The subject of this memoir is developing into a long-term programme by the
authors, also in collaboration with Kim and Roush [13, 14, 15]. The results are of-
ten somewhat technical. The paper [92] provides a beautiful survey of the Williams
problem, its history, and its resolution. The subject of equivalences, strong, shift
equivalence, or weak equivalence, is situated neither directly in operator algebras,
nor perhaps precisely in dynamical systems or wavelet analysis, but rather inter-
mingles these, and more, areas of mathematics.
Let us now add more specific definitions and go into more details:
2. Three ways of measuring isomorphism. During the sixties and seven-
ties it was established that there is a one-to-one canonical correspondence between
the following three sets [6, 30, 32, 33, 34, 47]:
(i) the isomorphism classes of AF-algebras,
(ii) the isomorphism classes of certain ordered abelian groups, called dimen-
sion groups,
and finally
(iii) the equivalence classes of certain combinatorial objects, called Bratteli
In more recent times, this has led to an undercurrent of misunderstanding that
AF-algebras, which are complex objects, are classified by dimension groups, which
are easy objects, and that this is the end of the story. However, as anyone who has
worked with these matters knows, although for special subsets it may be easier to
work with one of the three sets mentioned above rather than another, in general the
computation of isomorphism classes in any of the three categories is equally difficult.
Although dimension groups are easy objects, their isomorphism classes in general
are not! One may even be tempted to flip the coin around and say that dimension
groups are classified by AF-algebras. If one thinks about isomorphism classes, this
is logically true, but the only completely general method to decide isomorphism
classes in all the cases is to resort to the computation of the equivalence relation
for the associated Bratteli diagrams. This problem is not only hard in general, it is
even undecidable: There is no general recursive algorithm to decide if two effective
presentations of Bratteli diagrams yield equivalent diagrams, see [67] and [66]. In
this memoir, we will encounter this problem in a very special situation, and try
to resolve it in a modest way by introducing various numerical invariants which
are easily computable from the diagram. In the situation that the AF-embeddings
are given by a constant primitive nonsingular matrix, the classification problem
has, after the writing of this monograph, been proved in general to be decidable
[13, 14, 15].
Recall that an AF-algebra is a separable C*-algebra with the property that for
any e 0, any finite subset of the algebra can be approximated with elements of
some finite-dimensional *-subalgebra with the precision given by e. An AF-algebra
is stable if it is isomorphic to its tensor product with the compact operators on a
separable Hilbert space. A dimension group is a countable abelian group with an
order satisfying the Riesz interpolation property and which is unperforated. The
Bratteli diagram is described in [6], [29], and [38], and the equivalence relation is
also described in detail in [7] and in Remark 5.6. (All these concepts will be treated
in some detail in Chapter 5, where it is also explained that the stability assumption
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