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