Simplicial cohomology of band semigroup algebras

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.

arXiv commentv1: AMS-LaTeX, 24 pages, 1 figure. v2: some typos corrected; a few minor adjustments made for clarity; references updated. Accepted June 2011 by Proc. Royal Soc. Edinburgh Sect. A

References in corpus (2)

• Simplicial homology of strong semilattices of Banach algebras
• Splitting maps and norm bounds for the cyclic cohomology of biflat Banach algebras

Cited by in corpus (3)

• Splitting maps and norm bounds for the cyclic cohomology of biflat Banach algebras
• Module and Hochschild cohomology of certain semigroup algebras
• The cyclic and simplicial cohomology of the Cuntz semigroup algebra