On the topological centre problem for weighted convolution algebras and semigroup compactifications

Author:
Matthias Neufang

Journal:
Proc. Amer. Math. Soc. **136** (2008), 1831-1839

MSC (2000):
Primary 22D15, 43A10, 43A20, 43A22, 46H40, 54D35

DOI:
https://doi.org/10.1090/S0002-9939-08-08908-9

Published electronically:
January 30, 2008

MathSciNet review:
2373615

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Abstract: Let be a locally compact, non-compact group (we make the non-compactness 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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Additional Information

**Matthias Neufang**

Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, K1S 5B6 Canada

Email:
mneufang@math.carleton.ca

DOI:
https://doi.org/10.1090/S0002-9939-08-08908-9

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. Tomczak-Jaegermann

Article copyright:
© Copyright 2008
American Mathematical Society