| Authors | Ebrahim Nasrabadi |
|---|---|
| Journal | Boletim da Sociedade Paranaense de Matematica |
| Page number | 1-8 |
| Serial number | 41 |
| Volume number | 3 |
| Paper Type | Full Paper |
| Published At | 2022 |
| Journal Type | Typographic |
| Journal Country | Albania |
| Journal Index | Scopus |
Abstract
Let $S$ be a discrete semigroup and $T$ be a left multiplier on $S$. A new product on $S$ defined by $T$ creates a new induced semigroup $S _{T} $. In this paper, we show that if $T$ is bijective, then the first module cohomology groups $ \HH_{\ell^1(E)}^{1}(\ell^1(S), \ell^{\infty}(S))$ and $ \HH_{\ell^1(E_{T})}^{1}(\ell^1({S_{T}}), \ell^{\infty}(S_{T})) $ are equal, where $E$ and $E_{T}$ are sets of idempotent elements in $S$ and $S _{T}$, respectively. Which in particular means that $\ell^1(S)$ is weak $\ell^1(E)$-module amenable if and only if $\ell^1(S_T)$ is weak $\ell^1(E_T)$-module amenable. Finally, by giving an example, we show that the bijectivity of $T$, is necessary.