×

Affine semigroups and second duals. (English) Zbl 0782.22002

This note proves that an affine semigroup \(K\) has a universal affine semigroup compactification. As a topological space, this compactification is just the state space of the space of bounded real-valued affine functions on \(K\). When \(K\) is the unit ball of a Banach algebra, the compactification is the unit ball of the second dual in its weak\(^*\) topology with an Arens multiplication. The theories of affine semigroup compactifications and second duals of Banach algebras are essentially equivalent.

MSC:

22A15 Structure of topological semigroups
46L30 States of selfadjoint operator algebras
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Asimow, L., and A. J. Ellis, ”Convexity Theory and its Applications in Functional Analysis” Academic Press, London, 1980. · Zbl 0453.46013
[2] Berglund, J. F., H. D. Junghenn and P. Milnes, ”Analysis on Semigroups” Wiley, New York, 1989. · Zbl 0727.22001
[3] Bonsall, F.F., and J. Duncan, ”Complete Normed Algebras” Springer, Berlin, 1973. · Zbl 0271.46039
[4] Hewitt, E., and K. A. Ross, ”Abstract Harmonic Analysis” Vol I Springer, Berlin, 1963. · Zbl 0115.10603
[5] Kelley, J. L., and I. Namioka, ”Linear Topological Spaces” Van Nostrand, Princeton, 1963. · Zbl 0115.09902
[6] Pym, J. S.,Compact semigroups with one sided continuity in ”The Analytical and Topological Theory of Semigroups” ed. K.H. Hofmann et al., de Gruyter, 1990, pp. 197–217.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.