On the Symmetrizations of $\varepsilon$-Isometries on Positive Cones of Continuous Function Spaces

Let $K$ be a compact Hausdorff space, $C(K)$ be the real Banach space of all continuous functions on $K$ endowed with the supremum norm, and $C(K)^+$ be the positive cone of $C(K)$. A weak stability result for the symmetrization $\Theta=(f(\,\boldsymbol\cdot\,)-f(-\;\boldsymbol\cdot\,)/2$ of a general $\varepsilon$-isometry $f$ from $C(K)^+\cup-C(K)^+$ to a Banach space $Y$ is obtained: For any element $k\in K$, there exists a $\phi\in S_{Y^\ast}$ such that
\begin{equation*} |\langle\delta_k,x\rangle-\langle\phi,\Theta(x)\rangle|\le3\varepsilon/2\quad\text{for all }\,x\in C(K)^+\cup-C(K)^+. \end{equation*}
This result is used to prove new stability theorems for the symmetrization $\Theta$ of $f$.