For embeddable semigroups, denseness of finitely supported invariant measures is governed by extensibility to the free group, and this yields the periodic approximation property for left amenable residually finite semigroups and finite-rank free subsemigroups.
Ceccherini-Silberstein and M
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Extensibility and denseness of periodic semigroup actions
For embeddable semigroups, denseness of finitely supported invariant measures is governed by extensibility to the free group, and this yields the periodic approximation property for left amenable residually finite semigroups and finite-rank free subsemigroups.