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Distance to normal elements in $ C^*$-algebras of real rank zero


Authors: Ilya Kachkovskiy and Yuri Safarov
Journal: J. Amer. Math. Soc. 29 (2016), 61-80
MSC (2010): Primary 47A05; Secondary 47L30, 15A27
DOI: https://doi.org/10.1090/S0894-0347-2015-00823-2
Published electronically: January 8, 2015
MathSciNet review: 3402694
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Abstract: We obtain an order sharp estimate for the distance from a given bounded operator $ A$ on a Hilbert space to the set of normal operators in terms of $ \Vert[A,A^*]\Vert$ and the distance to the set of invertible operators. A slightly modified estimate holds in a general $ C^*$-algebra of real rank zero.


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  • [1] A. B. Aleksandrov, V. V. Peller, D. S. Potapov, and F. A. Sukochev, Functions of normal operators under perturbations, Adv. Math. 226 (2011), no. 6, 5216-5251. MR 2775899 (2012d:47055), https://doi.org/10.1016/j.aim.2011.01.008
  • [2] A. B. Aleksandrov and V. V. Peller, Estimates of operator moduli of continuity, J. Funct. Anal. 261 (2011), no. 10, 2741-2796. MR 2832580 (2012j:47026), https://doi.org/10.1016/j.jfa.2011.07.009
  • [3] A. B. Aleksandrov and V. V. Peller, Operator and commutator moduli of continuity for normal operators, Proc. Lond. Math. Soc. (3) 105 (2012), no. 4, 821-851. MR 2989805, https://doi.org/10.1112/plms/pds012
  • [4] I. David Berg and Kenneth R. Davidson, Almost commuting matrices and a quantitative version of the Brown-Douglas-Fillmore theorem, Acta Math. 166 (1991), no. 1-2, 121-161. MR 1088984 (92f:47015), https://doi.org/10.1007/BF02398885
  • [5] Richard Bouldin, Distance to invertible linear operators without separability, Proc. Amer. Math. Soc. 116 (1992), no. 2, 489-497. MR 1097336 (92m:47032), https://doi.org/10.2307/2159757
  • [6] L. G. Brown, R. G. Douglas, and P. A. Fillmore, Unitary equivalence modulo the compact operators and extensions of $ C^{\ast } $-algebras, Proceedings of a Conference on Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) Springer, Berlin, 1973, pp. 58-128. Lecture Notes in Math., Vol. 345. MR 0380478 (52 #1378)
  • [7] Lawrence G. Brown and Gert K. Pedersen, $ C^*$-algebras of real rank zero, J. Funct. Anal. 99 (1991), no. 1, 131-149. MR 1120918 (92m:46086), https://doi.org/10.1016/0022-1236(91)90056-B
  • [8] Man Duen Choi, Almost commuting matrices need not be nearly commuting, Proc. Amer. Math. Soc. 102 (1988), no. 3, 529-533. MR 928973 (89b:47021), https://doi.org/10.2307/2047216
  • [9] Kenneth R. Davidson, Almost commuting Hermitian matrices, Math. Scand. 56 (1985), no. 2, 222-240. MR 813638 (87e:47012)
  • [10] Kenneth R. Davidson, $ C^*$-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012 (97i:46095)
  • [11] Kenneth R. Davidson and Stanislaw J. Szarek, Local operator theory, random matrices and Banach spaces, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 317-366. MR 1863696 (2004f:47002a), https://doi.org/10.1016/S1874-5849(01)80010-3
  • [12] N. Filonov and Y. Safarov, On the relation between an operator and its self-commutator, J. Funct. Anal. 260 (2011), no. 10, 2902-2932. MR 2774059 (2012b:47051), https://doi.org/10.1016/j.jfa.2011.02.011
  • [13] Peter Friis and Mikael Rørdam, Almost commuting self-adjoint matrices--a short proof of Huaxin Lin's theorem, J. Reine Angew. Math. 479 (1996), 121-131. MR 1414391 (97i:46097), https://doi.org/10.1515/crll.1996.479.121
  • [14] Peter Friis and Mikael Rørdam, Approximation with normal operators with finite spectrum, and an elementary proof of a Brown-Douglas-Fillmore theorem, Pacific J. Math. 199 (2001), no. 2, 347-366. MR 1847138 (2002c:47043), https://doi.org/10.2140/pjm.2001.199.347
  • [15] P. R. Halmos, Some unsolved problems of unknown depth about operators on Hilbert space, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976/77), no. 1, 67-76. MR 0451002 (56 #9292)
  • [16] M. B. Hastings, Making almost commuting matrices commute, Comm. Math. Phys. 291 (2009), no. 2, 321-345. MR 2530163 (2010m:15032), https://doi.org/10.1007/s00220-009-0877-2
  • [17] Huaxin Lin, Almost commuting self-adjoint matrices and applications, Operator algebras and their applications (Waterloo, ON, 1994/1995), Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, 1997, pp. 193-233. MR 1424963 (98c:46121)
  • [18] Hua Xin Lin, Exponential rank of $ C^*$-algebras with real rank zero and the Brown-Pedersen conjectures, J. Funct. Anal. 114 (1993), no. 1, 1-11. MR 1220980 (95a:46079), https://doi.org/10.1006/jfan.1993.1060
  • [19] V. V. Peller, The behavior of functions of operators under perturbations, A glimpse at Hilbert space operators, Oper. Theory Adv. Appl., vol. 207, Birkhäuser Verlag, Basel, 2010, pp. 287-324. MR 2743424 (2011k:47022), https://doi.org/10.1007/978-3-0346-0347-8_16

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Additional Information

Ilya Kachkovskiy
Affiliation: Department of Mathematics, University of California, Irvine, Irvine, California 92697-3875
Email: ikachkov@uci.edu

Yuri Safarov
Affiliation: Department of Mathematics, King’s College London, Strand, London WC2R 2LS, United Kingdom
Email: yuri.safarov@kcl.ac.uk

DOI: https://doi.org/10.1090/S0894-0347-2015-00823-2
Keywords: Almost commuting operators, self-commutator, Brown-Douglas-Fillmore theorem.
Received by editor(s): April 15, 2014
Received by editor(s) in revised form: September 12, 2014
Published electronically: January 8, 2015
Additional Notes: The first author was supported by King’s Annual Fund and King’s Overseas ResearchStudentships, King’s College London, and partially by NSF Grant DMS-1101578.
Article copyright: © Copyright 2015 American Mathematical Society

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