Contents Abstract vii Preface ix Introduction xi Part A. Representation Theory 1 Chapter 1. General representations of Od on a separable Hilbert space 3 Chapter 2. The free group on d generators 12 Chapter 3. /3-KMS states for one-parameter subgroups of the action of Td on Od 18 Chapter 4. Subalgebras of Od 22 Part B. Numerical AF-Invariants 43 Chapter 5. The dimension group of 21^ 45 Chapter 6. Invariants related to the Perron-Frobenius eigenvalue 58 Chapter 7. The invariants A/", D, Prim(rajv), Prim (/?£), Prim(Qjv_^) 61 Chapter 8. The invariants KQ (21L) ^ Z ^n and (ker r) 0z Z n for n = 2, 3,4,... 74 Chapter 9. Associated structure of the groups K$ (21L) and kerr 80 Chapter 10. The invariant Ext (r (K0 (2lL)), ker r) 85 Chapter 11. Scaling and non-isomorphism 90 Chapter 12. Subgroups of G0 = \J™=0 J^nC 113 Chapter 13. Classification of the AF-algebras 21L with rank (AT0 (21L)) 2 116 Chapter 14. Linear algebra of J 126 Chapter 15. Lattice points 129 Chapter 16. Complete classification in the cases A = 2, N = 2, 3,4 131 Chapter 17. Complete classification in the case A = rajv 141 1. The case N = 1 149
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