On the dimension of left invariant means and left thick subsets

Abstract

IfSis a left amenable semigroup, letdim⁡⟨Ml(S)⟩\dim \langle Ml(S)\rangledenote the dimension of the set of left invariant means onS. Theorem.If S is left amenable, thendim⁡⟨Ml(S)⟩=n>∞\dim \langle Ml(S)\rangle = n > \inftyif and only if S contains exactly n disjoint finite left ideal groups. This result was proved by Granirer forScountable or left cancellative. Moreover, whenSis infinite, left amenable, and either left or right cancellative, we show thatdim⁡⟨Ml(S)⟩\dim \langle Ml(S)\rangleis at least the cardinality ofS. An application of these results shows that the radical of the second conjugate algebra ofl1(S){l_1}(S)is infinite dimensional whenSis a left amenable semigroup which does not contain a finite ideal.