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Algebras of real analytic functions; homorphisms and bounding sets


Biström, P. and Jaramillo Aguado, Jesús Ángel and Linsdtröm, M. (1993) Algebras of real analytic functions; homorphisms and bounding sets. Extracta Mathematicae, 8 (2-3). pp. 112-118. ISSN 0213-8743

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In this expository article, the authors investigate bounding sets and evaluating properties of homomorphisms for algebras A(E) of functions on a real Banach space E. A
subset B � E is said to be A-bounding if for all f 2 A(E), supx2B |f(x)| < 1. A(E) is said to be single-set evaluating if for every homomorphism � 2 Hom A(E) and every f 2 A(E), �(f) 2 f(E), and it is said to be sequentially evaluating if for every homomorphism � and every sequence (fn) � A(E), there is a 2 E such that �(fn) = fn(a) for every n.
The following are among the results described: If A(E) contains the algebra of rational functions on E, then each A-bounding set is relatively compact in E provided that there is a function f 2 A(`1) that is unbounded on the unit vectors of `1. As a consequence, for every E the C1-bounding sets are relatively compact. Let AE(E) denote the set of functions f : E ! R which are a pointwise (infinite) sum of polynomials, and let RAE(E) denote the smallest inverse-closed algebra containing AE(E). If E is weakly
Lindel¨of with the Dunford-Pettis property and E does not contain a copy of `1, then E = Hom(AE(E)) = HomRAE(E).

Item Type:Article
Uncontrolled Keywords:Homomorphisms; bounding sets; evaluating algebras
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:16765
Deposited On:19 Oct 2012 08:28
Last Modified:07 Feb 2014 09:35

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