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

Schulz, Eckart

doi:10.1090/S0002-9939-1991-1042272-3

The stable rank of crossed products of sectional $C^ *$-algebras by compact Lie groups
HTML articles powered by AMS MathViewer

by Eckart Schulz

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

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

PDF | Request permission

Abstract:

Let $X$ be a second countable, locally compact Hausdorff space, and let a compact Lie group $D$ act on a ${{\text {C}}^ * }$-bundle $(E,X)$, the fibres of which are of bounded, finite dimension. Denote by $A$ the ${{\text {C}}^ * }$-algebra of sections vanishing at infinity, and by $\hat \alpha$ the induced action of $D$ on $A$. The stable ranks of both, the fixed point algebra ${A^{\hat \alpha }}$ and the crossed product algebra $A{ \times _{\hat \alpha }}D$ are determined.

References

Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
Ryszard Engelking, Teoria wymiaru, Biblioteka Matematyczna, Tom 51. [Mathematics Library, Vol. 51], Państwowe Wydawnictwo Naukowe, Warsaw, 1977 (Polish). MR 0482696
Bruce D. Evans, $C^{\ast }$-bundles and compact transformation groups, Mem. Amer. Math. Soc. 39 (1982), no. 269, vii+63. MR 668809, DOI 10.1090/memo/0269
J. M. G. Fell, The structure of algebras of operator fields, Acta Math. 106 (1961), 233–280. MR 164248, DOI 10.1007/BF02545788

Representations of

-algabras, locally compact groups, and Banach

-algebraic bundles

Philip Green, The local structure of twisted covariance algebras, Acta Math. 140 (1978), no. 3-4, 191–250. MR 493349, DOI 10.1007/BF02392308
Ru Ying Lee, On the $C^*$-algebras of operator fields, Indiana Univ. Math. J. 25 (1976), no. 4, 303–314. MR 410400, DOI 10.1512/iumj.1976.25.25026
Lynn H. Loomis, An introduction to abstract harmonic analysis, D. Van Nostrand Co., Inc., Toronto-New York-London, 1953. MR 0054173
Calvin C. Moore, Group extensions and cohomology for locally compact groups. III, Trans. Amer. Math. Soc. 221 (1976), no. 1, 1–33. MR 414775, DOI 10.1090/S0002-9947-1976-0414775-X
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
Armand Borel, Seminar on transformation groups, Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, N.J., 1960. With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. MR 0116341
John Phillips and Iain Raeburn, Crossed products by locally unitary automorphism groups and principal bundles, J. Operator Theory 11 (1984), no. 2, 215–241. MR 749160
Marc A. Rieffel, Actions of finite groups on $C^{\ast }$-algebras, Math. Scand. 47 (1980), no. 1, 157–176. MR 600086, DOI 10.7146/math.scand.a-11882
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
Jonathan Rosenberg, Some results on cohomology with Borel cochains, with applications to group actions on operator algebras, Advances in invariant subspaces and other results of operator theory (Timişoara and Herculane, 1984) Oper. Theory Adv. Appl., vol. 17, Birkhäuser, Basel, 1986, pp. 301–330. MR 901077
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
Keith F. Taylor, $C^*$-algebras of crystal groups, The Gohberg anniversary collection, Vol. II (Calgary, AB, 1988) Oper. Theory Adv. Appl., vol. 41, Birkhäuser, Basel, 1989, pp. 511–518. MR 1038354