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,∥φ∥1= 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 φ1,φ2∈ P. Here NP(G) is the set of all f ∈𝒮 such that
inf{∥φ∗f∥∞: φ ∈ P} = 0 and d(φ1∗P,φ2∗P) =inf{∥φ1∗φ−φ2∗ψ∥1: 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.
