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

Kachkovskiy, Ilya; Safarov, Yuri

doi:10.1090/S0894-0347-2015-00823-2

Distance to normal elements in $C^*$-algebras of real rank zero
HTML articles powered by AMS MathViewer

by Ilya Kachkovskiy and Yuri Safarov;

J. Amer. Math. Soc. 29 (2016), 61-80

DOI: https://doi.org/10.1090/S0894-0347-2015-00823-2

Published electronically: January 8, 2015

PDF | Request permission

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 $\|[A,A^*]\|$ and the distance to the set of invertible operators. A slightly modified estimate holds in a general $C^*$-algebra of real rank zero.

References

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, DOI 10.1016/j.aim.2011.01.008
A. B. Aleksandrov and V. V. Peller, Estimates of operator moduli of continuity, J. Funct. Anal. 261 (2011), no. 10, 2741–2796. MR 2832580, DOI 10.1016/j.jfa.2011.07.009
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, DOI 10.1112/plms/pds012
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, DOI 10.1007/BF02398885
Richard Bouldin, Distance to invertible linear operators without separability, Proc. Amer. Math. Soc. 116 (1992), no. 2, 489–497. MR 1097336, DOI 10.1090/S0002-9939-1992-1097336-6
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) Lecture Notes in Math., Vol. 345, Springer, Berlin-New York, 1973, pp. 58–128. MR 380478
Lawrence G. Brown and Gert K. Pedersen, $C^*$-algebras of real rank zero, J. Funct. Anal. 99 (1991), no. 1, 131–149. MR 1120918, DOI 10.1016/0022-1236(91)90056-B
Man Duen Choi, Almost commuting matrices need not be nearly commuting, Proc. Amer. Math. Soc. 102 (1988), no. 3, 529–533. MR 928973, DOI 10.1090/S0002-9939-1988-0928973-3
Kenneth R. Davidson, Almost commuting Hermitian matrices, Math. Scand. 56 (1985), no. 2, 222–240. MR 813638, DOI 10.7146/math.scand.a-12098
Kenneth R. Davidson, $C^*$-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012, DOI 10.1090/fim/006
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, DOI 10.1016/S1874-5849(01)80010-3
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, DOI 10.1016/j.jfa.2011.02.011
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, DOI 10.1515/crll.1996.479.121
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, DOI 10.2140/pjm.2001.199.347
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 451002, DOI 10.1017/S0308210500019491
M. B. Hastings, Making almost commuting matrices commute, Comm. Math. Phys. 291 (2009), no. 2, 321–345. MR 2530163, DOI 10.1007/s00220-009-0877-2
Huaxin Lin, Almost commuting selfadjoint 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
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, DOI 10.1006/jfan.1993.1060
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, DOI 10.1007/978-3-0346-0347-8_{1}6