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Copyright of Human Review is the property of Eagora Science and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Investigaciones ANDINA is the property of Fundacion Universitaria del Area Andina and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]

The semigroups of unital extensions of separable C *-algebras come in two flavours: a strong and a weak version. By the unital Ext-groups, we mean the groups of invertible elements in these semigroups. We use the unital Ext-groups to obtain K -theoretic classification of both unital and non-unital extensions of C *-algebras, and in particular we obtain a complete K -theoretic classification of full extensions of UCT Kirchberg algebras by stable AF algebras. [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 prove that ample groupoids with σ-compact unit spaces are equivalent if and only if they are stably isomorphic in an appropriate sense, and relate this to Matui's notion of Kakutani equivalence. We use this result to show that diagonal-preserving stable isomorphisms of graph C*-algebras or Leavitt path algebras give rise to isomorphisms of the groupoids of the associated stabilised graphs. We deduce that the Leavitt path algebras Lℤ(E2) and Lℤ(E2-) are not stably *-isomorphic. [ABSTRACT FROM AUTHOR]

In 2006, Restorff completed the classification of all Cuntz-Krieger algebras with finitely many ideals (i.e., those that are purely infinite) up to stable isomorphism. He left open the questions concerning strong classification up to stable isomorphism and unital classification. In this paper, we address both questions. We show that any isomorphism between the reduced filtered K-theory of two Cuntz-Krieger algebras with finitely many ideals lifts to a *-isomorphism between the stabilized Cuntz-Krieger algebras. As a result, we also obtain strong unital classification. [ABSTRACT FROM AUTHOR]

The smallest primitive ideal spaces for which there exist counterexamples to the classification of non-simple, purely infinite, nuclear, separable C*-algebras using filtrated K-theory, are four-point spaces. In this article, we therefore restrict to real rank zero C*-algebras with four-point primitive ideal spaces. Up to homeomorphism, there are ten different connected T0-spaces with exactly four points. We show that filtrated K-theory classifies real rank zero, tight, stable, purely infinite, nuclear, separable C*-algebras that satisfy that all simple subquotients are in the bootstrap class for eight out of ten of these spaces. [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]

The structure of the automorphism group of a simple TAI algebra is studied. In particular, we show that $${\frac{\overline{ \rm{Inn}}(A) }{ \overline{\rm{Inn}}_{0} ( A ) }}$$ is isomorphic (as a topological group) to an inverse limit of discrete abelian groups for a unital, simple, AH algebra with bounded dimension growth. Consequently, $${\frac{\overline{\rm{Inn}}(A)}{\overline{\rm{Inn}}_{0}(A) }}$$ is totally disconnected. Another consequence of our results is the following: Suppose A is the transformation group C*-algebra of a minimal Furstenberg transformation with a unique invariant probability measure. Then the automorphism group of A is an extension of a simple topological group by the discrete group $${\rm{Aut}(\underline{K}(A))_{+,1}}$$ . [ABSTRACT FROM AUTHOR]