Biflatness of ${\ell }^1$-semilattice algebras

Abstract

Building on an old result of Duncan and Namioka, we show that the ${\ell }^1$-convolution algebra of a semilattice $S$ is biflat precisely when $S$ is uniformly locally finite. The proof shows in passing that for such $S$ the convolution algebra is isomorphic to ${\ell }^1(S)$ with pointwise multiplication. At the end we sketch how these techniques may be extended to prove an analogous characterisation of biflatness for Clifford semigroup algebras.

17 pages, accepted by Semigroup Forum. Some details have been added to clarify the closing remarks on the Clifford semigroup case