Uniform bounds for point cohomology of ${\mathrm{\ell }}^1({\mathbb{Z}}_{+})$ and related algebras

It is well-known that the point cohomology of the convolution algebra ${\mathrm{\ell }}^1({\mathbb{Z}}_{+})$ 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 ${\mathrm{\ell }}^1({\mathbb{Z}}_{+})$. Analogous results are obtained for some other $L^1$-algebras which arise from `rank one’ subsemigroups of ${\mathbb{R}}_{+}$.