Vol. 21, No. 3, 1967

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Invariant means and the Stone-Čech compactification

Carroll O. Wilde and Klaus G. Witz

Vol. 21 (1967), No. 3, 577–586

In the first part of this paper the Arens multiplication on a space of bounded functions is used to simplify and extend results by Day and Frey on amenability of subsemigroups and ideals of a semigroup. For example it is shown that if S is a left amenable cancellation semigroup then a subsemigroup A of S is left amenable if and only if each two right ideals of A intersect. The remainder and major portion of this paper is devoted to relations between Ieft invariant means on m(S) and left idea1s of βS(=the Stone-Čech compactification of S). We find: If μ is a left invariant mean on m(S) and if S has Ieft cancellation then 𝒮(μ), the support of μ considered as a Borel measure on β(S), is a Iefl ideal of β(S). An application is that if S is a left amenable semigroup and I is a Ieft ideaI of βS, then K(I), the w-closed convex hull of If contains an extreme left invariant mean; if in addition S has cancellation then K(I) contains a left invariant mean which is the w-limit of a net of unweighted finite averages.

Mathematical Subject Classification
Primary: 46.20
Secondary: 54.00
Received: 10 May 1965
Published: 1 June 1967
Carroll O. Wilde
Klaus G. Witz