Simplicial cohomology of band semigroup algebras

Details Simplicial cohomology of band semigroup algebras Simplicial cohomology of band semigroup algebras

2010-04-14

arxiv.org/abs/1004.2301v2

Mathematics

Functional Analysis

v1: AMS-LaTeX, 24 pages, 1 figure. v2: some typos corrected; a few minor adjustments made for clarity; references updated. Acc

Scientific paper

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.

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