Non-stationarity of isomorphism between AF algebras defined by stationary Bratteli diagrams

Abstract

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

Corrigendum to the paper ‘Decidability of the isomorphism problem for stationary AF-algebras and the associated ordered simple dimension groups’

“…This isomorphism is called C * -equivalence of the matrices in [BJKR98] and weak equivalence of the (transposed) matrices in [SwVo00]. In this paper we prove that the isomorphism problem in this setting is decidable, even when the assumption of nonsingularity is removed.…”

“…In [BJKR98] we studied isomorphism of the stable AF-algebras associated with constant square primitive nonsingular incidence matrices. This isomorphism is called C * -equivalence of the matrices in [BJKR98] and weak equivalence of the (transposed) matrices in [SwVo00].…”

Simple Bratteli diagrams with a Gödel-incomplete C*-equivalence problem

“…We prove the Gödel-incompleteness (i.e., the undecidability together with the effective enumerability) of the isomorphism problem for stable AF-algebras arising from abstract finite simplicial complexes. While the isomorphism problem is decidable for stable AF-algebras arising from the iteration of the same positive integer matrix [4], [5], our undecidability result holds for a class L of stable AF-algebras arising from very simple sequences of matrices ϕ i obtained by appending a suitable bottom row of zeros and ones to the (n + i) × (n + i) identity matrix. A stable AF-algebra is in L iff its K 0 -group, equipped with the natural order induced by the Murray-von Neumann order of equivalence classes of projections, is a finitely generated projective lattice-ordered abelian group.…”

“…Following [4] and [5], two square, non-singular, integer, primitive 13 matrices are said to be C * -equivalent iff their associated stable AF-algebras are isomorphic.…”

Ordered group invariants for one-dimensional spaces

“…A class of ordered groups, dimension groups, was used in the study of C * -algebras to classify AF-algebras ( [6]), and Giordano, Herman, Putnam and Skau ( [8,9]) defined (simple) dimension groups in terms of dynamical concepts to give complete information about the orbit structure of zero-dimensional minimal dynamical systems. Swanson and Volkmer ([15]) showed that the dimension group of a primitive matrix is a complete invariant for weak equivalence, which is called C * -equivalence by Bratteli, Jørgensen, Kim and Roush ( [5]). And Barge, Jacklitch and Vago ( [3]) showed that, for a certain class of one-dimensional inverse limit spaces, two spaces are homeomorphic if and only if their associated substitutions are weak equivalent, and that if two inverse limit spaces are homeomorphic and the squares of their connection maps are orientation preserving, then the dimension groups of the adjacency matrices associated with the substitutions are order isomorphic.…”