On te Algebra of All Local Observables

4) has a canonical reduction into factors according to the spectrum of its center. We prove that the canonical reduction of the algebra of all local observables also reduces the translation unitary operators. A vacuum sector is defined to be any reduced space in this reduction which contains at least one translationally invariant vector. We prove that the algebra of all local observables is a factor of type I with a unique vacuum state in any vacuum sector. Our main physical conclusion is essentially that the unique vacuum and irreducible algebra of all local observables are the most general situation of interest as far as the existence of a vacuum in the theory is desired.