First Hachschild Cohomology Group of Triangular Banach Algebras on Induced Semigroup Algebras

Abstract

‎Let $S$ be a discrete semigroup with a left multiplier operator $T$ on $S$‎. ‎A new product on $S$ defined by $T$ related to $S$ and $T$ creates a new induced semigroup $S _{T} $‎. ‎Suppose that $T$ is bijective and‎ ‎\begin{equation*}\mathcal{T}_1=\Mat{\ell^1({S})}{\ell^1({S})}{\ell^1({S})}\qquad \text{and} \qquad \mathcal{T}_2=\Mat{\ell^1({S_T})}{\ell^1({S_T})}{\ell^1({S_T})}‎. ‎\end{equation*}‎ ‎In this paper‎, ‎we show that the first cohomology groups $ \HH^{1}_{}(\mathcal{T}_1,\mathcal{T}_1^*) $ and $ \HH^{1}_{}(\mathcal{T}_2,\mathcal{T}_2^*) $ are equal‎. ‎Therefore $\mathcal{T}_1$ is weakly amenable if and only if $\mathcal{T}_2$ is weakly amenable‎.