for all $a\in A_{r(g)}$, $g\in\mathcal{G}$ and $i\in I$, we obtain a net
$\\{T_{i}\\}_{i\in I}$ of strict multipliers from
$\mathcal{A}^{(\alpha,u_{\alpha})}$ to $\mathcal{B}^{(\beta,u_{\beta})}$ such
that $m_{T_{i}}(a)$ converges uniformly on compact sets to $m_{T}(a)$ for
every $a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$. Moreover:

  1. (1)

if $T\in FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$, then for every $i\in I$
we have $T_{i}\in FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$,
$\|T_{i}\|_{FS}\leq\|T\|_{FS}$ and $\|T_{i}\|_{FS}\to\|T\|_{FS}$ (similar
statement holds with $FS$ replaced by $F$);

  2. (2)

if $T$ is positive-definite, then each $T_{i}$ is a positive-definite Fourier-
Stieljes multiplier. Moreover we have the estimate $\sup_{i\in
I}\|T_{i}\|_{FS}\leq\sup_{x\in X}\|T_{x}\|$.

###### Proof.

By definition, for $a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ and
$i\in I$ we have $m_{T_{i}}(a)=m_{T}(e_{i}ae_{i})$ and therefore

$\|m_{T_{i}}(a)-m_{T}(a)\|_{\infty}=\|m_{T}(e_{i}ae_{i}-a)\|_{\infty}\leq\operatorname{supp}_{g\in\mathcal{G}}\|T_{g}\|\cdot\|e_{i}ae_{i}-a\|_{\infty}\to
0.$

This proves the first part of the assertion, as each $T_{i}$ is a strict
multiplier by construction.

If $T=T_{\mathcal{E},L,\xi,\zeta}$, for an equivariant representation $L$ of
$\mathcal{G}$ on a $\mathrm{C}^{*}$-$\mathcal{A}$-$\mathcal{B}$-correspondence
bundle $\mathcal{E}$ and sections
$\xi,\zeta\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}))$ then for each $i\in I$
we have $T_{i}=T_{\mathcal{E},L,e_{i}\xi_{i},e_{i}\zeta}$. Clearly
$\|e_{i}\xi\|\leq\|\xi\|$, and $\|e_{i}\xi\|\to\|\xi\|$ because
$\mathrm{C_{0}}(\mathcal{E})=\mathrm{C_{0}}(\mathcal{A})\mathrm{C_{0}}(\mathcal{E})$,
as we assume $\mathrm{C}^{*}$-correspondences are non-degenerate. We have
similar relations for $\zeta$ and this readily implies the assertion in (1).

Further one can easily check that if $T$ is positive-definite, so is $T_{i}$
for each $i\in I$. Thus the assertion in (2) follows by Theorem 8.5. ∎

We now specify the context to twisted groupoid actions.

######  Lemma 10.2.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$ and let $L$ be an equivariant representation of $\mathcal{G}$ on
a $\mathrm{C}^{*}$-$\mathcal{A}$-$\mathcal{B}$-correspondence bundle
$\mathcal{E}$. If $T=T_{\ell^{2}(s^{*}\mathcal{E}),L^{\mathcal{G}},\xi,\zeta}$
for sections
$\xi,\zeta\in\mathrm{C_{b}}(\mathcal{M}(\ell^{2}(s^{*}\mathcal{E})))$, then
there are nets $\\{\xi_{i}\\}_{i\in I},\\{\zeta_{i}\\}_{i\in
I}\subseteq\mathrm{C_{c}}(s^{*}\mathcal{E})\subseteq\mathrm{C_{c}}(\ell^{2}(s^{*}\mathcal{E}))$
such that $\|\xi_{i}\|\leq\|\xi\|$, $\|\zeta_{i}\|\leq\|\zeta\|$, for every
$i\in I$, and putting
$T_{i}:=T_{\ell^{2}(s^{*}\mathcal{E}),L^{\mathcal{G}},\xi_{i},\zeta_{i}}$ it
follows that $T_{i}$ is compactly supported and $m_{T_{i}}(a)$ converges
uniformly on compact sets to $m_{T}(a)$ for every
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$.

###### Proof.

Using Lemma 10.1, and by passing to
$\xi_{i}:=e_{i}\xi\in\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))$ and
$\zeta_{i}:=e_{i}\zeta\in\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))$ for an
approximate unit $\\{e_{i}\\}_{i\in I}$ in $\mathrm{C_{0}}(\mathcal{A})$, we
may reduce the proof to the case when
$\xi,\zeta\in\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))$. Since
$\mathrm{C_{c}}(s^{*}\mathcal{E})$ is dense in
$\mathrm{C_{0}}(\ell^{2}(s^{*}\mathcal{E}))$ we may find nets
$\\{\xi_{i}\\}_{i\in I},\\{\zeta_{i}\\}_{i\in
I}\subseteq\mathrm{C_{c}}(s^{*}\mathcal{E})\subseteq\mathrm{C_{c}}(\ell^{2}(s^{*}\mathcal{E}))$
with $\|\xi_{i}\|\leq\|\xi\|$, $\|\zeta_{i}\|\leq\|\zeta\|$ for each $i\in I$
and $\|\xi-\xi_{i}\|\to 0$, $\|\zeta-\zeta_{i}\|\to 0$. Then putting
$T_{i}:=T_{\ell^{2}(s^{*}\mathcal{E}),L^{\mathcal{G}},\xi_{i},\zeta_{i}}$, we
have

$\|m_{T_{i}}(a)-m_{T}(a)\|_{\infty}\leq\|\xi-\xi_{i}\|\cdot\|a\|_{\infty}\cdot\|\zeta\|+\|\zeta-\zeta_{i}\|\cdot\|a\|_{\infty}\cdot\|\zeta_{i}\|\to
0.$

for every $a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$.

Finally, if $\xi$ and $\zeta$ are supported on precompact bisections $U$ and
$V\in\operatorname{Bis}(\mathcal{G})$, respectively, then (9.1) implies that
$T$ is supported on a precompact bisection $U^{*}V$. Since elements in
$\mathrm{C_{c}}(s^{*}\mathcal{E})$ are finite sums of elements with supports
in precompact bisections, this proves the compact support assertion, as
whenever $\tilde{\xi},\tilde{\zeta}\in\mathrm{C_{c}}(s^{*}\mathcal{E})$, the
multiplier
$T=T_{\ell^{2}(s^{*}\mathcal{E}),L^{\mathcal{G}},\tilde{\xi},\tilde{\zeta}}$
is compactly supported. ∎

The last lemma can be used to approximate Fourier multipliers by compactly
supported ones, but a priori only in the sense of uniform convergence on
compact sets.

######  Proposition 10.3.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. If $T\in F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$, then there
is a net $\\{T_{i}\\}_{i\in I}$ of compactly supported multipliers in
$F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ such that
$\|T_{i}\|_{F}\leq\|T\|_{F}$ for every $i\in I$ and $m_{T_{i}}(a)$ converges
uniformly on compact sets to $m_{T}(a)$ for every
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$.

###### Proof.

For each $k\in\mathbb{N}$ we can find an equivariant representation $L_{k}$ of
$\mathcal{G}$ on a $\mathrm{C}^{*}$-$\mathcal{A}$-$\mathcal{B}$-correspondence
bundle $\mathcal{E}_{k}$ and non-zero sections
$\xi_{k},\zeta_{k}\in\mathrm{C_{b}}(\mathcal{M}(\ell^{2}(s^{*}\mathcal{E}_{k})))$
such that
$T=T_{\ell^{2}(s^{*}\mathcal{E}_{k}),L_{k}^{\mathcal{G}},\xi_{k},\zeta_{k}}$
and $\|\xi_{k}\|\cdot\|\zeta_{k}\|\leq\|T\|_{F}+{1/k}$, so that
$\lambda_{k}:=\frac{\|T\|_{F}}{\|\xi_{k}\|\cdot\|\zeta_{k}\|}\leq 1$ converges
to $1$ as $k$ tends to infinity. Thus setting
$T_{k}:=\lambda_{k}T=T_{\ell^{2}(s^{*}\mathcal{E}_{k}),L_{k}^{\mathcal{G}},\lambda_{k}\xi_{k},\zeta_{k}}\in
F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$, for each $k\in\mathbb{N}$ we get
$\|T_{k}\|_{F}\leq\|\lambda_{k}\xi_{k}\|\cdot\|\zeta_{k}\|=\|T\|_{F}$ and
$m_{T_{k}}\to m_{T}$. Applying Lemma 10.2 to each $T_{k}$ one gets the desired
net. ∎

We are ready to define an analog of the property above for arbitrary
multipliers.

######  Definition 10.4.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. A multiplier $T$ from $\mathcal{A}^{(\alpha,u_{\alpha})}$ to
$\mathcal{B}^{(\beta,u_{\beta})}$ has the _(positive) Fourier approximation
property_ if there is a $\|\cdot\|_{F}$-bounded net $\\{T_{i}\\}_{i\in I}$ in
$F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ (resp. in
$F[(\alpha,u_{\alpha})],(\beta,u_{\beta})]^{+}$) such that $m_{T_{i}}(a)$
converges uniformly on compact sets to $m_{T}(a)$ for all
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$.

Later we will often use the acronym _AP_ for the approximation property. The
next proposition gathers several equivalent conditions for a given multiplier
to have the Fourier AP.

######  Proposition 10.5.

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 a multiplier $T$ from $\mathcal{A}^{(\alpha,u_{\alpha})}$
to $\mathcal{B}^{(\beta,u_{\beta})}$ the following are equivalent:

  1. (1)

$T$ has the (positive) Fourier approximation property;

  2. (2)

there is a $\|\cdot\|_{F}$-bounded net $\\{T_{i}\\}_{i\in I}$ in
$F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ (resp. in
$F[(\alpha,u_{\alpha})],(\beta,u_{\beta})]^{+}$) of compactly supported
multipliers such that $m_{T_{i}}(a)$ converges uniformly on compact sets to
$m_{T}(a)$ for every $a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$;

  3. (3)

there is a net $\\{L_{i}\\}_{i\in I}$ of equivariant representations of
$\mathcal{G}$ acting respectively on
$\mathrm{C}^{*}$-$\mathcal{A}$-$\mathcal{B}$-correspondence bundles
$\mathcal{E}_{i}$ and sections $\xi_{i}$,
$\zeta_{i}\in\mathrm{C_{c}}(s^{*}\mathcal{E}_{i})$, $i\in I$, (resp.
$\xi_{i}=\zeta_{i}$) such that

    1. (a)

$\sup_{i\in I}\sup_{x\in
X}\|\sum_{t\in\mathcal{G}_{x}}\langle\xi_{i}(t),\xi_{i}(t)\rangle_{B_{x}}\|\cdot\sup_{x\in
X}\|\sum_{t\in\mathcal{G}_{x}}\langle\zeta_{i}(t),\zeta_{i}(t)\rangle_{B_{x}}\|<\infty$,

    2. (b)

for every $a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ the
functions

$\mathcal{G}\ni
g\longmapsto\sum_{t\in\mathcal{G}_{r(g)}}\langle\xi_{i}(t),a(g)L_{g}\zeta_{i}(tg)\rangle_{B_{r(g)}}$

converge uniformly on compact sets to the function $\mathcal{G}\ni
g\longmapsto T_{g}(a)\in B_{r(g)}$.

###### Proof.

Combine Proposition 10.3 and Lemma 10.2. ∎

######  Definition 10.6.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. A multiplier $T$ from $\mathcal{A}^{(\alpha,u_{\alpha})}$ to
$\mathcal{B}^{(\beta,u_{\beta})}$ has the _Fourier–Stieltjes approximation
property_ if there is a $\|\cdot\|_{FS}$-bounded net $\\{T_{i}\\}_{i\in
I}\subseteq FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ of compactly supported
multipliers such that $m_{T_{i}}(a)$ converges uniformly on compact sets to
$m_{T}(a)$ for every $a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$.
If one can find $T_{i}$’s as above which are positive-definite, we speak of
the _positive Fourier–Stieltjes approximation property_.

######  Remark 10.7.

By Propositions 9.7 and 10.5(2), Fourier AP and Fourier–Stieltjes AP differ
only in the boundedness condition, formulated respectively in terms of the
Fourier and the Fourier–Stieltjes norm. Since
$\|\cdot\|_{FS}\leq\|\cdot\|_{F}$, the $FS$-boundedness is formally weaker
than $F$-boundedness. Hence the Fourier AP implies the Fourier–Stieltjes AP.
It seems reasonable to conjecture that the two notions coincide at least for
identity multipliers and positive approximation that we discuss below (see
Question 12.1). Note also that for the (twisted) actions of discrete groups on
unital $\mathrm{C}^{*}$-algebras the Fourier AP was introduced in [BC12] under
the name _weak approximation property_.

Theorem 8.5 implies the following characterisation of the positive
Fourier–Stieltjes AP.

######  Proposition 10.8.

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 $T$ from $\mathcal{A}^{(\alpha,u_{\alpha})}$
to $\mathcal{B}^{(\beta,u_{\beta})}$ the following are equivalent:

  1. (1)

$T$ has the positive Fourier-Stieltjes approximation property;

  2. (2)

there is a net $\\{T_{i}\\}_{i\in I}$ of compactly supported and positive-
definite (not necessarily strict) multipliers from
$\mathcal{A}^{(\alpha,u_{\alpha})}$ to $\mathcal{B}^{(\beta,u_{\beta})}$ such
that $\sup_{i\in I}\sup_{x\in X}\|(T_{i})_{x}\|<\infty$ and $m_{T_{i}}(a)$
converges uniformly on compact sets to $m_{T}(a)$ for every
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$;

  3. (3)

there is a net $\\{L_{i}\\}_{i\in I}$ of equivariant representations of
$\mathcal{G}$, acting respectively on
$\mathrm{C}^{*}$-$\mathcal{A}$-$\mathcal{B}$-correspondence bundles
$\mathcal{E}_{i}$, and sections $\xi_{i}\in\mathrm{C_{c}}(\mathcal{E}_{i})$,
$i\in I$, such that $\sup_{i\in I}\|\xi_{i}\|<\infty$, and for every
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ the functions

$\mathcal{G}\ni
g\longmapsto\sum_{t\in\mathcal{G}_{r(g)}}\langle\xi_{i}(t),a(g)L_{i,g}\big{(}\xi_{i}(tg)\big{)}\rangle_{B_{r(g)}}$

converge uniformly on compact sets to the function $\mathcal{G}\ni
g\longmapsto T_{g}(a)\in B_{r(g)}$.

###### Proof.

Theorem 8.5 implies the equivalence (1)$\iff$ (2) modulo strictness of the
$T_{i}$s, which can be ignored thanks to Lemma 10.1. The same theorem also
shows that both conditions are equivalent to a version of (3) where sections
$\xi_{i}$ are in $\mathrm{C_{b}}(\mathcal{M}(\mathcal{E}_{i}))$ rather than in
$\mathrm{C_{c}}(\mathcal{E}_{i})$, but then as in the proof of Lemma 10.2 we
may replace $\xi_{i}$ by $e_{j}\xi_{i}\in\mathrm{C_{c}}(\mathcal{E}_{i})$ for
an approximate unit $\\{e_{j}\\}_{j\in J}\subseteq\mathrm{C_{c}}(\mathcal{A})$
in $\mathrm{C_{0}}(\mathcal{A})$. ∎

One of the main consequences of the approximation property for a given
multiplier is the following result which shows that the multipliers enjoying
this property yield (completely) bounded maps between the appropriate
$\mathrm{C}^{*}$-algebras.

######  Lemma 10.9.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. Let $T$ and $\\{T_{i}\\}_{i\in I}$ be multipliers from
$\mathcal{A}^{(\alpha,u_{\alpha})}$ to $\mathcal{B}^{(\beta,u_{\beta})}$ such
that $m_{T_{i}}(a)$ converges uniformly on compact sets to $m_{T}(a)$ for
every $a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$.

  1. (1)

If $\\{T_{i}\\}_{i\in I}\subseteq F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$
and $M:=\sup_{i\in I}\|T_{i}\|_{F}<\infty$, then $m_{T}$ extends to a
completely bounded map
$m_{T}^{\operatorname{r},\operatorname{f}}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})$
such that $\|m_{T}^{\operatorname{r},\operatorname{f}}\|_{cb}\leq M$ and
$m_{T_{i}}^{\operatorname{r},\operatorname{f}}\to
m_{T}^{\operatorname{r},\operatorname{f}}$ pointwise.

  2. (2)

If $\\{T_{i}\\}_{i\in I}\subseteq FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$
and $M:=\sup_{i\in I}\|T_{i}\|_{FS}<\infty$, then $m_{T}$ extends to
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
$m_{T}^{f}:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})$,
where $\|m_{T}^{\operatorname{r}}\|_{cb}$,
$\|m_{T}^{\operatorname{f}}\|_{cb}\leq M$ and $m_{T_{i}}^{\operatorname{r}}\to
m_{T}^{\operatorname{r}}$, $m_{T_{i}}^{\operatorname{f}}\to
m_{T}^{\operatorname{f}}$ pointwise.

###### Proof.

Note first that the uniform convergence on compact sets in
$\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ implies convergence in the
maximal $\mathrm{C}^{*}$-norm. Indeed, for
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ we have $a=\sum_{U\in
F}a_{U}$ for some
$a_{U}\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})}|_{U})$, $U\in F$,
where $F\subseteq\operatorname{Bis}(\mathcal{G})$ is finite. The norm of
$\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})$ restricted to
$\mathrm{C_{c}}(\mathcal{B}^{(\beta,u_{\beta})}|_{U})$,
$U\in\operatorname{Bis}(\mathcal{G})$, coincides with the supremum norm.
Therefore
$\|m_{T}(a)-m_{T_{i}}(a)\|_{\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})}\leq\sum_{U\in
F}\|m_{T}(a_{U})-m_{T_{i}}(a_{U})\|_{\infty}\to 0$. Assume now that we are
given a net $\\{T_{i}\\}_{i\in I}\subseteq
F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ and $M:=\sup_{i}\|T_{i\in
I}\|_{F}<\infty$. Then
$\|m_{T_{i}}(a)\|_{\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})}\leq
M\cdot\|a\|_{\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})}$,
by Theorem 9.4, and thus
$\|m_{T}(a)\|_{\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})}\leq
M\cdot\|a\|_{\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})}$.
Hence $m_{T}$ yields a bounded operator
$m_{T}^{\operatorname{r},\operatorname{f}}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})$
with norm not greater than $M$. Applying this reasoning to matrices, we deduce
that $m_{T}^{\operatorname{r},\operatorname{f}}$ is in fact completely bounded
and $\|m_{T}^{\operatorname{r},\operatorname{f}}\|_{cb}\leq M$. This proves
(1). The proof of (2) is the same (instead of Theorem 9.4 use Theorem 7.13). ∎

The next two corollaries are immediate consequences of the above result and
appropriate definitions (in the second we also use Lemma 10.1).

######  Corollary 10.10.

Every multiplier with the Fourier approximation property is reduced to full.
Every multiplier with the Fourier–Stieltjes approximation property is both
reduced and full.

######  Corollary 10.11.

Every bounded positive-definite (not necessarily strict) multiplier $T$ from
$\mathcal{A}^{(\alpha,u_{\alpha})}$ to $\mathcal{B}^{(\beta,u_{\beta})}$ is
both reduced and full (and we have
$\|m_{T}^{\operatorname{r}}\|_{cb}=\|m_{T}^{\operatorname{f}}\|_{cb}=\sup_{x\in
X}\|T_{x}\|$).

### Fourier and Fourier–Stieltjes approximation properties for actions

We will now study the definitions considered above in the case of the identity
multiplier and view them as approximation properties of a given twisted
groupoid action.

######  Definition 10.12.

We say that an action $(\alpha,u_{\alpha})$ of a groupoid $\mathcal{G}$ on a
$\mathrm{C}^{*}$-bundle $\mathcal{A}$ has the _(positive) Fourier_ or
_Fourier–Stieltjes approximation property_ if the identity multiplier of
$\mathcal{A}^{(\alpha,u_{\alpha})}$ has this property.

######  Corollary 10.13.

A twisted groupoid action $(\alpha,u_{\alpha})$ has the positive
Fourier–Stieltjes approximation property if and only if there is a net
$\\{T_{i}\\}_{i\in I}$ of compactly supported and positive-definite
multipliers of $\mathcal{A}^{(\alpha,u_{\alpha})}$ such that $\sup_{i\in
I}\sup_{x\in X}\|(T_{i})_{x}\|<\infty$ and $m_{T_{i}}(a)\to a$ uniformly on
compact sets for every
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$.

###### Proof.

Apply Proposition 10.8. ∎

######  Remark 10.14.

We will focus on the Fourier–Stieltjes approximation property as it is (at
least formally) weaker than the Fourier approximation property. Exel’s
approximation property [Exe97, Definition 4.4], recently generalized to Fell
bundles over étale groupoids in [Kra23, Definition 3.1] and [BM23, Definition
4.10], when formulated for the bundle $\mathcal{A}^{(\alpha,u_{\alpha})}$
associated with a twisted groupoid action means that the identity multiplier
has the positive Fourier AP with the approximating net given by coefficients
of the regular equivariant representation of $\mathcal{G}$, see Examples 6.9,
6.5. More precisely, we say that the action $(\alpha,u_{\alpha})$ of a
groupoid $\mathcal{G}$ on a $\mathrm{C}^{*}$-bundle $\mathcal{A}$ has the
_Exel’s approximation property_ if there is a net $\\{\xi_{i}\\}_{i\in
I}\subseteq\mathrm{C_{c}}(r^{*}\mathcal{A})$ such that $\sup_{i\in
I}\sup_{x\in X}\|\sum_{g\in\mathcal{G}_{x}}\xi_{i}(g)^{*}\xi_{i}(g)\|<\infty$
and for every
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})=\mathrm{C_{c}}(r^{*}\mathcal{A})$,
the net of sections (compare to Example 9.2)

$\mathcal{G}\ni
g\longmapsto\sum_{t\in\mathcal{G}^{r(g)}}\xi_{i}(h)^{*}a(g)\alpha_{g}(\xi_{i}(g^{-1}h))\in
A_{r(g)}=\mathcal{A}^{(\alpha,u_{\alpha})}_{g}\subseteq\mathcal{A}^{(\alpha,u_{\alpha})}$

converges uniformly on compact sets to $a$. Following [BEW23, Definition 3.5],
[Tak14, Definition 2.2], [BO08, Theorem 5.6.18], one could say that the
twisted action $(\alpha,u_{\alpha})$ on $\mathcal{A}$ is _strongly amenable_
if there is a bounded net
$\\{\varphi_{i}\\}_{i}\subseteq\mathrm{C_{c}}(r^{*}Z\mathcal{M}(\mathcal{A}))$
of central sections of positive-type (see Remark 8.8) such that $\sup_{g\in
G}\|\varphi_{i}(g)a(g)-a(g)\|_{\infty}\longrightarrow 0$ for every
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$. By Corollary 8.7,
strong amenability implies the positive Fourier–Stieltjes AP. The diagram

strong amenabilityExel’s APpos. Fourier APpos. Fourier–Stieltjes APFourier
APFourier–Stieltjes APweak containment

summarizes the general relationships between the discussed approximation
properties. The last implication from Fourier–Stieltjes AP to the weak
containment property will be proved below (Theorem 10.15). For a brief
discussion of the converse we refer to Question 12.5.

The next result should be compared to [BC12, Theorem 5.8].

######  Theorem 10.15.

Let $\mathcal{A}$ be a $\mathrm{C}^{*}$-bundle over $X$, carrying a twisted
action $(\alpha,u_{\alpha})$ of $\mathcal{G}$. Assume $(\alpha,u_{\alpha})$
has the Fourier–Stieltjes approximation property. Then it has the weak
containment property, i.e.
$\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})=\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$,
and $\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})$ is nuclear if and only
if $A=\mathrm{C_{0}}(\mathcal{A})$ is nuclear.

###### Proof.

For the first part we cannot use Corollary 10.10, as the Fourier–Stieltjes AP
is (at least formally) weaker than the Fourier AP. We adapt the proof of
Bédos–Conti [BC16, Theorem 4.6] for crossed products, which was in turn
inspired by [BO08, Theorem 2.6] for groups.

Recall from Lemma 9.8 that if $V\subseteq\mathcal{G}$ is a precompact open set
then the subspace $\mathrm{C_{0}}(\mathcal{A}^{(\alpha,u_{\alpha})}|_{V})$ of
$\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ is closed in the reduced
and hence also in the universal (maximal) norm. Now let $\\{T_{i}\\}_{i\in
I}\subseteq FS[(\alpha,u_{\alpha})]$ be a net witnessing the approximation
property. For every $i\in I$ the support of each $T_{i}$ is contained in a
precompact open set $V_{i}$, and so
$m_{T}:\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C_{0}}(\mathcal{A}^{(\alpha,u_{\alpha})}|_{V_{i}})$.
By the above claim the extended maps
$m_{T_{i}}^{\operatorname{r}}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C_{0}}(\mathcal{A}^{(\alpha,u_{\alpha})}|_{V_{i}})$
and
$m_{T_{i}}^{\operatorname{f}}:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C_{0}}(\mathcal{A}^{(\alpha,u_{\alpha})}|_{V_{i}})$
(that exist by Theorem 7.13) take values in
$\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$. The regular
representation
$\Lambda:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$
is the identity on $\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$, and
therefore for every $a\in\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})$

$m_{T_{i}}^{\operatorname{r}}(\Lambda(a))=\Lambda(m_{T_{i}}^{\operatorname{f}}(a))=m_{T_{i}}^{\operatorname{f}}(a).$

Hence if $\Lambda(a)=0$, then $m_{T_{i}}^{\operatorname{f}}(a)=0$ for all
$i\in I$, which implies $a=0$ because $m_{T_{i}}^{\operatorname{f}}(a)\to a$,
see Lemma 10.9(2). Thus
$\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})=\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$.

If $\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})$ is nuclear, then so is
$A=\mathrm{C_{0}}(\mathcal{A})$ because we have a conditional expectation
$E:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\to A$. To prove the
converse implication we adapt Takeishi’s proof of [Tak14, Theorem 4.1] for
amenable groupoids, which in turn was based on [BO08, Theorem 5.6.18]. Assume
that $A$ is nuclear. Let $D$ be any $\mathrm{C}^{*}$-algebra. We need to show
that the canonical quotient map
$Q:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\otimes_{\max}D\rightarrow\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\otimes_{\min}D$
is injective. The first part of the proof of [Tak14, Theorem 4.1] shows (using
nuclearity of $A$) that for any precompact open set $V\subseteq\mathcal{G}$,
the quotient map $Q$ is injective on
$\mathrm{C_{0}}(\mathcal{A}^{(\alpha,u_{\alpha})}|_{V})\otimes_{\max}D\subseteq\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\otimes_{\max}D$.
For the approximating net $\\{T_{i}\\}_{i\in I}\subseteq
FS[(\alpha,u_{\alpha})]$, as above, the range of each
$m_{T_{i}}^{\operatorname{f}}$ is contained in the closed subspace
$\mathrm{C_{0}}(\mathcal{A}^{(\alpha,u_{\alpha})}|_{V_{i}})$, and therefore
${m}_{T_{i}}^{\operatorname{f}}\otimes_{\max}\mathrm{id}\left({\mathrm{C}^{*}}(\mathcal{A}^{(\alpha,u_{\alpha})})\otimes_{\max}D\right)$
is contained in
$\mathrm{C_{0}}(\mathcal{A}^{(\alpha,u_{\alpha})}|_{V_{i}})\otimes_{\max}D$.
Let $a\in{\mathrm{C}^{*}}(\mathcal{A}^{(\alpha,u_{\alpha})})\otimes_{\max}D$
with $Q(a)=0$. Using that the following diagram commutes:

$\textstyle{{\mathrm{C}^{*}}(\mathcal{A}^{(\alpha,u_{\alpha})})\otimes_{\max}D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q}$$\scriptstyle{{m}_{T_{i}}^{\operatorname{f}}\otimes_{\max}\mathrm{id}}$$\textstyle{{\mathrm{C}^{*}}(\mathcal{A}^{(\alpha,u_{\alpha})})\otimes_{\max}D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q}$$\textstyle{{\mathrm{C}^{*}}(\mathcal{A}^{(\alpha,u_{\alpha})})\otimes_{\min}D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{m}_{T_{i}}^{\operatorname{f}}\otimes_{\min}\mathrm{id}}$$\textstyle{{\mathrm{C}^{*}}(\mathcal{A}^{(\alpha,u_{\alpha})})\otimes_{\min}D}$,

we get
$Q\circ({m}_{T_{i}}^{\operatorname{f}}\otimes_{\max}\mathrm{id})(a)=({m}_{T_{i}}^{\operatorname{f}}\otimes_{\min}\mathrm{id})\circ
Q(a)=0$. Since $Q$ is injective on the range of
${m}_{T_{i}}^{\operatorname{f}}\otimes_{\max}\mathrm{id}$ this means that
${m}_{T_{i}}^{\operatorname{f}}\otimes_{\max}\mathrm{id}(a)=0$ for all $i\in
I$. Hence
$a=\lim_{i}{m}_{T_{i}}^{\operatorname{f}}\otimes_{\max}\mathrm{id}(a)=0$. This
proves that $Q$ is injective. ∎

We will now specify the above results to the context of (twisted) groupoid
$\mathrm{C}^{*}$-algebras.

######  Corollary 10.16.

Let $u$ be a continuous $2$-cocycle on a locally compact Hausdorff étale
groupoid $\mathcal{G}$. Then the following assertions are equivalent:

  1. (1)

the trivial twisted action $(\operatorname{id},u)$ on the trivial one-
dimensional $\mathrm{\mathrm{C}^{*}}$-bundle $\mathbb{C}\times X$ has the
Fourier–Stieltjes approximation property;

  2. (2)

$(\operatorname{id},u)$ or, equivalently, the trivial (untwisted) action
$\operatorname{id}$ on $\mathbb{C}\times X$ has the positive Fourier
approximation property.

  3. (3)

$C^{*}(\mathcal{G},u)$ or, equivalently, $C^{*}_{r}(\mathcal{G},u)$ is
nuclear;

  4. (4)

$C^{*}(\mathcal{G})$ or, equivalently, $C^{*}_{r}(\mathcal{G})$ is nuclear;

  5. (5)

$\mathcal{G}$ is amenable.

###### Proof.

If $(\operatorname{id},u)$ has the Fourier–Stieltjes approximation property,
then by the previous theorem $C^{*}(\mathcal{G},u)=C^{*}_{r}(\mathcal{G},u)$
is nuclear, and this is known to be equivalent to amenability of
$\mathcal{G}$, see [Tak14]. But amenability of $\mathcal{G}$ is, in turn,
equivalent to the existence of a net $\\{\varphi_{i}\\}_{i\in I}$ of positive-
type functions in $\mathrm{C_{c}}(\mathcal{G})$ that converge pointwise to $1$
uniformly on compacts, see [ADR00, Proposition 2.2.13] or [BO08, Theorem
5.6.18]. This is another way to say that $(\operatorname{id},u)$ or just
$\operatorname{id}$ has the positive Fourier–Stieltjes AP as
$FS(\mathcal{G},u)=FS(\mathcal{G})$, see Example 7.11. ∎

######  Remark 10.17.

Most of the statements in the above corollary were already known to be
equivalent before, as already indicated in the references throughout the
proof, except for the implication from (1) to the other statements (2)-(5).

##  11\. Some further applications: Haagerup trick and decomposable norms

The Haagerup trick is a way of producing multipliers of a dynamical system
from maps on the associated reduced algebra, applied first for groups by Uffe
Haagerup [Haa16, Lemma 2.5], later used in many different contexts and in
particular generalized to group actions in [BC16, Proposition 4.12] and
[MSTT18, Proposition 3.4]. A version of this trick for twisted groupoid
actions and bimodule maps was a main tool in [KLS22]. In general, for étale
groupoids and their actions it is not clear what the Haagerup trick should be.
However when either the groupoid $\mathcal{G}$ is discrete or when we consider
only bimodule maps the situation becomes much more transparent. We discuss
here these two cases and some of the applications, notably to identifying the
Fourier–Stieltjes norm of the associated multiplier. For the last purpose we
will also use the fact that maps coming from a Fourier–Stieltjes multiplier
$T=T_{\mathcal{E},L,\xi,\zeta}$ are _decomposable_ , that is they are finite
linear combinations of completely positive maps, which follows from the usual
polarization formula

$T=\frac{1}{4}\sum_{k=0}^{3}i^{k}T_{\mathcal{E},L,\xi+i^{k}\zeta,\xi+i^{k}\zeta}$

and the multipliers of the form $T_{\mathcal{E},L,\xi,\xi}$ induce completely
positive maps, cf. Theorem 8.5. Finally we will use the well-known fact that
for any Fell bundle $\mathcal{A}$ over $\mathcal{G}$, the inclusion
$\mathrm{C_{c}}(\mathcal{A})\subseteq\mathrm{C_{0}}(\mathcal{A})$ extends to a
contractive injective linear embedding
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A})\to\mathrm{C_{0}}(\mathcal{A})$,
cf. [KM21, Proposition 7.10].

###  11.1. Twisted actions of discrete groupoids

In this subsection we will assume that the groupoid $\mathcal{G}$ is discrete.

######  Lemma 11.1 (Haagerup Trick I).

Let $\mathcal{A}$ and $\mathcal{B}$ be Fell bundles over a discrete groupoid
$\mathcal{G}$. For any bounded linear map
$\Phi:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A})\to\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B})$
the formula

$T_{g}^{\Phi}(a)=\Phi(a\delta_{g})(g),\qquad a\in A_{g},\,\,g\in\mathcal{G},$

where $a\delta_{g}$ is a section in $\mathrm{C_{c}}(\mathcal{A}|_{\\{g\\}})$
with $a\delta_{g}(g)=a$, defines a multiplier
$T^{\Phi}=\\{T_{g}^{\Phi}\\}_{g\in\mathcal{G}}$ from $\mathcal{A}$ to
$\mathcal{B}$ with $\sup_{g\in\mathcal{G}}\|T_{g}^{\Phi}\|\leq\|\Phi\|$. If
$\Phi$ is completely positive, then
$T^{\Phi}=\\{T_{g}^{\Phi}\\}_{g\in\mathcal{G}}$ is positive-definite.

###### Proof.

It is clear that the formula above defines a multiplier. Moreover,
$\|T_{g}^{\Phi}(a)\|=\|\Phi(a\delta_{g})(g)\|\leq\|\Phi(a\delta_{g})\|\leq\|\Phi\|\cdot\|a\|$
for any $a\in A_{g}$, $g\in\mathcal{G}$. Hence
$\sup_{g\in\mathcal{G}}\|T_{g}^{\Phi}\|\leq\|\Phi\|$. Assume that $\Phi$ is
completely positive. Fix $x\in X$, $n\in\mathbb{N}$, elements
$g_{1},\ldots,g_{n}\in\mathcal{G}_{x}$, and collections $\\{a_{g_{i}}\in
A_{g_{i}}:1\leq i\leq n\\}$, $\\{b_{g_{i}}\in B_{g_{i}}:1\leq i\leq n\\}$. The
matrix
$(a_{g_{i}}a_{g_{j}}^{*}\delta_{g_{i}g_{j}^{-1}})_{i,j=1}^{n}=(a_{g_{i}}\delta_{g_{i}}*(a_{g_{j}}\delta_{g_{j}})^{*})_{i,j=1}^{n}$
is positive in
$\mathbb{M}_{n}(\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}))$ and so
$b:=\Phi^{(n)}((a_{g_{i}}a_{g_{j}}^{*}\delta_{g_{i}g_{j}^{-1}})_{i,j=1}^{n})$
is positive in $\mathcal{L}(\ell^{2}(\mathcal{B})_{x}^{\oplus n})$. Let
$\xi=\bigoplus_{i=1}^{n}\xi_{g_{i}}\in\ell^{2}(\mathcal{B})_{x}^{\oplus n}$ be
given by $\xi_{g}(h)=[h=g]b_{g}$, for all $h,g\in\mathcal{G}$. Then

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

Hence $T^{\Phi}$ is positive definite. ∎

######  Proposition 11.2.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of a
discrete groupoid $\mathcal{G}$. A multiplier $T$ from
$\mathcal{A}^{(\alpha,u_{\alpha})}$ to $\mathcal{B}^{(\beta,u_{\beta})}$ has
the positive Fourier–Stieltjes approximation property if and only if there is
a bounded net $\\{\Phi_{i}\\}_{i\in I}$ of completely positive maps
$\Phi_{i}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to
C(\mathcal{B}^{(\beta,u_{\beta})}|_{V_{i}})\subseteq\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B}^{(\beta,u_{\beta})})$
with ranges supported in finite sets $V_{i}\subseteq\mathcal{G}$, $i\in I$,
and such that $\Phi_{i}(a)\to m_{T}(a)$ for every
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$.

###### Proof.

If $\\{T_{i}\\}_{i\in I}\subseteq FS[(\alpha,u_{\alpha})]$ is a net witnessing
the positive Fourier–Stieltjes approximation property for $T$ then the
associated maps $\\{m_{T_{i}}^{\operatorname{r}}\\}_{i\in I}$, given by
Theorem 7.13, have the desired properties by Theorem 8.5 and Corollary 9.9.
Conversely, assume that $\\{\Phi_{i}\\}_{i\in I}$ is a bounded net of
completely positive maps
$\Phi_{i}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to
C(\mathcal{B}^{(\beta,u_{\beta})}|_{V_{i}})$ where $V_{i}$ is finite. Then the
associated net $\\{T^{\Phi_{i}}\\}_{i\in I}$ given by Lemma 11.1 consists of
positive-definite multipliers such that $\sup_{i\in I}\sup_{x\in
X}\|(T_{i})_{x}\|\leq\sup_{i\in I}\|\Phi_{i}\|<\infty$. Moreover,
$\Phi_{i}(a\delta_{g})\to m_{T}(a\delta_{g})=T_{g}(a)\delta_{g}$ for $a\in
A_{r(g)}$, $g\in\mathcal{G}$, implies

$m_{T^{\Phi_{i}}}(a\delta_{g})=T^{\Phi_{i}}(a)\delta_{g}=\Phi_{i}(a\delta_{g})(g)\delta_{g}\to
m_{T}(a\delta_{g})(g)\delta_{g}=T_{g}(a)\delta_{g}=m_{T}(a\delta_{g}).$

This implies that $m_{T^{\Phi_{i}}}(a)\to m_{T}(a)$ for every
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$. ∎

######  Corollary 11.3.

A twisted groupoid action $(\alpha,u_{\alpha})$ of a discrete groupoid
$\mathcal{G}$ on a bundle $\mathcal{A}$ has the positive Fourier–Stieltjes
approximation property if and only if there is a bounded net
$\\{\Phi_{i}\\}_{i\in I}$ of completely positive maps on
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ that
converge pointwise to the identity and that have ranges supported on finite
subsets of $\mathcal{G}$.

###### Proof.

Apply Proposition 11.2 to the identity multiplier. ∎

The above results make it more evident that the positive Fourier–Stieltjes AP
is a ‘nuclearity condition’ for multipliers. In particular, if in the
characterisation in Corollary 11.3 one replaces ‘ranges are supported on
finite subsets of $\mathcal{G}$’ with ‘ranges are finite-dimensional’ one gets
a characterisation of nuclearity of
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$, see
[CE78, Theorem 3.1]. We show that in this context ‘finite rank’ implies
‘supported on finite sets’ using the following lemma, which is probably well
known to experts, cf. the proof of [MSTT18, Theorem 4.3].

######  Lemma 11.4.

Let $A$ and $B$ be $C^{*}$-algebras. Assume $A$ is finite dimensional and let
$B_{0}$ be a dense $*$-subalgebra of $B$. For every completely positive map
$\Phi:A\to B$ there is a net $\\{\Phi_{i}\\}_{i\in I}$ of completely positive
maps $\Phi_{i}:A\to B_{0}\subseteq B$ such that $\|\Phi_{i}\|=\|\Phi\|$ and
$\Phi_{i}(a)\to\Phi(a)$ for $a\in A$ (equivalently $\Phi_{i}\to\Phi$ in norm).

###### Proof.

Since $A$ is finite dimensional we have
$A=\bigoplus_{k=1}^{N}\mathbb{M}_{n_{k}}(\mathbb{C})$. Then each map
$\Phi_{k}:=\Phi|_{\mathbb{M}_{n_{k}}(\mathbb{C})}$ (with $k=1,\ldots,N$) is
contractive and completely positive, and
$\Phi(\oplus_{k=1}^{N}a_{k})=\sum_{k=1}^{n}\Phi_{k}(a_{k})$ for
$\oplus_{k=1}^{n}a_{k}\in A$. Thus it suffices to prove the assertion for each
$\Phi_{k}$. Hence we may simply assume that $A=\mathbb{M}_{n}(\mathbb{C})$ for
some $n$. We may also assume that $\|\Phi\|=1$.

Let $e_{p,q}$, $p,q=1,\ldots,n$, be the canonical matrix units in
$\mathbb{M}_{n}(\mathbb{C})$. Since the matrix
$(e_{p,q})^{n}_{p,q=1}\in\mathbb{M}_{n}(\mathbb{M}_{n}(\mathbb{C}))$ is
positive and $\Phi$ is completely positive, we get that
$(\Phi(e_{p,q}))^{n}_{p,q=1}\in\mathbb{M}_{n}(B)^{+}$. As
$\mathbb{M}_{n}(B_{0})^{+}$ is dense in $\mathbb{M}_{n}(B)^{+}$, given
$\varepsilon_{i}>0$, $i\in I$, with $\varepsilon_{i}\to 0$, there are matrices
$(b^{i})_{p,q=1}^{n}\in\mathbb{M}_{n}(B_{0})^{+}$ such that
$\|(\Phi(e_{p,q}))^{n}_{p,q=1}-(b^{i})_{p,q=1}^{n}\|<\varepsilon_{i}/n$.
Define $\Phi_{i}:A\to B$ by

$\Phi_{i}\big{(}(\lambda_{p,q})^{n}_{p,q=1}\big{)}=\sum^{n}_{p,q=1}\lambda_{p,q}b^{i}_{p,q}.$

By Choi’s Theorem [Pau02, Theorem 3.14], $\Phi_{i}$ is completely positive.
Since
$\|\Phi(e_{p,q})-\Phi_{i}(e_{p,q})\|=\|\Phi(e_{p,q})-b^{i}_{p,q}\|<\varepsilon_{i}/n\stackrel{{\scriptstyle
i}}{{\to}}0$ and $\\{e_{p,g}\\}_{p,q=1}^{n}$ is a basis in
$\mathbb{M}_{n}(\mathbb{C})$, we conclude that $\Psi_{i}$ converges to $\Psi$
pointwise (and hence also in norm as $\dim(A)<\infty$). Moreover,

$|\|\Phi_{i}\|-1|=|\|\Phi_{i}(1)\|-\|\Phi(1)\||\leq\|\Phi_{i}(1)-\Phi(1)\|\leq\sum_{p=1}^{n}\|\Phi(e_{p,p})-b^{i}_{p,p}\|\leq\varepsilon_{i}/n\stackrel{{\scriptstyle
i}}{{\to}}0.$

Thus $\|\Phi_{i}\|\to 1$ and therefore replacing $\Phi_{i}$ by
$\frac{\Phi_{i}}{\|\Phi_{i}\|}$ we get the desired net. ∎

######  Corollary 11.5.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of a
discrete groupoid $\mathcal{G}$. If a map
$\Phi:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B}^{(\beta,u_{\beta})})$
is a pointwise limit of a bounded net $\\{\Phi_{i}\\}_{i\in I}$ of finite rank
completely positive maps
$\Phi_{i}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B}^{(\beta,u_{\beta})})$,
$i\in I$, then the multiplier $T^{\Phi}$ from Lemma 11.1 has the positive
Fourier–Stieltjes approximation property.

###### Proof.

Since
$\Phi_{i}(\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$
is finite dimensional, by Lemma 11.4 there is a net $\\{\phi^{i}_{j}\\}_{j\in
J}$ of norm one completely positive maps
$\phi_{j}^{i}:\Phi_{i}(\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C_{c}}(\mathcal{B}^{(\beta,u_{\beta})})\subseteq\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B}^{(\beta,u_{\beta})})$
that converge pointwise to the identity on
$\Phi_{i}(\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$.
Thus replacing the net $\\{\Phi_{i}\\}_{i\in I}$ with
$\\{\phi_{j}^{i}\circ\Phi_{i}\\}_{(i,j)\in I\times J}$ we may assume that each
map $\Phi_{i}$ takes values in the dense $*$-subalgebra
$\mathrm{C_{c}}(\mathcal{B}^{(\beta,u_{\beta})})$ of
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B}^{(\beta,u_{\beta})})$. Then
the finite rank for $\Phi_{i}$ implies that the range of $\Phi_{i}$ is
supported on a finite subset of $\mathcal{G}$. Hence the assertion follows by
Proposition 11.2. ∎

######  Remark 11.6.

The above results (i.e. Lemma 11.1, Proposition 11.2, and Corollaries 11.3 and
11.5, remain valid (with essentially the same proofs) when the reduced crossed
products are replaced with full crossed products. If, in addition, the crossed
product for $(\alpha,u_{\alpha})$ is separable, then nets
$\\{\Phi_{i}\\}_{i\in I}$ may be replaced by sequences, and then their
boundedness follow from their pointwise convergence.

######  Theorem 11.7.

Let $(\alpha,u_{\alpha})$ be a twisted groupoid action of a discrete groupoid
$\mathcal{G}$ on a bundle $\mathcal{A}$. Then the following conditions are
equivalent:

  1. (1)

$(\alpha,u_{\alpha})$ has the positive Fourier–Stieltjes approximation
property and $\mathrm{C_{0}}(\mathcal{A})$ is nuclear;

  2. (2)

$(\alpha,u_{\alpha})$ has the Fourier–Stieltjes approximation property and
$\mathrm{C_{0}}(\mathcal{A})$ is nuclear;

  3. (3)

the reduced crossed product
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ is
nuclear;

  4. (4)

the full crossed product $\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})$
is nuclear;

  5. (5)

there is a net of finitely supported positive-definite multipliers
$\\{T_{i}\\}_{i\in I}$ such that for each $i\in I$, $\sup_{x\in
X}\|(T_{i})_{x}\|\leq 1$, and for every for every $g\in\mathcal{G}$,
$(T_{i})_{g}$ is of finite rank and $(T_{i})_{g}(a)\to a$ for any $a\in
A_{r(g)}$.

If the above equivalent conditions are satisfied, then
$\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})=\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$.

###### Proof.

Clearly, (1) implies (2), and (2) implies (3) and (4), as well as
$\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})=\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$,
by Theorem 10.15. Suppose now that
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ is
nuclear. Equivalently, by [CE78, Theorem 3.1], the identity on
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ is
the pointwise limit of finite-rank completely positive contractions. Modifying
this net as in the proof of Corollary 11.5 one gets a net as in (5). The same
argument works for the full crossed product. Hence either (3) or (4) implies
(5). Now if $\\{T_{i}\\}_{i\in I}$ is a net as in (5), it witnesses the
positive Fourier–Stieltjes approximation property of $(\alpha,u_{\alpha})$ and
the maps $\\{m_{T_{i}}|_{\mathrm{C_{0}}(\mathcal{A})}$ witness nuclearity of
$\mathrm{C_{0}}(\mathcal{A})$. Hence (5) implies (1). ∎

######  Remark 11.8.

The problem of characterising nuclearity of the crossed product for actions on
nuclear $C^{*}$-algebras in terms of an amenability condition has received a
lot of attention, usually using von Neumann algebra techniques, see [AD87,
Théorème 4.5], [BEW23, Corollary 4.11], [Kra23, Theorem A]. In particular,
assuming that $\mathrm{C_{0}}(\mathcal{A})$ is nuclear, the aforementioned
results combined with Theorem 11.7 give that:

  1. (1)

If $\mathcal{G}=G$ is a discrete group and the twist is trivial, then the
Fourier–Stieltjes AP is equivalent to amenability of the action.

  2. (2)

If $\mathcal{G}=G$ is a discrete group, then the Fourier–Stieltjes AP is
equivalent to Exel’s AP for the twisted action.

  3. (3)

If $\mathcal{G}$ is countable, $\mathrm{C_{0}}(\mathcal{A})$ is separable and
the twist is trivial, then the Fourier–Stieltjes AP is equivalent to
measurewise amenability of the action.

The characterisation of nuclearity of
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ in
terms of condition (5) in Theorem 11.7 is a generalization of [MSTT18, Theorem
4.3] stated for discrete group actions on unital $C^{*}$-algebras.

###  11.2. Bimodule maps, Herz–Schur multipliers and strong amenability

In this section we consider again a general étale locally compact Haudorff
groupoid $\mathcal{G}$. We will consider however only bimodule maps on a
crossed product.

######  Proposition 11.9 (Haagerup Trick II).

Let $(\alpha,u)$ be a twisted action of an étale groupoid $\mathcal{G}$ on a
$\mathrm{C}^{*}$-bundle $\mathcal{A}$. To every bounded linear
$\mathrm{C_{0}}(\mathcal{A})$-bimodule map
$\Phi:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u)})\to\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u)})$
there is an associated bounded strictly continuous central section
$\varphi_{\Phi}\in\mathrm{C_{b}}(r^{*}Z\mathcal{M}(\mathcal{A}))$ which is
uniquely determined by the recipe

(11.1) $\varphi_{\Phi}(g)a(g)=\Phi(a)(g)\qquad\text{ for any
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u)})$ supported on a bisection}.$

In particular, $\sup_{g\in\mathcal{G}}\|\varphi_{\Phi}(g)\|\leq\|\Phi\|$. If
$\Phi$ is completely positive then the Herz–Schur multiplier
$\varphi_{\Phi}:\mathcal{G}\to r^{*}Z\mathcal{M}(\mathcal{A})$ is of positive
type.

###### Proof.

Fix $g\in\mathcal{G}$ and $U\in\operatorname{Bis}(\mathcal{G})$ containing
$g$. Let $a_{0}\in A_{r(g)}$ and take any
$a\in\mathrm{C_{c}}(\mathcal{A}|_{r(U)})$ and
$b\in\mathrm{C_{c}}(\mathcal{A}|_{s(U)})$ such that
$a_{0}=a(r(g))=\alpha_{g}(b(s(g))$. Choose an approximate unit
$\\{e_{i}^{U}\\}_{i\in I}\subseteq\mathrm{C_{c}}(\mathcal{A}|_{r(U)})$ in
$\mathrm{C_{0}}(\mathcal{A}|_{r(U)})$. We will use the notation from Lemma
4.8. Thus let $a\delta_{U}\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u)})$ be
given by $(a\delta_{U})(g)=a(r(g))$, $g\in U$. Clearly
$a*(e_{i}^{U}\delta_{U})\stackrel{{\scriptstyle i}}{{\to}}a\delta_{U}$ and
$(e_{i}^{U}\delta_{U})*\alpha_{U}^{-1}(a)\stackrel{{\scriptstyle
i}}{{\to}}a\delta_{U}$ uniformly. Using that both $\Phi$ and the embedding
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u)})\hookrightarrow\mathrm{C_{0}}(\mathcal{A}^{(\alpha,u)})$
(with convolution multiplication) are
$\mathrm{C_{0}}(\mathcal{A}^{(\alpha,u)})$-bimodule maps we get

$\lim_{i}a_{0}\Phi(e_{i}^{U}\delta_{U})(g)=\lim_{i}\big{(}a*\Phi(e_{i}^{U}\delta_{U})\big{)}(g)=\lim_{i}\Phi(a*(e_{i}^{U}\delta_{U})(g)=\Phi(a\delta_{U})(g)$

and similarly

$\lim_{i}\Phi(e_{i}^{U}\delta_{U})(g)a_{0}=\lim_{i}\big{(}\Phi(e_{i}^{U}\delta_{U})*\alpha_{U}^{-1}(a)\big{)}(g)=\lim_{i}\Phi\big{(}e_{i}^{U}\delta_{U}*\alpha_{U}^{-1}(a)\big{)}(g)=\Phi(a\delta_{U})(g).$

Hence the following strict limit in $\mathcal{M}(A_{r(g)})$

$\varphi_{\Phi}(g):=\text{s-}\lim_{i}\Phi(e_{i}^{U}\delta_{U})(g)$

exists, is central, and satisfies (11.1) (which clearly determines
$\varphi_{\Phi}(g)$ uniquely). We have
$\|\varphi_{\Phi}(g)\|=\|\text{s-}\lim_{i}\Phi(e_{i}^{U}\delta_{U})(g)\|\leq\sup_{i}\|\Phi(e_{i}^{U}\delta_{U})\|\leq\|\Phi\|$.

Assume now that $\Phi$ is completely positive. Fix $x\in X$, $n\in\mathbb{N}$
and elements $g_{i}\in\mathcal{G}_{x}$ and $a_{i}\in A_{x}$ for $1\leq i\leq
n$. Find sections $\\{f_{i}\\}_{i=1}^{n}\in\mathrm{C_{c}}(\mathcal{A})$
supported on bisections and such that $f_{i}(g_{i})=\alpha_{g_{i}}(a_{i})$,
$1\leq i\leq n$. Then $f_{i}*f_{j}^{*}$ is supported on a bisection and

$(f_{i}*f_{j}^{*})(g_{i}{g_{j}}^{-1})=\alpha_{g_{i}}(a_{i}a_{j}^{*})u(g_{i}g_{j}^{-1},g_{j})^{*}$

for every $1\leq i,j\leq n$, cf. the calculations in the proof of Lemma 8.4.
Hence by (11.1) we get

$\Phi(f_{i}*f_{j}^{*})(g_{i}g_{j}^{-1})u(g_{i}g_{j}^{-1},g_{j})=\varphi_{\Phi}(g_{i}g_{j}^{-1})\alpha_{g_{i}}(a_{g_{i}}a_{g_{j}}^{*}).$

Since the matrix
$(f_{i}*f_{j}^{*})_{i,j=1}^{n}\in\mathbb{M}_{n}(\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}))$
is positive, complete positivity of $\Phi$ implies that

$\sum_{i,j=1}^{n}\alpha^{-1}_{g_{i}}\Big{(}\Phi(f_{i}*f_{j}^{*})(g_{i}g_{j}^{-1})u(g_{i}g_{j}^{-1},g_{j})\Big{)}\geq
0$

(to see this replace $m_{T}$ in the proof of Lemma 8.3 by $\Phi$, see also the
proof of Lemma 8.4). Combining the last two displayed formulas we obtain

$\sum_{i,j=1}^{n}\alpha^{-1}_{g_{i}}\Big{(}\varphi_{\Phi}(g_{i}g_{j}^{-1})\alpha_{g_{i}}(a_{g_{i}}a_{g_{j}}^{*}))\Big{)}\geq
0,$

which means that the Herz–Schur multiplier $\varphi_{\Phi}$ is of positive
type, see Lemma 8.4. ∎

######  Remark 11.10.

A similar result for action of twisted groupoids was proved in [KLS22,
Proposition 4.1] under the assumption that $\mathcal{A}$ has a continuous
unital section. Note that the proposition mentioned above should have been
stated for bimodule maps (every $*$-preserving, so in particular every
completely positive, right $A$-module map is automatically an $A$-bimodule
map), and then the proof of Proposition 11.9 can be adapted to get a ‘non-
unital version’ of [KLS22, Proposition 4.1]. For group actions on unital
algebras a version of the above Haagerup trick appears also in the proof of
[DR12, Theorem 3.6].

######  Proposition 11.11.

Let $(\alpha,u)$ be a twisted action of an étale groupoid $\mathcal{G}$ on a
$\mathrm{C}^{*}$-bundle $\mathcal{A}$. The following conditions are
equivalent:

  1. (1)

The action $(\alpha,u)$ is _strongly amenable_ , i.e. there is a bounded net
of positive-type sections $\\{\varphi_{i}\\}_{i\in
I}\subseteq\mathrm{C_{c}}(r^{*}Z\mathcal{M}(\mathcal{A}))$ such that
$\sup_{g\in G}\|\varphi_{i}(g)a(g)-a(g)\|_{\infty}\longrightarrow 0$ for
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u)})$;

  2. (2)

the identity on
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u)})$ is the
pointwise limit of a bounded net $\\{\Phi_{i}\\}_{i\in I}$ of completely
positive $\mathrm{C_{0}}(\mathcal{A})$-bimodule maps
$\Phi_{i}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u)})\to
C(\mathcal{A}^{(\alpha,u)}|_{V_{i}})\subseteq\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u)})$
with ranges supported on precompact sets $V_{i}\subseteq\mathcal{G}$, $i\in
I$.

###### Proof.

If $\\{\varphi_{i}\\}_{i\in I}$ are as in (1) then the maps
$\\{m_{\varphi_{i}}^{\operatorname{r}}\\}_{i\in I}$ given by Corollary 8.7
satisfy the properties described in (2). Conversely, having a net
$\\{\Phi_{i}\\}_{i\in I}$ as in (2), the corresponding maps given by
Proposition 11.9 form a bounded net $\\{\varphi_{\Phi_{i}}\\}_{i\in
I}\subseteq\mathrm{C_{c}}(r^{*}Z\mathcal{M}(\mathcal{A}))$ of Herz–Schur
multipliers of positive type. Moreover, if
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u)})$ is supported on a bisection,
then $\sup_{g\in G}\|\varphi_{\Phi_{i}}(g)a(g)-a(g)\|_{\infty}=\sup_{g\in
G}\lim_{i}\|\Phi(a)(g)-a(g)\|_{\infty}=0$. Hence this holds for any
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u)})$. ∎

######  Remark 11.12.

For the trivial bundle $\mathcal{A}=\mathbb{C}\times X$, that is, the case of
just a twisted groupoid $(\mathcal{G},u)$ (see Example 4.12), the above result
reduces to amenability of the underlying groupoid $\mathcal{G}$, that is, the
existence of a net of positive-type functions $\\{\varphi_{i}\\}_{i\in
I}\subseteq\mathrm{C_{c}}(\mathcal{G})$ with $\varphi_{i}(g)\to 1$ uniformly
on compacts. This is just a re-statement of item (1) in the above proposition.
We know that amenability of $\mathcal{G}$ is equivalent to nuclearity of
$C^{*}(\mathcal{G},u)=C^{*}_{r}(\mathcal{G},u)$ (see Corollary 10.16), which
means that the identity map $C^{*}(\mathcal{G},u)\to C^{*}(\mathcal{G},u)$ can
be approximated (in the point-norm topology) by a composition of (contractive)
completely positive linear maps
$C^{*}(\mathcal{G},u)\to\mathbb{M}_{n_{i}}(\mathbb{C})\to
C^{*}(\mathcal{G},u)$. This is, in turn, equivalent the approximation property
appearing in item (2) of Proposition 11.11. We are not sure whether this has
already appeared in the literature in the way it is presented here.

###  11.3. Decomposable norm vs Fourier–Stieltjes norm

As we noted in the introduction to this section every map given by a
Fourier–Stieltjes multiplier, whether on the reduced or universal crossed
product, can be expressed as a linear combination of completely positive maps.
More interestingly, we can also relate the Fourier–Stieltjes norm of the
multiplier to the _decomposable_ norm of the associated map, as introduced in
[Haa85].

We first recall some basic relevant facts. Suppose that $A,B$ are
$\mathrm{C}^{*}$-algebras and $\Phi:A\to B$ is a linear map. Haagerup in
[Haa85] proved that following conditions are equivalent:

  1. (1)

$\Phi$ is _decomposable_ , i.e. it can be expressed as a linear combination of
completely positive maps from $A$ to $B$;

  2. (2)

$\Phi=\Phi_{1}-\Phi_{2}+i(\Phi_{3}-\Phi_{4})$ where $\Phi_{k}:A\to B$ are
completely positive for $k=1,\ldots,4$;

  3. (3)

there exists a completely positive map $\Psi:A\to\mathbb{M}_{2}(B)$ of the
form

$\Psi=\begin{pmatrix}\Psi_{1}&\Phi\\\ \Phi^{\dagger}&\Psi_{2}\end{pmatrix}$

Then the maps $\Psi_{1},\Psi_{2}:A\to B$ are necessarily completely positive.

Moreover, the linear space $DC(A,B)$ of decomposable maps from $A$ to $B$ with
the norm

$\|\Phi\|_{\mathrm{dec}}:=\sup\left\\{\max\\{\|\Phi_{1}\|,\|\Phi_{2}\|\\}:\Psi=\begin{pmatrix}\Phi_{1}&\Phi\\\
\Phi^{\dagger}&\Phi_{2}\end{pmatrix}\text{ is completely positive}\right\\}$

is a Banach space. The norm $\|\cdot\|_{\mathrm{dec}}$ dominates the
completely bounded norm, coincides with the latter on completely positive maps
and makes the set of decomposable maps a Banach space. It is also
submulticative with respect to composition, see [Haa85] for more details. Thus
$C^{*}$-algebras with decomposable maps as morphisms, equipped with
$\|\cdot\|_{\mathrm{dec}}$-norm, form a Banach category which contains the
category of $C^{*}$-algebras with morphisms being completely positive maps as
a Banach subcategory.

The next result can be viewed as an extension or continuation of Theorem 7.13.

######  Proposition 11.13.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of
$\mathcal{G}$. For every Fourier–Stieltjes multiplier $T\in
FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ the corresponding 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
$m_{T}^{\operatorname{f}}:\mathrm{C}^{*}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}(\mathcal{B}^{(\beta,u_{\beta})})$
are decomposable, and we have
$\|m_{T}\|_{cb}\leq\|m_{T}^{\operatorname{r}}\|_{\mathrm{dec}}\leq\|T\|_{FS}$,
$\|m_{T}\|_{cb}\leq\|m_{T}^{\operatorname{f}}\|_{\mathrm{dec}}\leq\|T\|_{FS}$.

If $T$ is positive-definite, then
$\|m_{T}\|_{cb}=\|m_{T}^{\operatorname{r}}\|_{\mathrm{dec}}=\|m_{T}^{\operatorname{f}}\|_{\mathrm{dec}}=\|T\|_{FS}$.

###### Proof.

Let $T:=T_{\mathcal{E},L,\xi,\zeta}=\\{T_{g}\\}_{g\in\mathcal{G}}$ be a
Fourier–Stieltjes multiplier. By rescaling the respective sections (passing to
$\sqrt{\frac{|\zeta\|}{\|\xi\|}}\xi$ and
$\sqrt{\frac{|\xi\|}{\|\zeta\|}}\zeta$) we can assume that
$\|\xi\|=\|\zeta\|$. Using Examples 3.6, 4.13 and 5.11 consider the
$\mathrm{C}^{*}$-correspondence $\mathbb{M}_{2}(\mathcal{E})$ from
$\mathcal{A}^{(\alpha^{(2)},u_{\alpha}^{(2)})}$ to
$\mathcal{B}^{(\beta^{(2)},u_{\beta}^{(2)})}$. Define the section
$\eta:=\xi\oplus\zeta\in\mathbb{M}_{2}(\mathcal{E})$ in the obvious ‘diagonal’
way and consider the associated Fourier–Stieltjes multiplier
$\tilde{T}:=T_{\mathbb{M}_{2}(\mathcal{E}),L^{(2)},\eta,\eta}$. The operator
$m_{\tilde{T}}^{\operatorname{r}}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha^{(2)},u_{\alpha}^{(2)})})\to\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B}^{(\beta^{(2)},u_{\beta}^{(2)})})$
is completely positive by Theorem 8.5. Recall that we have a natural
isomorphism

$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha^{(2)},u_{\alpha}^{(2)})})\cong\mathbb{M}_{2}(\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})}))$

(and its analogue for $\mathcal{B}^{(\beta,u_{\beta})}$). Define the map
$\Psi:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathbb{M}_{2}(\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B}^{(\beta,u_{\beta})}))$
via the formula

$\Psi(x)=m_{\tilde{T}}^{\operatorname{r}}\left(\begin{bmatrix}x&x\\\
x&x\end{bmatrix}\right),\qquad
x\in\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})}).$

As the map $x\mapsto\begin{bmatrix}x&x\\\ x&x\end{bmatrix}$ is completely
positive so is $\Psi$. It is now elementary to check that in fact

$\Psi=\begin{bmatrix}m_{T_{\mathcal{E},L,\xi,\xi}}^{\operatorname{r}}&m_{T}^{\operatorname{r}}\\\
({m_{T}^{\operatorname{r}}})^{\dagger}&m_{T_{\mathcal{E},L,\zeta,\zeta}}^{\operatorname{r}}\end{bmatrix}.$

Thus, since $m_{T_{\mathcal{E},L,\zeta,\zeta}}^{\operatorname{r}}$ and
$m_{T_{\mathcal{E},L,\xi,\xi}}^{\operatorname{r}}$ are completely positive,
$m_{T}^{\operatorname{r}}$ is decomposable. Moreover, as we rescaled to
$\|\xi\|=\|\zeta\|$, by Theorem 8.5 we have
$\|m_{T_{\mathcal{E},L,\zeta,\zeta}}^{\operatorname{r}}\|=\|m_{T_{\mathcal{E},L,\xi,\xi}}^{\operatorname{r}}\|=\|\xi\|\|\zeta\|$.
Thus $\|m_{T}^{\operatorname{r}}\|_{\mathrm{dec}}\leq\|\xi\|\cdot\|\zeta\|$,
which shows $\|m_{T}^{\operatorname{r}}\|_{\mathrm{dec}}\leq\|T\|_{FS}$. The
same argument gives
$\|m_{T}^{\operatorname{f}}\|_{\mathrm{dec}}\leq\|T\|_{FS}$. The other
statements follow easily from Theorem 7.13 and the discussion before the
proposition. ∎

######  Remark 11.14.

One could similarly show that Fourier multipliers yield decomposable ‘reduced-
to-full’ maps, as studied in Theorem 9.4.

The above result naturally raises a question whether we can identify the
decomposable norm of $m_{T}$ (say on the reduced crossed product) with the
norm $\|T\|_{FS}$. In the context of the usual Fourier–Stieltjes algebras of
discrete groups such an identification was noted only very recently in [AK22].
Below we will show that it remains true in the context of twisted crossed
products of _discrete_ groupoids. Before we do that we need a simple (and
likely well-known) lemma.

######  Lemma 11.15.

Let $A$, $B$ be $\mathrm{C}^{*}$-algebras, $n\in\mathbb{N}$. If $F$ is a
$\mathrm{C}^{*}$-correspondence from $A$ to $B$, then the direct sum
$E:=\bigoplus_{i=1}^{n}F$ is a $\mathrm{C}^{*}$-correspondence from $A$ to
$\mathbb{M}_{n}(B)$, when equipped with the diagonal left action of $A$ and
the scalar product and the action of $\mathbb{M}_{n}(B)$ given by

$\begin{gathered}\big{\langle}(\xi_{i})_{i=1}^{n},(\zeta_{i})_{i=1}^{n}\big{\rangle}_{\mathbb{M}_{n}(B)}:=\left(\langle\xi_{i},\zeta_{j}\rangle\right)_{i,j=1}^{n},\\\
(\xi_{i})_{i=1}^{n}\cdot\left(b_{ij}\right)_{i,j=1}^{n}:=\left(\sum_{j=1}^{n}\xi_{j}b_{ji}\right)_{i=1}^{n},\end{gathered}$

where $(\xi_{i})_{i=1}^{n},(\zeta_{i})_{i=1}^{n}\in\bigoplus_{i=1}^{n}F$,
$\left(b_{ij}\right)_{i,j=1}^{n}\in\mathbb{M}_{n}(B)$.

Conversely, every $\mathrm{C}^{*}$-correspondence from $A$ to
$\mathbb{M}_{n}(B)$ (up to unitary equivalence) arises in the way described
above from a $\mathrm{C}^{*}$-correspondence from $A$ to $B$.

###### Proof.

Assume first that $A$ and $B$ are unital. The proof amounts to a careful check
of algebraic formulas, so we will only indicate the main steps needed to
establish the second statement.

Suppose that $E$ is a $\mathrm{C}^{*}$-correspondence from $A$ to
$\mathbb{M}_{n}(B)$. Denote by $e_{ij}$ ($i,j=1,\ldots,n$) matrix units in
$\mathbb{M}_{n}(B)$. Define $F=Ee_{11}$, and let $\phi:B\to
e_{11}\mathbb{M}_{n}(B)e_{11}$ be the obvious isomorphism. It is easy to check
that $F$ becomes a $\mathrm{C}^{*}$-correspondence from $A$ to $B$, when
equipped with the scalar product

$\langle f_{1},f_{2}\rangle_{B}=\phi^{-1}(\langle f_{1},f_{2}\rangle),$

and the action

$f_{1}\cdot b:=f_{1}\phi(b),$

where $f_{1},f_{2}\in F,b\in B$; note that the left action of $A$ is
unchanged. When convenient, we will write the corresponding identification map
from $Ee_{11}$ to $F$ as $\psi_{1}$. Now given $j\in\\{2,\ldots,n\\}$ we
define a map $\psi_{j}:Ee_{jj}\to F$ simply by putting

$\psi_{j}(e)=ee_{j1},\qquad e\in Ee_{jj}.$

Finally we define the map $\Psi:E\to\bigoplus_{i=1}^{n}F$, with the right hand
side understood now as a $\mathrm{C}^{*}$-correspondence from $A$ to
$\mathbb{M}_{n}(B)$ defined in accordance with the first statement in the
lemma, setting

$\Psi(e)=\big{(}\psi_{i}(ee_{ii})\big{)}_{i=1}^{n}.$

We leave it to the diligent reader to verify that $\Psi$ is indeed a unitary
equivalence.

Consider now the case where $A$ and $B$ need not be unital. Let $E$ be again a
$\mathrm{C}^{*}$-correspondence from $A$ to $\mathbb{M}_{n}(B)$. Then
$\mathcal{M}(E)$ is a $\mathrm{C}^{*}$-correspondence from $\mathcal{M}(A)$ to
$\mathcal{M}(\mathbb{M}_{n}(B))\cong\mathbb{M}_{n}(\mathcal{M}(B))$. By the
unital case (and its proof) we have
$\mathcal{M}(E)\cong\bigoplus_{i=1}^{n}\tilde{F}$, where $\tilde{F}$ is a
$\mathrm{C}^{*}$-correspondence from $\mathcal{M}(A)$ to $\mathcal{M}(B)$. Set
$F=\tilde{F}B\subset\tilde{F}$. It is then a matter of careful check that the
explicit isomorphism from $\mathcal{M}(E)$ to $\bigoplus_{i=1}^{n}\tilde{F}$
restricts to an isomorphism from $E$ to $\bigoplus_{i=1}^{n}F$. ∎

We will now formulate a version of the above result for
$\mathrm{C}^{*}$-correspondence bundles.

######  Corollary 11.16.

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 $\mathcal{E}$ be a
$\mathrm{C}^{*}$-$\mathcal{A}$-$\mathbb{M}_{n}(\mathcal{B})$-correspondence
bundle and let $n\in\mathbb{N}$. Then $\mathcal{E}$ is unitarily equivalent to
the direct sum $\bigoplus_{i=1}^{n}\mathcal{F}$, where $\mathcal{F}$ is a
$\mathrm{C}^{*}$-$\mathcal{A}$-$\mathcal{B}$-correspondence bundle, with
pointwise operations described in Lemma 11.15. If moreover $\tilde{L}$ is an
equivariant representation of $\mathcal{G}$ on $\mathcal{E}$, then we may
assume that we also have an equivariant representation $L$ of $\mathcal{G}$ on
$\mathcal{F}$ and in the equivalence above $\tilde{L}_{g}$ corresponds to a
direct sum of $L_{g}$.

###### Proof.

For the first part it suffices to note that the isomorphisms introduced in
Lemma 11.15 are clearly continuous with respect to the relevant ‘bundle’
operations.

For the second we can argue as in the proof of Lemma 11.15, using the second
condition in Definition 5.1 and the form of matrix liftings described in
Example 4.13 (it essentially suffices to observe that in the unital case for
every $g\in\mathcal{G}$ the operator $\tilde{L}_{g}$ maps $E_{s(g)}e_{11}$
into $E_{r(g)}e_{11}$). Once again we leave the check to the reader. ∎

We are now ready to formulate the main decomposability result.

######  Theorem 11.17.

Let $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ be twisted actions of a
discrete groupoid $\mathcal{G}$ respectively on $\mathcal{A}$ and
$\mathcal{B}$. Then for every Fourier–Stieltjes multiplier $T\in
FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ we have

$\|m_{T}^{\operatorname{r}}\|_{\mathrm{dec}}=\|m_{T}^{\operatorname{f}}\|_{\mathrm{dec}}=\|T\|_{FS}.$

###### Proof.

In view of Proposition 11.13 it suffices to prove the inequalities
$\|m_{T}^{\operatorname{r}}\|_{\mathrm{dec}},\|m_{T}^{\operatorname{f}}\|_{\mathrm{dec}}\geq\|T\|_{FS}$,
$\|m_{T}^{\operatorname{r}}\|_{\mathrm{dec}}\geq\|T\|_{FS}$. We will prove
that $\|m_{T}^{\operatorname{r}}\|_{\mathrm{dec}}\geq\|T\|_{FS}$, as the
second inequality follows in the same way. Write then simply $m_{T}$ for
$m_{T}^{\operatorname{r}}$ and assume that

$\|m_{T}\|_{\mathrm{dec}}\leq 1.$

We will show that $\|T\|_{FS}\leq 1$.

By the definition of the decomposable norm the displayed inequality means that
there exists a completely positive map
$\Psi:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathbb{M}_{2}\left(\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B}^{(\beta,u_{\beta})})\right)$
such that

$\Psi=\begin{pmatrix}\Phi_{1}&m_{T}\\\ m_{T}^{\dagger}&\Phi_{2}\
\end{pmatrix},$

with
$\Phi_{1},\Phi_{2}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B}^{(\beta,u_{\beta})})$
completely positive and such that $\|\Phi_{1}\|\leq 1$, $\|\Phi_{2}\|\leq 1$.
Let us now apply Lemma 11.1 to the map $\Psi$, viewed now as a map from
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ to
$\mathrm{C}^{*}_{\operatorname{red}}(\mathbb{M}_{2}(\mathcal{B})^{(\beta^{(2)},u_{\beta}^{(2)})})$.
This produces a new completely positive map
$\tilde{\Psi}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}_{\operatorname{red}}(\mathbb{M}_{2}(\mathcal{B})^{(\beta^{(2)},u_{\beta}^{(2)})})$,
which comes from a multiplier. In fact, as the Haagerup trick acts on matrices
‘entrywise’, and does not change maps which are already multipliers, we have

$\tilde{\Psi}=\begin{pmatrix}\tilde{\Phi}_{1}&m_{T}\\\
m_{T}^{\dagger}&\tilde{\Phi}_{2}\end{pmatrix},$

with
$\tilde{\Phi}_{1},\tilde{\Phi}_{2}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B}^{(\beta,u_{\beta})})$
contractive completely positive maps. Applying Lemma 10.1 we may approximate
$\tilde{\Psi}$ by strict multipliers. Namely, for an approximate unit
$\\{e_{i}\\}_{i\in I}$ in $\mathrm{C_{0}}(\mathcal{A})$, Lemma 10.1 gives a
net $\\{\tilde{T}_{i}\\}_{i\in I}\in
FS[(\alpha,u_{\alpha}),(\beta^{(2)},u_{\beta}^{(2)})]$ such that for every
$i\in I$ we have
$\tilde{\Psi}(e_{i}ae_{i})=m_{\tilde{T}_{i}}^{\operatorname{r}}(a)$,
$a\in\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$.
Then

$m_{\tilde{T}_{i}}^{\operatorname{r}}=\begin{pmatrix}\tilde{\Phi}_{1,i}&m_{T_{i}}\\\
m_{T_{i}}^{\dagger}&\tilde{\Phi}_{2,i}\end{pmatrix},$

where
$\tilde{\Phi}_{1,i},\tilde{\Phi}_{2,i}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B}^{(\beta,u_{\beta})})$
are contractive completely positive maps (coming from Fourier–Stieltjes
multipliers) and $\\{T_{i}\\}_{i\in I}\in
FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ are such that
$\|T_{i}\|_{FS}\to\|T\|_{FS}$ (again by Lemma 10.1).

Fix for the moment $i\in I$. By Theorem 8.5 the map $\tilde{\Phi}_{i}$ is
given by a Fourier–Stieltjes multiplier $\tilde{T}_{i}\in
FS[(\alpha,u_{\alpha}),(\beta^{(2)},u_{\beta}^{(2)})]$,
$\tilde{T}=T_{\mathcal{E},L,\xi,\xi}$ for some
$\mathrm{C}^{*}$-$\mathcal{A}$-$\mathbb{M}_{2}(\mathcal{B})$-correspondence
$\mathcal{E}$ equipped with an equivariant representation $\tilde{L}$ of
$\mathcal{G}$ and section
$\xi\in\mathrm{C_{b}}(\mathcal{M}(\mathbb{M}_{2}(\mathcal{E})))$. Applying
Theorem 11.15 to $\mathcal{E}$, $\tilde{L}$ and $\xi$ we may identify
$\mathcal{E}$ with $\mathcal{F}\oplus\mathcal{F}$, $\tilde{L}$ with a direct
sum of two copies of an equivariant representation $L$ of $\mathcal{G}$ on
$\mathcal{F}$, and $\xi:=(\zeta,\eta)$ for two sections
$\zeta,\eta\in\mathrm{C_{b}}(\mathcal{M}(\mathcal{F})$. Using this
identification we obtain

$T_{i}=T_{L,F,\zeta,\eta},\;\;\tilde{\Phi}_{1,i}=m_{T_{L,F,\zeta,\zeta}},\qquad\tilde{\Phi}_{2,i}=m_{T_{L,F,\eta,\eta}}.$

Thus finally applying the last part of Theorem 8.5 respectively to
$\Phi_{1,i}$ and $\Phi_{2,i}$ we deduce that $\|\zeta\|,\|\eta\|\leq 1$. But
this means that $\|T_{i}\|_{FS}\leq 1$, which ends the proof, as
$\|T_{i}\|_{FS}\to\|T\|_{FS}$. ∎

######  Remark 11.18.

In the context of the last theorem we could define a version of Haagerup’s
decomposable norm for $\mathrm{C_{0}}(\mathcal{A})$-bimodule maps, say
$\|\cdot\|_{\mathrm{dec},\mathrm{bim}}$, requesting that all the entries of
the requested 2 by 2 matrix decomposition are bimodule maps. Then the
arguments of the above theorem together with the Haagerup trick from
Proposition 11.9 would show that even for not neccessarily discrete
$\mathcal{G}$ and arbitrary Herz–Schur type multiplier $T$ we have
$\|T\|_{FS}=\|m_{T}^{\operatorname{r}}\|_{\mathrm{dec},\mathrm{bim}}=\|m_{T}^{\operatorname{f}}\|_{\mathrm{dec},\mathrm{bim}}$.

######  Remark 11.19.

In contrast to Remark 7.14, where we only had contractive functors, assuming
that $\mathcal{G}$ is discrete we get isometric involutive
$\mathrm{C_{b}}(X)$-bimodule functors
$m^{\operatorname{f}}:FS_{TA}(\mathcal{G})\to DC_{TA}(\mathcal{G})$ and
$m^{\operatorname{r}}:FS_{TA}(\mathcal{G})\to
DC_{TA}^{\operatorname{r}}(\mathcal{G})$ by letting $DC_{TA}(\mathcal{G})$
(resp. $DC_{TA}^{\operatorname{r}}(\mathcal{G})$) to be the category whose
objects are twisted actions of $\mathcal{G}$ on $\mathrm{C}^{*}$-bundles and
morphisms are decomposable maps between full (resp. reduced) crossed products
that preserve fibers and are equipped with the norm
$\|\cdot\|_{\mathrm{dec}}$.

As an application (though formally we will use only that the norms
$\|\cdot\|_{\mathrm{dec}}$ and $\|\cdot\|_{FS}$ are ‘equivalent’) we
characterise the Fourier–Stieltjes AP for discrete groupoid actions in a
similar fashion as we did for the positive Fourier–Stieltjes AP in Proposition
11.2 and Corollary 11.3.

######  Proposition 11.20.

Let $\mathcal{A}$, $\mathcal{B}$ be $\mathrm{C}^{*}$-bundles over $X$,
carrying twisted actions $(\alpha,u_{\alpha})$ and $(\beta,u_{\beta})$ of a
discrete gropupoid $\mathcal{G}$. A multiplier $T$ from
$\mathcal{A}^{(\alpha,u_{\alpha})}$ to $\mathcal{B}^{(\beta,u_{\beta})}$ has
the Fourier–Stieltjes approximation property if and only if there is a
$\|\cdot\|_{\mathrm{dec}}$-bounded net $\\{\Phi_{i}\\}_{i\in I}$ of completely
bounded maps
$\Phi_{i}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to
C(\mathcal{B}^{(\beta,u_{\beta})}|_{V_{i}})\subseteq\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{B}^{(\beta,u_{\beta})})$
with ranges supported in finite sets $V_{i}\subseteq\mathcal{G}$, $i\in I$,
and such that $\Phi_{i}(a)\to m_{T}(a)$ for every
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$.

###### Proof.

If $\\{T_{i}\\}_{i\in I}\subseteq FS[(\alpha,u_{\alpha})]$ is a net witnessing
the positive Fourier–Stieltjes approximation property for $T$, then the
associated maps $\\{m_{T_{i}}^{\operatorname{r}}\\}_{i\in I}$, given by
Theorem 7.13, have the desired properties by Proposition 11.13 and Corollary
9.9. Conversely, assume that $\\{\Phi_{i}\\}_{i\in I}$ is a net of
decomposable maps such that $M:=\sup_{i\in
I}\|\Phi_{i}\|_{\mathrm{dec}}<\infty$ and
$\Phi_{i}:\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})\to
C(\mathcal{B}^{(\beta,u_{\beta})}|_{V_{i}})$ where $V_{i}$ is finite, $i\in
I$. We may write each $\Phi_{i}$ as a linear combination
$\sum_{k=0}^{3}i^{k}\Phi_{i,k}$ (here $i^{k}$ is the imaginary unit to the
$k$th power, not an index of the set $I$) where each $\Phi_{i,k}$ is
completely positive and $\|\Phi_{i,k}\|\leq\|\Phi_{i}\|_{\mathrm{dec}}\leq M$,
see the formulas in [Haa85, Page 175]. Applying Lemma 11.1 to $\Phi_{i}$ and
$\Phi_{i,k}$ we obtain completely positive maps $\widetilde{\Phi}_{i}$ and
$\widetilde{\Phi}_{i,k}$ coming from multipliers and such that
$\|\widetilde{\Phi}_{i,k}\|\leq\|\Phi_{i,k}\|\leq M$ and
$\widetilde{\Phi}_{i}=\sum_{k=0}^{3}i^{k}\widetilde{\Phi}_{i,k}^{(j)}$.
Choosing an approximate unit $\\{e_{j}\\}_{j\in J}$ in
$\mathrm{C_{0}}(\mathcal{A})$ and ‘strictifying’ the considered maps by
putting $\widetilde{\Phi}_{i}^{(j)}(a):=\widetilde{\Phi}_{i}(e_{j}ae_{j})$ and
$\widetilde{\Phi}_{i,k}^{(j)}(a):=\Phi_{i,k}(e_{j}ae_{j})$ for
$a\in\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$,
by Lemma 11.1 we get that $\widetilde{\Phi}_{i,k}^{(j)}=m_{T_{i,k}^{(j)}}$ for
$T_{i,k}^{(j)}\in FS((\alpha,u_{\alpha}),(\beta,u_{\beta}))^{+}$ and
$\|T_{i,k}^{(j)}\|_{FS}\leq\|\widetilde{\Phi}_{i,k}\|\leq M$. Thus putting
$T_{i}^{(j)}=\sum_{k=0}^{3}i^{k}T_{i,k}^{(j)}$ we obtain a bounded net
$\\{T_{i}^{(j)}\\}_{(i,j)\in I\times J}\subseteq
FS((\alpha,u_{\alpha}),(\beta,u_{\beta}))$ with $\sup_{(i,j)\in I\times
J}\|T_{i}^{(j)}\|_{FS}\leq 4M$. Moreover, for
$a\in\mathrm{C_{c}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ we have
$m_{T_{i}^{(j)}}(a)\to m_{T}(a)$ because $\Phi_{i}(a)\to m_{T}(a)$ and
$\widetilde{\Phi}_{i}^{(j)}(a)\to\Phi_{i}(a)$. ∎

######  Corollary 11.21.

A twisted groupoid action $(\alpha,u_{\alpha})$ of a discrete groupoid
$\mathcal{G}$ on a $\mathrm{C}^{*}$-bundle $\mathcal{A}$ has the
Fourier–Stieltjes approximation property if and only if there is a
$\|\cdot\|_{\mathrm{dec}}$-bounded net $\\{\Phi_{i}\\}_{i\in I}$ of
decomposable maps on
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ that
converge pointwise to the identity and that have ranges supported on finite
subsets of $\mathcal{G}$.

##  12\. Open questions and further perspectives

In this final section of the paper, we would like to discuss the key questions
left open in our work and present some of the perspectives for further
research regarding Fourier–Stieltjes algebras and their applications in the
context of crossed products by twisted groupoid actions.

### Open questions

For the first three questions we work with fixed $(\alpha,u_{\alpha})$ and
$(\beta,u_{\beta})$, twisted actions of a discrete groupoid $\mathcal{G}$ on
$\mathrm{C}^{*}$-bundles $\mathcal{A}$ and $\mathcal{B}$.

######  Question 12.1.

If $T\in F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ is a Fourier multiplier, is
it true that $\|T\|_{FS}=\|T\|_{F}$?

Note that we always have $\|T\|_{FS}\leq\|T\|_{F}$. The equality
$\|T\|_{FS}=\|T\|_{F}$ is known to hold in the classic case for discrete
groups, see [Eym64]; one of possible ways of approaching this question relates
to the ‘functional’ picture of Fourier–Stieltjes and Fourier algebras
mentioned in the introduction and missing in our context. In general we do not
also know if $F(\alpha,u_{\alpha})$ is closed in $FS(\alpha,u_{\alpha})$. Note
that the issue appears already in the context of groupoid
$\mathrm{C}^{*}$-algebras. To avoid that, in [Ren97, Definition 1.4], Renault
defines the Fourier algebra to be a closure in the Fourier–Stieltjes algebra
of the coefficients given by the regular representation, and this convention
is also adopted by Oty [Oty99]. We decided not to do this here.

######  Question 12.2.

Is it true that $F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]\cap
FS[(\alpha,u_{\alpha})(\beta,u_{\beta})]^{+}=F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]^{+}$?

If $T\in F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]^{+}$, then $T$ is positive-
definite, by Theorem 8.5. However this theorem does not imply that every
positive-definite Fourier multiplier is in
$F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]^{+}$, as a priori the associated
cyclic equivariant representation of $\mathcal{G}$ might not be regular. On
the other hand, if we denote by
$FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]^{+}$ the set of Fourier–Stieltjes
multipliers that can represented in the form $T_{\mathcal{E},L,\xi,\xi}$, then
Theorem 8.5 states that $FS[(\alpha,u_{\alpha}),(\beta,u_{\beta})]^{+}$
consists exactly of positive-definite, bounded, strict multipliers from
$\mathcal{A}^{(\alpha,u_{\alpha})}$ to $\mathcal{B}^{(\beta,u_{\beta})}$. Thus

$F[(\alpha,u_{\alpha}),(\alpha,u_{\alpha})]^{+}\subseteq
F[(\alpha,u_{\alpha}),(\alpha,u_{\alpha})]\cap
FS[(\alpha,u_{\alpha}),(\alpha,u_{\alpha})]^{+}$

and we do not know whether the reverse inclusion holds.

######  Question 12.3.

Is it true that $F[(\alpha,u_{\alpha}),(\beta,u_{\beta})]$ can be described as
the closure of compactly supported Fourier–Stieltjes multipliers? Either in
$\|\cdot\|_{F}$ or $\|\cdot\|_{FS}$?

The statement above is true in the context of discrete groups, as shown
already in [Eym64]. The analogous statement in the context of groupoid
$\mathrm{C}^{*}$-algebras appears in [Ren87], but without proof (only an
indication of an argument based on the partitions of unity; we do not see
however how to use it to obtain the desired convergence, although see
Proposition 10.3).

######  Question 12.4.

Can one find an example of a Fourier–Stieltjes multiplier $T$ such that
$\|m_{T}^{\operatorname{r}}\|_{\mathrm{dec}}<\|T\|_{FS}$?

Theorem 11.17 shows that this cannot happen for _discrete_ $\mathcal{G}$; the
proof of that theorem shows that any counterexample would imply a failure of
an even weaker form of the Haagerup trick. We do not however know the answer
to the question above even for groupoid $\mathrm{C}^{*}$-algebras.

Note that the problems above are clearly interrelated. For example a positive
answer to Question 12.2 would yield via Remark 11.18, a positive answer to
Question 12.1 for Herz–Schur type multipliers.

The following questions are related to the approximation properties of
actions/crossed products, and matters discussed in Section 10.

######  Question 12.5.

In what generality can one invert the arrows in the diagram in Remark 10.14?

Some of the reverse implications hold in certain special cases, for example
for discrete groupoids (see Theorem 11.7) but are not clear in general. For
instance, strong amenability implies Exel’s AP (which is in turn equivalent to
“amenability”) for group actions on $\mathrm{C}^{*}$-algebras, see [BEW23,
Theorem 4.9]. On the other hand, for group actions the implication from weak
containment to any of the approximation properties is generally a big mystery
although it holds in certain special cases, and there are examples of
groupoids that satisfy the weak containment property but are not amenable, see
[Wil15].

######  Question 12.6.

Does nuclearity of the reduced crossed product
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$ imply
one of the approximation properties in the diagram in Remark 10.14?

Strong amenability in general is not necessary for nuclearity of
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$,
which follows for instance from Suzuki’s examples of group actions on unital
simple $C^{*}$-algebras in [Suz19]. But it might be relevant when the algebra
$\mathrm{C_{0}}(\mathcal{A})$ is commutative, as for group actions, strong
amenability and amenability are equivalent, see [AD87, Théorème 4.9].

By Theorem 11.7 we know that nuclearity of
$\mathrm{C}^{*}_{\operatorname{red}}(\mathcal{A}^{(\alpha,u_{\alpha})})$
implies the positive Fourier–Stieltjes AP when $\mathcal{G}$ is discrete. When
$\mathcal{G}=G$ is a group nuclearity of the crossed product implies the
formally stronger Exel’s AP, see [BEW23, Corollary 4.11] and Remark 11.8. This
makes us tend to think that nuclearity may imply Exel’s AP for étale gropouid
actions.

### Perspectives

One of the deepest modern applications of the Fourier and Fourier–Stieltjes
algebras in the classical context of discrete groups is related to using these
algebras and their properties as means of distinguishing the corresponding
group von Neumann algebras. A key role is played here by the space of
_completely bounded_ multipliers of the Fourier algebra of $\Gamma$, denoted
$M_{cb}(A(\Gamma))$, which contains the Fourier–Stieltjes algebra $B(\Gamma)$
and can be used to define finer, often quantitative approximation properties
of $\Gamma$ (see for example [CH89], [HK94]). The starting point to study such
questions in our context (or even in the context of actions of discrete groups
on $\mathrm{C}^{*}$-algebras) one would need to

  * •

investigate the operator space structure of our Fourier and Fourier–Stieltjes
algebras;

  * •

analyze the corresponding space of (completely) bounded multipliers of the
Banach algebra $F[(\alpha,u),(\alpha,u)]$.

The first step is related to Question 12.1; we would expect the relevant cb-
multiplier norm to coincide both with the Fourier norm and Fourier–Stieltjes
norm of a given element of the Fourier algebra. Note also [Ren87, Theorem
3.1], which in the groupoid case (trivial action) expresses completely bounded
multipliers on the Fourier algebra of $\mathcal{G}$ via _invariant_ elements
of the Fourier–Stieltjes algebra of the product groupoid
$\mathcal{G}*\mathcal{G}$. We expect that in general it might be useful in
this context to use the language of _(dynamical) quantum groupoids_ , as
introduced for example in [ET16].

Another natural direction of study concerns attempting to reconstruct a given
groupoid dynamical system from the associated Fourier–Stieltjes or Fourier
algebra. It is well known that from the Fourier algebra $A(\Gamma)$ of a
discrete group $\Gamma$ one can reconstruct $\Gamma$ — as a set — via the
spectrum (space of characters) of $A(\Gamma)$. In the articles [Ren97] or
[RW97], dealing with the groupoid operator algebras, analogous questions are
often phrased as the study of _duality for étale groupoids_. This is closely
related to the issue of identifying the Fourier or Fourier–Stieltjes algebras
with the dual spaces (of some sort) associated with the relevant operator
algebras. As we mentioned in the introduction, in general this is far from
clear, and even the case of groupoid $\mathrm{C}^{*}$-algebras is not
straightforward, as exemplified by [Ren87, Theorem 2.3].

Concrete questions one can study are the following:

  * •

what equivalence relation between (pairs of) twisted groupoid actions leads to
the isometric isomorphism of the corresponding Fourier–Stieltjes
spaces/algebras?

  * •

Under what conditions one can for example indeed reconstruct a given twisted
groupoid action, up to the equivalence mentioned above, knowing its associated
Fourier–Stieltjes algebra?

In the first part one should mention the Morita equivalence of
$\mathrm{C}^{*}$-dynamical systems, discussed in [BC21, Section 4] and shown
to yield — under certain restrictive assumptions — isometric isomorphism of
the associated Fourier-Stieltjes algebras. In our context this can be of
course further enriched by looking at different, but equivalent, groupoids, in
the spirit of [Sim20, Section 3.4]. The second question is naturally far more
difficult, and even partial results would be of great interest.

The two broad topics mentioned above clearly do not exhaust the scope of
natural questions one can ask regarding our Fourier–Stieltjes algebras. Just
to quote significant body of work in the group case, as presented in [KL18],
one could also ask about the general form of (completely) bounded
homomorphisms between Fourier–Stieltjes algebras or about the behaviour of
these algebras with respect to the restriction to subgroupoids, related to
classical spectral synthesis questions. This brings us to the possibility of
dropping the étale assumption, and considering for example the crossed
products by actions of arbitrary locally compact groups. The last step
naturally would take us far beyond the scope of current work.

### Acknowledgements

AB was partially supported by CNPq/CAPES-Print - Brazil. AS was partially
supported by the National Science Center (NCN) grant no. 2020/39/I/ST1/01566.
BKK was supported by the National Science Center (NCN) grant no.
2019/35/B/ST1/02684.

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