Simplicial cohomology of band semigroup algebras
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 [\textit{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.
scientific article; zbMATH DE number 6076572