Spectral theorems for positive algebra homomorphisms

Let $X$ be a locally compact Hausdorff space, let $A$ be a partially ordered algebra, and let $\pi\colon \mathrm{C}_{\mathrm c}(X)\to A$ be a positive algebra homomorphism. Under conditions on $A$ that are satisfied in a good number of cases of practical interest, it is shown that $\pi$ is represented by a unique regular spectral measure $\mu$ on the Borel $\sigma$-algebra of $X$, taking its values in the positive idempotents in $A$. The measure $\mu$, which is $\sigma$-additive in an ordered sense, represents $\pi$ via the order integral (a generalisation of the Lebesgue integral) that goes back to J.D.M. Wright and which was investigated earlier by the authors. The positive algebra homomorphism $\pi$ can be extended from $\mathrm{C}_{\mathrm c}(X)$ to a positive linear map from the accompanying $L^1$-space of $\mu$ into $A$. It is shown that, quite often, this $L^1$-space is closed under multiplication, so that it is a vector lattice algebra, and that the extended map from $L^1$ into $A$ is not only an algebra homomorphism but, even when $A$ is not a vector lattice, also a vector lattice homomorphism in a sense that is explained in the paper. When $A$ has the countable sup property, the image of $L^1$ (or of its positive cone) is described in terms of consecutive ups and downs of the image of ${\mathrm C}_{\mathrm c}(X)$ (or of its positive cone). The general results are applied in three different contexts, showing how various spectral theorems have a common order-theoretical root: representations on Banach lattices, on Hilbert spaces, and (the algebra need not consist of operators) spectral theory for JBW-algebras.