Second Module Cohomology Group of Induced Semigroup Algebras

Abstract

‎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 $\HH_{\ell^1(E)}^{2}(\ell^1(S)‎, ‎\ell^{\infty}(S))$ and $\HH_{\ell^1(E_{T})}^{2}(\ell^1({S_{T}})‎, ‎\ell^{\infty}(S_{T}))$ 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 $n\in\mathbb{N}$‎, ‎$\HH_{\ell^1(E_{T})}^{2}(\ell^1(S_{T}),\ell^1(S_{T})^{(n)})$ is a Banach space‎, ‎when $S$ is a commutative inverse semigroup‎.