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Linear functionals on the Cuntz algebra

Authors:	Eui-Chai Jeong

Institution:	Department of Mathematics, Chung-Ang University, Dongjak-ku, Seoul, 156-756, South Korea

Abstract:	For a pure state on $\mathcal{O}_n$ , which is an extension of a pure state on $\mathrm{UHF}_n$ with the property that if $(\mathcal{H}_{p'},\pi_{p'},\omega_{p'})$ is a corresponding representation, then $\pi_{p'}(\mathrm{UHF}_n)=B(\mathcal{H}_{p'})$ , induces a unital shift of $B(\mathcal{H})$ of the Powers index . We describe states 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.

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