State extensions of states on $ \text{UHF}_n $ algebra to Cuntz algebra

Bull. Korean Math. Soc. 2002 Vol. 39, No. 3, 471-478 Printed September 1, 2002

Dong-Yun Shin University of Seoul

Abstract : Let $ \eta = \{\eta_m \}_m$ be an eventually constant sequence of unit vectors $\eta_m $ in $\Bbb C^n $ and let $\rho_\eta $ be the pure state on $\text{UHF}_n $ algebra which is defined by $\rho_{\eta} (v_{i_1} \cdots v_{i_k} v_{j_k}^* \cdots v_{j_1}^* )= \overline{\eta_1^{i_1}\cdots \eta_k^{i_k}} \eta_k^{j_k} \cdots \eta_1^{j_1}.$ We find infinitely many state extensions of $\rho_{\eta}$ to Cuntz algebra $\Cal O_n$ using representations and unitary operators. Also, we present their concrete expressions.