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We investigate dynamical systems consisting of a locally compact Hausdorff space equipped with a partially defined local homeomorphism. Important examples of such systems include self-covering maps, one-sided shifts of finite type and, more generally, the boundary-path spaces of directed and topological graphs. We characterize the topological conjugacy of these systems in terms of isomorphisms of their associated groupoids and C*-algebras. This significantly generalizes recent work of Matsumoto and of the second- and third-named authors. [ABSTRACT FROM AUTHOR]

Let E be a countable directed graph that is amplified in the sense that whenever there is an edge from v to w, there are infinitely many edges from v to w. We show that E can be recovered from C^*(E) together with its canonical gauge-action, and also from L_\mathbb {K}(E) together with its canonical grading. [ABSTRACT FROM AUTHOR]

We geometrically describe the relation induced on a set of graphs by isomorphism of their associated graph C*-algebras as the smallest equivalence relation generated by five types of moves. The graphs studied have finitely many vertices and finitely or countably infinitely many edges, corresponding to unital and separable C*-algebras. [ABSTRACT FROM AUTHOR]

Since their inception in the 1930s by von Neumann, operator algebras have been used to shed light on many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two has been sought since their emergence in the late 1960s. We connect these seemingly separate types of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and $C^{*}$ -algebras with additional $C^{*}$ -algebraic structure. Our approach naturally applies to algebras arising from $C^{*}$ -correspondences to resolve self-adjoint and non-self-adjoint isomorphism problems in the literature. We apply our strategy to completely elucidate this newly found hierarchy for operator algebras arising from directed graphs. [ABSTRACT FROM AUTHOR]

In this paper, a G-shift of finite type (G-SFT) is a shift of finite type together with a free continuous shift-commuting action by a finite group G. We reduce the classification of G-SFTs up to equivariant flow equivalence to an algebraic classification of a class of poset-blocked matrices over the integral group ring of G. For a special case of two irreducible components with G = Z2, we compute explicit complete invariants. We relate our matrix structures to the Adler-Kitchens-Marcus group actions approach. We give examples of G-SFT applications, including a new connection to involutions of cellular automata. [ABSTRACT FROM AUTHOR]

We characterize when there exists a diagonal-preserving $\ast$ -isomorphism between two graph $C^{\ast }$ -algebras in terms of the dynamics of the boundary path spaces. In particular, we refine the notion of ‘orbit equivalence’ between the boundary path spaces of the directed graphs $E$ and $F$ and show that this is a necessary and sufficient condition for the existence of a diagonal-preserving $\ast$ -isomorphism between the graph $C^{\ast }$ -algebras $C^{\ast }(E)$ and $C^{\ast }(F)$. [ABSTRACT FROM AUTHOR]

We classify certain sofic shifts (the irreducible Point Extension Type, or PET, sofic shifts) up to flow equivalence, using invariants of the canonical Fischer cover. There are two main ingredients. An extension theorem, for extending flow equivalences of subshifts to flow equivalent irreducible shifts of finite type which contain them.The classification of certain constant to one maps from SFTs via algebraic invariants of associated G-SFTs. [ABSTRACT FROM AUTHOR]

We show that the Cuntz splice induces stably isomorphic graph $$C^*$$ -algebras. This result is a key step towards the recent complete classification of unital graph $$C^*$$ -algebras both with respect to K-theoretical classification and with respect to geometrical classification in terms of an equivalence relation on the underlying graphs. [ABSTRACT FROM AUTHOR]

We study basic properties of flow equivalence on one-dimensional compact metric spaces with a particular emphasis on isotopy in the group of (self-) flow equivalences on such a space. In particular, we show that such an orbit-preserving map is not always an isotopy, but that this always is the case for suspension flows of irreducible shifts of finite type. We also provide a version of the fundamental discretization result of Parry and Sullivan which does not require that the flow maps are either injective or surjective. Our work is motivated by applications in the classification theory of sofic shift spaces, but has been formulated to supply a solid and accessible foundation for other purposes. [ABSTRACT FROM AUTHOR]

There are many classes of nonsimple graph C*-algebras that are classified by the six-term exact sequence in K-theory. In this paper we consider the range of this invariant and determine which cyclic six-term exact sequences can be obtained by various classes of graph C*-algebras. To accomplish this, we establish a general method that allows us to form a graph with a given sixterm exact sequence of K-groups by splicing together smaller graphs whose C*- algebras realize portions of the six-term exact sequence. As rather immediate consequences, we obtain the first permanence results for extensions of graph C*-algebras. We are hopeful that the results and methods presented here will also prove useful in more general cases, such as situations where the C*-algebras under investigation have more than one ideal and where there are currently no relevant classification theories available. [ABSTRACT FROM AUTHOR]

We detail the history of the problem of deciding how many ways one may combine n 2 x 4 LEGO bricks, and explain what is known--and not known--about the related question of how these numbers grow with n. [ABSTRACT FROM AUTHOR]

We propose the further study of the rate of growth of the number of contiguous buildings which may be made from n LEGO blocks of the same size and color. Specializing to blocks of dimension 2×4 we give upper and lower bounds, and speculate on the true value. [ABSTRACT FROM AUTHOR]

In this paper we show that the reducibility structure of several covers of sofic shifts is a flow invariant. In addition, we prove that for an irreducible subshift of almost finite type the left Krieger cover and the past set cover are reducible. We provide an example which shows that there are non almost finite type shifts which have reducible left Krieger covers. As an application we show that the Matsumoto algebra of an irreducible, strictly sofic shift of almost finite type is not simple. [ABSTRACT FROM AUTHOR]

At the cost of restricting the nature of the involved K-groups, we prove a classification result for a hitherto unexplored class of graph C*-algebras, allowing us to classify all graph C*-algebras on finitely many vertices with a finite linear ideal lattice if all pair of vertices are connected by infinitely many edges when they are connected at all. [ABSTRACT FROM AUTHOR]

We employ results from KK-theory, along with quasidiagonality techniques, to obtain wide-ranging classification results for nuclear C*-algebras. Using a new realization of the Cuntz picture of the Kasparov groups we show that two morphisms inducing equal KK-elements are approximately stably unitarily equivalent. Using K-theory with coefficients to associate a partial KK-element to an approximate morphism, our result is generalized to cover such maps. Conversely, we study the problem of lifting a (positive) partial KK-element to an approximate morphism. These results are employed to obtain classification results for certain classes of quasidiagonal C*-algebras introduced by H. Lin, and to reprove the classification of purely infinite simple nuclear C*-algebras of Kirchberg and Phillips. It is our hope that this work can be the starting point of a unified approach to the classification of nuclear C*-algebras. [ABSTRACT FROM PUBLISHER]

Close ties between approximation properties for relations on C*-algebra elements and lifting results for the universal C*-algebras the relations generate is a very widespread and useful phenomenon in C*-algebra theory. In this paper, we explore how to achieve results of this kind when the approximation properties and the lifting results are true only in special cases determined by K-theoretical contingencies. To interpolate between properties of these two basic types, we must investigate C*-algebras given by softened relations, in particular with emphasis on their K-theory. A surprisingly weak correlation between the K-theory of the C*-algebras given by exact and softened relations leads to delicate problems which must be treated with care. As an example of a set of relations which are prone to an analysis of this kind we study the pairs of unitaries commuting up to rational rotation. [ABSTRACT FROM AUTHOR]

We prove that a large and natural class of subhomogeneous C*-algebras with one-dimensional spectrum have the fragility property, that is, every morphism out of such a C*-algebra and into a C*-algebra of real rank zero can be approximated by a morphism with finite-dimensional range, provided that a K-theoretical obstruction vanishes. 1991 Mathematics Subject Classification 46L05. [ABSTRACT FROM AUTHOR]