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Closed Convex Hulls of Unitary Orbits in Certain Simple Real Rank Zero C* -algebras

Published online by Cambridge University Press:  20 November 2018

P.W. Ng  and
P. Skoufranis
P.W. Ng
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana, USA, 70504-3568 e-mail: [email protected]
P. Skoufranis
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3 e-mail: [email protected]
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Abstract

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In this paper, we characterize the closures of convex hulls of unitary orbits of self-adjoint operators in unital, separable, simple ${{\text{C}}^{*}}$ -algebras with non-trivial tracial simplex, real rank zero, stable rank one, and strict comparison of projections with respect to tracial states. In addition, an upper bound for the number of unitary conjugates in a convex combination needed to approximate a self-adjoint are obtained.

Keywords

46L05convex hull of unitary orbitsreal rank zero C* -algebras simpleeigenvalue functionmajorization
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 5 , 01 October 2017 , pp. 1109 - 1142
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Ando, T., Majorization, doubly stochastic matrices, and comparison of eigenvalues. Linear Algebra Appl. 118(1989), 163–248. http://dx.doi.Org/10.1016/0024-3795(89)90580-6 Google Scholar
[2] Argerami, M. and Massey, P., A Schur-Horn Theorem in IIl Factors. Indiana Univ. Math. J. 56(2007), no. 5, 2051–2060.http://dx.doi.org/10.1512/iumj.2007.56.3113 Google Scholar
[3] Argerami, M., The local form of doubly stochastic maps and joint majorization in IIl factors. Integral Equations Operator Theory 61(2008), 1–19. http://dx.doi.Org/10.1007/s00020-008-1569-6 Google Scholar
[4] Arveson, W. and Kadison, V., Diagonals of self-adjoint operators. In: Operator Theory, Operator Algebras, and Applications, 414, American Mathematical Society, Providence, RI, 2006, pp. 247–263.Google Scholar
[5] Birkhoff, G., Tres observaciones sobre el algebra lineal. Univ. Nac. Tucumn Rev. Ser. A 5(1946), 147–151.Google Scholar
[6] Blackadar, B., Comparison theory for simple C* -algebras. In: Operator Algebras and Applications, LMS Lecture Notes Series, 135, Cambridge University Press, Cambridge-New York, 1988, pp. 21–54.Google Scholar
[7] Blackadar, B., K-theoryfor operator algebras. MSRI Publications, 5, Cambridge University Press, Cambridge, 1998.Google Scholar
[8] Bownik, M. and Jasper, J., Spectra of frame operators with prescribed frame norms. In: Harmonic analysis and partial differential equations, Contemp. Math., 612, American Mathematical Society, Providence, RI, 2014, pp. 65–79.http://dx.doi.Org/10.1090/conm/612/12224 Google Scholar
[9] Bownik, M., The Schur-Horn theorem for operators with finite spectrum. Trans. Amer. Math. Soc. 367(2015), no. 7, 5009–5140.http://dx.doi.org/10.1090/S0002-9947-2015-06317-X Google Scholar
[10] Dykema, K. J. and Skoufranis, P., Numerical ranges in IIl factors. arxiv:1503.05766 Google Scholar
[11] Pack, T., Sur la notion de valuer caractéristique. J. Operator Theory 7(1982), no. 2, 207–333.Google Scholar
[12] Pack, T. and Kosaki, H., Generalized s-numbers of r-measurable operators. Pacific J. Math. 123(1986),no. 2, 269–300.http://dx.doi.org/10.2140/pjm.1986.123.269 Google Scholar
[13] Goldberg, M. and Straus, E., Elementary inclusion relations for generalized numerical ranges. Linear Algebra Appl. 18(1977), no. 1, 1–24.http://dx.doi.Org/10.101 6/0024-3795(77)90075-1 Google Scholar
[14] Hardy, G. H., Littlewood, J. E., and Pálya, G., Some simple inequalities satisfied by convex functions. Messenger Math. 58(1929), 145–152.Google Scholar
[15] Hardy, G. H., Inequalities. Second ed., Cambridge Univeristy Press, London-New York, 1952 Google Scholar
[16] Hiai, F., Majorization and stochastic maps in von Neumann algebras. J. Math. Anal. Appl. 127(1987), 18-48.http://dx.doi.Org/10.1016/0022-247X(87)90138-7 Google Scholar
[17] Hiai, F. and Nakamura, Y., Closed convex hulls of unitary orbits in von Neumann algebras. Trans. Amer. Math. Soc. 323(1991), no. 1, 1–38.http://dx.doi.org/10.1090/S0002-9947-1991-0984856-9 Google Scholar
[18] Horn, A., Doubly stochastic matrices and the diagonal of a rotation matrix. Amer. J. Math. 76(1954), no. 3, 620–630.http://dx.doi.org/10.2307/2372705 Google Scholar
[19] Hu, S. and Lin, H., Distance between unitary orbits of normal elements in simple C* -algebras of real rank zero. J. Funct. Anal. 269(2015), no. 2, 355–437.http://dx.doi.Org/10.1016/j.jfa.2015.02.006 Google Scholar
[20] Kamei, E.. Majorization in finite factors, Math. Japon. 28(1983), no. 4, 495–499.Google Scholar
[21] Kamei, E., Double stochasticity in finite factors, Math. Japon. 29(1984), no. 6, 903–907.Google Scholar
[22] Kamei, E., An order on statistical operators implicitly introduced by von Neumann. Math. Japon. 30(1985), 891–895.Google Scholar
[23] Kennedy, M. and Skoufranis, P., The Schur-Horn problem for normal operators. Proc. London Math. Soc. 111(2015), no. 2, 354–380.http://dx.doi.Org/10.1112/plms/pdvO3O Google Scholar
[24] Kennedy, M., Thompson-s theorem for IIlfactors. Trans. Amer. Math. Soc. 369(2017), no. 2,1495–1511.http://dx.doi.Org/10.1090/tran/6711 Google Scholar
[25] Lin, H., Tracially AF C* -algebras. Trans. Amer. Math. Soc. 353 (2000), no. 2, 693–722.http://dx.doi.org/10.1 090/S0002-9947-00-02680-5 Google Scholar
[26] Lin, H., Embedding an AH-algebra into a simple C* -algebra with prescribed KK-data. K-Theory 24(2001), no. 2, 135–156.http://dx.doi.Org/10.1023/A:1012752005485 Google Scholar
[27] Lin, H., An introduction to the classification of amenable C* -algebras. World Scientific Publishing Co., Inc., New Jersey, 2001.Google Scholar
[28] Lin, H., The tracial topological rank of C* -algebras. Proc. London Math. Soc. (3) 83(2001), no. 1, 199–234.http://dx.doi.Org/10.1112/plms/83.1.199 Google Scholar
[29] Lin, H., Classification of simple C* -algebras of tracial topological rank zero. Duke Math. J. 125(2004), no. 1, 91–119.http://dx.doi.org/10.1215/S0012-7094-04-12514-X Google Scholar
[30] Lin, H., The range of approximate unitary equivalence classes of homomorphisms from AH-algebras. Math. Z. 263(2009), no. 4, 903–922.http://dx.doi.Org/10.1007/s00209-008-0445-z Google Scholar
[31] Massey, P. and Ravichandran, M., Multivariable Schur-Horn theorems. Proc. London Math. Soc. 112(2016), no. 1, 206–234.http://dx.doi.Org/10.1112/plms/pdvO67 Google Scholar
[32] Murray, F. J. and Neumann, J., On rings of operators. Ann. of Math. 1936,116–229.Google Scholar
[33] Ng, P. W., Closed convex hulls of unitary orbits in . Integral Equations Operator Theory 86(2016), no. 1, 13–40.http://dx.doi.org/10.1007/s00020-016-2312-3 Google Scholar
[34] Pereraand, F. Rordam, M., AF embeddings into C* - algebras of real rank zero. J. Funct. Anal. 217(2004), no. 1, 142–170.http://dx.doi.Org/10.1016/j.jfa.2004.05.001 Google Scholar
[35] Petz, D., Spectral scale of self-adjoint operators and trace inequalities. J. Math. Anal. Appl. 109(1985), no. 1. 74–82.http://dx.doi.Org/10.1016/0022-247X(85)90176-3 Google Scholar
[36] Poon, Y. T., Another proof of a result of Westwick. Linear Algebra Appl. 9(1980), no. 1, 35–37.http://dx.doi.Org/10.1080/0308108800881 7347 Google Scholar
[37] Ravichandran, M., The Schur-Horn theorem in von Neumann algebras. arxiv:1209.0909 Google Scholar
[38] Schur, I., Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. Sitzungsber. Berl. Math. Ges. 22(1923), 9–20.Google Scholar
[39] Skoufranis, P., Closed convex hulls of unitary orbits in C* -algebras of real rank zero. J. Funct. Anal. 270(2016), no. 4,1319–1360.http://dx.doi.Org/10.1016/j.jfa.2015.09.018 Google Scholar
[40] Zhang, S., A Riesz decomposition property and ideal structure of multiplier algebras. J. Operator Theory 24(1990), no. 2, 209–225.Google Scholar