$\mathrm{C}^{*}$-algebras related to twisted groupoid actions. Once again we
first discuss the case of general Fell bundles.

Let $\mathcal{A}=\\{A_{g}\\}_{g\in\mathcal{G}}$ be a Fell bundle over
$\mathcal{G}$. We begin this section by introducing various forms of “regular
induced representations” for $\mathcal{A}$ using the
$\mathrm{C}^{*}(\mathcal{A})$-$\mathrm{C_{0}}(\mathcal{A}|_{X})$-$\mathrm{C}^{*}$-correspondence
$\mathrm{C_{0}}(\ell^{2}(\mathcal{A}))$, and a $\mathrm{C}^{*}$-correspondence
bundle from $\mathcal{A}|_{X}=\\{A_{x}\\}_{x\in X}$ to another
$\mathrm{C}^{*}$-bundle $\mathcal{B}=\\{B_{x}\\}_{x\in X}$. These do not seem
to have appeared explicitly in the literature.

######  Definition 6.1.

Let $\mathcal{E}=\\{E_{x}\\}_{x\in X}$ be a $\mathrm{C}^{*}$-correspondence
bundle from $\mathcal{A}|_{X}=\\{A_{x}\\}_{x\in X}$ to
$\mathcal{B}=\\{B_{x}\\}_{x\in X}$. The _regular representation of
$\mathrm{C}^{*}(\mathcal{A})$ induced by $\mathcal{E}$_ is given by the left
action
$\Lambda^{\mathcal{E}}:=\Lambda\otimes\mathrm{id}_{\mathrm{C_{0}}(\mathcal{E})}$
of $\mathrm{C}^{*}(\mathcal{A})$ on the
$\mathrm{C}^{*}(\mathcal{A})$-$\mathrm{C_{0}}(\mathcal{B})$-$\mathrm{C}^{*}$-correspondence
$\mathrm{C_{0}}(\ell^{2}(\mathcal{A}))\otimes_{\mathrm{C_{0}}(\mathcal{A}|_{X})}\mathrm{C_{0}}(\mathcal{E})$
(see Example 3.10).

######  Remark 6.2.

By construction $\ker\Lambda\subseteq\ker\Lambda^{\mathcal{E}}$ and we have
$\ker\Lambda=\ker\Lambda^{\mathcal{E}}$ whenever $\mathcal{E}$ is _faithful_ ,
i.e. the left action of $\mathrm{C_{0}}(\mathcal{A}|_{X})$ on
$\mathrm{C_{0}}(\mathcal{E})$ is faithful (this holds when there is a dense
set of points $X_{0}\subseteq X$ such that the
$A_{x}$-$B_{x}$-$\mathrm{C}^{*}$-correspondence $E_{x}$ is faithful for all
$x\in X_{0}$). Hence $\Lambda^{\mathcal{E}}$ factors through a representation
$\Lambda^{\mathcal{E}}_{\operatorname{red}}$ of
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A})$ and
$\Lambda^{\mathcal{E}}_{\operatorname{red}}$ is faithful if $\mathcal{E}$ is
so.

######  Remark 6.3.

We give an alternative description of $\Lambda^{\mathcal{E}}$ as follows. For
each $x\in X$, we have a $\mathrm{C}^{*}$-correspondence
$\ell^{2}(\mathcal{A}|_{\mathcal{G}_{x}})\otimes_{A_{x}}E_{x}$ from
$\mathrm{C}^{*}(\mathcal{A})$ to $B_{x}$. As a right Hilbert module it is
equal to the direct sum $\ell^{2}(\mathcal{A}\otimes_{A_{x}}E_{x})$ of Hilbert
$B_{x}$-modules
$\mathcal{A}\otimes_{A_{x}}E_{x}:=\\{A_{g}\otimes_{A_{x}}E_{x}\\}_{g\in\mathcal{G}_{x}}$.
The multiplication of $\mathcal{A}$ yields bounded linear maps
$A_{gh^{-1}}\otimes_{A_{x}}A_{h}\otimes_{A_{x}}E_{x}\to
A_{g}\otimes_{A_{x}}E_{x}$, $g,h\in\mathcal{G}_{x}$, that we write again as
‘multiplication’ maps whenever $s(g)=s(h)$. Then the formula

(6.1)
$[\Lambda_{x}^{\mathcal{E}}(a)\xi](g):=\sum_{h\in\mathcal{G}_{x}}a(gh^{-1})\cdot\xi(h),\qquad
a\in\mathrm{C_{c}}(\mathcal{A}),\
\xi\in\ell^{2}(\mathcal{A}\otimes_{A_{x}}E_{x}),\ g\in\mathcal{G}_{x},$

makes sense and gives the $*$-homomorphism
$\Lambda_{x}^{\mathcal{E}}:\mathrm{C}^{*}(\mathcal{A})\to\mathcal{L}(\ell^{2}(\mathcal{A}\otimes_{A_{x}}E_{x}))$
induced from $\Lambda_{x}$ by $E_{x}$. We equip the bundle
$\ell^{2}(\mathcal{A}\otimes_{\mathcal{A}|_{X}}\mathcal{E}):=\\{\ell^{2}(\mathcal{A}\otimes_{A_{x}}E_{x})\\}_{x\in
X}$ with the topology generated by declaring sections of the form
$x\mapsto\oplus_{g\in\mathcal{G}_{x}}\xi(g)$ to be continuous for all
$\xi\in\mathrm{C_{0}}(s^{*}\mathcal{E})$, i.e. requesting that $\mathcal{G}\ni
g\mapsto\xi(g)\in E_{s(g)}\subseteq\mathcal{E}$ is continuous. Then we get an
isomorphism of Hilbert $\mathrm{C_{0}}(\mathcal{B})$-modules

$\mathrm{C_{0}}(\ell^{2}(\mathcal{A}))\otimes_{\mathrm{C_{0}}(\mathcal{A}|_{X})}\mathrm{C_{0}}(\mathcal{E})\cong\mathrm{C_{0}}(\ell^{2}(\mathcal{A}\otimes_{\mathcal{A}|_{X}}\mathcal{E}))$

that maps
$a\otimes\xi\in\mathrm{C_{0}}(\ell^{2}(\mathcal{A}))\otimes_{\mathrm{C_{0}}(\mathcal{A}|_{X})}\mathrm{C_{0}}(\mathcal{E})$
to
$\widehat{a\otimes\xi}\in\mathrm{C_{0}}(\ell^{2}(\mathcal{A}\otimes_{\mathcal{A}|_{X}}\mathcal{E}))$
where $\widehat{a\otimes\xi}(x)=a(x)\otimes\xi(x)$. Under this isomorphism the
_induced regular representation_ is
$\Lambda^{\mathcal{E}}:\mathrm{C}^{*}(\mathcal{A})\to\mathcal{L}(\mathrm{C_{0}}(\ell^{2}(\mathcal{A}\otimes_{\mathcal{A}|_{X}}\mathcal{E})))$,
where

$(\Lambda^{\mathcal{E}}(a)\xi)(x)=\Lambda_{x}^{\mathcal{E}}(a)\xi(x)=\bigoplus_{g\in\mathcal{G}_{x}}\sum_{h\in\mathcal{G}_{x}}a(gh^{-1})\cdot\xi(h),$

$a\in\mathrm{C_{c}}(\mathcal{A}),\xi\in\mathrm{C_{0}}(\ell^{2}(\mathcal{A}\otimes_{\mathcal{A}|_{X}}\mathcal{E}))$,
$x\in X$.

We will now specify the context to Fell bundles coming from twisted groupoid
actions. Assume then that $\mathcal{A}$ carries a twisted action $(\alpha,u)$
of $\mathcal{G}$. Induced regular representations of
$\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u)})$ can be described as follows. To any
$\mathcal{A}$-$\mathcal{B}$-$\mathrm{C}^{*}$-correspondence bundle
$\mathcal{E}$ we associate the _regularized
$\mathcal{A}$-$\mathcal{B}$-$\mathrm{C}^{*}$-correspondence bundle_
$\ell^{2}(s^{*}\mathcal{E})=\\{\ell^{2}(s^{*}\mathcal{E})_{x}\\}_{x\in X}$
whose fibers are direct sums

$\ell^{2}(s^{*}\mathcal{E})_{x}=\bigoplus_{g\in\mathcal{G}_{x}}E_{x}$

of $A_{x}$-$B_{x}$-$\mathrm{C}^{*}$-correspondences $E_{x}$. We equip
$\ell^{2}(s^{*}\mathcal{E})$ with the unique topology making it a Banach
bundle and such that $\mathrm{C_{c}}(\ell^{2}(s^{*}\mathcal{E}))$ contains
$x\mapsto\bigoplus_{g\in\mathcal{G}_{x}}\xi(g)$ for
$\xi\in\mathrm{C_{c}}(s^{*}\mathcal{E})$. Then $\ell^{2}(s^{*}\mathcal{E})$ is
an $\mathcal{A}$-$\mathcal{B}$-$\mathrm{C}^{*}$-correspondence bundle.
Alternatively, the $\mathrm{C}^{*}$-correspondence bundle
$\ell^{2}(s^{*}\mathcal{E})$ is determined by the
$\mathrm{C_{0}}(X)$-$\mathrm{C}^{*}$-correspondence from
$\mathrm{C_{0}}(\mathcal{A})$ to $\mathrm{C_{0}}(\mathcal{B})$ given by the
completion of $\mathrm{C_{c}}(s^{*}\mathcal{E})$ with structure determined by
the operations

$\langle\xi,\zeta\rangle_{\mathrm{C_{0}}(\mathcal{B})}(x):=\sum_{g\in\mathcal{G}_{x}}\langle\xi(g),\zeta(g)\rangle_{B_{x}},\qquad
a\cdot\xi\cdot b(g):=a(s(g))\xi(g)b(s(g))$

for $\xi,\zeta\in\mathrm{C_{c}}(s^{*}\mathcal{E})$,
$a\in\mathrm{C_{0}}(\mathcal{A})$, $b\in\mathrm{C_{0}}(\mathcal{B})$, $x\in
X$, $g\in\mathcal{G}$.

######  Proposition 6.4.

Let $(\alpha,u)$ be a twisted action of $\mathcal{G}$ on $\mathcal{A}$ and let
$\mathcal{E}$ be an
$\mathcal{A}$-$\mathcal{B}$-$\mathrm{C}^{*}$-correspondence bundle for another
$\mathrm{C}^{*}$-bundle $\mathcal{B}$ over $X$. The induced regular
representation $\Lambda^{\mathcal{E}}$ of
$\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u)})$ is naturally unitarily equivalent
to a representation
$\pi^{\mathcal{E}}:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u)})\to\mathcal{L}(\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E})))$,
where

$(\pi^{\mathcal{E}}(a)\xi)(x)=\pi_{x}^{\mathcal{E}}(a)\xi(x),\qquad
a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u)}),\
\xi\in\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E})),\ x\in X,$

and
$\pi_{x}^{\mathcal{E}}:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u)})\to\mathcal{L}(\bigoplus_{g\in\mathcal{G}_{x}}E_{x})$
is a representation of $\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u)})$ on the
Hilbert $B_{x}$-module $\bigoplus_{g\in\mathcal{G}_{x}}E_{x}$, given by

$[\pi_{x}^{\mathcal{E}}(a)\xi](g):=\sum_{h\in\mathcal{G}_{x}}\alpha_{g}^{-1}\left(a(gh^{-1})u(gh^{-1},h)\right)\xi(h),\qquad
a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u)}),\
\xi\in\bigoplus_{h\in\mathcal{G}_{x}}E_{x},\ g\in\mathcal{G}_{x}.$

In particular, $\pi^{\mathcal{E}}$ descends to a representation of
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u)})$, which is
faithful if $\pi^{\mathcal{E}}$ is faithful on $\mathrm{C_{0}}(\mathcal{A})$.

###### Proof.

We use Remark 6.3 to treat $\Lambda^{\mathcal{E}}$ as a representation on the
bundle
$\ell^{2}(\mathcal{A}\otimes_{\mathcal{A}|_{X}}\mathcal{E}):=\\{\ell^{2}(\mathcal{A}\otimes_{A_{x}}E_{x})\\}_{x\in
X}$ where
$\mathcal{A}\otimes_{A_{x}}E_{x}:=\\{A_{g}\otimes_{A_{x}}E_{x}\\}_{g\in\mathcal{G}_{x}}$
for $x\in X$. For each $g\in\mathcal{G}_{x}$, we have an isomorphism
$A_{g}\otimes_{A_{x}}E_{x}\cong E_{x}$ of $B_{x}$-Hilbert modules given by
$a\otimes\xi\mapsto\alpha_{g}^{-1}(a)\xi$. These isomorphisms give a unitary
isomorphism
$\ell^{2}(\mathcal{A}\otimes_{A_{x}}E_{x})\cong\bigoplus_{g\in\mathcal{G}_{x}}E_{x}$
under which (6.1) translates to the formula for $\pi_{x}^{\mathcal{E}}(a)$ as
in the assertion. These unitaries yield an isomorphism
$\ell^{2}(\mathcal{A}\otimes_{\mathcal{A}|_{X}}\mathcal{E})\cong\ell^{2}(s^{*}\mathcal{E})$
of Hilbert module bundles that intertwines $\pi^{\mathcal{E}}$ and
$\Lambda^{\mathcal{E}}$. ∎

######  Example 6.5 (Regular representation revisited).

Suppose that $(\alpha,u)$ is a twisted $\mathcal{G}$-action on
$\mathcal{A}=\\{A_{x}\\}_{x\in X}$. Then the bundle
$\mathcal{A}=\\{A_{x}\\}_{x\in X}$ can be viewed as a faithful
$\mathrm{C}^{*}$-correspondence bundle from $\mathcal{A}$ to $\mathcal{A}$.
Hence, by Proposition 6.4, it induces a representation
$\pi^{\mathcal{A}}:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u)})\to\mathcal{L}(\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{A})))$,
where
$\ell^{2}(s^{*}\mathcal{A}):=\\{\bigoplus_{g\in\mathcal{G}_{x}}A_{x}\\}_{x\in
X}$, which factors through a faithful representation of
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u)})$. In fact, the
formula

$W(\xi)(x)=\bigoplus_{g\in\mathcal{G}_{x}}\alpha_{g}(\xi(g)),\qquad\xi\in\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{A})),\
x\in X,$

defines a Hilbert $A$-module unitary
$W:\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{A}))\to\mathrm{C_{0}}(\ell^{2}(\mathcal{A}^{(\alpha,u)}))$
that intertwines $\pi^{\mathcal{A}}$ and the regular representation
$\Lambda:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u)})\to\mathcal{L}(\mathrm{C_{0}}(\ell^{2}(\mathcal{A}^{(\alpha,u)})))$
from (4.1).

######  Example 6.6.

For the trivial bundle $\mathcal{A}=\\{\mathbb{C}\\}_{x\in X}$ we are treating
a twisted groupoid $(\mathcal{G},u)$ as in Example 4.12. In this case, the
induced regular representation $\pi^{\mathcal{E}}\colon
C^{*}(\mathcal{G},u)\to\mathcal{L}(\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E})))$
acting on the $\mathcal{A}$-$\mathcal{B}$-correspondence $\mathcal{E}$ is
given by:

$(\pi^{\mathcal{E}}_{x}(f)\xi)(g)=\sum_{h\in\mathcal{G}_{x}}f(gh^{-1})\xi(h)u(gh^{-1},h)=\sum_{h\in\mathcal{G}^{r(g)}}f(h)\xi(h^{-1}g)u(h,h^{-1}g)$

for $f\in\mathrm{C_{c}}(\mathcal{G})$,
$\xi\in\oplus_{g\in\mathcal{G}_{x}}E_{x}=\ell^{2}(\mathcal{G}_{x},E_{x})$ and
$g\in\mathcal{G}_{x}$. Here each $E_{x}$ is just a Hilbert module over $B_{x}$
because $A_{x}=\mathbb{C}$. Of course, this is a special case of Proposition
6.4 where the action $\alpha$ is trivial. The above representation generalizes
the ordinary regular representation $\lambda^{u}$ from (4.2) by adding the
coefficient bundle $\mathcal{E}$. For an equivariant representation $L$ on
$\mathcal{E}$, its associated regularized equivariant representation
$L^{\mathcal{G}}$ acts on $\ell^{2}(s^{*}\mathcal{E})$ by the formula (6.2).
There is no further simplification here, except that each $E_{x}$ is just a
Hilbert $B_{x}$-module. Further assuming that also
$\mathcal{B}=\\{\mathbb{C}\\}_{x\in X}$ is the one-dimensional trivial bundle,
then equivariant representations are described by continuous Hilbert
$\mathcal{G}$-bundles, that is, continuous Hilbert bundles
$\mathcal{H}=\\{H_{x}\\}_{x\in X}$ together with a continuous family of
unitaries $L_{g}\colon H_{s(g)}\xrightarrow{\sim}H_{r(g)}$ satisfying
$L_{g}L_{h}=L_{gh}$ for all $(g,h)\in\mathcal{G}^{(2)}$, as in Example 5.6.

Note that the (regularized)
$\mathrm{C_{0}}(\mathcal{A})$-$\mathrm{C_{0}}(\mathcal{B})$-$\mathrm{C}^{*}$-correspondence
$\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))$ contains
$\mathrm{C_{0}}(\mathcal{E})$ as a sub-$\mathrm{C}^{*}$-correspondence (the
embedding is given by the inclusions
$\mathrm{C_{c}}(\mathcal{E})\subseteq\mathrm{C_{c}}(s^{*}\mathcal{E})\subseteq\ell^{2}(s^{*}\mathcal{E})$).
Any equivariant representation of $\mathcal{G}$ on $\mathcal{E}$ extends
naturally to $\ell^{2}(s^{*}\mathcal{E})$ as follows.

######  Lemma 6.7.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over
$X:=\mathcal{G}^{(0)}$, carrying twisted actions $(\alpha,u_{\alpha})$ and
$(\beta,u_{\beta})$ of $\mathcal{G}$. An
$(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$ equivariant representation
$L=\\{L_{g}\\}_{g\in\mathcal{G}}$ of $\mathcal{G}$ on a
$\mathrm{C_{0}}(\mathcal{A})$-$\mathrm{C_{0}}(\mathcal{B})$-$\mathrm{C}^{*}$-correspondence
$\mathcal{E}$ induces an $(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$ equivariant
representation $L^{\mathcal{G}}:=\\{L^{\mathcal{G}}_{g}\\}_{g\in\mathcal{G}}$
of $\mathcal{G}$ on $\ell^{2}(s^{*}\mathcal{E})$, where for $g\in\mathcal{G}$
the map
$L^{\mathcal{G}}_{g}:\ell^{2}(s^{*}\mathcal{E})_{s(g)}\to\ell^{2}(s^{*}\mathcal{E})_{r(g)}$
is given by

(6.2)
$L^{\mathcal{G}}_{g}(\xi)(t)=L_{g}(\xi(tg)),\qquad\xi\in\ell^{2}(s^{*}\mathcal{E})_{s(g)},\,\,t\in\mathcal{G}_{r(g)}.$

###### Proof.

Given $g\in\mathcal{G}$ and
$\xi,\zeta\in\ell^{2}(s^{*}\mathcal{E})_{s(g)}=\bigoplus_{h\in\mathcal{G}_{s(g)}}E_{s(g)}$,
we have

$\begin{split}\langle
L^{\mathcal{G}}_{g}(\xi),L^{\mathcal{G}}_{g}(\zeta)\rangle_{B_{r(g)}}&=\sum_{t\in\mathcal{G}_{r(g)}}\langle
L^{\mathcal{G}}_{g}(\xi)(t),L^{\mathcal{G}}_{g}(\zeta)(t)\rangle_{B_{r(g)}}=\sum_{t\in\mathcal{G}_{r(g)}}\langle
L_{g}(\xi(tg)),L_{g}(\zeta(tg))\rangle_{B_{r(g)}}\\\
&=\beta_{g}\left(\sum_{t\in\mathcal{G}_{r(g)}}\langle\xi(tg),\zeta(tg)\rangle_{B_{s(g)}}\right)=\beta_{g}\left(\langle\xi,\zeta\rangle_{B_{s(g)}}\right).\end{split}$

This implies that
$L^{\mathcal{G}}_{g}:\ell^{2}(s^{*}\mathcal{E})_{s(g)}\to\ell^{2}(s^{*}\mathcal{E})_{r(g)}$
is a well-defined isometry. Furthermore, for $b\in B_{s(g)}$ and
$t\in\mathcal{G}_{r(g)}$, we have

$L^{\mathcal{G}}_{g}(\xi b)(t)=L_{g}((\xi
b)(tg))=L_{g}(\xi(tg)b)=L_{g}(\xi(tg))\beta_{g}(b)=L^{\mathcal{G}}_{g}(\xi)(t)\beta_{g}(b)=(L^{\mathcal{G}}_{g}(\xi)\beta_{g}(b))(t),$

so that $L^{\mathcal{G}}_{g}(\xi b)=L^{\mathcal{G}}_{g}(\xi)\beta_{g}(b)$.
Similarly, we get
$L^{\mathcal{G}}_{g}(a\xi)=\alpha_{g}(a)L^{\mathcal{G}}_{g}(\xi)$ for $a\in
A_{s(g)}$. If $g,h,t\in\mathcal{G}$ with $s(g)=r(h),s(t)=r(g)$ and
$\xi\in\ell^{2}(s^{*}\mathcal{B})_{s(h)}$ then we have

$\begin{split}L^{\mathcal{G}}_{g}(L^{\mathcal{G}}_{h}\xi)(t)&=L_{g}((L^{\mathcal{G}}_{h}\xi)(tg))=L_{g}\big{(}L_{h}(\xi(tgh))\big{)}=u_{\alpha}(g,h)L_{gh}(\xi(tgh))u_{\beta}(g,h)^{*}\\\
&=u_{\alpha}(g,h)L^{\mathcal{G}}_{gh}(\xi)(t)u_{\beta}(g,h)^{*}.\end{split}$

This shows conditions (1), (2) in Definition 5.1. To check condition (3), we
take sections $\xi_{0},\zeta_{0}\in\mathrm{C_{c}}(s^{*}\mathcal{E})$ and
consider the corresponding continuous sections
$\xi,\zeta\in\mathrm{C_{c}}(\ell^{2}(s^{*}\mathcal{E}))$ given by
$x\mapsto\xi_{0}|_{\mathcal{G}_{x}},\zeta_{0}|_{\mathcal{G}_{x}}$ as in the
definition of the topology of $\ell^{2}(s^{*}\mathcal{E})$. Then for
$g\in\mathcal{G}$ we get

$\langle\xi(r(g)),L^{\mathcal{G}}_{g}\zeta(s(g))\rangle_{B_{r(g)}}=\sum_{t\in\mathcal{G}_{r(g)}}\langle\xi_{0}(t),L_{g}(\zeta_{0}(tg))\rangle_{B_{r(g)}}$

This is a continuous section of the pullback $r^{*}\mathcal{B}$ because $L$ is
continuous and $\xi_{0},\zeta_{0}\in\mathrm{C_{c}}(s^{*}\mathcal{E})$, so in
particular the sum above is finite. ∎

######  Definition 6.8.

We call equivariant representations of $\mathcal{G}$ of the form
$L^{\mathcal{G}}$ _regular_ (or, if we want to stress the original equivariant
representation $L$, _regularized_).

######  Example 6.9.

Given a twisted action $(\alpha,u)$ of $\mathcal{G}$ on $\mathcal{A}$ we can
view the collection of maps $(\alpha_{g})_{g\in\mathcal{G}}$ as an equivariant
representation of $\mathcal{G}$ on $\mathcal{E}=\mathcal{A}$, see Example 5.9.
Its regularization $\alpha^{\mathcal{G}}$ acts on
$\ell^{2}(s^{*}\mathcal{A})=\\{\ell^{2}(s^{*}\mathcal{A})_{x}\\}_{x\in X}$ by
the formula $(\alpha^{\mathcal{G}}_{g}(a))_{t}=\alpha_{g}(a_{tg})$,
$a\xi\in\ell^{2}(s^{*}\mathcal{A})_{s(g)}$, $t\in\mathcal{G}_{r(g)}$.

######  Example 6.10.

For a twisted group action $(A,G,\alpha,u)$, i.e., when $\mathcal{G}=G$ is a
discrete group, the regular $(\alpha,u)$-$(\alpha,u)$ equivariant
representations of $\mathcal{G}$ coincide with induced equivariant
representations introduced in [BC12, Example 4.8], and $\alpha^{G}$ is called
the regular equivariant representation of $\alpha$ in [BC12, Example 4.7].

######  Definition 6.11.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. Two $(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$-equivariant
representations of $\mathcal{G}$, $L$ and $L^{\prime}$, defined respectively
on $\mathrm{C}^{*}$-correspondence bundles $\mathcal{E}$ and $\mathcal{F}$
from $\mathcal{A}$ to $\mathcal{B}$, are _unitarily equivalent_ if there is a
$\mathrm{C_{0}}(X)$-unitary
$\Gamma:\mathrm{C_{0}}(\mathcal{E})\to\mathrm{C_{0}}(\mathcal{F})$ such that
$\Gamma_{r(g)}L_{g}=L^{\prime}_{g}\Gamma_{s(g)}$ for all $g\in\Gamma$, where
$\\{\Gamma_{x}\\}_{x\in X}$ is the associated continuous family of unitaries
$\Gamma_{x}:E_{x}\to F_{x}$, $x\in X$, see Lemma 3.12.

######  Example 6.12 (Regularized equivariant representations revisited).

Instead of using the source map $s$ in the construction of a regular
equivariant representation, we could also use the range map $r$. More
specifically, let $(\mathcal{E},L)$ be an
$(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$-equivariant correspondence. We have
an $(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$-equivariant correspondence
$(\ell^{2}(r^{*}\mathcal{E}),{{}^{\mathcal{G}}L})$ where the fibers of
$\ell^{2}(r^{*}\mathcal{E})=\\{\ell^{2}(r^{*}\mathcal{E})_{x}\\}_{x\in X}$ are
direct sums
$\ell^{2}(r^{*}\mathcal{E})_{x}=\bigoplus_{g\in\mathcal{G}^{x}}E_{x}$ and for
each $g\in\mathcal{G}$ the map
${{}^{\mathcal{G}}L}_{g}:\ell^{2}(r^{*}\mathcal{E})_{s(g)}\to\ell^{2}(r^{*}\mathcal{E})_{r(g)}$
is given by

${}^{\mathcal{G}}L_{g}(\xi)(t)=L_{g}(\xi(g^{-1}t)),\qquad\xi\in\ell^{2}(r^{*}\mathcal{E})_{s(g)},\,\,t\in\mathcal{G}^{s(g)}.$

Then the regularized equivariant representations
$(\ell^{2}(s^{*}\mathcal{E}),L^{\mathcal{G}})$ and
$(\ell^{2}(r^{*}\mathcal{E}),{{}^{\mathcal{G}}L})$ are unitarily equivalent
with equivalence given by the bundle of unitaries
$\ell^{2}(s^{*}\mathcal{E})_{x}\xrightarrow{\sim}\ell^{2}(r^{*}\mathcal{E})_{x}$
where $\delta_{h}\mapsto\delta_{h^{-1}}$ for $h\in\mathcal{G}_{x}$. Also
unitary equivalence of Hilbert bundles mentioned in Example 5.6 is a special
case of the above notion of unitary equivalence, and tensoring
$(\mathcal{E},L)$ with left and right regular Hilbert bundles
$(\\{\ell^{2}(\mathcal{G}^{x})\\}_{x\in X},\lambda)$ and
$(\\{\ell^{2}(\mathcal{G}_{x})\\}_{x\in X},\rho)$, see Examples 5.6, 5.10, we
get equivariant correspondences
$(\mathcal{E}\otimes\\{\ell^{2}(\mathcal{G}^{x})\\}_{x\in X},L\otimes\lambda)$
and $(\mathcal{E}\otimes\\{\ell^{2}(\mathcal{G}_{x})\\}_{x\in
X},L\otimes\rho)$ where, for each $g\in\mathcal{G}$ and $\xi\in E_{s(g)}$, we
have

$\displaystyle L_{g}\otimes\lambda_{g}(\xi\otimes\delta_{h})$
$\displaystyle=L_{g}(\xi)\otimes\delta_{gh},\qquad\quad
h\in\mathcal{G}^{s(g)},$ $\displaystyle
L_{g}\otimes\rho_{g}(\xi\otimes\delta_{h})$
$\displaystyle=L_{g}(\xi)\otimes\delta_{hg^{-1}},\qquad
h\in\mathcal{G}_{s(g)}.$

Indeed, for instance the unitaries
$E_{x}\otimes\ell^{2}(\mathcal{G}_{x})\ni\sum_{h\in\mathcal{G}_{x}}\xi_{h}\otimes\delta_{h}\longmapsto\oplus_{h\in\mathcal{G}_{x}}\xi_{h}\in\ell^{2}(s^{*}\mathcal{E})=\bigoplus_{h\in\mathcal{G}_{x}}E_{x}$,
for $x\in X$, establish the equivalence
$(\ell^{2}(s^{*}\mathcal{E}),L^{\mathcal{G}})\cong(\mathcal{E}\otimes\\{\ell^{2}(\mathcal{G}_{x})\\}_{x\in
X},L\otimes\rho)$ Concluding, we have the following unitary equivalences

$(\ell^{2}(s^{*}\mathcal{E}),L^{\mathcal{G}})\cong(\mathcal{E}\otimes\\{\ell^{2}(\mathcal{G}_{x})\\}_{x\in
X},L\otimes\rho)\cong(\mathcal{E}\otimes\\{\ell^{2}(\mathcal{G}^{x})\\}_{x\in
X},L\otimes\lambda)\cong(\ell^{2}(r^{*}\mathcal{E}),{{}^{\mathcal{G}}L})$

yielding equivalent pictures of the regularization of the equivariant
correspondence $(\mathcal{E},L)$.

######  Proposition 6.13 (Absorption for equivariant representations).

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. Let $L$ be an $(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$
equivariant representation of $\mathcal{G}$ on a
$\mathrm{C}^{*}$-correspondence bundle $\mathcal{E}$ from $\mathcal{A}$ to
$\mathcal{B}$. The tensor product equivariant representations
$L\otimes\beta^{\mathcal{G}}$ and $\alpha^{\mathcal{G}}\otimes L$, acting on
$\mathcal{E}\otimes_{\mathcal{B}}\ell^{2}(s^{*}\mathcal{B})$ and
$\ell^{2}(s^{*}\mathcal{A})\otimes_{\mathcal{A}}\mathcal{E}$, respectively,
are both unitarily equivalent to the induced regular equivariant
representation $L^{\mathcal{G}}$ on $\ell^{2}(s^{*}\mathcal{E})$.

###### Proof.

We construct the desired $\mathrm{C_{0}}(X)$-unitary
$\Gamma^{\beta}:\mathrm{C_{0}}(\mathcal{E}\otimes_{\mathcal{B}}\ell^{2}(s^{*}\mathcal{B}))\to\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))$.
Given $x\in X$, there is a unique unitary
$\Gamma_{x}^{\beta}:E_{x}\otimes_{B_{x}}\ell^{2}(s^{*}\mathcal{B})_{x}\to\ell^{2}(s^{*}\mathcal{E})_{x}$
satisfying

(6.3) $\Gamma_{x}^{\beta}(\xi\otimes b)(t)=\xi\cdot b(t),\qquad\xi\in E_{x},\
b\in\ell^{2}(s^{*}\mathcal{B})_{x},\ t\in\mathcal{G}_{x}.$

Indeed, for $\xi,\xi^{\prime}\in E_{x}$,
$b,b^{\prime}\in\ell^{2}(s^{*}\mathcal{B})_{x}$ we have $\xi\otimes
b,\xi^{\prime}\otimes b^{\prime}\in
E_{x}\otimes_{B_{x}}\ell^{2}(s^{*}\mathcal{B})_{x}$ and

$\begin{split}\langle\Gamma_{x}^{\beta}(\xi\otimes
b),\Gamma_{x}^{\beta}(\xi^{\prime}\otimes
b^{\prime})\rangle_{B_{x}}&=\sum_{t\in\mathcal{G}_{x}}\langle\xi
b(t),\xi^{\prime}b^{\prime}(t)\rangle_{B_{x}}=\sum_{t\in\mathcal{G}_{x}}b(t)^{*}\langle\xi,\xi^{\prime}\rangle_{B_{x}}b^{\prime}(t)\\\
&=\sum_{t\in\mathcal{G}_{x}}\langle\xi\otimes b(t),\xi^{\prime}\otimes
b^{\prime}(t)\rangle_{B_{x}}=\langle\xi\otimes b,\xi^{\prime}\otimes
b^{\prime}\rangle_{B_{x}}.\end{split}$

This implies that (6.3) extends uniquely to an adjointable isometry, which is
invertible as its range clearly contains the dense subspace
$\mathrm{C_{c}}(s^{*}\mathcal{E}|_{\mathcal{G}_{x}})\subseteq\ell^{2}(s^{*}\mathcal{E})_{x}$.
To see that the bundle $\\{\Gamma_{x}^{\beta}\\}_{x\in X}$ of unitaries is
continuous take $\xi\in\mathrm{C_{c}}(\mathcal{E})$ and
$b\in\mathrm{C_{c}}(\ell^{2}(s^{*}\mathcal{B}))$ of the form
$b(x)=\oplus_{g\in\mathcal{G}_{x}}b_{0}(g)$ for some
$b_{0}\in\mathrm{C_{c}}(s^{*}\mathcal{B})$. Then the formula
$\zeta(g):=\xi(s(g))\cdot b_{0}(g)$ defines an element
$\zeta\in\mathrm{C_{c}}(s^{*}\mathcal{E})$ and so

$\Gamma^{\beta}(\xi\otimes b)(x):=\Gamma_{x}^{\beta}(\xi(x)\otimes
b(x))=\bigoplus_{g\in\mathcal{G}_{x}}\xi(x)\cdot b_{0}(g),\qquad x\in X,$

defines an element of $\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))$. Hence
$\\{\Gamma_{x}^{\beta}\\}_{x\in X}$ defines the $\mathrm{C_{0}}(X)$-unitary
$\Gamma^{\beta}$. It intertwines the equivariant representations as for
$\xi\in E_{x},b\in\ell^{2}(s^{*}\mathcal{B})_{x}$, $t,g\in\mathcal{G}_{x}$ we
have

$\begin{split}(\Gamma_{r(g)}^{\beta}\circ(L\otimes\beta^{\mathcal{G}})_{g})(\xi\otimes
b)(t)&=\Gamma_{r(g)}(L_{g}\xi\otimes\beta^{\mathcal{G}}_{g}(b))(t)=L_{g}(\xi)\cdot\beta^{\mathcal{G}}_{g}(b)(t)=L_{g}(\xi)\beta_{g}(b(tg))\\\
&=L_{g}(\xi b(tg))=L_{g}(\Gamma_{s(g)}^{\beta}(\xi\otimes
b(tg)))=(L^{\mathcal{G}}_{g}\circ\Gamma_{s(g)}^{\beta})(\xi\otimes
b)(t).\end{split}$

This proves the unitary equivalence between $L\otimes\beta^{\mathcal{G}}$ and
$L^{\mathcal{G}}$. The equivalence between $\alpha^{\mathcal{G}}\otimes L$ and
$L^{\mathcal{G}}$ can be proved analogously, and it is implemented by the
$\mathrm{C_{0}}(X)$-unitary
$\Gamma^{\alpha}:\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{A}\otimes_{\mathcal{A}}\mathcal{E})\to\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))$,
where for $x\in X$ we set

$\Gamma_{x}^{\alpha}(a\otimes\xi)(t)=a(t)\xi,\qquad\xi\in E_{x},\
a\in\ell^{2}(s^{*}\mathcal{A})_{x},\ t\in\mathcal{G}_{x}.$

We leave the details to the reader. ∎

######  Corollary 6.14.

Let $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ be $\mathrm{C}^{*}$-bundles
over $X$, carrying twisted actions $(\alpha,u_{\alpha})$, $(\beta,u_{\beta})$
and $(\gamma,u_{\gamma})$ of $\mathcal{G}$. Let $L$ and $K$ be two composable
equivariant representations of $\mathcal{G}$, so $L$ is
$(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$ equivariant and $K$ is
$(\beta,u_{\beta})$-$(\gamma,u_{\gamma})$ equivariant, see Example 5.12. Then

$L^{\mathcal{G}}\otimes K\cong L\otimes K^{\mathcal{G}}\cong(L\otimes
K)^{\mathcal{G}}.$

###### Proof.

Using the ‘associativity’ of the tensor product and applying Proposition 6.13
four times we get $L^{\mathcal{G}}\otimes K\cong
L\otimes\beta^{\mathcal{G}}\otimes K\cong L\otimes K^{\mathcal{G}}\cong
L\otimes K\otimes\gamma^{\mathcal{G}}\cong(L\otimes K)^{\mathcal{G}}$. ∎

We will use the above absorption statement to infer that our Fourier
multipliers form an ideal in the category of Fourier–Stieltjes multipliers (to
be defined in the next section). Also, Proposition 6.13 implies that

$\mathrm{C_{0}}(\mathcal{E})\otimes_{\mathrm{C_{0}}(\mathcal{B})}\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{B}))\cong\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E})).$

To show that Fourier–Stieltjes multipliers yield completely bounded maps on
reduced crossed products we will need yet another form of absorption, where we
change the left multiplication in the regular $\mathrm{C}^{*}$-correspondence
$\ell^{2}(s^{*}\mathcal{B})$ from the pointwise to the one given by the
regular representation. The following versions of the Fell absorption
principle are what we need; they are related to similar results appearing, for
instance, in [SW13, Lemma 2] and [BM23, Theorem 3.20]. The first one is a
generalization of [BC15, Theorem 4.10].

######  Theorem 6.15 (Fell’s absorption I).

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. Let $L$ be an $(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$
equivariant representation of $\mathcal{G}$ on a
$\mathrm{C}^{*}$-correspondence bundle $\mathcal{E}$ from $\mathcal{A}$ to
$\mathcal{B}$. The representation $L\operatorname{-Ind}(\pi^{\mathcal{B}})$
induced by $L$ (as in Proposition 5.14) from the regular representation
$\pi^{\mathcal{B}}:\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})\to\mathcal{L}(\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{B})))$
(see Example 6.5) is unitarily equivalent to the regular representation
$\pi^{\mathcal{E}}\cong\Lambda^{\mathcal{E}}$ induced by $\mathcal{E}$. This
equivalence is implemented by the isomorphism of right Hilbert
$\mathrm{C_{0}}(\mathcal{B})$-modules

$W:\mathrm{C_{0}}(\mathcal{E})\otimes_{\pi^{\mathcal{B}}}\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{B}))\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))$

where $W(\xi\otimes b)(x):=W_{x}(\xi\otimes b(x))$,
$\xi\in\mathrm{C_{0}}(\mathcal{E})$,
$b\in\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{B}))$, $x\in X$, and

$W_{x}:\mathrm{C_{0}}(\mathcal{E})\otimes_{\pi^{\mathcal{B}}_{x}}\ell^{2}(s^{*}\mathcal{B})_{x}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\ell^{2}(s^{*}\mathcal{E})_{x},\quad
W_{x}\left(\xi\otimes
b_{x}\right)(g):=L_{g}^{-1}\left(\xi(r(g)\right)b_{x}(g),$

for $\xi\in\mathrm{C_{0}}(\mathcal{E})$,
$b_{x}\in\ell^{2}(s^{*}\mathcal{B})_{x}$.

###### Proof.

Let $x\in X$. For any $\xi,\ \zeta\in E_{r(g)}$, $g\in\mathcal{G}_{x}$, and
$a,b\in B_{x}$, using Remark 5.2, we have

$\begin{split}\langle
L_{g}^{-1}(\xi)b,L_{g}^{-1}(\zeta)a\rangle_{B_{x}}&=\beta_{g}^{-1}(\langle\xi\beta_{g}(b),\zeta\beta_{g}(a)\rangle_{B_{r(g)}})=b^{*}\beta_{g}^{-1}(\langle\xi,\zeta\rangle_{B_{r(g)}})a.\end{split}$

Using this, for any $\xi,\zeta\in\mathrm{C_{0}}(\mathcal{E})$ and $a$,
$b\in\ell^{2}(s^{*}\mathcal{B})_{x}$ we get

$\begin{split}\left\langle W_{x}\left(\xi\otimes
b\right),W_{x}\left(\zeta\otimes
a\right)\right\rangle_{B_{x}}&=\sum_{g\in\mathcal{G}}b(g)^{*}\beta_{g}^{-1}(\langle\xi(r(g)),\zeta(r(g))\rangle_{B_{r(g)}})a(g)=\left\langle\xi\otimes
b,\zeta\otimes a\right\rangle_{B_{x}}.\end{split}$

This readily implies that $W_{x}$ is a well-defined isometry. It clearly has
dense range, hence in fact $W_{x}$ is an adjointable unitary. Now it suffices
to see that $W$ is well defined, i.e., that for
$\xi\in\mathrm{C_{c}}(\mathcal{E})$,
$b\in\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{B}))$ the formula $W(\xi\otimes
b)(x)=W_{x}(\xi\otimes b(x))$ defines an element of
$\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))$. We may assume here that $b$ is
of the form $x\mapsto\oplus_{g\in\mathcal{G}_{x}}b_{0}(g)$ for some
$b_{0}\in\mathrm{C_{0}}(s^{*}\mathcal{B})$, as such elements form a dense
subset of $\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{B}))$. The section
$\mathcal{G}\ni g\mapsto L_{g^{-1}}\left(\xi(r(g))\right)$ is in
$\mathrm{C_{b}}(s^{*}\mathcal{E})$ by continuity and boundedness of $\xi$ and
$L$, see Lemma 5.3. Therefore the section $\mathcal{G}\ni g\mapsto
L_{g^{-1}}\left(\xi(r(g)\right))u_{\beta}(g^{-1},g)b_{0}(g)$ is in
$\mathrm{C_{c}}(s^{*}\mathcal{E})$. Since $W(\xi\otimes
b)(x)=\oplus_{g\in\mathcal{G}_{x}}L_{g^{-1}}\left(\xi(r(g))\right)u_{\beta}(g^{-1},g)b_{0}(g)$
we conclude that $W(\xi\otimes
b)\in\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))$.

To prove that $W$ intertwines $L\operatorname{-Ind}(\pi^{\mathcal{B}})$ and
$\pi^{\mathcal{E}}$ we need to show that for every
$U\in\operatorname{Bis}(\mathcal{G})$,
$a\in\mathrm{C_{c}}(\mathcal{A}|_{r(U)})$,
$\xi\in\mathrm{C_{0}}(\mathcal{E}),\ b\in\ell^{2}(s^{*}\mathcal{B})$, we have

(6.4) $W^{*}\pi^{\mathcal{E}}(a)W(\xi\otimes
b)=aL_{U}(\xi)\otimes\pi^{\mathcal{B}}(\delta_{U})(b),$

where $L_{U}(\xi)(r(g)):=L_{g}(\xi(s(g))$ for $g\in U$ and

$\pi^{\mathcal{B}}(\delta_{U})(b)(x):=\oplus_{g\in\mathcal{G}_{x}}\sum_{h\in\mathcal{G}^{r(g)}\cap
U}\overline{\alpha}_{g}^{-1}\big{(}u(h,h^{-1}g)\big{)}b(h^{-1}g),\qquad x\in
X.$

Moreover, by linearity and continuity, it suffices to consider $b$ such that
for every $x\in X$, $b(x)\in\ell^{2}(s^{*}\mathcal{B})_{x}$, is supported on
single element of $\mathcal{G}_{x}$. Let us then fix $x\in X$ and assume that
$b_{x}:=b(x)\in\ell^{2}(s^{*}\mathcal{B})_{x}$, is supported on
$t\in\mathcal{G}_{x}$. There is at most one element in
$U\cap\mathcal{G}_{r(t)}$, that we denote by $h$ if it exists. We have

$\begin{split}\big{(}\pi^{\mathcal{E}}_{x}(a)W_{x}(\xi\otimes&b_{x})\big{)}(g)=[g=ht]\alpha_{g}^{-1}\left(a(h)u_{\alpha}(h,t)\right)L_{t}^{-1}\left(\xi(r(t))\right)b_{x}(t)\\\
&=[g=ht]\alpha_{g}^{-1}\left(a(h)u_{\alpha}(h,t)\right)L_{g}^{-1}L_{h^{-1}}^{-1}\left(u_{\alpha}(h^{-1},g)\xi(r(t))u_{\beta}(h^{-1},g)^{*}\right)b_{x}(t)\\\
&=[g=ht]L_{g}^{-1}\left(a(h)u_{\alpha}(h,t)L_{h^{-1}}^{-1}(u_{\alpha}(h^{-1},g)\xi(r(t))u_{\beta}(h^{-1},g)^{*}\right)b_{x}(t).\end{split}$

Consider
$c:=u_{\alpha}(h,t)L_{h^{-1}}^{-1}\left(u_{\alpha}(h^{-1},g)\xi(r(t))u_{\beta}(h^{-1},g)^{*}\right)$
where $g=ht$. Since $r(t)=s(h)$ and
$L_{h^{-1}}^{-1}(\cdot)=u_{\alpha}(h,h^{-1})^{*}L_{h}(\cdot)u_{\beta}(h,h^{-1})$
we get

$c=u_{\alpha}(h,t)u_{\alpha}(h,h^{-1})^{*}\alpha_{h}(u_{\alpha}(h^{-1},g))L_{h}\left(\xi(s(h))\right)\beta_{h}(u_{\beta}(h^{-1},g))^{*}u_{\beta}(h,h^{-1}).$

By the cocycle identity we have
$\alpha_{h}(u_{\alpha}(h^{-1},g))=u_{\alpha}(h,h^{-1})u_{\alpha}(h,t)^{*}$
(and similarly for $\beta$). Thus $c=L_{h}(\xi(s(h))u_{\beta}(h,t)$. Therefore
continuing the above calculations we get

$\begin{split}\big{(}\pi^{\mathcal{E}}_{x}(a)W_{x}(\xi\otimes
b_{x})\big{)}(g)&=[g=ht]L_{g}^{-1}\left(a(h)L_{h}(\xi(s(h))u_{\beta}(h,t)\right)b_{x}(t)\\\
&=W_{x}\left(aL_{U}(\xi)\otimes\pi^{\mathcal{B}}(\delta_{U})(b)\right)(g).\end{split}$

This implies the equality (6.4) and finishes the proof. ∎

The above theorem states that for any equivariant correspondence
$(\mathcal{E},L)$ the following diagram commutes

$\textstyle{\mathrm{\mathrm{C}^{*}}(\mathcal{A}^{(\alpha,u_{\alpha})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$L\operatorname{-Ind}(\pi^{\mathcal{B}})$$\scriptstyle{\pi^{\mathcal{E}}}$$\scriptstyle{\Lambda\otimes\mathrm{id}_{\mathrm{C_{0}}(\mathcal{E})\qquad\,\,}}$$\textstyle{\mathcal{L}\Big{(}\mathrm{C_{0}}(\ell^{2}(r^{*}\mathcal{A}))\otimes_{\mathrm{C_{0}}(\mathcal{A})}\mathrm{C_{0}}(\mathcal{E})\Big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathcal{L}\Big{(}\mathrm{C_{0}}(\mathcal{E})\otimes_{\pi^{\mathcal{B}}}\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{B}))\Big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\qquad\cong}$$\textstyle{\mathcal{L}\Big{(}\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))\Big{)}}$

where $\cong$ denotes an isomorphism implemented by a unitary between Hilbert
modules. Thus regular representations absorb the inducing functor in the sense
that composing $L\operatorname{-Ind}$ with the regular representation one gets
a regular representation. For groups the representation
$L\operatorname{-Ind}(\pi^{\mathcal{B}})$ is nothing but the tensor product of
a representation $L$ and the regular representation $\pi^{\mathcal{B}}$ and
the above theorem reduces to the classical Fell’s absorption.

Next we derive the second version of Fell’s absorption principle that will be
needed later, and which in particular generalizes [BC12, Theorem 4.11].

######  Theorem 6.16 (Fell’s absorption II).

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. Let $L$ be an $(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$
equivariant representation of $\mathcal{G}$ on a
$\mathrm{C}^{*}$-correspondence bundle $\mathcal{E}$ from $\mathcal{A}$ to
$\mathcal{B}$, and let
$\psi:\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})\to\mathcal{L}(\mathcal{H})$
be a representation of $\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})$ on a
Hilbert space $H$. The representation
$L^{\mathcal{G}}\operatorname{-Ind}(\psi):\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathcal{L}(\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))\otimes_{\psi}H)$
induced from $\psi$ by the regularization of $L$ is unitarily equivalent to
the amplification $\pi^{\mathcal{E}}\otimes
1_{\mathcal{H}}:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathcal{L}(\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))\otimes_{\psi}\mathcal{H})$
of the regular representation $\pi^{\mathcal{E}}$ induced by $\mathcal{E}$.

###### Proof.

Disintegrate $\psi$ into a covariant representation $(\pi,v)$ as in Subsection
4.1. So $\pi=\psi|_{\mathrm{C_{0}}(X)}$ and for any
$U\in\operatorname{Bis}(\mathcal{G})$ the partial isometry $v_{U}$ is the
strong limit of $\psi(e^{r(U)}_{i}\delta_{U})$, where for an open set
$V\subseteq X$, we choose an approximate unit $\\{e^{V}_{i}\\}_{i}$ in
$B_{V}=\mathrm{C_{0}}(\mathcal{B}|_{V})$. We will describe the action of the
desired unitary on the dense subspace
$\mathrm{C_{c}}(s^{*}\mathcal{E})\otimes_{\pi}\mathcal{H}$ of
$\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))\otimes_{\pi}H$. For
$U\in\operatorname{Bis}(\mathcal{G})$ and
$\xi\in\mathrm{C_{c}}(\mathcal{E}|_{s(U)})$, define
$\delta_{U}\xi\in\mathrm{C_{c}}(s^{*}\mathcal{E}|_{U})$ by

$(\delta_{U}\xi)(g):=\xi(s(g)),\qquad g\in U.$

For any $U\in\operatorname{Bis}(\mathcal{G})$ and any $\eta\in\mathcal{H}$ put

(6.5) $W(\delta_{U}\xi\otimes\eta):=\delta_{U^{*}}L_{U}(\xi)\otimes
v_{U}\eta,$

where $v_{U}$ is the partial isometry from the disintegration of $\psi$, see
Subsection 4.1. We first show that for any
$\xi^{\prime}\in\mathrm{C_{c}}(\mathcal{E}|_{s(U^{\prime})})$,
$U^{\prime}\in\operatorname{Bis}(\mathcal{G})$, $\eta^{\prime}\in H$ we have

(6.6) $\langle
W(\xi\delta_{U}\otimes\eta),W(\xi^{\prime}\delta_{U^{\prime}}\otimes\eta^{\prime})\rangle=\langle\xi\delta_{U}\otimes\eta,\xi^{\prime}\delta_{U^{\prime}}\otimes\eta^{\prime}\rangle.$

Let $U_{0}=U\cap U^{\prime}$. The section
$\langle\delta_{U^{*}}L_{U}(\xi),\delta_{U^{\prime*}}L_{U^{\prime}}(\xi^{\prime})\rangle_{\mathrm{C_{0}}(\mathcal{B})}$
vanishes outside $r(U_{0})$ and for $g\in U_{0}$ we have
$\langle\delta_{U^{*}}L_{U}(\xi),\delta_{U^{\prime*}}L_{U^{\prime}}(\xi^{\prime})\rangle_{\mathrm{C_{0}}(\mathcal{B})}(r(g))=\langle
L_{g}(\xi(s(g))),L_{g}(\xi^{\prime}(s(g)))\rangle_{B_{r(g)}}$. Thus

$\begin{split}\langle\delta_{U^{*}}L_{U}(\xi),\delta_{U^{\prime*}}L_{U^{\prime}}(\xi^{\prime})\rangle_{\mathrm{C_{0}}(\mathcal{B})}&=\lim_{i,j}e_{i}^{r(U_{0})}\cdot\langle\delta_{U^{*}}L_{U}(\xi),\delta_{U^{\prime*}}L_{U^{\prime}}(\xi^{\prime})\rangle_{\mathrm{C_{0}}(\mathcal{B})}\cdot
e_{j}^{r(U_{0})}\\\
&=\lim_{i,j}\langle\delta_{U^{*}}L_{U}(\xi)e_{i}^{r(U_{0})},\delta_{U^{\prime*}}L_{U^{\prime}}(\xi^{\prime})e_{j}^{r(U_{0})}\rangle_{\mathrm{C_{0}}(\mathcal{B})}\\\
&=\lim_{i,j}\langle\delta_{U^{*}}L_{U}(\xi
e_{i}^{s(U_{0})}),\delta_{U^{\prime*}}L_{U^{\prime}}(\xi^{\prime}e_{i}^{s(U_{0})})\rangle_{\mathrm{C_{0}}(\mathcal{B})}\\\
&=\lim_{i,j}\langle\delta_{U_{0}^{*}}L_{U_{0}}(\xi
e_{i}^{s(U_{0})}),\delta_{U_{0}^{*}}L_{U_{0}}(\xi^{\prime}e_{i}^{s(U_{0})})\rangle_{\mathrm{C_{0}}(\mathcal{B})}.\end{split}$

Similarly,
$\langle\delta_{U}\xi,\delta_{U^{\prime}}\xi^{\prime}\rangle_{\mathrm{C_{0}}(\mathcal{B})}=\lim_{i,j}\langle\delta_{U_{0}^{*}}\xi
e_{i}^{s(U_{0})},\delta_{U_{0}^{*}}\xi^{\prime}e_{j}^{s(U_{0})}\rangle_{\mathrm{C_{0}}(\mathcal{B})}$.
We also have $\psi(e_{i}^{r(U_{0})})v_{U}=\psi(e_{i}^{r(U_{0})})v_{U_{0}}$ and
$\psi(e_{i}^{r(U_{0})})v_{U^{\prime}}=\psi(e_{i}^{r(U_{0})})v_{U_{0}}$. Using
this and the covariance conditions in Subsection 4.1, we get

$\begin{split}\langle
W(\delta_{U}\xi\otimes\eta),W(\delta_{U^{\prime}}\xi^{\prime}\otimes&\eta^{\prime})\rangle=\langle
v_{U}\eta,\pi(\delta_{U^{*}}\langle
L_{U}(\xi),\delta_{U^{\prime*}}L_{U^{\prime}}(\xi^{\prime})\rangle)v_{U^{\prime}}\eta^{\prime}\rangle\\\
&=\lim_{i,j}\left\langle\eta,v_{U_{0}}^{*}\pi\left(\langle\delta_{U_{0}^{*}}L_{U_{0}}(\xi
e_{i}^{s(U_{0})}),\delta_{U_{0}^{*}}L_{U_{0}}(\xi^{\prime}e_{j}^{s(U_{0})})\rangle_{\mathrm{C_{0}}(\mathcal{B})}\right)v_{U_{0}}\eta^{\prime}\right\rangle\\\
&=\lim_{i,j}\left\langle\eta,\pi\left(\beta_{U_{0}}^{-1}\Big{(}\langle\delta_{U_{0}^{*}}L_{U_{0}}(\xi
e_{i}^{s(U_{0})}),\delta_{U_{0}^{*}}L_{U_{0}}(\xi^{\prime}e_{j}^{s(U_{0})})\rangle_{\mathrm{C_{0}}(\mathcal{B})}\Big{)}\right)\eta^{\prime}\right\rangle\\\
&=\lim_{i,j}\left\langle\eta,\pi\left(\langle\delta_{U_{0}}\xi
e_{i}^{s(U_{0})},\delta_{U_{0}}\xi^{\prime}e_{j}^{s(U_{0})}\rangle_{\mathrm{C_{0}}(\mathcal{B})}\right)\eta^{\prime}\right\rangle\\\
&=\left\langle\eta,\pi\left(\langle\xi\delta_{U},\xi^{\prime}\delta_{U^{\prime}}\rangle_{\mathrm{C_{0}}(\mathcal{B})}\right)\eta^{\prime}\right\rangle=\langle\xi\delta_{U}\otimes\eta,\xi^{\prime}\delta_{U^{\prime}}\otimes\eta^{\prime}\rangle.\end{split}$

Now by linearity, continuity and (6.6), we see that (6.5) determines an
isometry $W$ on $\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))\otimes_{\pi}H$.
Using that $\omega_{\alpha}(U,U^{*})a=\omega_{\alpha}(U_{0},U_{0}^{*})a$ for
any $U_{0}\subseteq U\in\operatorname{Bis}(\mathcal{G})$ and $a\in
A_{U^{0}}=\mathrm{C_{0}}(\mathcal{A}|_{U_{0}})$, one can adopt the above
arguments to see that the formula

$W^{*}(\delta_{U}\xi\otimes\eta):=\delta_{U^{*}}\omega_{\alpha}(U,U^{*})^{*}L_{U}(\xi)\otimes
v_{U}\eta,$

determines an isometry on
$\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))\otimes_{\pi}H$. Since
$\omega_{\alpha}(U,U^{*})^{*}L_{U}(\cdot)\omega_{\beta}(U,U^{*})=L_{U^{*}}^{-1}$,
$\overline{\pi}(\omega_{\beta}(U,U^{*})^{*})v_{U}=v_{U^{*}}^{*}$ and the
tensor product is balanced over $\overline{\pi}(\mathcal{M}(A))$, we also have

$W^{*}(\delta_{U}\xi\otimes\eta):=\delta_{U^{*}}L_{U^{*}}^{-1}(\xi)\otimes
v_{U^{*}}^{*}\eta.$

This readily implies that $WW^{*}=W^{*}W=1$, as for instance we have

$\begin{split}WW^{*}(\delta_{U}\xi\otimes\eta)&=\delta_{U}\xi\otimes
v_{U^{*}}v_{U^{*}}^{*}\eta=\text{s-}\lim_{i}\delta_{U}\xi\otimes\pi(e_{i}^{s(U)})\eta\\\
&=\text{s-}\lim_{i}\delta_{U}\xi\otimes\pi(e_{i}^{s(U)})\eta=\text{s-}\lim_{i}\delta_{U}\xi
e_{i}^{s(U)}\otimes\eta=\delta_{U}\xi\otimes\eta.\end{split}$

Hence $W$ is a unitary and $W^{*}$ is its adjoint.

For $V\in\operatorname{Bis}(G)$ and $a\in A_{r(V)}$ we define
$a\delta_{V}\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})}|_{V})$ by
$a\delta_{V}(g)=a(r(g))$ for $g\in V$. One computes that
$\pi^{\mathcal{E}}(a\delta_{V})\delta_{U^{*}}L_{U^{*}}^{-1}(\xi)=\delta_{VU^{*}}\zeta$
where $\zeta\in\mathrm{C_{c}}(\mathcal{E}|_{s(VU^{*})})$ for $gh\in VU^{*}$
where $g\in V$, $h\in U^{*}$ is given by

$\begin{split}\zeta(s(g))&=\alpha_{gh}^{-1}\Big{(}a(r(g))u_{\alpha}(g,h)\Big{)}L_{h}^{-1}\big{(}\xi(s(g))\big{)}\\\
&=\alpha_{h}^{-1}\Big{(}\alpha_{g}^{-1}\big{(}u_{\alpha}(g,h)a(r(g))\big{)}\Big{)}L_{h}^{-1}(\xi(s(g)))\\\
&=L_{h}^{-1}\Big{(}\alpha_{g}^{-1}\big{(}u_{\alpha}(g,h)a(r(g))\big{)}\xi(s(g)\Big{)}.\end{split}$

Thus on one hand we have

$(\pi^{\mathcal{E}}\otimes
1_{\mathcal{H}})(a\delta_{V})W^{*}(\delta_{U}\xi\otimes\eta)=\delta_{VU^{*}}\zeta\otimes
v_{U^{*}}^{*}\eta.$

On the other hand, a direct computation shows that
$aL_{V}^{\mathcal{G}}(\delta_{U}\xi)=\delta_{UV^{*}}\zeta_{0}$ where
$\zeta_{0}\in\mathrm{C_{c}}(\mathcal{E}|_{s(UV^{*})})=\mathrm{C_{c}}(\mathcal{E}|_{r(VU^{*})})$
is given by
$\zeta_{0}(r(g)):=L_{g}\Big{(}\alpha_{g}^{-1}(a(r(g))\xi(s(g)))\Big{)}$ for
$g\in V$. Moreover, for $gh\in VU^{*}$ where $g\in V$, $h\in U^{*}$, we have

$\begin{split}L_{VU^{*}}^{-1}(\zeta_{0})(s(g))&=L_{gh}^{-1}\Big{(}L_{g}\Big{(}\alpha_{g}^{-1}(a(r(g)))\xi(s(g)))\Big{)}\Big{)}\\\
&=L_{h}^{-1}L_{g}^{-1}L_{g}\Big{(}\alpha_{g}^{-1}(u_{\alpha}(g,h)a(r(g)))\xi(s(g))\beta_{g}^{-1}(u_{\beta}(g,h)^{*})\Big{)}\\\
&=L_{h}^{-1}\Big{(}\alpha_{g}^{-1}(u_{\alpha}(g,h)a(r(g)))\xi(s(g)))\Big{)}\beta_{h}^{-1}(\beta_{g}^{-1}(u_{\beta}(g,h)^{*})\\\
&=\zeta(s(gh))\beta_{gh}^{-1}(u_{\beta}(g,h)^{*}).\end{split}$

That is
$L_{VU^{*}}^{-1}(\zeta_{0})=\zeta\beta_{UV^{*}}^{-1}(\omega_{\beta}(V,U^{*})^{*})$.
Using this we get

$\begin{split}W^{*}L^{\mathcal{G}}\operatorname{-Ind}(\psi)(a\delta_{V})(\delta_{U}\xi\otimes\eta)&=W^{*}\big{(}aL_{V}^{\mathcal{G}}(\delta_{U}\xi)\otimes
v_{V}\eta\big{)}=W^{*}\big{(}\delta_{UV^{*}}\zeta_{0}\otimes
v_{V}\eta\big{)}\\\
&=\delta_{VU^{*}}\Big{(}\zeta\beta_{UV^{*}}^{-1}(\omega_{\beta}(V,U^{*})^{*})\Big{)}\otimes
v_{VU^{*}}^{*}v_{V}\eta\\\
&=\delta_{VU^{*}}\zeta\otimes\pi(\beta_{UV^{*}}^{-1}(\omega_{\beta}(V,U^{*})^{*}))v_{VU^{*}}^{*}v_{V}\eta.\end{split}$

Then using the covariance relations and the fact that $v_{V}^{*}v_{V}$ is a
strong limit of $\\{\pi(e_{i}^{s(V)})\\}$ we obtain that

$\begin{split}\pi(\beta_{UV^{*}}^{-1}(\omega_{\beta}(V,U^{*})^{*}))v_{VU^{*}}^{*}v_{V}\eta&=\lim_{i}\pi(e_{i}^{s(VU^{*})})v_{U^{*}}^{*}v_{V}^{*}v_{V}\eta=\lim_{i,j}v_{U^{*}}^{*}\pi(\beta_{U^{*}}^{-1}(e_{i}^{s(VU^{*})}))\pi(e_{j}^{s(V)})\eta\\\
&=\lim_{i}v_{U^{*}}^{*}\pi(\beta_{U^{*}}^{-1}(e_{i}^{s(VU^{*})}))\eta=\lim_{i}\pi(e_{i}^{s(VU^{*})})v_{U^{*}}^{*}\eta.\end{split}$

Further exploiting the fact that the tensor product is balanced we get

$W^{*}L^{\mathcal{G}}\operatorname{-Ind}(\psi)(a\delta_{V})(\delta_{U}\xi\otimes\eta)=\delta_{VU^{*}}\zeta\otimes
v_{U^{*}}^{*}\eta=(\pi^{\mathcal{E}}\otimes
1_{\mathcal{H}})(a\delta_{V})W^{*}(\delta_{U}\xi\otimes\eta).$

This finishes the proof. ∎

The above theorem states that for any equivariant correspondence
$(\mathcal{E},L)$ and for any representation $\psi$ of
$\mathrm{\mathrm{C}^{*}}(\mathcal{B}^{(\alpha,u_{\alpha})})$ on a Hilbert
space $H$ the following diagram commutes

$\textstyle{\mathrm{\mathrm{C}^{*}}(\mathcal{A}^{(\alpha,u_{\alpha})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$L^{\mathcal{G}}\operatorname{-Ind}(\psi)$$\scriptstyle{\pi^{\mathcal{E}}\otimes\mathrm{id}_{H}}$$\scriptstyle{\Lambda\otimes\mathrm{id}_{\mathrm{C_{0}}(\mathcal{E})}\otimes\mathrm{id}_{H}\qquad\qquad\quad}$$\textstyle{\mathcal{L}\Big{(}\mathrm{C_{0}}(\ell^{2}(r^{*}\mathcal{A}))\otimes_{\mathrm{C_{0}}(\mathcal{A})}\mathrm{C_{0}}(\mathcal{E})\otimes_{\psi}H\Big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathcal{L}\Big{(}\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))\otimes_{\psi}H\Big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathcal{L}\Big{(}\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))\otimes_{\psi}\mathcal{H}\Big{)}}$

where the isomorphisms $\cong$ are implemented by unitaries between Hilbert
modules. Thus the regular equivariant representation $L^{\mathcal{G}}$ absorbs
all representations in the sense that the induced representation is an
amplification of a regular representation, and hence descends to the reduced
crossed product. For groups the representation
$L^{\mathcal{G}}\operatorname{-Ind}(\psi)$ is nothing but the tensor product
of a regular representation $L^{\mathcal{G}}$ and the representation $\psi$,
and the above theorem recovers the classical Fell absorption principle.

###  6.1. Absorption principles for groupoids

Let us specialize the above results to the case of trivial one-dimensional
$\mathrm{\mathrm{C}^{*}}$-bundle $\mathcal{A}=\mathcal{B}=\mathbb{C}\times X$
over $X$. In this case, we are simply considering a twisted groupoid
$(\mathcal{G},u)$, and equivariant representations correspond to continuous
Hilbert $\mathcal{G}$-bundles $(\mathcal{H},L)$ as in Example 5.6. Recall the
identifications and notation introduced in Proposition 6.4. The regular
induced representation $\pi^{\mathcal{H}}\colon
C^{*}(\mathcal{G},u)\to\mathcal{L}(\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{H})))$
acts on $\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{H}))$, which is the Hilbert
module obtained as the completion of $\mathrm{C_{c}}(s^{*}\mathcal{H})$ with
respect to the obvious right action and inner product

$\langle\xi\,|\,\mathopen{}\eta\rangle(x):=\sum_{g\in\mathcal{G}_{x}}\langle\xi(g)\,|\,\mathopen{}\eta(g)\rangle,\qquad
x\in X,\,\xi,\,\eta\in\mathrm{C_{c}}(s^{*}\mathcal{H}).$

The representation
$\pi^{\mathcal{H}}\colon\mathrm{\mathrm{C}^{*}}(\mathcal{G},u)\to\mathcal{L}(\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{H})))$
is given by the formula

$\pi^{\mathcal{H}}(f)\xi(g)=\sum_{h\in\mathcal{G}^{r(g)}}f(h)\xi(h^{-1}g)u(h,h^{-1}g),$

where $f\in\mathrm{C_{c}}(\mathcal{G}),\xi\in\mathrm{C_{c}}(s^{*}\mathcal{H})$
and $g\in\mathcal{G}$.

The left regular representation $\lambda^{u}$ of $C^{*}(\mathcal{G},u)$ is
given by a similar formula, as it can be seen as regular representation
induced from the trivial Hilbert bundle $\mathbb{C}\times X$, which acts on
the $\mathrm{C_{0}}(X)$-Hilbert module which we will simply denote
$\ell^{2}(\mathcal{G})$: the completion of $\mathrm{C_{c}}(\mathcal{G})$ in
the $\mathrm{C_{0}}(X)$-valued inner product
$\langle\xi\,|\,\mathopen{}\eta\rangle(x):=\sum_{g\in\mathcal{G}_{x}}\xi(g)\overline{\eta(g)}$,
$x\in X,\,\xi,\,\eta\in\mathrm{C_{c}}(\mathcal{G})$. The induced
representation $L\operatorname{-Ind}(\lambda^{u})$ acts thus on
$\mathrm{C_{0}}(\mathcal{H})\otimes_{\mathrm{C_{0}}(X)}\ell^{2}(\mathcal{G})$
by the formula

$L\operatorname{-Ind}(\lambda^{u})(f\delta_{U})(\xi\otimes\eta)=f\cdot
L_{U}(\xi)\otimes\lambda^{u}(\delta_{U})\eta$

for an open bisection $U\subseteq\mathcal{G}$,
$f\in\mathrm{C_{c}}(r(U))\subseteq\mathrm{C_{c}}(X)$,
$\xi\in\mathrm{C_{0}}(\mathcal{H}|_{r(U)})\subseteq\mathrm{C_{0}}(\mathcal{H})$,
$\eta\in\mathrm{C_{c}}(\mathcal{G})\subseteq\ell^{2}(\mathcal{G})$, where
$f\delta_{U}(g)=[g\in U]f(r(g))$, $(f\cdot
L_{U})(\xi)(r(g))=f(r(g))L_{g}(\xi(s(g))$, and
$(\lambda^{u}(\delta_{U})\eta)(g)=[g\in\mathcal{G}^{r(h)},h\in
U]\eta(h^{-1}g)$ for $g\in\mathcal{G}$. With the above notation, Theorem 6.15
gives the following result.

######  Corollary 6.17.

The unitary
$W\colon\mathrm{C_{0}}(\mathcal{H})\otimes_{\mathrm{C_{0}}(X)}\ell^{2}(\mathcal{G})\xrightarrow{\sim}\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{H}))$
given by $W(\xi\otimes\eta)(g):=L_{g}^{-1}(\xi(r(g)))\eta(g)$, for
$\xi\in\mathrm{C_{0}}(\mathcal{H}),\eta\in\ell^{2}(\mathcal{G})$,
$g\in\mathcal{G}$, yields a unitary equivalence between the the $L$-induced
representation $L\operatorname{-Ind}(\lambda^{u})\colon
C^{*}(\mathcal{G},u)\to\mathcal{L}\big{(}\mathrm{C_{0}}(\mathcal{H})\otimes_{\mathrm{C_{0}}(X)}\ell^{2}(\mathcal{G})\big{)}$and
the regular induced representation $\pi^{\mathcal{H}}\colon
C^{*}(\mathcal{G},u)\to\mathcal{L}(\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{H})))$.

######  Remark 6.18.

By Proposition 6.4, $\pi^{\mathcal{H}}$ is naturally unitarily equivalent to
the representation
$\Lambda^{\mathcal{H}}:=\lambda^{u}\otimes\mathrm{id}_{\mathrm{C_{0}}(\mathcal{H})}$
of $\mathrm{C}^{*}(\mathcal{G},u)$ on
$\mathrm{C_{0}}(\ell^{2}(\mathcal{G}))\otimes_{\mathrm{C_{0}}(X)}\mathrm{C_{0}}(\mathcal{H})$.
Hence the above corollary amounts to saying that
$\lambda^{u}\otimes\mathrm{id}_{\mathrm{C_{0}}(\mathcal{H})}\cong
L\operatorname{-Ind}(\lambda^{u})$, and so the regular representation induced
by $\mathcal{H}$ from $\lambda^{u}$ is unitarily equivalent to the
representation induced by $L$ from $\lambda^{u}$. Pictorially, we can tensor
$\mathcal{H}$ by $\ell^{2}(\mathcal{G})$ on both sides and equivariance allows
us to move regular representations from one side to another.

We recall, see Example 6.12, that the regularization
$(\ell^{2}(s^{*}\mathcal{H}),L^{\mathcal{G}})$ of the Hilbert
$\mathcal{G}$-bundle $(\mathcal{H},L)$ is unitarily isomorphic to the
continuous Hilbert $\mathcal{G}$-bundle
$(\mathcal{H}\otimes\\{\ell^{2}(\mathcal{G}_{x})\\}_{x\in X},L\otimes\rho)$
where $(\\{\ell^{2}(\mathcal{G}_{x})\\}_{x\in X},\rho)$ is the right regular
continuous $\mathcal{G}$-bundle, cf. Examples 4.12, 5.6, 6.6. Thus for any
representation $\psi\colon C^{*}(\mathcal{G},u)\to B(H)$ on a Hilbert space
$H$, the induced representation $L^{\mathcal{G}}\operatorname{-Ind}(\psi)$
acts on
$\mathrm{C_{0}}(\mathcal{H})\otimes_{\mathrm{C_{0}}(X)}\ell^{2}(\mathcal{G})\otimes_{\psi}H$
by the formula

$L^{\mathcal{G}}\operatorname{-Ind}(\psi)(f\delta_{U})(\xi\otimes\eta\otimes\zeta)=\lim_{i}f\cdot
L_{U}(\xi)\otimes\rho_{U}(\eta)\otimes\psi(\delta_{U})\zeta$

for an open bisection $U\subseteq\mathcal{G}$,
$f\in\mathrm{C_{c}}(r(U))\subseteq\mathrm{C_{c}}(X)$,
$\xi\in\mathrm{C_{0}}(\mathcal{H}|_{r(U)})\subseteq\mathrm{C_{0}}(\mathcal{H})$,
$\eta\in\mathrm{C_{c}}(\mathcal{G})\subseteq\ell^{2}(\mathcal{G})$, where
$\rho_{U}(\eta)(g)=[g\in\mathcal{G}_{r(h)},h\in U]\eta(gh)$ and
$\psi(\delta_{U})\zeta$ is the strong limit of $\psi(e_{i}\delta_{U})\zeta$
where $\\{e_{i}\\}_{i\in I}$ an approximate unit of the ideal
$\mathrm{C_{0}}(r(U))\subseteq\mathrm{C_{0}}(X)$. Theorem 6.16 specialises in
this context to the following corollary.

######  Corollary 6.19.

Consider a continuous Hilbert $\mathcal{G}$-bundle $(\mathcal{H},L)$ over a
twisted groupoid $(\mathcal{G},u)$ and a representation $\psi\colon
C^{*}(\mathcal{G},u)\to B(H)$ of $C^{*}(\mathcal{G},u)$ on a Hilbert space
$H$. Then the representation
$L^{\mathcal{G}}\operatorname{-Ind}(\psi):C^{*}(\mathcal{G},u)\to\mathcal{L}(\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{H}))\otimes_{\psi}H)$
induced from $\psi$ by the regularization of $L$ is unitarily equivalent to
the amplification $\Lambda^{\mathcal{H}}\otimes
1_{H}:C^{*}(\mathcal{G},u)\to\mathcal{L}(\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{H}))\otimes_{\psi}H)$
of the regular representation $\Lambda^{\mathcal{H}}$ induced by
$\mathcal{H}$.

##  7\. Fourier–Stieltjes multipliers for groupoid twisted actions

This section is central for the work, as we are ready to exploit the results
established earlier in the paper to construct the Banach category of
Fourier–Stieltjes multipliers for twisted groupoid crossed products. Recall
that $\mathcal{G}$ is a fixed locally compact Hausdorff étale groupoid with
unit space $X=\mathcal{G}^{(0)}$. As in the last section we begin the
discussion in the general context of Fell bundles.

######  Definition 7.1.

Let $\mathcal{A}=\\{A_{g}\\}_{g\in\mathcal{G}}$ and
$\mathcal{B}=\\{B_{g}\\}_{g\in\mathcal{G}}$ be Fell bundles over
$\mathcal{G}$. A _multiplier_ from $\mathcal{A}$ to $\mathcal{B}$ is a
continuous bundle of fiberwise bounded operators
$T=\\{T_{g}\\}_{g\in\mathcal{G}}$, i.e. a family of maps
$T_{g}\in\mathbb{B}(A_{g},B_{g})$ such that for each
$a\in\mathrm{C_{c}}(\mathcal{A})$ the map

(7.1) $\mathcal{G}\ni g\longmapsto(m_{T}a)(g):=T_{g}(a(g))\in B_{g},$

is in $\mathrm{C_{c}}(\mathcal{B})$. If $A$ and $B$ are completions of
$\mathrm{C_{c}}(\mathcal{A})$ and $\mathrm{C_{c}}(\mathcal{B})$ respectively,
we say that the multiplier $T$ is $A$-$B$ _bounded_ (resp.  _completely
bounded_ or _completely positive_) if the map
$m_{T}:\mathrm{C_{c}}(\mathcal{A})\to\mathrm{C_{c}}(\mathcal{B})$ extends to a
bounded (resp. completely bounded, completely positive) map $m_{T}:A\to B$.
Usually, we will be interested in the case when $A$ and $B$ are reduced or
full section $\mathrm{C}^{*}$-algebras, in which case we will call $T$ a
_reduced_ or _full multiplier_ , respectively.

######  Remark 7.2.

There is a somewhat unfortunate long tradition of calling both the family
$T=\\{T_{g}\\}_{g\in\mathcal{G}}$ and the resulting map $m_{T}$ a multiplier.
As it is difficult to fight the tradition, we will rather follow it and hope
that it does not cause confusion to the readers.

######  Example 7.3 (Herz–Schur multipliers).

If $\mathcal{A}=\mathcal{B}$, then any bounded strictly continuous section
$\varphi\in\mathrm{C_{b}}(r^{*}\mathcal{M}(\mathcal{A}|_{X}))$ defines a
multiplier $T^{\varphi}$ from $\mathcal{A}$ to $\mathcal{A}$ where
$T^{\varphi}_{g}a:=\varphi(g)a$ for all $a\in A_{g}$, $g\in\mathcal{G}$. One
may call $T^{\varphi}$ a _Herz–Schur multiplier_ for $\mathcal{A}$ and write
$m_{\varphi}:=m_{T^{\varphi}}$, so

(7.2) $(m_{\varphi}a)(g):=\varphi(g)a(g)\in B_{g},\qquad
a\in\mathrm{C_{c}}(\mathcal{A}),\ g\in\mathcal{G}.$

Specialising to the case of the Fell bundle $\mathcal{A}^{(\alpha,u)}$ of a
twisted action $(\alpha,u)$ of $\mathcal{G}$, we have
$\varphi\in\mathrm{C_{b}}(r^{*}\mathcal{M}(\mathcal{A}))$ and the
corresponding multipliers were studied in [KLS22] (in the context of actions
of twisted groupoids). Note that in the case of trivial bundles, every
multiplier is a Herz-Schur multiplier (as fibres are one-dimensional). The
Herz–Schur multipliers studied in [MTT18], [MSTT18] for group actions, are
what we would call completely bounded reduced multipliers.

######  Example 7.4 (Exel multipliers).

Let $\mathcal{A}$ be a Fell bundle over $\mathcal{G}$ and let
$\mathcal{A}^{0}:=\mathcal{A}|_{X}$ be the restriction of $\mathcal{A}$ to
$X$. Consider two sections $\xi,\zeta\in\mathrm{C_{c}}(r^{*}\mathcal{A}^{0})$
and define

$T_{g}(a)=\sum_{h\in\mathcal{G}^{r(g)}}\xi(h)^{*}\cdot
a\cdot\zeta(g^{-1}h),\quad a\in A_{g},\ g\in\mathcal{G}.$

Here we use the natural $A_{r(g)}$-$A_{s(g)}$-bimodule structure of $A_{g}$.
The sum above is finite because $\xi$ has compact support and
$\mathcal{G}^{r(g)}$ is discrete. It is clear that $T_{g}$ is a bounded linear
map $A_{g}\to A_{g}$. Moreover, if $a\in\mathrm{C_{c}}(\mathcal{A})$, then
$g\mapsto T_{g}(a(g))$ is again a continuous section of $\mathcal{A}$ because
$\mathrm{C_{c}}(\mathcal{A})$ is a right module over
$\mathrm{C_{c}}(s^{*}(\mathcal{A}^{0}))$ in the canonical way.

For groups, multipliers of the above form first appeared in Exel’s work
[Exe97] and are used to define a notion of _approximation property_ for Fell
bundles over groups. More recently, the last notion has been generalized to
étale groupoids or inverse semigroups in [Kra23, BM23].

As mentioned in Remark 7.2, in the sequel we will often identify a multiplier
$T=\\{T_{g}\\}_{g\in\mathcal{G}}$ with the associated linear map
$m_{T}:\mathrm{C_{c}}(\mathcal{A})\to\mathrm{C_{c}}(\mathcal{B})$ given by
(7.1). _The set of all multipliers_ $m(\mathcal{A},\mathcal{B})$ from
$\mathcal{A}$ to $\mathcal{B}$, when viewed as maps on continuous sections, is
naturally an involutive $\mathrm{C_{b}}(X)$-bimodule. Namely, for
$T=\\{T_{g}\\}_{g\in\mathcal{G}},S=\\{S_{g}\\}_{g\in\mathcal{G}}\in
m(\mathcal{A},\mathcal{B})$ and $f\in\mathrm{C_{b}}(X)$ these structures are
given by $(\lambda T)_{g}:=\lambda T_{g}$, $(T+S)_{g}:=T_{g}+S_{g}$, $(f\cdot
T)_{g}:=f(r(g))T_{g}$, $(T\cdot f)_{g}:=f(s(g))T_{g}$, and

(7.3) $T^{\dagger}_{g}(a):=(T_{g^{-1}}(a^{*}))^{*},\qquad g\in\mathcal{G},\
a\in A_{g}.$

The involution $\dagger$ is conjugate-linear and for $f,T$ as above we have
$(f\cdot T)^{\dagger}=T^{\dagger}\cdot\overline{f}$, $(T\cdot
f)^{\dagger}=\overline{f}\cdot T^{\dagger}$. In terms of the map
$m_{T}:\mathrm{C_{c}}(\mathcal{A})\to\mathrm{C_{c}}(\mathcal{B})$ it is given
by $m_{T^{\dagger}}(a):=m_{T}(a^{*})^{*}$, $a\in\mathrm{C_{c}}(\mathcal{A})$.
We can also compose multipliers if the domain and codomain agree: for
$T=\\{T_{g}\\}_{g\in\mathcal{G}}\in m(\mathcal{A},\mathcal{B})$,
$S=\\{S_{g}\\}_{g\in\mathcal{G}}\in m(\mathcal{B},\mathcal{C})$ we put

$S\circ T:=\\{S_{g}\circ T_{g}\\}_{g\in\mathcal{G}}.$

In particular, all Fell bundles over $\mathcal{G}$ with multipliers as
morphisms form a category. Also $m(\mathcal{A}):=m(\mathcal{A},\mathcal{A})$
is an algebra and an involutive $\mathrm{C_{b}}(X)$-bimodule. The involution
(7.3) is multiplicative on the algebra $m(\mathcal{A})$, so it is not an
‘involution’ in the ordinary sense unless $m(\mathcal{A})$ is commutative.

When restricting to Fell bundles coming from twisted actions of $\mathcal{G}$
we have a distinguished subcategory with morphisms defined as follows. Let
$(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ be twisted actions of
$\mathcal{G}$ on $\mathrm{C}^{*}$-bundles $\mathcal{A}$ and $\mathcal{B}$ over
$X$, respectively, and consider the associated Fell bundles
$\mathcal{A}^{(\alpha,u_{\alpha})}=r^{*}\mathcal{A}$ and
$\mathcal{B}^{(\beta,u_{\beta})}=r^{*}\mathcal{B}$ over $\mathcal{G}$. For any
$(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$-equivariant representation $L$ of
$\mathcal{G}$ on a $\mathrm{C}^{*}$-$\mathcal{A}$-$\mathcal{B}$-correspondence
bundle $\mathcal{E}$ and any bounded strictly continuous sections
$\xi,\zeta\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))$ the formula

$T_{g}(a):=\langle\xi(r(g)),a\overline{L}_{g}\zeta(s(g))\rangle_{\mathcal{M}(B_{r(g)})}\in
B_{r(g)},\qquad a\in A_{r(g)},\ g\in\mathcal{G},$

defines a multiplier
$T_{\mathcal{E},L,\xi,\zeta}:=\\{T_{g}\\}_{g\in\mathcal{G}}$ from
$\mathcal{A}^{(\alpha,u_{\alpha})}$ to $\mathcal{B}^{(\beta,u_{\beta})}$, cf.
Proposition 5.4(3’). We can alternatively express $T_{g}$ without using the
extended action $\overline{L}$, as the following formula holds

$T_{g}(a)=\lim_{i}\left\langle
e_{i}\xi(r(g)),L_{g}\Big{(}\alpha_{g^{-1}}(a)\zeta(s(g))\Big{)}\right\rangle_{B_{r(g)}}$

where $\\{e_{i}\\}_{i}$ is an approximate unit in $A_{r(g)}$, $a\in A_{r(g)}$,
$g\in\mathcal{G}$.

######  Definition 7.5.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. We call $T_{\mathcal{E},L,\xi,\zeta}$ defined above the
_Fourier–Stieltjes multiplier_ associated to the
$(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$-equivariant representation $L$ of
$\mathcal{G}$ and sections
$\xi,\zeta\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))$. We denote by
$FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ the set of all Fourier–Stieltjes
multipliers from $(\alpha,u_{\alpha})$ to $(\beta,u_{\beta})$. We also write
$FS(\alpha,u_{\alpha}):=FS[(\alpha,u_{\alpha}),(\alpha,u_{\alpha})]$.

######  Example 7.6.

If $\mathcal{G}=G$ is a discrete group then $r^{*}\mathcal{A}\equiv G\times A$
and $A_{r(g)}=A$ for every $g\in G$. Thus if $(\alpha,u_{\alpha})$ and
$(\beta,u_{\beta})$ are twisted group actions of $\mathcal{G}$ respectively on
a $\mathrm{C}^{*}$-algebra $A$ and $B$, we may identify the Fourier–Stieltjes
multipliers in $FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ with maps
$T_{E,L,\xi,\zeta}:G\times A\to B$ given by
$T_{E,L,\xi,\zeta}(g,a)=\langle\xi,a\overline{L}_{g}\zeta\rangle_{\mathcal{M}(B)}$,
for $a\in A$, $\xi,\zeta\in\mathcal{M}(E)$, where $L$ is an equivariant
representation of $\mathcal{G}$ on an $A$-$B$-$\mathrm{C}^{*}$-correspondence
$E$. When $A=B$ is unital, these maps were called Fourier–Stieltjes
coefficients in [BC15]. In contrast to [BC15] we do not require the unitality
of the $\mathrm{C}^{*}$-algebras in question and also allow different ‘source’
and ‘target’ algebras.

######  Example 7.7.

If $\mathcal{A}$ is the trivial one-dimensional bundle over $X$, so that
$A=\mathrm{C_{0}}(X)$ and $r^{*}\mathcal{A}$ is the trivial one-dimensional
$\mathcal{G}$-bundle, we may identify the Fourier–Stieltjes multipliers with
Fourier–Stieltjes coefficients
$T_{\mathcal{H},\ell,\xi,\zeta}:\mathcal{G}\to\mathbb{C}$ given by
$T_{\mathcal{H},\ell,\xi,\zeta}(g)=\langle\xi(r(g)),\ell_{g}\zeta(s(g))\rangle$,
where $(\mathcal{H},\ell)$ is a $\mathcal{G}$-Hilbert bundle and
$\xi,\zeta\in\mathrm{C_{b}}(\mathcal{H})$, see for instance [Pat04] (in the
untwisted case) and compare to Examples 4.12 and 5.6.

######  Example 7.8.

If $\mathcal{A}$ is a $\mathrm{C}^{*}$-bundle over $X$, carrying a twisted
action $(\alpha,u_{\alpha})$ of $\mathcal{G}$, then the identity map on
$\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ arises as a
Fourier–Stieltjes multiplier. Namely, the unit section $1$ belongs to
$\mathrm{C_{b}}(\mathcal{M}(\mathcal{A}))$, for the multiplier
$\mathrm{C}^{*}$-correspondence bundle
$\mathcal{M}(\mathcal{A})=\\{\mathcal{M}(A_{x})\\}_{x\in X}$, and
$T_{\mathcal{A},\alpha,1,1}\in FS(\alpha,u_{\alpha})$ is a multiplier
consisting of identities on each fiber.

######  Lemma 7.9.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. Every Fourier–Stieltjes multiplier
$T:=T_{\mathcal{E},L,\xi,\zeta}\in FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$
is $\mathrm{C_{0}}$-completely bounded, meaning that the operator
$m_{T}:\mathrm{C_{0}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C_{0}}(\mathcal{B}^{(\beta,u_{\beta})})$
is completely bounded, where the spaces are equipped with the supremum norm.
In fact, $\|m_{T}\|_{cb}\leq\|\xi\|\cdot\|\zeta\|$. The same statement holds
if we view $m_{T}$ as an operator mapping
$\mathrm{C_{b}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ to
$\mathrm{C_{b}}(\mathcal{B}^{(\beta,u_{\beta})})$.

###### Proof.

Fix $T:=T_{\mathcal{E},L,\xi,\zeta}\in
FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$. The Schwarz inequality in the
$\mathrm{C_{0}}(\mathcal{B})$-Hilbert module $\mathrm{C_{0}}(\mathcal{E})$
implies that $\|m_{T}(a)\|\leq\|a\|\|\xi\|\cdot\|\zeta\|$ for
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$. Hence we have a
bounded operator
$m_{T}:\mathrm{C_{0}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C_{0}}(\mathcal{B}^{(\beta,u_{\beta})})$
with $\|m_{T}\|\leq\|\xi\|\cdot\|\zeta\|$. This generalizes to amplifications.
Indeed, for any $n\in\mathbb{N}$, we have a natural identification of matrix
amplifications
$\mathbb{M}_{n}(\mathrm{C_{0}}(r^{*}\mathcal{A}))\cong\mathrm{C_{0}}(r^{*}(\mathbb{M}_{n}(\mathcal{A}))$,
where $\mathbb{M}_{n}(\mathcal{A})$ is the Fell bundle over $X$ with fibers
given by $n$ by $n$ matrices over fibers of $\mathcal{A}$ (and the twisted
action of $\mathcal{G}$ extended to $\mathbb{M}_{n}(\mathcal{A})$ in the
obvious way), see Example 4.13. Thus to estimate the norm of
$m_{T}^{(n)}:\mathbb{M}_{n}(\mathrm{C_{0}}(\mathcal{A}^{(\alpha,u_{\alpha})}))\to\mathbb{M}_{n}(\mathrm{C_{0}}(\mathcal{B}^{(\beta,u_{\beta})}))$
it suffices to fix $g\in\mathcal{G}$ and estimate the norm of
$T_{g}^{(n)}:\mathbb{M}_{n}(A_{r(g)})\to\mathbb{M}_{n}(B_{r(g)})$. A
straightforward computation shows that given a matrix
$a=[a_{ij}]_{i,j=1}^{n}\in\mathbb{M}_{n}(A_{r(g)})$ we have

$T_{g}^{(n)}(a)=\langle\tilde{\xi},a\cdot\tilde{\zeta}\rangle,$

where
$\tilde{\xi}=[\xi(r(g))\delta_{ij}]_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathcal{E}_{r(g)})$,
$\tilde{\zeta}=[\overline{L}_{g}\zeta(s(g))\delta_{ij}]_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathcal{E}_{r(g)})$.
As before, the Schwarz inequality shows that

$\|T_{g}^{(n)}(a)\|\leq\|a\|\|\tilde{\xi}\|\|\tilde{\zeta}\|,$

and it remains to note that by [Ble97, Section 3, Formula (1’)] we have

$\displaystyle\|\tilde{\xi}\|$
$\displaystyle=\Big{\|}\Big{[}\sum_{k=1}^{n}\langle\tilde{\xi}_{ki},\tilde{\xi}_{kj}\rangle\Big{]}_{i,j=1}^{n}\Big{\|}^{\frac{1}{2}}=\Big{\|}\big{[}\langle\xi(r(g)),\xi(r(g))\rangle\delta_{i,j}\big{]}_{i,j=1}^{n}\Big{\|}^{\frac{1}{2}}$
$\displaystyle=\|\langle\xi(r(g)),\xi(r(g))\rangle\|^{\frac{1}{2}}=\|\xi(r(g))\|\leq\|\xi\|$

and similarly $\|\tilde{\zeta}\|=\|\zeta(s(g))\|\leq\|\zeta\|$. Hence
$\|m_{T}^{(n)}\|=\sup_{g\in\mathcal{G}}\|T_{g}^{(n)}\|\leq\|\xi\|\cdot\|\zeta\|$.
∎

For fixed $\mathcal{G}$, the class of twisted actions with (positive-definite)
Fourier–Stieltjes multipliers as morphisms forms a Banach
$\mathrm{C_{b}}(X)$-bimodule category.

######  Proposition 7.10.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. The Fourier–Stieltjes multipliers
$FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ form an involutive Banach
$\mathrm{C_{b}}(X)$-bimodule with the algebraic structure inherited from
$m(\mathcal{A},\mathcal{B})$ and the norm

(7.4) $\|T\|_{FS}:=\inf\\{\|\xi\|\|\zeta\|:T=T_{\mathcal{E},L,\xi,\zeta}\text{
for some }(\mathcal{E},L)\text{ and
}\xi,\zeta\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))\\},$

Moreover, the composition of Fourier–Stieltjes multipliers is a
Fourier–Stieltjes multiplier, the norm $\|\cdot\|_{FS}$ is submultiplicative
and the identities are Fourier–Stieltjes multipliers. In particular,
$FS(\alpha,u_{\alpha})$ is a unital Banach algebra and involutive Banach
$\mathrm{C_{b}}(X)$-bimodule with a multiplicative, conjugate-linear
involution.

###### Proof.

The fact that the collection of Fourier–Stieltjes multipliers
$FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ is closed under linear
combinations, and in fact forms a $\mathrm{C_{b}}(\mathcal{G}^{(0)})$-bimodule
follows from the following straightforward relations

$f\cdot
T_{\mathcal{E},L,\xi,\zeta}=T_{\mathcal{E},L,\overline{f}\xi,\zeta},\quad
T_{\mathcal{E},L,\xi,\zeta}\cdot f=T_{\mathcal{E},L,\xi,f\zeta},\quad
T_{\mathcal{E},L,\xi_{1},\zeta_{1}}+T_{\mathcal{F},K,\xi_{2},\zeta_{2}}=T_{\mathcal{E}\oplus\mathcal{F},L\oplus
K,\xi_{1}\oplus\xi_{2},\zeta_{1}\oplus\zeta_{2}}.$

These also imply that $\|\cdot\|_{FS}$ is a seminorm. Since $\|\cdot\|_{FS}$
majorizes the operator norm in
$\mathbb{B}(\mathrm{C_{0}}(r^{*}\mathcal{A}),\mathrm{C_{0}}(r^{*}\mathcal{B}))$,
see Lemma 7.9, and clearly $m_{T}=0$ if and only if $T=0$, the expression
$\|\cdot\|_{FS}$ is in fact a norm. That
$FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ is closed under the involution,
and that the involution is isometric, follows from the relation

$(T_{\mathcal{E},L,\xi,\zeta})^{\dagger}=T_{\mathcal{E},L,\zeta,\xi}.$

Indeed, set $T_{\mathcal{E},L,\xi,\zeta}=\\{T_{g}\\}_{g\in\mathcal{G}}$ and
let $g\in\mathcal{G}$, $a\in A_{r(g)}$. Using relations in Remarks 4.7, 5.2
and the fact that
$\alpha_{g}(u_{\alpha}(g^{-1},g)^{*})u_{\alpha}(g,g^{-1})=1_{\mathcal{M}(A_{r(g)})}$
we get:

$\begin{split}T_{g^{-1}}(a^{*})^{*}&=\left(\langle\xi(r(g^{-1})),u_{\alpha}(g^{-1},g)^{*}\alpha_{g^{-1}}(a^{*})\overline{L}_{g^{-1}}\zeta(s(g^{-1}))\rangle\right)^{*}\\\
&=u_{\beta}(g,g^{-1})^{*}\beta_{g}\left(\langle
u_{\alpha}(g^{-1},g)^{*}\alpha_{g^{-1}}(a^{*})\overline{L}_{g^{-1}}\zeta(r(g)),\xi(s(g))\rangle\right)\\\
&=u_{\beta}(g,g^{-1})^{*}\langle
L_{g}u_{\alpha}(g^{-1},g)^{*}L_{g^{-1}}a^{*}\zeta(r(g)),L_{g}\xi(s(g))\rangle\\\
&=u_{\beta}(g,g^{-1})^{*}\langle\alpha_{g}(u_{\alpha}(g^{-1},g)^{*})L_{g}L_{g^{-1}}a^{*}\zeta(r(g)),L_{g}\xi(s(g))\rangle\\\
&=u_{\beta}(g,g^{-1})^{*}\langle\alpha_{g}(u_{\alpha}(g^{-1},g)^{*})u_{\alpha}(g,g^{-1})a^{*}\zeta(r(g))u_{\beta}(g,g^{-1})^{*},L_{g}\xi(s(g))\rangle\\\
&=\langle
a^{*}\zeta(r(g)),L_{g}\xi(s(g))\rangle=\langle\zeta(r(g)),aL_{g}\xi(s(g))\rangle=T_{\mathcal{E},L,\zeta,\xi}(a)(g).\end{split}$

That Fourier–Stieltjes multipliers are closed under composition follows from
the relation

(7.5) $T_{\mathcal{F},K,\xi_{2},\zeta_{2}}\circ
T_{\mathcal{E},L,\xi_{1},\zeta_{1}}=T_{\mathcal{E}\otimes\mathcal{F},L\otimes
K,\xi_{1}\otimes\xi_{2},\zeta_{1}\otimes\zeta_{2}}.$

where we use notations from Examples 3.10 and 5.12. The displayed relation
holds because

$\begin{split}T_{\mathcal{F},K,\xi_{2},\zeta_{2}}(T_{\mathcal{E},L,\xi_{1},\zeta_{1}}(a))(g)&=\langle\xi_{2}(r(g)),T_{\mathcal{E},L,\xi_{1},\zeta_{1}}(a)\overline{K}_{g}\zeta_{2}(s(g))\rangle\\\
&=\langle\xi_{2}(r(g)),\langle\xi_{1}(r(g)),a(g)\overline{L}_{g}\zeta_{1}(s(g))\rangle_{\mathcal{M}(A_{r(g)})}\overline{K}_{g}\zeta_{2}(s(g))\rangle\\\
&=\langle(\xi_{1}\otimes\xi_{2})(r(g)),a(g)\overline{L}_{g}\zeta_{1}(s(g))\otimes\overline{K}_{g}\zeta_{2}(s(g))\rangle\\\
&=T_{\mathcal{E}\otimes\mathcal{F},L\otimes
K,\xi_{1}\otimes\xi_{2},\zeta_{1}\otimes\zeta_{2}}(a)(g).\end{split}$

Using this relation one sees that the norm $\|\cdot\|_{FS}$ is
submultiplicative. To prove that $FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$
is complete in this norm we take $\\{T_{i}\\}_{i=1}^{\infty}\subseteq
FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ such that
$\sum_{i=1}^{\infty}\|T_{i}\|_{FS}<\infty$ and we need to show that
$\sum_{i=1}^{\infty}T_{i}$ is convergent in $\|\cdot\|_{FS}$. We may pick
$(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$ equivariant representations of
$\mathcal{G}$ on $\mathrm{C}^{*}$-correspondences
$\\{(\mathcal{E}_{i},L_{i})\\}_{i=1}^{\infty}$ and non-zero sections
$\\{\xi_{i},\zeta_{i}\\}_{i=1}^{\infty}\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}_{i}))$
such that $T_{i}=T_{\mathcal{E}_{i},L_{i},\xi_{i},\zeta_{i}}$ and
$\|\xi_{i}\|\|\zeta_{i}\|\leq\|T_{i}\|+1/2^{i}$ for $i\in\mathbb{N}$. In
addition (by passing to $\sqrt{\frac{|\zeta_{i}\|}{\|\xi_{i}\|}}\xi_{i}$ and
$\sqrt{\frac{|\xi_{i}\|}{\|\zeta_{i}\|}}\zeta_{i}$) we may assume that
$\|\xi_{i}\|=\|\zeta_{i}\|$ for each $i\in\mathbb{N}$. Then it follows that
$\oplus_{i=1}^{\infty}\xi_{i}$ and $\oplus_{i=1}^{\infty}\zeta_{i}$ belong to
$\mathrm{C_{b}}(\mathcal{M}(\oplus_{i=1}^{\infty}\mathcal{E}_{i}))$ because
$\sum_{i=1}^{\infty}\|\xi_{i}\|^{2}=\sum_{i=1}^{\infty}\|\zeta_{i}\|^{2}<\sum_{i=1}^{\infty}\|T_{i}\|_{FS}+1$.
Let
$T:=T_{\oplus_{i=1}^{\infty}\mathcal{E}_{i},\oplus_{i=1}^{\infty}L_{i},\oplus_{i=1}^{\infty}\xi_{i},\oplus_{i=1}^{\infty}\zeta_{i}}$.
Then

$\begin{split}\|T-\sum_{i=1}^{n}T_{i}\|_{FS}&\leq\|\oplus_{i=n+1}^{\infty}\xi_{i}\|\cdot\|\oplus_{i=n+1}^{\infty}\zeta_{i}\|\leq\sum_{i=n+1}^{\infty}\|\xi_{i}\|^{2}\sum_{i=n+1}\|\zeta_{i}\|^{2}\\\
&<\left(\sum_{i=n+1}^{\infty}\|T_{i}\|_{FS}+1/2^{i}\right)^{2}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}0.\end{split}$

By Example 7.8, $T_{\mathcal{A},\alpha,1,1}$ is the unit in
$FS(\alpha,u_{\alpha})$ and clearly its FS-norm equals $1$. ∎

######  Example 7.11.

Applying the above construction to actions on a trivial line bundle
$\\{\mathbb{C}\\}_{x\in X}$, cf. Example 5.6, we get the _Fourier–Stieltjes
algebra $FS(\mathcal{G})$ of the groupoid $\mathcal{G}$_, similar to what was
considered in [RW97] or [Oty99]. In our setting, $FS(\mathcal{G})$ is the
space of all (continuous, bounded) functions
$\varphi\colon\mathcal{G}\to\mathbb{C}$ of the form
$\varphi(g)=\langle\xi(r(g)),\ell_{g}(\zeta(s(g)))\rangle$, for
$g\in\mathcal{G}$ and sections $\xi,\zeta\in\mathrm{C_{b}}(\mathcal{H})$ of a
(continuous) $\mathcal{G}$-Hilbert bundle $(\mathcal{H},\ell)$. By the above
proposition, $FS(\mathcal{G})$ is an involutive Banach algebra with respect to
the pointwise product of functions $\mathcal{G}\to\mathbb{C}$, the conjugation
as involution, and the norm given as in (7.4). By Remark 5.7, we get the same
algebra $FS(\mathcal{G},u)=FS(\mathcal{G})$ for any (continuous) $2$-cocycle
$u$ on $\mathcal{G}$.

######  Proposition 7.12.

For a twisted action $(\alpha,u)$ of $\mathcal{G}$ on $\mathcal{A}$, we have
an injective contractive involutive algebra homomorphism
$FS(\mathcal{G})\ni\varphi\mapsto T^{\varphi}\in FS(\alpha,u)$ where
$T^{\varphi}_{g}(a):=\varphi(g)a$, for $a\in A_{r(g)}$, $g\in\mathcal{G}$.

###### Proof.

Take any continuous $\mathcal{G}$-Hilbert bundle $(\mathcal{H},\ell)$ and
sections $\xi,\zeta\in\mathrm{C_{b}}(\mathcal{H})$ that define $\varphi$, i.e.
$\varphi(g)=\langle\xi(r(g)),\ell_{g}(\zeta(s(g)))\rangle$, for
$g\in\mathcal{G}$. Tensor $(\mathcal{H},\ell)$ with the trivial equivariant
representation $(\mathcal{A},\alpha)$ of $(\alpha,u)$, see Examples 5.9, 5.10.
For $a\in A_{r(g)}$, $g\in\mathcal{G}$ we have

$\begin{split}T_{\mathcal{A}\otimes\mathcal{H},\alpha\otimes\ell,1\otimes\xi,1\otimes\zeta}(g)(a)&=\langle(1\otimes\xi)\big{(}r(g)\big{)},a\cdot\overline{\alpha\otimes\ell}_{g}(1\otimes\zeta)\big{(}s(g)\big{)}\rangle=a\langle\xi\big{(}r(g)),\ell_{g}\big{(}\zeta(s(g))\big{)}\rangle\\\
&=\varphi(g)a=T^{\varphi}_{g}(a).\end{split}$

Hence $T^{\varphi}\in FS(\alpha,u)$ and
$\|T^{\varphi}\|_{FS}\leq\|1\otimes\xi\|\|1\otimes\zeta\|\leq\|\xi\|\|\zeta\|$,
which implies $\|T^{\varphi}\|_{FS}\leq\|\varphi\|_{FS}$. The map
$\varphi\mapsto T^{\varphi}$ is clearly injective, and it is easily checked
that it is an involutive algebra homomorphism. ∎

The next theorem is one of the central results in this paper. Its proof uses
the absorption principles established in Sections 5 and 6 and the inverse
semigroup picture introduced in Section 4.

######  Theorem 7.13.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. Every Fourier–Stieltjes multiplier $T\in
FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ yields strict completely bounded
maps
$m_{T}^{\operatorname{r}}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B}^{(\beta,u_{\beta})})$
and
$m_{T}^{\operatorname{f}}:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})$
between reduced and full crossed products, and
$\|m_{T}^{\operatorname{r}}\|_{cb},\|m_{T}^{\operatorname{f}}\|_{cb}\leq\|T\|_{FS}$.
If $T=T_{\mathcal{E},L,\xi,\xi}$ for some equivariant representation $L$ of
$\mathcal{G}$ on a $\mathrm{C}^{*}$-$\mathcal{A}$-$\mathcal{B}$-correspondence
bundle $\mathcal{E}$ and section
$\xi\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))$, then $m_{T}^{r}$ and
$m_{T}^{f}$ are completely positive and
$\|m_{T}^{\operatorname{r}}\|_{cb}=\|m_{T}^{\operatorname{f}}\|_{cb}=\|T\|_{FS}=\|\xi\|^{2}$.

###### Proof.

Let us fix an equivariant representation $L$ of $\mathcal{G}$ on a
$\mathrm{C}^{*}$-$\mathcal{A}$-$\mathcal{B}$-correspondence bundle
$\mathcal{E}$, sections $\xi,\zeta\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))$
and the corresponding Fourier–Stieltjes multiplier
$T:=T_{\mathcal{E},L,\xi,\zeta}=\\{T_{g}\\}_{g\in\mathcal{G}}$. To deal with
reduced crossed products we will use our first version of the Fell absorption
principle. Using the notation from Example 3.10 and Theorem 6.15, we define
the (creation) operator
$\theta_{\xi}:\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{B}))\to\mathrm{C_{0}}(\mathcal{E})\otimes_{\pi^{\mathcal{B}}}\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{B}))$
by the formula $\theta_{\xi}(b):=\xi\otimes b$,
$b\in\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{B}))$; for an approximate unit
$\\{e_{i}\\}_{i\in I}$ in $\mathrm{C_{0}}(\mathcal{B})$ we have $\xi\otimes
b=\lim_{i}\xi\otimes\pi^{\mathcal{B}}(e_{i})b=\lim_{i}\xi e_{i}\otimes
b\in\mathrm{C_{0}}(\mathcal{E})\otimes_{\pi^{\mathcal{B}}}\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{B}))$.
The operator $\theta_{\xi}$ is adjointable with norm dominated by $\|\xi\|$
and adjoint given by the formula

(7.6) $\theta_{\xi}^{*}\big{(}a\zeta\otimes
b)=\pi_{\mathcal{B}}(\langle\xi,a\zeta\rangle_{\mathrm{C_{0}}(\mathcal{B})})b,$

for $a\in\mathrm{C_{0}}(\mathcal{A})$ and
$b\in\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{B}))$ (then
$a\zeta\in\mathrm{C_{0}}(\mathcal{E})$, so the inner product takes values in
$\mathrm{C_{0}}(\mathcal{B})$ and the formula makes sense). Using the unitary
$W:\mathrm{C_{0}}(\mathcal{E})\otimes_{\pi^{\mathcal{B}}}\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{B}))\to\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))$
from Theorem 6.15 we define a map
$\Psi:\mathcal{L}(\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E})))\to\mathcal{L}(\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{B})))$
by the formula

$\Psi(S):=\theta_{\xi}^{*}W^{*}SW\theta_{\zeta},\qquad
S\in\mathcal{L}(\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))).$

By its form, $\Psi$ is completely bounded and strict, with
$\|\Psi\|_{cb}\leq\|\xi\|\|\zeta\|$. We claim that
$\pi^{\mathcal{B}}(m_{T}(a))=\Psi(\pi^{\mathcal{E}}(a))$ for
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$. Indeed, by (6.4) it
suffices to check that for every $U\in\operatorname{Bis}(\mathcal{G})$,
$a\in\mathrm{C_{c}}(\mathcal{A}|_{r(U)})$, $b\in\ell^{2}(s^{*}\mathcal{B})$,
we have
$\pi^{\mathcal{B}}(m_{T}(a))b=\theta_{\xi}^{*}(aL_{U}(\zeta)\otimes\pi^{\mathcal{B}}(\delta_{U})(b))$.
But this readily follows from (7.6). Since $\pi^{\mathcal{B}}$ extends to a
faithful representation of
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B}^{(\beta,u_{\beta})})$, and
similarly for $\pi^{\mathcal{E}}$ and
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ (see
the last part of Proposition 6.4), we conclude that
$m_{T}:\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C_{c}}(\mathcal{B}^{(\beta,u_{\beta})})$
extends to a map
$m_{T}^{\operatorname{r}}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B}^{(\beta,u_{\beta})})$
with

$\pi^{\mathcal{B}}(m_{T}^{\operatorname{r}}(a))=\Psi(\pi^{\mathcal{E}}(a))=\theta_{\xi}^{*}W^{*}\pi^{\mathcal{E}}(a)W\theta_{\zeta},\qquad
a\in\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})}).$

This implies that $m_{T}^{\operatorname{r}}$ is strict, completely bounded and

$\|m_{T}^{\operatorname{r}}\|_{cb}=\|\Psi\circ\pi^{\mathcal{E}}\|_{cb}\leq\|\Psi\|_{cb}\leq\|\xi\|\|\zeta\|.$

In particular, $\|m_{T}^{\operatorname{r}}\|_{cb}\leq\|T\|_{FS}$ (since the
presentation of $T$ was arbitrary). If $\xi=\zeta$, then $\Psi$ and hence also
$m_{T}^{\operatorname{r}}$ is completely positive and for any approximate unit
$\\{e_{i}\\}_{i\in I}$ in $\mathrm{C_{0}}(\mathcal{A})$ we get

$\|m_{T}^{\operatorname{r}}\|_{cb}=\|\Psi\circ\pi^{\mathcal{E}}\|_{cb}=\lim_{i}\|\Psi\circ\pi^{\mathcal{E}}(e_{i})\|=\|\theta_{\xi}^{*}W^{*}W\theta_{\xi}\|=\|\xi\|^{2}.$

Thus we have $\|m_{T}^{\operatorname{r}}\|_{cb}=\|T\|_{FS}=\|\xi\|^{2}$.

To deal with full crossed products let
$\psi:\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})\to\mathbb{B}(\mathcal{H})$
be a faithful nondegenerate representation of
$\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})$ on a Hilbert space $H$. We
view $\psi=\pi\rtimes v$ as the integrated form of a covariant representation
$(\pi,v)$ of the associated inverse semigroup action, see Subsection 4.1.
Consider the induced representation
$L\operatorname{-Ind}(\psi):\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathbb{B}(\mathrm{C_{0}}(\mathcal{E})\otimes_{\pi}H)$
as in Proposition 5.14. Similarly to above we define the (creation) operator
$\theta_{\xi}:H\to\mathrm{C_{0}}(\mathcal{E})\otimes_{\pi}H$ by
$\theta_{\xi}(\pi(b)h):=(\xi\cdot b)\otimes h$ for $b\in
B=\mathrm{C_{0}}(\mathcal{B})$ and $h\in H$. In other words,
$\theta_{\xi}(h)=\lim_{i}\xi e_{i}^{B}\otimes h$ where $\\{e_{i}^{B}\\}$ is an
approximate unit in $B$. Its adjoint is determined by
$\theta_{\xi}^{*}\big{(}a\zeta\otimes h)=\pi(\langle
a^{*}\xi,\zeta\rangle_{B})h$, where $\zeta\in E$, $a\in A$, and $h\in H$. So
$\theta_{\xi}^{*}(\zeta\otimes h)=\lim_{i}\pi(\langle
e_{i}^{A}\xi,\zeta\rangle_{B})h$ where $\\{e_{i}^{A}\\}$ is an approximate
unit in $A$. We define a completely bounded map
$\Psi:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\to B(H)$ by the
formula

$\Psi(b):=\theta_{\xi}^{*}(L\operatorname{-Ind}(\psi))(b)\theta_{\zeta}.$

We claim that $\Psi(b)=\psi(T_{\mathcal{E},L,\xi,\zeta}(b))$ for all
$b\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$. It suffices to check
that for $b=a\delta_{U}$ where $a\in A_{r(U)}$,
$U\in\operatorname{Bis}(\mathcal{G})$. Then for any $g\in U$ we have

$T_{\mathcal{E},L,\xi,\zeta}(b)(g)=\langle\xi(r(g)),L_{g}(\alpha_{g}^{-1}(a(r(g))\zeta(s(g))\rangle_{\mathcal{M}(B_{r(g)})}=\lim_{i}\langle
e_{i}^{A}\xi,L_{U}(\alpha_{U}^{-1}(a)\zeta)\rangle_{B}(r(g)).$

That is $T_{\mathcal{E},L,\xi,\zeta}(b)\circ r|_{U}^{-1}=\lim_{i}\langle
e_{i}^{A}\xi,L_{U}(\alpha_{U}^{-1}(a)\zeta)\rangle_{B}$. Using this we get

$\begin{split}\Psi(b)h&=\lim_{i}\theta_{\xi}^{*}(L\operatorname{-Ind}(\psi))(b)(\zeta\cdot
e_{i}^{B})\otimes
h=\lim_{i}\theta_{\xi}^{*}\big{(}L_{U}(\alpha_{U}^{-1}(a)\zeta)\otimes
v_{U}h\big{)}\\\ &=\lim_{i}\pi(\langle
e_{i}^{A}\xi,L_{U}(\alpha_{U}^{-1}(a)\zeta)\rangle_{B})v_{U}h=\psi(T_{\mathcal{E},L,\xi,\zeta}(b))h.\end{split}$

This proves our claim and implies that the completely bounded map $\Psi$ maps
$\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})$ to
$\psi(\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})}))$. Thus, putting
$m_{T}^{\operatorname{f}}:=\psi^{-1}\circ\Psi$, we get the desired multiplier
map
$m_{T}^{\operatorname{f}}:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})$.
The assertions concerning norms follow as in the first part. ∎

Note that the above theorem will be later enriched by the discussion of the
_decomposable_ norms of the maps $m_{T}^{r}$ and $m_{T}^{s}$, see Proposition
11.13 and Theorem 11.17.

The set $CB[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ of completely bounded
$\mathrm{C_{0}}(X)$-bimodule maps
$\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})$
has a similar structure to that of Fourier–Stieltjes mulipliers described in
Proposition 7.10. Namely, it is a Banach space with pointwise operations and
the norm $\|\cdot\|_{cb}$. It is also an involutive Banach
$\mathrm{C_{b}}(X)$-bimodule, where the bimodule structure is determined by

$f\cdot\Psi(a):=\Psi((f\circ r)\cdot a),\qquad
f\cdot\Psi(a):=\Psi(a\cdot(f\circ s)),\qquad
f\in\mathrm{C_{b}}(X),\,a\in\mathrm{C_{c}}(\mathcal{B}^{(\beta,u_{\beta})}),$

with $\Psi\in CB[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$, and the involution
is given by

$\Psi^{\dagger}(a):=\Psi(a^{*})^{*},\qquad
a\in\mathrm{C_{c}}(\mathcal{B}^{(\beta,u_{\beta})}),$

cf. [Wal86, Proposition 1]. This involution is called _conjugation_ in
[Wal86], [BC15]. In the same way, the set
$CB^{\operatorname{red}}[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ of completely
bounded $\mathrm{C_{0}}(X)$-bimodule maps
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B}^{(\beta,u_{\beta})})$
forms an involutive Banach $\mathrm{C_{b}}(X)$-bimodule.

######  Remark 7.14.

Viewing the class of all twisted actions of $\mathcal{G}$ as objects, denote
by $FS_{TA}(\mathcal{G})$ the category where morphisms are Fourier–Stieltjes
multipliers, and by $CB_{TA}(\mathcal{G})$ (resp.
$CB_{TA}^{\operatorname{r}}(\mathcal{G})$) the category where morphisms are
completely bounded, strict $\mathrm{C_{0}}(X)$-bimodule maps between full
(resp. reduced) crossed products, equipped with the norm $\|\cdot\|_{cb}$. The
extensions in Theorem 7.13 give contractive involutive
$\mathrm{C_{b}}(X)$-bimodule functors
$m^{\operatorname{f}}:FS_{TA}(\mathcal{G})\to CB_{TA}(\mathcal{G})$ and
$m^{\operatorname{r}}:FS_{TA}(\mathcal{G})\to
CB_{TA}^{\operatorname{r}}(\mathcal{G})$.

##  8\. Positive-definiteness and multipliers on section C*-algebras

In this section we will specify our considerations to the completely positive
multipliers.

######  Definition 8.1.

Let $\mathcal{A}=\\{A_{g}\\}_{g\in\mathcal{G}}$ and
$\mathcal{B}=\\{B_{g}\\}_{g\in\mathcal{G}}$ be Fell bundles over
$\mathcal{G}$. We say that a multiplier $T=\\{T_{g}\\}_{g\in\mathcal{G}}$ from
$\mathcal{A}$ to $\mathcal{B}$ is _positive-definite_ if for any $x\in X$, any
$n\in\mathbb{N}$, any $g_{1},\ldots,g_{n}\in\mathcal{G}_{x}$ and any
collections $\\{a_{g_{i}}\in A_{g_{i}}:1\leq i\leq n\\}$ and $\\{b_{g_{i}}\in
B_{g_{i}}:1\leq i\leq n\\}$ we have

$\sum_{i,j=1}^{n}b_{g_{i}}^{*}\left(T_{g_{i}g_{j}^{-1}}(a_{g_{i}}a_{g_{j}}^{*})\right)b_{g_{j}}\geq
0$

(note that the last sum takes values in $B_{x}$). Then each of the maps
$T_{x}\in\mathbb{B}(A_{x},B_{x})$, $x\in X$, is completely positive, and we
say that $T$ is _strict_ if each map $T_{x}:A_{x}\to B_{x}$, $x\in X$, is
strict, and we call $T$ _bounded_ if $\sup_{x\in X}\|T_{x}\|<\infty$.

######  Remark 8.2.

By passing to adjoints and inverses (replacing $g_{i}$ by $g_{i}^{-1}$ and
putting $\widetilde{a}_{g_{i}}:=a_{g_{i}^{-1}}^{*}$,
$\widetilde{b}_{g_{i}}:=b_{g_{i}^{-1}}^{*}$) one sees that a multiplier
$T=\\{T_{g}\\}_{g\in\mathcal{G}}$ from $\mathcal{A}$ to $\mathcal{B}$ is
positive-definite if and only if any $n\in\mathbb{N}$, any
$g_{1},\ldots,g_{n}\in\mathcal{G}^{x}$ and any collections $\\{a_{g_{i}}\in
A_{g_{i}}:1\leq i\leq n\\}$ and $\\{b_{g_{i}}\in B_{g_{i}}:1\leq i\leq n\\}$
we have

$\sum_{i,j=1}^{n}b_{g_{i}}\left(T_{g_{i}^{-1}g_{j}}(a_{g_{i}}^{*}a_{g_{j}})\right)b_{g_{j}}^{*}\geq
0.$

######  Lemma 8.3.

Let $\mathcal{A}=\\{A_{g}\\}_{g\in\mathcal{G}}$ and
$\mathcal{B}=\\{B_{g}\\}_{g\in\mathcal{G}}$ be Fell bundles over
$\mathcal{G}$. Let $T=\\{T_{g}\\}_{g\in\mathcal{G}}$ be a reduced or full
multiplier from $\mathcal{A}$ to $\mathcal{B}$. If the corresponding map
$m_{T}$ between cross-sectional $\mathrm{C}^{*}$-algebras is completely
positive, then $T$ is positive-definite and bounded (in fact we have
$\|m_{T}\|=\sup_{x\in X}\|T_{x}\|$), and $T$ is strict if $m_{T}$ is strict.

###### Proof.

Fix $x\in X$, $n\in\mathbb{N}$, a family
$g_{1},\ldots,g_{n}\in\mathcal{G}_{x}$ and collections $\\{a_{g_{i}}\in
A_{g_{i}}:1\leq i\leq n\\}$, $\\{b_{g_{i}}\in B_{g_{i}}:1\leq i\leq n\\}$. We
may find sections $\\{f_{i}\\}_{i=1}^{n}\in\mathrm{C_{c}}(\mathcal{A})$
supported on open bisections and such that $f_{i}(g_{i})=a_{g_{i}}$, $1\leq
i\leq n$. Then $(f_{i}*f_{j}^{*})(g_{i}{g_{j}}^{-1})=a_{g_{i}}a_{g_{j}}^{*}$
for $1\leq i,j\leq n$. Since the matrix $(f_{i}*f_{j}^{*})_{i,j=1}^{n}$ is
positive both as an element of $\mathbb{M}_{n}(\mathrm{C}^{*}(\mathcal{A}))$
and of $\mathbb{M}_{n}(\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}))$, and
$m_{T}^{(n)}$ is positive by assumption, we conclude that the operator

$b:=\Lambda_{x}^{(n)}\circ
m_{T}^{(n)}\big{(}(f_{i}*f_{j}^{*})_{i,j=1}^{n}\big{)}\in\mathcal{L}(\ell^{2}(\mathcal{B})_{x}^{\oplus
n})$

is positive. Choose
$\xi_{i}\in\ell^{2}(\mathcal{B})_{x}=\bigoplus_{h\in\mathcal{G}_{x}}B_{h}$
such that $\xi_{i}(g_{j})=[i=j]b_{g_{i}}$ for $1\leq i,j\leq n$, and let
$\xi=\bigoplus_{i=1}^{n}\xi_{i}\in\ell^{2}(\mathcal{B})_{x}^{\oplus
n}=\bigoplus_{i=1}^{n}\bigoplus_{h\in\mathcal{G}_{x}}B_{h}$. Then

$0\leq\langle\xi,b\xi\rangle_{B_{x}}=\sum_{i,j=1}^{n}\langle\xi_{i},\Lambda_{x}(m_{T}(f_{i}*f_{j}^{*}))\xi_{j}\rangle_{B_{x}}=\sum_{i,j=1}^{n}b_{g_{i}}^{*}\left(T_{g_{i}g_{j}^{-1}}(a_{g_{i}}a_{g_{j}}^{*})\right)b_{g_{j}}.$

Hence $T$ is a positive-definite multiplier. Since any approximate unit
$\\{e_{i}\\}_{i\in I}$ in $\mathrm{C_{0}}(\mathcal{A}|_{X})$ is also an
approximate unit in $\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A})$ and in
$\mathrm{C}^{*}(\mathcal{A})$, and $m_{T}$ is a completely positive map, we
get

$\|m_{T}\|=\lim_{i\in I}\|m_{T}(e_{i})\|=\lim_{i\in I}\sup_{x\in
X}\|T_{x}(e_{i}(x))\|.$

Similarly, since, for any $x\in X$, the net $\\{e_{i}(x)\\}_{i\in I}$ is an
approximate unit in $A_{x}$, and $T_{x}:A_{x}\to B_{x}$ is a completely
positive map, we also get $\|T_{x}\|=\lim_{i\in I}\|T_{x}(e_{i}(x))\|$.
Combining these equalities one gets $\|m_{T}\|=\sup_{x\in X}\|T_{x}\|$.
Finally, if $m_{T}$ is strict, then the net $\\{T(e_{i})\\}_{i\in I}$
converges strictly to a positive element
$T(1)\in\mathcal{M}(\mathrm{C_{0}}(\mathcal{B}|_{X}))$. For any $x\in X$, the
net $\\{e_{i}(x)\\}_{i\in I}$ is an approximate unit in $A_{x}$. For any $a\in
B_{x}$ we may find
$b\in\mathrm{C_{c}}(\mathcal{B})\subseteq\mathrm{C}^{*}_{r}(\mathcal{B},\mathcal{G},\alpha)$
with $b(x)=a$, and then

$\lim_{i\in I}T_{x}(e_{i}(x))a=\lim_{i\in I}T_{x}(e_{i}(x))b(x)=\lim_{i\in
I}\big{(}T(e_{i})b\big{)}(x)=\big{(}T(1)b\big{)}(x).$

Thus the map $T_{x}:A_{x}\to B_{x}$ is strict. ∎

We will now pass to the context of twisted groupoid actions.

######  Lemma 8.4.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. For a multiplier $T=\\{T_{g}\\}_{g\in\mathcal{G}}$ from
$\mathcal{A}^{(\alpha,u_{\alpha})}$ to $\mathcal{B}^{(\beta,u_{\beta})}$ the
following are equivalent:

  1. (1)

$T$ is positive-definite;

  2. (2)

for any $x\in X$, $g_{1},\ldots,g_{n}\in\mathcal{G}_{x}$, $n\in\mathbb{N}$,
and any collection $\\{a_{g_{i}}\in A_{x}:1\leq i\leq n\\}$ the following
matrix is positive:

$\left(\beta_{g_{i}}^{-1}\left(T_{g_{i}g_{j}^{-1}}\big{(}\alpha_{g_{i}}(a_{g_{i}}a_{g_{j}}^{*})u_{\alpha}(g_{i}g_{j}^{-1},g_{j})^{*}\big{)}u_{\beta}(g_{i}g_{j}^{-1},g_{j})\right)\right)_{i,j=1}^{n}\in\mathbb{M}_{n}(B_{x});$

  3. (3)

for any $x\in X$, $g_{1},\ldots,g_{n}\in\mathcal{G}^{x}$, $n\in\mathbb{N}$,
and any collection $\\{a_{g_{i}}\in A_{x}:1\leq i\leq n\\}$ the following
matrix is positive:

$\left(\beta_{g_{i}}\left(T_{g_{i}^{-1}g_{j}}\big{(}\alpha_{g_{i}}^{-1}(a_{g_{i}}^{*}a_{g_{j}}u_{\alpha}(g_{i},g_{i}^{-1}g_{j})^{*})\big{)}\right)u_{\beta}(g_{i},g_{i}^{-1}g_{j})\right)_{i,j=1}^{n}\in\mathbb{M}_{n}(B_{x}).$

###### Proof.

(1)$\iff$(2) Fix $x\in X$, $n\in\mathbb{N}$ and a
$g_{1},\ldots,g_{n}\in\mathcal{G}_{x}$. Choosing $\tilde{a}_{g_{i}}\in
A_{g_{i}}^{(\alpha,u_{\alpha})}=A_{r(g_{i})}$ and $\tilde{b}_{g_{i}}\in
B_{g_{i}}^{(\beta,u_{\beta})}=B_{r(g_{i})}$, for $1\leq i\leq n$, is
equivalent to choosing $a_{g_{i}}\in A_{x}$ and $b_{g_{i}}\in B_{x}$, for
$1\leq i\leq n$, by putting
$\tilde{a}_{g_{i}}=\alpha_{g_{i}}(a_{g_{i}})\delta_{g_{i}}$ and
$\tilde{b}_{g_{i}}=\beta_{g_{i}}(b_{g_{i}})\delta_{g_{i}}$. Using the fact
that for all $1\leq i,j\leq n$

$\alpha_{g_{i}}(u_{\alpha}(g_{j}^{-1},g_{j}))=\alpha_{g_{i}}(u_{\alpha}(g_{j}^{-1},g_{j}))u_{\alpha}(g_{i},x)=u_{\alpha}(g_{i},g_{j}^{-1})u_{\alpha}(g_{i}g_{j}^{-1},g_{j})$

one gets
$\tilde{a}_{g_{i}}\cdot(\tilde{a}_{g_{j}})^{*}=\alpha_{g_{i}}(a_{g_{i}}a_{g_{j}}^{*})u_{\alpha}(g_{i}g_{j}^{-1},g_{j})^{*}\delta_{g_{i}g_{j}^{-1}}$.
Similarly for $1\leq i,j\leq n$

$\tilde{b}_{g_{i}}^{*}\left(T_{g_{i}g_{j}^{-1}}(\tilde{a}_{g_{i}}\tilde{a}_{g_{j}}^{*})\right)\tilde{b}_{g_{j}}=b_{g_{i}}^{*}\cdot\beta_{g_{i}}^{-1}\left(T_{g_{i}g_{j}^{-1}}\big{(}\alpha_{g_{i}}(a_{g_{i}}a_{g_{j}}^{*})u_{\alpha}(g_{i}g_{j}^{-1},g_{j})^{*}\big{)}u_{\beta}(g_{i}g_{j}^{-1},g_{j})\right)\cdot
b_{g_{j}}.$

Hence positivity of the elements
$\sum_{i,j=1}^{n}\tilde{b}_{g_{i}}^{*}\left(T_{g_{i}g_{j}^{-1}}(\tilde{a}_{g_{i}}\tilde{a}_{g_{j}}^{*})\right)\tilde{b}_{g_{j}}$
for all choices of elements $\tilde{b}_{g_{i}}\in
B_{g_{i}}^{(\beta,u_{\beta})}$ is equivalent to positivity of the matrix in
the assertion.

(1)$\iff$(3) We will use the version of (1) described in Remark 8.2. Fix $x\in
X$, $n\in\mathbb{N}$, and $g_{1},\ldots,g_{n}\in\mathcal{G}^{x}$. Choosing
$a_{g_{i}}\in A_{x}$ and $b_{g_{i}}\in B_{x}$, for $1\leq i\leq n$, is
equivalent to choosing $\tilde{a}_{g_{i}}\in A_{g_{i}}^{(\alpha,u_{\alpha})}$
and $\tilde{b}_{g_{i}}\in B_{g_{i}}^{(\beta,u_{\beta})}$, by putting
$\tilde{a}_{g_{i}}=a_{g_{i}}\delta_{g_{i}}$ and
$\tilde{b}_{g_{i}}=\beta_{g_{i}}(b_{g_{i}})\delta_{g_{i}}$. Fix then $i,j$.
The equality
$\alpha_{g_{i}}(u_{\alpha}(g_{i}^{-1},g_{j}))u(g_{i},g_{i}^{-1}g_{j})=u_{\alpha}(g_{i},g_{i}^{-1})=\alpha_{g}(u_{\alpha}(g_{i}^{-1},g_{i}))$
is equivalent to

$u_{\alpha}(g_{i}^{-1},g_{i})^{*}u_{\alpha}(g_{i}^{-1},g_{j})=\alpha_{g_{i}}^{-1}(u_{\alpha}(g_{i},g_{i}^{-1}g_{j})^{*}).$

Using the last equality one gets
$\tilde{a}_{g_{i}}^{*}\cdot(\tilde{a}_{g_{j}})=\alpha_{g_{i}}^{-1}\big{(}a_{g_{i}}^{*}a_{g_{j}}u_{\alpha}(g_{i},g_{i}^{-1}g_{j})^{*}\big{)}\delta_{g_{i}^{-1}g_{j}}$.
Similarly,

$\tilde{b}_{g_{i}}\left(T_{g_{i}^{-1}g_{j}}(\tilde{a}_{g_{i}}^{*}\tilde{a}_{g_{j}})\right)\tilde{b}_{g_{j}}^{*}=b_{g_{i}}\beta_{g_{i}}\left(T_{g_{i}^{-1}g_{j}}\big{(}\alpha_{g_{i}}^{-1}(a_{g_{i}}^{*}a_{g_{j}}u_{\alpha}(g_{i},g_{i}^{-1}g_{j})^{*})\big{)}\right)u_{\beta}(g_{i},g_{i}^{-1}g_{j})\cdot
b_{g_{j}}^{*}.$

Hence positivity of the elements
$\sum_{i,j=1}^{n}\tilde{b}_{g_{i}}\left(T_{g_{i}^{-1}g_{j}}(\tilde{a}_{g_{i}}^{*}\tilde{a}_{g_{j}})\right)\tilde{b}_{g_{j}}^{*}$
for all choices of elements $\tilde{b}_{g_{i}}\in
B_{g_{i}}^{(\beta,u_{\beta})}$ is equivalent to positivity of the matrix in
the assertion. ∎

We are ready to formulate and prove the main result of this section.

######  Theorem 8.5.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. For any multiplier $T$ from $\mathcal{A}^{(\alpha,u_{\alpha})}$
to $\mathcal{B}^{(\beta,u_{\beta})}$, the following are equivalent:

  1. (1)

$T$ is strict, bounded and positive-definite;

  2. (2)

$m_{T}$ extends to a strict completely positive map
$m_{T}^{\operatorname{r}}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B}^{(\beta,u_{\beta})})$;

  3. (3)

$m_{T}$ extends to a strict completely positive map
$m_{T}^{\operatorname{f}}:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})$;

  4. (4)

$T=T_{\mathcal{E},L,\xi,\xi}$ for some equivariant representation $L$ of
$\mathcal{G}$ on a $\mathrm{C}^{*}$-$\mathcal{A}$-$\mathcal{B}$-correspondence
bundle $\mathcal{E}$ and section
$\xi\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))$.

If the above equivalent conditions hold, then
$\|m_{T}^{\operatorname{f}}\|_{cb}=\|m_{T}^{\operatorname{r}}\|_{cb}=\|m_{T}^{\operatorname{f}}\|=\|m_{T}^{\operatorname{r}}\|=\|T\|_{FS}=\|\xi\|^{2}$
and the triple $(\mathcal{E},L,\xi)$ in (4) can be chosen to be cyclic, in the
sense that the set

$\\{L_{U}(a\xi b):a\in\mathrm{C_{0}}(\mathcal{A}|_{s(U)}),\
b\in\mathrm{C_{0}}(\mathcal{B}|_{s(U)}),\
U\in\operatorname{Bis}(\mathcal{G})\\}$

is linearly dense in $\mathrm{C_{0}}(\mathcal{E})$. Moreover, a cyclic triple
$(\mathcal{E},L,\xi)$ in (4) is unique up to a unitary equivalence in the
sense that if $(\mathcal{E}^{\prime},L^{\prime},\xi^{\prime})$ is another
cyclic triple with
$T=T_{\mathcal{E}^{\prime},L^{\prime},\xi^{\prime},\xi^{\prime}}$, then there
is an $A$-$B$-bimodule $\mathrm{C_{0}}(X)$-unitary
$W:\mathrm{C_{0}}(\mathcal{E})\to\mathrm{C_{0}}(\mathcal{E}^{\prime})$
establishing equivalence between $L$ and $L^{\prime}$, and sending $\xi$ to
$\xi^{\prime}$.

###### Proof.

The implications (4)$\implies$(2),(3) follow from Theorem 7.13 and the
implications (2),(3)$\implies$(1) are a consequence of Lemma 8.3.

Thus we only need to prove that (1) implies (4). Fix a strict, bounded and
positive-definite multiplier $T$. For each $x\in X$ define
$k_{x}:\mathcal{G}^{x}\times\mathcal{G}^{x}\to\mathcal{B}(A_{x},B_{x})$ via

$k_{x}(g,h)(a):=\beta_{g}\left(T_{g^{-1}h}\big{(}\alpha_{g}^{-1}(au_{\alpha}(g,g^{-1}h)^{*})\big{)}\right)u_{\beta}(g,g^{-1}h)\qquad
g,h\in\mathcal{G}_{x}.$

Then by Lemma 8.4, $k_{x}$ is positive-definite and strict in the sense of
Corollary 2.4. Thus, by this latter result, there is a
$\mathrm{C}^{*}$-correspondence $E_{x}$ from $A_{x}$ to $B_{x}$ and a map
$\xi_{x}:\mathcal{G}^{x}\to\mathcal{M}(E_{x})$ such that the set
$E_{x}^{0}:=\\{a\xi_{x}(g)b:g\in\mathcal{G}^{x},\ a\in A_{x},b\in B_{x}\\}$ is
linearly dense in $E_{x}$, and for $g,h\in\mathcal{G}^{x}$, $a\in A_{x}$, we
have

(8.1)
$\langle\xi_{x}(g),a\cdot\xi_{x}(h)\rangle_{\mathcal{M}(B_{x})}=\beta_{g}\left(T_{g^{-1}h}\big{(}\alpha_{g}^{-1}(au_{\alpha}(g,g^{-1}h)^{*})\big{)}\right)u_{\beta}(g,g^{-1}h)\in
B_{x}.$

In this way, we get a family of $\mathrm{C}^{*}$-correspondences
$\mathcal{E}:=\\{E_{x}\\}_{x\in X}$. Define the section
$\xi:\mathcal{G}\to\mathcal{M}(\mathcal{E})=\\{\mathcal{M}(E_{x})\\}_{x\in X}$
by setting $\xi(g):=\xi_{r(g)}(g)$, $g\in\mathcal{G}$. To topologize
$\mathcal{E}$ we let $\Gamma$ be the linear span of sections of the form
$a\xi_{U}b$ where $U\in\operatorname{Bis}(\mathcal{G})$,
$a\in\mathrm{C_{c}}(\mathcal{A}|_{r(U)})$,
$b\in\mathrm{C_{c}}(\mathcal{B}|_{r(U)})$ and

$(a\xi_{U}b)(x):=a(x)\xi(r|_{U}^{-1}(x))b(x),\qquad x\in X.$

If $c\xi_{V}d\in\Gamma$ is another element of this form, with
$V\in\operatorname{Bis}(\mathcal{G})$,
$c\in\mathrm{C_{c}}(\mathcal{A}|_{r(V})$,
$d\in\mathrm{C_{c}}(\mathcal{B}|_{r(V)})$, then $\langle
a\xi_{U}b,c\xi_{V}d\rangle_{B}(x):=\langle
a\xi_{U}b(x),c\xi_{V}d(x)\rangle_{B_{x}}$ is zero outside $r(U)\cap r(V)$ and
writing $g:=r|_{U}^{-1}(x)$ and $h:=r|_{V}^{-1}(x)$ for $x\in r(U)\cap r(V)$,
by (8.1) we get

$\langle
a\xi_{U}b,c\xi_{V}d\rangle_{B}(x)=b(x)^{*}\Big{(}\beta_{g}\Big{(}T_{g^{-1}h}\big{(}\alpha_{g}^{-1}(a(x)^{*}c(x)u_{\alpha}(g,g^{-1}h)^{*})\big{)}\Big{)}u_{\beta}(g,g^{-1}h)\Big{)}d(x).$

By continuity of all the operations involved, we see that the map $X\ni
x\mapsto\langle a\xi_{U}b,c\xi_{V}d\rangle_{B}(x)\in B_{x}$ belongs to
$\mathrm{C_{c}}(\mathcal{B})$. It extends uniquely to the (positive-definite)
conjugate-linear map
$\langle\cdot,\cdot\rangle_{B}:\Gamma\times\Gamma\to\mathrm{C_{c}}(\mathcal{B})\subseteq
B$. In particular, for any $a\in\Gamma$ the map
$X\mapsto\|a(x)\|^{2}=\|\langle a,a\rangle_{B}(x)\|$ is upper semicontinuous.
The fact that for each $x\in X$ the values of prescribed sections are dense in
the fibre $E_{x}$ follows immediately from the minimality assumption on each
$E_{x}$ and the above construction of $\Gamma$. Hence $\mathcal{E}$ admits a
unique topology making it a Banach bundle such that
$\Gamma\subseteq\mathrm{C_{c}}(\mathcal{E})$. The left $\mathcal{A}$ and right
$\mathcal{B}$-module actions on $\mathcal{E}$ are continuous by the very
definition of sections, and the $B$-valued inner product
$\langle\cdot,\cdot\rangle_{B}$ extends uniquely to
$\mathrm{C_{0}}(\mathcal{E})$ so that $\mathrm{C_{0}}(\mathcal{E})$ becomes a
$\mathrm{C_{0}}(X)$-$\mathrm{C}^{*}$-correspondence from $A$ to $B$.

We claim that there is an
$(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$-equivariant representation of
$\mathcal{G}$ on $\mathcal{E}$ determined by

(8.2)
$L_{g}(a\xi(h)b):=\alpha_{g}(a)u_{\alpha}{(g,h)}\xi(gh)u_{\beta}(g,h)^{*}\beta_{g}(b),$

where $g\in\mathcal{G}$, $h\in\mathcal{G}^{s(g)},a\in A_{s(g)},b\in B_{s(g)}$.
The most difficult (technically involved) part of the proof is showing that
for $h,f\in\mathcal{G}^{s(g)}$, $a,c\in A_{s(g)}$, $b,d\in B_{s(g)}$ we have

(8.3) $\langle
L_{g}(a\xi(h)b),L_{g}(c\xi(f)d)\rangle_{B_{r(g)}}=\beta_{g}(\langle
a\xi(h)b,c\xi(f)d\rangle_{B_{s(g)}}).$

We denote by $L$ and $R$ the left and right hand side of (8.3). To make the
calculations easier to read and write we abbreviate
$u_{g,h}:=u_{\alpha}{(g,h)}$ and $u_{g,h}:=u_{\beta}{(g,h)}$ as it is clear
from the context whether we twist $\alpha$ or $\beta$. Then

$\begin{split}R&=\beta_{g}\Big{(}b^{*}\beta_{h}\Big{(}T_{h^{-1}f}\big{(}\alpha_{h}^{-1}(a^{*}c\cdot
u^{*}_{h,h^{-1}f}\big{)}\Big{)}u_{h,h^{-1}f}\,d\Big{)}\\\
&=\beta_{g}(b^{*})u_{g,h}\,\beta_{gh}\Big{(}T_{h^{-1}f}\big{(}\alpha_{h}^{-1}(a^{*}c\cdot
u^{*}_{h,h^{-1}f})\big{)}\Big{)}u^{*}_{g,h}\beta_{g}(u^{\beta}_{h,h^{-1}f})\beta_{g}(d)\\\
&=\beta_{g}(b^{*})u_{g,h}\beta_{gh}\Big{(}T_{h^{-1}f}\big{(}\alpha_{h}^{-1}(a^{*}c\cdot
u^{*}_{h,h^{-1}f})\big{)}\Big{)}u_{gh,h^{-1}f}u^{*}_{g,h}\beta_{g}(d),\end{split}$

where in the last equality we used the cocycle identity. The left hand side is
equal to

$\begin{split}L&=\Big{\langle}\alpha_{g}(a)u_{g,h}\xi(gh)u^{*}_{g,h}\beta_{g}(b),\alpha_{g}(c)u_{g,f}\xi(gf)u^{*}_{g,f}\beta_{g}(d)\Big{\rangle}_{B_{r(g)}}\\\
&=\beta_{g}(b^{*})u_{g,h}\beta_{gh}\Big{(}T_{h^{-1}f}\big{(}\alpha_{gh}^{-1}(u^{*}_{g,h}\alpha_{g}(a^{*}c)u_{g,f}u^{*}_{gh,h^{-1}f})\big{)}\Big{)}u_{gh,h^{-1}f}u^{*}_{g,h}\beta_{g}(d).\end{split}$

Thus we need to check that the terms that $T_{h^{-1}f}$ acts on are equal. We
compute

$\begin{split}\alpha_{gh}^{-1}\big{(}u^{*}_{g,h}\alpha_{g}(a^{*}c)u_{g,f}u^{*}_{gh,h^{-1}f}\big{)}&=\alpha_{h}^{-1}\alpha_{g}^{-1}\operatorname{Ad}_{u_{g,h}}\big{(}u^{*}_{g,h}\alpha_{g}(a^{*}c)u_{g,f}u^{*}_{gh,h^{-1}f}\big{)}\\\
&=\alpha_{h}^{-1}\Big{(}\alpha_{g}^{-1}\big{(}\alpha_{g}(a^{*}c)u_{g,f}u^{*}_{gh,h^{-1}f}u^{*}_{g,h}\big{)}\big{)}\\\
&=\alpha_{h}^{-1}\Big{(}a^{*}c\cdot\alpha_{g}^{-1}\big{(}u_{g,f}u^{*}_{gh,h^{-1}f}u^{*}_{g,h}\big{)}\Big{)}\\\
&=\alpha_{h}^{-1}(a^{*}c\cdot u^{*}_{h,h^{-1}f})\end{split}$

where the equality
$\alpha_{g}^{-1}(u_{g,f}u^{*}_{gh,h^{-1}f}u^{*}_{g,h})=u^{*}_{h,h^{-1}f}$ at
the last step follows from the cocycle identity, as we have
$\alpha_{g}(u_{h,h^{-1}f})u_{g,f}=u_{g,h}u_{gh,h^{-1}f}$. This proves that
$L=R$, that is (8.3) holds. This last equality implies that the linear
extension of the prescription in (8.2) defines consistently a linear isometric
map $L_{g}:E_{s(g)}^{0}\to E_{r(g)}^{0}$ such that, for all
$\xi_{1},\xi_{2}\in E_{s(g)}^{0}$,

(8.4) $\langle
L_{h}(\xi_{1}),L_{h}(\xi_{2})\rangle_{B_{r(h)}}=\beta_{h}(\langle\xi_{1},\xi_{2}\rangle_{B_{s(h)}}),$

By continuity we get an isometry $L_{g}\in\mathcal{B}(E_{s(g)},E_{r(g)})$
where (8.4) holds for all $\xi_{1},\xi_{2}\in E_{s(g)}$. Next note that (8.2)
readily implies that

$L_{g}(a\cdot a_{0}\xi(h)b_{0}\cdot b)=\alpha_{g}(a)\cdot
L_{g}(a_{0}\xi(g)b_{0})\cdot\beta_{g}(b),$

for all $a,a_{0}\in A_{s(g)}$, $b,b_{0}\in B_{s(g)}$,
$h\in\mathcal{G}^{s(g)}$. Thus by linearity and continuity

$L_{g}(a\cdot\xi\cdot b)=\alpha_{g}(a)\cdot
L_{g}(\xi)\cdot\beta_{g}(b),\qquad\xi\in E_{s(g)},\ a\in A_{s(g)},\ b\in
B_{s(g)}.$

Similarly, to see that for $(f,g)\in\mathcal{G}^{(2)}$ we have $L_{f}\circ
L_{g}(\cdot)=u_{\alpha}(f,g)L_{fg}(\cdot)u_{\beta}(f,g)^{*}$ on $E_{s(g)}$, we
only need to check it on $a\xi(h)b$ where $f\in\mathcal{G}^{s(g)}$, $a\in
A_{s(g)}$, $b\in B_{s(g)}$. We compute using our shorthand notation for
twists:

$\begin{split}L_{f}(L_{g}(a\xi(h)b))&=L_{f}\Big{(}\alpha_{g}(a)\,u_{g,h}\cdot\xi(gh)\cdot
u_{g,h}^{*}\,\beta_{g}(b)\Big{)}\\\
&=\alpha_{f}\big{(}\alpha_{g}(a)u_{g,h}\big{)}u_{f,gh}\cdot\xi(fgh)\cdot
u_{f,gh}^{*}\,\beta_{f}\big{(}u_{g,h}^{*}\beta_{g}(b)\big{)}\\\
&=\alpha_{f}\big{(}\alpha_{g}(a)\big{)}\big{(}\alpha_{f}(u_{g,h})u_{f,gh}\big{)}\cdot\xi(fgh)\cdot\big{(}u_{f,gh}^{*}\,\beta_{f}(u_{g,h}^{*})\big{)}\beta_{f}\big{(}\beta_{g}(b)\big{)}\\\
&=\alpha_{f}\big{(}\alpha_{g}(a)\big{)}(u_{f,g}u_{fg,h})\cdot\xi(fgh)\cdot(u_{fg,h}^{*}u_{f,g}^{*})\beta_{f}\big{(}\beta_{g}(b)\big{)}\\\
&=u_{f,g}\alpha_{fg}(a)u_{fg,h}\cdot\xi(fgh)\cdot
u_{fg,h}^{*}\,\beta_{fg}(b)u_{f,g}^{*}\\\
&=u_{f,g}\,L_{fg}(a\xi(h)b)\,u_{f,g}^{*}.\end{split}$

Hence $L=\\{L_{g}\\}_{g\in\mathcal{G}}$ is an
$(\alpha,u_{\alpha})$-$(\beta,u_{\beta})$-equivariant representation of
$\mathcal{G}$ on $\mathcal{E}$. Moreover, for any $g\in\mathcal{G}$ and $a\in
A_{r(g)}$, (8.1) implies that
$T_{g}(a)=\langle\xi(r(g)),a\xi(g)\rangle_{\mathcal{M}(B_{r(g)})}\in B_{r(g)}$
and (8.2) implies that $\overline{L}_{g}(\xi(s(g))=\xi(g)$. Thus

$T_{g}(a)=\langle\xi(r(g)),a\overline{L}_{g}\big{(}\xi(s(g))\big{)}\rangle_{\mathcal{M}(B_{r(g)})},$

so that $T=T_{\mathcal{E},L,\xi|_{X},\xi|_{X}}$ where
$\xi|_{X}\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))$ is bounded because $T$
is bounded, and in view of the above we have $\|\xi(x)\|^{2}=\|T_{x}\|$ for
every $x\in X$. This finishes the proof of (1)$\implies$(4). Moreover, in view
of (8.2) we have $L_{U}(a\xi|_{X}b)=\alpha_{U}(a)\xi_{U}\beta_{U}(b)$ for
every $a\in A_{s(U)}$, $b\in A_{s(U)}$, $U\in\operatorname{Bis}(\mathcal{G})$.
The linear span of such elements is equal to $\Gamma$, which by construction
is dense in $\mathrm{C_{0}}(\mathcal{E})$. Hence the triple
$(\mathcal{E},L,\xi|_{X})$ is cyclic as in the second part of the assertion.

Finally, let $(\mathcal{E},L,\xi)$,
$(\mathcal{E}^{\prime},L^{\prime},\xi^{\prime})$, be two cyclic triples such
that
$T_{\mathcal{E}^{\prime},L^{\prime},\xi^{\prime},\xi^{\prime}}=T_{\mathcal{E},L,\xi,\xi}$.
We claim that for $a\in\mathrm{C_{0}}(\mathcal{A}|_{s(U)})$,
$U\in\operatorname{Bis}(\mathcal{G})$, and
$c\in\mathrm{C_{0}}(\mathcal{A}|_{s(V)})$,
$V\in\operatorname{Bis}(\mathcal{G})$

(8.5) $\langle L_{U}(a\xi),L_{V}(b\xi)\rangle=\langle
L_{U}^{\prime}(a\xi^{\prime}),L_{V}^{\prime}(c\xi^{\prime})\rangle.$

To prove this we express the left hand side only in terms of $a,c$, $U,V$ and
$T$. If $x\notin r(U)\cap r(V)$, then $\langle
L_{U}(a\xi),L_{V}(b\xi)\rangle(x)=0$. Assume then that $x\in r(U)\cap r(V)$.
There are unique $g\in U$ and $h\in V$ such that $r(h)=r(g)=x$. Put
$a_{0}=a(s(g))^{*}\alpha_{g}^{-1}(u_{\beta}(g,g^{-1}h)^{*})$ and
$c_{0}=\alpha_{g^{-1}{h}}(c(s(h)))$ and compute

$\begin{split}\langle L_{U}(a\xi),L_{V}(c\xi)\rangle(x)&=\langle
L_{g}(a\xi(s(g))),L_{h}(c\xi(s(h))\rangle\\\ &=\langle
L_{g}(a\xi(s(g))),L_{h}(c\xi(s(h)))u_{\beta}(g,g^{-1}h)^{*}\rangle
u_{\beta}(g,g^{-1}h)\\\ &=\langle
L_{g}(a\xi(s(g))),u_{\beta}(g,g^{-1}h)^{*}L_{g}L_{g^{-1}h}(c\xi(s(h)))\rangle
u_{\beta}(g,g^{-1}h)\\\
&=\beta_{g}\left(\langle\xi(s(g)),a(s(g))^{*}\alpha_{g}^{-1}(u_{\beta}(g,g^{-1}h)^{*})L_{g^{-1}h}(c\xi(s(h)))\rangle\right)u_{\beta}(g,g^{-1}h)\\\
&=\beta_{g}\left(\langle\xi(s(g)),a_{0}c_{0}L_{g^{-1}h}(\xi(s(h)))\rangle\right)u_{\beta}(g,g^{-1}h)\\\
&=\beta_{g}\left(T_{gh^{-1}}(a_{0}c_{0})\right)u_{\beta}(g,g^{-1}h).\end{split}$

This proves (8.5), which implies that for
$a\in\mathrm{C_{0}}(\mathcal{A}|_{s(U)})$,
$b\in\mathrm{C_{0}}(\mathcal{B}|_{s(U)})$,
$U\in\operatorname{Bis}(\mathcal{G})$, and
$c\in\mathrm{C_{0}}(\mathcal{A}|_{s(V)})$,
$d\in\mathrm{C_{0}}(\mathcal{B}|_{s(V)})$,
$V\in\operatorname{Bis}(\mathcal{G})$ we have

$\langle L_{U}(a\xi b),L_{V}(c\xi d)\rangle=\langle
L_{U}^{\prime}(a\xi^{\prime}b),L_{V}^{\prime}(c\xi^{\prime}d)\rangle.$

Thus if the triples $(\mathcal{E},L,\xi)$,
$(\mathcal{E}^{\prime},L^{\prime},\xi^{\prime})$ are cyclic, then by linearity
and continuity the formula $W(L_{U}(a\xi b)):=L_{U}^{\prime}(a\xi^{\prime}b)$
determines a unitary
$W:\mathrm{C_{0}}(\mathcal{E})\to\mathrm{C_{0}}(\mathcal{E}^{\prime})$ with
the properties described in the last part of the assertion. ∎

######  Remark 8.6.

Extending Remark 7.14, let $FS_{TA}^{+}(\mathcal{G})$, $CP_{TA}(\mathcal{G})$,
$CP_{TA}^{\operatorname{r}}(\mathcal{G})$ be the subcategories of
$FS_{TA}(\mathcal{G})$, $CB_{TA}(\mathcal{G})$, $CB_{TA}(\mathcal{G})$
consisting of positive-definite multipliers, and completely positive maps,
respectively. By Theorem 8.5, the functors from Remark 7.14 restrict to
isometric functors $m^{\operatorname{f}}:FS_{TA}^{+}(\mathcal{G})\to
CP_{TA}(\mathcal{G})$ and $m^{\operatorname{r}}:FS_{TA}^{+}(\mathcal{G})\to
CP_{TA}^{\operatorname{r}}(\mathcal{G})$, and the ranges of these functors
consist of fiber preserving maps in $CP_{TA}(\mathcal{G})$ and
$CP_{TA}^{\operatorname{r}}(\mathcal{G})$, respectively.

The next corollary extends the characterisation obtained in the above theorem
further in the case of Herz–Schur multipliers.

######  Corollary 8.7.

Let $\mathcal{A}$ be a $\mathrm{C}^{*}$-bundle over $X$, carrying a twisted
action $(\alpha,u_{\alpha})$ of $\mathcal{G}$ and let
$\varphi\in\mathrm{C_{b}}(r^{*}\mathcal{M}(\mathcal{A}))$. Denote by
$T^{\varphi}$ the Herz–Schur multiplier associated with $\varphi$ and by
$m_{\varphi}$ the associated multiplier map defined as in (7.2). The following
statements are equivalent:

  1. (1)

$T^{\varphi}$ is a positive-definite multiplier for
$\mathcal{A}^{(\alpha,u_{\alpha})}$;

  2. (2)

$\varphi$ takes values in central elements, i.e. $\varphi(g)\in
Z\mathcal{M}(A_{r(g)})$, $g\in\mathcal{G}$, and for every $x\in X$ and every
$g_{1},\ldots,g_{n}\in\mathcal{G}_{x}$, $n\in\mathbb{N}$, the matrix
$\left(\alpha_{g_{i}}^{-1}(\varphi(g_{i}g_{j}^{-1}))\right)_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathcal{M}(A_{x}))$
is positive;

  3. (3)

$\varphi$ takes values in central elements, i.e. $\varphi(g)\in
Z\mathcal{M}(A_{r(g)})$, $g\in\mathcal{G}$, and for every $x\in X$ and every
$g_{1},\ldots,g_{n}\in\mathcal{G}^{x}$, $n\in\mathbb{N}$, the matrix
$\left(\alpha_{g_{i}}(\varphi(g_{i}^{-1}g_{j}))\right)_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathcal{M}(A_{x}))$
is positive;

  4. (4)

$m_{\varphi}$ extends to a completely positive map
$m_{\varphi}^{\operatorname{r}}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$;

  5. (5)

$m_{\varphi}$ extends to a completely positive map
$m_{\varphi}^{\operatorname{f}}:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})$;

  6. (6)

$T^{\varphi}=T_{\mathcal{E},L,\xi,\xi}$ for some $\mathcal{E}$, $L$ and
$\xi\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))$, which is central in the
sense that $a\xi=\xi a$ for all $a\in A=\mathrm{C_{0}}(\mathcal{A})$.

If the above equivalent conditions hold, then
$\|\varphi\|_{\infty}=\|\varphi|_{X}\|_{\infty}=\|m_{\varphi}^{\operatorname{r}}\|=\|m_{\varphi}^{\operatorname{f}}\|=\|\xi\|$.

###### Proof.

By its form, $T^{\varphi}$ is strict and bounded (we have $\sup_{x\in
X}\|T_{x}^{\varphi}\|=\|\varphi|_{X}\|_{\infty}\leq\|\varphi\|_{\infty}<\infty$).
By Lemma 8.4(2), $T^{\varphi}$ is positive-definite if and only if for every
$x\in X$ and every $g_{1},\ldots,g_{n}\in\mathcal{G}_{x}$ and
$\\{a_{g_{i}}:1\leq i\leq n\\}\subseteq A_{x}$, $n\in\mathbb{N}$, the matrix
$\left(\alpha_{g_{i}}^{-1}(\varphi(g_{i}g_{j}^{-1}))a_{g_{i}}a_{g_{j}}^{*}\right)_{i,j=1}^{n}\in\mathbb{M}_{n}(A_{x})$
is positive. This condition implies that for any $g\in\mathcal{G}$, and any
$a_{g}\in A_{s(g)}$ the matrix

$\left(\begin{array}[]{cc}\varphi(s(g))&\varphi(g^{-1})a_{g}^{*}\\\
\alpha_{g}^{-1}(\varphi(g))a_{g}&\alpha_{g}^{-1}(\varphi(s(g)))a_{g}a_{g}^{*}\end{array}\right)$

is positive and, in particular, self-adjoint. This implies
$\alpha_{g}^{-1}(\varphi(g))=\varphi(g^{-1})^{*}\in
Z\mathcal{M}(\mathcal{A}_{s(g)})$, and so $\varphi$ is central. When $\varphi$
is central, positivity of
$\left(\alpha_{g_{i}}^{-1}(\varphi(g_{i}g_{j}^{-1}))a_{g_{i}}a_{g_{j}}^{*}\right)_{i,j=1}^{n}\in\mathbb{M}_{n}(A_{x})$
for any set $\\{a_{g_{1}},\ldots,a_{g_{n}}\\}\subseteq A_{x}$ is clearly
equivalent to positivity of
$\left(\alpha_{g_{i}}^{-1}(\varphi(g_{i}g_{j}^{-1}))\right)_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathcal{M}(A_{x}))$.
This proves the equivalence (1)$\iff$(2). The equivalence (1)$\iff$(3) can be
checked analogously, cf. also the considerations before [BC16, Proposition
4.3].

Finally, Theorem 8.5 gives the implications (6)$\implies$(1)$\iff$(4)$\iff$(5)
and further shows that (1) implies that
$T^{\varphi}=T_{\mathcal{E},L,\xi,\xi}$ for some equivariant representation
$L$ of $\mathcal{G}$ on a
$\mathrm{C}^{*}$-$\mathcal{A}$-$\mathcal{A}$-correspondence bundle
$\mathcal{E}$ and section $\xi\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))$,
which is (6) except for the centrality of $\xi$. To show the last fact assume
(1) and invoke the construction from the proof of Theorem 8.5. By (8.1) for
any $a\in\mathrm{C_{0}}(\mathcal{A})$ we have

$\langle\xi,a\xi\rangle_{A}(x)=\langle\xi,\xi\rangle_{\mathcal{M}(A)}(x)a(x),$

where
$\langle\xi,\xi\rangle_{\mathcal{M}(A)}(x)=\sum_{h,g\in\mathcal{G}^{x}}\alpha_{g}(\varphi(g^{-1}h))$
is central in $\mathcal{M}(A_{x})$ because $\varphi$ is central. This implies
that

$\langle\xi,a^{*}a\xi\rangle_{A}=a^{*}\langle\xi,a\xi\rangle_{A}=\langle\xi,a^{*}\xi\rangle_{A}a=a^{*}\langle\xi,\xi\rangle_{A}a=a^{*}a\langle\xi,\xi\rangle_{A},$

and consequently $\langle a\xi-\xi a,a\xi-\xi a\rangle_{A}=0$. Hence $a\xi=\xi
a$. ∎

######  Remark 8.8.

The positivity condition in (2) implies the one in [KLS22, Definition 3.4]. In
the context of group actions, the positivity condition in (3) appears in
[AD87], [AD02] and [BC16], where the corresponding $\varphi$ is called of
_positive-type_ or _AD-positive-definite_ , respectively.

##  9\. Fourier multipliers

In this section we introduce Fourier multipliers as those Fourier–Stieltjes
multipliers which arise from regular equivariant representations.

######  Definition 9.1.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. We say that a multiplier $T\in
FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ is a _Fourier multiplier_ if it can
be represented by a regular equivariant representation of $\mathcal{G}$, i.e.
if there exists an equivariant representation $L$ of $\mathcal{G}$ on a
$\mathrm{C}^{*}$-$\mathcal{A}$-$\mathcal{B}$-correspondence bundle
$\mathcal{E}$ and sections
$\xi,\zeta\in\mathrm{C_{b}}(\mathcal{M}(\ell^{2}(s^{*}\mathcal{E})))$ such
that for any $a\in A_{r(g)}$ and $g\in\mathcal{G}$

(9.1)
$T_{g}(a)=\sum_{t\in\mathcal{G}_{r(g)}}\langle\xi(t),a\overline{L}_{g}(\zeta(tg)\rangle_{\mathcal{M}(B_{r}(g))}.$

We denote by $F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ the set of all such
Fourier multipliers and by $F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]^{+}$ the
set of Fourier multipliers that can represented in the form (9.1) with
$\xi=\zeta$.

Note that the last definition is related to a certain subtlety regarding
various possible notions of positivity — see Question 12.2.

######  Example 9.2.

Let $\mathcal{A}$ be a $\mathrm{C}^{*}$-bundle over $X$, carrying a twisted
action $(\alpha,u_{\alpha})$ of $\mathcal{G}$. The multipliers described in
Example 7.4, given by sections in $\mathrm{C_{c}}(r^{*}\mathcal{A})$ applied
to a Fell bundle $\mathcal{A}^{(\alpha,u_{\alpha})}$, give rise to Fourier
multipliers in $F[(\alpha,u_{\alpha}),(\alpha,u_{\alpha})]$. More concretely,
given $\xi,\zeta\in\mathrm{C_{c}}(r^{*}(\mathcal{A}))$, we get the Fourier
multiplier

$T_{g}(a)=\sum_{h\in\mathcal{G}^{r(g)}}\xi(h)^{*}a\alpha_{g}(\zeta(g^{-1}h)),\quad
a\in A_{r(g)}.$

Specializing further to the case of the trivial bundle
$\mathcal{A}=\mathbb{C}\times X$, that is, the case where we have just a
twisted groupoid $(\mathcal{G},u)$ (see Example 4.12), given functions
$\xi,\zeta\in\mathrm{C_{c}}(\mathcal{G})$, we get from the above formula a
multiplier $T$ of $(\mathcal{G},u)$, that we view again as the function in
$\mathrm{C_{c}}(\mathcal{G})$ given by

$T_{g}=\sum_{h\in\mathcal{G}^{r(g)}}\overline{\xi(h)}\zeta(g^{-1}h),\quad g\in
G.$

Notice that the above function equals the (untwisted) convolution product
$\tilde{\xi}^{*}*\tilde{\zeta}\in\mathrm{C_{c}}(\mathcal{G})$, where
$\tilde{\xi}(g):=\xi(g^{-1})$. In particular, if $\xi=\zeta$, this is a
prototype example of a positive-type function, see [BO08, Definition 5.6.15].

The next proposition shows that the collection of Fourier multipliers has a
natural ‘ideal’ property.

######  Proposition 9.3.

Let $\mathcal{A}$, $\mathcal{B}$, $\mathcal{D}$ be $\mathrm{C}^{*}$-bundles
over $X$, carrying twisted actions $(\alpha,u_{\alpha})$, $(\beta,u_{\beta})$
and $(\gamma,u_{\gamma})$ of $\mathcal{G}$. Fourier multipliers
$F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ form an involutive
$\mathrm{C_{b}}(X)$-submodule of $FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$.
Moreover, $F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ equipped with the norm

$\|T\|_{F}:=\inf\big{\\{}\|\xi\|\|\zeta\|:T=T_{\ell^{2}(s^{*}\mathcal{E}),L^{\mathcal{G}},\xi,\zeta}\
\text{for some $(\mathcal{E},L)$ and
$\xi,\zeta\in\mathrm{C_{b}}(\mathcal{M}(\ell^{2}(s^{*}\mathcal{E})))$}\big{\\}},$

is a Banach space. Further

$\displaystyle FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]\cdot
F[(\beta,u_{\beta}),(\gamma,u_{\gamma})]\subseteq
F[(\alpha,u_{\alpha}),(\gamma,u_{\gamma})],$ $\displaystyle
F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]\cdot
FS[(\beta,u_{\beta}),(\gamma,u_{\gamma})]\subseteq
F[(\alpha,u_{\alpha}),(\gamma,u_{\gamma})].$

In particular, $F(\alpha,u_{\alpha})$ is a two-sided ideal of the Banach
algebra $FS(\alpha,u_{\alpha})$.

###### Proof.

The same arguments as in the proof of Proposition 7.10 show that
$F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ is a Banach
$\mathrm{C_{b}}(X)$-bimodule with norm $\|\cdot\|_{F}$. For instance, the
direct sum of regular equivariant representations of $\mathcal{G}$ is a
regular equivariant representation, i.e. $(L\oplus
K)^{\mathcal{G}}=L^{\mathcal{G}}\oplus K^{\mathcal{G}}$, or more generally
$(\oplus_{i\in I}L_{i})^{\mathcal{G}}=\oplus_{i\in I}L_{i}^{\mathcal{G}}$.
Also, by Corollary 6.14, we have $L^{\mathcal{G}}\otimes K\cong L\otimes
K^{\mathcal{G}}\cong(L\otimes K)^{\mathcal{G}}$, which implies the inclusions
in the assertion. Indeed, if
$T_{\ell^{2}(\mathcal{G},\mathcal{E}),L^{\mathcal{G}},\xi_{1},\zeta_{1}}\in
FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$,
$T_{\mathcal{F},K,\xi_{2},\zeta_{2}}\in
F[(\beta,u_{\beta}),(\gamma,u_{\gamma})]$, and $\Gamma:L^{\mathcal{G}}\otimes
K\to(L\otimes K)^{\mathcal{G}}$ is the unitary from Corollary 6.14, then using
(7.5) we get

$\begin{split}T_{\mathcal{F},K,\xi_{2},\zeta_{2}}\circ
T_{\ell^{2}(\mathcal{G},\mathcal{E}),L^{\mathcal{G}},\xi_{1},\zeta_{1}}&=T_{\ell^{2}(\mathcal{G},\mathcal{E})\otimes\mathcal{F},L^{\mathcal{G}}\otimes
K,\xi_{1}\otimes\xi_{2},\zeta_{1}\otimes\zeta_{2}}\\\
&=T_{\ell^{2}(\mathcal{G},\mathcal{E}\otimes\mathcal{F}),(L\otimes
K)^{\mathcal{G}},\Gamma(\xi_{1}\otimes\xi_{2}),\Gamma(\zeta_{1}\otimes\zeta_{2})}\in
F[(\alpha,u_{\alpha}),(\gamma,u_{\gamma})].\end{split}$

Hence $FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]\cdot
F[(\beta,u_{\beta}),(\gamma,u_{\gamma})]\subseteq
F[(\alpha,u_{\alpha}),(\gamma,u_{\gamma})]$, and the other inclusion is proved
analogously. ∎

The equality of the norms $\|\cdot\|_{F}$ and $\|\cdot\|_{FS}$ on the set of
Fourier multipliers remains unclear. See Question 12.1 for further discussion.

######  Theorem 9.4.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. Every Fourier multiplier $T\in
F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ extends to a strict completely
bounded map
$m_{T}^{\operatorname{r},\operatorname{f}}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})$
from reduced to full crossed products, and
$\|m_{T}^{\operatorname{r},\operatorname{f}}\|_{cb}\leq\|T\|_{F}$. If
$T=T_{s^{*}\mathcal{E},L^{\mathcal{G}},\xi,\xi}\in
F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]^{+}$, then
$m_{T}^{\operatorname{r},\operatorname{f}}$ is completely positive and
$\|m_{T}^{\operatorname{r},\operatorname{f}}\|_{cb}=\|T\|_{FS}=\|T\|_{F}=\|\xi\|^{2}$.

###### Proof.

Write
$T:=T_{s^{*}\mathcal{E},L^{\mathcal{G}},\xi,\zeta}=\\{T_{g}\\}_{g\in\mathcal{G}}$
where $\xi,\zeta\in\mathrm{C_{b}}(\mathcal{M}(s^{*}\mathcal{E}))$. Let
$\psi:\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})\to\mathbb{B}(\mathcal{H})$
be a faithful nondegenerate representation on a Hilbert space $H$. As in the
second part of the proof of Theorem 7.13, we have

(9.2)
$m_{T}(a):=\theta_{\xi}^{*}(L^{\mathcal{G}}\operatorname{-Ind}(\psi))(a)\theta_{\zeta},\qquad
a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})}),$

where
$L^{\mathcal{G}}\operatorname{-Ind}(\psi):\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathbb{B}(\mathrm{C_{0}}(s^{*}\mathcal{E})\otimes_{\psi}H)$
is the induced representation and
$\theta_{\xi},\theta_{\zeta}:H\to\mathrm{C_{0}}(s^{*}\mathcal{E})\otimes_{\psi}H$
are creation operators. By Fell absorption (Theorem 6.16)
$L^{\mathcal{G}}\operatorname{-Ind}(\psi)$ descends to a representation of the
reduced crossed product, and by abuse of notation we may write
$L^{\mathcal{G}}\operatorname{-Ind}(\psi):\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathbb{B}(\mathrm{C_{0}}(s^{*}\mathcal{E})\otimes_{\psi}H)$.
Then (9.2) implies that $m_{T}$ extends to a strict completely bounded map
$m_{T}^{\operatorname{r},\operatorname{f}}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})$
and
$\|m_{T}^{\operatorname{r},\operatorname{f}}\|_{cb}\leq\|\xi\|\cdot\|\zeta\|$.
In particular,
$\|m_{T}^{\operatorname{r},\operatorname{f}}\|_{cb}\leq\|T\|_{F}$. Also if
$\xi=\zeta$, then (9.2) implies that
$m_{T}^{\operatorname{r},\operatorname{f}}$ is completely positive and
$\|m_{T}^{\operatorname{r},\operatorname{f}}\|_{cb}=\|\xi\|^{2}$, which also
implies $\|m_{T}^{\operatorname{r},\operatorname{f}}\|_{cb}=\|T\|_{F}$. ∎

######  Remark 9.5.

To use the categorical language, as in Remarks 7.14 and 8.6, denote by
$F_{TA}(\mathcal{G})$ and $CB_{TA}^{\operatorname{f},\operatorname{r}}(G)$ the
non-unital categories of all twisted actions of $\mathcal{G}$ where morphisms
in $F_{TA}(\mathcal{G})$ are Fourier multipliers, and morphisms in
$CB_{TA}^{\operatorname{f},\operatorname{r}}(G)$ are completely bounded strict
$\mathrm{C_{0}}(X)$-bimodule maps from full to reduced crossed products whose
composition is defined by $TS:=T\circ\Lambda\circ S$ where $\Lambda$ is the
relevant regular representation. The extension in Theorem 7.13 gives a
contractive involutive $\mathrm{C_{b}}(X)$-bimodule functor
$m^{\operatorname{r},f}:F_{TA}(\mathcal{G})\to
CB_{TA}^{\operatorname{f},\operatorname{r}}(\mathcal{G})$. It restricts to an
isometric functor $m^{\operatorname{r},f}:F_{TA}^{+}(\mathcal{G})\to
CP_{TA}^{\operatorname{r},f}(\mathcal{G})$, where morphisms in
$F_{TA}^{+}(\mathcal{G})\subseteq F_{TA}(\mathcal{G})$ are elements of
$F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]^{+}$ and morphisms in
$CP_{TA}^{\operatorname{r},f}(\mathcal{G})\subseteq
CB_{TA}^{\operatorname{f},\operatorname{r}}(\mathcal{G})$ are completely
positive maps.

######  Definition 9.6.

Given two Fell bundles $\mathcal{A}$, $\mathcal{B}$ over $\mathcal{G}$ we
define the _support_ of a multiplier $T=\\{T_{g}\\}_{g\in\mathcal{G}}$ as
$\operatorname{supp}(T):=\overline{\\{g\in\mathcal{G}:T_{g}\neq 0\\}}$.

######  Proposition 9.7.

Compactly supported Fourier–Stieltjes multiplier are Fourier multipliers, i.e.
if $\mathcal{A}$, $\mathcal{B}$ are $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$, and $T\in FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ has
compact support, then $T\in F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$.

###### Proof.

Let $K$ be a compact subset of $\mathcal{G}$. We claim that there exist
$\xi,\zeta\in\mathrm{C_{b}}(\mathcal{M}(\ell^{2}(s^{*}\mathcal{A})))$ such
that $T_{\ell^{2}(s^{*}\mathcal{A}),\alpha^{\mathcal{G}},\xi,\zeta}(g)(a)=a$
for all $a\in A_{r(g)}$ and $g\in K$. Indeed, pick
$\xi_{0}\in\mathrm{C_{c}}(X)\subseteq\mathrm{C_{c}}(\mathcal{G})$ such that
$\xi_{0}(x)=1$ for all $x\in r(K)$, and $\zeta\in\mathrm{C_{c}}(\mathcal{G})$
such that $\zeta(g)=1$ for all $g\in K$. Consider the corresponding strictly
continuous sections
$\xi,\zeta\in\mathrm{C_{c}}(\mathcal{M}(\ell^{2}(s^{*}\mathcal{A})))$ given by
$x\mapsto\xi_{0}|_{\mathcal{G}_{x}},\zeta_{0}|_{\mathcal{G}_{x}}$, cf.
Proposition 7.12. Then for $a\in A_{r(g)}$ and $g\in K$ we have

$\begin{split}T_{\ell^{2}(s^{*}\mathcal{A}),\alpha^{\mathcal{G}},\xi,\zeta}(g)(a)&=\langle\xi(r(g)),a\overline{\alpha^{\mathcal{G}}}_{g}(\zeta(s(g)))\rangle_{\mathcal{M}(A_{r(g)})}\\\
&=\sum_{t\in\mathcal{G}_{r(g)}}\xi_{0}(t)^{*}a\alpha_{g}\big{(}\zeta_{0}(tg)\big{)}=\xi_{0}(r(g))^{*}a\zeta_{0}(g)=a.\end{split}$

This proves our claim.

Suppose now that $T\in FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ has compact
support. The claim above yields
$T_{\ell^{2}(s^{*}\mathcal{A}),\alpha^{\mathcal{G}},\xi,\zeta}\in
F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ such that
$T_{\ell^{2}(s^{*}\mathcal{A}),\alpha^{\mathcal{G}},\xi,\zeta}\circ T=T$. Thus
$T\in F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$, by Proposition 9.3. ∎

In terms of maps on crossed products “compactness of the support” means that
the range of the map is supported on a precompact open set. We formalize it as
follows.

######  Lemma 9.8.

For any Fell bundle $\mathcal{A}$ over $\mathcal{G}$ and every precompact open
set $V\subseteq\mathcal{G}$ the subspace $\mathrm{C_{0}}(\mathcal{A}|_{V})$ of
$\mathrm{C_{c}}(\mathcal{A})$ is closed both in the reduced and (hence) also
in the universal (maximal) norm.

###### Proof.

Since the reduced norm is dominated by the universal norm it suffices to show
that $\mathrm{C_{0}}(\mathcal{A}|_{V})$ is closed in the reduced norm. We use
the known fact that the inclusion
$\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})\subseteq\mathrm{C_{0}}(\mathcal{A}^{(\alpha,u_{\alpha})})$
extends to a contractive linear map
$j:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C_{0}}(\mathcal{A}^{(\alpha,u_{\alpha})})$,
where the codomain is equipped with the supremum norm, see [KM21, Proposition
7.10]. This implies the assertion because $j$ is the identity on
$\mathrm{C_{0}}(\mathcal{A}^{(\alpha,u_{\alpha})}|_{V})$ and
$\mathrm{C_{0}}(\mathcal{A}^{(\alpha,u_{\alpha})}|_{V})$ is closed in
$\mathrm{C_{0}}(\mathcal{A}^{(\alpha,u_{\alpha})})$. ∎

######  Corollary 9.9.

Let $T=\\{T_{g}\\}_{g\in\mathcal{G}}$ be a reduced (or full) multiplier from a
Fell bundle $\mathcal{A}$ to a Fell bundle $\mathcal{B}$ over $\mathcal{G}$.
Then $T$ has compact support if and only if the range of
$m_{T}^{\operatorname{r}}$ (resp. $m_{T}^{\operatorname{f}}$) is contained in
$\mathrm{C_{0}}(\mathcal{B}|_{V})$ for some precompact open
$V\subseteq\mathcal{G}$.

###### Proof.

If the range of $m_{T}^{\operatorname{r}}$ (resp. $m_{T}^{\operatorname{f}}$)
is contained in $\mathrm{C_{0}}(\mathcal{B}|_{V})$, then clearly
$\operatorname{supp}(T)\subseteq V$. Hence precompactnes of $V$ implies
compactness of $\operatorname{supp}(T)$. Conversely, if
$\operatorname{supp}(T)$ is compact then there is a precompact open set
$V\subseteq\mathcal{G}$ containing $\operatorname{supp}(T)$. Since
$m_{t}(\mathrm{C_{c}}(\mathcal{A}))\subseteq\mathrm{C_{0}}(\mathcal{B}|_{V})$
and $\mathrm{C_{0}}(\mathcal{B}|_{V})$ is closed in both
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B})$ and
$\mathrm{C}^{*}(\mathcal{B})$ one concludes that the range of the continuous
extension of $m_{T}$ is also in $\mathrm{C_{0}}(\mathcal{B}|_{V})$. ∎

##  10\. Approximation properties for multipliers

In this section we turn to the discussion of approximation properties. We
begin in the general context of abstract Fell bundle multipliers, proving a
lemma that shows one can always approximately ‘strictify’ a given multiplier.

######  Lemma 10.1.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. Let $T$ be a multiplier from
$\mathcal{A}^{(\alpha,u_{\alpha})}$ to $\mathcal{B}^{(\beta,u_{\beta})}$ such
that $\operatorname{supp}_{g\in\mathcal{G}}\|T_{g}\|<\infty$. Choose an
approximate unit $\\{e_{i}\\}_{i\in I}$ in $\mathrm{C_{0}}(\mathcal{A})$.
Putting

$T_{i,g}(a):=T_{g}\big{(}e_{i}(r(g))a\alpha_{g}(e_{i}(s(g)))\big{)}$