$f$-Representation of a function algebra

Abstract

We investigate representations $\Phi :A\longrightarrow \mathcal {L}_{b}(X)$, where $A$ is a unital function algebra and $\mathcal {L}_{b}(X)$ is the space of all order bounded operators on a vector lattice $X$. Given an element $x\in X$, the orbit space $\Phi \left[ x\right]$ generated by $\Phi$ at $x$ is the subspace

$$\begin{aligned} \Phi \left[ x\right] =\left\{ \Phi (a)(x):a\in A\right\} . \end{aligned}$$

In this paper we make a detailed study of the orbite space $\Phi \left[ x\right] , x\in X$. It turn out that they are vector lattices with a weak order unit. Moreover, It is proved that for any representation $\Phi :A\longrightarrow \mathcal {L}_{b}(X)$ can be extended to a representation of the order bidual $A^{\sim }{}^{\sim }$ of $A$.