\(G\)-compactifications of pseudocompact \(G\)-spaces (Q2469572)

A \(G\)-space (a topological space \(X\) together with a continuous action \(\alpha:G\times X\to X\) of a topological group \(G\) on \(X\)) is said to be \(G\)-Tychonoff if it admits an equivariant embedding into a compact \(G\)-space. After reviewing earlier related positive and negative results, the author constructs examples of a pseudocompact (even countably compact) \(G\)-space which is not \(G\)-Tychonoff and a locally compact pseudocompact (even countably compact) \(G\)-Tychonoff space \(X\) with \(\beta_G X\not= \beta X\), where \(\beta_G X\) is the maximal \(G\)-space compactification.