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

## TL;DR

This paper analyzes the structure of a specific $C^*$-algebra generated by field operators, revealing its grading by subspaces, and describes the spectral properties of operators affiliated to it, akin to N-body Hamiltonians.

## Contribution

It provides a detailed structural and spectral analysis of the $C^*$-algebra generated by field operators, including a grading by subspaces and an HVZ-type theorem for spectral characterization.

## Key findings

- The $C^*$-algebra is graded by finite dimensional subspaces of the symplectic space.
- Self-adjoint operators affiliated to the algebra exhibit a many-channel structure similar to N-body Hamiltonians.
- The essential spectrum of these operators is characterized by an HVZ-type theorem.

## Abstract

We show that the $C^*$-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.