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

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I'm not an expert of groupoids and not sure what "Fourier coefficients" exactly means, but perhaps Wassermann's classical counterexample (mr1127480) 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.

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  • $\begingroup$ Thanks a lot for your answer and also for the suggestion to look at the case of group bundles. This took me back to Rufus Willet's example of a non-amenable groupoid whose maximal and reduced C$^*$-algebras coincide (Münster J. of Math. 8 (2015), 241--252), which also provides a counter example to my question. However Willet's example is based on $\mathbb F_2$, and his Remark 2.9.(i) seems to indicate that his construction will fail for $\text{SL}(3,{\mathbb Z})$, so I am still puzzling if a residually finite group with property (T) is the right choice. $\endgroup$
    – Ruy
    Apr 19, 2021 at 15:57
  • $\begingroup$ In any case I am still studying your example which is perhaps not exactly along the same lines. $\endgroup$
    – Ruy
    Apr 19, 2021 at 15:58
  • $\begingroup$ Regarding Fourier coefficients I guess I shouldn't have used that terminology since it is perhaps not standard. All I mean is that the evaluation maps $f\in C_c(\mathscr G) \mapsto f(\gamma )\in {\mathbb C}$ extend continuously to $C^*_{\text{red}}(\mathscr G)$. $\endgroup$
    – Ruy
    Apr 19, 2021 at 16:54

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