A COMPLETE SET OF ISOMETRIC IMPLEMENTERS FOR A QUASI-FREE ENDOMORPHISM OF THE CAR ALGEBRA VIA UNITARY DILATIONS

Abstract

In this paper we concern ourselves with non-surjective quasi-free endomorphisms of the CAR algebra. More precisely we give a construction of a complete set of isometric implementers for such an endomorphism in a fixed pure quasi-free state, the approach to which is more natural than has previously been proposed. The key idea in this construction is the use of the unitary implementers of an associated quasi-free automorphism in a set of carefully chosen quasi-free states. Our treatment is also novel in that we provide a sequence of implementers within the framework of the complex formalism of the CAR algebra.