Representation Theory and Numerical AF-invariants

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.