Two-sided multiplication operators on the space of regular operators

Let $W$, $X$, $Y$ and $Z$ be Dedekind complete Riesz spaces. For $A\in L^{r}(Y, Z)$and$B\in L^{r}(W, X)$let$M_{A,,B}$ be the two-sided multiplication operator from $L^r(X,Y)$ into $L^r(W,Z)$ defined by $M_{A,B}(T)=ATB$. We show that for every $0\le A_0\in L_n^r(Y,Z)$, $\mathrm{\mid }{M}_{A_0,B}\mathrm{\mid }(T)=M_{A_0,\mathrm{\mid }B\mathrm{\mid }}(T)$ holds for all $B\in L^r(W,X)$ and all $T\in L_n^r(X,Y)$. Furthermore, if $W$, $X$, $Y$ and $Z$ are Dedekind complete Banach lattices such that $X$ and $Y$ have order continuous norms, then $\mathrm{\mid }{M}_{A,B}\mathrm{\mid }=M_{\mathrm{\mid }A\mathrm{\mid },\mathrm{\mid }B\mathrm{\mid }}$ for all $A\in L^r(Y,Z)$ and all $B\in L^{r}(W, X)$. Our results generalize the related results of Synnatzschke and Wickstead, respectively.