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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]

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]

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]

Abstract: We consider the problem of identifying exactly which AF-algebras are isomorphic to a graph -algebra. We prove that any separable, unital, Type I -algebra with finitely many ideals is isomorphic to a graph -algebra. This result allows us to prove that a unital AF-algebra is isomorphic to a graph -algebra if and only if it is a Type I -algebra with finitely many ideals. We also consider nonunital AF-algebras that have a largest ideal with the property that the quotient by this ideal is the only unital quotient of the AF-algebra. We show that such an AF-algebra is isomorphic to a graph -algebra if and only if its unital quotient is Type I, which occurs if and only if its unital quotient is isomorphic to for some natural number k. All of these results provide vast supporting evidence for the conjecture that an AF-algebra is isomorphic to a graph -algebra if and only if each unital quotient of the AF-algebra is Type I with finitely many ideals, and bear relevance for the extension problem for graph -algebras. [Copyright &y& Elsevier]