information of the gauge structures required, and its representation space contained the physical representations, it suffered from the following defects.

-   The true local algebras (i.e. the kinematics algebras for the model on finite sublattices) were not subalgebras of the constructed kinematics algebra. The kinematics algebra of [@GrRu] did contain isomorphic copies of the local algebras, whose multiplier algebras contained the local algebras, but these did not satisfy local graded-commutativity. That is, they did not graded-commute if they corresponded to disjoint parts of the lattice, unlike the true local algebras. This was due to a novel form of the infinite tensor product of nonunital algebras, where approximate identities replaced the identity in the infinite “tails" of the tensor products.

-   As a consequence of this infinite tensor product, current methods of defining dynamics on lattice systems by suitable limits of the local dynamics did not apply, which made it very hard to construct dynamics for the full system.

Due to these problems, especially the latter one, we will here extend our focus to the multiplier algebra of our previous kinematics field algebra, and build an appropriate new kinematics field algebra in this setting. Our strategy will be to define a 