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Simplicial cohomology of some semigroup algebras. (English) Zbl 1136.43001
The authors investigate the higher simplicial cohomology groups of the convolution algebra \(\ell^{1}(S)\) for various semigroups \(S\). Let \(\mathcal A\) be a Banach algebra and let \(\mathcal X\) be a Banach \(\mathcal A\)-module. Then \(\mathcal H ^n(\mathcal A, \mathcal X)\) and \(\mathcal H \mathcal C^n(\mathcal A, \mathcal X)\) denote the \(n\)-th simplicial cohomology and \(n\)-th cyclic cohomology groups. The following main results are obtained:
(i) If \(S\) is a semilattice, i.e. a commutative semigroup such that \(s^2 =s\) for every \(s \in S\), and \(\mathcal X\) is a commutative \(\ell^{1} (S)\)-module, then \({\mathcal H}^{3} (\ell^{1} (S), {\mathcal X})\) is a Banach space.
(ii) If \(S\) is a semilattice, then \({\mathcal H}^{3} (\ell^{1} (S), \ell^{\infty} (S))=0\).
(iii) For \(a > 0\), consider the semigroup N\(_{a}=\{n \in \mathbb Z \colon n \geq a\}\). Then \({\mathcal H}^{2} (\ell^{1} (\)N\(_{a}), \ell^{\infty} (\)N\(_{a}))\) is a Banach space.
(iv) If \(S\) is a Clifford semigroup, then \({\mathcal H}^{2} (\ell^{1} (S), \ell^{\infty} (S))\) is a Banach space.
(v) If \(S\) is a regular Rees semigroup, then \({\mathcal H}^{2} (\ell^{1} (S), \ell^{\infty} (S))\) and \({\mathcal H \mathcal C}^{2} (\ell^{1} (S), \ell^{\infty} (S))\) are Banach spaces.
There are several reasons for establishing that a cohomology group is a Banach space, e.g. it is a first step toward the identification of the cohomology group itself, and this additional structure may allow the application of more involved techniques.

MSC:
43A20 \(L^1\)-algebras on groups, semigroups, etc.
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
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