June 11, 2026
Let \(p \colon \mathcal{A} \to G\) be a Fell bundle over a locally compact Hausdorff second countable groupoid \(G\) equipped with a Haar system, and let \(\Gamma\) be a discrete group. Given a continuous \(1\)-cocycle \(c \colon G \to \Gamma\), we show that the \(\mathrm{C}^*\)-algebra of the restricted Fell bundle \(\mathcal{A}|_{G_e}\) embeds isometrically into \(\mathrm{C}^*(G;\mathcal{A})\), where \(G_e = c^{-1}(e)\) is the clopen subgroupoid corresponding to the identity element. We exploit this embedding to show that \(\mathrm{C}^*(G;\mathcal{A})\) admits a natural structure of a topologically graded \(\mathrm{C}^*\)-algebra in the sense of Exel. As a consequence, we obtain a canonical coaction \(\delta\) of \(\Gamma\) on \(\mathrm{C}^*(G; \mathcal{A})\). We further show that the associated coaction crossed product \(\mathrm{C}^*(G; \mathcal{A})\rtimes_\delta \Gamma\) is naturally isomorphic to the \(\mathrm{C}^*\)-algebra of a Fell bundle constructed from the cocycle data.
Fell bundles over groupoids introduced by Kumjian [1], provide a unified framework for encoding a wide range of constructions in operator algebras, including twisted groupoid \(\mathrm{C}^*\)-algebras, higher-rank graph \(\mathrm{C}^*\)-algebras and \(\mathrm{C}^*\)-dynamical systems.
Let \(G\) be a locally compact Hausdorff second countable groupoid equipped with a Haar system, and let \(\Gamma\) be a discrete group. A continuous \(1\)-cocycle \(c \colon G \to \Gamma\) induces a decomposition \(G = \bigsqcup_{\gamma \in \Gamma} G_\gamma\), where \(G_\gamma = c^{-1}(\gamma)\). The identity fibre \(G_e\) plays a distinguished role: its \(\mathrm{C}^*\)-algebra embeds canonically into \(\mathrm{C}^*(G)\) ([2]), and this inclusion realizes \(\mathrm{C}^*(G)\) as a topologically graded \(\mathrm{C}^*\)-algebra in the sense of Exel [3].
In this paper, we extend this picture to Fell bundles. Let \(p \colon \mathcal{A}\to G\) be a Fell bundle and let \(c \colon G \to \Gamma\) be a continuous \(1\)-cocycle. Restricting the Fell bundle \(\mathcal{A}\) to the fibres \(G_{\gamma}\) yields upper semicontinuous Banach bundles \(\mathcal{A}|_{G_\gamma}\), and in particular a Fell bundle \(\mathcal{A}|_{G_e}\) over \(G_e\). Our first result shows that the natural inclusion of compactly supported sections extends to an isometric embedding \(\mathrm{C}^*(G_e;\mathcal{A}|_{G_e}) \hookrightarrow \mathrm{C}^*(G;\mathcal{A})\) (see Theorem 2). The proof proceeds via the construction of a canonical \(\mathrm{C}^*\)-correspondence from \(\mathrm{C}^*(G;\mathcal{A})\) to \(\mathrm{C}^*(G_e;\mathcal{A}|_{G_e})\) ( see Theorem 4), which allows us to identify the \(\mathrm{C}^*\)-norms of sections supported in \(G_e\) inside both the algebras.
We then show that the subspaces \(\mathrm{C_c}(G_\gamma;\mathcal{A}|_{G_\gamma})\) determine a \(\Gamma\)-grading of \(\mathrm{C}^*(G;\mathcal{A})\). Moreover, this is a topological grading in Exel’s sense: restriction to the clopen subgroupoid \(G_e\) defines a canonical conditional expectation onto \(\mathrm{C}^*(G_e;\mathcal{A}|_{G_e})\).
This topological grading enables us to define a coaction \(\delta\) of the discrete group \(\Gamma\) on \(\mathrm{C}^*(G;\mathcal{A})\). We study the associated coaction crossed product and show that it admits a concrete realization in terms of a Fell bundle naturally associated to the cocycle data \(c\). More precisely, we prove that the coaction crossed product is isomorphic to the \(\mathrm{C}^*\)-algebra of a pull-back Fell bundle \(\varphi^*\mathcal{A}\) over the skew-product groupoid \(G(c)\), where \(\varphi\colon G(c)\to G\) is a canonical groupoid homomorphism (see Theorem 4).
As an application, we consider twisted higher-rank graph \(\mathrm{C}^*\)-algebras [4], [5], which can be realized as Fell line bundles over path groupoids. We show that our construction yields a natural coaction on twisted higher-rank graph algebras and identify the corresponding coaction crossed product with the \(\mathrm{C}^*\)-algebra of the twisted skew-product higher-rank graph (Proposition 1). This extends a result of Kaliszewski–Quigg–Raeburn [6] to the setting of twisted higher-rank graphs.
Section 2 recalls background on Fell bundles, their \(\mathrm{C}^*\)-algebras, and coactions of discrete groups. In Section 3, we prove the inclusion theorem and establish the topological grading. Section 4 contains the construction of the coaction and the proof of the coaction crossed product identification. Section 5 is devoted to applications to twisted higher-rank graph \(\mathrm{C}^*\)-algebras.
In this section, we recall the necessary background on groupoids, Fell bundles over groupoids, their \(\mathrm{C}^*\)-algebras, and crossed products by coactions of discrete groups. In this article, we shall consider Fell bundles that are saturated and separable.
Throughout this article, \(\mathcal{H}\) denotes an infinite dimensional separable Hilbert space, \(\mathbb{B}(\mathcal{H})\) the algebra of bounded linear operators on \(\mathcal{H}\), and, for a \(\mathrm{C}^*\)-algebra \(A\), \(M(A)\) its multiplier algebra.
A groupoid is a small category \(G\) in which every element is invertible. The collection of units of \(G\) is denoted by \({G}^{(0)}\), while the set of all composable pairs is denoted by \(G^{(2)}\). The multiplication is given by a map \(G^{(2)} \to G\). Additionally, there are two maps associated with \(G\): the range map \(r\) and the source map \(s\), defined by \(r,s\colon G \to {G}^{(0)}\) by \(r(g) = g g^{-1}\) and \(s(g) = g^{-1}g\). A groupoid is called locally compact Hausdorff if \(G\) equipped with a locally compact Hausdorff topology such that the structure maps are continuous. A Haar system on \(G\) is a fully supported left invariant continuous family of measures \(\lambda=\{\lambda^u\}_{u\in {G}^{(0)}}\) along the range map \(r\) (see [7]). The groupoid \(G\) is called étale if the range (or equivalently, source) map is a local homeomorphism.
A continuous \(\mathbb{T}\)-valued \(2\)-cocycle \(\sigma\) on \(G\) is a continuous map \(\sigma\colon G^{(2)} \to \mathbb{T}\) satisfying \(\sigma(r(\gamma), \gamma) = \sigma(\gamma, s(\gamma)) =1\) and \(\sigma(\alpha, \beta)\sigma(\alpha\beta, \gamma) = \sigma(\alpha, \beta\gamma)\sigma(\beta, \gamma)\) for all composable triples \((\alpha, \beta, \gamma)\). The set of all continuous \(2\)-cocycle is denoted by \(Z^2(G, \mathbb{T})\). Let \(\sigma\) be a continuous \(2\)-cocycle on a locally compact Hausdorff groupoid \(G\) equipped with a Haar system \(\lambda\). Then the space of all compactly supported continuous function on \(G\) is denoted by \(\mathrm{C_c}(G, \sigma)\) and forms a \(^*\)-algebra with respect to the operations \[\xi*\eta(g) = \int_{G} \xi(h)\eta(h^{-1}g) \sigma(h,h^{-1}g) \mathrm{d}\lambda^{r(g)}(h)\quad \mathrm{and} \quad \xi^*(g) = \overline{\sigma(g,g^{-1})\xi(g)}\] for \(\xi, \eta\in \mathrm{C_c}(G, \sigma)\). The \(I\)-norm on \(\mathrm{C_c}(G,\sigma)\) is given by \[\lvert\!\lvert \xi\rvert\!\rvert_I = \max \Big\{\sup_{u\in {G}^{(0)}}\int_{r^{-1}(u)}|\xi(g)| \mathrm{d}\lambda^u(g), \sup_{u\in {G}^{(0)}}\int_{s^{-1}(u)}|\xi(g)| \mathrm{d}\lambda^u(g)\Big\}.\] A representation \(\pi\colon \mathrm{C_c}(G, \sigma) \to \mathbb{B}(\mathcal{H})\) is called \(I\)-bounded if \(\lvert\!\lvert \pi(\xi)\rvert\!\rvert\leq \lvert\!\lvert \xi\rvert\!\rvert_I\) for all \(\xi \in \mathrm{C_c}(G,\sigma)\). The twisted groupoid \(\mathrm{C}^*\)-algebra \(\mathrm{C}^*(G,\sigma)\) is the completion of \(\mathrm{C_c}(G, \sigma)\) with respect to the universal norm \[\lvert\!\lvert \xi\rvert\!\rvert=\sup\big\{\lvert\!\lvert \pi(\xi)\rvert\!\rvert : \pi\mathrm{ is a \(I\)-norm bounded representation of }\mathrm{C_c}(G, \sigma)\big\}.\]
Our reference for Fell bundles and upper semicontinuous bundle of Banach spaces are [8], [9] and [1]. An upper semicontinuous Banach bundle over a topological space \(X\) is a topological space \(\mathcal{A}\) with a continuous bundle map \(p\colon \mathcal{A}\to X\) such that each fibre \(\mathcal{A}_x\mathrel{\vcentcolon=}p^{-1}(x)\) is a Banach space and the map \(a\mapsto \lvert\!\lvert a\rvert\!\rvert\) from \(\mathcal{A}\to \mathbb{R}\) is an upper semicontinuous function. Let \(G\) be a locally compact groupoid and \(p\colon \mathcal{A} \to G\) an upper semicontinuous Banach bundle over \(G\). We define \[\mathcal{A}^{(2)} = \big\{ (a, b) \in \mathcal{A} \times \mathcal{A} : (s(p(a)), r(p(b))) \in G^{(2)} \big\}.\]
Definition 1 (Fell bundle [9]). A Fell bundle over a groupoid \(G\) is an upper semicontinuous bundle of Banach spaces \(p\colon \mathcal{A}\to G\) equipped with a continuous ‘multiplication’ map \(\mathcal{A}^{(2)}\to \mathcal{A}, (a,b) \mapsto ab\) and an ‘involution’ map \(\mathcal{A}\to \mathcal{A}, a\mapsto a^*\) for \(a\in \mathcal{A}\) which satisfy the following axioms:
\(p(ab) =p(a)p(b)\) for all \((a,b) \in \mathcal{A}^{(2)}\);
\(p(a^*)=p(a)^{-1}\) for all \(a\in \mathcal{A}\);
for each \(u\in {G}^{(0)}\), \(\mathcal{A}_u\) is a \(\mathrm{C}^*\)-algebra with respect to the inherited multiplication and involution on \(\mathcal{A}_u\);
for \(\gamma\in G\), \(\mathcal{A}_\gamma\) is a \(\mathcal{A}_{r(\gamma)}\)-\(\mathcal{A}_{s(\gamma)}\)-imprimitivity bimodule equipped with the inherited actions and inner products given by \[{}_{\mathcal{A}_{r(\gamma)}}\langle a, b\rangle\mathrel{\vcentcolon=}ab^* \quad \mathrm{and} \quad \langle a, b\rangle_{\mathcal{A}_{s(\gamma)}} \mathrel{\vcentcolon=}a^*b.\]
For the rest of this paper, we fix a Fell bundle \(p\colon \mathcal{A}\to G\) over a locally compact, Hausdorff, second countable groupoid \(G\) equipped with a Haar system \(\lambda\). Let \(\mathrm{C_c}(G;\mathcal{A})\) be the space of all compactly supported continuous sections of the bundle \(p\colon \mathcal{A}\to G\). Define \(\mathrm{C_c}(G;\mathcal{A})\) as a \(^*\)-algebra using the following convolution and involution operations: \[\xi*\eta(g) = \int_{G} \xi(h)\eta(h^{-1}g) \mathrm{d}\lambda^{r(g)}(h) \quad \text{and} \quad \xi^*(g) = \xi(g^{-1})^*\] where \(\xi, \eta \in \mathrm{C_c}(G;\mathcal{A})\). The \(I\)-norm on \(\mathrm{C_c}(G;\mathcal{A})\) is given by \[\lvert\!\lvert \xi\rvert\!\rvert_I = \max \Big\{\sup_{u\in {G}^{(0)}}\int_{r^{-1}(u)}\|\xi(g) \| \mathrm{d}\lambda^u(g), \sup_{u\in {G}^{(0)}}\int_{s^{-1}(u)}\|\xi(g)\| \mathrm{d}\lambda^u(g)\Big\}.\]
The Fell bundle \(\mathrm{C}^*\)-algebra, denoted by \(\mathrm{C}^*(G,\mathcal{A})\), is the completion of \(\mathrm{C_c}(G;\mathcal{A})\) with respect to the universal norm \[\lvert\!\lvert \xi\rvert\!\rvert=\sup\{\lvert\!\lvert \pi(\xi)\rvert\!\rvert : \pi\mathrm{ is a \(I\)-norm bounded representation of }\mathrm{C_c}(G;\mathcal{A})\}.\]
Observation 1. Using a similar argument as [10] one concludes that \(I\)-norm is sub-multiplicative for Fell bundles over groupoids.
Remark 1. Let \(G, H\) be two groupoids and \(p\colon \mathcal{A}\to H\) a Fell bundle over \(H\). Suppose \(\varphi\colon G \to H\) is a groupoid homomorphism. Then the pull-back \[\varphi^*\mathcal{A}=\{(g,a)\in G\times \mathcal{A}: \varphi(g) =p(a) \}\] is a Fell bundle over \(G\) with respect to the operations \((g,a)(h,b) = (gh,ab)\) and \((g,a)^* =(g^{-1},a^*)\) (see [11]). Note that the fibre at \(g\in G\) can be identified with \(\mathcal{A}_{\varphi(g)}\).
We refer the reader to [12], [13], [14] for a details treatment on coactions of groups. Let \(\Gamma\) be a discrete group. The comultiplication of \(\mathrm{C}^*(\Gamma)\) is the homomorphism \[\delta_{\Gamma}\colon \mathrm{C}^*(\Gamma) \to \mathrm{C}^*(\Gamma)\otimes \mathrm{C}^*(\Gamma)\] given by the integrated form of the unitary homomorphism from \(\Gamma \to \mathrm{C}^*(\Gamma)\otimes \mathrm{C}^*(\Gamma)\) by \(\gamma\mapsto \gamma\otimes \gamma\). Let \(A\) be a \(\mathrm{C}^*\)-algebra. A coaction of \(\Gamma\) on \(A\) is an injective homomorphism \(\delta \colon A \to M(A\otimes\mathrm{C}^*(\Gamma))\) satisfying coaction identity \[(\delta \otimes \mathrm{id}) \circ \delta = (\mathrm{id}\otimes \delta_{\Gamma}) \circ \delta\] together with the nondegeneracy condition \[\overline{\mathrm{span}}\{\delta(A)\cdot (1\otimes \mathrm{C}^*(\Gamma))\} = A\otimes \mathrm{C}^*(\Gamma).\] Note that the above nondegeneracy condition implies that \(\delta\) is nondegenerate as a map from \(A\to M(A\otimes \mathrm{C}^*(\Gamma))\).
A covariant representation of a coaction \((A, \Gamma, \delta)\) in \(M(B)\) is a pair \((\pi, \mu)\) consisting of nondegenerate homomorphisms \(\pi\colon A\to M(B)\) and \(\mu\colon \Gamma\to M(B)\) satisfies the covariance condition, that is, the diagram is commute \[\begin{tikzcd}[ column sep=3cm, row sep=normal ] A \arrow[r, "\delta"] \arrow[d, "\pi"'] & M\bigl(A \otimes \mathrm{C}^*(\Gamma)\bigr) \arrow[d, "\pi \otimes \mathrm{id}"] \\ M(B) \arrow[r, "\mathrm{Ad}(\mu \otimes \mathrm{id})(w_\Gamma) \circ (\cdot \otimes 1)"'] & M\bigl(B \otimes \mathrm{C}^*(\Gamma)\bigr). \end{tikzcd}\]
A coaction crossed product \(A\rtimes_{\delta}\Gamma\) is the universal \(\mathrm{C}^*\)-algebra generated by the universal covariant representation \((j_A, j_{\Gamma})\) of \((A, \Gamma,\delta)\), that is, for any covariant representation \((\pi, \mu)\) on \(M(B)\) there is a unique nondegenerate homomorphism \(\pi\times \mu \colon A\rtimes_{\delta} \Gamma \to M(B)\) such that \((\pi\times \mu)\circ j_A= \pi\) and \((\pi\times \mu)\circ j_{\Gamma}= \mu\). And \[A\rtimes_{\delta}\Gamma = \overline{\mathrm{span}}\{j_A(a)j_{\Gamma}(1_h): a\in A, h\in \Gamma\}\] where \(1_h\) is the characteristic function of \(\{h\}\).
In this section, we prove under certain hypothesis on the underlying groupoid of a Fell bundle, the \(\mathrm{C}^*\)-algebra of the Fell bundle can be realized as a topologically graded \(\mathrm{C}^*\)-algebra. We shall use this graded structure in Section 4 to obtain a coaction on the Fell bundle algebra \(\mathrm{C}^*(G;\mathcal{A})\). Most of the techniques used in the current section are motivated by [2].
Let \(G\) be a locally compact Hausdorff second countable groupoid with a Haar system \(\lambda\) and let \(\Gamma\) be a discrete group with a continuous \(1\)-cocycle \(c\colon G \to \Gamma\). For \(\gamma \in \Gamma\), set \(G_{\gamma} \mathrel{\vcentcolon=}c^{-1}(\gamma)\). Then \(G_{e}\) is a clopen subgroupoid of \(G\), where \(e\) is the identity element of \(\Gamma\). We equip \(G_e\) with the Haar system obtained by restricting \(\lambda\); that is, \(\{\lambda^{u}|_{G_e} : u\in {G}^{(0)}\}\). For simplicity, we continue to denote the same notation \(\lambda^u\) for \(\lambda^{u}|_{G_{e}}\).
Consider a Fell bundle \(p\colon \mathcal{A}\to G\) over \(G\). Let \(\xi \in \mathrm{C_c}(G;\mathcal{A})\), then \({\mathrm{supp}}(\xi) \subseteq \bigcup_{\gamma \in \Gamma} c^{-1}(\gamma)\). Since \({\mathrm{supp}}(\xi)\) is a compact subset of \(G\), there exists a finite subset \(F\subseteq \Gamma\) such that \({\mathrm{supp}}(\xi) \subseteq \bigcup_{\gamma \in F} c^{-1}(\gamma)\). We define \(\xi_{\gamma}\mathrel{\vcentcolon=}\xi|_{c^{-1}(\gamma)}\) for \(\gamma\in \Gamma\). Then \(\xi = \sum_{\gamma\in \Gamma} \xi_{\gamma}\). Since \(\xi_{\gamma} =0\) if \(\gamma \notin F\), the sum survives only for a finitely many \(\gamma\). As \(c^{-1}(\gamma)\) is a clopen subset of \(G\), for \(\gamma\in \Gamma\), we can extend \(\xi_{\gamma}\) on \(\mathrm{C_c}(G;\mathcal{A})\) by declaring zero outside \(G_{\gamma}\); we denote this extension by \(\widetilde{\xi_{\gamma}}\).
For \(\xi, \eta\in \mathrm{C_c}(G;\mathcal{A})\) and \(g\in \mathrm{C_c}(G_{e}; \mathcal{A}|_{G_e})\), we defined \[\begin{align} \tag{1} \xi*_Rg(x) &\mathrel{\vcentcolon=}\int_{G_e} \xi(xt)g(t^{-1}) \mathrm{d}\lambda^{s(x)}(t) \\ \tag{2} \xi*_L\eta (y) &\mathrel{\vcentcolon=}\xi*\eta (y) = \int_{G} \xi(x)\eta(x^{-1}y)\mathrm{d}\lambda^{r(y)}(x) \\ \tag{3} \langle\xi, \eta\rangle (x) &\mathrel{\vcentcolon=}\sum_{\gamma \in \Gamma} \xi^*_{\gamma} \bullet \eta_{\gamma}(x) \end{align}\] where \(x\in G_e, y\in G\) and \[\xi^*_{\gamma} \bullet \eta_{\gamma}(x) = \int_{G} \widetilde{\xi_{\gamma}}^*(z) \widetilde{\eta_{\gamma}}(z^{-1}x) \mathrm{d}\lambda^{r(x)}(z).\]
We will show that Equations 1 and 2 make \(\mathrm{C_c}(G;\mathcal{A})\) a bimodule over \(\mathrm{C_c}(G;\mathcal{A})\) and \(\mathrm{C_c}(G_e;\mathcal{A}|_{G_e})\), respectively; and that Equation 3 endows \(\mathrm{C_c}(G;\mathcal{A})\) with the structure of a pre-Hilbert module over \(\mathrm{C_c}(G_e;\mathcal{A}|_{G_e})\), carrying a left action of \(\mathrm{C_c}(G;\mathcal{A})\) given by Equation 2 . Finally, we complete the bimodule to obtain a \(\mathrm{C}^*\)-correspondence from \(\mathrm{C}^*(G;\mathcal{A})\to \mathrm{C}^*(G_e;\mathcal{A}|_{G_e})\). Before proving this we have the following observation, which will be useful in subsequent computations.
Observation 1. Let \(\xi,\eta\in \mathrm{C_c}(G;\mathcal{A})\) and \(g\in \mathrm{C_c}(G_e; \mathcal{A}|_{G_e})\). Then, we have
\((\eta*_Rg)_{\gamma} = \eta_{\gamma}*_Rg\);
\(\xi^*_{\gamma}\bullet (\eta*_Rg)_{\gamma} = (\xi^*_\gamma\bullet\eta_{\gamma})*_R g\)
for \(\gamma \in \Gamma\).
Proof. (1). For \(x\in G_{\gamma}\), we have \[(\eta*_Rg)_{\gamma}(x) = (\eta*_Rg)|_{c^{-1}(\gamma)} (x) = \int_{G_e} \eta(xt)g(t^{-1}) \mathrm{d}\lambda^{s(x)}(t).\] Since \(c\) is a \(1\)-cocycle, \(xt\in c^{-1}(\gamma)\) for \(t\in G_e\). Thus the last term can be written as \[\int_{G_e} \eta|_{c^{-1}(\gamma)}(xt)g(t^{-1}) \mathrm{d}\lambda^{s(x)}(t) = \eta_{\gamma}*_Rg(x).\]
(2). Let \(x\in G_{\gamma}\). Now using Part (1), we get \[\begin{gather} \xi^*_{\gamma}\bullet (\eta*_Rg)_{\gamma}(x) = \xi^*_{\gamma}\bullet (\eta_{\gamma}*_Rg)(x) = \int_{G} \widetilde{\xi_{\gamma}}^*(z)\widetilde{(\eta_{\gamma}*_Rg)}(z^{-1}x) \mathrm{d}\lambda^{r(x)}(z) \\ = \int_{G} \widetilde{\xi_{\gamma}}^*(z) \int_{G_e} \widetilde{\eta_{\gamma}}(z^{-1}xt)g(t^{-1}) \mathrm{d}\lambda^{s(x)}(t) \mathrm{d}\lambda^{r(x)}(z). \end{gather}\] Using Fubini theorem the last term can be written as \[\begin{gather} \int_{G_e}\int_{G} \widetilde{\xi_{\gamma}}^*(z)\widetilde{\eta_{\gamma}}(z^{-1}xt) g(t^{-1}) \mathrm{d}\lambda^{r(x)}(z) \mathrm{d}\lambda^{s(x)}(t) \\ = \int_{G_e} (\xi^*_{\gamma}\bullet\eta_{\gamma})(xt)g(t^{-1}) \mathrm{d}\lambda^{s(x)}(t) = (\xi^*_\gamma\bullet\eta_{\gamma})*_R g(x). \end{gather}\] ◻
Lemma 1. For \(\xi, \eta, \zeta \in \mathrm{C_c}(G;\mathcal{A})\) and \(g,h\in \mathrm{C_c}(G_e;\mathcal{A}|_{G_e})\), we have \[\begin{align} (\xi+\eta)*_Rg &= \xi*_Rg + \eta*_R g \tag{4}\\ \xi*_R(g+h) &= \xi*_Rg + \xi*_Rh \tag{5}\\ \xi*_R(g*h) &= (\xi*_Rg)*_Rh \tag{6}\\ \xi*_L(\eta +\zeta) &= \xi*_L\eta + \xi*_L\zeta \tag{7}\\ (\xi+\eta)*_L\zeta &= \xi*_L\eta + \xi*_L \zeta \tag{8}\\ (\xi*\eta)*_L\zeta &= \xi*_L(\eta*_L\zeta) \tag{9}\\ (\xi*_L\eta)*_Rg &=\xi*_L(\eta*_Rg) \tag{10}\\ \langle\xi, \eta+\zeta\rangle &= \langle\xi, \eta\rangle +\langle\xi, \zeta\rangle \tag{11}\\ \langle\xi, \eta*_Rg\rangle &=\langle\xi, \eta\rangle*g \tag{12} \\ \langle\xi, \eta\rangle^* &= \langle\eta, \xi\rangle \tag{13}\\ \langle\xi*_L\eta, \zeta\rangle &= \langle\eta, \xi^**_L\zeta\rangle. \tag{14} \end{align}\]
Proof. Equations 4 , 5 , 7 , 8 and 11 follows from the definition. Equation 10 follows from the associativity of the convolution.
The proof of Equations 6 and 9 are similar. Therefore, we only prove Equation 6 . \[\begin{align} \xi*_R(g*h)(x) &= \int_{G_e} \xi(xt)(g*h)(t^{-1}) \mathrm{d}\lambda^{s(x)}(t) \\ &= \int_{G_e} \xi(xt) \int_{G_e} g(y)h(y^{-1}t^{-1}) \mathrm{d}\lambda^{s(t)}(y) \mathrm{d}\lambda^{s(x)}(t) \\ &= \int_{G_e} \int_{G_e} \xi(xt) g(y)h(y^{-1}t^{-1}) \mathrm{d}\lambda^{s(t)}(y) \mathrm{d}\lambda^{s(x)}(t). \end{align}\] Using the change of variables \(ty\mapsto z\), the last term above becomes \[\int_{G_e} \int_{G_e} \xi(xt)g(t^{-1}z) h(z^{-1}) \mathrm{d}\lambda^{s(x)}(z) \mathrm{d}\lambda^{s(x)}(t).\] We now use Fubini theorem to rewrite the last term as \[\int_{G_e} \int_{G_e} \xi(xt)g(t^{-1}z) h(z^{-1}) \mathrm{d}\lambda^{s(x)}(t) \mathrm{d}\lambda^{s(x)}(z).\] Using the change of variables \(t\mapsto zt\), we obtain \[\begin{gather} \int_{G_e} \int_{G_e} \xi(xzt)g(t^{-1})g(t^{-1}) h(z^{-1}) \mathrm{d}\lambda^{s(z)}(t) \mathrm{d}\lambda^{s(x)}(z)\\ = \int_{G_e} (\xi*_Rg) (xz)h(z^{-1}) \mathrm{d}\lambda^{s(x)}(z) = (\xi*_Rg)*_Rh(x). \end{gather}\]
For Equation 12 , \[\begin{align} \langle\xi, \eta*_Rg\rangle(x) &= \sum_{\gamma\in \Gamma} \xi^*_{\gamma}\bullet (\eta*_Rg)_{\gamma}(x) = \sum_{\gamma\in \Gamma} (\xi^*_{\gamma}\bullet \eta_{\gamma})*_Rg(x)\\ &= \sum_{\gamma\in \Gamma} (\xi^*_{\gamma}\bullet \eta_{\gamma})*g(x) = \Big(\sum_{\gamma\in \Gamma} \xi^*_{\gamma}\bullet \eta_{\gamma}\Big)*g(x) = \langle\xi, \eta\rangle*g(x). \end{align}\] The second equality in the first line follows form Observation 1(2) and second equality in the second line follows because the sum is finite.
For Equation 13 , \[\begin{align} \langle\xi, \eta\rangle^*(x) &= (\langle\xi, \eta\rangle(x^{-1}))^* = \Big(\sum_{\gamma\in \Gamma}\xi^*_{\gamma}\bullet \eta_{\gamma}(x^{-1})\Big)^* \\ &= \Big(\sum_{\gamma\in \Gamma} \int_{G} \widetilde{\xi_{\gamma}}^*(z) \widetilde{\eta_{\gamma}}(z^{-1}x^{-1}) \mathrm{d}\lambda^{s(x)}(z)\Big)^*\\ &= \sum_{\gamma \in \Gamma} \int_{G} \bigl(\widetilde{\eta_{\gamma}}(z^{-1}x^{-1})\bigr)^* \widetilde{\xi_{\gamma}}(z^{-1}) \mathrm{d}\lambda^{s(x)}(z) \\ &= \sum_{\gamma \in \Gamma} \int_{G} \widetilde{\eta_{\gamma}}^*(xz) \widetilde{\xi_{\gamma}}(z^{-1}) \mathrm{d}\lambda^{s(x)}(z). \end{align}\] Using the change of variables \(z\mapsto x^{-1}z\), the last term can be written as \[\sum_{\gamma \in \Gamma} \int_{G} \widetilde{\eta_{\gamma}}^*(z) \widetilde{\xi_{\gamma}}(z^{-1}x) \mathrm{d}\lambda^{r(x)}(z) = \sum_{\gamma \in \Gamma} \eta_{\gamma}^*\bullet \xi_{\gamma}(x) = \langle\eta, \xi\rangle(x).\] For Equation 14 , we assume \({\mathrm{supp}}(\xi)\subseteq c^{-1}(\delta)\) for some \(\delta \in \Gamma\), \[\begin{align} \langle\xi*_L\eta, \zeta\rangle (x) &= \sum_{\gamma \in \Gamma} (\xi*_L\eta)_{\gamma}^*\bullet \widetilde{\zeta_{\gamma}}(x) = \sum_{\gamma \in \Gamma} \int_{G} \widetilde{(\xi*_L\eta)_{\delta\gamma}}^*(z) \widetilde{\zeta_{\delta\gamma}}(z^{-1}x) \mathrm{d}\lambda^{r(x)}(z)\\ &= \sum_{\gamma \in \Gamma} \int_{G} \int_{G} \bigl( \widetilde{\xi_{\delta}}(y) \widetilde{\eta_{\gamma}}(y^{-1}z^{-1})\bigr)^* \mathrm{d}\lambda^{s(z)}(y) \widetilde{\zeta_{\delta\gamma}}(z^{-1}x) \mathrm{d}\lambda^{r(x)}(z) \\ &= \sum_{\gamma \in \Gamma} \int_{G} \int_{G} \widetilde{\eta_{\gamma}}^*(zy) \widetilde{ \xi_{\delta}}^*(y^{-1}) \widetilde{\zeta_{\delta\gamma}}(z^{-1}x) \mathrm{d}\lambda^{s(z)}(y) \mathrm{d}\lambda^{r(x)}(z) \end{align}\] Using the change of variables \(y\mapsto z^{-1}y\), the last term can be written as \[\sum_{\gamma \in \Gamma} \int_{G} \int_{G} \widetilde{\eta_{\gamma}}^*(y) \widetilde{\xi_{\delta}}^*(y^{-1}z) \widetilde{\zeta_{\delta\gamma}}(z^{-1}x) \mathrm{d}\lambda^{r(z)}(y) \mathrm{d}\lambda^{r(x)}(z).\] Again, using the change of variables \(z\mapsto yz\) and Fubini theorem, the above term becomes, \[\begin{align} &\sum_{\gamma \in \Gamma} \int_{G} \int_{G} \widetilde{\eta_{\gamma}}^*(y) \widetilde{\xi_\delta}^*(z) \widetilde{\zeta_{\delta\gamma}}(z^{-1}y^{-1}x) \mathrm{d}\lambda^{s(y)}(z) \mathrm{d}\lambda^{r(x)}(y) \\ &= \sum_{\gamma \in \Gamma} \int_{G} \widetilde{\eta_{\gamma}}^*(y) (\widetilde{\xi^**_L\zeta})_{\gamma}(y^{-1}x) \mathrm{d}\lambda^{r(x)}(y) \\ &= \sum_{\gamma \in \Gamma} \eta^*_{\gamma}\bullet (\xi^**_L\zeta)_{\gamma} = \langle\eta, \xi^**_L\zeta\rangle. \end{align}\] If \(\xi\in \mathrm{C_c}(G;\mathcal{A})\), then we can cover the compact set \({\mathrm{supp}}(\xi)\) by the open cover \(\{c^{-1}(\gamma)\}_{\gamma\in \Gamma}\) and hence admits a finite subcover. Thus, using a partition of unity argument, we conclude that Equation 14 holds. ◻
Lemma 2.
Suppose \((\xi_n)_{n\in \mathbb{N}}\) is a sequence of sections in \(\mathrm{C_c}(G;\mathcal{A})\) that converges to the zero section in the inductive limit topology. Then \(\langle\xi_n, \xi_n\rangle\to 0\) in \(\mathrm{C_c}(G_e; \mathcal{A}|_{G_e})\) in the inductive limit topology, hence in \(\mathrm{C}^*(G_e;\mathcal{A}|_{G_e})\) as well.
Let \(g\in \mathrm{C_c}(G;\mathcal{A})\). If \(f_n\to f\) in \(\mathrm{C_c}(G;\mathcal{A})\) in the inductive limit topology, then \(f_n*_Lg \to f*_Lg\) in \(\mathrm{C_c}(G;\mathcal{A})\) in the inductive limit topology.
Proof. (1). Since \(\xi_n \to 0\) in the inductive limit topology, there is a fixed compact set \(K\) such that \({\mathrm{supp}}(\xi_n) \subseteq K\) for all \(n\). We write \(\xi_n =\sum_{\gamma \in \Gamma} (\xi_n)_\gamma\), with \({\mathrm{supp}}((\xi_n)_\gamma) \subseteq G_{\gamma}\). Therefore, \[\langle\xi_n, \xi_n\rangle = \sum_{\gamma \in \Gamma} (\xi_n)^*_\gamma\bullet (\xi_n)_\gamma \to 0\] uniformly as \(n\to \infty\) because each \((\xi_n)^*_\gamma\bullet (\xi_n)_\gamma \to 0\) uniformly as \(n\to \infty\) and supported in the compact set \((K\cap G_\gamma)^{-1}(K\cap G_\gamma)\). Since \({\mathrm{supp}}(\langle\xi_n, \xi_n\rangle)\) is in the compact set \(\bigcup_{\mathrm{finite}}(K\cap G_\gamma)^{-1}(K\cap G_\gamma)\), \(\langle\xi_n, \xi_n\rangle \to 0\) in the inductive limit topology.
(2). This part follows from a similar argument as Lemma 3.19(2) of [15]. ◻
Lemma 3. Let \(g, h\in \mathrm{C_c}(G;\mathcal{A})\). Then the map \(f\mapsto \langle g, f*_Lh\rangle\) from \(\mathrm{C_c}(G;\mathcal{A})\) to \(\mathrm{C_c}(G_e;\mathcal{A}|_{G_e})\) is continuous in the inductive limit topology.
Proof. Let \((f_n)_{n\in \mathbb{N}}\) be a sequence in \(\mathrm{C_c}(G;\mathcal{A})\) converges to \(f\) in the inductive limit topology. Let \(K\) be a fixed compact set with \({\mathrm{supp}}(f_n)\) and \({\mathrm{supp}}(f)\) contained in \(K\). Then \({\mathrm{supp}}(\langle g, f_n*_Lh\rangle)\) and \({\mathrm{supp}}(\langle g, f*_Lh\rangle)\) are also contained in some fixed compact set. Using Lemma 1 and Cauchy–Schwarz inequality, we have \[\begin{align} \lvert\!\lvert \langle g, f_n*_Lh\rangle-\langle g, f*_Lh\rangle\rvert\!\rvert^2 &= \lvert\!\lvert \langle g, (f_n-f)*_Lh\rangle\rvert\!\rvert^2\\ &= \lvert\!\lvert \langle g, (f_n-f)*_Lh\rangle^*\langle g, (f_n-f)*_Lh\rangle\rvert\!\rvert \\ &\leq \lvert\!\lvert \langle g, g\rangle\rvert\!\rvert \lvert\!\lvert \langle(f_n-f)*_Lh, (f_n-f)*_Lh\rangle\rvert\!\rvert. \end{align}\] Since \(f_n\to f\) in the inductive limit topology, Lemma 2(2) ensures that \(f_n*_Lh \to f*_Lh\) in the inductive limit topology and Lemma 2(1), gives us \[\langle(f_n-f)*_Lh, (f_n-f)*_Lh\rangle \to 0\] in the inductive limit topology. This completes the proof. ◻
Lemma 4.
For \(\xi\in \mathrm{C_c}(G;\mathcal{A})\), \(\langle\xi, \xi\rangle\geq 0\) in \(\mathrm{C_c}(G_{e}; \mathcal{A}|_{G_e})\).
\(\mathrm{span}\{\langle\xi, \eta\rangle : \xi, \eta \in \mathrm{C_c}(G;\mathcal{A})\}\) is dense in \(\mathrm{C}^*(G_{e}; \mathcal{A}|_{G_e})\).
Let \(\eta, \zeta \in \mathrm{C_c}(G;\mathcal{A})\). If \(\xi_n\to \xi\) in \(\mathrm{C_c}(G;\mathcal{A})\) in the inductive limit topology, then \(\langle\xi_n*_L\eta, \zeta\rangle \to \langle\xi*_L\eta, \zeta\rangle\) in \(\mathrm{C_c}(G_{e};\mathcal{A}|_{G_e})\) in the inductive limit topology.
Proof. (1). We have \(\langle\xi, \xi\rangle = \sum_{\gamma \in F} \xi^*_{\gamma}\bullet\xi_{\gamma}\) for a finite subset \(F\subseteq \Gamma\). As sum of positive elements is positive, \(\langle\xi, \xi\rangle\) is positive in \(\mathrm{C_c}(G;\mathcal{A})\). Since \(\langle\xi, \xi\rangle \in \mathrm{C_c}(G_e; \mathcal{A}|_{G_e})\) and \(\overline{\mathrm{C_c}(G_e;\mathcal{A}|_{G_e})}^{I} \subseteq \overline{\mathrm{C_c}(G;\mathcal{A})}^{I}\), spectral permanence allows us to conclude that \(\langle\xi, \xi\rangle\) is positive in \(\mathrm{C_c}(G_e;\mathcal{A}|_{G_e})\).
(2). Let \(f\in \mathrm{C_c}(G_e; \mathcal{A}|_{G_e})\). Choose a self-adjoint approximate identity of \((u_\alpha)_{\alpha}\) in \(\mathrm{C}^*(G_e; \mathcal{A}|_{G_e})\) such that \(u_\alpha \in \mathrm{C_c}(G_e; \mathcal{A}|_{G_e})\) with each \(u_\alpha\) has norm al most \(1\) (since the groupoid \(G_e\) is second countable the Fell bundle is saturated such approximate identity always exists see [9]). Let \(f\in \mathrm{C_c}(G_e; \mathcal{A}|_{G_e})\). Then \[\langle u_\alpha, f\rangle = (u_\alpha)^*_e\bullet (f)_e= u_\alpha\bullet f = u_\alpha * f \to f.\]
(3). The proof follows from a similar argument as Lemma 3. ◻
Lemma 1 shows that \(\mathrm{C_c}(G;\mathcal{A})\) is a right module over \(\mathrm{C_c}(G_{e}; \mathcal{A}|_{G_e})\), and Lemma 4(2) shows that the \(\mathrm{C_c}(G_e;\mathcal{A}|_{G_e})\)-valued inner product \(\langle\cdot, \cdot\rangle\) defined by Equation 3 is positive definite. Let \(Y\) be the Hilbert \(\mathrm{C}^*(G_e; \mathcal{A}|_{G_e})\)-module obtained by completing the pre-Hilbert \(\mathrm{C}^*(G_e;\mathcal{A}|_{G_e})\)-module \(\mathrm{C_c}(G;\mathcal{A})\). Lemma 4(2) implies that \(Y\) is a full Hilbert module.
Lemma 5. The \(\mathrm{span}\{\xi*_L\eta: \xi, \eta\in \mathrm{C_c}(G,\mathcal{A})\}\) is dense in the right Hilbert-\(\mathrm{C}^*(G_e; \mathcal{A}|_{G_e})\) module \(Y\).
Proof. Let \((u_\alpha)_{\alpha}\) be a self-adjoint approximate identity of \(\mathrm{C}^*(G_e;\mathcal{A}|_{G_e})\) such that \(u_\alpha \in \mathrm{C_c}(G_e;\mathcal{A}|_{G_e})\) and \(\lvert\!\lvert u_\alpha\rvert\!\rvert \leq 1\). Let \(f\in \mathrm{C_c}(G;\mathcal{A})\) with \(f =\sum_{\gamma \in F} f_\gamma\), where \(F\) is a finite subset of \(\Gamma\). Now, we have \[\begin{align} \lvert\!\lvert f*_L\widetilde{u_\alpha} - f\rvert\!\rvert^2_{\mathrm{C}^*(G_e; \mathcal{A}|_{G_e})} &= \lvert\!\lvert (f*_L\widetilde{u_\alpha} - f)^*(f*_L\widetilde{u_\alpha} - f)\rvert\!\rvert \\ &= \bigr\|\sum_{\gamma \in F}(f_\gamma *u_\alpha - f_\gamma)^*(f_\gamma*u_\alpha - f_{\gamma}) \bigr \|\\ &= \bigr\| \sum_{\gamma \in F} u^*_\alpha f^*_{\gamma} f_{\gamma} u_{\alpha} - u^*_{\alpha} f^*_{\gamma} f_{\gamma} - f^*_{\gamma} f_{\gamma} u_{\alpha} -f^*_{\gamma} f_{\gamma} \bigr\| \\ &\leq \sum_{\gamma \in F} \big(\lvert\!\lvert u_\alpha\rvert\!\rvert \lvert\!\lvert f^*_{\gamma} f_{\gamma} u_{\alpha}- f^*_{\gamma} f_{\gamma}\rvert\!\rvert + \lvert\!\lvert f^*_{\gamma} f_{\gamma} - f^*_{\gamma} f_{\gamma} u_{\alpha}\rvert\!\rvert \big)\\ &\leq 2\sum_{\gamma \in F} \lvert\!\lvert f^*_{\gamma} f_{\gamma} u_{\alpha}- f^*_{\gamma} f_{\gamma}\rvert\!\rvert. \end{align}\] The last term goes to zero as \(\alpha \to \infty\). ◻
We now show that the left action \(*_L\) defined by Equation 2 extends to a nondegenerate representation of \(\mathrm{C}^*(G;\mathcal{A})\) on \(Y\) by adjointable operators. Equation 14 of Lemma 1 ensures that \(\mathrm{C_c}(G;\mathcal{A})\) acts on itself by adjointable operators. To show that the left action is bounded, let \(\varphi\) be a state on \(\mathrm{C}^*(G_e; \mathcal{A}|_{G_e})\). Consider the Hilbert space completion of the inner product space \((Y, \langle\cdot, \cdot\rangle_{\varphi})\), where \(\langle f, g\rangle_{\varphi} \mathrel{\vcentcolon=}\varphi(\langle f, g\rangle)\) for \(f,g\in Y\). We denote this Hilbert space completion by \(Y_{\varphi}\). Define the vector subspace \(W\subseteq Y_{\varphi}\) generated by \(\{\xi*_L\eta : \xi, \eta \in \mathrm{C_c}(G;\mathcal{A})\}\). Lemma 5 shows that \(W\) is dense subspace of \(Y\). Define a representation \(\pi\) of \(\mathrm{C_c}(G;\mathcal{A})\) on \(W\) by \[\pi(f)(g) = f*_Lg\] for \(f,g \in \mathrm{C_c}(G;\mathcal{A})\). We now show that \(\pi\) satisfies the three conditions required to be a pre-representation (see Definition 4.1 of [9]) of \(\mathrm{C_c}(G;\mathcal{A})\) on \(W\).
\(\pi\) is nondegenerate follows from Lemma 5.
Lemma 3 ensures that, the map \(f \mapsto \langle\xi, \pi(f)\eta\rangle_{\varphi}\) is continuous in the inductive limit topology for \(\xi, \eta\) and \(f \in\mathrm{C_c}(G;\mathcal{A})\).
By Lemma 1, we have \(\langle f*_L\xi, \eta\rangle_{\varphi} = \langle\xi, f^**_L\eta\rangle_{\varphi}\) for \(\xi, \eta, f\in \mathrm{C_c}(G;\mathcal{A})\).
Therefore, using the disintegration theorem for Fell bundles [9] we extends \(\pi\) to be a nondegenerate (bounded) representation of \(\mathrm{C}^*(G;\mathcal{A})\) on \(Y_{\varphi}\). Thus, \[\varphi(\langle f*_L\xi, f*_L\xi\rangle) \leq \lvert\!\lvert f\rvert\!\rvert^2_{\mathrm{C}^*(G;\mathcal{A})} \varphi(\langle\xi, \xi\rangle)\] for all \(\xi, f\in \mathrm{C_c}(G;\mathcal{A})\). Since \(\varphi\) was an arbitrary state on \(\mathrm{C}^*(G_e; \mathcal{A}|_{G_e})\), we have \[\langle f*_L\xi, f*_L\xi\rangle \leq \lvert\!\lvert f\rvert\!\rvert^2_{\mathrm{C}^*(G;\mathcal{A})} \langle\xi, \xi\rangle.\] Hence, the left action \(*_L\) of \(\mathrm{C_c}(G; \mathcal{A})\) on itself extends to a nondegenerate representation of \(\mathrm{C}^*(G;\mathcal{A})\) on \(Y\).
Lemmas 1, 4 and the above discussion gives us the following result.
Theorem 1. Let \(p\colon \mathcal{A}\to G\) be a Fell bundle over a locally compact Hausdorff second countable groupoid equipped with a Haar system. Let \(\Gamma\) be a discrete group and \(c\colon G\to \Gamma\) a continuous \(1\)-cocycle. Then the \(\mathrm{C_c}(G;\mathcal{A})\)-\(\mathrm{C_c}(G_e;\mathcal{A}|_{G_e})\)-bimodule \(\mathrm{C_c}(G;\mathcal{A})\) completes to a \(\mathrm{C}^*\)-correspondence \[Y\colon \mathrm{C}^*(G;\mathcal{A}) \to \mathrm{C}^*(G_e;\mathcal{A}|_{G_e}).\]
Remark 1. We denote the left action of \(\mathrm{C}^*(G;\mathcal{A})\) on \(Y\) in Theorem 1 by \(L\), that is, \(L\colon \mathrm{C}^*(G;\mathcal{A}) \to \mathbb{B}(Y)\) is a homomorphism, where \(\mathbb{B}(Y)\) denotes the set of all adjointable operators on \(Y\).
We denote \[\mathcal{B}_e =\overline{\{f\in \mathrm{C_c}(G;\mathcal{A}) : {\mathrm{supp}}(f) \subset G_{e}\}} \subset \mathrm{C}^*(G;\mathcal{A})\] where \(e\) is the identity element of \(\Gamma\) and the above closure is taken with respect to the \(\mathrm{C}^*\)-norm of \(\mathrm{C}^*(G;\mathcal{A})\). We now prove the main theorem of this section.
Theorem 2. Let \(L\colon \mathrm{C}^*(G;\mathcal{A}) \to \mathbb{B}(Y)\) be the homomorphism of Remark 1. Then \(L|_{B_e}\colon B_e\to \mathbb{B}(Y)\) is injective and the inclusion map \(\iota\colon \mathrm{C_c}(G_e; \mathcal{A}|_{G_e}) \to \mathrm{C_c}(G;\mathcal{A})\) given by \[\iota(f)(\gamma) = \begin{cases} f(\gamma) & \mathrm{if } \gamma \in G_e,\\ 0 & \mathrm{ otherwise } \end{cases}\] extends to an isomorphism from \(\mathrm{C}^*(G_e;\mathcal{A}|_{G_e}) \to \mathcal{B}_e\).
Proof. Note that the map \(\iota\colon \mathrm{C_c}(G_e; \mathcal{A}|_{G_e}) \to \mathrm{C_c}(G;\mathcal{A})\) is a homomorphism of \(^*\)-algebras and it preserve the \(I\)-norm. Therefore, \(\iota\) can be extended to a norm decreasing \(^*\)-homomorphism \(\bar{\iota}\colon \mathrm{C}^*(G_e;\mathcal{A}|_{G_e}) \to \mathrm{C}^*(G;\mathcal{A})\). Thus, for \(f\in \mathrm{C_c}(G_e;\mathcal{A}|_{G_e})\), we have \(\lvert\!\lvert \iota(f)\rvert\!\rvert \leq \lvert\!\lvert f\rvert\!\rvert\). Now \[\lvert\!\lvert \iota(f)\rvert\!\rvert^2_{\mathrm{C}^*(G_e;\mathcal{A}|_{G_e})} = \lvert\!\lvert \langle\iota(f), \iota(f)\rangle\rvert\!\rvert =\lvert\!\lvert \iota(f)^*_e\iota(f)_e\rvert\!\rvert = \lvert\!\lvert f^*f\rvert\!\rvert =\lvert\!\lvert f\rvert\!\rvert^2\] for \(f\in \mathrm{C_c}(G_e;\mathcal{A}|_{G_e})\). The second equality above follows because the support of \(\iota(f)\) is contained in \(G_e\). If \(f\) is a nonzero section in \(\mathrm{C_c}(G_e;\mathcal{A}|_{G_e})\), then \[\begin{align} \lvert\!\lvert L(\iota(f))\rvert\!\rvert &=\sup\big\{\lvert\!\lvert \iota(f)g\rvert\!\rvert_{\mathrm{C}^*(G_e;\mathcal{A}|_{G_e})} : g\in Y \mathrm{ with } \lvert\!\lvert g\rvert\!\rvert\leq 1\big \}\\ &\geq \lvert\!\lvert \iota(f)\iota(f)^*\rvert\!\rvert_{\mathrm{C}^*(G_e;\mathcal{A}|_{G_e})}/\lvert\!\lvert f\rvert\!\rvert = \lvert\!\lvert ff^*\rvert\!\rvert/\lvert\!\lvert f\rvert\!\rvert =\lvert\!\lvert f\rvert\!\rvert. \end{align}\] Since \(L\) is norm-decreasing (by Theorem 1), we have \[\lvert\!\lvert f\rvert\!\rvert\leq \lvert\!\lvert L(\iota(f))\rvert\!\rvert \leq \lvert\!\lvert \iota(f)\rvert\!\rvert \leq \lvert\!\lvert f\rvert\!\rvert.\] Thus, the above line ensures that \(\iota\) can be extended to an isometric map \(\bar{\iota}\) and the restriction of \(L\) on \(\mathcal{B}_e\) is an isometry. ◻
Our next goal is to show that the Fell bundle \(\mathrm{C}^*\)-algebra \(\mathrm{C}^*(G;\mathcal{A})\) is a topological grading in the sense of Exel [3]. We now recall the definition of topological grading from [3]. A \(\mathrm{C}^*\)-algebra \(A\) is called topological grading over a discrete group \(\Gamma\) if there exists a collection of closed subspace \(\{A_{\gamma} : \gamma \in \Gamma\}\) of \(A\) such that
\(A_\gamma A_{\eta} \subset A_{\gamma \eta}\) for \(\gamma, \eta\in \Gamma\);
\(A_{\gamma}^* = A_{\gamma^{-1}}\) for all \(\gamma\in \Gamma\);
the linear span of \(\{A_{\gamma} : \gamma \in \Gamma\}\) is dense in \(A\);
there is a bounded linear map \(E \colon A \to A\) such that \(E = \mathrm{id}\) on \(A_{e}\) and \(E = 0\) on \(A_{\gamma}\) for \(\gamma\neq e\).
If \(A\) is a topological grading over a discrete group, then Exel proved [3] that the map in Condition [cond:top-grading-4] of the above definition is a conditional expectation from \(A\) onto \(A_e\) and the collection \(\{A_{\gamma} : \gamma \in \Gamma\}\) is linearly independent.
Proposition 1. Let \(p\colon \mathcal{A}\to G\) be a Fell bundle over a locally compact Hausdorff second countable groupoid equipped with a Haar system. If \(c\colon G\to \Gamma\) is a continuous \(1\)-cocycle, then there exists a linear contraction \(E\colon \mathrm{C}^*(G;\mathcal{A}) \to \mathrm{C}^*(G_{e}; \mathcal{A}|_{G_e})\) given by \(E(f) = f|_{G_e}\) for \(f\in \mathrm{C_c}(G;\mathcal{A})\).
Proof. For \(f\in \mathrm{C_c}(G;\mathcal{A})\), we have \[\label{liner-cont-eq-1} \lvert\!\lvert f\rvert\!\rvert^2_{\mathrm{C}^*(G_e;\mathcal{A}|_{G_e})} = \bigr \| \sum_{\gamma \in \Gamma} f^*_{\gamma} f_{\gamma} \bigr\| \geq \lvert\!\lvert f^*_ef_e\rvert\!\rvert = \lvert\!\lvert f|_{G_e}\rvert\!\rvert^2,\tag{15}\] where the inequality of the above line follows from \(\sum_{\gamma \in \Gamma}f^*_{\gamma}f_{\gamma} \geq f^*_ef_e \geq 0\). Choose a self-adjoint approximation identity \((u_\alpha)_{\alpha}\) for \(\mathrm{C}^*(G_e;\mathcal{A}|_{G_e})\) in \(\mathrm{C_c}(G_e;\mathcal{A}|_{G_e})\) (such approximation identity exists by [9]). Since \(\lvert\!\lvert \iota(u_{\alpha})\rvert\!\rvert \leq 1\), we have \[\begin{gather} \lvert\!\lvert L(f)\rvert\!\rvert^2 \geq \lvert\!\lvert L(f)(\iota(u_{\alpha}))\rvert\!\rvert_{\mathrm{C}^*(G_e;\mathcal{A}|_{G_e})}^2 = \lvert\!\lvert f*_L\iota(u_{\alpha})\rvert\!\rvert_{\mathrm{C}^*(G_e;\mathcal{A}|_{G_e})}^2\\ = \lvert\!\lvert u^*_{\alpha}\langle f, f\rangle u_{\alpha}\rvert\!\rvert \to \lvert\!\lvert f\rvert\!\rvert^2_{\mathrm{C}^*(G_e;\mathcal{A}|_{G_e})} \quad \mathrm{as } \alpha \to \infty. \end{gather}\] Equation 15 , Theorem 1 and the above computation gives us \[\lvert\!\lvert f|_{G_e}\rvert\!\rvert \leq \lvert\!\lvert f\rvert\!\rvert_{\mathrm{C}^*(G_e;\mathcal{A}|_{G_e})} \leq \lvert\!\lvert L(f)\rvert\!\rvert\leq \lvert\!\lvert f\rvert\!\rvert.\] Therefore, the restriction extends to a bounded linear contraction \(E\). ◻
Proposition 1. Let \(p\colon \mathcal{A}\to G\) be a Fell bundle over a locally compact Hausdorff second countable groupoid equipped with a Haar system. Let \(\Gamma\) be a discrete group with a continuous \(1\)-cocycle \(c\colon G\to \Gamma\). For \(\gamma \in \Gamma\), set \[\mathcal{B}_{\gamma}= \overline{\{f\in \mathrm{C_c}(G;\mathcal{A}) : {\mathrm{supp}}(f) \subseteq G_{\gamma}\}} \subseteq \mathrm{C}^*(G;\mathcal{A}),\] where the closure is taken with respect to the \(\mathrm{C}^*\)-norm. Then \(\mathrm{C}^*(G;\mathcal{A})\) is a topologically graded \(\mathrm{C}^*\)-algebra. The grading subspaces are \(\mathcal{B}= \{\mathcal{B}_{\gamma}\}_{\gamma \in \Gamma}\) and the conditional expectation \(E\colon \mathrm{C}^*(G;\mathcal{A}) \to \mathrm{C}^*(G;\mathcal{A})\) induced by the restriction of sections to \(G_e\).
Proof. Let \(\gamma, \eta \in \Gamma\) and let \(f,g\in \mathrm{C_c}(G;\mathcal{A})\) with \({\mathrm{supp}}(f)\subseteq G_\gamma\) and \({\mathrm{supp}}(g) \subseteq G_{\eta}\). Since \(c\) is a \(1\)-cocycle, it follows that \({\mathrm{supp}}(f){\mathrm{supp}}(g) \subseteq G_{\gamma} G_{\eta} \subseteq G_{\gamma\eta}\). Thus, \(\mathcal{B}_{\gamma}\mathcal{B}_{\eta} \subseteq \mathcal{B}_{\gamma \eta}\). Moreover, since \({\mathrm{supp}}(f^*)\subseteq G_{\gamma^{-1}}\), we have \(\mathcal{B}^*_{\gamma} = \mathcal{B}_{\gamma^{-1}}\).
Let \(f\in \mathrm{C_c}(G;\mathcal{A})\). We write \(f=\sum_{\gamma \in F}f_{\gamma}\) for some finite subset \(F\) of \(\Gamma\). Now we can represent \(f\) as \(f= \sum_{\gamma \in F}f_{\gamma} =\sum_{\gamma \in F}\iota_{\gamma}(f_{\gamma})\), where \(\iota_{\gamma}\colon \mathrm{C_c}(G_{\gamma};\mathcal{A}|_{G_\gamma}) \hookrightarrow \mathrm{C_c}(G;\mathcal{A})\) is the inclusion. Therefore, \(\mathrm{C_c}(G;\mathcal{A})\) is contained in \(\mathrm{span}\{\mathcal{B}_{\gamma} : \gamma \in \Gamma\}\), and hence \(\mathrm{span}\{\mathcal{B}_{\gamma} : \gamma \in \Gamma\}\) is dense in \(\mathrm{C}^*(G;\mathcal{A})\).
Recall from Proposition 1, that the map \(E\colon \mathrm{C}^*(G;\mathcal{A}) \to \mathrm{C}^*(G_e;\mathcal{A}|_{G_e})\) defined by \(E(f) =f|_{G_e}\) is a linear contraction. Composing the isometric isomorphism \(\overline{\iota}\colon \mathrm{C}^*(G_e;\mathcal{A}|_{G_e}) \to \mathrm{C}^*(G;\mathcal{A})\) obtained in Theorem 2 with \(E\), we obtain a contraction \(\widetilde{E} = \overline{\iota}\circ E\colon \mathrm{C}^*(G;\mathcal{A}) \to \mathrm{C}^*(G;\mathcal{A})\). The map \(\widetilde{E}\) satisfies \(\widetilde{E} =0\) on \(\mathcal{B}_{\gamma}\) for \(\gamma \neq e\) and \(\widetilde{E} =\mathrm{id}\) on \(\mathcal{B}_e\). Thus, \(\widetilde{E}\) is a conditional expectation, and hence \(\mathrm{C}^*(G;\mathcal{A})\) is a topologically graded \(\mathrm{C}^*\)-algebra. ◻
Theorem 3. Let \(p\colon \mathcal{A}\to G\) be a Fell bundle over a locally compact Hausdorff second countable groupoid equipped with a Haar system. Let \(\Gamma\) be a discrete group with a continuous \(1\)-cocycle \(c\colon G\to \Gamma\). Consider the group Fell bundle \(\mathcal{B}=\{\mathcal{B}_{\gamma}\}_{\gamma \in \Gamma}\) as described above. Then the group Fell bundle algebra \(\mathrm{C}^*(\Gamma; \mathcal{B})\) is isomorphic to \(\mathrm{C}^*(G;\mathcal{A})\).
Proof. Define a representation (see [3]) \(\{\mathrm{id}_{\gamma}\}_{\gamma \in \Gamma}\) of \(\mathcal{B}\) on \(\mathrm{C}^*(G;\mathcal{A})\) given by the identity maps \(\mathrm{id}_{\gamma} \colon \mathcal{B}_{\gamma} \to \mathrm{C}^*(G;\mathcal{A})\). By Proposition 1, the algebra \(\mathrm{C}^*(G;\mathcal{A})\) generated by \(\mathrm{span}\{\mathrm{id}_{\gamma}(f) : f\in \mathcal{B}_{\gamma}, \gamma \in \Gamma\}\). To show \(\mathrm{C}^*(G;\mathcal{A})\) satisfies the universal property of \(\mathrm{C}^*(\Gamma; \mathcal{B})\), let \(\{\pi_{\gamma}\}_{\gamma \in \Gamma}\) be a representation of the group Fell bundle \(\mathcal{B}\) on a \(\mathrm{C}^*\)-algebra \(M\). We aim to construct a representation \(\pi\colon \mathrm{C}^*(G;\mathcal{A}) \to M\) such that \(\pi\circ \mathrm{id}_{\gamma} = \pi_{\gamma}\) for \(\gamma \in \Gamma\). Define \(\pi\colon \mathrm{C_c}(G;\mathcal{A}) \to M\) by \[\pi(f) =\sum_{\gamma \in \Gamma}\pi_{\gamma}(f_{\gamma})\] for \(f = \sum_{\gamma \in \Gamma}f_{\gamma} \in \mathrm{C_c}(G;\mathcal{A})\). By definition, \(\pi\) is multiplicative and preserves the involution. We now show that \(\pi\) is continuous in the inductive limit topology, and hence extends to a \(^*\)-homomorphism from \(\mathrm{C}^*(G;\mathcal{A}) \to M\). Since we can identify \(\mathrm{C}^*(G_e;\mathcal{A}|_{G_e})\) with \(\mathcal{B}_e\) by the isometric isomorphism \(\overline{\iota}\colon \mathrm{C}^*(G_e;\mathcal{A}|_{G_e}) \to \mathcal{B}_e\) of Theorem 2, we have a \(^*\)-homomorphism \(\pi_e\colon \mathrm{C}^*(G_e; \mathcal{A}|_{G_e})\to M\). Since \(\mathrm{C}^*(G_e; \mathcal{A}|_{G_e})\) is a Fell bundle algebra, \(\pi_e\) is \(I\)-norm bounded on \(\mathrm{C_c}(G_e; \mathcal{A}|_{G_e})\) by [9]. For \(f\in \mathrm{C_c}(G;\mathcal{A})\cap \mathcal{B}_{\gamma}\), we have \[\lvert\!\lvert \pi_{\gamma}(f)\rvert\!\rvert^2= \lvert\!\lvert \pi_{\gamma}(f)^*\pi_{\gamma}(f)\rvert\!\rvert =\lvert\!\lvert \pi_e({f^*f})\rvert\!\rvert \leq \lvert\!\lvert f^*f\rvert\!\rvert_I\leq \lvert\!\lvert f^*\rvert\!\rvert_I\lvert\!\lvert f\rvert\!\rvert=\lvert\!\lvert f\rvert\!\rvert^2_I\] The second inequality of the above line follows from Observation 1. Since \(\mathrm{span}\{\mathcal{B}_{\gamma} :\gamma \in \Gamma\}\) is dense in \(\mathrm{C_c}(G;\mathcal{A})\), \(\pi\) is \(I\)-norm bounded on \(\mathrm{C_c}(G;\mathcal{A})\). By the disintegration theorem ([9]), \(\pi\) is continuous in the inductive limit topology. Since \(\pi\circ \mathrm{id}_{\gamma} =\pi_{\gamma}\) holds on \(\mathrm{C_c}(G;\mathcal{A})\), the equality extends to \(\mathrm{C}^*(G;\mathcal{A})\). Therefore, \(\mathrm{C}^*(G;\mathcal{A})\) has the universal property of \(\mathrm{C}^*(\Gamma; \mathcal{B})\), and hence the two \(\mathrm{C}^*\)-algebras \(\mathrm{C}^*(G;\mathcal{A})\) and \(\mathrm{C}^*(\Gamma; \mathcal{B})\) are isomorphic. ◻
Remark 1. Let \(p\colon \mathcal{A}\to G\) be a Fell bundle over an amenable étalegroupoid \(G\) (see [16] for the definition of amenable groupoid) with a continuous \(1\)-cocycle \(c\colon G \to \mathbb{Z}\). Assume that the cocycle \(c\) is unperforated in the sense of [17], that is, in some sense \(c^{-1}(e)\) generates \(G\). Then [17] shows that \(\mathrm{C}^*(G;\mathcal{A})\) can be realized as the Cuntz–Pimsner algebra of a \(\mathrm{C}^*\)-correspondence constructed from the cocycle date \(c\). Hence, Theorem 3 yields a topological grading of the Cuntz–Pimsner algebra constructed in [17].
In this section, we exploit the topological grading (equivalently, the group Fell bundle structure) of the section algebra \(\mathrm{C}^*(G;\mathcal{A})\) established in Section 3 to construct a coaction of the underlying discrete group on \(\mathrm{C}^*(G;\mathcal{A})\). We then identify the crossed product by this coaction with the \(\mathrm{C}^*\)-algebra of a Fell bundle over the skew-product groupoid (see Theorem 4). For background on coactions of discrete groups and their crossed products, we refer the reader to Section 2.3.
Proposition 1. Let \(p\colon \mathcal{A}\to G\) be a Fell bundle over a locally compact Hausdorff second countable groupoid equipped with a Haar system. Let \(\Gamma\) be a discrete group with a continuous \(1\)-cocycle \(c\colon G\to \Gamma\). Then there exists a coaction \(\delta\) of \(\Gamma\) on \(\mathrm{C}^*(G;\mathcal{A})\) such that \[\label{eq-coaction-prop-1} \delta(f_{\gamma}) = f_{\gamma}\otimes \gamma\qquad{(1)}\] where \({\mathrm{supp}}(f_{\gamma}) \subseteq G_{\gamma}\).
Proof. Note that the group Fell bundle algebra \(\mathrm{C}^*(\Gamma, \mathcal{B})\) is isomorphic to \(\mathrm{C}^*(G;\mathcal{A})\) by Theorem 3. Thus, the map \(\delta \colon \mathrm{C_c}(\Gamma, \mathcal{B}) \to \mathrm{C}^*(\Gamma, \mathcal{B})\otimes \mathrm{C}^*(\Gamma)\) given by Equation ?? gives a \(^*\)-homomorphism. Since the enveloping \(\mathrm{C}^*\)-algebra of \(\mathrm{C_c}(\Gamma; \mathcal{B})\) is \(\mathrm{C}^*(\Gamma; \mathcal{B}) \cong \mathrm{C}^*(G;\mathcal{A})\), \(\delta\) extends to a unique \(^*\)-homomorphism \(\delta \colon \mathrm{C}^*(G;\mathcal{A}) \to \mathrm{C}^*(G;\mathcal{A})\otimes \mathrm{C}^*(\Gamma)\). We now verify the coaction identity on the generating elements \(f_{\gamma}\): \[\begin{gather} (\delta\otimes \mathrm{id}_{G})\circ \delta (f_\gamma) = (\delta\otimes \mathrm{id}_{G})(f_{\gamma} \otimes \gamma) = (f_{\gamma}\otimes \gamma)\otimes \gamma = f_{\gamma} \otimes (\gamma\otimes \gamma) \\ =\mathrm{id}_{\mathrm{C}^*(G;\mathcal{A})}\otimes \delta_{G}(f_\gamma \otimes \gamma) = (\mathrm{id}_{\mathrm{C}^*(G;\mathcal{A})} \otimes \delta_{G}) \circ \delta (f_\gamma). \end{gather}\] Therefore, the coaction identity holds on \(\mathrm{C}^*(G;\mathcal{A})\). Choose an approximate identity \((u_\alpha)_{\alpha}\) of \(\mathrm{C}^*(G_e;\mathcal{A}|_{G_e})\) in \(\mathrm{C_c}(G_e;\mathcal{A}|_{G_e})\), then we have \[\delta(f_\gamma) (u_\alpha\otimes \gamma^{-1}\eta) = f_{\gamma}u_{\alpha} \otimes \eta \to f_{\gamma}\otimes \eta \quad \mathrm{as } \alpha \to \infty.\] Therefore, the map \(\delta\) is nondegenerate in the sense \[\overline{\mathrm{span}} \{\delta(\mathrm{C}^*(G; \mathcal{A}))(1 \otimes \mathrm{C}^*(\Gamma))\} = \mathrm{C}^*(G;\mathcal{A}) \otimes \mathrm{C}^*(\Gamma),\] where \(1\) is the unit of the multiplier algebra \(M(\mathrm{C}^*(G;\mathcal{A}))\). Let \(1_G\colon G \to \mathbb{C}\) be the trivial representation of \(G\). Then for \(f_{\gamma} \in \mathrm{C_c}(G;\mathcal{A})\), we have \[(\mathrm{id}_{\mathrm{C}^*(G;\mathcal{A})}\otimes 1_G)\circ \delta(f_\gamma) = (\mathrm{id}_{\mathrm{C}^*(G;\mathcal{A})}\otimes 1_G)(f_{\gamma} \otimes \gamma) = f_{\gamma}\otimes 1 = f_{\gamma} =\mathrm{id}_{\mathrm{C}^*(G;\mathcal{A})}(f_{\gamma}).\] Thus, we have \((\mathrm{id}_{\mathrm{C}^*(G;\mathcal{A})}\otimes 1_G)\circ \delta =\mathrm{id}_{\mathrm{C}^*(G;\mathcal{A})}\). Hence, [12] ensures that \(\delta\) is injective. ◻
Let \(c\colon G\to \Gamma\) be a continuous \(1\)-cocycle. We now consider the skew-product groupoid from [7]. However, for skew-product, we follow the convention of [6] rather than that of [7]. Nevertheless, these conventions yield isomorphic groupoids (see [6]).
The underlying set of the skew-product groupoid \(G(c)\) is \(G\times \Gamma\). The groupoid operations are as follows: \((g,\gamma)\) and \((h, \eta)\) are composable if and only if \((g,h) \in G^{(2)}\) and \(\gamma = c(h)\eta\) and \[(g,c(h)\eta)(h, \eta) = (gh, \eta), \quad (g,\gamma)^{-1} =(g^{-1}, c(g)\gamma).\] The range and source maps are given by \(r(g,\gamma) = (r(g), c(g)\gamma)\) and \(s(g, \gamma) = (s(g), \gamma)\). Note that \(G(c)\) is a locally compact Hausdorff groupoid. Moreover, if \(G\) has a Haar system, then \(G(c)\) also has a Haar system given by \[\int f\mathrm{d}\lambda^{(u,\gamma)} = \int_{G^{u}} f(g, c(g)^{-1}\gamma) \mathrm{d}\lambda^u(x).\] If \(G\) is an étalegroupoid, then \(G(c)\) is also étale. Define a groupoid homomorphism \(\varphi\colon G(c) \to G\) by \(\varphi(g, \gamma) =g\) for \((g,\gamma) \in G(c)\).
Let \(p\colon \mathcal{A}\to G\) be a Fell bundle and \(c\colon G \to \Gamma\) a continuous \(1\)-cocycle. Consider the pull-back bundle \(q\colon \varphi^*\mathcal{A}\to G(c)\) over \(G(c)\) along the homomorphism \(\varphi\) (recall the pull-back Fell bundle from Remark 1) \[\begin{tikzcd} \varphi^*\mathcal{A}\arrow[r, dashed] \arrow[d, "q"] & \mathcal{A}\arrow[d, "p"] \\ G(c) \arrow[r, "\varphi"] & G. \end{tikzcd}\] Our main result in this section (Theorem 4) establishes an isomorphism between the coaction crossed product \(\mathrm{C}^*(G;\mathcal{A})\rtimes_{\delta} \Gamma\) and the Fell bundle algebra \(\mathrm{C}^*(G(c); \varphi^*\mathcal{A})\).
Theorem 4. Let \(p\colon \mathcal{A}\to G\) be a Fell bundle over a locally compact Hausdorff second countable groupoid equipped with a Haar system. Let \(\Gamma\) be a discrete group with a continuous \(1\)-cocycle \(c\colon G\to \Gamma\). Consider the pull-back Fell bundle as described above, and the coaction \(\delta\) as in Proposition 1. Then the associated coaction crossed product \(\mathrm{C}^*(G;\mathcal{A})\rtimes_{\delta}\Gamma\) is isomorphic to \(\mathrm{C}^*(G(c);\varphi^*\mathcal{A})\).
Proof. Recall the group Fell bundle \(\mathcal{B}\) over \(\Gamma\) from Theorem 3. Let \(\Gamma\ltimes \Gamma\) be the groupoid with underlying set \(\Gamma\times \Gamma\) and the operations are given by \((\gamma, \eta\zeta) (\eta, \zeta) = (\gamma\eta, \zeta)\) and \((\gamma, \eta)^{-1} = (\gamma^{-1}, \gamma\eta)\). Consider the product Fell bundle \(\mathcal{B}\times \Gamma\) over \(\Gamma\ltimes \Gamma\) whose fiber at \((\gamma, \eta)\) is \(\mathcal{B}_{\gamma}\times \{\eta\}\) and the operations on \(\mathcal{B}\times \Gamma\) are given by \[(f_{\gamma}, \eta \zeta)(g_{\eta}, \zeta) = (f_{\gamma}g_{\eta}, \zeta) \quad \mathrm{and} \quad (f_{\gamma}, \eta)^* = (f^*_{\gamma}, \gamma\eta)\] where \(f_{\gamma} \in \mathcal{B}_{\gamma}, g_{\eta} \in \mathcal{B}_{\eta}\) and \(\gamma, \eta, \zeta \in \Gamma\). Our next claim is to show that \(\mathrm{C}^*(G;\mathcal{A})\rtimes_{\delta}\Gamma\) is the enveloping \(\mathrm{C}^*\)-algebra of \(\mathrm{C_c}(\Gamma\ltimes \Gamma; \mathcal{B}\times \Gamma)\). Since \(\mathrm{C}^*(G;\mathcal{A})\) is the enveloping \(\mathrm{C}^*\)-algebra of the group Fell bundle \(\mathcal{B}\) (by Theorem 3). Using [18] it is enough to show that the unit fiber algebra \(\mathrm{C}^*(G;\mathcal{A})_{e}\) of the Fell bundle associated to the coaction \(\delta\) is isomorphic to the \(\mathrm{C}^*\)-algebra \(\mathcal{B}_e\). But note that \(\mathrm{C}^*(G;\mathcal{A})_{e} = \{f\in \mathrm{C}^*(G;\mathcal{A}) : \delta(f) =f\otimes e\}\) is the closure of \(\mathrm{C_c}(G_e;\mathcal{A}|_{G_e})\) inside \(\mathrm{C}^*(G;\mathcal{A})\). Thus, Theorem 2 ensures that \(\mathrm{C}^*(G;\mathcal{A})_{e}\) is isomorphic to \(\mathcal{B}_e\). This proves the claim.
We now show that \(\mathrm{C}^*(\Gamma\ltimes \Gamma; \mathcal{B}\times \Gamma)\) is isomorphic to \(\mathrm{C}^*(G(c); \varphi^*\mathcal{A})\), \(\mathrm{C}^*\)-algebra of the pull-back Fell bundle over \(G(c)\). For \(\gamma, \eta \in \Gamma\), set \[D_{\gamma, \eta} = \big\{f\in \mathrm{C_c}(G(c);\varphi^*\mathcal{A}) : {\mathrm{supp}}(f) \subseteq G_{\gamma}\times \{\eta\}\big\}.\] Since \(\Gamma\) is discrete, the sets \(G_\gamma\times\{\eta\}\) are clopen in \(G(c)\). Every compact subset of \(G(c)\) intersects only finitely many such sets, and therefore every compactly supported section decomposes as a finite sum of elements supported in individual \(G_\gamma\times\{\eta\}\). Thus, \[\mathrm{C_c}(G(c);\varphi^*\mathcal{A}) = \mathrm{span}\{D_{\gamma,\eta}:\gamma,\eta\in\Gamma\}.\] Define a linear map \(\Phi\colon \mathrm{C_c}(\Gamma\ltimes \Gamma ;\mathcal{B}\times \Gamma)\to \mathrm{C_c}(G(c);\varphi^*\mathcal{A})\) by \[\Phi(f,\gamma)(g,\eta) = \begin{cases} f(g) & \mathrm{if } \gamma =\eta,\\ 0 & \mathrm{ otherwise. } \end{cases}\] Then \(\Phi\) can be extended to a linear bijection from \(\mathrm{C_c}(\Gamma\ltimes \Gamma ;\mathcal{B}\times \Gamma)\to \mathrm{C_c}(G(c);\varphi^*\mathcal{A})\). We now show that it preserves the multiplication and involution. Fix \(\gamma, \eta, \zeta, \xi, \alpha \in \Gamma\) and \(f_{\gamma} \in \mathcal{B}_{\gamma}, g_{\zeta}\in \mathcal{B}_{\zeta}\) and \(h\in G\). Then we have \[\begin{gather} \Phi(f_{\gamma}, \eta)*\Phi(g_{\zeta}, \xi)(h, \alpha) \\= \int_{G(c)^{r(h, \alpha)}} \Phi(f_{\gamma}, \eta) (k,\beta)\Phi(g_{\zeta}, \xi)((k, \beta)^{-1}(h, \alpha))\mathrm{d}\lambda^{(r(h), c(h)\alpha)} (k,\beta). \end{gather}\] Now \((k, \beta) \in G(c)^{r(h, \alpha)}\) if and only if \(r(k) = r(h)\) and \(c(k)\beta = c(h)\alpha\), that is, \(\beta = c(k^{-1}h)\alpha\). Thus, the last integral become \[\label{eq-Main-thm-mult-comp-1} \int_{G^{r(h)}} \Phi(f_{\gamma}, \eta) (k,c(k)^{-1}c(h)\alpha)\Phi(g_{\zeta}, \xi)(k^{-1}h, \alpha)\mathrm{d}\lambda^{r(h)} (k).\tag{16}\] Using the definition of \(\Phi\), we have \[\Phi(f_{\gamma}, \eta) (k,c(k)^{-1}c(h)\alpha) = \begin{cases} f_{\gamma}(k) & \mathrm{if } c(k)^{-1}c(h)\alpha =\eta,\\ 0 & \mathrm{ otherwise } \end{cases}\] and \[\Phi(g_{\zeta}, \xi)(k^{-1}h, \alpha) = \begin{cases} g_{\zeta}(k^{-1}h) & \mathrm{if } \alpha =\xi,\\ 0 & \mathrm{ otherwise. } \end{cases}\] Since \({\mathrm{supp}}(f_{\gamma})\subseteq G_{\gamma}\) and \({\mathrm{supp}}(g_{\zeta})\subseteq G_{\zeta}\), the integral in Equation 16 is nonzero only when \(c(k) = \gamma\) and \(c(k^{-1}h) = \zeta\), that is, \(c(h) =s(\zeta)\) and \(c(k)^{-1}c(h)\xi=\zeta\xi\). This implies that \(\eta =\zeta\xi\). Therefore, the integral in Equation 16 can be written as \[\begin{align} &\begin{cases} \int_{G^{r(h)}} f_{\gamma}(k)g_{\zeta}(k^{-1}h)\mathrm{d}\lambda^{r(h)}(k) & \mathrm{if } \alpha=\xi \mathrm{ and } \eta = \zeta\xi,\\ 0 & \mathrm{ otherwise } \end{cases}\\ =& \begin{cases} f_{\gamma}*g_{\zeta}(h) & \mathrm{if } \alpha=\xi \mathrm{ and } \eta = \zeta\xi,\\ 0 & \mathrm{ otherwise } \end{cases}\\ =& \begin{cases} \Phi(f_{\gamma}*g_{\zeta},\xi)(h,\alpha) & \mathrm{if } \alpha=\xi \mathrm{ and } \eta = \zeta\xi,\\ 0 & \mathrm{ otherwise } \end{cases}\\ =&\Phi((f_{\gamma}, \eta)(g_{\zeta}, \xi))(h,\alpha). \end{align}\] For involution, we have \[\begin{align} (\Phi(f_{\gamma}, \eta))^*(h,\zeta) &= \bigr(\Phi(f_\gamma, \eta)(h,\zeta)^{-1}\bigr)^* = \bigr(\Phi(f_\gamma, \eta)(h^{-1}, c(h)\zeta)\bigr)^*\\ &= \begin{cases} \big(f_{\gamma}(h^{-1})\big)^* & \mathrm{if } t=c(h)\zeta,\\ 0 & \mathrm{ otherwise} \end{cases}\\ &= \begin{cases} f^*_{\gamma}(h) & \mathrm{if } \gamma\eta=\zeta,\\ 0 & \mathrm{ otherwise } \end{cases}\\ &=\Phi(f^*_{\gamma}, \gamma\eta)(h,\zeta)\\ & = \Phi\big(f_{\gamma}, \eta\big)^*(h,\zeta). \end{align}\] Thus, \(\Phi\) is a \(^*\)-algebra isomorphism from \(\mathrm{C_c}(\Gamma\ltimes \Gamma ;\mathcal{B}\times \Gamma)\to \mathrm{C_c}(G(c);\varphi^*\mathcal{A})\). Therefore, \(\Phi\) gives a bijection between the nondegenerate representations of \(\mathrm{C_c}(\Gamma\ltimes \Gamma ;\mathcal{B}\times \Gamma)\) and \(\mathrm{C_c}(G(c);\varphi^*\mathcal{A})\) and hence \(\Phi\) can be extended to be an isomorphism \(\mathrm{C}^*(\Gamma\ltimes\Gamma;\mathcal{B}\times\Gamma) \cong \mathrm{C}^*(G(c);\varphi^*\mathcal{A})\). Therefore, the coaction crossed product \(\mathrm{C}^*(G;\mathcal{A})\rtimes_{\delta}\Gamma\) is isomorphic to \(\mathrm{C}^*(G(c);\varphi^*\mathcal{A})\). ◻
Let \(G\) be a locally compact Hausdorff second countable groupoid equipped with a Haar system and \(\sigma \in Z^2(G,\mathbb{T})\). Let \(\mathcal{A}= G\times \mathbb{C}\) equip with the product topology. Define \(p\colon \mathcal{A}\to G\) by \(p(g, z) =g\). Then \(\mathcal{A}\) is a Fell bundle, called Fell line bundle, with respect to the operations \[(g,z)(h,w) =(gh, \sigma(g,h)zw) \quad \mathrm{and} \quad (g,z)^*=(g^{-1}, \overline{\sigma(g,g^{-1})z}).\] Then the Fell bundle algebra \(\mathrm{C}^*(G;\mathcal{A})\) is isomorphic to the twisted groupoid \(\mathrm{C}^*\)-algebra \(\mathrm{C}^*(G,\sigma)\) (see [19]). Let \(c\colon G \to \Gamma\) be a continuous \(1\)-cocycle, where \(\Gamma\) is a discrete group. The restriction of \(\sigma\) on \(G_e\) gives a \(\mathbb{T}\)-valued \(2\)-cocycle on \(G_e\), and the corresponding Fell line bundle over \(G_e\) can be identified with the restricted Fell bundle \(\mathcal{A}|_{G_e}\). Therefore, applying Theorem 2 on the Fell line bundle and identifying the Fell bundle \(\mathrm{C}^*\)-algebras with the corresponding twisted groupoid \(\mathrm{C}^*\)-algebras, we obtain the following corollary.
Corollary 1. Let \(G\) be a locally compact Hausdorff second countable groupoid equipped with a Haar system. Let \(\sigma \in Z^2(G,\mathbb{T})\) and \(c\colon G\to \Gamma\) a continuous \(1\)-cocycle. The map \(\iota\colon \mathrm{C_c}(G_e, \sigma) \to \mathrm{C_c}(G, \sigma)\) defined by \[\iota(f)(\gamma) = \begin{cases} f(\gamma) & \mathrm{if } \gamma \in G_e,\\ 0 & \mathrm{ otherwise } \end{cases}\] extends to an injective homomorphism from \(\mathrm{C}^*(G_e,\sigma) \to \mathrm{C}^*(G,\sigma)\).
Corollary 1 extends Theorem 1.1 of [2] to the setting of twisted groupoid \(\mathrm{C}^*\)-algebras.
Proposition 1. Let \(G\) be a locally compact Hausdorff second countable groupoid equipped with a Haar system. Let \(\sigma \in Z^2(G,\mathbb{T})\) and let \(\Gamma\) be a discrete group with a continuous \(1\)-cocycle \(c\colon G\to \Gamma\). Then there exists a coaction \(\delta\) of \(\Gamma\) on \(\mathrm{C}^*(G,\sigma)\) and the associated coaction crossed product \(\mathrm{C}^*(G,\sigma)\rtimes_{\delta}\Gamma\) is isomorphic to \(\mathrm{C}^*(G(c),\sigma)\).
Proof. Consider the Fell line bundle \(\mathcal{A}=G\times \mathbb{C}\) over \(G\) associate to \(\sigma\). Proposition 1 ensures that there exists a coaction of \(\Gamma\) on \(\mathrm{C}^*(G;\mathcal{A})\) and hence on \(\mathrm{C}^*(G,\sigma)\). Moreover, Theorem 4 allows us to identify the coaction crossed product \(\mathrm{C}^*(G,\sigma)\rtimes_{\delta}\Gamma\) with \(\mathrm{C}^*(G(c); \varphi^*\mathcal{A})\), where \(\varphi^*\mathcal{A}\) is the pull-back Fell bundle over the skew-product \(G(c)\). Note that the \(2\)-cocycle \(\sigma\) can be lifted to the skew-product as follows: \(\widetilde{\sigma}\colon G(c)^{(2)}\to \mathbb{T}\) by \(\widetilde{\sigma}((g,\gamma), (h, \eta)) =\sigma(g,h)\). And the pull-back bundle \(\varphi^*\mathcal{A}\) is a Fell line bundle as \(\varphi^*\mathcal{A}_{(g,\gamma)} \cong \mathcal{A}_{\varphi(g,\gamma)} = \mathcal{A}_g\cong \mathbb{C}\) and the operations are given by \[((g,\gamma),z)((h,\eta), w) = ((g,\gamma)(h,\eta), \sigma(g,h)zw)\] and \[((g,\gamma),z)^*=((g, \gamma)^{-1}, \overline{\sigma(g,g^{-1})z}).\] Therefore, this pull-back bundle \(\varphi^*\mathcal{A}\) can be identify the Fell line bundle over \(G(c)\) associated to the twist (2-cocycle) \(\widetilde{\sigma}\). Hence, we have \[\mathrm{C}^*(G,\sigma)\rtimes_{\delta}\Gamma \cong \mathrm{C}^*(G(c);\varphi^*\mathcal{A})\cong\mathrm{C}^*(G(c), \widetilde{ \sigma}) = \mathrm{C}^*(G(c), \sigma).\] ◻
Remark 1. Proposition 1 applies to \(\sigma=1\) recovers a result of Kaliszewski, Quigg and Raeburn in [6]. This proposition gives a non-étale version of their result.
Higher-rank graphs or \(k\)-graphs are introduced by Kumjian and Pask in [4] as a higher dimensional generalization of Cuntz–Krieger algebras. A \(k\)-graph is a countable category \(\Lambda\) together with a functor \(d\colon \Lambda \to \mathbb{N}^{k}\) satisfying the factorization property, that is, for every \(\lambda \in \Lambda\) with \(d(\lambda) = m+n\) there are unique elements \(\mu, \gamma \in \Lambda\) such that \[s(\mu) =r(\gamma),\quad d(\mu) = m, \quad d(\gamma) =n \quad \mathrm{and}\quad \lambda = \mu\gamma.\] The set of vertices is denoted by \(\Lambda^{0}\) and can be identified with \(d^{-1}(0)\).
Let \(\Lambda^2=\{(\lambda, \mu) \in \Lambda \times \Lambda : s(\lambda) =r(\mu)\}\). A map \(\sigma\colon \Lambda^2 \to \mathbb{T}\) is called a \(\mathbb{T}\)-valued \(2\)-cocycle on \(\Lambda\) if \(\sigma(r(\lambda),\lambda) =\sigma(\lambda, s(\lambda))=1\) and \[\sigma(\lambda, \mu)\sigma(\lambda\mu, \nu) = \sigma(\lambda, \mu\nu)\sigma(\mu, \nu)\] for all composable triples \((\lambda, \mu,\nu)\). The set of all \(\mathbb{T}\)-valued \(2\)-cocycle is denoted by \(Z^2(\Lambda, \mathbb{T})\). For a row finite \(k\)-graph with a \(2\)-cocycle \(\sigma\), the twisted \(k\)-graph \(\mathrm{C}^*\)-algebra will be denoted by \(\mathrm{C}^*(\Gamma, \sigma)\) (see [5] for a definition of twisted \(k\)-graph algebra).
Let \(\Omega_k = \{(m,n)\in \mathbb{N}^k\times \mathbb{N}^k : m\leq n\}\). Then \(\Omega_k\) forms a \(k\)-graph, where the structure maps are given by \(r(m,n) = (m,m), s(m,n) =(n,n)\), \((m,n)(n,k) = (m,k)\) and \(d(m,n) = n-m\). The vertex set \(\Omega^0_k\) can be identified with \(\mathbb{N}^k\).
The infinite-path space of a \(k\)-graph \(\Lambda\) is given by \[\Lambda^{\infty} = \{x\colon \Omega_k \to \Lambda \mid x \mathrm{ is a functor that intertwines the degree maps}\}.\] For \(l\in \mathbb{N}^{k}\), the shift map \(\rho^l\colon \Lambda^{\infty} \to \Lambda^{\infty}\) is defined by \[\rho^l(x)(m,n) \mathrel{\vcentcolon=}x(m+l, n+l).\] Let \(\Lambda\) be a row finite \(k\)-graph with no source. Then \[G_{\Lambda} :=\big\{(x,l,y) \in \Lambda^{\infty}\times \mathbb{Z}^k\times \Lambda^{\infty} : l=m-n, m,n\in \mathbb{N}^k \mathrm{ and } \rho^m(x) =\rho^n(y)\big\}\] is a groupoid, called infinite-path groupoid. The structure maps are given by \(r(x,l,y)= x, s(x,l,y) =y, (x,l,y) (y,l^{\prime}, z) = (x, l+l^{\prime}, z)\) and \((x,l,y)^{-1} = (y,-l,x)\). The unit space of \(G_{\Lambda}\) can be identified with \(\Lambda^{\infty}\). For \(\lambda, \mu \in \Lambda\) satisfying \(s(\lambda) = r(\mu)\), define \[Z(\lambda,\mu) \mathrel{\vcentcolon=}\big\{(\lambda x,\, d(\lambda) - d(\mu),\, \mu x) \in \mathcal{G}_\Lambda : x \in \Lambda^\infty \mathrm{ with } r(x) = s(\lambda)\big\}.\] The collection \(\{Z(\lambda,\mu) : \lambda, \mu \in \Lambda\}\) forms a basis for a locally compact Hausdorff topology on \(G_\Lambda\). Moreover, \(G_{\Lambda}\) is an étale groupoid ([4]).
Let \(\Lambda{_s *_s} \Lambda := \{(\lambda,\mu) \in \Lambda \times \Lambda : s(\lambda)=s(\mu)\}.\) Fix a subset \(\mathcal{P} \subseteq \Lambda {_s *_s} \Lambda\) such that \((\lambda,s(\lambda)) \in \mathcal{P}\) for all \(\lambda \in \Lambda\) and \[G_{\Lambda} = \bigsqcup_{(\lambda,\mu)\in \mathcal{P}} Z(\lambda,\mu).\] Such a choice is always possible by [5]. For \(\alpha \in G_{\Lambda}\), let \((\lambda_\alpha,\mu_\alpha)\) denote the unique element of \(\mathcal{P}\) with \(\alpha \in Z(\lambda_\alpha,\mu_\alpha)\), and define \(f\colon G_{\Lambda} \to \mathbb{Z}^k\) by \(f(x,n,y)=n\). Let \(\omega\) be a \(2\)-cocycle on \(\Lambda\). Then [5] ensures that for any composable pair \((\alpha,\beta) \in G^{(2)}_{\Lambda}\), there exist \(\nu,\xi,\zeta \in \Lambda\) and \(y \in \Lambda^\infty\) satisfying \(\mu_\alpha \nu = \lambda_\beta \xi, \lambda_\alpha \nu = \lambda_{\alpha\beta} \zeta, \mu_\beta \xi = \mu_{\alpha\beta} \zeta\), and \[\alpha = (\lambda_\alpha \nu y,f(\alpha),\mu_\alpha \nu y),\quad \beta = (\lambda_\beta \xi y,f(\beta),\mu_\beta \xi y),\quad \alpha\beta = (\lambda_{\alpha\beta} \zeta y, f(\alpha\beta), \mu_{\alpha\beta} \zeta y).\] Then the map \(\sigma_{\omega}\colon G^{(2)}_{\Lambda} \to \mathbb{T}\) by \[\label{eq-2-cocycle-gpd} \sigma_{\omega}(\alpha,\beta) = \omega(\lambda_\alpha,\nu)\overline{\omega(\mu_\alpha,\xi)}\omega(\lambda_\beta,\xi)\overline{\omega(\mu_\beta,\xi)}\overline{\omega(\lambda_{\alpha\beta},\zeta)}\omega(\mu_{\alpha\beta},\zeta)\tag{17}\] defines a continuous \(2\)-cocycle on \(G_{\Lambda}\), independent of the choice of \(\nu,\xi,\zeta\). Moreover, by [5], the cocycles arising from different choices of \(\mathcal{P}\) are cohomologous. And the twisted \(k\)-graph \(\mathrm{C}^*\)-algebra \(\mathrm{C}^*(\Lambda, \omega)\) is isomorphic to the twisted groupoid algebra \(\mathrm{C}^*(G_{\Lambda}, \sigma_{\omega})\) (by [5]).
A \(\Gamma\)-valued \(1\)-cocycle \(\eta\) on \(\Lambda\) is a functor \(\eta \colon \Lambda \to \Gamma\). Then the skew-product \(k\)-graph \(\Lambda\times_{\eta}\Gamma\) is defined by \((\Lambda\times_{\eta}\Gamma)^0 = \Lambda^0\times \Gamma,\) where \(r,s\colon \Lambda\times_{\eta} \Gamma \to (\Lambda\times_{\eta} \Gamma)^0\) by \(r(\lambda, t) = (r(\lambda), t), s(\lambda,t) = (s(\lambda), \eta(\lambda)t)\) and \(d\colon \Lambda\times_{\eta} \Gamma \to \mathbb{N}^k\) by \(d(\lambda, t) = d(\lambda)\). The map \(\eta\) induces a \(1\)-cocycle \(c_\eta\colon G_{\Lambda} \to \Gamma\) by \[\label{eq-cocycle-gpd} c_{\eta}(x,m-n,y) = \eta(x(m,0))\eta(y(n,0))^{-1}.\tag{18}\]
Proposition 1. Let \(\Lambda\) be a row-finite \(k\)-graph with no sources and let \(\omega \in Z^2(\Lambda,\mathbb{T})\). Let \(\eta\colon \Lambda \to \Gamma\) be a \(1\)-cocycle, where \(\Gamma\) is a discrete group. Then there exists a coaction \(\delta\) of \(\Gamma\) on \(\mathrm{C}^*(\Lambda,\omega)\). Moreover, \[\mathrm{C}^*(\Lambda,\omega)\rtimes_\delta \Gamma \;\cong\; \mathrm{C}^*(\Lambda\times_\eta \Gamma,\widetilde{\omega}),\] where \(\widetilde{\omega}\) is the \(2\)-cocycle on \(\Lambda\times_\eta \Gamma\) defined by \(\widetilde{\omega}((\lambda,g),(\mu,\eta(\lambda)g)) := \omega(\lambda,\mu).\)
Proof. By [5], there exists an isomorphism \[\mathrm{C}^*(\Lambda,\omega) \;\cong\; \mathrm{C}^*(G_\Lambda,\sigma_\omega),\] where \(\sigma_\omega\) is the groupoid \(2\)-cocycle defined by Equation 17 . The \(1\)-cocycle \(\eta\colon \Lambda \to \Gamma\) induces a continuous groupoid \(1\)-cocycle \(c_\eta\colon G_\Lambda \to \Gamma\) given by Equation 18 . Hence, by Proposition 1, there exists a coaction \(\delta\) of \(\Gamma\) on \(\mathrm{C}^*(G_\Lambda,\sigma_\omega)\) such that \[\mathrm{C}^*(G_\Lambda,\sigma_\omega)\rtimes_\delta \Gamma \;\cong\; \mathrm{C}^*(G_\Lambda(c_\eta),\sigma_\omega),\] where \(G_\Lambda(c_\eta)\) is the skew-product groupoid.
Our next claim is to establishes an isomorphism of twisted groupoid \(\mathrm{C}^*\)-algebras \[\label{eq-identifiaction-gpd-skew-prd} \mathrm{C}^*(G_\Lambda(c_\eta),\sigma_\omega) \;\cong\; \mathrm{C}^*(G_{\Lambda\times_\eta \Gamma},\sigma_{\widetilde{\omega}}).\tag{19}\] Proof of the claim. We can identify the infinite-path space of \(\Lambda\times_{\eta}\Gamma\) with \(\Lambda^{\infty}\times \Gamma\). Under this identification, the shift map satisfies the following: \[\rho^r(x,g) = \big(\rho^r(x),\, g\,\eta(x(r,0))\big),\] for \(r\in \mathbb{N}^k\). Let \(((x,g),\, l,\, (y,h)) \in G_{\Lambda\times_\eta \Gamma}\). Then \(\rho^m(x,g)=\rho^n(y,h)\) for \(m,n\in\mathbb{N}^k\) with \(l=m-n\). This give us \[\rho^m(x)=\rho^n(y) \quad \text{and} \quad g\,\eta(x(m,0)) = h\,\eta(y(n,0)).\] The last equality gives us \[h = g\,\eta(x(m,0))\,\eta(y(n,0))^{-1} = g\,c_\eta(x,m-n,y).\] Therefore, \[G_{\Lambda\times_\eta \Gamma} = \Big\{((x,g),l,(y,h)) : (x,l,y)\in G_\Lambda,\;h = g\,c_\eta(x,l,y)\Big\}.\] Define \(\Phi: G_\Lambda(c_\eta) \to G_{\Lambda\times_\eta \Gamma}\) by \[\Phi((x,l,y),g) = \big((x,g),\,l,\,(y, g\,c_\eta(x,l,y))\big).\] Using the definition of \(c_\eta\), one checks that \(\Phi\) is a well-defined bijection preserving multiplication and inversion. Hence \(\Phi\) is a groupoid isomorphism. Since the cocycle \(\widetilde{\omega}\) is defined by \(\widetilde{\omega}((\lambda,g),(\mu,g\eta(\lambda)))=\omega(\lambda,\mu)\), the associated groupoid cocycle \(\sigma_{\widetilde{\omega}}\) depends only on the underlying \(\Lambda\)-paths. Therefore, \(\Phi\) intertwines the cocycles \(\sigma_\omega\) and \(\sigma_{\widetilde{\omega}}\) and so induces an isomorphism \[\mathrm{C}^*(G_\Lambda(c_\eta),\sigma_\omega) \;\cong\; \mathrm{C}^*(G_{\Lambda\times_\eta \Gamma},\sigma_{\widetilde{\omega}}).\] This proves the claim. Combining the above identifications, we obtain \[\begin{align} \mathrm{C}^*(\Lambda\times_\eta \Gamma,\widetilde{\omega}) &\cong \mathrm{C}^*(G_{\Lambda\times_\eta \Gamma},\sigma_{\widetilde{\omega}}) \quad (\mathrm{by \cite{Kumjian-Pask-Sims2015Twisted-Higher-rank-graph-alg}})\\ &\cong \mathrm{C}^*(G_\Lambda(c_\eta),\sigma_\omega) \quad \mathrm{ (by Equation~\eqref{eq-identifiaction-gpd-skew-prd})}\\ &\cong \mathrm{C}^*(G_\Lambda,\sigma_\omega)\rtimes_\delta \Gamma \quad \mathrm{(by Proposition~\ref{prop-coation-twisted-ver})} \\ &\cong \mathrm{C}^*(\Lambda,\omega)\rtimes_\delta \Gamma \quad \mathrm{(by \cite{Kumjian-Pask-Sims2015Twisted-Higher-rank-graph-alg})}, \end{align}\] which completes the proof. ◻
Remark 1. In particular, Proposition 1 extends a result of Kaliszewski–Quigg–Raeburn [6] from graph \(\mathrm{C}^*\)-algebras to the broader setting of twisted higher-rank graph \(\mathrm{C}^*\)-algebras.
We thank Rohit Dilip Holkar and Ralf Meyer for fruitful discussions.