Let $mathcal{O}_{d}$ be the Cuntz algebra on generators $S_{1},dots,S_{d}$, $2leq d{}}}^{{}}S_{alpha_{{}}}^{ast}=S_{alpha_{1}}^{{}}cdot s S_{alpha_{k}}^{{}}S_{alpha_{k}}^{ast}cdots S_{alpha_{1}}^{ast}$ where $alpha=left(alpha_{1}dotsalpha _{k}right)$ ranges over all multi-indices formed from $left{1,dots,dright}$. In any representation of $mathcal{O}_{d}$, $mathcal{D}_{d}$ may be simultaneously diagonalized.Using $S_{i}^{{}}left(S_{alpha}^{{}}S_{alpha}^{ast}right)=left(S_{ialpha}^{{}}S_{ialpha}^{ast}right) S_{i}^{{}}$, 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.
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