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A general approach to the notion of Silov boundary. (English) Zbl 0661.46049

Suppose that X is a locally convex complex space and that K is a compact subset of X such that the closed convex hull \(\overline{conv K}\) of K is compact. It is shown that the closure of extreme points of \(\overline{conv K}\) is the smallest closed subset of K on which the supremum of the modulus of every functional f from the dual of X is equal to the supremum of the modulus of f on the whole of K. The above is shown under the assumption that K lies on some hyperplane away from the origin. The result is applied to get an immediate existence proof of the Shilov boundary of a commutative unital Banach algebra.
Reviewer: P.Kajetanowicz

MSC:

46J20 Ideals, maximal ideals, boundaries
46A55 Convex sets in topological linear spaces; Choquet theory
46A50 Compactness in topological linear spaces; angelic spaces, etc.
52A07 Convex sets in topological vector spaces (aspects of convex geometry)
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References:

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