  Let O_d be the Cuntz algebra on generators S_1,...,S_d, 2 \leq d < \infty,
and let D_d \subset O_d be the abelian subalgebra generated by monomials
S_\alpha S_\alpha^*
=S_{\alpha_{1}}...S_{\alpha_{k}}S_{\alpha_{k}}^*...S_{\alpha_{1}}^* where
\alpha=(\alpha_1...\alpha_k) ranges over all multi-indices formed from
{1,...,d}. In any representation of O_d, D_d may be simultaneously
diagonalized. Using S_i(S_\alpha S_\alpha^*) =(S_{i\alpha}S_{i\alpha}^*)S_i, we
show that the operators S_i from a general representation of O_d may be
expressed directly in terms of the spectral representation of D_d. We use this
in describing a class of type III representations of O_d and corresponding
endomorphisms, and the heart of the paper 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.
