Abstract. Given square matrices B and B with a poset-indexed block structure (for which an ij block is zero unless i j), when are there invertible matrices U and V with this required-zero-block structure such that UBV = B ? We give complete invariants for the existence of such an equivalence for matrices over a principal ideal domain R. As one application, when R is a field we classify such matrices up to similarity by matrices respecting the block structure. We also give complete invariants for equivalence under the additional requirement that the diagonal blocks of U and V have determinant 1. The invariants involve an associated diagram (the "K-web") of R-module homomorphisms. The study is motivated by applications to symbolic dynamics and C * -algebras.
There are four Bowen-Franks groups associated to each shift of finite type. For an irreducible shift of finite type, we show that a 4-tuple of automorphisms corresponding to the four Bowen-Franks groups can be induced simultaneously by a specific path of flow equivalence from the shift to itself, if and only if it is F-compatible. The Fcompatibleness defined in this paper describes completely the intrinsic relations among the four automorphisms induced by a flow equivalence. This result is one of the key ingredients in classifying reducible shifts of finite type up to flow equivalence. In the mean time, it also discloses a new and sharp difference between the invariants of flow equivalence for an irreducible shift of finite type, and the invariants of stable isomorphism for the associated simple Cuntz-Krieger algebra.
Abstract. Cuntz-Krieger algebras with exactly one nontrivial closed ideal are classified up to stable isomorphism by the Cuntz invariant. The proof relies on Rørdam's classification of simple Cuntz-Krieger algebras up to stable isomorphism and the author's classification of two-component reducible topological Markov chains up to flow equivalence.
PreliminariesLet A ∈ M n ({0, 1}) be nondegenerate (no zero rows or columns) such that no irreducible component is a permutation matrix. The Cuntz-Krieger C * -algebra (CK-algebra) O A associated to A is defined in [CK] as the C * -algebra generated by partial isometries s 1 , s 2 , . . . , s n , satisfying the relationsThe assumption ruling out permutation components is needed to guarantee that O A is uniquely defined up to isomorphism [CK], [C1].Cuntz showed in [C1] that there is a bijective correspondence between the closed ideals of O A and the hereditary subsets of the poset Γ A of irreducible components of A. In particular, O A is simple if and only if A is irreducible, and O A has exactly one nontrivial closed ideal if and only if A is indecomposable (that is, Γ A is not a union of two order-disconnected proper subsets) and has exactly two irreducible components. In this note, we will discuss exclusively the latter case. For A decomposable with two irreducible components, the associated algebra O A is simply the direct sum of its two simple CK-subalgebras (ideals). In this case, the classification of two-component CK-algebras is trivially reduced to the classification of simple CK-algebras which has been accomplished just recently in [R], [C3].
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