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 ' ' ' SakSji ' ' ' 57i 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 siasi 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