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Linear functionals on the Cuntz algebra


Author: Eui-Chai Jeong
Journal: Proc. Amer. Math. Soc. 134 (2006), 99-104
MSC (2000): Primary 46L05; Secondary 46L40
DOI: https://doi.org/10.1090/S0002-9939-05-07886-X
Published electronically: August 22, 2005
MathSciNet review: 2170548
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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.


References [Enhancements On Off] (What's this?)

  • [Arv] W. B. Arveson, Continuous analogues of Fock space I, Mem. Amer. Math. Soc. 80, no. 409. MR 0987590 (90f:47061)
  • [BrJo] O. Bratteli, P. E. T. Jorgensen, Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale N, Integral Equations Operator Theory 28 (1997), 382-432. MR 1465320 (99k:46094b)
  • [BJP] O. Bratteli, P. E. T. Jorgensen, and G. L. Price, Endomorphisms of $B(\mathcal{H})$, Proc. Symposi. Pure Math. 59 (1996), 93-138. MR 1392986 (97h:46107)
  • [Cun] J. Cuntz, Simple $C^*$-algebras generated by isometries, Commun. Math. Phys. 57 (1977), 173-185. MR 0467330 (57:7189)
  • [Lac] M. Laca, Endomorphisms of $B(\mathcal{H})$ and Cuntz algebras, J. Oper. Theory 30 (1993), 85-108. MR 1302609 (95m:46100)
  • [Jon] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. Math. 126 (1987), 335-388. MR 0908150 (89c:46092)
  • [Pow] R. T. Powers, An index theory for semigroups of $\ast$-endomorphisms of $B(\mathcal{H})$ and type II$_1$ factors, Canad. J. Math. 40 (1988), 46-114. MR 0928215 (89f:46116)
  • [Wor] S. L. Woronowicz, Compact matrix pseudogroups, Commun. Math. Phys. 111 (1987), 613-655. MR 0901157 (88m:46079)

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Additional Information

Eui-Chai Jeong
Affiliation: Department of Mathematics, Chung-Ang University, Dongjak-ku, Seoul, 156-756, South Korea
Email: jeong@cau.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-05-07886-X
Received by editor(s): October 25, 2000
Published electronically: August 22, 2005
Additional Notes: This work was supported by the Brain Korea 21 Project
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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