semilattices, Clifford semigroups, Möbius function, Banach algebras
Appeared as Semigroup Forum 75 (2007), no. 2, 253--271.
Preprint version available at arXiv math.FA/0606366
We show that if L is a semilattice then the l1-convolution algebra of L is biflat precisely when L is "uniformly locally finite". Our proof technique shows in passing that if this convolution algebra is biflat then it is isomorphic as a Banach algebra to the Banach space l1(L) equipped 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.
In the preprint versions, the footnotes on the first page (which usually house MSC data and the suchlike) included an acknowledgment that the paper uses Paul Taylor's diagrams.sty macros. This acknowledgment appears to have gone missing in the proof stages, and I apologise for not picking up on this at the time.
The results here have been extended in two slightly different directions. Work of Grønbæk and Habibian (Biflatness and biprojectivity of Banach algebras graded over a semilattice., Glasgow Math. J. 52 (2010), no. 3, 479--495) characterizes the biflatness of l1(S) when S is commutative; while the case of general inverse semigroups has now been treated thoroughly by P. Ramsden (Biflatness of semigroup algebras, Semigroup Forum 79 (2009), no. 3, 515--530). Ramsden's results use a more careful version of the arguments in the present paper, together with slightly more sophisticated tools from the representation theory of finite semigroups.
The present article uses some crude estimates of the `amenability constant' of l1(S) when S is a finite semilattice. More precise computations, with extensions to other examples, can be found in subsequent work of other authors (M. Ghandehari, H. Hatami, N. Spronk. Amenability constants for semilattice algebras, Semigroup Forum 79 (2009), no.2, 279--297).Yemon Choi Last modified: Tue Feb 1 17:09:19 CT 2011