2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78783Let 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_αS_α^* =S_{α_{1}}...S_{α_{k}}S_{α_{k}}^*...S_{α_{1}}^* where α=(α_1...α_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_αS_α^*) =(S_{iα}S_{iα}^*)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.xiv+178 pages, AMS-LaTeX, wncyr font; changes in versions 2 and 3 only in Introduction, except minor typos; more sources in Bibliography (+17 in v2, +3 in v3); accepted for publication in Mem. Amer. Math. SocOperator AlgebrasDynamical SystemsFunctional AnalysisRepresentation Theory46L30, 46L55, 46L89, 47A13, 47A67 (Primary) 47A20, 47D25, 43A65 (Secondary)text