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# The Haar theorem for lattice-ordered abelian groups with order-unit

• Given a finite union $P$ of rational simplexes, we assign to $P$ numerical invariants $\lambda_{0}, \lambda_{1},\ldots,\lambda_{\dim P};$ each $\lambda_{i}$ is the suitably normalized volume of the $i$-dimensional part of $P$. We then prove that every finitely generated projective lattice-ordered abelian group $G$ with order-unit $u$ has a faithful invariant positive linear functional $s: G\to \mathbb R$. For each $g\in G$, $s(g)$ is the integral of $g$ over the maximal spectrum of $G$, the latter being canonically identified with a rational polyhedron $P$. Volume elements are measured by the $\lambda_{i}$'s. The proof uses the polyhedral versions of the Włodarczyk-Morelli theorem on decompositions of birational toric maps in blow-ups and blow-downs, and of the De Concini-Procesi theorem on elimination of points of indeterminacy.
Mathematics Subject Classification: Primary: 28C05, 52B70; Secondary: 06F20, 28C10,52A38, 14E05, 26B15.

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