First module cohomology group of induced semigroup algebras
Abstract
Let $S$ be a discrete semigroup and $T$ be a left multiplier operator 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 condition of bijectivity for $T$, is necessary.
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References
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