On pseudo-amenability of commutative semigroup algebras and their second duals (Q1644736)

A Banach algebra \(A\) is called pseudo-amenable if there is a net \((m_{\alpha})\) in \(A\otimes_{p}A\) such that \(a\cdot m_{\alpha}-m_{\alpha}\cdot a\rightarrow 0\) and \(\pi_{A}(m_{\alpha})a\rightarrow a,\) where \(A\otimes_{p}A\) denotes the projective tensor product of \(A\) with \(A\) and \(\pi_{A}:A\otimes_{p}A\rightarrow A\) is given by \(\pi_{A}(a\otimes b)=ab,\) for each \(a,b\in A\). Among the Banach algebras, semigroup algebras are important and complicated examples. The present work studies the pseudo-amenability of Archimedean semigroup algebras. In fact, the authors show that \(\ell^{1}(S)\) is pseudo-amenable if and only if \(S\) is an amenable group, provided that \(S\) is an Archimedean semigroup. Also, for a commutative semigroup \(S\), the authors show that, if \(\ell^{1}(S)\) pseudo-amenable, then \(S\) is a Clifford semigroup.