On purely infinite 𝐶*-algebras

Jeong, Ja A; Lee, Sa Ge

doi:10.1090/S0002-9939-1993-1113642-1

On purely infinite $C\sp *$ -algebras

Authors: Ja A Jeong and Sa Ge Lee
Journal: Proc. Amer. Math. Soc. 117 (1993), 679-682
MSC: Primary 46L05
DOI: https://doi.org/10.1090/S0002-9939-1993-1113642-1
MathSciNet review: 1113642
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Abstract: We find conditions under which a quotient ${C^{\ast}}$ -algebra of a purely infinite ${C^{\ast}}$ -algebra becomes purely infinite. S. Zhang proved that if is a $\sigma$ -unital, simple, purely infinite ${C^{\ast}}$ -algebra, then a hereditary ${C^{\ast}}$ -subalgebra ${A_x}$ is a stable or unital for each of ${A_ + }$ . We prove the converse for a completely $\sigma$ -unital infinite simple ${C^{\ast}}$ -algebra.

[1] Bruce Blackadar, Comparison theory for simple 𝐶*-algebras, Operator algebras and applications, Vol. 1, London Math. Soc. Lecture Note Ser., vol. 135, Cambridge Univ. Press, Cambridge, 1988, pp. 21–54. MR 996438
[2] Lawrence G. Brown, Philip Green, and Marc A. Rieffel, Stable isomorphism and strong Morita equivalence of 𝐶*-algebras, Pacific J. Math. 71 (1977), no. 2, 349–363. MR 0463928
[3] Lawrence G. Brown and Gert K. Pedersen, 𝐶*-algebras of real rank zero, J. Funct. Anal. 99 (1991), no. 1, 131–149. MR 1120918, https://doi.org/10.1016/0022-1236(91)90056-B
[4] Mikael Rørdam, On the structure of simple 𝐶*-algebras tensored with a UHF-algebra, J. Funct. Anal. 100 (1991), no. 1, 1–17. MR 1124289, https://doi.org/10.1016/0022-1236(91)90098-P
[5] Jun Tomiyama, Invitation to 𝐶*-algebras and topological dynamics, World Scientific Advanced Series in Dynamical Systems, vol. 3, World Scientific Publishing Co., Singapore, 1987. MR 910298
[6] Shuang Zhang, On the structure of projections and ideals of corona algebras, Canad. J. Math. 41 (1989), no. 4, 721–742. MR 1012625, https://doi.org/10.4153/CJM-1989-033-4
[7] Shuang Zhang, Stable isomorphism of hereditary 𝐶*-subalgebras and stable equivalence of open projections, Proc. Amer. Math. Soc. 105 (1989), no. 3, 677–682. MR 961418, https://doi.org/10.1090/S0002-9939-1989-0961418-7
[8] Shuang Zhang, 𝐶*-algebras with real rank zero and the internal structure of their corona and multiplier algebras. III, Canad. J. Math. 42 (1990), no. 1, 159–190. MR 1043517, https://doi.org/10.4153/CJM-1990-010-5
[9] Shuang Zhang, A property of purely infinite simple 𝐶*-algebras, Proc. Amer. Math. Soc. 109 (1990), no. 3, 717–720. MR 1010004, https://doi.org/10.1090/S0002-9939-1990-1010004-X
[10] Shuang Zhang, 𝐾₁-groups, quasidiagonality, and interpolation by multiplier projections, Trans. Amer. Math. Soc. 325 (1991), no. 2, 793–818. MR 998130, https://doi.org/10.1090/S0002-9947-1991-0998130-8
[11] -, Certain ${C^{\ast}}$ -algebras with real rank zero and their corona and multiplier algebras, Part I, preprint.