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Functional calculus and $*$-regularity of a class of Banach algebras


Authors: Chi-Wai Leung and Chi-Keung Ng
Journal: Proc. Amer. Math. Soc. 134 (2006), 755-763
MSC (2000): Primary 47A60, 32A65
DOI: https://doi.org/10.1090/S0002-9939-05-08025-1
Published electronically: July 19, 2005
MathSciNet review: 2180894
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Abstract: Suppose that $(A,G,\alpha)$ is a $C^*$-dynamical system such that $G$ is of polynomial growth. If $A$ is finite dimensional, we show that any element in $K(G;A)$ has slow growth and that $L^1(G, A)$is $*$-regular. Furthermore, if $G$ is discrete and $\pi$ is a ``nice representation'' of $A$, we define a new Banach $*$-algebra $l^1_{\pi}(G, A)$ which coincides with $l^1(G;A)$ when $A$ is finite dimensional. We also show that any element in $K(G;A)$ has slow growth and $l^1_{\pi}(G, A)$ is $*$-regular.


References [Enhancements On Off] (What's this?)

  • 1. M. Baillet, Analyse spectrale des opérateurs hermitiens d'une espace de Banach, J. London Math. Soc. (2) 19 (1979), 497-508. MR 0540066 (80j:46080)
  • 2. J. Boidol, H. Leptin, J. Schürman and D. Vahle, Räume primitiver Ideale von Gruppenalgebren, Math. Ann. 236 (1978), 1-13. MR 0498971 (58:16959)
  • 3. J. Dixmier, Opérateurs de rang fini dans les représentations unitaires, Inst. Hautes Études Sci. Publ. Math. No. 6 (1960), 13-25. MR 0136684 (25:149)
  • 4. R. Exel, Amenability for Fell bundles, J. Reine Angew. Math. 492 (1997), 41-73. MR 1488064 (99a:46131)
  • 5. U. Haagerup, On the dual weights for crossed products of von Neumann algebras I - Removing separability conditions, Math. Scand. 43 (1978/79), 99-118. MR 0523830 (81e:46048a)
  • 6. A. Kishimoto, Ideals of $C^*$-crossed products by locally compact abelian groups, Proc. Symp. Pure Math. 38 (1982), 365-368. MR 0679718 (84b:46086)
  • 7. T. W. Palmer, Banach algebras and the general theory of $*$-algebras. Vol. 2, Encyclopedia of Mathematics and its Applications 79, Camb. Univ. Press (2001).MR 1819503 (2002e:46002)

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Additional Information

Chi-Wai Leung
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
Email: cwleung@math.cuhk.edu.hk

Chi-Keung Ng
Affiliation: School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
Email: ckng@nankai.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-05-08025-1
Keywords: Banach algebras, functional calculus, $*$-regular
Received by editor(s): June 23, 2004
Received by editor(s) in revised form: August 19, 2004, and October 13, 2004
Published electronically: July 19, 2005
Additional Notes: This work was jointly supported by Hong Kong RGC Direct Grant and the National Natural Science Foundation of China (10371058)
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society

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