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

Abstract

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

This is a deeply revised version of the paper (55 pages instead of 44). The title is modified. Definitions (1.3) and (3.32) of the first version are wrong (this plays no role in later arguments there), they are corrected here, see Remark 2.26 of this version