Amenability properties of Figa-Talamanca-Herz algebras on inverse semigroups

This paper continues the joint work with A. R. Medghalchi (2012) and the author's recent work (2015). For an inverse semigroup S, it is shown that A(p)(S) has a bounded approximate identity if and only if l(1)(S) is amenable (a generalization of Leptin's theorem) and that A(S), the Fourier algebra of S, is operator amenable if and only if l(1)(S) is amenable (a generalization of Ruan's theorem).