Duality and Conditional Expectations in the Nakajima-Mori-Zwanzig Formulation

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.