Biflatness of certain semigroup algebras (Q459281)

\textit{Y. Choi} et al. [Proc. R. Soc. Edinb., Sect. A, Math. 142, No. 4, 715--744 (2012; Zbl 1263.46060)] studied the simplicial cohomology of band semigroup algebras and showed that for every band semigroup \(S\), the semigroup algebra of \(\ell^1(S)\) is simplicially trivial. Since every biflat Banach algebra is simplicially trivial it is natural to ask when band semigroup algebras are biflat. The authors of the paper under review investigate the above inquiry. In fact, they give a necessary condition for the biflatness of band semigroup algebras, by using the fact that each band semigroup is a semilattice of rectangular band semigroups. They show, however, that this condition is not sufficient. They also study the biflatness of the second dual of semigroup algebras and show that for a certain class of inverse semigroups \(S\), the biflatness of \(\ell^1(S)^{''}\) is equivalent to the biprojectivity of \(\ell^1(S)\).