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# Cuntz Algebra States Defined by Implementers of Endomorphisms of the $\CAR$ Algebra

Published:2002-08-01
Printed: Aug 2002
• Michael J. Gabriel
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## Abstract

We investigate representations of the Cuntz algebra $\mathcal{O}_2$ on antisymmetric Fock space $F_a (\mathcal{K}_1)$ defined by isometric implementers of certain quasi-free endomorphisms of the CAR algebra in pure quasi-free states $\varphi_{P_1}$. We pay corresponding to these representations and the Fock special attention to the vector states on $\mathcal{O}_2$ vacuum, for which we obtain explicit formulae. Restricting these states to the gauge-invariant subalgebra $\mathcal{F}_2$, we find that for natural choices of implementers, they are again pure quasi-free and are, in fact, essentially the states $\varphi_{P_1}$. 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 $\mathcal{F}_2$. The endomorphisms of $B \bigl( F_a (\mathcal{K}_1) \bigr)$ associated with these representations of $\mathcal{O}_2$ are also considered.
 MSC Classifications: 46L05 - General theory of $C^*$-algebras 46L30 - States

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