Based-free involutions

Let B denote the closed unit ball of Hilbert space ℓ₂ endowed with the standard involution σ:x↦−x. It is proved that for every separable metrizable space (X,τ) with a based-free involution (i.e., an involution that has a unique fixed point), the equivariant maps (X,τ)→(B,σ) separate points and closed sets in X. This is applied to give a direct proof of a result by the second two authors [10] that (ℓ₂,σ) is universal for all based-free involutions. Two new characterizations of equivariant based-free compactifications are given.