On the structure of the field C⁎-algebra of a symplectic space and spectral analysis of the operators affiliated to it

Abstract

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.