Amenability properties of Figà-Talamanca-Herz algebras on inverse semigroups (Q2809351)

Let \(S\) be an inverse semigroup. \textit{A.R. Medghalchi} and the author [J. Math. Anal. Appl. 395, No. 2, 473--485 (2012; Zbl 1257.43004)] extended the Figà-Talamanca-Herz algebras \(A_p(G)\) on locally compact groups \(G\) to algebras \(A_p(S)\) on inverse semigroups. In addition, pseudomeasures and pseudofunctions on inverse semigroups were studied by the author [Semigroup Forum 90, No. 3, 632-647 (2015; Zbl 1325.43001)]. In the paper under review, the author continues his studies of the papers above. He indeed obtains generalizations of Leptin's theorem and Ruan's theorem to (discrete) inverse semigroups by showing that \(A_p(S)\) has a bounded approximate identity if and only if the semigroup algebra \(l_1(S)\) is amenable, and that the Fourier algebra \(A(S)\) of \(S\) is operator amenable if and only if \(l_1(S)\) is amenable. Recall that \textit{H. Leptin}'s theorem asserts that for a locally compact group \(G\), the algebra \(A_p(G)\) has a bounded approximate identity if and only if \(G\) is amenable (see [C. R. Acad. Sci., Paris, Sér. A 266, 1180--1182 (1968; Zbl 0169.46501)]) and \textit{Z.-J. Ruan}'s theorem states that the operator amenability of the Fourier algebra \(A(G)\) is equivalent to the amenability of \(G\) (see [Am. J. Math. 117, No. 6, 1449--1474 (1995; Zbl 0842.43004)]). He also characterizes the amenability of \(A_p(S)\) in the sense of the amenability of Figà-Talamanca-Herz algebras of maximal subgroups of \(S\).