  We develop a new operator algebraic formulation of the Nakajima-Mori-Zwanzig
(NMZ) method of projections. The new theory is built upon rigorous mathematical
foundations, and it can be applied to both classical and quantum systems. We
show that a duality principle between the NMZ formulation in the space of
observables and in the state space can be established, analogous to the
Heisenberg and Schr\"odinger pictures in quantum mechanics. Based on this
duality we prove that, under natural assumptions, the projection operators
appearing in the NMZ equation must be conditional expectations. The proposed
formulation is illustrated in various examples.
