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 GBAR 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 GBAR, then the following conditions are equivalent: (i) G is locally pseudocompact; (ii) G is C* -embedded in GBAR (i.e., betaG = betaGBAR; (iii) G is C-embedded in GBAR (i.e., upsilonG = upsilonGBAR); (iv) G is M-embedded in GBAR (i.e., gammaG = GBAR); (v) some nonempty open subset U of G satisfies beta(cl(G)U) = clGBARU; (vi) every bounded open subset U of G satisfies beta(cl(G)U) = cl(GBAR)U.