On the structure of the $C^*$-algebra generated by the field operators and spectral analysis of the operators affiliated to it

We show that the $C^*$-algebra generated by the field operators associated to a symplectic space $Ξ$ is graded by the semilattice of all finite dimensional subspaces of $Ξ$. If $Ξ$ 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.

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