On the dimension of left invariant means and left thick subsets

Abstract:

If S is a left amenable semigroup, let $\dim \langle Ml(S)\rangle$ denote the dimension of the set of left invariant means on S. Theorem. If S is left amenable, then $\dim \langle Ml(S)\rangle = n < \infty$ if and only if S contains exactly n disjoint finite left ideal groups. This result was proved by Granirer for S countable or left cancellative. Moreover, when S is infinite, left amenable, and either left or right cancellative, we show that $\dim \langle Ml(S)\rangle$ is at least the cardinality of S. An application of these results shows that the radical of the second conjugate algebra of ${l_1}(S)$ is infinite dimensional when S is a left amenable semigroup which does not contain a finite ideal.