The gauge-invariant (particle number preserving) observable algebra generated by a non-relativistic Bose field is studied in the C*-algebraic framework of the resolvent algebra. It is shown that this algebra is isomorphic to the inverse limit of a system of approximately finite dimensional C*-algebras. Using this result, it is proven that the algebra is compatible with the Heisenberg picture in the sense that it is stable under the dynamics induced by Hamiltonians involving pair interactions. The argument does not require any approximations, it deals from the outset with the full dynamics. It is outlined how these results shed new light on several topics in many body theory, ranging from causality aspects over the construction of ground states and collision theory up to the determination of thermal equilibrium states. The present approach leads to conceptual simplifications and admits a unified field theoretic treatment of small and large bosonic systems.