Remarks on actions on compacta by some infinite-dimensional groups

We discuss some techniques related to equivariant compactifications of uniform spaces and amenability of topological groups. In particular, we give a new proof of a recent result by Glasner and Weiss describing the universal minimal flow of the infinite symmetric group ${\mathfrak{S}}_{\mathrm{\infty }}$ with the standard Polish topology, and extend Bekka’s concept of an amenable representation, enabling one to deduce non-amenability of the Banach–Lie groups $\text{GL}(L_p)$ and $\text{GL}({\mathrm{\ell }}_p)$, $1\le p<\mathrm{\infty }$.