Let $S$ be a commutative inverse semigroup with idempotent set $E$. In this paper, we show that for every $n\in \mathbb{N}_0$, $n$-th Hochschild cohomology group of semigroup algebra $\ell^1(S)$ with coefficients in $\ell^\infty(S)$ and its $n$-th $\ell^1(E)$-module cohomology group, are equal. Indeed, we prove that \[ \HH^{n}(\ell^1(S),\ell^\infty(S))=\HH^{n}_{\ell^1(E)}(\ell^1(S),\ell^\infty(S)),\] for all $n\geq 0$.
In this paper, we study Connes amenability of $l^1$-Munn algebras. We apply this results to semigroup algebras. We show that for a weakly cancellative semigroup $S$ with finite idempotents, amenability and Connes amenability are equivalent.
Abstract
We give a new and conceptually straightforward proof of the well-known presentation for the Temperley–Lieb algebra, via an alternative new presentation. Our method involves twisted semigroup algebras, and we make use of two apparently new submonoids of the Temperley–Lieb monoid.
The equality of Hochschild cohomology group and module cohomology group for semigroup algebras
