The stable rank of crossed products of sectional -algebras by compact Lie groups

Author:
Eckart Schulz

Journal:
Proc. Amer. Math. Soc. **112** (1991), 733-744

MSC:
Primary 46L55; Secondary 22D25, 46L05

DOI:
https://doi.org/10.1090/S0002-9939-1991-1042272-3

MathSciNet review:
1042272

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a second countable, locally compact Hausdorff space, and let a compact Lie group act on a -bundle , the fibres of which are of bounded, finite dimension. Denote by the -algebra of sections vanishing at infinity, and by the induced action of on . The stable ranks of both, the fixed point algebra and the crossed product algebra are determined.

**[Bre]**G. E. Bredon,*Introduction to compact transformation groups*, Academic Press, 1972. MR**0413144 (54:1265)****[En]**R. Engelking,*Dimension theory*, North-Holland, 1978. MR**0482697 (58:2753b)****[Ev]**B. Evans,*-bundles and compact transformation groups*, Mem. Amer. Math. Soc., no. 269, Amer. Math. Soc., Providence RI, 1982. MR**668809 (84a:46148)****[Fe]**J. M. G. Fell,*The structure of algebras of operator fields*, Acta Math.**106**(1961), 233-280. MR**0164248 (29:1547)****[Fe1]**J. M. G. Fell and R. S. Doran,*Representations of**-algabras, locally compact groups, and Banach**-algebraic bundles*, vol. 1, Academic Press, 1988.**[Gr]**P. Green,*The local structure of twisted covariance algebras*, Acta Math.**140**(1978), 191-250. MR**0493349 (58:12376)****[Le]**R. Y. Lee,*On the*-*algebras of operator fields*, Indiana Univ. Math. J.**25**(1976), 303-314. MR**0410400 (53:14150)****[Lo]**L. H. Loomis,*An introduction to abstract harmonic analysis*, Van Nostrand, 1953. MR**0054173 (14:883c)****[Mo]**C. C. Moore,*Group extensions and cohomology for locally compact groups*III, Trans. Amer. Math. Soc.**221**(1976), 1-33. MR**0414775 (54:2867)****[Ni]**V. Nistor,*Stable rank for a certain class of type I**-algebras*, J. Operator Theory**17**(1987), 365-373. MR**887231 (88h:46110)****[Pa]**R. S. Palais,*Slices and equivariant embeddings, seminar on transformation groups*(A. Borel, ed.), Ann. of Math. Studies, no. 46, Princeton Univ. Press, 1960. MR**0116341 (22:7129)****[PhR]**J. Phillips and I. Raeburn,*Crossed products by locally unitary automorphism groups and principal bundles*, J. Operator Theory**11**(1984), 215-241. MR**749160 (86m:46058)****[Ri]**M. Rieffel,*Actions of finite groups on**-algebras*, Math. Scand.**47**(1980), 157-176. MR**600086 (83c:46062)****[Ri2]**-,*Dimension and stable rank in the**-theory of**-algebras*, Proc. London Math. Soc. (3)**46**(1983), 301-333. MR**693043 (84g:46085)****[Ro]**J. Rosenberg,*Some results On cohomology with Borel cochains, with applications to group actions on operator algebras*, Operator Theory: Advances and Appl.**17**(1986), 301-330. MR**901077 (88k:46081)****[She]**A.J.L. Sheu,*The cancellation property for modules over the group**-algebras of certain nilpotent Lie groups*, Canad. J. Math.**39**, 365-427. MR**899843 (88i:46093)****[Ta]**K. F. Taylor,*-algebras of crystal groups*, Operator Theory: Advances and Appl.**41**(1989), 571-578. MR**1038354 (91e:46080)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
46L55,
22D25,
46L05

Retrieve articles in all journals with MSC: 46L55, 22D25, 46L05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1991-1042272-3

Keywords:
-crossed product,
stable rank,
-bundle

Article copyright:
© Copyright 1991
American Mathematical Society