$\newcommand{\Cstar}{C^\*\_{\text{red}}}\newcommand{\G}{\mathscr G}\newcommand{\H}{\mathscr H}$Let
$\G$ be an etale groupoid, let $U$ be an open subset of $\G^{(0)}$, and let
$$
 \H = \{\gamma \in \G: r(\gamma ), s(\gamma )\in U\}
 $$
be the reduction of $\G$ to $U$.


It is well known that any element of $\Cstar(\G)$ may be seen as a function defined on $\G$, via a process resembling
the Fourier transform (ask me if you'd like more details on this).


Viewing elements of $\Cstar(\G)$ as functions, according to this,
it is not hard to see that every $f$ in the range of the natural embedding
$$
 \Cstar(\H) \hookrightarrow \Cstar(\G)
 $$
vanishes off $\H$.


**Question**. If $f$ is an element in $\Cstar(\G)$ vanishing off $\H$, can we guarantee that $f$ lies in the range of
the natural embedding above?


I am mostly interested in the case in which $U$ is the complement of a point in $\G^{(0)}$, so you are more than
welcome to assume this is the case.


I believe I know how to prove this in case $\G$ is amenable, or even if we simply assume that the full and reduced
groupoid C$^\*$-algebras of $\G$ coincide. I am therefore really looking for a proof in the non-amenable case.


Just like my previous question ([Restricting a function defined on an étale groupoid to an isotropy group](https://mathoverflow.net/questions/389504/restricting-a-function-defined-on-an-%c3%a9tale-groupoid-to-an-isotropy-group)) whose bounty expired recently without any answer $\ddot\frown$, I'd be happier if the answer covered
the case in which the groupoid is twisted and not necessarily Hausdorff (with a Hausdorff unit space).



Is it possible to characterize the elements of the C$^*$-algebra of an open subgroupoid?
I'm not an expert of groupoids and not sure what "Fourier coefficients" exactly means, but perhaps Wassermann's classical counterexample ([mr1127480](https://mathscinet.ams.org/mathscinet-getitem?mr=1127480)) serve (against) the purpose?
Take an infinite residually finite Kazhdan (T) group $\Gamma$ with finite quotients denoted by $\Gamma\_n$ and consider the unitary representation $\pi:=\bigoplus\lambda\_n$ on $\bigoplus\ell\_2\Gamma\_n$.
Then, groupoids $\mathcal G$ and $\mathcal H$ can be constructed in such a way that
$$C^\*\_r{\mathcal G}=C^\*(\pi(\Gamma))+\bigoplus B(\ell\_2\Gamma\_n)\subset B(\bigoplus\ell\_2\Gamma\_n)$$ and $C^\*\_r({\mathcal H})=\bigoplus B(\ell\_2\Gamma\_n)$
(or whole compact operators in place of the block-diagonal compact operators $\bigoplus B(\ell\_2\Gamma\_n)$).
Denote by $p\_n$ the rank-one projection corresponding to $1\_{\Gamma\_n} \in \ell\_2\Gamma\_n$, which is the $\Gamma\_n$-by-$\Gamma\_n$ matrix with all entries $|\Gamma\_n|^{-1}$.
Then the element $p:=\sum\_n^\oplus p\_n$ belongs to $C^\*\_r({\mathcal G})$ by property (T),
but not to $C^\*\_r({\mathcal H})$ although each "fiber" of $p$ does.


