as before for a finite lattice.

A maximal C\*-algebra for dynamics construction. {#GFA}
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In this section we wish to define a concrete C\*-algebra which is “maximally large” in the sense that it contains all the true local C\*-algebras, as well as the (bounded) terms of the local Hamiltonians. In the next section we will then define the dynamics automorphisms on it, and finally we will then take our new field C\*-algebra to be the smallest subalgebra which contains all the true local C\*-algebras, and is preserved w.r.t. the dynamics. Thus, the dynamics itself will create the field algebra for the system. The gauge transformations will be added later.

Recall from above that for every link $\ell$ we have a generalised Weyl algebra ${C(G)\rtimes_\lambda G}
\cong\cl K.\big(L^2(G)\big)$, for which we have a generalized Schrödinger representation $\pi_0:C(G)\rtimes_\lambda G\to\cl B.\big(L^2(G)\big)$ such that $\pi_0\big(C(G)\rtimes_\lambda G\big)=\cl K.\big(L^2(G)\big)$. Explicitly, it is given by $\pi_0(\varphi\cdot f)=\pi_1(\varphi)\pi_2( f)$ for $\varphi\in L^1(G)$ and $f\in C(G)$ where $$(\pi_1(\varphi)\psi)(g):=\int\varphi(h)\psi(h^{-1}g)\,{\rm d}h \qquad\hbox{and}\qquad
 (\pi_2(f)\psi)(g):=f(g)\psi(g)$$ for all $\psi\in L^2(G)$. Note that by irreducibility, the constant vector $\psi_0=1$ is cyclic and normalized (assuming normalized Haar measure on $G$).

We start by taking an infinite product of generalized Schrödinger representations, one for each $\ell\in\Lambda^1$, w.r.t. 