Locally pseudocompact topological groups

A topological group is said to be locally pseudocompact if the identity has a pseudocompact neighborhood (equivalently: if the identity has a local basis of pseudocompact neighborhoods). Such groups are locally bounded in the sense of A. Weil, so each such group G is densely embedded in an essentially unique locally compact group G (called its Weil completion). The authors present necessary and sufficient conditions of local and global nature for a locally bounded group to be locally pseudocompact, as follows. Theorem. If G is a locally bounded group with Weil completionG, then the following conditions are equivalent: 1. s locally pseudocompact; is C∗-embedded in G (i.e., βG = βG); G is C-embedded in G (i.e., υG = υG); is M-embedded in G (i.e., γG = G); me nonempty open subsetU of G satisfies β(clGU) = clGU; very bounded open subset U of G satisfies β(clGU) = clGU.