Abstract
Let S be a discrete semigroup, m (S) the space of all bounded real functions on S with
the usualsupremum norm. Let Acm (S) be a uniformly closed left invariant subalgebra
of m (S) with 1 c A. We say that A is extremely left amenable if there isamultiplicative
left invariant meanon A. Let P = {h ?A: h =|g-1,g | forsome g ?A, s ?S}. It isshown that .
A is extremely left amenable if and only if there is a mean ? on A such that ?(PA) = 0.
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