Asymptotic quasi--local structure induced by Magnetic field

We introduce a magnetic coherent–state frame over the Landau levels of the two–dimensional Landau Hamiltonian, indexed by a discrete metric space combining spatial and energy degrees of freedom. Although overcomplete, the frame is almost–orthogonal, and we use it to endow the CAR $C^{\ast }$–algebra over the one–particle Hilbert space with an \emph{asymptotic quasi–local} (AQL) structure, in which graded commutators of observables localized on disjoint index sets decay exponentially rather than vanish exactly. By means of a family of conditional expectations onto the resulting local subalgebras, we construct a Fréchet $\ast$–subalgebra of almost–local observables, invariant under the discrete group of lattice translations, which act on it by continuous automorphisms. For the full continuous group of magnetic translations we establish a quantitative control on how local observables spread through this almost–local algebra, and we show that this control suffices to extend graded asymptotic abelianness from the lattice to the entire continuous translation group. The results provide a discrete–continuum dictionary intended as the starting point for extending Lieb–Robinson–type techniques from lattice systems to interacting continuum electrons in a magnetic field.