Xll

REPRESENTATION THEORY AND NUMERICAL AF-INVARIANTS

subshifts. Section 7.5 in the book [63] serves as an excellent background refer-

ence. In particular, it describes how substitution matrices and dimension groups

are derived from symbolic systems in dynamics, and it gives the basics on some

of the numerical invariants used there. The matrices that we encounter in the

context of AF-C*-algebras are a subclass of those used for the general dynamics

problem, and our notion of equivalence is weaker. As we show in [13], our notion

of C*-equivalence for two dimension groups corresponds to a weaker concept of

equivalence than the ones of strong shift equivalence and shift equivalence. The

latter are nicely surveyed in [92], which also recounts the history of the Williams

problem, and its resolution.

Let us give a short rundown of this problem. In the rest of this paragraph, let

the term 'matrix' mean 'matrix over the nonnegative integers7. Then two square

matrices J, K are elementary shift equivalent if there exist matrices A, B such

that J = BA and K — AB. We say that J, K are shift equivalent of lag k if

A J = KA, BK = JB, BA =

Jk,

and AB =

Kk

hold for some matrices A, B, and

they are shift equivalent if they are shift equivalent of some lag k in N. Thus shift

equivalence of lag 1 is the same as elementary shift equivalence [63, Proposition

7.3.2]. The matrices J, K are strongly shift equivalent if they can be connected

by a finite chain of elementary shift equivalent matrices. (Note that elementary

shift equivalence is not an equivalence relation; it is not transitive.) Strong shift

equivalence trivially implies shift equivalence [63, Theorem 7.3.3], but the converse

is the longstanding Williams conjecture, and it is not true [57], even when J and K

are irreducible [58]. The conjecture was stated in [94]. The properly weaker notion

of C*-equivalence or weak equivalence of two square matrices A, B is defined by

the existence of a sequence A\, B\, A2, £?2? ... of matrices and sequences (n^),

(mk) of natural numbers such that

Jnk

= BkAk

Km* = Ak+1Bk

for k = 1,2,....

One notes that the numerical invariants that are known in the subject have

a history based on computation, and that applies to the issue of C*-equivalence,

also called weak equivalence, as well. And the invariants for weak equivalence are

different from those used in [92] and the papers cited there. The latter are based

on sign-gyration conditions, cohomology, and the if-theory group K

2

: see [92,

Section 1]. A part of the paper [92] further describes a class of one-dimensional

systems for which equivalence up to homeomorphism is decided by weak equivalence

(C*-equivalence) for the corresponding dimension groups.

Chapter 1 is a discussion of the spectral decomposition of representations of

the Cuntz algebras relative to canonical MASAs (maximal abelian subalgebras).

It serves as a pointer to how the particular AF-algebras in Chapter 4, which form

the basis for our analysis, arise. But the dimension groups of the AF-algebras

which result as fixed-point subalgebras from our operator-theoretic construct turn

out also to come up in other areas of mathematics, and readers primarily inter-

ested in dynamics may wish to consult [63], or [5] for this alternative use of the

same special dimension groups. These problems have a flavor of algebraic num-

ber theory; see also [14]. Chapter 2 specialises the general setup of Chapter 1 to