Maps which preserve norms of non-symmetrical quotients between groups of exponentials of Lipschitz functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:415

Article

Hatori, Osamu¹  Jimenez-Vargas, A.²  Villegas-Vallecillos, Moises³ 

[1] Niigata Univ, Fac Sci, Dept Math, Niigata 9502181, Japan

[2] Univ Almeria, Dept Matemat, Almeria 04120, Spain

[3] Univ Cadiz, Fac Ciencias, Dept Matemat, Puerto Real 11510, Spain

关键词: Lipschitz algebra;    Peaking function;    Algebra isomorphism;    Isometric isomorphism;   

DOI  :  10.1016/j.jmaa.2014.01.088

来源: Elsevier

【 摘 要 】

Let Phi : expLip(X-1) -> expLip(X-2) be a surjective mapping where X-1 and X-2 are compact metric spaces. We prove that if Phi satisfies the non-symmetric-quotient norm condition for the uniform norm: parallel to g/f - 1 parallel to(infinity) = parallel to Phi(g)/Phi(f) - 1 parallel to(infinity) (f,g is an element of exp Lip(X-1) ), then Phi is of the form Phi(f)(y) = {Phi(1)(y)f(phi(y)) if y is an element of K, (f is an element of exp Lip(X-1)), Phi(1)(y)<(f(phi(y)))over bar> if y is an element of X-2 backslash K where Phi : X-2 -> X-1 is a homeomorphism and K is a closed open subset of X-2. On the other hand, if Phi satisfies the non-symmetric-quotient norm condition for the Lipschitz algebra norm: parallel to g/f - 1 parallel to(infinity) + parallel to g/f - 1 parallel to(L) = parallel to Phi(g)/Phi(f) - 1 parallel to(infinity) + parallel to Phi(g)/Phi(f) - 1 parallel to(L) (f,g is an element of exp Lip(X-1)), we show that Phi is of the form Phi(f)(y) = Phi(1)(y)f(phi(y)) (y is an element of X-2, f is an element of exp Lip(X-1)), or Phi(f)(y) = Phi(1)(y)<(f(phi(y)))over bar> (y is an element of X-2, f is an element of exp Lip(X-1)), where Phi : X-2 -> X-1 is a surjective isometry. (C) 2014 Elsevier Inc. All rights reserved.

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