  Classical mechanics is presented here in a unary operator form, constructed
using the binary multiplication and Poisson bracket operations that are given
in a phase space formalism, then a Gibbs equilibrium state over this unary
operator algebra is introduced, which allows the construction of a Hilbert
space as a representation space of a Heisenberg algebra, giving a
noncommutative operator algebraic variant of the Koopman-von Neumann approach.
In this form, the measurement theory for unary classical mechanics can be the
same as and inform that for quantum mechanics, expanding classical mechanics to
include noncommutative operators so that it is close to quantum mechanics,
instead of attempting to squeeze quantum mechanics into a classical mechanics
mold. The measurement problem as it appears in unary classical mechanics
suggests a classical signal analysis approach that can also be successfully
applied to the measurement problem of quantum mechanics. The development offers
elementary mathematics that allows a formal reconciliation of "collapse" and
"no-collapse" interpretations of quantum mechanics.
