Cyclic Amenability of Induced Semigroup Algebras

Abstract

‎For a discrete semigroup algebra $ 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 for an inverse semigroup $S$‎, ‎under the certain conditions on $T$‎, ‎the first cyclic cohomology groups $ \HH\CC^{1}(\ell^1(S)‎, ‎\ell^{\infty}(S))$ and $\HH\CC^{1}(\ell^1({S_{T}})‎, ‎\ell^{\infty}(S_{T})) $ are equal‎. ‎Which in particular means that semigroup algebra $\ell^1(S)$ is cyclic amenable if and only if induced semigroup algebra $\ell^1(S_T)$ is cyclic amenable‎.