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Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Approximating spectral invariants of Harper operators on graphs II

Author(s): Varghese Mathai; Thomas Schick; Stuart Yates
Journal: Proc. Amer. Math. Soc. 131 (2003), 1917-1923.
MSC (2000): Primary 58J50, 39A12
Posted: September 20, 2002
MathSciNet review: 1955281
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Abstract: We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada. The spectral density function of the DML is defined using the von Neumann trace associated with the free action of a discrete group on a graph. The main result in this paper states that when the group is amenable, the spectral density function is equal to the integrated density of states of the DML that is defined using either Dirichlet or Neumann boundary conditions. This establishes the main conjecture in a paper by Mathai and Yates. The result is generalized to other self adjoint operators with finite propagation speed.

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T. Adachi, A note on the Følner condition of amenability, Nagoya Math. J. 131 (1993) 67-74. MR 94h:43001

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J. Dodziuk, P. Linnell, V. Mathai, T. Schick and S. Yates, Approximating $L^2$-invariants, and the Atiyah conjecture, Commun. in Pure and Applied Math. (to appear).

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B. Eckmann, Approximating $\ell_2$-Betti numbers of an amenable covering by ordinary Betti numbers, Comment. Math. Helv. 74 (1999) 150-155. MR 2000c:57050

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G. Elek, On the analytic zero divisor conjecture of Linnell, math.GR/0111180.

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V. Mathai and S. Yates, Approximating spectral invariants of Harper operators on graphs, J. Functional Analysis 188 (2002), no. 1, 111-136.

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M. Shubin, Discrete Magnetic Laplacian, Commun. Math. Phys. 164 (1994), no. 2, 259-275. MR 97b:82010

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Additional Information:

Varghese Mathai
Affiliation: Department of Mathematics, University of Adelaide, Adelaide 5005, Australia
Email: vmathai@maths.adelaide.edu.au

Thomas Schick
Affiliation: FB Mathematik, Universität Göttingen, Bunsenstrasse~3, 37073 Göttingen, Germany
Email: schick@uni-math.gwdg.de

Stuart Yates
Affiliation: Department of Mathematics, University of Adelaide, Adelaide 5005, Australia
Email: syates@maths.adelaide.edu.au

DOI: 10.1090/S0002-9939-02-06739-4
PII: S 0002-9939(02)06739-4
Keywords: Harper operator, discrete magnetic Laplacian, DML, approximation theorems, amenable groups, von Neumann algebras, graphs, integrated density of states
Received by editor(s): January 12, 2002
Posted: September 20, 2002
Additional Notes: The first and third authors acknowledge support from the Australian Research Council.
Communicated by: Jozef Dodziuk
Copyright of article: Copyright 2002, American Mathematical Society




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