Kaplansky Theorem for Completely Regular Spaces
نویسنده
چکیده
Let X,Y be realcompact spaces or completely regular spaces consisting of Gδ-points. Let φ be a linear bijective map from C(X) (resp. C(X)) onto C(Y ) (resp. C(Y )). We show that if φ preserves nonvanishing functions, that is, f(x) 6= 0,∀x ∈ X, ⇐⇒ φ(f)(y) 6= 0,∀ y ∈ Y, then φ is a weighted composition operator φ(f) = φ(1) · f ◦ τ, arising from a homeomorphism τ : Y → X. This result is applied also to other nice function spaces, e.g., uniformly or Lipschitz continuous functions on metric spaces.
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