The positivity of linear functionals on Cuntz algebras associated to unit vectors
HTML articles powered by AMS MathViewer
- by Jung-Rye Lee and Dong-Yun Shin PDF
- Proc. Amer. Math. Soc. 132 (2004), 2115-2119 Request permission
Abstract:
We study the linear functional $\rho$ on the Cuntz algebra $\mathcal {O}_{n}$ associated to a sequence $\langle \eta _{m} \rangle$ of unit vectors $\eta _{m}$ in $\mathbb {C}^{n}$ that is a generalization of the Cuntz state. We prove that $\rho$ is positive if and only if $\langle \eta _{m} \rangle$ is a constant sequence.References
- Ola Bratteli, Palle E. T. Jorgensen, and Geoffrey L. Price, Endomorphisms of ${\scr B}({\scr H})$, Quantization, nonlinear partial differential equations, and operator algebra (Cambridge, MA, 1994) Proc. Sympos. Pure Math., vol. 59, Amer. Math. Soc., Providence, RI, 1996, pp. 93–138. MR 1392986, DOI 10.1090/pspum/059/1392986
- Joachim Cuntz, Simple $C^*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), no. 2, 173–185. MR 467330, DOI 10.1007/BF01625776
- James G. Glimm, On a certain class of operator algebras, Trans. Amer. Math. Soc. 95 (1960), 318–340. MR 112057, DOI 10.1090/S0002-9947-1960-0112057-5
- Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
- Robert T. Powers, Representations of uniformly hyperfinite algebras and their associated von Neumann rings, Ann. of Math. (2) 86 (1967), 138–171. MR 218905, DOI 10.2307/1970364
Additional Information
- Jung-Rye Lee
- Affiliation: Department of Mathematics, Daejin University, Kyeonggi, 487-711, Korea
- Email: jrlee@daejin.ac.kr
- Dong-Yun Shin
- Affiliation: Department of Mathematics, University of Seoul, Seoul, 130-743, Korea
- Email: dyshin@uos.ac.kr
- Received by editor(s): February 25, 2003
- Received by editor(s) in revised form: April 17, 2003
- Published electronically: February 12, 2004
- Additional Notes: The second author was supported by UOS-2002
- Communicated by: David R. Larson
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 2115-2119
- MSC (2000): Primary 46L30; Secondary 46L05
- DOI: https://doi.org/10.1090/S0002-9939-04-07395-2
- MathSciNet review: 2053984