s)^{*}\in
D_{s^{*}},$ cf. [BuE1, page 250]. In particular, every twisted groupoid
$C^{*}$-dynamical system $(\mathcal{A},G,\Sigma,\alpha)$ yields a saturated
Fell bundle $\bigsqcup_{U\in S}C_{0}(\mathcal{A}|_{r(U)})$ with operations
coming from the twisted action $(A,S,\beta,\omega)$ of an inverse semigroup
$S$ on $A:=C_{0}(\mathcal{A})$ as described in Lemma 2.8.

###  2.4. The associated convolution $C^{*}$-algebras

Let $\mathcal{B}=\bigsqcup_{\gamma\in G}B_{\gamma}$ be a Fell bundle over an
étale locally compact Hausdorff groupoid $G$. The space of compactly supported
continuous sections $C_{c}(\mathcal{B})$ is naturally a $*$-algebra with
operations

(2.2) $(f*g)(\gamma):=\sum_{r(\eta)=r(\gamma)}f(\eta)\cdot
g(\eta^{-1}\cdot\gamma),\qquad(f^{*})(\gamma):=f(\gamma^{-1})^{*},$

$f,g\in C_{c}(\mathcal{B})$, $\gamma\in G$. The _full section $C^{*}$-algebra_
$C^{*}(\mathcal{B})$ is the maximal $C^{*}$-completion of the $*$-algebra
$C_{c}(\mathcal{B})$. Let $X:=G^{(0)}$ be the unit space. The
$C_{0}(X)$-$C^{*}$-algebra $A:=C_{0}(\mathcal{B}|_{X})$ embeds naturally into
$C^{*}(\mathcal{B})$, and the restriction map $C_{c}(\mathcal{B})\ni f\to
f|_{X}\in A$ extends to a conditional expectation $C^{*}(\mathcal{B})\to A$.
The _reduced section $C^{*}$-algebra_ $C_{r}^{*}(\mathcal{B})$ can be defined
as the completion of the $*$-algebra $C_{c}(\mathcal{B})$ with respect to the
minimal $C^{*}$-norm such that the restriction map $C_{c}(\mathcal{B})\to A$
is contractive. Thus the restriction extends to a faithful conditional
expectation $E:C_{r}^{*}(\mathcal{B})\to A$. Note that if this conditional
expectation is tracial, then $A$ is commutative; but this condition is itself
not sufficient. We will also need an explicit faithful representation of
$C_{r}^{*}(\mathcal{B})$, as described for example in [Kum2]. Given $x\in
G^{(0)}$ put $G_{x}:=\\{\gamma\in G:s(\gamma)=x\\}$ and consider the Hilbert
$B_{x}$-module $V_{x}:=\bigoplus_{\gamma\in G_{x}}B_{\gamma}$ (recall that the
scalar product comes from the bundle operations: if $a,b\in B_{\gamma}$, then
$a^{*}b\in B_{\gamma^{-1}}B_{\gamma}\subseteq
B_{\gamma^{-1}\gamma}=B_{s(\gamma)}$). Then we have a representation
$\pi_{x}:C_{r}^{*}(\mathcal{B})\to B(V_{x})$ determined by the formula

(2.3) $\pi_{x}(f)(\bigoplus_{\gamma\in
G_{x}}\xi_{\gamma})=\bigoplus_{\gamma\in G_{x}}\left(\sum_{\beta\in
G_{x}}f(\gamma\beta^{-1})\xi_{\beta}\right),$

where $f\in C_{c}(\mathcal{B})$, $\xi_{\gamma}\in B_{\gamma}$ for each
$\gamma\in G_{x}$, and the series $\sum_{\gamma\in
G_{x}}\langle\xi_{\gamma},\xi_{\gamma}\rangle$ converges in $B_{x}$. The
direct sum of the representations $\pi_{x}$ over $x\in G^{(0)}$ is faithful.

######  Remark 2.12.

When equipped with the supremum norm, $C_{0}(\mathcal{B})$ is a Banach space.
The embedding $C_{c}(\mathcal{B})\to C_{0}(\mathcal{B})$ extends uniquely to
an injective and contractive linear map $C_{r}^{*}(\mathcal{B})\to
C_{0}(\mathcal{B})$ which allows us to assume the identification

$C_{r}^{*}(\mathcal{B})\subseteq C_{0}(\mathcal{B}).$

Under this identification the $*$-algebraic operations in
$C_{r}^{*}(\mathcal{B})$ are still given by (2.2), see [KwM1, Proposition
7.10]. Moreover, for any $C^{*}$-completion $B=\overline{C_{c}(\mathcal{B})}$
in a $C^{*}$-norm $\|\cdot\|$ that extends the supremum norm on
$C_{c}(\mathcal{B}|_{{G^{(0)}}})$ we have natural isometric embeddings
$C_{0}(\mathcal{B}|_{U})\hookrightarrow B$ for all open bisections $U\subseteq
G$. Indeed, if $f\in C_{c}(\mathcal{B}|_{U})$, then
$\|f\|^{2}=\|f^{*}*f\|_{C_{0}(\mathcal{B}|_{{G^{(0)}}})}=\max_{x\in{G^{(0)}}}\|(f^{*}*f)(x)\|=\max_{\gamma\in
U}\|f^{*}(\gamma)f(\gamma)\|=\max_{\gamma\in U}\|f(\gamma)\|^{2}.$

######  Example 2.13.

Let $(\mathcal{A},G,\alpha)$ be a groupoid $C^{*}$-dynamical system and let
$\mathcal{A}\rtimes_{\alpha}G$ be the associated Fell bundle, cf. Example 2.3.
The $*$-algebra $C_{c}(\mathcal{A}\rtimes_{\alpha}G)$ consists of compactly
supported continuous sections of $\mathcal{A}*_{r}G=\bigsqcup_{\gamma\in
G}A_{r(\gamma)}$ with operations given by

$(f*g)(\gamma):=\sum_{r(\eta)=r(\gamma)}f(\eta)\cdot\alpha_{\eta}\big{(}g(\eta^{-1}\cdot\gamma)\big{)},\qquad(f^{*})(\gamma):=\alpha_{\gamma}(f(\gamma^{-1}))^{*},\;\;\;\gamma\in
G.$

The _full_ and _reduced crossed product_ , denoted by
$C^{*}(\mathcal{A},G,\alpha)$ and $C^{*}_{r}(\mathcal{A},G,\alpha)$, is by
definition $C^{*}(\mathcal{A}\rtimes_{\alpha}G)$ and
$C^{*}_{r}(\mathcal{A}\rtimes_{\alpha}G)$, respectively.

Let now $(\mathcal{A},G,\Sigma,\alpha)$ be a twisted groupoid
$C^{*}$-dynamical system and let $\mathcal{B}$ be the associated Fell bundle
over $G$, cf. Example 2.5. By definition, the _full_ and _reduced crossed
product_ , denoted by $C^{*}(\mathcal{A},G,\Sigma,\alpha)$ and
$C^{*}_{r}(\mathcal{A},G,\Sigma,\alpha)$, is $C^{*}(\mathcal{B})$ and
$C^{*}_{r}(\mathcal{B})$, respectively. Using the fact that
$\pi^{-1}(X)=X\times\mathbb{T}\subseteq\Sigma$ is a trivial bundle, we get
that $\mathcal{A}\supseteq A_{x}\ni a\mapsto[a,(x,1)]\in
B_{x}\subseteq\mathcal{B}|_{X}$ is the isomorphism of $C^{*}$-bundles
$\mathcal{A}\cong\mathcal{B}|_{X}$. Therefore we may treat

$A:=C_{0}(\mathcal{A})\cong C_{0}(\mathcal{B}|_{X})$

as a $C^{*}$-subalgebra of both $C^{*}(\mathcal{A},G,\Sigma,\alpha)$ and
$C^{*}_{r}(\mathcal{A},G,\Sigma,\alpha)$. In particular, we have a faithful
conditional expectation $E:C^{*}_{r}(\mathcal{A},G,\Sigma,\alpha)\to A$. In a
similar manner we may identify the spaces of sections of $\mathcal{B}$
restricted to open bisections on which the twist $\Sigma\twoheadrightarrow G$
is trivial.

######  Lemma 2.14.

Let $(\mathcal{A},G,\Sigma,\alpha)$ be a twisted groupoid $C^{*}$-dynamical
system. Let $\mathcal{B}$ be the associated Fell bundle over $G$ (see Example
2.5), and let $(A,S,\beta,\omega)$ be the associated twisted action of an
inverse semigroup $S$ on $A:=C_{0}(\mathcal{A})$ described in Lemma 2.8. Put
$B:=C^{*}_{r}(\mathcal{A},G,\Sigma,\alpha)$. We have the following natural
actions of the inverse semigroup $S$ by Hilbert bimodules on $A$, which are
all naturally isomorphic:

  1. (1)

$\bigsqcup_{U\in S}B_{U}$ where
$B_{U}:=\overline{C_{c}(\mathcal{B}|_{U})}\subseteq B$ and all the operations
are inherited from $B$ (the closure is with respect to the supremum norm).

  2. (2)

$\bigsqcup_{U\in S}C_{0}(\mathcal{B}|_{U})$ with the multiplication
$C_{0}(\mathcal{B}|_{U})\times C_{0}(\mathcal{B}|_{V})\ni(f,g)\mapsto f*g\in
C_{0}(\mathcal{B}|_{UV})$ and the involution $C_{0}(\mathcal{B}|_{U})\ni
f\mapsto f^{*}\in C_{0}(\mathcal{B}|_{U^{*}})$ given by

$f*g\;(\gamma\eta):=f(\gamma)\cdot g(\eta),\qquad
f^{*}(\gamma^{-1})=f(\gamma)^{*},\quad\text{ for $\gamma\in U$ and $\eta\in
V$}.$

  3. (3)

$\bigsqcup_{U\in S}C_{0}(\mathcal{A}*_{r}G|_{U})$ with the multiplication
$C_{0}(\mathcal{A}*_{r}G|_{U})\times
C_{0}(\mathcal{A}*_{r}G|_{V})\ni(f,g)\mapsto f*g\in
C_{0}(\mathcal{A}*_{r}G|_{UV})$ and the involution
$C_{0}(\mathcal{A}*_{r}G|_{U})\ni f\mapsto f^{*}\in
C_{0}(\mathcal{A}*_{r}G|_{U^{*}})$ given by

$f*g\;(\gamma\eta):=f(\gamma)\alpha_{c_{U}(\gamma)}(g(\eta))u(c_{U}(\gamma)c_{V}(\eta)c_{UV}(\gamma\eta)^{-1}),$
$f^{*}(\gamma^{-1}):=\alpha_{c_{U}(\gamma)}^{-1}(f(\gamma)^{*})u(c_{U^{*}}(\gamma^{-1})c_{U}(\gamma))^{*},$

where $\gamma\in U$ and $\eta\in V$.

  4. (4)

$\bigsqcup_{U\in S}D_{U}$ associated to $(A,S,\beta,\omega)$ as in Example
2.11, so that the Hilbert $A$-bimodule structure on
$D_{U}=C_{0}(\mathcal{A}|_{r(U)})$ is given by

$a\cdot f:=af,\quad f\cdot
a:=\beta_{U}(\beta^{-1}_{U}(f)a),\quad{{}_{A}\langle}f,g\rangle:=fg^{*},\quad\langle
f,g\rangle_{A}:=\beta_{U}^{-1}(f^{*}g),$

and the multiplication $D_{U}\times
D_{V}\ni(f,g)\longmapsto\beta_{U}(\beta^{-1}_{U}(f)g)\omega(U,V)\in D_{UV}$.

The isomorphisms are given by $B_{U}\supseteq C_{c}(\mathcal{B}|_{U})\ni
f\stackrel{{\scriptstyle id}}{{\mapsto}}f\in C_{0}(\mathcal{B}|_{U})$,
$C_{0}(\mathcal{A}|_{r(U)})\ni f\mapsto f\circ r\in
C_{0}(\mathcal{A}*_{r}G|_{U})$ and $C_{c}(\mathcal{A}*_{r}G|_{U})\ni
f\mapsto[f]\in C_{c}(\mathcal{B}|_{U})\subseteq B_{U}$ where
$[f](\gamma):=[f(\gamma),c_{U}(\gamma)]$.

###### Proof.

For the isomorphism between (1) and (2) recall that the identity map
$B_{U}\supseteq C_{c}(\mathcal{B}|_{U})\ni f\stackrel{{\scriptstyle
id}}{{\mapsto}}f\in C_{0}(\mathcal{B}|_{U})$ is isometric and hence extends to
the isomorphism $B_{U}\cong C_{0}(\mathcal{B}|_{U})$, see Remark 2.12. That
this isomorphism is consistent with the multiplication and the involution is
obvious.

The isomorphism between the actions in (3) and (4) is straightforward, since
for each $U\in S$ the homeomorphism $r:U\to r(U)$ induces an isomorphism
$\mathcal{A}*_{r}G|_{U}\cong\mathcal{A}|_{r(U)}$ of bundles (recall also that
$c_{U}(x)=(x,1)$ for $U\subseteq X$, by definition). We will show below that
these actions are isomorphic to the action (2).

Let $U\in S$. Recall that $U\times\mathbb{T}\ni(\gamma,z)\mapsto z\cdot
c_{U}(\gamma)\in\pi^{-1}(U)$ is a homeomorphism
$U\times\mathbb{T}\cong\pi^{-1}(U)$. This homeomorphism transfers
$\mathcal{A}*_{r}\Sigma|_{\pi^{-1}(U)}$ into a bundle over $U\times\mathbb{T}$
whose quotient under the $\mathbb{T}$-action (2.1) is isomorphic to
$\mathcal{A}*_{r}G|_{U}$. Thus
$\mathcal{B}|_{U}=\mathcal{A}*_{r}\Sigma|_{\pi^{-1}(U)}/\mathbb{T}\cong\mathcal{A}*_{r}G|_{U}$
and the isomorphism between algebras presented in items (3) and (2) of the
lemma is given by $C_{c}(\mathcal{A}*_{r}G|_{U})\ni f\mapsto[f]\in
C_{c}(\mathcal{B}|_{U})\subseteq B_{U}$ where
$[f](\gamma):=[f(\gamma),c_{U}(\gamma)]$. In particular, for any $f\in
C_{c}(\mathcal{A}*_{r}G|_{U})$, $g\in C_{c}(\mathcal{A}*_{r}G|_{V})$,
$\gamma\in U$ and $\eta\in V$we have

$\displaystyle[f][g](\gamma\eta)$
$\displaystyle=[f](\gamma)[g](\eta)=[f(\gamma),c_{U}(\gamma)][g(\eta),c_{V}(\eta)]=[f(\gamma)\alpha_{c_{U}(\gamma)}(g(\eta)),c_{U}(\gamma)c_{V}(\eta)]$
$\displaystyle=[f(\gamma)\alpha_{c_{U}(\gamma)}(g(\eta))u(c_{UV}(\gamma\eta)c_{V}(\eta)^{-1}c_{U}(\gamma)^{-1})^{-1},c_{UV}(\gamma\eta)]$
$\displaystyle=[f(\gamma)\alpha_{c_{U}(\gamma)}(g(\eta))u(c_{U}(\gamma)c_{V}(\eta)c_{UV}(\gamma\eta)^{-1}),c_{UV}(\gamma\eta)]$
$\displaystyle=[f*g(\gamma\eta),c_{UV}(\gamma\eta)].$

Thus we have a semigroup isomorphism $\bigsqcup_{U\in
S}C_{c}(\mathcal{A}*_{r}G|_{U})\to C_{c}(\mathcal{B}|_{U})$. It preserves the
involution as we have

$\displaystyle[f]^{*}(\gamma^{-1})$
$\displaystyle=[f](\gamma)^{*}=[f(\gamma),c_{U}(\gamma)]^{*}=[\alpha_{c_{U}(\gamma)^{-1}}(f(\gamma)^{*}),c_{U}(\gamma)^{-1}]$
$\displaystyle=[\alpha_{c_{U}(\gamma)}^{-1}(f(\gamma)^{*}),\big{(}c_{U^{*}}(\gamma^{-1})c_{U}(\gamma)\big{)}^{-1}c_{U^{*}}(\gamma^{-1})]$
$\displaystyle=[\alpha_{c_{U}(\gamma)}^{-1}(f(\gamma)^{*})u(c_{U^{*}}(\gamma^{-1})c_{U}(\gamma))^{-1},c_{U^{*}}(\gamma^{-1})]$
$\displaystyle=[f^{*}](\gamma^{-1}).$

∎

######  Remark 2.15.

The last lemma could be equally well formulated with
$B=C^{*}(\mathcal{A},G,\Sigma,\alpha)$ or more generally with $B$ being any
quotient of $C^{*}(\mathcal{A},G,\Sigma,\alpha)$ by an ideal that intersects
$A$ trivially, cf. Remark 2.12.

If $\mathcal{C}=\bigsqcup_{t\in S}C_{t}$ is an action of an inverse semigroup
$S$ by Hilbert bimodules on $A=C_{1}$, then $\bigoplus_{t\in S}C_{t}$ with
operations from $\mathcal{C}$ is naturally a $*$-algebra and using the induced
inclusion maps one can define a quotient $*$-algebra of $\bigoplus_{t\in
S}C_{t}$ whose full completion and a completion allowing a faithul weak
conditional expectation are respectively the full and reduced $C^{*}$-algebras
of $\mathcal{C}$, see [BEM]. They coincide with the $C^{*}$-algebras
associated to $\mathcal{C}$ treated as a Fell bundle over $S$ as defined in
[Exe], [BuE1] or [BuE2]. The _full $A\rtimes_{\beta}^{\omega}S$_ and _reduced
crossed product $A\rtimes_{\beta,r}^{\omega}S$ of an inverse semigroup twisted
action_ $(A,S,\beta,\omega)$ are by definition the full and reduced
$C^{*}$-algebra of the associated Fell bundle over $S$, described in Example
2.11, see [BuE1].

As explained below Remark 2.6, the following result can be viewed as a
generalisation of [PaR, Proposition 5.1] from group actions to groupoid
actions. It also generalises a part of [BuE1, Theorem 7.2] that characterises
$C^{*}$-algebras of groupoid twists as crossed products by twisted actions.

######  Theorem 2.16.

Let $(\mathcal{A},G,\Sigma,\alpha)$ be a twisted groupoid $C^{*}$-dynamical
system with $G$ étale and let $(A,S,\beta,\omega)$ be the twisted inverse
semigroup action defined in Lemma 2.8, so that $S$ denotes the family of open
bisections $U$ of $G$ on which the $\mathbb{T}$-bundle $\pi:\Sigma\to G$ is
trivial. Let $S_{0}\subseteq S$ be any inverse semigroup which is wide in $G$,
as defined before Lemma 2.8 (one may take $S_{0}=S)$. Then the corresponding
twisted crossed products are canonically isomorphic:

$C^{*}(\mathcal{A},G,\Sigma,\alpha)\cong A\rtimes_{\beta}^{\omega}S_{0},\qquad
C^{*}_{r}(\mathcal{A},G,\Sigma,\alpha)\cong A\rtimes_{\beta,r}^{\omega}S_{0}.$

###### Proof.

Let $\mathcal{C}=\bigsqcup_{U\in S}C_{0}(\mathcal{B}|_{U})$ be the inverse
semigroup action by Hilbert bimodules on $A$ described in Lemma 2.14(2). This
is a standard inverse semigroup action associated to the Fell bundle
$\mathcal{B}$ over $G$, see [BuE2, Example 2.9] or [KwM1, section 7.1]. Thus
restricting $\mathcal{C}$ to any inverse subsemigroup $S_{0}\subseteq S$ which
is wide in $G$ we get $C^{*}(\bigsqcup_{U\in
S_{0}}C_{0}(\mathcal{B}|_{U}))\cong C^{*}(\mathcal{B})$ and
$C^{*}_{r}(\bigsqcup_{U\in S_{0}}C_{0}(\mathcal{B}|_{U}))\cong
C^{*}_{r}(\mathcal{B})$ by [BuE2, Theorem 2.13], or [BuM, Corollary 5.6], and
[BEM, Theorem 8.11], cf. also [KwM1, Propositions 7.6, 7.9]. This gives the
assertion as $\mathcal{B}$ is isomorphic to the bundle associated to
$(\mathcal{A},G,\Sigma,\alpha)$ by Lemma 2.14. ∎

##  3\. Multipliers for twisted groupoid dynamical systems

In this section we discuss the notion of mapping multipliers for Fell bundles
over groupoids, and we characterise a natural class of completely positive
multipliers for Fell bundles coming from twisted groupoid dynamical systems.
We consider the maps acting on the reduced $C^{*}$-algebras, as opposed to the
universal ones, studied in detail for example in [RaW], [Ren4] or [Ren6]
(albeit only in the case of trivial line bundles).

######  Definition 3.1.

Let $\mathcal{B}$ be a Fell bundle over the groupoid $G$. A _mapping
$\mathcal{B}$-multiplier_ is a function $G\ni\gamma\mapsto h(\gamma)\in
B(B_{\gamma})$ which is continuous in the sense that the formula

(3.4) $(m_{h}f)(\gamma)=h(\gamma)f(\gamma)\;\;\;\;f\in
C_{c}(\mathcal{B}),\gamma\in G,$

defines a map (a priori with no continuity requirement)
$m_{h}:C_{c}(\mathcal{B})\to C_{c}(\mathcal{B})$. We say that a mapping
$\mathcal{B}$-multiplier $h$ is _bounded_ (resp._completely bounded_ or
_completely positive_) if the map $m_{h}$ extends to to a bounded (resp.
completely bounded, completely positive) map on $C_{r}^{*}(\mathcal{B})$.

######  Example 3.2 (Scalar multipliers).

Every continuous function $h:G\to\mathbb{C}$ can be treated as a mapping
$\mathcal{B}$-multiplier for every Fell bundle $\mathcal{B}$ over $G$, as then
(3.4) defines $m_{h}:C_{c}(\mathcal{B})\to C_{c}(\mathcal{B})$. We call such
multipliers _scalar $\mathcal{B}$-multipliers_. Recall that a function
$h:G\to\mathbb{C}$ is called _positive-definite_ if for any $x\in{G^{(0)}}$
and a finite set $F\subseteq G_{x}$ the matrix

$\left[h(\gamma\eta^{-1})\right]_{\eta,\gamma\in F}$

is positive-definite. Takeishi showed in [Tak, Lemma 4.2] that every compactly
supported positive-definite function is a completely positive scalar
$\mathcal{B}$-multiplier for any $G$-bundle $\mathcal{B}$ and remarked that
the compact support condition can be dropped; we record the corresponding
fact, together with its converse, in Proposition 3.7.

######  Example 3.3 ($M(\mathcal{B}|_{{G^{(0)}}})$-valued multipliers).

Let again $\mathcal{B}$ be a Fell bundle over $G$. Then
$\mathcal{B}|_{{G^{(0)}}}$ is a continuous field of $C^{*}$-algebras and so we
have a pullback bundle $\mathcal{B}|_{{G^{(0)}}}*_{r}G=\bigsqcup_{\gamma\in
G}B_{r(\gamma)}$. Every fiber $B_{\gamma}$ is naturally a left
$B_{r(\gamma)}$-Hilbert module, and hence also a left
$M(B_{r(\gamma)})$-Hilbert module. Hence every _strictly continuous_ section
$h$ of the multiplier bundle
$M(\mathcal{B}|_{{G^{(0)}}}*_{r}G):=\bigsqcup_{\gamma\in G}M(B_{r(\gamma)})$
can be treated as a mapping $\mathcal{B}$-multiplier. _Strict continuity_ here
means that for any continuous section $a$ of $\mathcal{B}|_{{G^{(0)}}}*_{r}G$
the pointwise product $ha$ is also continuous as a section of
$\mathcal{B}|_{{G^{(0)}}}*_{r}G$. If $\mathcal{B}$ is saturated then such
multipliers coincide with mapping $\mathcal{B}$-multipliers with the values in
adjointable operators on the fibers $B_{\gamma}$, $\gamma\in G$. Sometimes we
will call mapping multipliers of this type simply $\mathcal{B}$-multipliers.

Recall that if $\mathcal{B}$ is a Fell bundle associated to a twisted groupoid
$C^{*}$-dynamical system $(\mathcal{A},G,\Sigma,\alpha)$, then
$\mathcal{B}|_{{G^{(0)}}}\cong\mathcal{A}$ with the isomorphism given by
$\mathcal{A}\ni a_{\gamma}\mapsto[a_{\gamma},1]$. So every strictly continuous
section $h$ of $M(\mathcal{A}*_{r}G)=\bigsqcup_{\gamma\in G}M(A_{r(\gamma)})$
is a $\mathcal{B}$-multiplier; we have
$(m_{h}f)(\dot{\sigma})=[h(\dot{\sigma})a,\sigma]$ for each $\sigma\in\Sigma$,
$f\in C_{c}(\mathcal{B})$ and $a\in A_{r(\sigma)}$ such that
$f(\dot{\sigma})=[a,\sigma]$.

In general it is not clear how to describe explicitly completely positive
mapping $\mathcal{B}$-multipliers. But we can do it for
$\mathcal{B}$-multipliers on Fell bundles coming from twisted groupoid
$C^{*}$-dynamical systems.

######  Definition 3.4.

Let $(\mathcal{A},G,\Sigma,\alpha)$ be a twisted groupoid $C^{*}$-dynamical
system. We say that a section $h$ of $M(\mathcal{A}*_{r}G)$ is _positive-
definite_ if for any $x\in{G^{(0)}}$ and any finite set $F\subseteq G_{x}$,
for some section $c:F\to\Sigma$ the matrix

$\left[\alpha_{c(\gamma)}^{-1}\left(h(\gamma\eta^{-1})\right)\right]_{\eta,\gamma\in
F}\in M_{|F|}(M(A_{x}))$

is positive.

We will now show that the definition above in fact does not depend on the
choice of the section, and discuss the central $\mathcal{B}$-multipliers.

######  Lemma 3.5.

Let $(\mathcal{A},G,\Sigma,\alpha)$ be a twisted groupoid $C^{*}$-dynamical
system and let $h$ be a section of $M(\mathcal{A}*_{r}G)$.

  1. (1)

Let $x\in{G^{(0)}}$ and let $F\subseteq G_{x}$ be a finite set. If there is a
section $c:F\to\Sigma$ such that the matrix
$\left[\alpha_{c(\gamma)}^{-1}\left(h(\gamma\eta^{-1})\right)\right]_{\eta,\gamma\in
F}$ is positive, then the corresponding matrix is positive for any section
$c^{\prime}:F\to\Sigma$.

  2. (2)

The map $m_{h}$ is a $C_{c}(\mathcal{A})$-bimodule map if and only if
$h(\gamma)\in ZM(A_{r(\gamma)})$, for all $\gamma\in G.$

###### Proof.

(1) Consider two sections $c,c^{\prime}:F\to\Sigma$ for $x,F$ as above. Then
there is a function $f:F\to\mathbb{T}$ such that $c=fc^{\prime}$. Note that
using Definition 2.4(2) we obtain for any element $z\in\mathbb{T}$ and
$\sigma\in\Sigma$ the following equality:
$\alpha_{z\sigma}(\cdot)=u_{r(\sigma)}(z)\alpha_{\sigma}(\cdot)u_{r(\sigma)}(z)^{-1}$,
and further by Definition 2.4(3) we obtain
$\alpha_{z\sigma}(\cdot)=\alpha_{\sigma}(u_{s(\sigma)}(z)\cdot
u_{s(\sigma)}(z)^{-1})$. This is in turn equivalent to
$\alpha_{z\sigma}^{-1}(\cdot)=u_{s(\sigma)}(z)^{-1}\alpha_{\sigma}^{-1}(\cdot)u_{s(\sigma)}(z)$.
Thus for any $\\{a_{\gamma}\\}_{\gamma\in F}\subseteq M(A_{x})$ putting
$b_{\gamma}:=a_{\gamma}u_{x}(f(\gamma))$, $\gamma\in F$ we get

$\sum_{\gamma,\eta\in
F}a_{\gamma}\alpha_{c^{\prime}(\gamma)}^{-1}\left(h(\gamma\eta^{-1})\right)a_{\eta}^{*}=\sum_{\gamma,\eta\in
F}b_{\gamma}\alpha_{c(\gamma)}^{-1}\left(h(\gamma\eta^{-1})\right)b_{\eta}^{*}.$

This proves that positivity of
$[\alpha_{c(\gamma)}^{-1}(h(\gamma\eta^{-1}))]_{\eta,\gamma\in F}$ implies
that of
$[\alpha_{c^{\prime}(\gamma)}^{-1}(h(\gamma\eta^{-1}))]_{\eta,\gamma\in F}.$

(2) The formula (3.4) implies that each $m_{h}$ is a right
$C_{c}(\mathcal{A})$-module map. Hence it is also a
$C_{c}(\mathcal{A})$-bimodule map if $h(\gamma)\in ZM(A_{r(\gamma)})$, for all
$\gamma\in G$. Conversely, assume $m_{h}$ is a left $A$-module map. For
arbitrary $a,b\in A_{r(\sigma)}$, $\sigma\in\Sigma$, there are functions $f\in
C_{c}(\mathcal{A})$ and $g\in C_{c}(\mathcal{B})$ supported on open bisections
and such that $f(r(\sigma))=a$, $g(\dot{\sigma})=[b,\sigma]$. Then, denoting
the canonical image of $f$ in $C_{c}(\mathcal{B})$ by $\dot{f}$ (so that
$\dot{f}(\dot{\sigma})=[f(\sigma),r(\sigma)]$), we have

$[h(\dot{\sigma})ab,\sigma]=m_{h}(\dot{f}\cdot g)(\dot{\sigma})=(\dot{f}\cdot
m_{h}(g))(\dot{\sigma})=[a,r(\sigma)][h(\dot{\sigma})b,\sigma]=[ah(\dot{\sigma})b,\sigma].$

Hence $h(\dot{\sigma})ab=ah(\dot{\sigma})b$, for every $a,b\in A_{r(\sigma)}$.
This implies $h(\dot{\sigma})\in ZM(A_{r(\sigma)})$. ∎

Following the classical argument from the context of Herz-Schur multipliers
(see for example [BrO, Subsection 5.6]) we obtain the next result.

######  Proposition 3.6.

Let $\mathcal{B}$ be a Fell bundle associated to a twisted groupoid
$C^{*}$-dynamical system $(\mathcal{A},G,\Sigma,\alpha)$, and let $h$ be a
section of $M(\mathcal{A}*_{r}G)$. The following conditions are equivalent:

  1. (1)

$h$ is a completely positive $\mathcal{B}$-multiplier, i.e (3.4) defines a
completely positive map $m_{h}:C_{r}^{*}(\mathcal{B})\to
C_{r}^{*}(\mathcal{B})$;

  2. (2)

$h$ is strictly continuous, bounded and positive-definite.

If these equivalent conditions hold then $h$ is central (i.e. $h(\gamma)\in
ZM(A_{r(\gamma)})$, for all $\gamma\in G$), $m_{h}$ is a
$C_{0}(\mathcal{A})$-bimodule map, and $\|m_{h}\|=\sup_{\gamma\in
G}\|h(\gamma)\|=\sup_{x\in{G^{(0)}}}\|h(x)\|.$

###### Proof.

(1)$\Rightarrow$(2) Clearly, $h$ is a mapping $\mathcal{B}$-multiplier, i.e.
it defines $m_{h}:C_{c}(\mathcal{B})\to C_{c}(\mathcal{B})$, if and only if
$h$ is strictly continuous. Since $m_{h}:C_{r}^{*}(\mathcal{B})\to
C_{r}^{*}(\mathcal{B})$ is completely positive, it is automatically bounded
and $\|m_{h}\|=\sup_{\lambda}\|m_{h}(\mu_{\lambda})\|$ for an approximate unit
$(\mu_{\lambda})$ in $A:=C_{0}(\mathcal{A})$. This readily implies that
$\|m_{h}\|=\sup_{x\in{G^{(0)}}}\|h(x)\|\leq\sup_{\gamma\in G}\|h(\gamma)\|$.
Moreover, for any $\gamma\in G$ and $\varepsilon>0$ we may find an open
bisection $U$ containing $\gamma$ and a section $a\in
C_{c}(\mathcal{B}|_{U})\subseteq C_{c}(\mathcal{B})\subseteq
C^{*}_{r}(\mathcal{B})$ such that
$\|h(\gamma)\|\leq\|h(\gamma)a(\gamma)\|+\varepsilon$ and $\|a\|=1$ (cf. the
second part of Remark 2.12). Hence
$\|h(\gamma)\|\leq\|m_{h}(a)(\gamma)\|+\varepsilon\leq\|m_{h}\|+\varepsilon$.
This implies that $\sup_{\gamma\in G}\|h(\gamma)\|=\|m_{h}\|$, and so $h$ is
bounded. Furthermore, since $m_{h}:C_{r}^{*}(\mathcal{B})\to
C_{r}^{*}(\mathcal{B})$ is (completely) positive it is $*$-preserving, and by
its very definition (3.4), $m_{h}$ is a right $C_{0}(\mathcal{A})$-module map.
Hence $m_{h}$ is a $C_{0}(\mathcal{A})$-bimodule map:
$m_{h}(ab)=m_{h}(b^{*}a^{*})^{*}=am_{h}(b)$, $a\in C_{c}(\mathcal{A})$, $b\in
C_{c}(\mathcal{B})$. Thus $h$ is central by Lemma 3.5 (2).

To see that $h$ is positive-definite, fix $x\in G^{(0)}$ and a finite set
$F\subseteq G_{x}$. For every $\gamma\in F$ let $f_{\gamma}\in
C_{c}(\mathcal{B})$ be a section supported on an open bisection. The matrix
$(f_{\gamma}f_{\eta}^{*})_{\gamma,\eta\in F}\in
M_{|F|}(C^{*}_{r}(\mathcal{B}))$ is positive (recall that the representation
$\pi_{x}$ and the module $V_{X}$ were introduced in the formula (2.3) and the
discussion preceding it). So also the operator

$B:=\pi_{x}^{(|F|)}\circ
m_{h}^{(|F|)}((f_{\gamma}f_{\eta}^{*})_{\gamma,\eta\in F})\in
B(V_{x}^{\oplus|F|})$

is positive. Let us choose a section $c:F\to\Sigma$ and write
$f_{\gamma}(\gamma)=[a_{\gamma},c(\gamma)]$ for $a_{\gamma}\in A_{r(\gamma)}$
and $\gamma\in F$. Note that

$(f_{\gamma}f_{\eta}^{*})(\gamma\eta^{-1})=f_{\gamma}(\gamma)f_{\eta}(\eta)^{*}=[a_{\gamma},c(\gamma)][\alpha_{c(\eta)^{-1}}(a_{\eta}^{*}),c(\eta)^{-1}]=[a_{\gamma}\alpha_{c(\gamma)c(\eta)^{-1}}(a_{\eta}^{*}),c(\gamma)c(\eta)^{-1}].$

Let then $\xi=\bigoplus_{\eta\in F}\xi_{\eta}=\bigoplus_{\eta\in
F}\bigoplus_{\beta\in G_{x}}\xi_{\eta}(\beta)\in V_{x}^{\oplus|F|}$ be such
that $\xi_{\eta}(\beta)\in B_{\beta}$ is zero unless $\beta=\eta$. Write
$\xi_{\eta}(\eta)=[b_{\eta},c(\eta)]$ for $b_{\eta}\in A_{r(\eta)}$ and
$\eta\in F$. We have

$\displaystyle 0\leq\langle\xi,B\xi\rangle$
$\displaystyle=\sum_{\gamma,\eta\in
F}\langle\xi_{\gamma},\pi_{x}(m_{h}(f_{\gamma}f_{\eta}^{*}))\xi_{\eta}\rangle=\sum_{\gamma,\eta\in
F}\langle\xi_{\gamma}(\gamma),m_{h}(f_{\gamma}f_{\eta}^{*})(\gamma\eta^{-1})\xi_{\eta}(\eta)\rangle$
$\displaystyle=\sum_{\gamma,\eta\in
F}\left\langle[b_{\gamma},c(\gamma)],[h(\gamma\eta^{-1})a_{\gamma}\alpha_{c(\gamma)c(\eta)^{-1}}(a_{\eta}^{*}),c(\gamma)c(\eta)^{-1}]\cdot[b_{\eta},c(\eta)]\right\rangle$
$\displaystyle=\sum_{\gamma,\eta\in
F}\alpha_{c(\gamma)}^{-1}\left(b_{\gamma}^{*}h(\gamma\eta^{-1})a_{\gamma}\alpha_{c(\gamma)c(\eta)^{-1}}(a_{\eta}^{*})\alpha_{c(\gamma)c(\eta)^{-1}}(b_{\eta})\right)$
$\displaystyle=\sum_{\gamma,\eta\in
F}\alpha_{c(\gamma)}^{-1}(b_{\gamma}^{*})\alpha_{c(\gamma)}^{-1}(h(\gamma\eta^{-1}))\alpha_{c(\gamma)}^{-1}(a_{\gamma})\alpha_{c(\eta)^{-1}}(a_{\eta}^{*}b_{\eta}).$

It now remains to note that we can choose the functions $f^{i}_{\gamma}$ so
that the corresponding $a^{i}_{\gamma}$ approximate $1_{M(A_{r(\gamma)})}$ in
the strict topology. Then taking the limit over $i$ we obtain

$0\leq\sum_{\gamma,\eta\in
F}\alpha_{c(\gamma)}^{-1}(b_{\gamma}^{*})\alpha_{c(\gamma)}^{-1}(h(\gamma\eta^{-1}))\alpha_{c(\eta)}^{-1}(b_{\eta}).$

Now as the choice of elements $b_{\gamma}$ was arbitrary, the matrix
$\left[\alpha_{c(\gamma)}^{-1}\left(h(\gamma\eta^{-1})\right)\right]_{\gamma,\eta\in
F}$ is positive in $M_{|F|}(M(A_{x}))$.

(2)$\Rightarrow$(1) Since $h$ is bounded we may normalize it so that it is a
strictly continuous section of $M(\mathcal{A}*_{r}G)$ bounded by $1$. Fix
$x\in G^{(0)}$ and choose a section $c:G_{x}\to\Sigma_{x}\subseteq\Sigma$. By
Lemma 3.5, the formula
$k_{x}(\gamma,\eta)=\alpha_{c(\gamma)^{-1}}h(\gamma\eta^{-1})$,
$\gamma,\eta\in G_{x}$ defines a positive-definite kernel $k_{x}:G_{x}\times
G_{x}\to M(A_{x})$. Hence by [Mur, Theorem 2.3] there exists a Hilbert
$A_{x}$-module $H_{x}$ and a map $\zeta:G_{x}\to\mathcal{L}(A_{x},H_{x})$
(with values in the unit ball) such that for each $\kappa,\beta\in G_{x}$ we
have

(3.5)
$\alpha_{c(\kappa)}^{-1}(h(\kappa\beta^{-1}))=\zeta(\kappa)^{*}\zeta(\beta).$

Note that we can view $\mathcal{L}(A_{x},H_{x})$ as a Hilbert module
$\tilde{H}_{x}$ over $M(A_{x})\cong\mathcal{L}(A_{x})$, cf. [RaT, Proposition
1.2].

The isomorphisms $A_{r(\gamma)}\ni a_{\gamma}\mapsto[a_{\gamma},c(\gamma)]\in
B_{\gamma}$, extended to direct sums, allow us to identify the right
$A_{x}$-module $V_{x}=\bigoplus_{\gamma\in G_{x}}B_{\gamma}$ (described in the
construction of the regular representation) with the $A_{x}$-module
$\bigoplus_{\gamma\in G_{x}}A_{r(\gamma)}$ with operations given by

$\left(\bigoplus_{\gamma\in G_{x}}a_{\gamma}\right)\cdot
a=\bigoplus_{\gamma\in
G_{x}}a_{\gamma}\alpha_{c(\gamma)}(a),\qquad\left\langle\bigoplus_{\gamma\in
G_{x}}b_{\gamma},\bigoplus_{\gamma\in
G_{x}}d_{\gamma}\right\rangle_{A_{x}}=\sum_{\gamma\in
G_{x}}\alpha_{c(\gamma)}^{-1}(b_{\gamma}^{*}d_{\gamma}).$

We equip $V_{x}$ with a natural left action
$\phi_{x}:M(A_{x})\to\mathcal{L}(V_{x})$, given by the formula:

$\phi_{x}(m)\cdot\bigoplus_{\gamma\in G_{x}}a_{\gamma}:=\bigoplus_{\gamma\in
G_{x}}\alpha_{c(\gamma)}(m)a_{\gamma},\qquad m\in
M(A_{x}),\bigoplus_{\gamma\in G_{x}}a_{\gamma}\in V_{x}.$

Thus we can form a new $A_{x}$-Hilbert module
$W_{x}=\tilde{H}_{x}\otimes_{M(A_{x})}V_{x}$. Consider the map
$\Theta_{x}:V_{x}\to W_{x}$ given by

$\Theta_{x}\left(\bigoplus_{\gamma\in
G_{x}}b_{\gamma}\right)=\bigoplus_{\gamma\in G_{x}}\zeta(\gamma)\otimes
b_{\gamma}.$

One readily verifies that $\Theta_{x}$ is a contraction and we can define a
completely positive contractive map $\Psi_{x}:B(V_{x})\to B(V_{x})$ by the
formula

$\Psi_{x}(T)=\Theta_{x}^{*}(1_{\tilde{H}_{x}}\otimes T)\Theta_{x},\;\;\;T\in
B(V_{x}).$

The formula (2.3) allows us to check that we have

(3.6) $\pi_{x}(m_{h}f)=\Psi_{x}(\pi_{x}(f)),\;\;\;f\in C_{c}(\mathcal{B}).$

Indeed, by linearity and continuity, to establish the above equality it
suffices to verify that for each $\eta,\gamma\in G_{x}$ and $b_{\gamma}\in
B_{\gamma},d_{\eta}\in B_{\eta}$ we have $\langle
b_{\gamma},\pi_{x}(m_{h}f)d_{\eta}\rangle_{A_{x}}=\langle
b_{\gamma},\Psi_{x}(\pi_{x}(f))d_{\eta}\rangle_{A_{x}}$. Using (3.5) we get
$(m_{h}f)(\gamma\eta^{-1})d_{\eta}=\phi_{x}(\langle\zeta(\gamma),\zeta(\eta)\rangle_{M(A_{x})})f(\gamma\eta^{-1})d_{\eta}$.
Therefore, using the definition of the internal tensor product of Hilbert
modules and of the representation $\pi_{x}$ we obtain

$\displaystyle\langle
b_{\gamma},(m_{h}f)(\gamma\eta^{-1})d_{\eta}\rangle_{A_{x}}$
$\displaystyle=\langle
b_{\gamma},\phi_{x}(\langle\zeta(\gamma),\zeta(\eta)\rangle_{M(A_{x})})\cdot(f(\gamma\eta^{-1})\cdot
d_{\eta})\rangle_{A_{x}}$ $\displaystyle=\langle\zeta(\gamma)\otimes
b_{\gamma},\zeta(\eta)\otimes\pi_{x}(f)d_{\eta}\rangle_{A_{x}}$
$\displaystyle=\langle\Theta_{x}(b_{\gamma}),(I\otimes\pi_{x}(f))\Theta_{x}(d_{\eta})\rangle_{A_{x}}$
$\displaystyle=\langle
b_{\gamma},\Psi_{x}(\pi_{x}(f))d_{\eta}\rangle_{A_{x}}.$

Now, as the direct sum of the representations $\pi_{x}$ is faithful on
$C_{r}^{*}(\mathcal{B})$, (3.6) implies that the map $m_{h}$ is contractive
and completely positive. ∎

A very similar proof gives also the following result: note that we allow below
arbitrary $G$-bundles, but consider only scalar multipliers.

######  Proposition 3.7.

Let $h:G\to\mathbb{C}$ be a bounded continuous function and let $\mathcal{B}$
be a $G$-bundle. Consider the following conditions:

  * (i)

$h$ is positive-definite;

  * (ii)

$h$ is a completely positive scalar $\mathcal{B}$-multiplier.

Then (i) implies (ii), and if $\mathcal{B}$ is “strongly saturated” in the
sense that for every $\gamma\in G$ there is an approximate unit for
$B_{s(\gamma)}$ contained in $\langle
B_{\gamma},B_{\gamma}\rangle_{B_{s(\gamma)}}$, then (ii) implies (i). Thus
(i)$\Leftrightarrow$(ii) for continuous line bundles and for bundles coming
from twisted groupoid actions.

######  Remark 3.8.

If we consider a transformation groupoid $G:=\Gamma\times X$ (where a discrete
group $\Gamma$ acts on a compact space $X$), and consider the trivial line
bundle over $G$, then a function $h\in C(G)$ can be canonically identified
with a map $\tilde{h}:\Gamma\to C(X)$. The positive-definiteness condition for
$h$ reads then as follows: for any $x\in X$ and a finite set
$F\subseteq\Gamma$ we have that the matrix

$\left[\tilde{h}(f^{-1}g)(f\cdot)\right]_{f,g\in F}\in M_{|F|}(C(X))$

is positive. This is the condition appearing for example in [DoR, Equation
(3.2)] (up to a change of convention) or in [BC2, Definition 4.2]; see also a
discussion in [MSTT, Section 2].

##  4\. The Haagerup trick for twisted groupoid dynamical systems

Throughout the section we fix a twisted groupoid $C^{*}$-dynamical system
$(\mathcal{A},G,\Sigma,\alpha)$ where $G$ is an étale locally compact
Hausdorff groupoid. We let $\mathcal{B}$ be the Fell bundle associated to it
and consider the following $C^{*}$-inclusion

$A:=C_{0}(\mathcal{A})\subseteq B=C^{*}_{r}(\mathcal{A},G,\Sigma,\alpha)$

equipped with the canonical faithful conditional expectation $E:B\to A$. In
Proposition 3.6 we have seen that strictly continuous positive-definite
sections of $M(\mathcal{A}*_{r}G)$ give rise to completely positive maps on
$B$. In this section we prove the converse. To this end, we develop a
generalisation of ‘the Haagerup Trick’, cf. [Haa2, Lemma 2.5]. To avoid
technicalities, and also because the approximation properties we want to study
in sequel sections seem to be designed for the unital context, from now on we
will assume here that

> the bundle $\mathcal{A}$ has a _continuous unital section_ , i.e. each fiber
> $A_{x}$, $x\in{G^{(0)}}$ is unital and the unit section ${G^{(0)}}\ni
> x\mapsto 1_{x}\in A_{x}\subseteq\mathcal{A}$ is continuous.

Here and below $1_{x}$ denotes the unit in the algebra $A_{x}$,
$x\in{G^{(0)}}$. Our standing assumption implies that $A$ is unital if and
only if ${G^{(0)}}$ is compact. It also implies that
$M(\mathcal{A}*_{r}G)=\mathcal{A}*_{r}G$ and for sections of this bundle
strict continuity is just continuity. Further we may assume that
$C_{0}(G)\subseteq Z(A)$ by identifying functions in $C_{0}(G)$ with their
product by the unit section.

In the following proposition we use the notation introduced in Lemma 2.8, (see
also Lemma 2.14), we denote by $\mathcal{B}$ the Fell bundle associated to
$(\mathcal{A},G,\Sigma,\alpha)$ in Example 2.5, and use the identification
$C_{r}^{*}(\mathcal{B})\subseteq C_{0}(\mathcal{B})$ from Remark 2.12.

######  Proposition 4.1 (Haagerup Trick).

Suppose $\Phi:B\to B$ is a bounded right $A$-module map. Fix an element $a\in
A$. For each $\gamma\in G$ choose a bisection $U\in S$ containing $\gamma$ and
a section $f\in C_{c}(\mathcal{A}|_{r(U)})$ which attains units at a
neighborhood of $r(\gamma)$, and define

$f_{\gamma}(\eta):=[f(r(\eta)),c_{U}(\eta)],\qquad\eta\in U.$

Then $f_{\gamma}\in C_{c}(\mathcal{B}|_{U})\subseteq B$ and we define

(4.7)
$h^{\Phi}(\gamma):=\Phi(f_{\gamma})(\gamma)f_{\gamma}(\gamma)^{*}=E(\Phi(f_{\gamma})f_{\gamma}^{*})(r(\gamma))\in
A_{r(\gamma)}.$

The formula above does not depend on the choice of $U$, $f$ and $c_{U}$, and
defines a bounded continuous section $h^{\Phi}$ of $\mathcal{A}*_{r}G$ (we
have $\|h^{\Phi}\|_{\infty}\leq\|\Phi\|$). Moreover, if $\Phi$ is completely
positive then $h^{\Phi}$ is positive-definite.

###### Proof.

The element $f_{\gamma}$ corresponds to the section $f$ under the isomorphism
$C_{0}(\mathcal{B}|_{U})\cong C_{0}(\mathcal{A}*_{r}G|_{U})\cong
C_{0}(\mathcal{A}|_{r(U)})$ from Lemma 2.14. In particular, $f_{\gamma}\in
C_{c}(\mathcal{B}|_{U})\subseteq B$. To see that the definition of
$h^{\Phi}(\gamma)$ does not depend on the choice of the section
$c_{U}:U\to\Sigma$, let $c_{U}^{\prime}:U\to\Sigma$ be another continuous
section that trivializes the $\mathbb{T}$-bundle, and put
$f_{\gamma}^{\prime}(\eta):=[f(r(\eta)),c_{U}^{\prime}(\eta)]$, $\eta\in G$.
Then $c(r(\eta)):=c_{U}^{\prime}(\eta)c_{U}(\eta)^{-1}$, $\eta\in U$, defines
a continuous section of $\Sigma|_{r(U)}\cong
r(U)\times\mathbb{T}\twoheadrightarrow r(U)$. Thus identifying $c(x)$ with the
uniquely determined element of $\mathbb{T}$ we see that $s(U)\ni
s(\eta)\mapsto u_{s(\eta)}(c(r(\eta)))\in A_{x}\subseteq\mathcal{A}$ is also a
continuous section. Let $h\in C_{0}(s(U))$ be any function which is equal to
one on $s(\operatorname{supp}(f_{\gamma}))$. Then
$u(s(\eta)):=h(s(\eta))u_{s(\eta)}(c(r(\eta)))$, $\eta\in U$, defines an
element of $C_{0}(\mathcal{A}|_{s(U)})\subseteq A$. For any
$\eta\in\operatorname{supp}(f_{\gamma})\subseteq U$ we have

$\displaystyle f_{\gamma}^{\prime}(\eta)$
$\displaystyle=[f(r(\eta)),c_{U}^{\prime}(\eta)]=[f(r(\eta)),c(r(\eta))c_{U}(\eta)]=[f(r(\eta))u_{r(\eta)}(c(r(\eta))),c_{U}(\eta)]$
$\displaystyle=[f(r(\eta))\alpha_{c_{U}(\eta)}\big{(}u_{s(\eta)}(c(r(\eta)))\big{)},c_{U}(\eta)]=[f(r(\eta))\alpha_{c_{U}(\eta)}\big{(}u(s(\eta))\big{)},c_{U}(\eta)]$
$\displaystyle=(f_{\gamma}u)(\eta).$

Hence $f_{\gamma}^{\prime}=f_{\gamma}u$ where $u\in A$ is such that $uu^{*}$
is the unit section on $s(\operatorname{supp}(f_{\gamma}))$. As $\Phi$ is a
right $A$-module map and $(uu^{*})(x)=1_{x}$ for all $x\in
s(\operatorname{supp}(f_{\gamma}))$, we obtain

$\Phi(f_{\gamma}^{\prime})(f_{\gamma}^{\prime})^{*}=\Phi(f_{\gamma})uu^{*}f_{\gamma}^{*}=\Phi(f_{\gamma})f_{\gamma}^{*}.$

Independence of $h^{\Phi}(\gamma)$ from the choice of $U$ will follow once we
show its independence from the choice of $f$. To show the latter, let
$f^{\prime}\in C_{c}(\mathcal{A}|_{r(U^{\prime})})$, $U^{\prime}\in S$, be
another section which is unital at a neighborhood of $r(\gamma)$, and put
$(f_{\gamma}^{\prime})(\eta):=[f^{\prime}(r(\eta)),c_{U}(\eta)]$. Then
$f_{\gamma}|_{V}=f_{\gamma}^{\prime}|_{V}$ for some neighborhood $V$ of
$\gamma$ (find $V$ such that both $f$ and $f^{\prime}$ are unital on
$r(V)\subseteq r(U)\cap r(U^{\prime})$). Let $h\in C_{c}(s(V))$ be such that
$h(s(\gamma))=1$. We may treat $h$ as an element of $A$ by multiplying it by
the unit section of $\mathcal{A}$. Then $f_{\gamma}h=f_{\gamma}^{\prime}h$ and
using that $\Phi$ is a right $A$-module map we get

$\displaystyle\Phi(f_{\gamma})(\gamma)(f_{\gamma})^{*}(\gamma)$
$\displaystyle=\Phi(f_{\gamma})(\gamma)h(s(\gamma))\overline{h(s(\gamma))}(f_{\gamma}^{*}(\gamma))=\Phi(f_{\gamma}h)(\gamma)(f_{\gamma}h)^{*}(\gamma)$
$\displaystyle=\Phi(f_{\gamma}^{\prime}h)(\gamma)(f_{\gamma}^{\prime}h)^{*}(\gamma)=\Phi(f_{\gamma}^{\prime})(\gamma)h(s(\gamma))\overline{h(s(\gamma))}(f_{\gamma}^{\prime*}(\gamma))$
$\displaystyle=\Phi(f_{\gamma}^{\prime})(\gamma)(f_{\gamma}^{\prime})^{*}(\gamma).$

Hence the formula (4.7) gives a well defined section $h^{\Phi}$ of
$\mathcal{A}*_{r}G$; the fact that we also have the expression involving the
conditional expectation follows as $f_{\gamma}$ is supported on a bisection.
For any $\gamma\in G$ the section $f_{\gamma}$ is given by $f$ with
$f(r(\eta))=1_{r(\eta)}$ for all $\eta\in V$ for some open neighbourhood $V$
of $\gamma$. Hence by the independence on the choice of a right $f$
established above we have
$h^{\Phi}(\eta)=E(\Phi(f_{\gamma})f_{\gamma}^{*})(r(\eta))$ for any $\eta\in
V$. As $E(\Phi(f_{\gamma})f_{\gamma}^{*})\in A=C_{0}(\mathcal{A})$ is
continuous, this implies that $h^{\Phi}$ is continuous at $\gamma$. Also we
may choose $f$ so that $\|f\|=1$, equivalently $\|f_{\gamma}\|=1$, and this
implies that
$\|h^{\Phi}(\gamma)\|=\|E(\Phi(f_{\gamma})f_{\gamma}^{*})(r(\gamma))\|\leq\|\Phi\|$.
Thus the section $h^{\Phi}$ is bounded by $\|\Phi\|$.

Now suppose that $\Phi$ is completely positive. To see that $h^{\Phi}$ is
positive-definite fix $x\in{G^{(0)}}$ and a finite set $F\subseteq G_{x}$. For
each $\gamma\in F$ choose $f_{\gamma}$ as in (4.7) supported on $U_{\gamma}\in
S$. Define a section $c:F\to\Sigma$ by $c(\gamma):=c_{U_{\gamma}}(\gamma)$,
$\gamma\in F$. Note that for any $\eta,\gamma\in F$, the product
$f_{\gamma}*f_{\eta}^{*}$ is supported on an open bisection in $S$ and it is
the unit section on some neighbourhood of $\gamma\eta^{-1}$. Hence

$\displaystyle h^{\Phi}(\gamma\eta^{-1})$
$\displaystyle=\Phi(f_{\gamma}*f_{\eta}^{*})(\gamma\eta^{-1})(f_{\gamma}*f_{\eta}^{*})(\gamma\eta^{-1})^{*}=\Phi(f_{\gamma}*f_{\eta}^{*})(\gamma\eta^{-1})f_{\eta}(\eta)f_{\gamma}(\gamma)^{*}.$

A simple computation using the formulas from Example 2.5 and the fact that
$f_{\gamma}(\gamma)=[1_{A_{r(\gamma)}},c_{U}(\gamma)]$ shows that
$f_{\gamma}(\gamma)^{*}h^{\Phi}(\gamma\eta^{-1})f_{\gamma}(\gamma)=\alpha_{c(\gamma)}^{-1}\big{(}h^{\Phi}(\gamma\eta^{-1})\big{)}$
and $f_{\gamma}(\gamma)^{*}f_{\gamma}(\gamma)=1_{s(\gamma)}$. Thus we get

$\alpha_{c(\gamma)}^{-1}\big{(}h^{\Phi}(\gamma\eta^{-1})\big{)}=f_{\gamma}(\gamma)^{*}\Phi(f_{\gamma}*f_{\eta}^{*})(\gamma\eta^{-1})f_{\eta}(\eta)=E\left(f_{\gamma}^{*}*\Phi(f_{\gamma}*f_{\eta}^{*})*f_{\eta}\right)(x),$

where in the last equality we use again the fact that both $f_{\gamma}$ and
$f_{\eta}$ are supported on bisections. Thus the matrix

$\left[\alpha_{c(\gamma)}^{-1}\big{(}h^{\Phi}(\gamma\eta^{-1})\big{)}\right]_{\eta,\gamma\in
F}=\left[E\left(f_{\gamma}^{*}*\Phi(f_{\gamma}*f_{\eta}^{*})*f_{\eta}\right)(x)\right]_{\eta,\gamma\in
F}$

is positive as $\Phi$ and $E$ are completely positive. ∎

######  Remark 4.2.

In the case of a groupoid dynamical system $(\mathcal{A},G,\alpha)$ (without a
twist) we may replace $\mathcal{B}$ by $\mathcal{A}*_{r}G$ and then the map
(4.7) induced by $\Phi:B\to B$ could be defined by
$h^{\Phi}(\gamma)=\Phi(f)(\gamma)$ where $f\in C_{c}(\mathcal{A}*_{r}G|_{U})$
is supported on an open bisection $U$ and such that $f(\eta)=1_{r(\eta)}$ on
some neighbourhood of $\gamma$. We also invite the reader to check that when
the crossed product in question is just the usual group $C^{*}$-algebra (i.e.
$G$ is a discrete group and the bundle $\mathcal{A}$ is trivial) the formula
(4.7) reduces to the one from the classical Haagerup trick.

######  Remark 4.3.

Given any bounded mapping $\mathcal{B}$-multiplier $h\in
C_{b}(\mathcal{A}*_{r}G)$ the map $m_{h}:B\to B$ is a bounded right $A$-module
map and $h^{m_{h}}=h$. But for a general bounded right $A$-module $\Phi:B\to
B$ it may happen that $m_{h^{\Phi}}\neq\Phi$.

We now wish to characterise when a section obtained by the Haagerup Trick is
of $C_{0}$ or $C_{c}$ class. To this end, we use the right Hilbert $A$-module
$B_{E}$ associated to the canonical expectation $E:B\to A$. Thus

(4.8) $B_{E}:=\overline{B}^{\|\cdot\|_{E}},\quad\langle
x,y\rangle_{E}:=E(x^{*}y),\qquad x,y\in B\subseteq B_{E}.$

As is common, we will write $\mathcal{L}(B_{E})$ for the algebra of all
adjointable operators on $B_{E}$, for $\xi,\eta\in B_{E}$ the symbol
$\Theta_{\xi,\eta}\in\mathcal{L}(B_{E})$ will denote the obvious ‘rank-one
operator’ and $\mathcal{K}(B_{E})$ will denote the closed linear span of rank-
one operators inside $\mathcal{L}(B_{E})$ (see for example [Lan]). Denoting by
$\|\cdot\|_{E}$ the norm in $B_{E}$, by $\|\cdot\|_{\infty}$ the supremum norm
on $C_{0}(\mathcal{B})$ and by $\|\cdot\|_{r}$ the norm in
$B=C^{*}_{r}(\mathcal{B})$ we have $\|f\|_{\infty}\leq\|f\|_{E}\leq\|f\|_{r}$
for all $f\in C_{c}(\mathcal{B})$. In particular, we may assume the
identifications

$C_{c}(\mathcal{B})\subseteq B\subseteq B_{E}=\\{f\in
C_{0}(\mathcal{B}):\sup_{x\in{G^{(0)}}}\|\sum_{\gamma\in
G_{x}}f^{*}(\gamma)f(\gamma)\|_{A_{x}}<\infty\\}\subseteq C_{0}(\mathcal{B}),$

cf. Remark 2.12.

In general, for an element $h\in C_{b}(\mathcal{A}*_{r}G)$ the formula (3.4),
does not define a bounded map $m_{h}$ on $B=C^{*}_{r}(\mathcal{B})$, and a
bounded right $A$-module map $\Phi:B\to B$ does not need to extend to a map
$\tilde{\Phi}\in\mathcal{L}(B_{E})$. This can be seen already if one considers
$\mathcal{A}$ to be trivial and $G$ to be a group, so that we are in the
context of Herz-Schur multipliers. However, we have the following result.

######  Proposition 4.4.

The space $C_{b}(\mathcal{A}*_{r}G)$ of continuous bounded sections equipped
with pointwise operations and supremum norm is a $C^{*}$-algebra. We have an
injective $*$-homomorphism $C_{b}(\mathcal{A}*_{r}G)\ni
h\to\widetilde{m}_{h}\in\mathcal{L}(B_{E})$ where

(4.9) $(\widetilde{m}_{h}(a))(\gamma):=h(\gamma)\cdot a(\gamma),\qquad a\in
C_{c}(\mathcal{B})\subseteq B_{E},\,\,\gamma\in G.$

Moreover, if $h\in C_{b}(\mathcal{A}*_{r}G)$ then

  1. (1)

$h\in C_{c}(\mathcal{A}*_{r}G)$ if and only if
$\widetilde{m}_{h}\in\mathcal{F}(C_{c}(\mathcal{B})):=\text{span}\\{\Theta_{f,g}:f,g\in
C_{c}(\mathcal{B})\\}$.

  2. (2)

$h\in C_{0}(\mathcal{A}*_{r}G)$ if and only if
$\widetilde{m}_{h}\in\mathcal{K}(B_{E})$.

Similarly, if $\Phi:B\to B$ is a bounded right $A$-module map that extends to
a bounded map $\tilde{\Phi}:B_{E}\to B_{E}$, then we have

  1. (3)

$h^{\Phi}\in C_{c}(\mathcal{A}*_{r}G)$ if
$\tilde{\Phi}\in\mathcal{F}(C_{c}(\mathcal{B}))$.

  2. (4)

$h^{\Phi}\in C_{0}(\mathcal{A}*_{r}G)$ if $\tilde{\Phi}\in\mathcal{K}(B_{E})$.

###### Proof.

The first claim is immediate. Let $h\in C_{b}(\mathcal{A}*_{r}G)$. Clearly,
(4.9) defines a map $\widetilde{m}_{h}:C_{c}(\mathcal{B})\to
C_{c}(\mathcal{B})$. Let $a,b\in C_{c}(\mathcal{B})$ and $x\in{G^{(0)}}$.
Since $E(a^{*}b)(x)=(a^{*}*b)(x)=\sum_{\gamma\in
G_{x}}a(\gamma)^{*}b(\gamma)$, we get

(4.10) $E(\widetilde{m}_{h}(a)^{*}b)(x)=\sum_{\gamma\in
G_{x}}a(\gamma)^{*}h(\gamma)^{*}b(\gamma)=E(a^{*}\widetilde{m}_{h^{*}}(b))(x).$

That is, $\langle\widetilde{m}_{h}(a),b\rangle_{E}=\langle
a,\widetilde{m}_{h^{*}}(b)\rangle_{E}$. Similarly,

$\displaystyle E(\widetilde{m}_{h}(a)^{*}\widetilde{m}_{h}(a))(x)$
$\displaystyle=\sum_{\gamma\in
G_{x}}a(\gamma)^{*}|h(\gamma)|^{2}a(\gamma)\leq\|h\|^{2}\sum_{\gamma\in
G_{x}}a(\gamma)^{*}a(\gamma)=\|h\|^{2}E(a^{*}a)(x).$

This implies that
$\langle\widetilde{m}_{h}(a),\widetilde{m}_{h}(a)\rangle_{E}\leq\|h\|\langle
a,a\rangle_{E}$ in $A$, and hence
$\|\widetilde{m}_{h}(a)\|^{2}_{E}\leq\|h\|^{2}\|a\|^{2}_{E}$. So
$\widetilde{m}_{h}$ is bounded, and $\|\widetilde{m}_{h}\|\leq\|h\|$.
Accordingly, $\widetilde{m}_{h}:C_{c}(\mathcal{B})\to C_{c}(\mathcal{B})$
extends to a bounded map on $B_{E}$. By (4.10) this map is adjointable, with
the adjoint equal to $\widetilde{m}_{h^{*}}$. Thus the map
$C_{b}(\mathcal{A}*_{r}G)\ni h\to\widetilde{m}_{h}\in\mathcal{L}(B_{E})$ is
well defined and $*$-preserving. By (4.9) it is immediate that this map is
also linear, multiplicative and injective.

(1) and (3): To show that $h\in C_{c}(\mathcal{A}*_{r}G)$ implies
$\widetilde{m}_{h}\in\mathcal{F}(C_{c}(\mathcal{B}))$ it suffices to consider
the case when $h$ is supported on an open bisection $U\in S$ where $S$ is as
in Lemma 2.8, cf. also Lemma 2.14. Using that $C_{0}(\mathcal{A}|_{r(U)})$ is
a $C^{*}$-algebra and $r$ induces isomorphisms
$\mathcal{B}|_{U}\cong\mathcal{A}|_{r(U)}$ and $C_{c}(\mathcal{B}|_{U})\cong
C_{c}(\mathcal{A}|_{r(U)})$, we can find elements $f,g\in
C_{c}(\mathcal{A}*_{r}G|_{U})$ such that $h(\gamma)=f(\gamma)g(\gamma)^{*}\in
A_{r(\gamma)}$ for every $\gamma\in G$. Then for any $a\in
C_{c}(\mathcal{A}*_{r}G|_{V})$, $V\in S$, and $\gamma\in U\cap V$ we get

$\displaystyle h(\gamma)[a](\gamma)$
$\displaystyle=[f(\gamma)g(\gamma)^{*}a(\gamma),c_{U}(\gamma)]=[f(\gamma),c_{U}(\gamma)][\alpha_{c_{U}(\gamma)}^{-1}\big{(}g(\gamma)^{*}a(\gamma)\big{)},s(\gamma)]$
$\displaystyle=[f](\gamma)\langle[g],[a]\rangle_{A}(s(\gamma))=(\Theta_{[f],[g]}[a])(\gamma).$

Hence $\widetilde{m}_{h}=\Theta_{[f],[g]}\in\mathcal{F}(C_{c}(G))$.

Suppose now that $h\in C_{b}(\mathcal{A}*_{r}G)$ is any section such that
$\widetilde{m}_{h}\in\mathcal{F}(C_{c}(\mathcal{B}))$ or that $h=h^{\Phi}$
where $\tilde{\Phi}\in\mathcal{F}(C_{c}(\mathcal{B}))$. Then either
$\widetilde{m}_{h}$ or $\tilde{\Phi}$ is of the form
$T=\sum_{i=1}^{n}\Theta_{f_{i},g_{i}}$ for $f_{i}\in
C_{c}(\mathcal{B}|_{U_{i}})$, $g_{i}\in C_{c}(\mathcal{B}|_{V_{i}})$, where
$U_{i},V_{i}\subseteq G$ are open bisections, $i=1,...,n$. For each $\gamma\in
G$ choose $f_{\gamma}$ is as in Proposition 4.1. Then

$\displaystyle(Tf_{\gamma})(\gamma)$
$\displaystyle=\left(\sum_{i=1}^{n}f_{i}\cdot E(g_{i}^{*}\cdot
f_{\gamma})\right)(\gamma)=\sum_{i=1}^{n}f_{i}(\gamma)g_{i}(\gamma)^{*}f_{\gamma}(\gamma)=\left(\sum_{i=1}^{n}f_{i}(\gamma)g_{i}(\gamma)^{*}\right)f_{\gamma}(\gamma).$

If $T=\widetilde{m}_{h}$ we get
$h(\gamma)f_{\gamma}(\gamma)=\left(\sum_{i=1}^{n}f_{i}(\gamma)g_{i}(\gamma)^{*}\right)f_{\gamma}(\gamma)$.
And if $T=\tilde{\Phi}$ we get
$h(\gamma)=h^{\Phi}(\gamma)=\left(\sum_{i=1}^{n}f_{i}(\gamma)g_{i}(\gamma)^{*}\right)$.
In both cases, we conclude that $h(\gamma)\neq 0$ if and only if
$\sum_{i=1}^{n}f_{i}(\gamma)g_{i}(\gamma)^{*}\neq 0$, so the support of $h$ is
compact, that is $h\in C_{c}(\mathcal{A}*_{r}G)$.

(2) and (4): By what we have proved $C_{c}(\mathcal{A}*_{r}G)\ni
h\to\widetilde{m}_{h}\in\mathcal{F}(C_{c}(\mathcal{B}))\subseteq\mathcal{K}(B_{E})$
is a contractive linear map. Hence it extends uniquely to a contractive map
$C_{0}(\mathcal{A}*_{r}G)\ni h\to\widetilde{m}_{h}\in\mathcal{K}(B_{E})$, and
it is easy to check that this extension is compatible with the formula
defining the map in Proposition 4.4. Thus if $h\in C_{0}(\mathcal{A}*_{r}G)$,
then $\widetilde{m}_{h}\in\mathcal{K}(B_{E})$.

Conversely, assume that $h\in C_{b}(\mathcal{A}*_{r}G)$ is such that
$\widetilde{m}_{h}\in\mathcal{K}(B_{E})$ or that $h=h^{\Phi}$ where
$\tilde{\Phi}\in\mathcal{K}(B_{E})$. Let $T\in\mathcal{K}(B_{E})$ stand either
for $\widetilde{m}_{h}$ or for $\tilde{\Phi}$ and let $\varepsilon>0$. Since
$\overline{\mathcal{F}(C_{c}(\mathcal{B}))}=\mathcal{K}(B_{E})$, there is
$R=\sum_{i=1}^{n}\Theta_{f_{i},g_{i}}$ with $f_{i}\in
C_{c}(\mathcal{B}|_{U_{i}})$, $g_{i}\in C_{c}(\mathcal{B}|_{V_{i}})$, where
$U_{i},V_{i}\in S$, $i=1,...,n$, and
$\|T-R\|_{\mathcal{L}(B_{E})}<\varepsilon/2$. It suffices to show that the
closed set $\\{\gamma\in G:\|h(\gamma)\|\geq\varepsilon\\}$ is contained in
the following compact subset of $G$:

$K:=\bigcup_{i=1}^{n}(s|_{V_{i}})^{-1}(s(\operatorname{supp}f_{i})).$

To see this, note that for any $\gamma\notin K$ we may find $U$ and
$f_{\gamma}\in C_{c}(\mathcal{B}|_{U})$ as in Proposition 4.1 such that
$U\subseteq G\setminus K$ and $\|f_{\gamma}\|_{\infty}=1$. Then
$\|f_{\gamma}\|_{E}^{2}=\|E(f_{\gamma}^{*}f_{\gamma})\|_{\infty}=\|f_{\gamma}^{*}f_{\gamma}\|_{\infty}=1$.
Moreover, $R(f_{\gamma})=0$, as for any $\eta\in G$,

$(R(f_{\gamma}))(\eta)=\sum_{i=1}^{n}f_{i}(\eta)g_{i}(s|_{V_{i}}^{-1}(s(\eta)))^{*}f_{\gamma}(s|_{V_{i}}^{-1}(s(\eta)))=0.$

If $T=\tilde{\Phi}$ then
$\|h^{\Phi}(\gamma)\|=\|\Phi(f_{\gamma})(\gamma)f_{\gamma}(\gamma)^{*}\|\leq\|\Phi(f_{\gamma})(\gamma)\|\leq\|\tilde{\Phi}(f_{\gamma})\|_{E}=\|\tilde{\Phi}(f_{\gamma})-R(f_{\gamma})\|_{E}<\varepsilon.$
If $T=\widetilde{m}_{h}$, then
$\|h(\gamma)\|=\|h(\gamma)f_{\gamma}(\gamma)\|\leq\|\widetilde{m}_{h}(f_{\gamma})\|_{E}=\|\widetilde{m}_{h}(f_{\gamma})-R(f_{\gamma})\|_{E}<\varepsilon.$
Thus in both cases we get that $\\{\gamma\in
G:\|h^{\Phi}(\gamma)\|\geq\varepsilon\\}\subseteq K$. Hence $h\in
C_{0}(\mathcal{B})$. ∎

The uniform convergence on compact subsets of $G$ can be rephrased in terms of
pointwise convergence in the $\|\cdot\|_{E}$ norm.

######  Lemma 4.5.

A net $(h_{i})_{i\in\mathcal{I}}\subseteq C_{b}(\mathcal{A}*_{r}G)$ converges
to the unit section uniformly on compact sets, i.e. $\sup_{\gamma\in
K}\|h_{i}(\gamma)-1_{r(\gamma)}\|\to 0$ for every compact $K\subseteq G$, if
and only if $(\widetilde{m}_{h_{i}})_{i\in\mathcal{I}}$ converges pointwise on
$C_{c}(\mathcal{B})\subseteq B_{E}$ to the identity operator (so that if
$(\widetilde{m}_{h_{i}})_{i\in\mathcal{I}}$ is bounded it converges pointwise
to the identity operator on $B_{E}$). Also if $(\Phi_{i})_{i\in I}$ is a net
of bounded $A$-bimodule maps on $C_{r}^{*}(\mathcal{B})$ which converges to
identity pointwise on $C_{c}(\mathcal{B})\subseteq C_{r}^{*}(\mathcal{B})$ in
the $\|\cdot\|_{E}$ norm, then $(h^{\Phi_{i}})_{i\in\mathcal{I}}$ converges to
the unit section uniformly on compact subsets.

###### Proof.

Suppose that the net $(h_{i})_{i\in\mathcal{I}}$ as above converges to the
unit section uniformly on compact sets. To show that
$\lim_{i\in\mathcal{I}}\|\widetilde{m}_{h_{i}}(b)-b\|_{E}=0$ for $b\in
C_{c}(\mathcal{B})$ we may assume that $b\in C_{c}(\mathcal{B}|_{U})$ where
$U\in S$ is an open bisection. Thus there is $a\in
C_{c}(\mathcal{A}*_{r}G|_{U})$ such that $b(\gamma)=[a(\gamma),c_{U}(\gamma)]$
for all $\gamma\in G$. Since $K:=\overline{\\{\gamma:b(\gamma)\neq 0\\}}$ is
compact we then get

$\displaystyle\|\widetilde{m}_{h_{i}}(b)-b\|_{E}^{2}$
$\displaystyle=\sup_{x\in{G^{(0)}}}\|\left((\widetilde{m}_{h_{i}}(b)-b)^{*}(\widetilde{m}_{h_{i}}(b)-b)\right)(x)\|$
$\displaystyle=\sup_{\gamma\in
K}\|a(\gamma)^{*}(h_{i}(\gamma)-1_{r(\gamma)})^{*}(h_{i}(\gamma)-1_{r(\gamma)})a(\gamma)\|$
$\displaystyle\leq\sup_{\gamma\in
K}\|a(\gamma)^{*}a(\gamma)\|\|h_{i}(\gamma)-1_{r(\gamma)}\|^{2}$
$\displaystyle\leq\|a\|^{2}_{\infty}\cdot\sup_{\gamma\in
K}\|h_{i}(\gamma)-1_{r(\gamma)}\|^{2}\to 0.$

Now suppose conversely that $(\widetilde{m}_{h_{i}})_{i\in\mathcal{I}}$
converges pointwise on $C_{c}(\mathcal{B})\subseteq B_{E}$ to the identity
operator, or that $(h_{i})_{i\in\mathcal{I}}=(h^{\Phi_{i}})_{i\in\mathcal{I}}$
where $(\Phi_{i})_{i\in I}$ converges to identity pointwise on
$C_{c}(\mathcal{B})\subseteq C_{r}^{*}(\mathcal{B})$ in the $\|\cdot\|_{E}$
norm. Let $K\subseteq G$ be a compact set. We need to show that
$\sup_{\gamma\in K}\|h(\gamma)-1_{r(\gamma)}\|\to 0$. There is a finite open
cover $\\{V_{j}\\}_{j=1}^{n}$ of $K$ such that $\overline{V_{j}}$ is compact
and contained in an open bisection $U_{j}\in S$, for each $j=1,...,n$.
Accordingly, there are sections $f_{j}\in C_{c}(\mathcal{B}|_{U_{j}})$ such
that $f_{j}(\gamma)=[1_{r(\gamma)},c_{U_{j}}(\gamma)]$ for
$\gamma\in\overline{V_{j}}$ and $j=1,...,n$. If
$(h_{i})_{i\in\mathcal{I}}=(h^{\Phi_{i}})_{i\in\mathcal{I}}$, then

$\displaystyle\sup_{\gamma\in\overline{V_{j}}}\|h^{\Phi_{i}}(\gamma)-1_{r(\gamma)}\|$
$\displaystyle=\sup_{\gamma\in\overline{V_{j}}}\|\Phi_{i}(f_{j})(\gamma)f_{j}(\gamma)^{*}-f_{j}(\gamma)f_{j}(\gamma)^{*}\|$
$\displaystyle\leq\sup_{\gamma\in\overline{V_{j}}}\|\Phi_{i}(f_{j})(\gamma)-f_{j}(\gamma)\|\leq\|\Phi_{i}(f_{j})-f_{j}\|_{E}\to
0.$

In the remaining case we have
$\sup_{\gamma\in\overline{V_{j}}}\|h_{i}(\gamma)-1_{r(\gamma)}\|=\sup_{\gamma\in\overline{V_{j}}}\|\widetilde{m}_{h_{i}}(f_{j})(\gamma)-f_{j}(\gamma)\|\leq\|\widetilde{m}_{h_{i}}(f_{j})-f_{j}\|_{E}\to
0.$ Thus in both cases $\sup_{\gamma\in
K}\|h(\gamma)-1_{r(\gamma)}\|\leq\max_{j=1,...,n}\sup_{\gamma\in\overline{V_{j}}}\|h(\gamma)-1_{r(\gamma)}\|\to
0$. ∎

##  5\. The Haagerup property for twisted groupoid dynamical systems

In this section we analyse the relations between the geometric Haagerup
property for groupoids, based on conditions introduced in [Tu], and its
$C^{*}$-algebraic counterpart from [DoR], working in the context of general
twisted groupoid $C^{*}$-dynamical systems. For second countable groups the
Haagerup property can be formulated using various equivalent conditions, see
[CCJJV, Definition 1.1.1]. These conditions can be naturally generalised to
étale topological groupoids. However, as we explain below, the relationship
between some of them is subtle when the space of units is not compact.

######  Definition 5.1 ([Tu]).

A _locally proper negative type function_ on an étale groupoid $G$ is a
continuous function $\psi:G\to\mathbb{R}$ which is

  1. (i)

_normalised_ : $\psi|_{{G^{(0)}}}=0$;

  2. (ii)

_symmetric_ : $\psi(g)=\psi(g^{-1})$ for all $g\in G$;

  3. (iii)

_conditionally negative-definite_ , i.e. for each $x\in{G^{(0)}}$,
$n\in\mathbb{N}$, $g_{1},\ldots g_{n}\in G_{x}$ and
$\lambda_{1},\ldots,\lambda_{n}\in\mathbb{R}$ such that
$\sum_{i=1}^{n}\lambda_{i}=0$ we have

$\sum_{i,j=1}^{n}\overline{\lambda_{i}}\psi(g_{i}g_{j}^{-1})\lambda_{j}\leq
0,$

  4. (iv)

_locally proper_ : the function
$(\psi,r,s):G\to\mathbb{R}\times{G^{(0)}}\times{G^{(0)}}$ is proper.

######  Remark 5.2.

Suppose that the groupoid $G$ is $\sigma$-compact. If $G$ acts properly on a
continuous field $H=(H_{x})_{x\in{G^{(0)}}}$ of affine Euclidean spaces, see
[Tu, Definition 3.2, 3.3], then the field admits a continuous section that
allows us to write the action as $\gamma\xi=\alpha_{\gamma}(\xi)+b(\gamma)$
where $\alpha$ is the linear part and $\gamma\to b(\gamma)\in H_{r(\gamma)}$
is a cocycle. Putting $\psi(\gamma):=\|b(\gamma)\|^{2}$ we obtain a locally
proper negative type function. Moreover, every locally proper function of
negative type has this form, see [Tu, Proposition 3.8]. Hence, $G$ admits a
locally proper negative type function if and only if $G$ acts properly on a
continuous field of affine Euclidean spaces. In particular, this holds when
$G$ is amenable, by [Tu, Lemma 3.5].

It is standard that if $h:G\to\mathbb{C}$ is positive-definite then for each
$g\in G$ we have $h(g)=\overline{h(g^{-1})},|h(g)|\leq h(r(g))$. Further if
the function as above is real-valued, then to verify positive-definiteness it
suffices to show the corresponding condition working only with real
coefficients. Normalised symmetric conditionally negative-definite functions
take only non-negative values. Finally if ${G^{(0)}}$ is compact then a
function $\psi:G\to\mathbb{R}$ as above is locally proper if and only if it is
proper in the usual sense (as a function from $G$ to $\mathbb{R}$). We will
make an extensive use of countable exhaustions by compact sets, as discussed
in the lemma below (which is likely well-known, but we could not locate a
reference).

######  Lemma 5.3.

A locally compact Hausdorff space is $\sigma$-compact if and only if it admits
a countable exhaustion by compact sets, i.e. there is a sequence
$(L_{n})_{n\in\mathbb{N}}$ of compact sets whose union is the whole space and
$L_{n}$ is contained in the interior of $L_{n+1}$, for each $n\in\mathbb{N}$.

###### Proof.

The ‘if’ part is clear. To show the ‘only if’ part let
$(K_{n})_{n\in\mathbb{N}}$ be a sequence of compact sets that cover the space.
Put $L_{1}:=K_{1}$. By [Rud, Theorem 2.7], for any open $U$ containing
$L_{1}\cup K_{2}$ there is an open set $V$ with compact closure such that
$L_{1}\cup K_{2}\subseteq V\subseteq\overline{V}\subseteq U$. Put
$L_{2}=\overline{V}$. Apply the same fact to $L_{2}\cup K_{3}$ to find
$L_{3}$, and proceed by induction. ∎

The proof of the next fact uses standard ideas, appearing in different setups
for example in [A-D] in the setting of measured groupoids, in [Jol] for
measured equivalence relations.

######  Proposition 5.4.

Let $G$ be an étale locally compact Hausdorff groupoid. Consider the following
conditions:

  1. (1)

$G$ admits a locally proper negative type function;

  2. (2)

there exists a sequence of continuous positive-definite functions
$(h_{n})_{n\in\mathbb{N}}$ such that

    1. (a)

each $h_{n}$ is _normalised_ : $h_{n}|_{{G^{(0)}}}=1$,

    2. (b)

each $h_{n}$ is _locally $C_{0}$_: for each compact set $K\subseteq{G^{(0)}}$
we have $h_{n}|_{G_{K}^{K}}\in C_{0}(G_{K}^{K})$ where
$G_{K}^{K}:=r^{-1}(K)\cap s^{-1}(K)$,

    3. (c)

the sequence converges to $1$ uniformly on compact sets;

  3. (3)

there is a net $(h_{i})_{i\in\mathcal{I}}\subseteq C_{0}(G)$ of positive-
definite functions, with values in the unit disk and converging to $1$
uniformly on compact subsets of $G$.

Then (1)$\Rightarrow$(2)$\Rightarrow$(3). If $G$ is $\sigma$-compact, then
(1)$\Leftrightarrow$(2) and a net in (3) may be replaced by a sequence. If $G$
is $\sigma$-compact and ${G^{(0)}}$ is a union of compact open sets, then all
the conditions (1),(2),(3) are equivalent.

###### Proof.

(1)$\Rightarrow$(2): Assume that $\psi$ is a locally proper negative type
function, and define

$h_{n}(\gamma)=\exp\left(-\frac{1}{n}\psi(\gamma)\right),\;\;\;\;n\in\mathbb{N},\gamma\in
G.$

Then $h_{n}$ is continuous and normalised. It is positive-definite by well-
known arguments (here we use the fact it is real-valued). Moreover the
sequence $(h_{n})_{n\in\mathbb{N}}$ converges to $1$ (uniformly on compacts,
as $\psi$ is continuous, so bounded on compact sets). It remains to note that
each $h_{n}$ is locally $C_{0}$: without loss of generality we consider $n=1$.
Fix $\epsilon\in(0,1)$ and note that the pre-image of the set
$[0,-\log\epsilon]\times K\times K$ with respect to $(\psi,r,s)$ is compact –
denote it by $L$. Then for $\gamma\in G$ such that $r(\gamma),s(\gamma)\in K$
we have that $|h_{1}(\gamma)|<\epsilon$ if and only if $\gamma\notin L$.

(2)$\Rightarrow$(3): Let $(h_{n})_{n\in\mathbb{N}}$ be as described in (2).
Then it is bounded by $1$ as using positive-definiteness we have
$|h(\gamma)|\leq|h(r(\gamma))|=1$ for $\gamma\in G$. We can modify
$(h_{n})_{n\in\mathbb{N}}$, dropping the normalisation condition, so that its
members are $C_{0}$, and not just locally $C_{0}$. To this end, we fix an
approximate unit $(f_{i})_{i\in I}\subseteq C_{c}({G^{(0)}})^{+}$ for
$C_{0}({G^{(0)}})$. Then the net $((f_{i}\circ r)h_{n}(f_{i}\circ
s))_{(n,i)\in\mathbb{N}\times I}$ satisfies the conditions in (3); in
particular each of the functions is positive-definite and is bounded by $1$.

From now on _we assume $G$ is $\sigma$-compact_. By Lemma 5.3, $G$ admits a
countable exhaustion $(L_{n})_{n\in\mathbb{N}}$ by compact sets. In
particular, if $(h_{i})_{i\in\mathcal{I}}$ is a net as in (3) for each
$n\in\mathbb{N}$ we may choose $i_{n}\in\mathcal{I}$ so that
$\|h_{i_{n}}-1_{L_{n}}\|_{\infty}<1/n$. Then the sequence $h_{i_{n}}$
converges to $1$ uniformly on compact subsets of $G$. Hence the net
$(h_{i})_{i\in\mathcal{I}}$ might be replaced by a sequence.

(2)$\Rightarrow$(1): Fix two sequences of strictly positive numbers
$(\alpha_{n})_{n\in\mathbb{N}}$ and $(\epsilon_{n})_{n\in\mathbb{N}}$ such
that the first one is increasing to infinity, and the series
$\sum_{n\in\mathbb{N}}\alpha_{n}\epsilon_{n}$ is convergent. Then proceed as
follows: for each $n\in\mathbb{N}$ find a function $\phi_{n}:G\to\mathbb{C}$
which is positive-definite, normalised, vanishes at infinity, and such that
$|1-\phi_{n}(\gamma)|<\epsilon_{n}$ for $\gamma\in L_{n}$ (these are naturally
just chosen by picking a subsequence of $(h_{n})_{n\in\mathbb{N}}$). Then
define the function $\psi:G\to\mathbb{R}$ by the formula:

$\psi(\gamma)=\sum_{n=1}^{\infty}\alpha_{n}\textup{Re}(1-\phi_{n}(\gamma)),\;\;\;\gamma\in
G.$

The series above converges uniformly on compact subsets (as any of these is
contained in almost all $L_{n}$), so yields a continuous function, which is
conditionally negative-definite as a sum of conditionally negative-definite
functions. It is obviously normalised.

It remains to check that it is locally proper. Another exhaustion argument
(for example) guarantees that any compact subset of
$\mathbb{R}\times{G^{(0)}}\times{G^{(0)}}$ is contained in a set of the form
$[-M,M]\times K\times K$ for some $M>0$ and $K\subseteq{G^{(0)}}$ compact. Fix
then such $M$ and $K$. Let $k\in\mathbb{N}$ be such that
$\sum_{n=1}^{k}\alpha_{n}>2M$ and for each $n=1,\ldots,k$ find a compact set
$L_{n}\subseteq G$ such that for $\gamma\in G\setminus L_{n}$ such that
$r(\gamma),s(\gamma)\in K$ we have $|\phi_{n}(\gamma)|\leq\frac{1}{2}$. Then
put $L=\bigcup_{n=1}^{k}L_{n}$. If $\gamma\in G\setminus L$,
$r(\gamma),s(\gamma)\in K$, then

$\psi(\gamma)\geq\sum_{n=1}^{k}\alpha_{n}\textup{Re}(1-\phi_{n}(\gamma))\geq\sum_{n=1}^{k}\alpha_{n}(1-|\phi_{n}(\gamma)|)\geq
M.$

Thus the pre-image of the set $[-M,M]\times K\times K$ with respect to
$(\psi,r,s)$ is contained in $L$. This ends the proof of (2)$\Rightarrow$(1).

Assume now that $G$ is $\sigma$-compact and ${G^{(0)}}$ is a union of compact
open sets. We will then show that (3)$\Rightarrow$(1). There is an exhaustion
$(L_{n})_{n\in\mathbb{N}}$ of $G$ such that $r(L_{n})=s(L_{n})$ and this set,
denoted by $K_{n}$, is compact open in ${G^{(0)}}$, for all $n\in\mathbb{N}$.
Indeed, our assumptions clearly imply that there is an exhaustion
$(K_{n})_{n\in\mathbb{N}}$ of ${G^{(0)}}$ by compact open sets and an
exhaustion $(\tilde{L}_{n})_{n\in\mathbb{N}}$ of $G$ by compact sets. Then it
suffices to find an increasing sequence $(m_{n})_{n\in\mathbb{N}}$ such that
$K_{n}\subseteq\tilde{L}_{m_{n}}$ for each $n\in\mathbb{N}$ and put
$L_{n}:=\tilde{L}_{m_{n}}\cap s^{-1}(K_{n})\cap r^{-1}(K_{n})$.

Let $(h_{n})_{n=1}^{\infty}\subseteq C_{0}(G)$ be an approximate unit
consisting of positive-definite functions. We can modify this sequence so that
for each $n\in\mathbb{N}$ the $n$-th element of the sequence is normalised on
$K_{n}$. Indeed, we may find an increasing sequence of natural numbers
$(k_{n})_{n\in\mathbb{N}}$ such that
$\|h_{k_{n}}|_{K_{n}}-1_{K_{n}}\|<\frac{1}{2}$. Consider the new functions
defined by the formula

$\widetilde{h_{n}}(\gamma)=\begin{cases}(h_{k_{n}}(r(\gamma))^{-\frac{1}{2}}h_{k_{n}}(\gamma)(h_{k_{n}}(s(\gamma))^{-\frac{1}{2}},&\gamma\in
s^{-1}(K_{n})\cap r^{-1}(K_{n}),\\\ 0,&\gamma\notin s^{-1}(K_{n})\cap
r^{-1}(K_{n}).\end{cases}$

The function $\widetilde{h_{n}}$ is continuous, as $K_{n}$ is clopen in $G$,
and explicit verification shows it is positive-definite. Thus the sequence
$(\widetilde{h_{n}})_{n=1}^{\infty}$ satisfies all the requirements, including
the normalization condition: $\widetilde{h_{n}}|_{K_{n}}=1$, $n\in\mathbb{N}$.
Choose sequences $(\alpha_{n})_{n\in\mathbb{N}}$ and
$(\epsilon_{n})_{n\in\mathbb{N}}$ as in the proof of the implication
(2)$\Rightarrow$(1). By choosing a subsequence of
$(\widetilde{h_{n}})_{n=1}^{\infty}$ we get a sequence
$(\phi_{n})_{n\in\mathbb{N}}$ such that for each $n\in\mathbb{N}$,
$\phi_{n}\in C_{0}(G_{K_{n}}^{K_{n}})$, $\phi_{n}|_{K_{n}}=1$ and
$|1-\phi_{n}(\gamma)|<\epsilon_{n}$ for $\gamma\in L_{n}$. Then define the
function $\psi:G\to\mathbb{R}$ by the formula:

$\psi(\gamma)=\sum_{n=1}^{\infty}\alpha_{n}\textup{Re}(1_{G_{K_{n}}^{K_{n}}}(\gamma)-\phi_{n}(\gamma)),\;\;\;\gamma\in
G.$

The series above as before converges uniformly on compact subsets so yields a
continuous function, which is conditionally negative-definite as a sum of
conditionally negative-definite functions; the fact that the individual
factors are conditionally negative-definite can be verified directly, and uses
the fact that each $\phi_{n}$ is supported on $G_{K_{n}}^{K_{n}}$. The
function $\psi$ is obviously normalised and symmetric. ∎

######  Remark 5.5.

It follows from the proof above that if $G$ is $\sigma$-compact and
${G^{(0)}}$ is a union of compact open sets, then the net in (3), if it
exists, can be arranged so that
$\sup_{i\in\mathcal{I},x\in{G^{(0)}}}|h_{i}(x)|\leq 1$.

Let $G$ be a $\sigma$-compact étale Hausdorff groupoid. Then ${G^{(0)}}$ is a
union of compact open sets if and only if the $C^{*}$-inclusion
$C_{0}({G^{(0)}})\subseteq C_{r}^{*}(G)$ is _relative $\sigma$-unital_ in the
sense of [Mat, Definition 2.1]. This condition is automatically satisfied when
${G^{(0)}}$ is compact or totally disconnected (equivalently, the inclusion is
unital or the groupoid $G$ is ample). We do not know whether this condition is
necessary to deduce the equivalence between conditions (3) and (1) in
Proposition 5.4. On the other hand, we would like to view the Haagerup
property as an approximation property. Therefore in this paper we choose the
condition (2) as the most convenient one. It generalises to groupoid actions
as follows.

######  Definition 5.6.

Let $(\mathcal{A},G,\Sigma,\alpha)$ be a twisted groupoid $C^{*}$-dynamical
system such that the bundle $\mathcal{A}$ has a continuous unit section. We
say that $(\mathcal{A},G,\Sigma,\alpha)$ has the _Haagerup property_ if if
there is a net $(h_{i})_{i\in\mathcal{I}}$ of continuous positive-definite
sections of $\mathcal{A}*_{r}G$ such that

  1. (i)

each $h_{i}$ is _normalised_ : $h_{i}|_{{G^{(0)}}}$ is the unit section of
$\mathcal{A}$,

  2. (ii)

each $h_{i}$ is _locally $C_{0}$_: for each compact set $K\subseteq{G^{(0)}}$
we have $h_{i}|_{G^{K}_{K}}\in C_{0}(\mathcal{A}|_{K}*_{r}G^{K}_{K})$,

  3. (iii)

the net converges to the unit section uniformly on compact sets: for every
compact $K\subseteq G$ we have $\sup_{\gamma\in
K}\|h_{i}(\gamma)-1_{r(\gamma)}\|\stackrel{{\scriptstyle
i\in\mathcal{I}}}{{\longrightarrow}}0$.

The groupoid $G$ itself has the _Haagerup property_ if the above holds for the
trivial bundle $\mathcal{A}={G^{(0)}}\times\mathbb{C}$ (i.e. there is a net
with properties listed in Proposition 5.4 (2)).

######  Remark 5.7.

We could define the Haagerup property above without the assumption that
$\mathcal{A}$ has a continuous unit section by replacing the last condition by
the following one: for every compact $a\in C_{c}(\mathcal{A}*_{r}G)$ we have
$\|h_{i}a-a\|_{\infty}\stackrel{{\scriptstyle
i\in\mathcal{I}}}{{\longrightarrow}}0.$ However, this would itself imply that
$A:=C_{0}(\mathcal{A})$ has a central approximate unit. Indeed, by Proposition
3.6 positive-definite sections $(h_{i})_{i\in\mathcal{I}}$ are necessarily
central, that is we necessarily have $h_{i}(\gamma)\in Z(M(A_{r(\gamma)}))$
for each $\gamma$ and $i\in\mathcal{I}$. Thus
$(m_{h_{i}}|_{A})_{i\in\mathcal{I}}$ would be a central approximate unit in
$A$.

One of the benefits from the continuous unit section assumption on
$\mathcal{A}$ is that we immediately obtain the desirable fact that if $G$ has
the Haagerup property then every twisted $C^{*}$-dynamical systems
$(\mathcal{A},G,\Sigma,\alpha)$ the Haagerup property.

######  Proposition 5.8.

A $C^{*}$-dynamical system $(\mathcal{A},G,\alpha)$ with ${G^{(0)}}$ compact
and $A:=C_{0}(\mathcal{A})$ unital has the Haagerup property if and only if
there is a net $(h_{i})_{i\in\mathcal{I}}\subseteq C_{0}(\mathcal{A}*_{r}G)$
of positive-definite sections that converge to the unit section uniformly on
compact sets.

###### Proof.

Since ${G^{(0)}}$ is compact, being locally $C_{0}$ is the same as being
$C_{0}$. Hence the “only if” part is trivial. So assume
$(h_{i})_{i\in\mathcal{I}}\subseteq C_{0}(\mathcal{A}*_{r}G)$ is a net of
positive-definite sections that converge to the unit section uniformly on
compact sets. We need to modify the elements of this net so that they are
normalised. Since ${G^{(0)}}$ is compact this is easy to observe: for big
enough $i\in\mathcal{I}$, say $i\geq i_{0}$, we have that
$\|h_{i}|_{{G^{(0)}}}-1_{{G^{(0)}}}\|<\frac{1}{2}$. Consider for $i\geq i_{0}$
the new functions defined by the formula

$\widetilde{h_{i}}(\gamma)=(h_{i}(r(\gamma))^{-\frac{1}{2}}h_{i}(\gamma)\alpha_{\gamma}\Big{(}(h_{i}(s(\gamma))^{-\frac{1}{2}}\Big{)},\qquad\gamma\in
G.$

The function $\widetilde{h_{i}}$ is continuous, as ${G^{(0)}}$ is clopen in
$G$, and in fact $\widetilde{h_{i}}\in C_{0}(\mathcal{A}*_{r}G)$. By
construction $\widetilde{h_{i}}|_{{G^{(0)}}}=1_{{G^{(0)}}}$. Since $h_{i}$ is
positive-definite for any finite set $F\subseteq G_{x}$, $x\in{G^{(0)}}$, we
have
$\left[\alpha_{\gamma}^{-1}\left(h_{i}(\gamma\eta^{-1})\right)\right]_{\eta,\gamma\in
F}\geq 0$ in $M_{|F|}(A_{x})$. Putting
$a_{\gamma}:=\alpha_{\gamma}^{-1}\big{(}(h_{i}(r(\gamma))^{-\frac{1}{2}}\big{)}$,
$\gamma\in F$ we get positive elements in $A_{x}$ and

$\left[\alpha_{\gamma}^{-1}\left(\widetilde{h_{i}}(\gamma\eta^{-1})\right)\right]_{\eta,\gamma\in
F}=\left[a_{\gamma}\alpha_{\gamma}^{-1}\big{(}h_{i}(\gamma\eta^{-1})\big{)}a_{\eta}\right]_{\eta,\gamma\in
F}\geq 0.$

Hence $\widetilde{h_{i}}$ is positive-definite. Let $K\subseteq G$ be a
compact set. The net $(\widetilde{h_{i}}|_{K})_{i\in\mathcal{I}}$ converges
uniformly to the unit section $1_{K}$, as the net $h_{i}$ converges uniformly
to the unit section on $K\cup r(K)\cup s(K)$, and therefore the nets of
sections $K\in\gamma\to h_{i}(r(\gamma))^{-\frac{1}{2}}$ and
$K\in\gamma\to\alpha_{\gamma}\Big{(}h_{i}(s(\gamma))^{-\frac{1}{2}}\Big{)}$
converge uniformly to the unit section on $K$. ∎

######  Remark 5.9.

By Proposition 5.8, Definition 5.6 generalises the Haagerup property
introduced in [DoR, Section 3] for discrete group actions on unital
$C^{*}$-algebras. Indeed, if $G=\Gamma$ is a discrete group and the twist is
trivial, then the groupoid action $(\mathcal{A},G,\alpha)$ is nothing but a
group action $\alpha:\Gamma\curvearrowright A$ on a unital $C^{*}$-algebra
$A=\mathcal{A}_{e}$. This action has the the Haagerup property if and only if
there exists a net $(h_{i})_{i\in\mathcal{I}}$ of functions $h_{i}:\Gamma\to
Z(A)$ which are positive-definite, in the sense of Section 3 of [DoR], and
such that $h_{i}\to 1$ pointwise on $\Gamma$. A similar interpretation is true
for actions of discrete groups twisted by $\mathbb{T}$-valued cocycles: again
our definition means that there exists a net $(h_{i})_{i\in\mathcal{I}}$ of
functions $h_{i}:\Gamma\to Z(A)$ which are positive-definite in a twisted
sense (see for example Definition 4.2 of [BC2]) and converge pointwise to $1$.
Note however that even for non-twisted actions our notion differs from the
Haagerup property of actions of discrete groups on $C^{*}$-algebras as defined
in [MSTT]. Roughly speaking, in [MSTT] the authors allowed arbitrary mapping
$\mathcal{B}$-multipliers.

######  Lemma 5.10.

Let $(G,\Sigma)$ be a twisted groupoid where $G$ is étale locally compact
Hausdorff. Treated as a twisted action on $C_{0}(G^{(0)})$, $(G,\Sigma)$ has
the Haagerup property if and only if $G$ has the Haagerup property.

###### Proof.

We identify $(G,\Sigma)$ with $(\mathcal{A},G,\Sigma,\alpha)$ where
$\mathcal{A}={G^{(0)}}\times\mathbb{C}$, $\alpha_{\gamma}=\textup{id}$ and
$u_{x}(z)=1$ for $\gamma\in G$, $x\in{G^{(0)}}$ and $z\in\mathbb{T}$. Hence
$\mathcal{A}*_{r}G\cong G\times\mathbb{C}$ and therefore
$C_{0}(\mathcal{A}*_{r}G)\cong C_{0}(G)$. It is immediate that the respective
notions of positive-definiteness coincide. ∎

We will now modify the definition of the Haagerup property for a
$C^{*}$-inclusion introduced in [DoR]. Recall that if $E:B\to A\subseteq B$ is
a faithful conditional expectation from a $C^{*}$-algebra $B$ onto $A$, then
$B_{E}$ denotes the right Hilbert $A$-module associated to $E$ via (4.8). In
fact $B_{E}$ is a $C^{*}$-correspondence over $A$ with natural left action. To
define “locally compact” maps on $B_{E}$ we will use the notion of a
_Pedersen’s ideal_ , i.e. a minimal hereditary dense ideal (see [Ped, 5.6]).
Recall that every $C^{*}$-algebra $A$ contains such an ideal; we denote it by
$K(A)$. The ideal $K(A)$ is a linear span of $K(A)_{+}:=\\{a\in
A_{+}:a\leq\sum_{i=1}^{n}a_{k},a_{k}\in K(A)_{0}\\}$ where
$K(A)_{0}:=\\{f(a):a\in A_{+},\,\,f\in C_{c}(0,\infty),f\geq 0\\}$.

######  Lemma 5.11.

Let $\mathcal{A}$ be a $C^{*}$-bundle over a locally compact Hausdorff space
$X$ that admits a continuous unit section. The Pedersen’s ideal of
$A=C^{*}(\mathcal{A})$ is $C_{c}(\mathcal{A})$.

###### Proof.

Recall that Since $\mathcal{A}$ is unital, we may identify $C_{c}(X)$ with a
subalgebra of $C_{c}(\mathcal{A})\subseteq A$. Then obviously
$C_{c}(X)_{+}\subseteq K(A)_{0}$ and every $a\in C_{c}(\mathcal{A})_{+}$ is
dominated by some element in $C_{c}(X)_{+}$. Hence
$C_{c}(\mathcal{A})\subseteq K(A)$. The reverse inclusion follows because
$C_{c}(\mathcal{A})$ is clearly a hereditary ideal in $A$. ∎

######  Definition 5.12.

Let $A\subseteq B$ be a nondegenerate $C^{*}$-inclusion equipped with a
faithful conditional expectation $E:B\to A$. We say that $B$ has the _$E$
-Haagerup property_ if there exists a net $(\Phi_{i})_{i\in\mathcal{I}}$ of
completely positive maps on $B$ such that

  1. (i)

$\Phi_{i}|_{A}=id|_{A}$ for each $i\in\mathcal{I}$;

  2. (ii)

for each $i\in\mathcal{I}$, $\Phi_{i}$ extends to a bounded map
$\widetilde{\Phi}_{i}:B_{E}\to B_{E}$ which is “locally compact”, that is
$k\widetilde{\Phi}_{i}k\in\mathcal{K}(B_{E})$ for every $k\in K(A)$ (here
$(k\widetilde{\Phi}_{i}k)(b):=k\widetilde{\Phi}_{i}(b)k$ for all $b\in
B_{E}$);

  3. (iii)

$\lim_{i\in\mathcal{I}}\|\Phi_{i}(b)-b\|_{B_{E}}=0$ for every $b\in B$.

We will say that a _$C^{*}$ -inclusion $A\subseteq B$ has the Haagerup
property_ if the inclusion has the unique conditional expectation, denoted
$E$, and $B$ has the $E$-Haagerup property.

######  Remark 5.13.

Condition (i) above implies that $\Phi_{i}$ is a contractive $A$-bimodule map.
Indeed, since we assume that $A$ and $B$ have a common (contractive)
approximate unit $(\mu_{\lambda})_{\lambda\in\Lambda}$ we get
$\|\Phi\|=\lim_{\lambda\in\Lambda}\|\Phi_{i}(\mu_{\lambda})\|=\lim_{\lambda\in\Lambda}\|\mu_{\lambda}\|=1$.
Hence the multiplicative domain of $\Phi_{i}$ is equal to $\\{b\in
B:\Phi_{i}(bb^{*})=\Phi_{i}(b)\Phi_{i}(b^{*}),\,\,\Phi_{i}(b^{*}b)=\Phi_{i}(b^{*})\Phi_{i}(b)\\}$.
This together with (i) implies that $\Phi_{i}$ is an $A$-bimodule map.

######  Remark 5.14.

A sufficient condition for a completely positive map $\Phi:B\to B$ to extend
to a bounded map $\widetilde{\Phi}:B_{E}\to B_{E}$ with
$\|\widetilde{\Phi}\|\leq m$ (where $m>0$), is that $E\circ\Phi\leq mE$.
Indeed, for every $b\in B$ we then have

$E(\Phi(b)^{*}\Phi(b))\leq\|\Phi\|E(\Phi(b^{*}b))\leq m\|\Phi\|E(b^{*}b),$

cf. [DoR]. In fact, in the definition of the Haagerup property it is usually
assumed (cf. for instance [DoR], [Don]) that $E\circ\Phi\leq E$. We did not
include it in our definition as in the context of twisted groupoid crossed
products this can always be arranged, see Remark 5.18.

######  Remark 5.15.

Dong and Ruan speak simply of the $A$-Haagerup property, but this is not quite
precise, as shown by Suzuki in [Suz]: in general the property depends not only
on the pair $(A,B)$ but also on the choice of a conditional expectation $E$.
On the other hand, there are natural situations when the inclusion $A\subseteq
B$ has a unique conditional expectation. This holds for instance when $B$ is
the reduced cross-sectional $C^{*}$-algebra of a Fell bundle $\mathcal{B}$
over a topologically free étale locally compact Hausdorff groupoid $G$, and
more generally for noncommutative Cartan subalgebras $A\subseteq B$, cf.
[Exe], [KwM2].

It is immediate that our definition is consistent with the one in [Don]. It is
also consistent with the one given by Dong and Ruan [DoR], concerning unital
inclusions, if we assume further that the conditional expectation is tracial –
as is the case for the canonical expectation related to crossed products of
actions of discrete groups on commutative algebras, being the main focus of
[DoR].

######  Proposition 5.16.

Let $A\subseteq B$ be a unital inclusion of $C^{*}$-algebras and let $E:B\to
A\subseteq B$ be a faithful tracial conditional expectation. Then $B$ has the
$E$-Haagerup property if and only if there exists a net
$(\Phi_{i})_{i\in\mathcal{I}}$ of completely positive $A$-bimodule maps on $B$
such that, for every $i\in\mathcal{I}$, $\Phi_{i}$ extends to a compact map
$\widetilde{\Phi}_{i}\in\mathcal{K}(B_{E})$ and
$\lim_{i\in\mathcal{I}}\|\Phi_{i}(b)-b\|_{B_{E}}=0$ for every $b\in B$.

###### Proof.

Since $1\in A\subseteq B$ “locally compact” and “compact” on $B_{E}$ is the
same. Hence the “only if” part is clear by Remark 5.13. Let then
$(\Phi_{i})_{i\in\mathcal{I}}$ be completely positive $A$-bimodule maps on $B$
such that for every $i\in\mathcal{I}$ we have
$\widetilde{\Phi}_{i}\in\mathcal{K}(B_{E})$ and
$\lim_{i\in\mathcal{I}}\|\Phi_{i}(b)-b\|_{B_{E}}=0$ for $b\in B$. Find
$i_{0}\in\mathcal{I}$ such that $\|\Phi_{i}(1)-1\|<\frac{1}{2}$ for $i\geq
i_{0}$. Note that $\Phi_{i}(1)$ is a positive element commuting with $A$ (as
$\Phi_{i}$ is an $A$-bimodule map). Thus putting, for $i\geq i_{0}$,

$\Psi_{i}(b):=\Phi_{i}(1)^{-\frac{1}{2}}\Phi_{i}(b)\Phi_{i}(1)^{-\frac{1}{2}},\qquad
b\in B,$

we get a completely positive map $\Psi_{i}$ on $B$ such that
$\Psi_{i}|_{A}=id|_{A}$ and $\Psi_{i}$ extends to a compact map
$\widetilde{\Psi}_{i}=\Phi_{i}(1)^{-\frac{1}{2}}\widetilde{\Phi}_{i}\Phi_{i}(1)^{-\frac{1}{2}}$
on $B_{E}$. Moreover, for $b\in B$ we have $\lim_{i\geq
i_{0}}\|\Psi_{i}(b)-b\|_{B_{E}}=0$ by a $3\epsilon$-argument. Note that this
is the place where we use the tracial property of $E$, so that we have the
‘right’ estimate $\|bc\|_{B_{E}}\leq\|c\|_{\infty}\|b\|_{B_{E}}$ for $b,c\in
B$. ∎

Using the facts from previous sections we get the following far reaching
generalisation of [DoR, Theorem 3.6], one of the main results of our paper.

######  Theorem 5.17.

Let $B:=C^{*}_{r}(\mathcal{A},G,\Sigma,\alpha)$ be the reduced crossed product
of the twisted action $(\mathcal{A},G,\Sigma,\alpha)$ of an étale locally
compact Hausdorff groupoid $G$ on the $C^{*}$-algebra $A:=C_{0}(\mathcal{A})$,
and let $E:B\to A$ be the canonical conditional expectation. Then $B$ has the
$E$-Haagerup property if and only if $(\mathcal{A},G,\Sigma,\alpha)$ has the
Haagerup property.

###### Proof.

Suppose that $(\mathcal{A},G,\Sigma,\alpha)$ has the Haagerup property. Let
$(h_{i})_{i\in\mathcal{I}}$ be a net as in Definition 5.6. By Proposition 3.7
we have the corresponding completely positive contractive multiplier maps
$m_{h_{i}}:C_{r}^{*}(\mathcal{B})\to C_{r}^{*}(\mathcal{B})$. By this
proposition $\Phi_{i}:=m_{h_{i}}$ is an $A$-bimodule map. By Definition 5.6
(i) we have $\Phi_{i}|_{A}=id|_{A}$ and the form of $E$ implies that
$E\circ\Phi_{i}=E$. Hence $(\widetilde{\Phi}_{i})_{i\in\mathcal{I}}$ is
uniformly bounded by $1$, and therefore it converges pointwise to the identity
operator on $B_{E}$, by Lemma 4.5. Finally, let $k\in
K(A)=C_{c}(\mathcal{A})$. Take any positive function $h\in C_{c}({G^{(0)}})$
which is equal to $1$ on the support of $k$. Then
$\widetilde{h}_{i}(\gamma):=h(r(\gamma))h_{i}(\gamma)h(s(\gamma))$, $\gamma\in
G$, is a positive-definite section of $\mathcal{A}*_{r}G$. Denote the support
of $h$ by $K$. Since $\widetilde{h}_{i}$ vanishes outside $G_{K}^{K}$ and
$h_{i}|_{K}^{K}\in C_{0}(\mathcal{A}|_{K}*_{r}G^{K}_{K})$ (by Definition 5.6
(ii)) we get $\widetilde{h}_{i}\in C_{0}(\mathcal{A}*_{r}G)$. Therefore
$\widetilde{m}_{\widetilde{h}_{i}}\in\mathcal{K}(B_{E})$ by Proposition 4.4
(2). Hence
$k\Phi_{i}k=k\widetilde{m}_{\widetilde{h}_{i}}k\in\mathcal{K}(B_{E})$.

Now let $(\Phi_{i})_{i\in\mathcal{I}}$ be any net as in Definition 5.12. Let
$h^{\Phi_{i}}$, $i\in\mathcal{I}$, be given by Proposition 4.1. The formula
(4.7) and the equality $\Phi_{i}|_{A}=id|_{A}$ imply that
$h^{\Phi_{i}}|_{{G^{(0)}}}=1_{{G^{(0)}}}$. By Lemma 4.5,
$(h^{\Phi_{i}})_{i\in\mathcal{I}}$ converges to the unit section uniformly on
compact sets. To see that $h_{i}|_{G^{K}_{K}}\in
C_{0}(\mathcal{A}|_{K}*_{r}G^{K}_{K})$, for any compact set
$K\subseteq{G^{(0)}}$, fix such a $K$ and take a positive function $h\in
C_{c}({G^{(0)}})$ equal to $1$ on $K$. Multiplying it by the unit section of
$\mathcal{A}$ we get an element $k\in K(A)$. It follows from (4.7) that
$h^{k\Phi_{i}k}(\gamma)=h(r(\gamma))h^{\Phi_{i}}(\gamma)h(s(\gamma))$,
$\gamma\in G$. Since $k\Phi_{i}k\in\mathcal{K}(B_{E})$, we get
$h^{k\Phi_{i}k}\in C_{0}(\mathcal{A}*_{r}{G^{(0)}})$ by Proposition 4.4 (4).
Therefore $h^{\Phi_{i}}|_{G^{K}_{K}}=h^{k\Phi_{i}k}|_{G^{K}_{K}}\in
C_{0}(\mathcal{A}|_{K}*_{r}G^{K}_{K})$. This verifies that
$(\mathcal{A},G,\Sigma,\alpha)$ has the Haagerup property. ∎

######  Remark 5.18.

It follows from the proof above that
$B:=C^{*}_{r}(\mathcal{A},G,\Sigma,\alpha)$ has the $E$-Haagerup property if
and only if we can find a net $(\Phi_{i})_{i\in\mathcal{I}}$ as in Definition
5.12 with the additional property that $E\circ\Phi_{i}=E$ for each
$i\in\mathcal{I}$.

Theorem 5.17, apart from generalising the main result of [DoR], has an
immediate consequence for the relative Haagerup property with respect to a
Cartan subalgebra.

######  Corollary 5.19.

Let $A\subseteq B$ be a (commutative) Cartan subalgebra of a $C^{*}$-algebra
$B$. Then $A\subseteq B$ has the Haagerup property if and only if the
associated Weyl groupoid has the Haagerup property.

###### Proof.

We may assume identifications $A=C_{0}({G^{(0)}})$ and $B=C_{r}^{*}(G,\Sigma)$
for a twisted groupoid $(G,\Sigma)$ (this is well known when $B$ is separable
[Ren5] but holds in general, see [KwM2] or [Raa]). The twisted groupoid
$(G,\Sigma)$ is uniquely determined by the inclusion $A\subseteq B$, and $G$
is called the Weyl groupoid of $A\subseteq B$. Thus it suffices to combine
Theorem 5.17 and Lemma 5.10. ∎

As another immediate consequence, we obtain a generalisation of [DoR, Theorem
4.2] from countable groups to metric spaces with bounded geometry.

######  Corollary 5.20.

Let $X$ be a metric space with bounded geometry and $G(X)$ be its coarse
groupoid. Then $X$ coarsely embeds into a Hilbert space if and only if there
is a net $(h_{i})_{i\in\mathcal{I}}\subseteq C_{0}(G(X))$ of positive-definite
functions, converging to $1$ uniformly on compact subsets of $G(X)$ if and
only if for all (equivalently, some) twist $\Sigma$ over $G(X)$, the
$C^{*}$-inclusion $C(G(X)^{(0)})\subseteq C_{r}^{*}(G(X),\Sigma)$ has the
Haagerup property.

###### Proof.

By [STY, Proposition 3.2], $G(X)$ is a locally compact Hausdorff, principal
étale groupoid. Moreover, $G(X)$ is $\sigma$-compact and has a compact unit
space. By [STY, Theorem 5.4], $X$ coarsely embeds into a Hilbert space if and
only if $G(X)$ admits a proper negative type function. Hence the assertion
follows from Proposition 5.4, Lemma 5.10 and Corollary 5.19. ∎

######  Remark 5.21.

Recall that the coarse groupoid $G(X)$ of a bounded geometry metric space $X$
is amenable if and only if $X$ has property A if and only if
$C_{r}^{*}(G(X),\Sigma)$ is nuclear for all (equivalently, some) twist
$\Sigma$ over $G(X)$ (see [STY, Theorem 5.3] and [Tak, Theorem 5.4]). On the
other hand, there exist metric spaces with bounded geometry which do not have
property A but coarsely embed into a Hilbert space (see e.g. [AGS, Osa]).

##  6\. The Haagerup property and the Universal Coefficient Theorem

In this section we discuss the consequences of the results of the last section
for the questions related to the Universal Coefficient Theorem. We recall that
the Universal Coefficient Theorem (UCT) for $C^{*}$-algebras was introduced by
Rosenberg and Schochet in [RoS]. A separable $C^{*}$-algebra $A$ is said to
satisfy the UCT if for every separable $C^{*}$-algebra $B$ the following
natural sequence

$0\rightarrow\text{Ext}(K_{*}(A),K_{*-1}(B))\rightarrow
KK_{*}(A,B)\rightarrow\text{Hom}(K_{*}(A),K_{*}(B))\rightarrow 0$

is exact. A separable $C^{*}$-algebra satisfies the UCT if and only if it is
$KK$-equivalent to a commutative $C^{*}$-algebra (see [Bla, Theorem 23.10.5]).
Although there exist exact non-nuclear $C^{*}$-algebras that do not satisfy
the UCT (see [Ska])111It is also worth noting that there exist separable non-
exact $C^{*}$-algebras which satisfy the UCT. Indeed, this is the case for the
reduced and full group $C^{*}$-algebras of $\Gamma$, where $\Gamma$ is a
finitely generated non-exact group with the Haagerup property (as exhibited in
[Osa, Theorem 2])., it is still open whether all separable nuclear
$C^{*}$-algebras satisfy the UCT. This question is often refereed to as the
UCT problem, and it is receiving renewed interest due to the recent
breakthrough results in the classification program of separable simple nuclear
$C^{*}$-algebras satisfying the UCT (see e.g. [Win1, Win2]).

It follows from Tu’s remarkable paper [Tu] that all $C^{*}$-algebras
associated to groupoids with the Haagerup property satisfy the UCT. Building
on Tu’s techniques, Barlak and Li have proved that this also holds for twisted
amenable étale groupoids (see [BaL]), using recent articles of Takeishi [Tak]
and of van Erp and Williams [ErW]. In this section, we generalise the key
results of [BaL] concerning UCT, following the same idea. But instead of [ErW]
we exploit the Stabilization Theorem [IKSW] of Ionescu, Kumjian, Sims and
Williams, and instead of amenability we assume the Haagerup property. This
enables us to obtain a stronger assertion under weaker assumptions.

Recall that a Fell bundle is said to be _separable_ if each of its fibers is
separable.

######  Theorem 6.1.

Let $\mathcal{B}$ be a continuous saturated and separable Fell bundle over a
second countable locally compact Hausdorff étale groupoid $G$. Assume that
$C^{*}(\mathcal{B}|_{{G^{(0)}}})$ is of type I and $G$ has the Haagerup
property. Then both $C^{*}_{r}(\mathcal{B})$ and $C^{*}(\mathcal{B})$ satisfy
the UCT.

###### Proof.

We claim that we may reduce the situation to the case where $\mathcal{B}$ is