INTRODUCTIO N xvii Z^ = { 1 , . . . , d} (see [26]). We relate general representations of Od with the spectral resolution of the restriction of the representation t o Td- From this, we read off cocycle formulations of the factor property, irreducibility, and of equivalence of representations. We then specialize to the representations associated with the GNS construction from states UJ — UJP on Od indexed by pi 0, J2i=iPi = ^ given by ^ iSai ' ' ' S akSji ' ' ' 5 7i j = ^fcZ"ai7i ' ' ' OoLklkP&i ' ' ' P^k where a and 7 are multi-indices formed from Z^. The cyclic space 1~LU , for (pi) fixed, is shown in Chapter 2 to have a bundle structure over the set of all Z^-multi- indices with fiber £2 (S^) where S^ is the free semigroup on d generators. Let L = ( L i , . . . , Ld) G Md, and consider the one-parameter group JL of ^ a u - tomorphisms of Od defined by °t(sj) =exp(itLj)sj. It can be shown ([35], or Proposition 3.1) that oL admits a J L - K M S state, at some value /?, if and only if all Lj's are nonzero and have the same sign. This value (3 is then unique and is defined as the solution of and the (crL,/?)-KMS state is then also unique, and is the state defined in the previous paragraph with pk — e~^Lk. Note that the group uL is periodic if and only if any pair Lj, L^ is rationally dependent. In that case, let 21/, be the fixed- point subalgebra in Od under the action aL. We show in Chapter 4 that %L is an AF-algebra ([6], [32]) if and only if all L^'s have the same sign, and furthermore the Sl^'s are then simple with a unique trace state (namely the restriction of the state in the previous paragraph t o 21L). We compute the Bratteli diagrams of the 21L in this case, and show (using a result from [25]) that the endomorphism P (a) : = J2i=i s iasi restricts to a shift (in the sense of Powers [78]) on each of the algebras » L , i.e., fl^Li Pk ( » L ) = CI . While the dimension group £)(2l^), described in [6], [34], [32], and [29] in principle is a complete AF-invariant, we have mentioned that its structure is not immediately transparent. For the present AF-algebras 2 1 ^ the classification is facilitated by the display of specific numerical invariants, derived from D (21L)5 but at the same time computable directly in terms of the given data ( L i , . . . , Ld). These invariants are described in Chapters 6-17 where their connection to Ext is partially explained. Let us give a short road map to the various invariants introduced and where it proved that they are invariants (sometimes in restricted settings): KQ (21L) in (5.6), (5.19) r (K0 (2lL)) in (5.22) D ( a L ) = (K0 ( a L ) , K0 ( » L ) + , [11]) in (5.30) ker r in (5.31) Q [A] together with the prime factors of A before (6.1) N, D, Prim(rajv), Prim (QN-D), Prim (RD) m Theorem 7.8 KQ (2l.c)(8)z^n a n d ker r ^ z ^ n in Chapter 8, M in (8.26) rank£fc in Corollary 9.5 class in Ext (r (K0 (21^)), ker r ) in Chapter 10 Dx (K0 (a L ) ) in (11.57)-(11.58) I {J) in (17.12) and Corollary 17.6. In general it is very hard to find complete invariants apart from D (21L), even for special sub- classes but if the Perron-Frobenius eigenvalue A of J is rational (and thus integral) and N — 2 and Prim (A) = Prim(ra2/A), then Prim (A) is a complete invariant by

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