which has as representation space all covariant representations of $\alpha:\gauc \Lambda \to\aut{\mathfrak A}_{\Lambda}$. As it is convenient to have an identity in the algebra, our full field algebra for the system will be taken to be: $$\al F._e:=({\mathfrak A}_{\Lambda}\oplus\C)\rtimes_\alpha (\gauc \Lambda)$$ where ${\mathfrak A}_{\Lambda}\oplus\C$ denotes ${\mathfrak A}_{\Lambda}$ with an identity adjoined. This has a unique faithful representation on ${\mathcal{H}}$ corresponding to the covariant representation ${(\pi,{\widehat}{W})}$.

We consider two types of gauge invariant observables for lattice QCD which appeared in the literature (cf. [@KS]).

-   We start with gauge invariant variables of pure gauge type, and consider the well-known Wilson loops cf. [@Wilson]. To construct a Wilson loop, we choose an oriented loop $L=\{\ell_1,\ell_2,\ldots,
     \ell_m\}\subset\Lambda^1$, $\ell_j=(x_j,y_j)$, such that $y_j=x_{j+1}$ for $j=1,\ldots,{m-1}$ and $y_m=x_1$. Let $G_k=G$ be the configuration space of $\ell_k$. Denoting the components of a gauge potential $\Phi$ by $\Phi_{ij}(\ell_k)\in C(G_k)$ as in equation , the matrix components of the quantum connection at $\ell_k$ are given by $T_{\Phi_{ij}(\ell_k)}$. To construct the gauge invariant observable associated with the loop, define (summing over repeated indices): $$\begin{aligned}
     W(L)&:=&\Phi_{i_1i_2}(\ell_1)(g_1)\,\Phi_{i_2i_3}(\ell_3)(g_2)\cdots \Phi_{i_{m-1}i_1}(\ell_m)(g_m)\\[1mm]
     &=&(e_{i_1},g_1g_2\cdots g_me_{i_1})= {\rm Tr}(g_1g_2\cdots g_m)\,.
      \end{aligned}$$ (Note that to perform the product we need to fix identifications of $G_i$ with $G$.) This defines a 