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Simplicial cohomology of band semigroup algebras. (English) Zbl 1263.46060
A semigroup formed solely by idempotent elements is called a band semigroup. The \(\ell^1\)-convolution algebra on such a semigroup is a Banach algebra. Its simplicial homology, Hochschild and cyclic cohomology are computed with great ingenuity by the three authors of the present article. Partial results were obtained by one of them [Y. Choi, Glasg. Math. J. 48, No. 2, 231–245 (2006; Zbl 1112.46056); Houston J. Math. 36, No. 1, 237–260 (2010; Zbl 1217.46031)]. Specifically, the main results are, for a band semigroup \(S\): The cyclic cohomology of \(\ell^1(S)\) is isomorphic in even degrees to the space of continuous traces on \(\ell^1(S)\), and vanishes in odd degrees. On the other hand, the simplicial cohomology of \(\ell^1(S)\) vanishes in positive degrees.

MSC:
46M18 Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)
43A10 Measure algebras on groups, semigroups, etc.
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
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