the reference sequence ${(\psi_0,\psi_0,\ldots)}$ where $\psi_0=1$ is the constant vector (cf.[@vN]). Thus the space $\al H._\infty$ is the completion of the pre–Hilbert space spanned by finite combinations of elementary tensors of the type $$\varphi_1\otimes\cdots\otimes \varphi_k\otimes\psi_0\otimes\psi_0\otimes\cdots,\quad\varphi_i\in \al H._i= L^2(G),\;
k\in\N$$ w.r.t. the pre–inner product given by $$\left(\varphi_1\otimes\cdots\otimes \varphi_k\otimes\psi_0\otimes\psi_0\otimes\cdots,\;
\varphi'_1\otimes\cdots\otimes \varphi'_k\otimes\psi_0\otimes\psi_0\otimes\cdots\right)_\infty
:=\prod_{i=1}^k(\varphi_i,\varphi'_i)\,,$$ and we assume the usual entry-wise tensorial linear operations for the elementary tensors. Denote the reference vector by $\psi_0^\infty:=\psi_0\otimes\psi_0\otimes\cdots$.

Fix a finite nonempty connected subgraph $S$ of $(\Lambda^0, \Lambda^1)$, and let $$\Lambda_S:=(\Lambda^0_S, \Lambda^1_S, \Lambda^2_S).$$ Then $\al L._S:={\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}\big(C(G)\rtimes_\lambda G\big)}$ acts on $\al H._\infty$ as a product representation of $\pi_0$ where each factor $\al L._\ell:=C(G)\rtimes_\lambda G$ acts on the factor of $\al H._\infty$ corresponding to $\ell$. In fact if $[\cdot]$ denotes closed span, then $$[\al L._S\psi_0^\infty]=\Big(\mathop{\bigotimes}\limits_{\ell\in\Lambda^1_S}\al H._\ell\Big)\otimes
 \mathop{\bigotimes}\limits_{\ell\not\in\Lambda^1_S}\psi_0\subset\al H._\infty.$$ Thus all $\al L._S$ are faithfully imbedded in $\al B.(\al H._\infty)$, and if $S$ and $S'$ are disjoint, $\al L._S$ and $\al L._{S'}$ commute.

Now consider the Fock representation $\pi_{\rm Fock}:{\mathfrak F}_{\Lambda}\to\al B.(\al H._{\rm Fock})$ of the CAR–algebra with vacuum vector $\Omega$. The Hilbert space on which we define our infinite lattice model is $$\al H.:=\al H._{\rm Fock}\otimes\al H._\infty.$$ Then by ${\mathfrak F}_{S}\subset{\mathfrak F}_{\Lambda}$, we also have a product representation of the local 