point cohomology, semigroup algebra, uniformly bounded solution of cohomology problems, flat Banach module, maximal ideal
Appeared as J. Math. Anal. Appl. 358 (2009), no. 2, 249--260.
Preprint version available at arXiv math.0808.2265 (last updated March 2009. Warning: not the final version; some references are different.)
[ Math Reviews | Zentralblatt (summary) ]
It is well-known that the point cohomology of the convolution algebra l1(Z+) vanishes in degrees 2 and above. We sharpen this result by obtaining splitting maps whose norms are bounded independently of the choice of point module. Our construction is a by-product of new estimates on projectivity constants of maximal ideals in l1(Z+). Analogous results are obtained for some other L1-algebras which arise from `rank one' subsemigroups of R+.
The main result of this article (the uniform solution of point cohomology problems for l1(Z+)) originally had a messier proof, which was in turn based on a direct and even messier proof for the special case of degree 2 cohomology. It was while trying to extend the messy argument to handle the `continuous analogue' L1(R+), and examining some of the Laplace transforms that came up, that I arrived at the present argument.
This article is dedicated to the memory of Graham R. Allan (1936-2007).
Some personal comments: Graham was still lecturing when I was an undergraduate, and I am deeply indebted not only to the lectures he gave on various aspects of (functional) analysis, at varying levels and in different courses, but also to the generosity and solicitude which he extended to students. While the results of this paper are somewhat removed from his own work and interests, I like to think that in one or two places the arguments used would not have displeased him.