On the structure of the $C^{\ast }$-algebra generated by the field operators and spectral analysis of the operators affiliated to it

We show that the $C^{\ast }$-algebra generated by the field operators associated to a symplectic space $\mathrm{\Xi }$ is graded by the semilattice of all finite dimensional subspaces of $\mathrm{\Xi }$. If $\mathrm{\Xi }$ is finite dimensional we give a simple intrinsic description of the components of the grading, we show that the self-adjoint operators affiliated to the algebra have a many channel structure similar to that of N-body Hamiltonians, in particular their essential spectrum is described by a kind of HVZ theorem, and we point out a large class of operators affiliated to the algebra.