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1. CJM 2002 (vol 54 pp. 694)
Cuntz Algebra States Defined by Implementers of Endomorphisms of the $\CAR$ Algebra 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.
Categories:46L05, 46L30 |