  We present an algebraic theory of the states of the infinite classical
lattices. The construction follows the Haag-Kastler axioms from quantum field
theory. By comparison, the *-algebras of the quantum theory are replaced here
with the Banach lattices (MI-spaces) to have real-valued measurements, and the
Gelfand-Naimark-Segal construction with the structure theorem for MI-spaces to
represent the Segal algebra as C(X). The theory represents any compact convex
set of states as a decomposition problem of states on an abstract Segal algebra
C(X), where X is isomorphic with the space of extremal states of the set. Three
examples are treated, the study of groups of symmetries and symmetry breakdown,
the Gibbs states, and the set of all stationary states on the lattice. For
relating the theory to standard problems of statistical mechanics, it is shown
that every thermodynamic-limit state is uniquely identified by expectation
values with an algebraic state.
