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The ideal property in crossed products


Author: Cornel Pasnicu
Journal: Proc. Amer. Math. Soc. 131 (2003), 2103-2108
MSC (2000): Primary 46L05; Secondary 46L55
DOI: https://doi.org/10.1090/S0002-9939-03-07032-1
Published electronically: February 5, 2003
MathSciNet review: 1963756
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Abstract: We describe the lattice of the ideals generated by projections and prove a characterization of the ideal property for ``large" classes of crossed products of commutative $C^*$-algebras by discrete, amenable groups; some applications are also given. We prove that the crossed product of a $C^*$-algebra with the ideal property by a group with the ideal property may fail to have the ideal property; this answers a question of Shuzhou Wang.


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  • [Bl] B. Blackadar, $K$-theory for operator algebras, Second edition, Mathematical Sciences Research Institute Publications, 5, Cambridge University Press, Cambridge, 1998. MR 99g:46104
  • [Br] O. Bratteli, Inductive limits of finite dimensional $C^*$-algebras, Trans. Amer. Math. Soc. 171 (1972), 195-234. MR 47:844
  • [BPe] L. G. Brown and G. K. Pedersen, $C^*$-algebras of real rank zero, J. Funct. Anal. 99 (1991), no. 1, 131-149. MR 92m:46086
  • [D] K. Davidson, $C^*$-algebras by example, Fields Institute Monographs, 6, American Mathematical Society, Providence, RI, 1996. MR 97i:46095
  • [EH] E. G. Effros and F. Hahn, Locally compact transformation groups and $C^*$-algebras, Memoirs of the American Mathematical Society, No. 75, American Mathematical Society, Providence, RI, 1967. MR 37:2895
  • [Ell] G. A. Elliott, The classification problem for amenable $C^*$-algebras, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 922-932, Birkhäuser, Basel, 1995. MR 97g:46072
  • [GL] E. C. Gootman and A. J. Lazar, Applications of non-commutative duality to crossed product $C^*$-algebras determined by an action or coaction, Proc. London Math. Soc. (3) 59 (1989), no. 3, 593-624. MR 91b:46069
  • [Pa1] C. Pasnicu, $AH$ algebras with the ideal property, Operator algebras and operator theory (Shanghai, 1997), 277-288, Contemp. Math., 228, American Mathematical Society, Providence, RI, 1998. MR 99m:46141
  • [Pa2] C. Pasnicu, Extensions of $AH$ algebras with the ideal property, Proc. Edinburgh Math. Soc. (2) 42 (1999), no. 1, 65-76. MR 2001c:46112
  • [Pa3] C. Pasnicu, Shape equivalence, nonstable $K$-theory and $AH$ algebras, Pacific J. Math. 192 (2000), no. 1, 159-182. MR 2001d:46102
  • [Pa4] C. Pasnicu, On the $AH$ algebras with the ideal property, J. Operator Theory 43 (2000), no. 2, 389-407. MR 2001c:46111
  • [Pa5] C. Pasnicu, On the (strong) $GAH$algebras, Rev. Roumaine Math. Pures Appl. 46 (2001), no. 4, 489-498.
  • [Pa6] C. Pasnicu, The ideal property and traces, Math. Nachr. 227 (2001), 127-132. MR 2002g:46101
  • [Pa7] C. Pasnicu, Ideals generated by projections and inductive limit $C^*$-algebras, Rocky Mountain J. Math. 31 (2001), no. 3, 1083-1095.
  • [Pa8] C. Pasnicu, The projection property, Glasg. Math. J. 44 (2002), no. 2, 293-300.
  • [Pa9] C. Pasnicu, $LB$ algebras, J. Operator Theory (to appear).
  • [PaR] C. Pasnicu and M. Rørdam, Tensor products of $C^*$-algebras with the ideal property, J. Funct. Anal. 177 (2000), no. 1, 130-137. MR 2001m:46124
  • [Pe] G. K. Pedersen, $C^*$-algebras and their automorphism groups, London Mathematical Society Monographs, 14, Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers), London-New York, 1979. MR 81e:46037
  • [S] K. H. Stevens, The classification of certain non-simple approximate interval algebras, Fields Inst. Commun., 20, American Mathematical Society, Providence, RI, 1998. MR 2002d:46052
  • [T] H. Takai, On a duality for crossed products of $C^*$-algebras, J. Funct. Anal. 19 (1975), 25-39. MR 51:1413
  • [Z] G. Zeller - Meier, Products croisés d'une $C^*$-algèbre par un groupe d'atuomorphismes, J. Math. Pures Appl. (9) 47 (1968), 101-239. MR 39:3329

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Additional Information

Cornel Pasnicu
Affiliation: Department of Mathematics and Computer Science, University of Puerto Rico, Box 23355, San Juan, Puerto Rico 00931-3355
Email: cpasnic@upracd.upr.clu.edu

DOI: https://doi.org/10.1090/S0002-9939-03-07032-1
Keywords: $C^*$-algebra, the ideal property, crossed product
Received by editor(s): February 1, 2002
Published electronically: February 5, 2003
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society

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