Geodesic
Approximately finite-dimensional $C^*$-algebras with projective Hilbert modules, their Bratteli diagrams, and $K_0$-groups
Sbornik. Mathematics, Tome 188 (1997) no. 10, pp. 1543-1560 
A. Ya. Helemskii. Approximately finite-dimensional $C^*$-algebras with projective Hilbert modules, their Bratteli diagrams, and $K_0$-groups. Sbornik. Mathematics, Tome 188 (1997) no. 10, pp. 1543-1560. http://geodesic.mathdoc.fr/item/SM_1997_188_10_a5/
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     title = {Approximately finite-dimensional $C^*$-algebras with projective {Hilbert} modules, their {Bratteli} diagrams, and $K_0$-groups},
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Voir la notice de l'article provenant de la source Math-Net.Ru

This paper is devoted to the homological classification of approximately finite-dimensional $C^*$-algebras. The algebras in this class for which there exists at least one non-trivial Hilbert module and those for which there exists at least one faithful Hilbert module are described. The description is given in terms of the Bratteli diagrams of the algebras in question and in terms of the ordered $K_0$-groups of these algebras.

[1] Connes A., “On the cohomology of operator algebras”, J. Funct. Anal., 28 (1978), 248–253 | DOI | MR | Zbl

[2] Haagerup U., “All nuclear $C^*$-algebras are amenable”, Invent. Math., 74 (1983), 305–319 | DOI | MR | Zbl

[3] Khelemskii A. Ya., Gomologiya v banakhovykh i topologicheskikh algebrakh, Izd-vo MGU, M., 1986 | MR

[4] Khelemskii A. Ya., “Gomologicheskaya kharakterizatsiya faktorov tipa I”, Dokl. AN, 344 (1995), 454–456 | MR

[5] Helemskii A. Ya., “Homology in algebras of analysis”, Handbook of Algebra, V. II, Elsevier (to appear)

[6] Kadison R. V., Ringrose J. R., Fundamentals of the theory of operator algebras. V. II. Advanced theory, Academic Press, London, 1986 | MR | Zbl

[7] Connes A., Noncommutative geometry, Academic Press, London, 1994 | MR | Zbl

[8] Khelemskii A. Ya., Banakhovy i polinormirovannye algebry: obschaya teoriya, predstavleniya, gomologii, Nauka, M., 1989 | MR

[9] Douglas R. G., Paulsen V. I., Hilbert modules over function algebras, Longman, New York, 1989 | MR

[10] Lance E. C., Hilbert $C^*$-modules, Cambridge Univ. Press, Cambridge, 1995 | MR | Zbl

[11] Bratteli O., “Inductive limits of finite dimensional $C^*$-algebras”, Trans. Amer. Math. Soc., 171 (1972), 195–234 | DOI | MR | Zbl

[12] Murphy G. J., $C^*$-algebras and operator theory, Acad. Press, London, 1990 | MR | Zbl

[13] Effros E. G., Dimensions and $C^*$-algebras, Amer. Math. Soc., Providence, 1980 | MR | Zbl

[14] Wegge-Olsen N. E., $K$-theory and $C^*$-akgebras, Oxford Univ. Press, Oxford, 1993 | MR | Zbl

[15] Diksme Zh., $C^*$-algebry i ikh predstavleniya, Nauka, M., 1974 | MR

[16] Takesaki M., “On the conjugate space of an operator algebra”, Tohoku Math. J., 10 (1958), 194–203 | DOI | MR | Zbl

[17] Selivanov Yu. V., “Biproektivnye banakhovy algebry”, Izv. AN SSSR. Ser. matem., 43 (1979), 1159–1174 | MR | Zbl

[18] Semadeni Z., Banach Spaces of Continuous Functions, PWN, Warszawa, 1971 | Zbl

[19] Kaliman Sh. I., Selivanov Yu. V., “O kogomologiyakh operatornykh algebr”, Vestn. MGU. Ser. 1. Matem., mekh., 1974, no. 5, 24–27 | MR | Zbl

[20] Khelemskii A. Ya., “O gomologicheskoi razmernosti normirovannykh modulei nad banakhovymi algebrami”, Matem. sb., 81 (1970), 430–444 | MR