Linear functionals on the Cuntz algebra

Abstract:

For a pure state $p’$ on $\mathcal {O}_n$, which is an extension of a pure state $p$ on $\mathrm {UHF}_n$ with the property that if $(\mathcal {H}_{p’},\pi _{p’},\omega _{p’})$ is a corresponding representation, then $\pi _{p’}(\mathrm {UHF}_n)=B(\mathcal {H}_{p’})$, $p’$ induces a unital shift of $B(\mathcal {H})$ of the Powers index $n$. We describe states $p$ on $\mathrm {UHF}_n$ by using sequences of unit vectors in $\mathbb {C}^n$. We study the linear functionals on the Cuntz algebra $\mathcal {O}_n$ whose restrictions are the product pure state on $\mathrm {UHF}_n$. We find conditions on the sequence of unit vectors for which the corresponding linear functionals on $\mathcal {O}_n$ become states under these conditions.