Topological Cones: Functional Analysis in a T0-Setting

Already in his PhD Thesis on compact Abelian semigropups under the direction of Karl Heinrich Hofmann the author was lead to investigate locally compact cones [18]. This happened in the setting of Hausdorff topologies. The theme of topological cones has been reappearing in the author’s work in a non-Hausdorff setting motivated by the needs of mathematical models for a denotational semantics of languages combining probabilistic and nondeterministic choice. This is in the line of common work with Karl Heinrich Hofmann in Continuous Lattices and Domains [12]. Domain Theory is based on order theoretical notions from which intrinsic non-Hausdorff topologies are derived. Along these lines, domain theoretical variants of (sub-) probability measures have been introduced by Jones and Plotkin [15, 16]. Kirch [22] and Tix [36] have extended this theory to a domain theoretical version of measures and they have introduced and studied directed complete partially ordered cones as appropriate structures. Driven by the needs of a semantics for languages combining probabilty and nondeterminism, Tix [38, 37] and later on Plotkin and the author [39] developed basic functional analytic tools for these structures. In this paper we extend this theory to topological cones the topologies of which are strongly non-Hausdorff. We carefully introduce these structures and their elementary properties. We prove Hahn-Banach type separation theorems under appropriate local convexitiy hypotheses. We finally construct a monad assigning to every topological cone C another topological cone S(C) the elements of which are nonempty compact convex subsets of C. For proving that this construction has good properties needed for the application in semantics we use the functional analytic tools developed before. ∗Thanks to Gordon Plotkin for numerous discussions. Preliminary results have been announced at MFPS XXIII [20]. In his Master’s thesis supervised by the author, B. Cohen [6] has worked out some of those results.