Generalizations of weakly peripherally multiplicative maps between uniform algebras

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:375

Article

Lee, Kristopher¹  Luttman, Aaron¹ 

[1] Clarkson Univ, Dept Math, Potsdam, NY 13676 USA

关键词: Spectral preserver problems;    Uniform algebra;    Weak peripheral multiplicativity;    Algebra isomorphism;   

DOI  :  10.1016/j.jmaa.2010.08.051

来源: Elsevier

【 摘 要 】

Let A and B be uniform algebras on first-countable, compact Hausdorff spaces X and Y, respectively. For f is an element of A, the peripheral spectrum of f, denoted by sigma(pi)(f) = {lambda is an element of sigma(f): vertical bar lambda vertical bar = parallel to f parallel to}, is the set of spectral values of maximum modulus. A map T : A -> B is weakly peripherally multiplicative if sigma(pi) (T(f)T(g)) boolean AND sigma(pi) (fg) not equal empty set for all f, g is an element of A. We show that if T is a surjective, weakly peripherally multiplicative map, then T is a weighted composition operator, extending earlier results. Furthermore, if T-1, T-2 : A -> B are surjective mappings that satisfy sigma(pi) (T-1(f)T-2(g)) boolean AND sigma(pi)(fg) not equal (empty set) for all f, g is an element of A, then T-1(f)T-2(1) = T-1(1)T-2(f) for all f is an element of A, and the map f bar right arrow (f)T-2(1) is an isometric algebra isomorphism. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

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