a Hilbert space $\cl H.$ which has represented on it all the local algebras ${\mathfrak A}_S$ (each associated with a finite sublattice $S$), with the correct (graded) commutation relations. We then define a conveniently large C\*-algebra $\al A._{\rm max}$ acting on this Hilbert space, and containing all the local algebras. In Sect. \[LDA\], we then define on $\al A._{\rm max}$ the “local automorphism groups" $\alpha_t^S$, i.e. those produced by the Hamiltonians of the finite sublattices $S$. Using a Lieb–Robinson bounds argument, we then prove in Sect. \[DMFA\] that for each $A\in \al A._{\rm max}$, that $\alpha_t^S(A)$ converges in norm as $S$ increases to the full lattice, to an element $\alpha_t(A)$, and this defines a one-parameter automorphism group $t\mapsto\alpha_t\in{\rm Aut}(\al A._{\rm max})$. This is the global automorphism group, and we use it to define in Sect. \[KARR\] our chosen minimal field algebra by $${\mathfrak A}_{\Lambda}:=C^*\Big(  \bigcup_{S\in\al S.}\alpha_{\R}({\mathfrak A}_S   )    \Big)
\subset\cl A._{\rm max}    \subset\cl B.(\cl H.).$$ We also clarify the relation of ${\mathfrak A}_{\Lambda}$ with the kinematics algebra previously constructed in [@GrRu]. We prove the existence of regular ground states in Sect. \[PGDA\], and in Sect. \[GTGL\] we define gauge transformations and consider enforcement of the Gauss law constraint.

The Kinematics 