Composition in Modulus Maps on Semigroups of Continuous Functions

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For locally compact Hausdorff spaces $X$ and $Y$, and function algebras $A$ and $B$ on $X$ and $Y$, respectively, surjections $T:A \longrightarrow B$ satisfying norm multiplicative condition $\|Tf\, Tg\|_Y =\|fg\|_X$, $f,g\in A$, with respect to the supremum norms, and those satisfying $\||Tf|+|Tg|\|_Y=\||f|+|g|\|_X$ have been extensively studied. Motivated by this, we consider certain (multiplicative or additive) subsemigroups $A$ and $B$ of $C_0(X)$ and $C_0(Y)$, respectively, and study surjections $T$ from $A$ onto $B$ satisfying the norm condition $\rho(Tf, Tg)=\rho(f,g)$, $f,g \in A$, for some classes of two variable positive functions $\rho$. It is shown that such a map $T$ is also a composition in modulus map.