On the semiring $L^ +(C_ 0(X))$

Abstract:

Let $X$ and $Y$ be locally compact Hausdorff spaces. Let ${C_0}(X)$ (resp. ${C_0}(Y)$) denote the Banach space of all continuous functions on $X$ vanishing at infinity on $X$ (resp. $Y$) and ${L^ + }({C_0}(X))$ (resp. ${L^ + }({C_0}(Y))$) the semiring of positive operators on ${C_0}(X)$ (resp. ${C_0}(Y)$). We prove that if there exists a semiring isomorphism $\varphi$ from ${L^ + }({C_0}(X))$ onto ${L^ + }({C_0}(Y))$, then $X$ and $Y$ are homeomorphic. If $X$ and $Y$ are assumed to be compact then the same conclusion holds under the milder condition that $\varphi$ is an affine isomorphism and $\varphi ({I_{C(X)}})$ is order bounded.