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

Gabriel, Michael J.

Canadian journal of mathematics, Tome 54 (2002) no. 4, pp. 694-708

Résumé

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 special attention to the vector states on ${{\mathcal{O}}_{2}}$ corresponding to these representations and the Fock 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 $B\left( {{F}_{a}}({{\mathcal{K}}_{1}}) \right)$ associated with these representations of ${{\mathcal{O}}_{2}}$ are also considered.

Détail

DOI : 10.4153/CJM-2002-026-5

Mots-clés : 46L05, 46L30

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