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

TL;DR

This paper improves bounds on the point cohomology of certain $L^1$-algebras, providing uniform splitting maps and new estimates on projectivity constants, enhancing understanding of their algebraic structure.

Contribution

It introduces uniform bounds for point cohomology and constructs splitting maps with norms independent of the point module, based on new projectivity estimates.

Findings

Point cohomology vanishes in degrees 2 and above for $ell^1(bZ_+)$.

Constructs splitting maps with uniformly bounded norms.

Extends results to related $L^1$-algebras from rank one subsemigroups.

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 ${\mathbb R}_+$.