$\newcommand{\Cstar}{C^*_{\text{red}}}\newcommand{\G}{\mathscr G}\newcommand{\H}{\mathscr H}$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) 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).