# Representation Theory and Numerical AF-invariants: The Representations and Centralizers of Certain States on $\mathcal{O}_d$ (Memoirs of the American

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### Description

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 $S_{\alpha_{ {}}}^{ {}}S_{\alpha_{ {}}}^{\ast}=S_{\alpha_{1}}^{ {}}\cdot s S_{\alpha_{k}}^{ {}}S_{\alpha_{k}}^{\ast}\cdots S_{\alpha_{1}}^{\ast}$ 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 of this title are devoted to finding effective methods to decide isomorphism and non-isomorphism in this class of AF-algebras.

### Contents

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

### Product Details

• ISBN13: 9780821834916
• Format: Paperback
• ID: 9780821834916
• ISBN10: 0821834916

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