C*-embeddings of spaces of probability measures, nonnegative measures and unit balls

For a Tychonoff space X and p∈,β,R,τ,σ,a\ let Mₚ(X) be the space of discrete, compactly supported, Radon, $τ$-additive, $σ$-additive or finitely additive signed measures on X, regarded as a subspace of M(βX)=C(βX)^* with the weak^* topology, and let M⁺ₚ(X), Uₚ(X) and Pₚ(X) be its subsets of nonnegative measures, measures of norm at most one, and probability measures. In the companion paper it was shown that Mₚ(X) is C-embedded in Mq(X) iff it is C^*-embedded iff it is G_δ-dense. We show that G_δ-density of Qₚ(X) in Qq(X) is the same condition for all Q∈,M⁺,U,P\, and that C^*-embedding of Qₚ(X) in Qq(X) implies G_δ-density, while the converse fails for Q∈⁺,U,P\. We prove that M⁺ₚ(X) is C^*-embedded in M⁺(βX) iff Pₚ(X) is C^*-embedded in $P(βX)$, that pseudocompactness of Pₚ(X) implies pseudocompactness of all its powers, and that the unit ball UR(X) is C^*-embedded in $U(βX)$ iff PR(X) is pseudocompact. If $βX$ is $κ$-metrizable (e.g., a product of metrizable compact spaces or a compact group), then pseudocompactness of Pₚ(X) and the C^*-embeddings of Pₚ(X), Uₚ(X) and M⁺ₚ(X) in $P(βX)$, $U(βX)$ and M⁺(βX) are equivalent to the C-embedding of Mₚ(X) in $M(βX)$. Consequently, P_τ(X) is pseudocompact for every pseudocompact dense subspace X of a Tychonoff cube. This gives a space X for which P_τ(X) is pseudocompact while the space PR(X) of Radon probability measures is not, and answers a question of the first author.