The ideal property in crossed products

Pasnicu, Cornel

doi:10.1090/S0002-9939-03-07032-1

The ideal property in crossed products
HTML articles powered by AMS MathViewer

by Cornel Pasnicu PDF

Proc. Amer. Math. Soc. 131 (2003), 2103-2108 Request permission

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.

References

Bruce Blackadar, $K$-theory for operator algebras, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. MR 1656031
Ola Bratteli, Inductive limits of finite dimensional $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 171 (1972), 195–234. MR 312282, DOI 10.1090/S0002-9947-1972-0312282-2
Lawrence G. Brown and Gert K. Pedersen, $C^*$-algebras of real rank zero, J. Funct. Anal. 99 (1991), no. 1, 131–149. MR 1120918, DOI 10.1016/0022-1236(91)90056-B
Kenneth R. Davidson, $C^*$-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR 1402012, DOI 10.1090/fim/006
Edward G. Effros and Frank Hahn, Locally compact transformation groups and $C^{\ast }$- algebras, Memoirs of the American Mathematical Society, No. 75, American Mathematical Society, Providence, R.I., 1967. MR 0227310
George A. Elliott, The classification problem for amenable $C^*$-algebras, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 922–932. MR 1403992
Elliot C. Gootman and Aldo J. Lazar, Applications of noncommutative duality to crossed product $C^*$-algebras determined by an action or coaction, Proc. London Math. Soc. (3) 59 (1989), no. 3, 593–624. MR 1014872, DOI 10.1112/plms/s3-59.3.593
Cornel Pasnicu, $AH$ algebras with the ideal property, Operator algebras and operator theory (Shanghai, 1997) Contemp. Math., vol. 228, Amer. Math. Soc., Providence, RI, 1998, pp. 277–288. MR 1667665, DOI 10.1090/conm/228/03292
Cornel Pasnicu, Extensions of AH algebras with the ideal property, Proc. Edinburgh Math. Soc. (2) 42 (1999), no. 1, 65–76. MR 1669385, DOI 10.1017/S0013091500020010
Cornel Pasnicu, Shape equivalence, nonstable $K$-theory and $AH$ algebras, Pacific J. Math. 192 (2000), no. 1, 159–182. MR 1741023, DOI 10.2140/pjm.2000.192.159
Cornel Pasnicu, On the AH algebras with the ideal property, J. Operator Theory 43 (2000), no. 2, 389–407. MR 1753416
C. Pasnicu, On the (strong) $GAH$ algebras, Rev. Roumaine Math. Pures Appl. 46 (2001), no. 4, 489-498.
Cornel Pasnicu, The ideal property and traces, Math. Nachr. 227 (2001), 127–132. MR 1840559, DOI 10.1002/1522-2616(200107)227:1<127::AID-MANA127>3.3.CO;2-O
C. Pasnicu, Ideals generated by projections and inductive limit $C^*$-algebras, Rocky Mountain J. Math. 31 (2001), no. 3, 1083-1095.
C. Pasnicu, The projection property, Glasg. Math. J. 44 (2002), no. 2, 293-300.
C. Pasnicu, $LB$ algebras, J. Operator Theory (to appear).
Cornel Pasnicu and Mikael Rørdam, Tensor products of $C^\ast$-algebras with the ideal property, J. Funct. Anal. 177 (2000), no. 1, 130–137. MR 1789946, DOI 10.1006/jfan.2000.3630
Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
Kenneth H. Stevens, The classification of certain non-simple approximate interval algebras, Operator algebras and their applications, II (Waterloo, ON, 1994/1995) Fields Inst. Commun., vol. 20, Amer. Math. Soc., Providence, RI, 1998, pp. 105–148. MR 1643184
Hiroshi Takai, On a duality for crossed products of $C^{\ast }$-algebras, J. Functional Analysis 19 (1975), 25–39. MR 0365160, DOI 10.1016/0022-1236(75)90004-x
G. Zeller-Meier, Produits croisés d’une $C^{\ast }$-algèbre par un groupe d’automorphismes, J. Math. Pures Appl. (9) 47 (1968), 101–239 (French). MR 241994