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 »
, 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
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
) = (K0 ( a
) , K0 ( »
, [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
) ) 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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