On the topological centre problem for weighted convolution algebras and semigroup compactifications
Author:
Matthias Neufang
Journal:
Proc. Amer. Math. Soc. 136 (2008), 18311839
MSC (2000):
Primary 22D15, 43A10, 43A20, 43A22, 46H40, 54D35
Published electronically:
January 30, 2008
MathSciNet review:
2373615
Fulltext PDF Free Access
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Abstract: Let be a locally compact, noncompact group (we make the noncompactness assumption, for the most part, simply to avoid trivialities). We show that under a very mild assumption on the weight function , the weighted group algebra is strongly Arens irregular in the sense of Dales and Lau; i.e., both topological centres of equal . Also, we show that the topological centre of the algebra equals the weighted measure algebra . Moreover, still in the same situation, we prove that every linear (left) module map on is automatically bounded, and even continuous, hence given by convolution with an element in . To this end, we derive a general factorization theorem for bounded families in the module . Finally, using this result in the case where , we give a short proof of a theorem due to Protasov and Pym, stating that the topological centre of the semigroup is empty, where denotes the compactification of . This sharpens an earlier result by Lau and Pym; moreover, our method of proof gives a partial answer to a problem raised by Lau and Pym in 1995.
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 DALES, H. G.; LAU, A. T.M., The second duals of Beurling algebras, Mem. Amer. Math. Soc. 177 (2005), no. 836. MR 2155972 (2006k:43002)
 [2]
 DALES, H. G.; LAU, A. T.M.; STRAUSS, D., Banach algebras on semigroups and their compactifications, preprint, submitted to the Memoirs of the American Mathematical Society.
 [3]
 GHAHRAMANI, F., Weighted group algebra as an ideal in its second dual space, Proc. Amer. Math. Soc. 90 (1984), no. 1, 7176. MR 722417 (85i:43007)
 [4]
 GHAHRAMANI, F.; MCCLURE, J.P., Module homomorphisms of the dual modules of convolution Banach algebras, Canad. Math. Bull. 35 (1992), no. 2, 180185. MR 1165166 (93f:43004)
 [5]
 GRøNBæK, N., Amenability of weighted convolution algebras on locally compact groups, Trans. Amer. Math. Soc. 319 (1990), no. 2, 765775. MR 962282 (90j:43003)
 [6]
 HOFMEIER, H.; WITTSTOCK, G., A bicommutant theorem for completely bounded module homomorphisms, Math. Ann. 308 (1997), no. 1, 141154. MR 1446204 (98h:46065)
 [7]
 LAU, A. T.M., Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups, Math. Proc. Camb. Phil. Soc. 99 (1986), 273283. MR 817669 (87i:43001)
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 LAU, A. T.M.; LOSERT, V., On the second conjugate algebra of of a locally compact group, J. London Math. Soc. (2) 37 (1988), no. 3, 464470. MR 939122 (89e:43007)
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 LAU, A. T.M.; MILNES, P.; PYM, J. S., Locally compact groups, invariant means and the centres of compactifications, J. London Math. Soc. (2) 56 (1997), no. 1, 7790. MR 1462827 (98k:22021)
 [10]
 LAU, A. T.M.; PYM, J., The topological centre of a compactification of a locally compact group, Math. Z. 219 (1995), no. 4, 567579. MR 1343662 (96e:22010)
 [11]
 LAU, A. T.M.; ÜLGER, A., Topological centers of certain dual algebras, Trans. Amer. Math. Soc. 348 (1996), no. 3, 11911212. MR 1322952 (96h:43003)
 [12]
 NEUFANG, M., A unified approach to the topological centre problem for certain Banach algebras arising in abstract harmonic analysis, Arch. Math. 82 (2004), no. 2, 164171. MR 2047670 (2005g:22004)
 [13]
 NEUFANG, M., Solution to a conjecture by HofmeierWittstock, J. Funct. Anal. 217 (2004), no. 1, 171180. MR 2097611 (2005i:43003)
 [14]
 NEUFANG, M., On a conjecture by GhahramaniLau and related problems concerning topological centres, J. Funct. Anal. 224 (2005), no. 1, 217229. MR 2139110 (2006b:46063)
 [15]
 NEUFANG, M., On the Mazur property and property , to appear in: Journal of Operator Theory.
 [16]
 PALMER, T. W., Banach algebras and the general theory of algebras. Vol. I. Algebras and Banach algebras, Encyclopedia of Mathematics and its Applications, 49. Cambridge University Press, Cambridge, 1994. MR 1270014 (95c:46002)
 [17]
 PROTASOV, I. V.; PYM, J. S., Continuity of multiplication in the largest compactification of a locally compact group, Bull. London Math. Soc. 33 (2001), no. 3, 279282. MR 1817766 (2002h:22006)
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Additional Information
Matthias Neufang
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, K1S 5B6 Canada
Email:
mneufang@math.carleton.ca
DOI:
http://dx.doi.org/10.1090/S0002993908089089
PII:
S 00029939(08)089089
Keywords:
Locally compact group,
weighted group algebra,
left uniformly continuous function,
Arens product,
topological centre,
semigroup compactification.
Received by editor(s):
June 26, 2006
Received by editor(s) in revised form:
August 31, 2006
Published electronically:
January 30, 2008
Additional Notes:
The present work was partly supported by NSERC. This support is gratefully acknowledged.
Communicated by:
N. TomczakJaegermann
Article copyright:
© Copyright 2008
American Mathematical Society
