A solution of the maximality problem for one-parameter dynamical systems

We prove a maximality theorem for one-parameter dynamical systems that include W*-, C*- and multiplier one-parameter dynamical systems. Our main result is new even for one-parameter actions on commutative multiplier algebras including the algebra Cb(R) of bounded continuous functions on R acted upon by translations. The methods we develop and use in our characterization of maximality include harmonic analysis, topological vector spaces and operator algebra techniques.