Let O d be the Cuntz algebra on generators S 1 , . . . , S d , 2 ≤ d < ∞, and let D d ⊆ O d be the abelian subalgebra generated by monomials S α S * 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 α S * 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 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.

We study * -representations of certain algebras which can be described in terms of graphs and positive functions (weights) on the set of their vertices, continuing an earlier investigation of the case where the graph is a Dynkin graph or an extended Dynkin graph with a weight of a special kind.For the cases where the graph is one of the extended Dynkin graphsD 4 ,Ẽ 6 ,Ẽ 7 orẼ 8 , we prove that all irreducible * -representations of the corresponding algebras are finite-dimensional.In the case of a graph which properly contains an extended Dynkin graph, we study the evolution of weights under the action of the Coxeter functors, in particular, we show that there exist two linearly independent p-invariant weights. We also prove that there exists a weight which makes the corresponding algebra to have an infinite-dimensional irreducible * -representation.

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We prove spectral theorems for families of linearly connected self-adjoint operators with given special spectra associated with extended Dynkin graphs. We establish that all irreducible families of linearly connected operators with arbitrary spectra associated with extended Dynkin graphs are finite-dimensional.

REPRESENTATION OF AN ALGEBRA ASSOCIATED WITH THE DYNKIN GRAPHWe describe the structure of pairs of self-adjoint operators A and B whose spectra belong to the set { ± 1 / 2, ± 3 / 2 } and for which ( A + B ) 2 = I. Such pairs of operators determine a representation of a * -algebra A Ẽ 7 associated with the extended Dynkin graph Ẽ 7 .