Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Majorization in C*-algebras


Authors: Ping Wong Ng, Leonel Robert and Paul Skoufranis
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 46L05
DOI: https://doi.org/10.1090/tran/7163
Published electronically: March 16, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the closed convex hull of unitary orbits of selfadjoint elements in arbitrary unital C*-algebras. Using a notion of majorization against unbounded traces, a characterization of these closed convex hulls is obtained. Furthermore, for C*-algebras satisfying Blackadar's strict comparison of positive elements by traces or for collections of C*-algebras with a uniform bound on their nuclear dimension, an upper bound for the number of unitary conjugates in a convex combination required to approximate an element in the closed convex hull within a given error is shown to exist. This property, however, fails for certain ``badly behaved'' simple nuclear C*-algebras.


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

  • [And94] T. Ando, Majorizations and inequalities in matrix theory, Linear Algebra Appl. 199 (1994), 17-67. MR 1274407, https://doi.org/10.1016/0024-3795(94)90341-7
  • [BRT$^+$12] Bruce Blackadar, Leonel Robert, Aaron P. Tikuisis, Andrew S. Toms, and Wilhelm Winter, An algebraic approach to the radius of comparison, Trans. Amer. Math. Soc. 364 (2012), no. 7, 3657-3674. MR 2901228, https://doi.org/10.1090/S0002-9947-2012-05538-3
  • [CP79] Joachim Cuntz and Gert Kjaergaard Pedersen, Equivalence and traces on $ C^{\ast}$-algebras, J. Funct. Anal. 33 (1979), no. 2, 135-164. MR 546503, https://doi.org/10.1016/0022-1236(79)90108-3
  • [Day57] Mahlon M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509-544. MR 0092128
  • [ERS11] George A. Elliott, Leonel Robert, and Luis Santiago, The cone of lower semicontinuous traces on a $ C^*$-algebra, Amer. J. Math. 133 (2011), no. 4, 969-1005. MR 2823868, https://doi.org/10.1353/ajm.2011.0027
  • [FHL$^+$16] Ilijas Farah, Bradd Hart, Martino Lupini, Leonel Robert, Aaron Tikuisis, Alessandro Vignati, and Wilhelm Winter, The Model theory of C*-algebras (2016), available at https://arxiv.org/abs/1602.08072.
  • [HZ84] Uffe Haagerup and László Zsidó, Sur la propriété de Dixmier pour les $ C^{\ast} $-algèbres, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 8, 173-176 (French, with English summary). MR 741088
  • [HN91] Fumio Hiai and Yoshihiro Nakamura, Closed convex hulls of unitary orbits in von Neumann algebras, Trans. Amer. Math. Soc. 323 (1991), no. 1, 1-38. MR 984856, https://doi.org/10.2307/2001613
  • [KR97] Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Advanced theory; Corrected reprint of the 1986 original. Graduate Studies in Mathematics, vol. 16, American Mathematical Society, Providence, RI, 1997. MR 1468230
  • [Kam83] Eizaburo Kamei, Majorization in finite factors, Math. Japon. 28 (1983), no. 4, 495-499. MR 717521
  • [KR02] Eberhard Kirchberg and Mikael Rørdam, Infinite non-simple $ C^*$-algebras: absorbing the Cuntz algebras $ \mathcal{O}_\infty$, Adv. Math. 167 (2002), no. 2, 195-264. MR 1906257, https://doi.org/10.1006/aima.2001.2041
  • [KR14] Eberhard Kirchberg and Mikael Rørdam, Central sequence $ C^*$-algebras and tensorial absorption of the Jiang-Su algebra, J. Reine Angew. Math. 695 (2014), 175-214. MR 3276157, https://doi.org/10.1515/crelle-2012-0118
  • [NR16] Ping Wong Ng and Leonel Robert, Sums of commutators in pure $ \rm C^*$-algebras, Münster J. Math. 9 (2016), no. 1, 121-154. MR 3549546
  • [NS16] Ping Wong Ng and Paul Skoufranis, Closed convex hulls of unitary orbits in certain simple real rank zero C*-algebras (2016), available at http://arxiv.org/abs/1603.07059.
  • [RT17] Leonel Robert and Aaron Tikuisis, Nuclear dimension and $ \mathcal{Z}$-stability of non-simple $ \rm C^*$-algebras, Trans. Amer. Math. Soc. 369 (2017), no. 7, 4631-4670. MR 3632545, https://doi.org/10.1090/tran/6842
  • [Rob09] Leonel Robert, On the comparison of positive elements of a $ C^*$-algebra by lower semicontinuous traces, Indiana Univ. Math. J. 58 (2009), no. 6, 2509-2515. MR 2603757, https://doi.org/10.1512/iumj.2009.58.3704
  • [Rob15] Leonel Robert, Nuclear dimension and sums of commutators, Indiana Univ. Math. J. 64 (2015), no. 2, 559-576. MR 3344439, https://doi.org/10.1512/iumj.2015.64.5472
  • [SS08] Allan M. Sinclair and Roger R. Smith, Finite von Neumann algebras and masas, London Mathematical Society Lecture Note Series, vol. 351, Cambridge University Press, Cambridge, 2008. MR 2433341
  • [Sko16] Paul Skoufranis, Closed convex hulls of unitary orbits in $ C^*$-algebras of real rank zero, J. Funct. Anal. 270 (2016), no. 4, 1319-1360. MR 3447713, https://doi.org/10.1016/j.jfa.2015.09.018
  • [WZ09] Wilhelm Winter and Joachim Zacharias, Completely positive maps of order zero, Münster J. Math. 2 (2009), 311-324. MR 2545617
  • [WZ10] Wilhelm Winter and Joachim Zacharias, The nuclear dimension of $ C^\ast$-algebras, Adv. Math. 224 (2010), no. 2, 461-498. MR 2609012, https://doi.org/10.1016/j.aim.2009.12.005

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 46L05

Retrieve articles in all journals with MSC (2010): 46L05


Additional Information

Ping Wong Ng
Affiliation: Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana
Email: png@louisiana.edu

Leonel Robert
Affiliation: Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana
Email: lrobert@louisiana.edu

Paul Skoufranis
Affiliation: Department of Mathematics and Statistics, York University, Toronto, Canada
Email: pskoufra@yorku.ca

DOI: https://doi.org/10.1090/tran/7163
Received by editor(s): August 26, 2016
Received by editor(s) in revised form: December 15, 2016
Published electronically: March 16, 2018
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society