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

Abstract

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

19 pages, submitted. Uses diagrams.sty macros. v2: 16 pp. same mathematical content, with some reorganization and trimming of the presentation. v3: correction of a few typos. v4: some more minor corrections of typos. Slightly shorter version to appear in J. Math. Anal. Appl