Vol. 16, No. 2, 1966

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Invariant means on topological semigroups

Loren N. Argabright

Vol. 16 (1966), No. 2, 193–203

This paper is concerned with the existence and structure of invariant means on the space C(S) of all (including unbounded) continuous real-valued functions on a topological semigroup S. The main result is that for realcompact semigroups every left invariant mean (if any exist) arises as an integral over a compact left invariant subset of S. The question of existence of noncompact group G such that C(G) admits a left invariant mean is also considered. If G is a realcompact (or discrete, or locally compact abelian) group, then C(G) admits a left invariant mean only if G is compact.

Mathematical Subject Classification
Primary: 46.80
Secondary: 42.50
Received: 17 May 1964
Published: 1 February 1966
Loren N. Argabright