Cuntz Algebra States Defined by Implementers of Endomorphisms of the CAR Algebra

Abstract

AbstractWe investigate representations of the Cuntz algebra on antisymmetric Fock space Fa(𝒦1) defined by isometric implementers of certain quasi-free endomorphisms of the CAR algebra in pure quasi-free states φP1. We pay special attention to the vector states on corresponding to these representations and the Fock vacuum, for which we obtain explicit formulae. Restricting these states to the gauge-invariant subalgebra , we find that for natural choices of implementers, they are again pure quasi-free and are, in fact, essentially the states φP1. We proceed to consider the case for an arbitrary pair of implementers, and deduce that these Cuntz algebra representations are irreducible, as are their restrictions to .The endomorphisms of B(Fa(𝒦)) associated with these representations of are also considered.