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Annales Henri Poincaré

, Volume 19, Issue 9, pp 2575–2597 | Cite as

Traces of Random Operators Associated with Self-Affine Delone Sets and Shubin’s Formula

  • Scott Schmieding
  • Rodrigo Treviño
Article
  • 23 Downloads

Abstract

We study operators defined on a Hilbert space defined by a self-affine Delone set \(\Lambda \) and show that the usual trace of a restriction of the operator to finite-dimensional subspaces satisfies a certain \(\limsup \) law controlled by traces on a certain subalgebra. The asymptotic traces are defined through asymptotic cycles, or \({\mathbb {R}}^d\)-invariant distributions of a dynamical system defined by \(\Lambda \). We use this to refine Shubin’s trace formula for certain self-adjoint operators acting on \(\ell ^2(\Lambda )\) and show that the errors of convergence in Shubin’s formula are given by these traces.

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Notes

Acknowledgements

We would like to thank J. Kellendonk for discussing questions related to Shubin’s formula and I. Putnam for discussing traces on \(*\)-algebras. Part of R.T.’s travel funding for this project came from a AMS-Simons Travel Grant.

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics DepartmentEvanstonUSA
  2. 2.Department of MathematicsCollege ParkUSA

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