Representation Theory and Numerical AF-invariants

Let $\mathcal{O}{d}$ be the Cuntz algebra on generators $S{1},\dots,S{d}$, $2\leq d{}}}^{{}}S{\alpha{{}}}^{\ast}=S{\alpha{1}}^{{}}\cdot s S{\alpha{k}}^{{}}S{\alpha{k}}^{\ast}\cdots S{\alpha{1}}^{\ast}$ where $\alpha=\left(\alpha{1}\dots\alpha {k}\right)$ 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.