Abstract
In this paper we give a description of the duality between closed submonoids of topological groups (Lie groups) and certain cones of continuous (smooth, analytic) functions in a categorial setting. We consider the categories of topological spaces, smooth manifolds and analytic manifolds. A direct approach in the category of smooth manifolds can already be found in [4]. We use the properties of heat kernels on Lie groups to obtain the results in the analytic category. Finally we give a characterization of those Lie wedges which admit strictly positive analytic functions which is a stronger version of results in [3] and [4]. This characterization has various applications in the globality theory of Lie wedge ([3], [4], [7], [9]) and also to ordered homogeneous spaces ([6]) and harmonic analysis on ordered symmetric spaces ([9]).
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