Geodesic
A~criterion for points to belong to the same Gleason part of the algebra~$R(X)$
Izvestiya. Mathematics , Tome 11 (1977) no. 5, pp. 1109-1117.

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In this paper necessary and sufficient conditions in terms of analytic capacity are found in order that two points of a compact set $X$ belong to the same Gleason part of the algebra $R(X)$. Bibliography: 9 titles. 
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     title = {A~criterion for points to belong to the same {Gleason} part of the algebra~$R(X)$},
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M. S. Mel'nikov. A~criterion for points to belong to the same Gleason part of the algebra~$R(X)$. Izvestiya. Mathematics , Tome 11 (1977) no. 5, pp. 1109-1117. http://geodesic.mathdoc.fr/item/IM2_1977_11_5_a10/

[1] Wilken D. R., “Lebesgue measure for parts on $R(X)$”, Proc. Amer. Math. Soc., 18:3 (1967), 508–512 | DOI | MR | Zbl

[2] Melnikov M. S., “O stroenii dolei Glisona algebry $R(E)$”, Funkts. analiz i ego prilozh., 1:1 (1967), 97–100 | MR

[3] Wilken D. R., “The support of representing measures for $R(X)$”, Pacif. J. Math., 26:3 (1968), 621–625 | MR

[4] Melnikov M. S., “O dolyakh Glisona algebry $R(X)$”, Matem. sb., 101(143) (1976), 293–300 | MR

[5] Davie A. M., Wilken D. R., “An extension of Melnikov's theorem”, J. Funct. Anal., 11:2 (1972), 179–183 | DOI | MR | Zbl

[6] Gamelin T., Ravnomernye algebry, Mir, M., 1973 | Zbl

[7] Melnikov M. S., “Otsenka integrala Koshi po analiticheskoi krivoi”, Matem. sb., 71:4 (1966), 503–514 | MR

[8] Valdes Kastro K., Melnikov M. S., “Ob interpolyatsionnykh mnozhestvakh algebry $R(X)$”, Izv. AN SSSR. Ser. matem., 41 (1977), 325–333 | MR

[9] Davie A. M., “Analytic capacity and approximation problems”, Trans. Amer. Math. Soc., 171 (1972), 409–444 | DOI | MR | Zbl