Vol. 108, No. 2, 1983

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Characterisations of amenable locally compact semigroups

James Chin-Sze Wong and Abdolhamid Riazi

Vol. 108 (1983), No. 2, 479–496

Let G be a locally compact group, 𝒮 = 𝒮(G) the space of all (real) measurable simple functions on G and P = P(G) = {φ L1(G) : φ O,φ1 = 1}. W. R. Emerson recently proved that the following conditions on G are equivalent: (a) G is amenable (i.e. L(G) or 𝒮 has a left invariant mean; (b) NP(G) is closed under addition; (c) d(φ1 P,φ2 P) = 0 for φ12 P. Here NP(G) is the set of all f ∈𝒮 such that inf{∥φf : φ P} = 0 and d(φ1 P,φ2 P) = inf{∥φ1 φφ2 ψ1 : φ,ψ P}. He also demonstrated that some well-known results and characterisations of amenability follow this as simple consequences. The main purpose of this paper is to show that similar (and more) results hold true for locally compact semigroups S and invariant means on subspaces of M(S), where M(S) is the Banach algebra of all bounded regular Borel measures on S. We also provide an answer to a problem raised by Emerson. However, Emerson’s arguments do not carry over in the absence of a Haar integral.

Mathematical Subject Classification 2000
Primary: 43A07
Secondary: 22A20
Received: 4 June 1980
Published: 1 October 1983
James Chin-Sze Wong
Abdolhamid Riazi