Decidability of the isomorphism problem for stationary AF-algebras and the associated ordered simple dimension groups (Q2773052)

Bratteli diagrams were introduced in the seminal paper [\textit{O. Bratteli}, Trans. Am. Math. Soc. 171, 195-234 (1972; Zbl 0264.46057)] for a better understanding of the structure and classification of those \(C^*\)-algebras which arise as inductive limits of finite-dimensional \(C^*\)-algebras, the so-called AF- (approximately finite-dimensional) algebras.NEWLINENEWLINENEWLINEIn the paper under review, the notion of isomorphism on stable AF-algebras is considered in the case when the corresponding Bratteli diagram is stationary, i.e., it is associated with a single square primitive incidence matrix \(A\) (i.e., sufficiently high powers \(A^k\) have only strictly positive matrix entries). A \(C^*\)-isomorphism induces an equivalence relation on these matrices, called \(C^*\)-equivalence. Remember that two matrices \(A, B\) with non-negative matrix entries are \(C^*\)-equivalent if there are two sequences \(n_1,n_2,\dots\) and \(m_1,m_2,\dots\) of natural numbers and two sequences of matrices \(J(1), J(2),\dots\) and \(K(1), K(2),\dots\) with non-negative integer matrix entries such that the identities \(A^{n_k}=K(k)J(k)\) and \(B^{m_k}=J(k+1)K(k)\) are satisfied for \(k=1,2,\dots\). The present paper provides an algorithm that can be used to check in a finite number of steps whether two given primitive matrices \(A, B\) are \(C^*\)-equivalent or not.