%0 Journal Article %T Second Module Cohomology Group of Induced Semigroup Algebras %J Sahand Communications in Mathematical Analysis %I University of Maragheh %Z 2322-5807 %A Miri, Mohammad Reza %A Nasrabadi, Ebrahim %A Kazemi, Kianoush %D 2021 %\ 05/01/2021 %V 18 %N 2 %P 73-84 %! Second Module Cohomology Group of Induced Semigroup Algebras %K second module cohomology group‎ %K ‎inverse semigroup‎ %K ‎induced semigroup %K semigroup algebra %R 10.22130/scma.2020.130935.826 %X For a discrete semigroup $ S $ and a left multiplier operator  $T$ on  $S$, there is a new induced semigroup $S_{T}$, related to $S$ and $T$. In this paper, we show that if $T$ is multiplier and bijective,  then the second module cohomology groups $\mathcal{H}_{\ell^1(E)}^{2}(\ell^1(S), \ell^{\infty}(S))$ and $\mathcal{H}_{\ell^1(E_{T})}^{2}(\ell^1({S_{T}}), \ell^{\infty}(S_{T}))$ are equal, where $E$ and  $E_{T}$ are subsemigroups of idempotent elements in $S$ and $S_{T}$,   respectively.  Finally, we show thet, for every odd $n\in\mathbb{N}$,  $\mathcal{H}_{\ell^1(E_{T})}^{2}(\ell^1(S_{T}),\ell^1(S_{T})^{(n)})$ is a Banach space, when $S$ is a commutative inverse semigroup. %U https://scma.maragheh.ac.ir/article_242308_d3a184fdd9d1327f1e96dc7532c37ac9.pdf