Abstract and Applied Analysis

The Cuntz Comparison in the Standard C * -Algebra

Xiaochun Fang, Nung-Sing Sze, and Xiao-Ming Xu

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The Cuntz comparison, introduced by Cuntz in early 1978, associates each C * -algebra with an abelian semigroup which is an invariant for the classification of the nuclear C * -algebras and called the Cuntz semigroup. In this paper, we study the Cuntz comparison in the standard C * -algebra. We characterize the Cuntz comparison in terms of the dimension of the operator range. Also, we consider the structure of the semilinear map which preserves the Cuntz comparison.

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Abstr. Appl. Anal., Volume 2014 (2014), Article ID 623520, 4 pages.

First available in Project Euclid: 2 October 2014

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Fang, Xiaochun; Sze, Nung-Sing; Xu, Xiao-Ming. The Cuntz Comparison in the Standard ${\mathbf{C}}^{\mathbf{\ast}}$ -Algebra. Abstr. Appl. Anal. 2014 (2014), Article ID 623520, 4 pages. doi:10.1155/2014/623520. https://projecteuclid.org/euclid.aaa/1412277038

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  • J. Cuntz, “Dimension functions on simple C$^{\ast\,\!}$-algebras,” Mathematische Annalen, vol. 233, no. 2, pp. 145–153, 1978.
  • P. Ara, F. Perera, and A. S. Toms, “K-theory for operator algebras. Classification of C$^{\ast\,\!}$-algebras,” submitted, http://arxiv.org/ abs/0902.3381V1.
  • N. P. Brown, F. Perera, and A. S. Toms, “The Cuntz semigroup, the Elliott conjecture, and dimension functions on C$^{\ast\,\!}$-algebras,” Journal für die Reine und Angewandte Mathematik, vol. 621, pp. 191–211, 2008.
  • E. Ortega, M. Rørdam, and H. Thiel, “The Cuntz semigroup and comparison of open projections,” Journal of Functional Analysis, vol. 260, no. 12, pp. 3474–3493, 2011.
  • F. Perera and A. S. Toms, “Recasting the Elliott conjecture,” Mathematische Annalen, vol. 338, no. 3, pp. 669–702, 2007.
  • A. S. Toms, “Stability in the Cuntz semigroup of a commutative C$^{\ast\,\!}$-algebra,” Proceedings of the London Mathematical Society, vol. 96, no. 1, pp. 1–25, 2008.
  • A. S. Toms, “Comparison theory and smooth minimal C$^{\ast\,\!}$-dynamics,” Communications in Mathematical Physics, vol. 289, no. 2, pp. 401–433, 2009.
  • S. Clark, C.-K. Li, J. Mahle, and L. Rodman, “Linear preservers of higher rank numerical ranges and radii,” Linear and Multilinear Algebra, vol. 57, no. 5, pp. 503–521, 2009.
  • G. Dolinar and L. Molnár, “Maps on quantum observables preserving the Gudder order,” Reports on Mathematical Physics, vol. 60, no. 1, pp. 159–166, 2007.
  • J. C. Hou, “Rank-preserving linear maps on $\mathcal{B}(X)$,” Science in China (Scientia Sinica). A. Mathematics, Physics, Astronomy, vol. 32, no. 8, pp. 929–940, 1989.
  • J. C. Hou and J. L. Cui, “Linear maps preserving essential spectral functions and closeness of operator ranges,” Bulletin of the London Mathematical Society, vol. 39, no. 4, pp. 575–582, 2007.
  • J. C. Hou and X. F. Qi, “Linear maps preserving separability of pure states,” Linear Algebra and Its Applications, vol. 439, no. 5, pp. 1245–1257, 2013.
  • C.-K. Li, E. Poon, and N.-S. Sze, “Preservers for norms of lie product,” Operators and Matrices, vol. 3, no. 2, pp. 187–203, 2009.
  • C.-K. Li and L. Rodman, “Preservers of spectral radius, numerical radius, or spectral norm of the sum on nonnegative matrices,” Linear Algebra and Its Applications, vol. 430, no. 7, pp. 1739–1761, 2009.
  • X. L. Zhang, J. C. Hou, and K. He, “Maps preserving numerical radius and cross norms of operator products,” Linear and Multilinear Algebra, vol. 57, no. 5, pp. 523–534, 2009.
  • J. B. Conway, A Course in Functional Analysis, vol. 96 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1985. \endinput