2026/08/04 by Natsumi Shibata, Izuho Matsuzaki, Takeshi Miura
Mathematics · #math.FA #msc:46E25 #msc:46B04 #msc:46J10
arxiv created 2026/08/04 · arxiv updated 2026/08/05
Let X and Y be locally compact Hausdorff spaces and let \mathbbK∈ \ℝ,ℂ\. We say that a bijection T\colon C0(X,\mathbbK)→ C0(Y,\mathbbK) is a ring isomorphism in norm if ‖T(f+g)‖=‖T(f)+T(g)‖, ‖T(fg)‖=‖T(f)T(g)‖ for every f,g∈ C0(X,\mathbbK). We determine the form of such maps. When \mathbbK=ℂ, under the additional assumption that ‖T( f)‖=‖T(f)‖ for every f∈ C0(X,ℂ), there exist a continuous function w\colon Y→\λ∈ℂ:|λ|=1\, a homeomorphism φ\colon Y→ X, and a closed and open subset Y0⊂ Y such that T(f)(y)= \begincases w(y)f(φ(y)), y∈ Y0,
w(y)f(φ(y)), y∈ Y∖ Y0, \endcases for every f∈ C0(X,ℂ) and y∈ Y. When \mathbbK=ℝ, there exist a continuous function w\colon Y→\±1\ and a homeomorphism φ\colon Y→ X such that T(f)(y)=w(y)f(φ(y)) for every f∈ C0(X,ℝ) and y∈ Y. In particular, the real case extends the norm version of the Gelfand--Kolmogoroff theorem to the locally compact setting.