Abstract
Let K be a convex compactum in a complex locally convex space E, P(K) be the uniform algebra of functions on K generated by the restrictions of complex-affine continuous functions on E. For x, yε E, we set H (x,y)={(1−λ)x+λy∶λε ℂ}. It is proved that: (a) the space of maximal ideals of the algebra P(K) coincides with K; (b) distinct points x,y from K belong to the same Gleason part if and only if x and y are relatively interior points of the set H(x,y)∩K (as a subset of H(x,y); (c) the Choquet boundary of the algebra P (K) coincides with the set of complex-extreme points of the compactum K (that is, of points x not belonging to the relative interior of any set of the form H(x,y) ∩ K for y≠x).
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Literature cited
S. N. Bychkov, Izv. Akad. Nauk SSSR, Ser. Mat.,44, No. 1, 46–62 (1980).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 113, pp. 204–207, 1981.
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Arenson, E.L. Gleason parts and Choquet boundary of a function algebra on a convex compactum. J Math Sci 22, 1832–1834 (1983). https://doi.org/10.1007/BF01882582
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DOI: https://doi.org/10.1007/BF01882582