Locally pseudocompact topological groups

Abstract

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 completion G , then the following conditions are equivalent: 1. (i) G is locally pseudocompact; 2. (ii) G is C ∗- embedded in G (i.e., βG = β G) ; 3. (iii) G is C- embedded in G (i.e., υG = υ G) ; 4. (iv) G is M- embedded in G (i.e., γG = G) ; 5. (v) some nonempty open subset U of G satisfies β(cl GU) = cl G U ; 6. (vi) every bounded open subset U of G satisfies β(cl GU) = cl G U .