Injectivity/surjectivity/bijectivity of the Ellis-compactification map 𝔈_θ

Determine conditions under which the map 𝔈_θ: 𝒞𝔾(X) → 𝔈(G), which assigns to each G-compactification bX of the G-space (G, X, θ) its Ellis compactification e_{bX}G, is injective, surjective, or bijective, for a topological group G that is τ_p-representable in a Tychonoff space X. Here 𝒞𝔾(X) denotes the poset of G-compactifications of X and 𝔈(G) denotes the poset of Ellis compactifications of G.

Background

The paper defines a canonical map 𝔈θ from the poset of G-compactifications of a phase space X to Ellis compactifications of the acting group G, assuming G is τ_p-representable in X. Corollary 3 (‘order’) shows that 𝔈θ preserves the partial order, and the surrounding discussion establishes how maps of compactifications induce maps of Ellis compactifications.

Despite these structural properties, the authors highlight that it is not known in general when this map is injective, surjective, or bijective. Clarifying these conditions would characterize how tightly G-compactifications of X control Ellis compactifications of G and would provide a sharper correspondence between dynamical compactifications and semigroup-theoretic completions.