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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.

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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