On isometries of symmetric products of metric spaces

Abstract

By Fn(X), n≥1, we denote the n-th symmetric product of a metric space (X,d) as the space of the nonempty finite subsets of X with at most n elements endowed with the Hausdorff metric dH. By Iso(X) we denote the group of all isometries from X onto itself with the topology of pointwise convergence. In this paper, we show that, under the certain hypothesis, Iso(Fn(X)) is topologically isomorphic to the semidirect product group Iso(Fn(X),F1(X))⋊Iso(X). We apply those results to ℓpq, (p,q)∈[1,∞]×N≥2⁎, as particular spaces and prove the following statements:(1)If p∈{1,∞}, then Iso(F2(ℓp2)) is topologically isomorphic to Z2×Iso(ℓp2).(2)If 3≤q<∞, then Iso(F2(ℓ∞q)) is topologically isomorphic to ∏i=1q−1(Z2)i⋊Iso(ℓ∞q).(3)In other cases except (n,p,q)∈N≥2×{1,∞}×{∞}, the canonical homomorphism χn:Iso(ℓpq)→Iso(Fn(ℓpq)) is a topological isomorphism.