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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
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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.

Readership

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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