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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