Type I 𝐶*-algebras of real rank zero

Lin, Huaxin

doi:10.1090/S0002-9939-97-03890-2

Type I $C^*$-algebras of real rank zero
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Proc. Amer. Math. Soc. 125 (1997), 2671-2676 Request permission

Abstract:

We show that a separable $C^*$-algebra $A$ of type I has real rank zero if and only if $d({\hat A})=0,$ where $d$ is a modified dimension. We also show that a separable $C^*$-algebra of type I has real rank zero if and only if it is an AF-algebra.

References

Ola Bratteli and George A. Elliott, Structure spaces of approximately finite-dimensional $C^{\ast }$-algebras. II, J. Functional Analysis 30 (1978), no. 1, 74–82. MR 513479, DOI 10.1016/0022-1236(78)90056-3
L. G. Brown, Extensions of AF algebras: The projection lifting problem, Proceedings Symposia in Pure Mathematics 38 (American Mathematical Society, Providence, 1982),
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
Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
M. Dadarlat and G. Gong, A classification result for approximately homogeneous $C^*$-algebras of real rank zero, preprint.
G. A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1876), 29-44.
George A. Elliott, On the classification of $C^*$-algebras of real rank zero, J. Reine Angew. Math. 443 (1993), 179–219. MR 1241132, DOI 10.1515/crll.1993.443.179
G. A. Elliott, The classification problem for amenable $C^*$-algebras, Proc. ICM ’94, to appear.
G. A. Elliott and G. Gong, On the classification of $C^*$-algebras of real rank zero, II, Ann. Math. 144 (1996), 497–610.
H. Lin and H. Su, Classification of direct limits of generalized Toeplitz algebras, Pacific J. Math., to appear.
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