Norm-linear and norm-additive operators between uniform algebras

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:357

Article

Tonev, Thomas¹  Yates, Rebekah¹ 

[1] Univ Montana, Dept Math Sci, Missoula, MT 59812 USA

关键词: Uniform algebra;    Peaking function;    Peak set;    Generalized peak point;    Choquet boundary;    Shilov boundary;    Peripheral spectrum;    Homeomorphism;    Algebra isomorphism;    Norm-additive operator;    Norm-linear operator;   

DOI  :  10.1016/j.jmaa.2009.03.039

来源: Elsevier

【 摘 要 】

Let A subset of (CX) and B subset of C(Y) be uniform algebras with Choquet boundaries delta A and delta B. A map T : A -> 8 is called norm-linear if parallel to lambda Tf + mu Tg parallel to = parallel to lambda f + mu g parallel to; norm-additive, if parallel to Tf + Tg parallel to = parallel to f + g parallel to, and norm-additive in modulus, if parallel to vertical bar Tf vertical bar + vertical bar Tg vertical bar parallel to = parallel to vertical bar f vertical bar + vertical bar g vertical bar parallel to for each lambda, mu is an element of C and all algebra elements f and g. We show that for any norm-linear surjection T : A -> B there exists a homeomorphism psi : delta A -> delta B such that vertical bar(Tf)(y)vertical bar = vertical bar f(psi(y))vertical bar for every f is an element of A and y is an element of delta B. Sufficient conditions for norm-additive and norm-linear surjections, not assumed a priori to be linear, or continuous, to be unital isometric algebra isomorphisms are given. We prove that any unital norm-linear surjection T for which T(i) = i, or which preserves the peripheral spectra of C-peaking functions of A. is a unital isometric algebra isomorphism. In particular, we show that if a linear operator between two uniform algebras, which is surjective and norm-preserving, is unital, or preserves the peripheral spectra of C-peaking functions,. then it is automatically multiplicative and, in fact, an algebra isomorphism. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

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