Innerness of $\sigma$-derivations, innerness of $\sigma$-higher derivations and $W^{*}$-algebra

Let $\mathcal{M}$ be a $W^{*}$-algebra and $\mathcal{A} \subseteq \mathcal{M}$ a commutative $W^{*}$-subalgebra of $\mathcal{M}$ containing the identity element of $\mathcal{M}$. Let $\sigma:\mathcal{M} \to \mathcal{M}$ be a continuous homomorphism and $\delta: \mathcal{M} \to \mathcal{M}$ a $\sigma$-derivation. In this paper, it is proved that there exists a $x_{0} \in \mathcal{M}$ such that $\delta(a)=\sigma(a)x_{0}-x_{0}\sigma(a)$, for each $a \in \mathcal{A}$. Also, it is shown that if $\{d_n\}$ is a continuous strongly $\sigma$-higher derivationon on $\mathcal{M}$, then there exist $x_{1}, x_{2}, x_{3}, ... x_{n} \in \mathcal{M}$ such that \begin{align*} d_{n}(a)= \sigma^{n}(a)x_{n}-x_{n}\sigma^{n}(a) -x_{n-1}d_{1}(\sigma^{n-1}(a))\\ -x_{n-2}d_{2}(\sigma^{n-2})(a)- \ldots -x_{1}d_{n-1}(\sigma(a)) \end{align*} for each $a \in \mathcal{A}$.