The cyclic and simplicial cohomology of the Cuntz semigroup algebra

TL;DR

This paper computes the cyclic and simplicial cohomology groups of the Cuntz semigroup algebra, revealing specific vanishing and dimensionality patterns, and introduces general results applicable to Banach algebra cohomology.

Contribution

It establishes new general results for Banach algebra cohomology and determines the cohomology groups of the Cuntz semigroup algebra and related tensor algebras.

Findings

Cyclic cohomology groups vanish for odd degrees and are one-dimensional for even degrees (n≥2).

Simplicial cohomology groups vanish for all degrees n≥1.

Results apply to tensor algebras of Banach spaces, including free semigroup algebras.

Abstract

The main objective of this paper is to determine the simplicial and cyclic cohomology groups of the Cuntz semigroup algebra $\ell^1(\Cuntz)$. In order to do so, we first establish some general results which can be used when studying simplicial and cyclic cohomology of Banach algebras in general. We then turn our attention to $\ell^1(\Cuntz)$, showing that the cyclic cohomology groups of degree $n$ vanish when $n$ is odd and are one-dimensional when $n$ is even ($n\ge 2$). Using the Connes-Tzygan exact sequence, these results are used to show that the simplicial cohomology groups of degree $n$ vanish for $n\ge 1$. We also determine the simplicial and cyclic cohomology of the tensor algebra of a Banach space, a class which includes the algebra on the free semigroup on $m$-generators $\ell^1(\FS)$.