is finite, where $\cl L.$ is a generic infinite dimensional separable Hilbert space. So $${\mathfrak A}_{\Lambda} ={\mathfrak F}_{\Lambda} \otimes\mathop{\bigotimes}\limits_{\ell\in\Lambda^1}\big(C(G)\rtimes_\lambda G\big)\cong
{\mathfrak F}_{\Lambda} \otimes\cl K.\big(\mathop{\otimes}\limits_{\ell\in\Lambda^1}L^2(G)\big)\cong\cl K.(\cl L.)$$ as ${\mathfrak F}_{\Lambda}$ is a full matrix algebra. This shows that for a finite lattice there will be only one irreducible representation, up to unitary equivalence. Also, ${\mathfrak A}_{\Lambda}$ is simple, so all representations are faithful.

The algebra ${\mathfrak A}_{\Lambda}$ is faithfully and irreducibly represented on $\cl H.=\cl H._F\otimes\mathop{\bigotimes}\limits_{\ell\in\Lambda^1}L^2(G)$ by $\pi=\pi_F\otimes\big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1}\pi_{\ell}\big)$ where $\pi_\ell:C(G)\rtimes_\lambda G\to L^2(G)$ is the generalized Schrödinger representation for the $\ell^{\rm th}$ entry. Then $\pi\big({\mathfrak A}_{\Lambda}\big)$ contains in its multiplier algebra the operators ${\widehat}{T}^{(\ell)}_{\Phi_{ij}(\ell)},\;{\widehat}{U}^{(\ell)}_g$ for all $\ell\in \Lambda^1$.

To complete the picture, we also need to define the action of the local gauge group on ${\mathfrak A}_{\Lambda}$. Recall that in $\pi$ it is given by $\zeta\to{\rm Ad}({\widehat}{W}_\zeta)$, and this clearly preserves $\pi\big({\mathfrak A}_{\Lambda}\big)=\cl K.(\cl H.)$ and defines a strongly continuous action $\alpha$ of $\gauc \Lambda$ on $\pi\big({\mathfrak A}_{\Lambda}\big)$ (hence on ${\mathfrak A}_{\Lambda} $) as $\zeta\to{\widehat}{W}_\zeta$ is strong operator continuous. By construction ${(\pi,{\widehat}{W})}$ is a covariant representation for the C\*-dynamical system given by $\alpha:\gauc \Lambda \to\aut{\mathfrak A}_{\Lambda}$. As $\gauc \Lambda=\prod\limits_{x\in \Lambda^0}G$ is compact, we can construct the crossed product ${\mathfrak A}_{\Lambda}\rtimes_\alpha \gauc \Lambda$ 