Algebras of the extended probabilistic powerdomain monad

We investigate the Eilenberg-Moore algebras of the extended probabilistic powerdomain monad $\mathcal V_w$ over the category $\mathbf{TOP}_0$ of $T_0$ topological spaces and continuous maps. We prove that every $\mathcal V_w$-algebra in our setting is a weakly locally convex sober topological cone, and that a map is the structure map of a $\mathcal V_w$-algebra if and only if it is continuous and sends every continuous valuation to its unique barycentre. Conversely, for locally linear sober cones (a strong form of local convexity), the mere existence of barycentres entails that the barycentre map is the structure map of a $\mathcal V_w$-algebra; moreover the algebra morphisms are exactly the linear continuous maps in that case. We also examine the algebras of two related monads, the simple valuation monad $\mathcal V_{\mathrm f}$ and the point-continuous valuation monad $\mathcal V_{\mathrm p}$. In $\mathbf{TOP}_0$ their algebras are fully characterised as weakly locally convex topological cones and weakly locally convex sober topological cones, respectively. In both cases, the algebra morphisms are continuous linear maps between the corresponding algebras.