2001). Decidability of the isomorphism problem for stationary AF-algebras and the associated ordered simple dimension groups.Abstract. The notion of isomorphism on stable AF-C * -algebras is considered in this paper in the case when the corresponding Bratteli diagram is stationary, i.e. is associated with a single square primitive incidence matrix. A C * -isomorphism induces an equivalence relation on these matrices, called C * -equivalence. We show that the associated isomorphism equivalence problem is decidable, i.e. there is an algorithm that can be used to check in a finite number of steps whether two given primitive matrices are C * -equivalent or not. Special cases of this problem will be considered in a forthcoming paper. The isomorphism problem for stationary AF-algebras 1629of the associated dimension group [BJO99]. (We will argue in [BJKR99b, §5] that the class of AF-algebras we obtain in this manner will no longer be the same if A and B are merely required to be primitive but not necessarily non-singular. This does not contradict the results in §2, because the matrices replacing A, B there no longer have positive matrix entries, and the order is defined in a different manner.) In this case we note that J (1) and the sequences n 1 , . . . and m 1 , . . . determine all other K(k) and J (j) from (1.1), i.e.

Abstract. We first study situations where the stable AF-algebras defined by two square primitive nonsingular incidence matrices with nonnegative integer matrix elements are isomorphic even though no powers of the associated automorphisms of the corresponding dimension groups are isomorphic. More generally we consider neccessary and sufficient conditions for two such matrices to determine isomorphic dimension groups. We give several examples. This paper was motivated by attempts in [BJO98] to classify certain AF algebras defined by constant incidence matrices. The specific incidence matrices considered in [BJO98] are of the form (18) below, and we shall see there that the first problem referred to in the abstract is most interesting for those matrices. The second problem referred to in the abstract is significant not only for AF algebras but also for e.g.-classification of substitution minimal systems up to strong orbit equivalence,-homeomorphism classification of domains of certain inverse limit hyperbolic systems, [BD95], [SV98]. The latter paper, which was written independently of this paper, and which was pointed out to us by the referee, makes contributions in the same direction as our paper. Our C * -equivalence of matrices correspond to weak equivalence of (the transposed) matrices in that paper. Theorem 2.3 (which is [BD95, Corollary 3.5]) and 2.4 in [SV98] corresponds more or less to our Theorem 10. Their Theorem 3.2 is similar to our Theorem 6. While the latter part of their paper is focused on a class of incidence matrices arising from periodic kneading sequences, our focus here and in [BJO98] is on matrices of the form (18) below which arises in the representation theory of Cuntz algebras.-cohomology of subshifts of finite type,In a forthcoming paper we will show that the isomorphism problem for stationary AF algebras is decidable. It is already known that shift equivalence is decidable in this setting, [KR79],[KR88]. This is interesting in view of the fact that isomorphism between two AF algebras is known not to be decidable in general, i.e. there is no recursive algorithm to decide if two given effective presentations of Bratteli diagrams yield equivalent diagrams in the general (non-stationary) case, see [MP98].First we will survey some terminology and basic facts in the fields of operator algebras and symbolic dynamics. Recall from [Bra72] that a C * -algebra A is called AF (approximately finite dimensional) if it is the closure of the union an increasing sequence A n of finite dimensional subalgebras. It is known from [Bra72, Theorem 2.7] that two AF algebras A = n A n , B = n B n are isomorphic if and only if there are increasing sequences k i , l i of natural numbers and injections 1

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