We introduce the notion of the crossed product A X Z of a C *algebra A by a Hilbert C * -bimodule X. It is shown that given a C * -algebra B which carries a semi-saturated action of the circle group (in the sense that B is generated by the spectral subspaces B 0 and B 1 ), then B is isomorphic to the crossed product B 0 B 1 Z. We then present our main result, in which we show that the crossed products A X Z and B Y Z are strongly Morita equivalent to each other, provided that A and B are strongly Morita equivalent under an imprimitivity bimodule M satisfying X ⊗ A M M ⊗ B Y as A − B Hilbert C * -bimodules. We also present a six-term exact sequence for K-groups of crossed products by Hilbert C * -bimodules.
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. 2000 Mathematical Subject Classification: primary 46L35; secondary 19K14, 19K35, 46L80.
Abstract. We address the classi cation problem for graph C * -algebras of nite graphs ( nitely many edges and vertices), containing the class of Cuntz-Krieger algebras as a prominent special case. Contrasting earlier work, we do not assume that the graphs satisfy the standard condition (K), so that the graph C * -algebras may come with uncountably many ideals.We nd that in this generality, stable isomorphism of graph C * -algebras does not coincide with the geometric notion of Cuntz move equivalence. However, adding a modest condition on the graphs, the two notions are proved to be mutually equivalent and equivalent to the C * -algebras having isomorphic K-theories. is proves in turn that under this condition, the graph C * -algebras are in fact classi able by K-theory, providing in particular complete classi cation when the C * -algebras in question are either of real rank zero or type I/postliminal. e key ingredient in obtaining these results is a characterization of Cuntz move equivalence using the adjacency matrices of the graphs.Our results are applied to discuss the classi cation problem for the quantum lens spaces de ned by Hong and Szymański, and to complete the classi cation of graph C * -algebras associated to all simple graphs with four vertices or less.
A description of the Kasparov group KK(A, B) is given in terms of Cuntz pairs of representations and the notion of proper asymptotic unitary equivalence that we introduce here. The use of the word 'proper' reflects the crucial fact that all unitaries implementing the equivalence can be chosen to be compact perturbations of identity. The result has significant applications to the classification theory of nuclear C *-algebras.
We study completions of diagrams of extensions of C*-algebras of the form 0 wwÄ v wwÄ v wwÄ v wwÄ 0 0 wwÄ v wwÄ v wwÄ v wwÄ 0 in which all three C*-algebras in one of the rows and either the ideal or the quotient in the other are given, along with the three morphisms between them. We find universal solutions to all four of these problems under restrictions of varying severity, on the given vertical maps and describe the solutions in terms of push-outs and pull-backs of certain diagrams. Our characterization of the universal solution to one of the diagrams yields a concrete description of various amalgamated free products. This leads to new results about the K-theory of amalgamated free products, verifying the Cuntz conjecture in certain cases. We also obtain new results about extensions of matricial field C*-algebras, verifying partially a conjecture of Blackadar and Kirchberg. Finally, we show that almost commuting unitary matrices can be uniformly approximated by commuting unitaries when an index obstruction vanishes.
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