Rights statement: http://journals.cambridge.org/action/displayJournal?jid=PRM The final, definitive version of this article has been published in the Journal, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 142 (4), pp 715-744 2012, © 2012 Cambridge University Press.
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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
}
TY - JOUR
T1 - Simplicial cohomology of band semigroup algebras
AU - Choi, Yemon
AU - Gourdeau, Frédéric
AU - White, Michael C.
N1 - http://journals.cambridge.org/action/displayJournal?jid=PRM The final, definitive version of this article has been published in the Journal, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 142 (4), pp 715-744 2012, © 2012 Cambridge University Press
PY - 2012/8
Y1 - 2012/8
N2 - We establish the simplicial triviality of the convolution algebra $\ell^1(S)$, where $S$ is a band semigroup. This generalizes some results of Choi (Glasgow Math. J. 48 (2006), 231–245; Houston J. Math. 36 (2010), 237–260). 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.
AB - We establish the simplicial triviality of the convolution algebra $\ell^1(S)$, where $S$ is a band semigroup. This generalizes some results of Choi (Glasgow Math. J. 48 (2006), 231–245; Houston J. Math. 36 (2010), 237–260). 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.
U2 - 10.1017/S0308210510000648
DO - 10.1017/S0308210510000648
M3 - Journal article
VL - 142
SP - 715
EP - 744
JO - Proceedings of the Royal Society of Edinburgh: Section A Mathematics
JF - Proceedings of the Royal Society of Edinburgh: Section A Mathematics
SN - 1473-7124
IS - 4
ER -