A note on the stability of isometries on the positive cones of continuous function spaces

Let K be a compact Hausdorff space, Y be a Banach space, C(K) be the real Banach spaces of all continuous functions on K endowed with the supremum norm and $$C_+(K)=\{f\in C(K): f(k)\ge 0\ \mathrm{for\ all }\ k\in K\}$$ C + ( K ) = { f ∈ C ( K ) : f ( k ) ≥ 0 for all k ∈ K } be the positive cone of C(K). In this note, We show that for every standard isometry $$F: C_+(K)\rightarrow Y$$ F : C + ( K ) → Y , there exists a bounded linear operator $$T:\overline{\mathrm{span}}[F(C_+(K))]\rightarrow C(K)$$ T : span ¯ [ F ( C + ( K ) ) ] → C ( K ) with $$\Vert T\Vert =1$$ ‖ T ‖ = 1 such that $$\begin{aligned} TF=Id_{C_+(K)}. \end{aligned}$$ T F = I d C + ( K ) . This result can be regarded as a generalization of Figiel’s theorem on the positive cones of continuous functions spaces. Furthermore, if $$T^{*}(C(K)^{*})$$ T ∗ ( C ( K ) ∗ ) is $$w^{*}$$ w ∗ -1-complemented in $$\overline{\mathrm{span}}[F(C_+(K))]^{*}$$ span ¯ [ F ( C + ( K ) ) ] ∗ , then there exists a linear isometry $$U: C(K)\rightarrow \overline{\mathrm{span}}[F(C_+(K))]$$ U : C ( K ) → span ¯ [ F ( C + ( K ) ) ] such that $$TU=Id_{C(K)}$$ T U = I d C ( K ) . Making use of this result, we show that if $$F(C_+(K))$$ F ( C + ( K ) ) contains a closed reproducing wedge of Y, then $$F=U|_{C_+(K)}$$ F = U | C + ( K ) .