New topological models for Banach–Mazur compacta

Sufficient conditions are given for an action of the orthogonal group $\mathrm{O}(n)$ on the Hilbert cube $Q$ in order that the corresponding orbit space $Q/\mathrm{O}(n)$ be homeomorphic to the Banach–Mazur compactum $\mathrm{BM}(n)$. This result is applied to obtain simple topological models for $\mathrm{BM}(2)$.