gauge invariant element $W(L)\in C(G_1)\otimes\cdots\otimes C(G_m)={C(G_1\times\cdots G_m)}$. To see the gauge invariance, just note that $$\big(\zeta\cdot\Phi)(\ell)\big(\zeta\cdot\Phi)(\ell')=
    \zeta(x_\ell)\,\Phi(\ell)\,\zeta(y_\ell)^{-1}\zeta(x_{\ell'})\,\Phi(\ell')\,\zeta(y_{\ell'})^{-1}
    =\zeta(x_\ell)\,\Phi(\ell)\,\Phi(\ell')\,\zeta(y_{\ell'})^{-1}$$ if $y_\ell=x_{\ell'}$ (i.e. $\ell'$ follows $\ell$), and use the trace property for $W(L)$.

    Wilson loops of particular importance are those where the paths are plaquettes, i.e. $L={(\ell_1,\ell_2,\ell_3,\ell_4)}\in\Lambda^2$ as such $W(L)$ occur in the lattice Hamiltonian. As remarked above, as $C(G_j)\subset M({C(G_j)\rtimes_\lambda G_j})$, it is not actually contained in $\cl L._{\ell_j}$. We embed $C(G_1)\otimes\cdots\otimes C(G_m)$ (hence $W(L)$) in $M({\mathfrak A}_{\Lambda})$ by letting it act as the identity in entries not corresponding to $\{\ell_1,\ell_2,\ell_3,\ell_4\}.$

-   Another method of constructing gauge invariant observables, is by Fermi bilinears connected with a Wilson line (cf. [@KS]). Consider a path $C=\{\ell_1,\ell_2,\ldots,
     \ell_m\}\subset\Lambda^1$, $\ell_j=(x_j,y_j)$, such that $y_j=x_{j+1}$ for $j=1,\ldots,m-1$. We take notation as above, so $G_k=G$ is the configuration space of $\ell_k,$ and $\Phi_{ij}(\ell_k)(g_k):=(e_i,g_ke_j)$, $g_k\in G_k$. To construct a gauge invariant observable associated with the path, consider (with summation convention): $$\begin{aligned}
     Q(C)&:=& \psi^*_{i_1}(x_1)\,\Phi_{i_1i_2}(\ell_1)\,\Phi_{i_2i_3}(\ell_3)\cdots \Phi_{i_{m-1}i_m}(\ell_m)\, \psi_{ i_m}(y_m)\\[1mm]
    &\in& {\mathfrak F}_{S}\otimes C(G_1)\otimes\cdots\otimes C(G_m)
      \end{aligned}$$ where $S\subseteq\Lambda^0$ contains the path and we assume $\Cn=\C^k$ (otherwise $\Cn=\C^k\times{\bf W}$ and there are more indices). Then $ Q(C)$ is gauge invariant. As above, we embed ${\mathfrak F}_{S}\otimes C(G_1)\otimes\cdots\otimes C(G_m)$ in $M({\mathfrak A}_{\Lambda})$ in the natural 