Majorization in C*-algebras

Ng, Ping; Robert, Leonel; Skoufranis, Paul

doi:10.1090/tran/7163

Majorization in C*-algebras
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by Ping Wong Ng, Leonel Robert and Paul Skoufranis PDF

Trans. Amer. Math. Soc. 370 (2018), 5725-5759 Request permission

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

T. Ando, Majorizations and inequalities in matrix theory, Linear Algebra Appl. 199 (1994), 17–67. MR 1274407, DOI 10.1016/0024-3795(94)90341-7
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, DOI 10.1090/S0002-9947-2012-05538-3
Joachim Cuntz and Gert Kjaergȧrd Pedersen, Equivalence and traces on $C^{\ast }$-algebras, J. Functional Analysis 33 (1979), no. 2, 135–164. MR 546503, DOI 10.1016/0022-1236(79)90108-3
Mahlon M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509–544. MR 92128
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, DOI 10.1353/ajm.2011.0027
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.
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
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, DOI 10.1090/S0002-9947-1991-0984856-9
Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Graduate Studies in Mathematics, vol. 16, American Mathematical Society, Providence, RI, 1997. Advanced theory; Corrected reprint of the 1986 original. MR 1468230, DOI 10.1090/gsm/016/01
Eizaburo Kamei, Majorization in finite factors, Math. Japon. 28 (1983), no. 4, 495–499. MR 717521
Eberhard Kirchberg and Mikael Rørdam, Infinite non-simple $C^*$-algebras: absorbing the Cuntz algebras $\scr O_\infty$, Adv. Math. 167 (2002), no. 2, 195–264. MR 1906257, DOI 10.1006/aima.2001.2041
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, DOI 10.1515/crelle-2012-0118
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, DOI 10.17879/35209721075
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.
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, DOI 10.1090/tran/6842
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, DOI 10.1512/iumj.2009.58.3704
Leonel Robert, Nuclear dimension and sums of commutators, Indiana Univ. Math. J. 64 (2015), no. 2, 559–576. MR 3344439, DOI 10.1512/iumj.2015.64.5472
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, DOI 10.1017/CBO9780511666230
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, DOI 10.1016/j.jfa.2015.09.018
Wilhelm Winter and Joachim Zacharias, Completely positive maps of order zero, Münster J. Math. 2 (2009), 311–324. MR 2545617
Wilhelm Winter and Joachim Zacharias, The nuclear dimension of $C^\ast$-algebras, Adv. Math. 224 (2010), no. 2, 461–498. MR 2609012, DOI 10.1016/j.aim.2009.12.005