Ola Bratteli 
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 $S_{/alpha_{{}}}^{{}}S_{/alpha_{{}} }^{/ast}=S_{/alpha_{1}}^{{}}/cdots S_{/alpha_{k}}^{{}}S_{/alpha_{k}}^{/ast }/cdots S_{/alpha_{1}}^{/ast}$ 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.