Memoirs of the American Mathematical Society 2004; 178 pp; softcover Volume: 168 ISBN10: 0821834916 ISBN13: 9780821834916 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 multiindices 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 AFalgebras arising as the fixedpoint algebras of the associated modular automorphism groups. Chapters 518 are devoted to finding effective methods to decide isomorphism and nonisomorphism in this class of AFalgebras. Readership Graduate students and research mathematicians interested in functional analysis and operator theory. Table of Contents  Part A. Representation theory
 Part B. Numerical AFinvariants
 Bibliography
 List of figures
 List of tables
 List of terms and symbols
