Simplicial cohomology of band semigroup algebras

TL;DR

This paper proves that the convolution algebra of a band semigroup is simplicially trivial by analyzing its cyclic cohomology, using a novel normalization technique and the semilattice structure.

Contribution

It introduces a new inductive normalization method in cyclic cohomology and extends the understanding of simplicial triviality for band semigroup algebras.

Findings

Cyclic cohomology vanishes in all odd degrees.

Even degrees' cohomology is isomorphic to continuous traces.

The structure semilattice is key to the analysis.

Abstract

We establish simplicial triviality of the convolution algebra $\ell^1(S)$, where $S$ is a band semigroup. This generalizes results of the first author [Glasgow Math. J. 2005, Houston J. Math. 2010]. To do so, we show that the cyclic cohomology of this algebra vanishes in all odd degrees, and is isomorphic in even degrees to the space of continuous traces on $\ell^1(S)$. Crucial to our approach is the use of the structure semilattice of $S$, and the associated grading of $S$, together with an inductive normalization procedure in cyclic cohomology; the latter technique appears to be new, and its underlying strategy may be applicable to other convolution algebras of interest.