# Rotation C*-Algebras and Almost Mathieu Operators

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*Florin-Petre Boca*

A publication of the Theta Foundation

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.