Abstract
Let G be a locally compact group and LUC(G) the C*-algebra of the bounded left uniformly continuous functions on G. The spectrum G LUC of LUC(G) is the universal semigroup compactification of G with respect to the joint continuity property: the multiplication on G×G LUC is jointly continuous. The paper studies the joint weak* continuity of multiplication on LUC(G)* and, in particular, the question how the joint continuity property of G LUC can be related to a property concerning the whole algebra LUC(G)*. The group G is naturally replaced by the measure algebra M(G), and LUC(G)* can be identified with M(G LUC), the space of regular Borel measures on G LUC. It is shown that the joint weak* continuity can fail even on bounded sets of M(G)×M(G LUC), but, on the other hand, the multiplication on M(G)×M(G LUC) is positive continuous in the sense of Jewett.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berglund, J.F., Junghenn, H.D., Milnes, P.: Analysis on Semigroups. Wiley, New York (1989)
Dales, H.G., Lau, A.T.-M., Strauss, D.: Banach algebras on semigroups and on their compactifications. Mem. Am. Math. Soc. (to appear)
Dunkl, C.F.: The measure algebra of a locally compact hypergroup. Trans. Am. Math. Soc. 179, 331–348 (1973)
Filali, M., Salmi, P.: Slowly oscillating functions in semigroup compactifications and convolution algebras. J. Funct. Anal. 250, 144–166 (2007)
Grosser, M., Losert, V.: The norm-strict bidual of a Banach algebra and the dual of C u (G). Manuscr. Math. 45, 127–146 (1984)
Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, vol. 2. Springer, Berlin (1970)
Holmes, R.B.: Geometric Functional Analysis and Its Applications. Springer, New York (1975)
Jewett, R.I.: Spaces with an abstract convolution of measures. Adv. Math. 18, 1–101 (1975)
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. Cambr. Philos. Soc. 99, 273–283 (1986)
Lau, A.T.-M., Pym, J.: The topological centre of a compactification ofa locally compact group. Math. Z. 219, 567–579 (1995)
Neufang, M.: A unified approach to the topological centre problem for certain Banach algebras arising in abstract harmonic analysis. Arch. Math. (Basel) 82, 164–171 (2004)
Pym, J.: A note on \({G}^{\mathcal{LUC}}\) and Veech’s theorem. Semigroup Forum 59, 171–174 (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jimmie D. Lawson.
Supported by Academy of Finland.
Rights and permissions
About this article
Cite this article
Salmi, P. Joint continuity of multiplication on the dual of the left uniformly continuous functions. Semigroup Forum 80, 155–163 (2010). https://doi.org/10.1007/s00233-009-9189-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-009-9189-2