On higher real and stable ranks for $CCR$ $C^*-$algebras
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Abstract:
We calculate the real rank and stable rank of $CCR$ algebras which either have only finite dimensional irreducible representations or have finite topological dimension. We show that either rank of $A$ is determined in a good way by the ranks of an ideal $I$ and the quotient $A/I$ in four cases: when $A$ is $CCR$; when $I$ has only finite dimensional irreducible representations; when $I$ is separable, of generalized continuous trace and finite topological dimension, and all irreducible representations of $I$ are infinite dimensional; or when $I$ is separable, stable, has an approximate identity consisting of projections, and has the corona factorization property. We also present a counterexample on higher ranks of $M(A)$, $A$ subhomogeneous, and a theorem of P. Green on generalized continuous trace algebras.References
- Robert J. Archbold and Eberhard Kaniuth, Stable rank and real rank for some classes of group $C^*$-algebras, Trans. Amer. Math. Soc. 357 (2005), no. 6, 2165–2186. MR 2140436, DOI 10.1090/S0002-9947-05-03835-3
- Edwin J. Beggs and David E. Evans, The real rank of algebras of matrix valued functions, Internat. J. Math. 2 (1991), no. 2, 131–138. MR 1094700, DOI 10.1142/S0129167X91000089
- 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
- Lawrence G. Brown, Stable isomorphism of hereditary subalgebras of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 335–348. MR 454645
- Lawrence G. Brown, Semicontinuity and multipliers of $C^*$-algebras, Canad. J. Math. 40 (1988), no. 4, 865–988. MR 969204, DOI 10.4153/CJM-1988-038-5
- 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
- Lawrence G. Brown and Gert K. Pedersen, Limits and $C^\ast$-algebras of low rank or dimension, J. Operator Theory 61 (2009), no. 2, 381–417. MR 2501012, DOI 10.1140/epjc/s10052-009-1024-0
- Robert C. Busby, Double centralizers and extensions of $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 132 (1968), 79–99. MR 225175, DOI 10.1090/S0002-9947-1968-0225175-5
- J. Dixmier, Points séparés dans le spectre d’une $C^*$-algèbre, Acta Sci. Math. (Szeged) 22 (1961), 115–128 (French). MR 126736
- Jacques Dixmier, Traces sur les $C^*$-algèbres, Ann. Inst. Fourier (Grenoble) 13 (1963), no. fasc. 1, 219–262 (French). MR 149317
- Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR 0171173
- Jacques Dixmier and Adrien Douady, Champs continus d’espaces hilbertiens et de $C^{\ast }$-algèbres, Bull. Soc. Math. France 91 (1963), 227–284 (French). MR 163182
- Philip Green, letter dated October 14, 1976.
- Nawfal Elhage Hassan, Rang réel de certaines extensions, Proc. Amer. Math. Soc. 123 (1995), no. 10, 3067–3073 (French, with French summary). MR 1264814, DOI 10.1090/S0002-9939-1995-1264814-9
- Richard H. Herman and Leonid N. Vaserstein, The stable range of $C^{\ast }$-algebras, Invent. Math. 77 (1984), no. 3, 553–555. MR 759256, DOI 10.1007/BF01388839
- Dale Husemoller, Fibre bundles, 2nd ed., Graduate Texts in Mathematics, No. 20, Springer-Verlag, New York-Heidelberg, 1975. MR 0370578
- G. G. Kasparov, Hilbert $C^{\ast }$-modules: theorems of Stinespring and Voiculescu, J. Operator Theory 4 (1980), no. 1, 133–150. MR 587371
- Dan Kucerovsky and P. W. Ng, The corona factorization property and approximate unitary equivalence, Houston J. Math. 32 (2006), no. 2, 531–550. MR 2219330
- Hua Xin Lin, Generalized Weyl-von Neumann theorems, Internat. J. Math. 2 (1991), no. 6, 725–739. MR 1137095, DOI 10.1142/S0129167X91000405
- Masaru Nagisa, Hiroyuki Osaka, and N. Christopher Phillips, Ranks of algebras of continuous $C^\ast$-algebra valued functions, Canad. J. Math. 53 (2001), no. 5, 979–1030. MR 1859764, DOI 10.4153/CJM-2001-039-8
- Gabriel Nagy, Some remarks on lifting invertible elements from quotient $C^*$-algebras, J. Operator Theory 21 (1989), no. 2, 379–386. MR 1023322
- V. Nistor, Stable range for tensor products of extensions of ${\scr K}$ by $C(X)$, J. Operator Theory 16 (1986), no. 2, 387–396. MR 860355
- Victor Nistor, Stable rank for a certain class of type $\textrm {I}$ $C^\ast$-algebras, J. Operator Theory 17 (1987), no. 2, 365–373. MR 887231
- Hiroyuki Osaka, Real rank of crossed products by connected compact groups, Bull. London Math. Soc. 27 (1995), no. 3, 257–264. MR 1328702, DOI 10.1112/blms/27.3.257
- Hiroyuki Osaka, Non-commutative dimension for $C^*$-algebras, Interdiscip. Inform. Sci. 9 (2003), no. 2, 209–220. MR 2038012, DOI 10.4036/iis.2003.209
- 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
- Marc A. Rieffel, Dimension and stable rank in the $K$-theory of $C^{\ast }$-algebras, Proc. London Math. Soc. (3) 46 (1983), no. 2, 301–333. MR 693043, DOI 10.1112/plms/s3-46.2.301
- Albert Jeu-Liang Sheu, A cancellation theorem for modules over the group $C^\ast$-algebras of certain nilpotent Lie groups, Canad. J. Math. 39 (1987), no. 2, 365–427. MR 899843, DOI 10.4153/CJM-1987-018-7
- Wilhelm Winter, Decomposition rank of subhomogeneous $C^*$-algebras, Proc. London Math. Soc. (3) 89 (2004), no. 2, 427–456. MR 2078703, DOI 10.1112/S0024611504014716
Additional Information
- Lawrence G. Brown
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- MR Author ID: 42165
- Email: lgb@math.purdue.edu
- Received by editor(s): May 6, 2014
- Received by editor(s) in revised form: October 21, 2014
- Published electronically: January 27, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 7461-7475
- MSC (2010): Primary 46L05; Secondary 46M20
- DOI: https://doi.org/10.1090/tran/6616
- MathSciNet review: 3471097