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
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