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Representation Theory and Numerical AF-Invariants: The Representations and Centralizers of Certain States on \(\mathcal{O}_d\)
Ola Bratteli, Mathematics Institute, Oslo, Norway, Palle E. T. Jorgensen, University of Iowa, Iowa City, IA, and Vasyl' Ostrovs'kyĭ, National Academy of Sciences of Ukraine, Kiev, Ukraine

Memoirs of the American Mathematical Society
2004; 178 pp; softcover
Volume: 168
ISBN-10: 0-8218-3491-6
ISBN-13: 978-0-8218-3491-6
List Price: US$69
Individual Members: US$41.40
Institutional Members: US$55.20
Order Code: MEMO/168/797
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Let \(\mathcal{O}_{d}\) be the Cuntz algebra on generators \(S_{1},\dots,S_{d}\), \(2\leq d<\infty\). Let \(\mathcal{D}_{d}\subset\mathcal{O}_{d}\) be the abelian subalgebra generated by monomials \(S_{\alpha_{{}}}^{{}}S_{\alpha_{{}} }^{\ast}=S_{\alpha_{1}}^{{}}\cdots S_{\alpha_{k}}^{{}}S_{\alpha_{k}}^{\ast }\cdots S_{\alpha_{1}}^{\ast}\) where \(\alpha=\left(\alpha_{1}\dots\alpha _{k}\right)\) ranges over all multi-indices formed from \(\left\{ 1,\dots,d\right\}\). In any representation of \(\mathcal{O}_{d}\), \(\mathcal{D}_{d}\) may be simultaneously diagonalized. Using \(S_{i}^{{}}\left( S_{\alpha}^{{}}S_{\alpha}^{\ast}\right) =\left( S_{i\alpha}^{{}}S_{i\alpha }^{\ast}\right) S_{i}^{{}}\), we show that the operators \(S_{i}\) from a general representation of \(\mathcal{O}_{d}\) may be expressed directly in terms of the spectral representation of \(\mathcal{D}_{d}\). We use this in describing a class of type \(\mathrm{III}\) representations of \(\mathcal{O}_{d}\) and corresponding endomorphisms, and the heart of the memoir is a description of an associated family of AF-algebras arising as the fixed-point algebras of the associated modular automorphism groups. Chapters 5-18 are devoted to finding effective methods to decide isomorphism and non-isomorphism in this class of AF-algebras.


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

Table of Contents

  • Part A. Representation theory
  • Part B. Numerical AF-invariants
  • Bibliography
  • List of figures
  • List of tables
  • List of terms and symbols
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