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Rotation C*-Algebras and Almost Mathieu Operators
Florin-Petre Boca, Cardiff University, UK
A publication of the Theta Foundation.
International Book Series of Mathematical Texts
2001; 172 pp; hardcover
ISBN-10: 973-99097-7-9
ISBN-13: 978-973-99097-7-8
List Price: US$28
Member Price: US$22.40
Order Code: THETA/2
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This book delivers a swift, yet concise, introduction to some aspects of rotation \(C^*\)-algebras and almost Mathieu operators. The two topics come from different areas of analysis: operator algebras and the spectral theory of Schrödinger operators, but can be approached in a unified way. The book does not try to be the definitive treatise on the subject, but rather presents a survey highlighting the important results and demonstrating this unified approach.

For each real number \(\alpha\), the rotation \(C^*\)-algebra \(A_\alpha\) can be abstractly defined as the universal \(C^*\)-algebra generated by two elements \(U\) and \(V\) subject to the relation \(UV = e^{2\pi i \alpha} VU\). When \(\alpha\) is an integer, \(A_\alpha\) is isomorphic to the commutative \(C^*\)-algebra of continuous functions on a two-dimensional torus. When \(\alpha\) is not an integer, the algebra is sometimes called a non-commutative 2-torus. In this respect, some of the methods you will find here can be regarded as a sort of non-commutative Fourier analysis. An almost Mathieu operator is a type of self-adjoint operator on the Hilbert space \(\ell^2 = \ell^2(\mathbf{Z})\).

The exposition is geared toward a wide audience of mathematicians: researchers and advanced students interested in operator algebras, operator theory and mathematical physics. Readers are assumed to be acquainted with some functional analysis, such as definitions and basic properties of \(C^*\)-algebras and von Neumann algebras, some general results from ergodic theory, as well as the Fourier transform (harmonic analysis) on elementary abelian locally compact groups of the form \(\mathbf{R}^d \times \mathbf{Z}^k \times \mathbf{T}^1 \times F\), where \(F\) is a finite group.

Much progress has been made on these topics in the last twenty years. The present book will introduce you to the subjects and to the significant results.

A publication of the Theta Foundation. Distributed worldwide, except in Romania, by the AMS.

Readership

Graduate students and research mathematicians interested in functional analysis, quantum theory, and operator theory.

Table of Contents

  • Prerequisites on rotation \(C^*\)-algebras
  • Almost Mathieu operators and automorphisms of \(A_\alpha\)
  • Perturbations of the spectrum of \(H_{\alpha, \lambda}\)
  • The spectrum of almost Mathieu operators for rational \(\alpha\)
  • The absence of isolated points in the spectrum of \(H_{\alpha, \lambda}\)
  • Lyapunov exponents and pure point spectrum
  • The Lebesgue measure of \(\mathrm{spec}_{(p/q,\lambda)}\)
  • Some estimates for the Lebesgue measure of \(\mathrm{spec}(H_{(p/q,\lambda)})\)
  • Spectral computations for certain non-self-adjoint operators
  • Projections in rotation \(C^*\)-algebras
  • The approximation of irrational rotation \(C^*\)-algebras
  • The approximation of irrational non-commutative spheres
  • Subject index
  • Notation
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