Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Stable rank and real rank for some classes of group $C^\ast$ -algebras

Author(s): Robert J. Archbold; Eberhard Kaniuth
Journal: Trans. Amer. Math. Soc. 357 (2005), 2165-2186.
MSC (2000): Primary 46L05, 46L35, 22D25
Posted: January 21, 2005
MathSciNet review: 2140436
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: 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.

References:

1.: E.J. Beggs and D.E. Evans, The real rank of algebras of matrix valued functions, Intern. J. Math. 2 (1991), 131-137. MR 1094700 (92e:46114)
2.: L.G. Brown and G.K. Pedersen, $C^\ast$ -algebras of real rank zero, J. Funct. Anal. 99 (1991), 131-149. MR 1120918 (92m:46086)
3.: K.R. Davidson, $C^\ast$ -algebras by example, Fields Institute Monographs, American Mathematical Society, 1996. MR 1402012 (97i:46095)
4.: J. Dixmier, $C^\ast$ -algebras, North-Holland, 1977.MR 0458185 (56:16388)
5.: K.J. Dykema, U. Haagerup and M. Rørdam, The stable rank of some free product $C^\ast$ -algebras, Duke Math. J. 90 (1997), 95-121.MR 1478545 (99g:46077a)
6.: K.J. Dykema and P. de la Harpe, Some groups whose reduced $C^\ast$ -algebras have stable rank one, J. Math. Pures Appl. 78 (1999), 591-608.MR 1708667 (2000f:46070)
7.: S. Echterhoff and E. Kaniuth, Certain group extensions and twisted covariance algebras with generalized continuous trace, In: Harmonic Analysis (Eds. P. Eymard and J.P. Pier), LNM 1359, 159-169, Springer-Verlag, 1987.MR 0974312 (90c:46083)
8.: N. Elhage Hassan, Rang reel de certaines extensions, Proc. Amer. Math. Soc. 123 (1995), 3067-3073. MR 1264814 (95m:46117)
9.: J.M.G. Fell, The dual spaces of $C^\ast$ -algebras, Trans. Amer. Math. Soc. 94 (1960), 365-403. MR 0146681 (26:4201)
10.: J.M.G. Fell and R.S. Doran, Representations of $\ast$ -algebras, locally compact groups, and Banach $\ast$ -algebraic bundles, Vol. 2, Academic Press, 1988. MR 0936629 (90c:46002)
11.: S. Grosser and M. Moskowitz, Compactness conditions in topological groups, J. Reine Angew. Math. 246 (1971), 1-40.MR 0284541 (44:1766)
12.: E. Hewitt and K.A. Ross, Abstract harmonic analysis I, Springer-Verlag, 1963. MR 0156915 (28:158)
13.: E. Kaniuth, Primitive ideal spaces of groups with relatively compact conjugacy classes, Arch. Math. 32 (1979), 16-24.MR 0532844 (81f:22009)
14.: E. Kaniuth, Group $C^\ast$ -algebras of real rank zero or one, Proc. Amer. Math. Soc. 119 (1993), 1347-1354. MR 1164146 (94a:46074)
15.: E. Kaniuth and K.F. Taylor, Kazhdan constants and the dual space topology, Math. Ann. 293 (1992), 495-508. MR 1170523 (93i:22003)
16.: C.C. Moore, Groups with finite dimensional irreducible representations, Trans. Amer. Math. Soc. 166 (1972), 401-410. MR 0302817 (46:1960)
17.: K. Nagami, Dimension theory, Academic Press, New York/London, 1970. MR 0271918 (42:6799)
18.: M. Nagisa, Stable rank of some full group $C^\ast$ -algebras of groups obtained by the free product, Intern. J. Math. 8 (1997), 375-382. MR 1454479 (98i:46057)
19.: M. Nagisa, H. Osaka and N.C. Phillips, Ranks of algebras of continuous $C^\ast$ -algebra valued functions, Canad. J. Math. 53 (2001), 979-1030. MR 1859764 (2002h:46094)
20.: V. Nistor, Stable range for tensor products of extensions of by , J. Oper. Theory 16 (1986), 387-396.MR 0860355 (88b:46085)
21.: V. Nistor, Stable rank for a certain class of type I $C^\ast$ -algebras, J. Oper. Theory 17 (1987), 365-373.MR 0887231 (88h:46110)
22.: H. Osaka, Real rank of crossed products by connected compact groups, Bull. London Math. Soc. 27 (1995), 257-264.MR 1328702 (96e:46076)
23.: A.R. Pears, Dimension theory of general spaces, Cambridge University Press, Cambridge, 1975. MR 0394604 (52:15405)
24.: M.A. Rieffel, Dimension and stable rank in the -theory of $C^\ast$ -algebras, Proc. London Math. Soc.(3) 6 (1983), 301-333. MR 0693043 (84g:46085)
25.: E. Schulz, The stable rank of crossed products of sectional $C^\ast$ -algebras by compact Lie groups, Proc. Amer. Math. Soc. 112 (1991), 733-744. MR 1042272 (91j:46082)
26.: A.J.-L. Sheu, A cancellation theorem for projective modules over the group $C^\ast$ -algebras of certain nilpotent Lie groups, Canad. J. Math. 39 (1987), 365-427. MR 0899843 (88i:46093)
27.: T. Sudo, Stable rank of the reduced $C^\ast$ -algebras of non-amenable Lie groups of type I, Proc. Amer. Math. Soc. 125 (1997), 3647-3654.MR 1415371 (98b:46093)
28.: T. Sudo, Stable rank of the $C^\ast$ -algebras of amenable Lie groups of type I, Math. Scand. 84 (1999), 231-242.MR 1700530 (2000g:46078)
29.: T. Sudo, Dimension theory of group $C^\ast$ -algebras of connected Lie groups of type I, J. Math. Soc. Japan 42 (2000), 583-590. MR 1760606 (2001h:22007)
30.: T. Sudo and H. Takai, Stable rank of the $C^\ast$ -algebras of nilpotent Lie groups, Intern. J. Math. 6 (1995), 439-446.MR 1327158 (96b:46083)
31.: T. Sudo and H. Takai, Stable rank of the $C^\ast$ -algebras of solvable Lie groups of type I, J. Oper. Theory 38 (1997), 67-86.MR 1462016 (99a:46125)
32.: J. Tomiyama and M. Takesaki, Applications of fibre bundles to the certain class of $C^\ast$ -algebras, Tohoku Math. J. 13 (1963), 498-523.MR 0139025 (25:2465)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|