Finite sums and products of commutators in inductive limit C * -algebras
Thomsen, Klaus
Annales de l'Institut Fourier, Tome 43 (1993), p. 225-249 / Harvested from Numdam
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Des résultats de T. Fack, P. de La Harpe et G. Skandalis sur la structure interne des AF-algèbres simples sont généralisés à des C * -algèbres qui sont limites inductives de sommes directes finies de C * -algèbres homogènes. Les généralisations sont obtenues sous diverses hypothèses concernant les C * -algèbres dont les constructions dépendent; mais tous les résultats sont valables pour les limites inductives (avec unité) de sommes directes finies d’algèbres de matrices à coefficients dans les fonctions continues sur le cercle.

Results of T. Fack, P. de la Harpe and G. Skandalis concerning the internal structure of simple AF-algebras are extended to C * -algebras that are inductive limits of finite direct sums of homogeneous C * -algebras. The generalizations are obtained with slightly varying assumptions on the building blocks, but all results are applicable to unital simple inductive limits of finite direct sums of circle algebras.

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     author = {Thomsen, Klaus},
     title = {Finite sums and products of commutators in inductive limit $C^*$-algebras},
     journal = {Annales de l'Institut Fourier},
     volume = {43},
     year = {1993},
     pages = {225-249},
     doi = {10.5802/aif.1328},
     mrnumber = {94g:46062},
     zbl = {0766.46043},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1993__43_1_225_0}
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Thomsen, Klaus. Finite sums and products of commutators in inductive limit $C^*$-algebras. Annales de l'Institut Fourier, Tome 43 (1993) pp. 225-249. doi : 10.5802/aif.1328. http://gdmltest.u-ga.fr/item/AIF_1993__43_1_225_0/

[1] B. Blackadar, O. Bratteli, G.A. Elliott and A. Kumjian, Reduction or real rank in inductive limits of C*-algebras, Math. Ann., 292 (1992), 111-126. | MR 93a:46112 | Zbl 0738.46027

[2] M. Choi, G. Elliott, Density of the selfadjoint elements with finite spectrum in an irrational rotation C*-algebra, Math. Scand., 67 (1990), 73-86. | MR 92a:46062 | Zbl 0743.46070

[3] J. Cuntz and G.K. Pedersen, Equivalence and traces on C*-algebras, J. Func. Anal., 33 (1979), 135-164. | MR 80m:46053 | Zbl 0427.46042

[4] M. Dadarlat, G. Nagy, A. Nemethi, C. Pasnicu, Reduction of topological stable rank in inductive limits of C*-algebras, preprint, 1990.

[5] G. Elliott, On the classification of C*-algebras of real rank zero, preprint. | Zbl 0867.46041

[6] G. Elliott and M. Rordam, The automorphisms of inductive limits of circle algebras, Comm. Math. Phys., to appear.

[7] R. Engelking, Dimension theory, North-Holland, Amsterdam, Oxford, New York, 1978.

[8] D. Evans, G. Elliott, The structure of the irrational rotation C*-algebras, Ann. Math., to appear. | Zbl 0847.46034

[9] T. Fack, Finite sums of commutators in C*-algebras, Ann. Inst. Fourier, Grenoble, 32-1 (1982), 129-137. | Numdam | MR 83g:46051 | Zbl 0464.46047

[10] P.G. Ghatage and W.J. Philips, C*-algebras generated by weighted shifts II, Indiana Univ. Math. J., 30 (1981), 539-545. | MR 83h:46072 | Zbl 0437.46048

[11] K. R. Goodearl, Riesz decomposition in inductive limit C*-algebras, preprint. | Zbl 0820.46064

[12] P. De La Harpe, G. Skandalis, Déterminant associé à une trace sur une algèbre de Banach, Ann. Inst. Fourier, Grenoble, 34-1 (1984), 241-260. | Numdam | MR 87i:46146a | Zbl 0521.46037

[13] P. De La Harpe, G. Skandalis, Produits finis de commutateurs dans les C*-algèbres, Ann. Inst. Fourier, Grenoble, 34-4 (1984), 169-202. | Numdam | MR 87i:46146b | Zbl 0536.46044

[14] P. De La Harpe, G. Skandalis, Sur la simplicité essentielle du groupe des inversibles et du groupe unitaire dans une C*-algèbre simple, J. Func. Anal., 62 (1985), 354-378. | MR 87i:46146c | Zbl 0573.46033

[15] V. Nistor, On the homotopy groups of the automorphism group of AF — C*-algebras, J. Operator Theory, 19 (1988), 319-340. | MR 90g:46089 | Zbl 0674.46042

[16] G.K. Pedersen, C*-algebras and their automorphism groups, Academic Press, London, New York, Sans Francisco, 1979. | MR 81e:46037 | Zbl 0416.46043

[17] I. Putmam, On the topological stable rank of certain transformation group C*-algebras, Ergod. Th. & Dynam. Sys., 10 (1990), 197-207. | MR 91f:46090 | Zbl 0667.46045

[18] M. A. Rieffel, Dimension and stable rank in the K-theory of C*-algebras, Proc. London Math. Soc., (3), 46 (1983), 301-333. | MR 84g:46085 | Zbl 0533.46046

[19] M. A. Rieffel, The homotopy groups of the unitary groups of non-commutative tori, J. Oper. Th., 17 (1987), 237-254. | MR 88f:22018 | Zbl 0656.46056

[20] K. Thomsen, Nonstable K-theory for operator algebras, K-theory, 4 (1991), 245-267. | MR 92h:46102 | Zbl 0744.46069

[21] K. Thomsen, Inductive limits of interval algebras: the tracial state space, Amer. J. Math., to appear. | Zbl 0814.46050

[22] K. Thomsen, Homomorphisms between finite direct sums of circle algebras, Linear and Multilinear algebra, 32 (1992), 33-50. | MR 94a:46080 | Zbl 0783.46029