Pseudocompactness from separately continuous extension of multiplication to the Stone–Čech compactification of a topological group

Determine whether every topological group G whose multiplication extends to a separately continuous operation on the Stone–Čech compactification βG is necessarily pseudocompact. Equivalently, ascertain whether under this hypothesis βG is a topological group, or whether for any semitopological group G whose multiplication extends to a separately continuous operation on βG one must have that G is pseudocompact and βG is a topological group.

Background

The paper establishes several equivalences for topological groups regarding the extension of operations to their Stone–Čech compactification βG and pseudocompactness (Theorem 3.1). In particular, for groups whose operations extend continuously to βG, pseudocompactness is equivalent to βG being a Dugundji compactum.

The authors then pose the case when only separate continuity of the extended multiplication on βG is assumed. They formulate a cluster of equivalent conjectures capturing this uncertainty, the most direct of which asks whether such a group must be pseudocompact. Resolving this would bridge the gap between separate continuity of the extension and the full equivalences in Theorem 3.1.