Abstract Let Od be the Cuntz algebra on generators S i , . . . , S^, 2 d oo. Let Vd C (9d be the abelian subalgebra generated by monomials SaS^ = Sai SakS^k *S*X where a = ( a i . . . a^) ranges over all multi-indices formed from { 1 , . . . , d}. In any representation of Od-, Vd may be simultaneously diagonalized. Using Si (SaS^) = (Sialic*) ^ w e s n o w that the operators Si from a general representation of Od may be expressed directly in terms of the spectral representation of Td- We use this in describing a class of type III representations of Od and corresponding endomor- phisms, 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 automor- phism groups. Chapters 5-18 are devoted to finding effective methods to decide isomorphism and non-isomorphism in this class of AF-algebras. Key words and phrases. C*-algebras, Fourier basis, irreducible representations, Hilbert space, wavelets, radix-representations, lattices, iterated function systems. 2000 Mathematics Subject Classification. Primary 37A55, 46L30, 46L55, 46L89, 47A13, 47A67 Secondary 47A20, 43A65 Work supported by the Norwegian Research Council, the University of Oslo, the U.S. National Science Foundation, and the CRDF research grant UM1-311. Received by the editor July 21, 1999, and in revised form January 4, 2003.
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