Stable rank and real rank for some classes of group \(C^*\)-algebras. (English) Zbl 1067.46044
Summary: We investigate the real and stable rank of the \(C^\ast\)-algebras of locally compact groups with relatively compact conjugacy classes or finite-dimensional irreducible representations. Estimates and formulae are given in terms of the group-theoretic rank.
MSC:
| 46L05 | General theory of \(C^*\)-algebras |
| 46L35 | Classifications of \(C^*\)-algebras |
| 22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
References:
| [1] | 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. · Zbl 0748.46030 · doi:10.1142/S0129167X91000089 |
| [2] | Lawrence G. Brown and Gert K. Pedersen, \?*-algebras of real rank zero, J. Funct. Anal. 99 (1991), no. 1, 131 – 149. · Zbl 0776.46026 · doi:10.1016/0022-1236(91)90056-B |
| [3] | Kenneth R. Davidson, \?*-algebras by example, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. · Zbl 0958.46029 |
| [4] | Jacques Dixmier, \?*-algebras, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett; North-Holland Mathematical Library, Vol. 15. · Zbl 0372.46058 |
| [5] | Ken Dykema, Uffe Haagerup, and Mikael Rørdam, The stable rank of some free product \?*-algebras, Duke Math. J. 90 (1997), no. 1, 95 – 121. , https://doi.org/10.1215/S0012-7094-97-09004-9 Ken Dykema, Uffe Haagerup, and Mikael Rørdam, Correction to: ”The stable rank of some free product \?*-algebras”, Duke Math. J. 94 (1998), no. 1, 213. · Zbl 0943.46034 · doi:10.1215/S0012-7094-98-09410-8 |
| [6] | Kenneth J. Dykema and Pierre de la Harpe, Some groups whose reduced \?*-algebras have stable rank one, J. Math. Pures Appl. (9) 78 (1999), no. 6, 591 – 608 (English, with English and French summaries). · Zbl 0929.22008 · doi:10.1016/S0021-7824(99)00015-X |
| [7] | Siegfried Echterhoff and Eberhard Kaniuth, Certain group extensions and twisted covariance algebras with generalized continuous trace, Harmonic analysis (Luxembourg, 1987) Lecture Notes in Math., vol. 1359, Springer, Berlin, 1988, pp. 159 – 169. · Zbl 0687.43004 · doi:10.1007/BFb0086596 |
| [8] | Nawfal Elhage Hassan, Rang réel de certaines extensions, Proc. Amer. Math. Soc. 123 (1995), no. 10, 3067 – 3073 (French, with French summary). · Zbl 0848.46036 |
| [9] | J. M. G. Fell, The dual spaces of \?*-algebras, Trans. Amer. Math. Soc. 94 (1960), 365 – 403. · Zbl 0090.32803 |
| [10] | J. M. G. Fell and R. S. Doran, Representations of *-algebras, locally compact groups, and Banach *-algebraic bundles. Vol. 2, Pure and Applied Mathematics, vol. 126, Academic Press, Inc., Boston, MA, 1988. Banach *-algebraic bundles, induced representations, and the generalized Mackey analysis. · Zbl 0652.46050 |
| [11] | Siegfried Grosser and Martin Moskowitz, Compactness conditions in topological groups, J. Reine Angew. Math. 246 (1971), 1 – 40. · Zbl 0219.22011 · doi:10.1515/crll.1971.246.1 |
| [12] | Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der mathematischen Wissenschaften, Bd. 115, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. · Zbl 0416.43001 |
| [13] | Eberhard Kaniuth, Primitive ideal spaces of groups with relatively compact conjugacy classes, Arch. Math. (Basel) 32 (1979), no. 1, 16 – 24. · Zbl 0388.22003 · doi:10.1007/BF01238463 |
| [14] | Eberhard Kaniuth, Group \?*-algebras of real rank zero or one, Proc. Amer. Math. Soc. 119 (1993), no. 4, 1347 – 1354. · Zbl 0822.22002 |
| [15] | Eberhard Kaniuth and Keith F. Taylor, Kazhdan constants and the dual space topology, Math. Ann. 293 (1992), no. 3, 495 – 508. · Zbl 0787.22006 · doi:10.1007/BF01444731 |
| [16] | Calvin C. Moore, Groups with finite dimensional irreducible representations, Trans. Amer. Math. Soc. 166 (1972), 401 – 410. · Zbl 0236.22010 |
| [17] | Keiô Nagami, Dimension theory, With an appendix by Yukihiro Kodama. Pure and Applied Mathematics, Vol. 37, Academic Press, New York-London, 1970. · Zbl 0224.54060 |
| [18] | Masaru Nagisa, Stable rank of some full group \?*-algebras of groups obtained by the free product, Internat. J. Math. 8 (1997), no. 3, 375 – 382. · Zbl 0885.46049 · doi:10.1142/S0129167X97000184 |
| [19] | Masaru Nagisa, Hiroyuki Osaka, and N. Christopher Phillips, Ranks of algebras of continuous \?*-algebra valued functions, Canad. J. Math. 53 (2001), no. 5, 979 – 1030. · Zbl 0991.46029 · doi:10.4153/CJM-2001-039-8 |
| [20] | V. Nistor, Stable range for tensor products of extensions of \? by \?(\?), J. Operator Theory 16 (1986), no. 2, 387 – 396. · Zbl 0638.46041 |
| [21] | Victor Nistor, Stable rank for a certain class of type \? \?*-algebras, J. Operator Theory 17 (1987), no. 2, 365 – 373. · Zbl 0647.46056 |
| [22] | Hiroyuki Osaka, Real rank of crossed products by connected compact groups, Bull. London Math. Soc. 27 (1995), no. 3, 257 – 264. · Zbl 0831.46075 · doi:10.1112/blms/27.3.257 |
| [23] | A. R. Pears, Dimension theory of general spaces, Cambridge University Press, Cambridge, England-New York-Melbourne, 1975. · Zbl 0312.54001 |
| [24] | Marc A. Rieffel, Dimension and stable rank in the \?-theory of \?*-algebras, Proc. London Math. Soc. (3) 46 (1983), no. 2, 301 – 333. · Zbl 0533.46046 · doi:10.1112/plms/s3-46.2.301 |
| [25] | Eckart Schulz, The stable rank of crossed products of sectional \?*-algebras by compact Lie groups, Proc. Amer. Math. Soc. 112 (1991), no. 3, 733 – 744. · Zbl 0744.46053 |
| [26] | Albert Jeu-Liang Sheu, A cancellation theorem for modules over the group \?*-algebras of certain nilpotent Lie groups, Canad. J. Math. 39 (1987), no. 2, 365 – 427. · Zbl 0692.46064 · doi:10.4153/CJM-1987-018-7 |
| [27] | Takahiro Sudo, Stable rank of the reduced \?*-algebras of non-amenable Lie groups of type I, Proc. Amer. Math. Soc. 125 (1997), no. 12, 3647 – 3654. · Zbl 0888.46040 |
| [28] | Takahiro Sudo, Stable rank of the \?*-algebras of amenable Lie groups of type I, Math. Scand. 84 (1999), no. 2, 231 – 242. · Zbl 0957.22012 · doi:10.7146/math.scand.a-13877 |
| [29] | Takahiro Sudo, Dimension theory of group \?*-algebras of connected Lie groups of type I, J. Math. Soc. Japan 52 (2000), no. 3, 583 – 590. · Zbl 0971.46039 · doi:10.2969/jmsj/05230583 |
| [30] | Takahiro Sudo and Hiroshi Takai, Stable rank of the \?*-algebras of nilpotent Lie groups, Internat. J. Math. 6 (1995), no. 3, 439 – 446. · Zbl 0836.22009 · doi:10.1142/S0129167X95000158 |
| [31] | Takahiro Sudo and Hiroshi Takai, Stable rank of the \?*-algebras of solvable Lie groups of type I, J. Operator Theory 38 (1997), no. 1, 67 – 86. · Zbl 0892.46074 |
| [32] | Jun Tomiyama and Masamichi Takesaki, Applications of fibre bundles to the certain class of \?*-algebras, Tôhoku Math. J. (2) 13 (1961), 498 – 522. · Zbl 0113.09701 · doi:10.2748/tmj/1178244253 |
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