The weak compactification of locally compact groups (Q2077300)

For a locally compact group \(G\), the authors study the weak topology generated by its irreducible unitary representations; denote by \(G^w\) the group \(G\) endowed with this weak topology. First, they give a proof of a result by \textit{J. Ernest} [Trans. Am. Math. Soc. 156, 287--307 (1971; Zbl 0218.22011)] and \textit{R. Hughes} [Bull. Am. Math. Soc. 79, 122--123 (1973; Zbl 0263.22006)]: the weak topology of a locally compact group \(G\) respects compactness, that is, every subset of \(G\) that is compact in \(G^w\) is compact also in \(G\) (with its given locally compact topology); more precisely, \(G\) and \(G^w\) have the same compact subsets. Then, they show that also countable compactness and pseudocompactness are respected in the above sense. In particular, the preservation of pseudocompacness is found as a direct consequence of the main theorem of the present paper, which states that, for a locally compact group \(G\), the group \(G^w\) is a \(\mu\)-space, namely every closed functionally bounded subset of \(G^w\) is compact.