  We construct C*-dynamical systems for the dynamics of classical infinite
particle systems describing harmonic oscillators interacting with arbitrarily
many neighbors on lattices, as well on more general structures. Our approach
allows particles with varying masses, varying frequencies, irregularly placed
lattice sites and varying interactions subject to a simple summability
constraint. A key role is played by the commutative resolvent algebra, which is
a C*-algebra of bounded continuous functions on an infinite dimensional vector
space, and in a strong sense the classical limit of the Buchholz--Grundling
resolvent algebra, which suggests that quantum analogs of our results are
likely to exist. For a general class of Hamiltonians, we show that the
commutative resolvent algebra is time-stable, and admits a time-stable
sub-algebra on which the dynamics is strongly continuous, therefore obtaining a
C*-dynamical system.
