Vol. 81, No. 1, 1979

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A note on measures on foundation semigroups with weakly compact orbits

Heneri Amos Murima Dzinotyiweyi and Gerard L. G. Sleijpen

Vol. 81 (1979), No. 1, 61–69

For an extensive class of locally compact semigroups S, foundation semigroups with identity element, we prove that two subalgebras of M(S) [the algebra of the bounded Radon measures on S] coincide. Namely, the algebra L(S), generated by the m M(S)+ for which the orbits on the compact subsets of S are weakly compact subsets of M(S), or, equivalently, for which the translations are weakly continuous, and the algebra Me(S), generated by the m M(S)+ for which the restrictions of the orbit of m on S to the compact subsets of S are weakly compact. In case S is a group, both these algebras consist of the bounded Radon measures that are absolutely continuous with respect to a Haar measure on S.

Mathematical Subject Classification 2000
Primary: 43A05
Secondary: 28C10
Received: 15 February 1978
Published: 1 March 1979
Heneri Amos Murima Dzinotyiweyi
Gerard L. G. Sleijpen