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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$73 Individual Members: US$43.80
Institutional Members: US\$58.40
Order Code: MEMO/168/797

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.