Linear Functionals on $\mathcal{O}_n$ Associated to Unit Vectors

Commun. Korean Math. Soc. 2000 Vol. 15, No. 4, 617-626

Eui-Chai Jeong, Jung-Rye Lee, Dong-Yun Shin Sung Kyun Kwan University, Daejin University, University of Seoul

Abstract : We study the vectors related to states on the Cuntz algebra $\Cal O _n$ and prove that, for two states $\omega $ and $\rho$ on $\Cal O_n $ with $\omega|_{ \text{UHF}_n }=\rho|_{ \text{UHF}_n }$, if $(\omega(s_1 ), \cdots, \omega (s_n ))$ and $(\rho (s_1 ), \cdots, \rho (s_n ))$ are unit vectors, then they are linearly dependent. We also study the linear functional on $\Cal O _n$ associated to a sequence of unit vectors in $\Bbb C^n $ which is the generalization of the Cuntz state. We show that if the linear functional associated to a sequence of unit vectors with a certain condition is a state, then it is just the Cuntz state.

Keywords : Cuntz algebra, $\text{UHF}_n$, associated linear functional, state