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Journal of Operator Theory

Volume 45, Issue 1, Winter 2001  pp. 3-18.

Hilbert $C^*$-bimodules and countably generated Cuntz-Krieger algebras

Authors T. Kajiwara (1), C. Pinzari (2), and Y. Watatani (3)
Author institution: (1) Department of Environmental and Mathematical Sciences, Okayama University, Tsushima, Okayama 700, Japan
(2) Dipartimento di Matematica, Universit\`a di Roma ``Tor Vergata'', 00133 Roma, Italy
(3) Graduate School of Mathematics, Kyushu University, Ropponmatsu, Fukuoka 810, Japan

Summary:  Results by Cuntz and Kreiger on uniqueness, simplicity and the ideal structure of the algebras $\cO_A$ associated with finite matrices with entries in $\{0, 1\}$ are generalized to the case where $A$ is an infinite matrix whose rows and columns are eventually zero, but not identically zero. Similar results have been recently obtained by Kumjian, Pask, Raeburn and Renault from the viewpoint of Renault's theory of groupoids. An alternative approach, based on the realization of $\cO_A$ as an algebra generated by a Hilbert $C^*$-bimodule introduced by Pimsner, is proposed and compared.

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