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Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups

Published online by Cambridge University Press:  24 October 2008

Anthony To-Ming Lau
Anthony To-Ming Lau
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alta. T6G 2G1, Canada
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Extract

Let G be a topological semigroup, i.e. G is a semigroup with a Hausdorff topology such that the map (a, b) → a.b from G × G into G is continuous when G × G has the product topology. Let C(G) denote the space of complex-valued bounded continuous functions on G with the supremum norm. Let LUC (G) denote the space of bounded left uniformly continuous complex-valued functions on G i.e. all f ε C(G) such that the map a → laf of G into C(G) is continuous when C(G) has a norm topology, where (laf )(x) = f (ax) (a, x ε G). Then LUC (G) is a closed subalgebra of C(G) invariant under translations. Furthermore, if m ε LUC (G)*, f ε LUC (G), then the function

is also in LUC (G). Hence we may define a product

for n, m ε LUC(G)*. LUC (G)* with this product is a Banach algebra. Furthermore, ʘ is precisely the restriction of the Arens product defined on the second conjugate algebra l∞(G)* = l1(G)** to LUC (G)*. We refer the reader to [1] and [10] for more details.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 99 , Issue 2 , March 1986 , pp. 273 - 283
Copyright
Copyright © Cambridge Philosophical Society 1986

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References

REFERENCES

[1]Berglund, J. F., Junghenn, H. D. and Milnes, P.. Compact right topological semigroups and generalizations of almost periodicity. Lecture Notes in Mathematics 663 (Springer-Verlag, 1978).CrossRefGoogle Scholar
[2]Chou, C.. On topologically invariant means on a locally compact group. Trans. Amer. Math. Soc. 151 (1970), 443–456.CrossRefGoogle Scholar
[3]Dunford, N. and Schwartz, J.. Linear Operators I (Interscience Publishers, 1957).Google Scholar
[4]Dzinotyiweyi, H. A. M.. Weakly almost periodic frunctions and the irregularity of multiplication in semigroup algebras. Math. Scand. 46 (1980), 157–172.CrossRefGoogle Scholar
[5]Granirer, E.. On amenable semigroups with a finite-dimensional set of invariant means 1. Illinois J. Math. 7 (1963), 32–48.Google Scholar
[6]Granirer, E.. On the invariant mean on the topological semigroups and on topological groups. Pacific J. Math. 15 (1965), 107–140.CrossRefGoogle Scholar
[7]Granirer, E.. Exposed points of convex sets and weak sequential convergence. Mem. Amer. Math. Soc. 123 (1972).Google Scholar
[8]Grosser, M. and Losert, V.. The norm-strict bidual of a Banach algebra and the dual of C u(G). Manuscripta Math. 45 (1984), 127–146.CrossRefGoogle Scholar
[9]Klawe, M.. On the dimension of left invariant means and left thick subsets. Trans. Amer. Math. Soc. 231 (1977), 507–518.CrossRefGoogle Scholar
[10]Mitchell, T.. Topological semigroups and fixed points. Illinois J. Math. 14 (1970), 630–641.CrossRefGoogle Scholar
[11]Parsons, D. J.. The centre of the second dual of a commutative semigroup algebra. Math. Proc. Cambridge Philos. Soc. 95 (1984), 71–92.CrossRefGoogle Scholar
[12]Ruppert, W.. On semigroup compactifications of topological groups. Proc. Roy. Irish Acad. 79 (1979), 179–200.Google Scholar
[13]Takesaki, M.. Theory of Operator Algebras I (Springer-Verlag, 1979).CrossRefGoogle Scholar
[14]Wong, J. C. S.. Invariant means on locally compact semigroups. Proc. Amer. Math. Soc. 31 (1972), 39–45.CrossRefGoogle Scholar
[15]Zappa, A.. The center of the convolution algebra C u(G)*. Rend. Sem. Mat. Univ. Padova 52 (1974), 71–83.Google Scholar