FIRST MODULE COHOMOLOGY OF TRIANGULAR BANACH ALGEBRAS ON INDUCED SEMIGROUP ALGEBRAS

Let $S$ be a discrete semigroup and $T$ be a left multiplier operator on $S$. The semigroup $S$, equipped with the new operation defined by $T,(s\circ t:=sT(t))$,
is called the induced semigroup and is denoted by $S _{T}$.
We consider the semigroup algebras $ \ell^1({S}) $ and $ \ell^1({S_T}) $, as well as the triangular Banach algebras:
\begin{equation*}\mathcal{T}=\Mat{\ell^1({S})}{M_{\delta_S}}{\ell^1({S})}\qquad \text{and} \qquad \mathcal{T}_T=\Mat{\ell^1({S_T})}{M_{\delta_{S_T}}}{\ell^1({S_T})}.
\end{equation*}

In this paper, we show that the first module cohomology groups of these triangular Banach algebras, $\HH^{1}_ \mathfrak{T}(\mathcal{T},\mathcal{T}^*) $ and $ \HH^{1}_{ \mathfrak{T}_{T}}(\mathcal{T}_T,\mathcal{T}_T^*)$, are equal, where

\begin{equation*}
\mathfrak{T}=\set{\Mat{\alpha}{}{\alpha},\alpha\in \ell^1({E})}\qquad \text{,} \qquad \mathfrak{T}_{T}=\set{\Mat{\beta}{}{\beta},\beta\in \ell^1({E_{T}})}.
\end{equation*}
Here, $ E $ and $ E_{T} $ denote the sets of idempotent elements in $ S $ and $ S_T $, respectively.
This result implies that, in a particular case, $\mathcal{T}$ is weakly $ \mathfrak{T}$-module amenable if and only if $\mathcal{T}_T$ is weakly $\mathfrak{T}_{T}$-module amenable.
Furthermore, $ M_{\delta_s} $ and $ M_{\delta_{S_T}} $ denote the canonical left modules over $ \ell^1({S}) $ and $ \ell^1({S_T}) $, respectively.