Gleason parts and point derivations for uniform algebras with dense invertible group

Izzo, Alexander

doi:10.1090/tran/7153

Gleason parts and point derivations for uniform algebras with dense invertible group
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by Alexander J. Izzo PDF

Trans. Amer. Math. Soc. 370 (2018), 4299-4321 Request permission

Abstract:

It is shown$\vphantom {\widehat {\widehat {\widehat {\widehat {\widehat {\widehat {\widehat X}}}}}}}$ that there exists a compact set $X$ in $\mathbb {C}^N$ ($N\geq 2$) such that $\widehat X\setminus X$ is nonempty and the uniform algebra $P(X)$ has a dense set of invertible elements, a large Gleason part, and an abundance of nonzero bounded point derivations. The existence of a Swiss cheese $X$ such that $R(X)$ has a Gleason part of full planar measure and a nonzero bounded point derivation at almost every point is established. An analogous result in $\mathbb {C}^N$ is presented. The analogue for rational hulls of a result of Duval and Levenberg on polynomial hulls containing no analytic discs is established. The results presented address questions raised by Dales and Feinstein.

References

Herbert Alexander and John Wermer, Several complex variables and Banach algebras, 3rd ed., Graduate Texts in Mathematics, vol. 35, Springer-Verlag, New York, 1998. MR 1482798
Richard F. Basener, On rationally convex hulls, Trans. Amer. Math. Soc. 182 (1973), 353–381. MR 379899, DOI 10.1090/S0002-9947-1973-0379899-1
Errett Bishop, A minimal boundary for function algebras, Pacific J. Math. 9 (1959), 629–642. MR 109305
Andrew Browder, Introduction to function algebras, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0246125
B. J. Cole, S. N. Ghosh, and A. J. Izzo, A Hull with no nontrivial Gleason parts, Indiana Univ. Math. J. (accepted).
Gustavo Corach and Fernando D. Suárez, Thin spectra and stable range conditions, J. Funct. Anal. 81 (1988), no. 2, 432–442. MR 971887, DOI 10.1016/0022-1236(88)90107-3
H. G. Dales and J. F. Feinstein, Banach function algebras with dense invertible group, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1295–1304. MR 2367103, DOI 10.1090/S0002-9939-07-09044-2
J. Duval and N. Levenberg, Large polynomial hulls with no analytic structure, Complex analysis and geometry (Trento, 1995) Pitman Res. Notes Math. Ser., vol. 366, Longman, Harlow, 1997, pp. 119–122. MR 1477444
C. M. Falcón Rodríguez, The denseness of the group of invertible elements of a uniform algebra, Cienc. Mat. (Havana) 9 (1988), no. 2, 11–17 (Spanish, with English summary). MR 1007646
Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
Robert C. Gunning, Introduction to holomorphic functions of several variables. Vol. I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990. Function theory. MR 1052649
Eva Kallin, Polynomial convexity: The three spheres problem, Proc. Conf. Complex Analysis (Minneapolis, 1964) Springer, Berlin, 1965, pp. 301–304. MR 0179383
G. Lumer, Analytic functions and Dirichlet problem, Bull. Amer. Math. Soc. 70 (1964), 98–104. MR 158283, DOI 10.1090/S0002-9904-1964-11036-3
Anthony G. O’Farrell, An isolated bounded point derivation, Proc. Amer. Math. Soc. 39 (1973), 559–562. MR 315452, DOI 10.1090/S0002-9939-1973-0315452-9
Anthony G. O’Farrell, A regular uniform algebra with a continuous point derivation of infinite order, Bull. London Math. Soc. 11 (1979), no. 1, 41–44. MR 535795, DOI 10.1112/blms/11.1.41
Hugo Rossi, Holomorphically convex sets in several complex variables, Ann. of Math. (2) 74 (1961), 470–493. MR 133479, DOI 10.2307/1970292
Lynn A. Steen, On uniform approximation by rational functions, Proc. Amer. Math. Soc. 17 (1966), 1007–1011. MR 199416, DOI 10.1090/S0002-9939-1966-0199416-0
Gabriel Stolzenberg, A hull with no analytic structure, J. Math. Mech. 12 (1963), 103–111. MR 0143061
Edgar Lee Stout, Polynomial convexity, Progress in Mathematics, vol. 261, Birkhäuser Boston, Inc., Boston, MA, 2007. MR 2305474, DOI 10.1007/978-0-8176-4538-0
John Wermer, Dirichlet algebras, Duke Math. J. 27 (1960), 373–381. MR 121671
John Wermer, Bounded point derivations on certain Banach algebras, J. Functional Analysis 1 (1967), 28–36. MR 0215105, DOI 10.1016/0022-1236(67)90025-0
John Wermer, On an example of Stolzenberg, Symposium on Several Complex Variables (Park City, Utah, 1970) Lecture Notes in Math., Vol. 184, Springer, Berlin, 1971, pp. 79–84. MR 0298428
Donald R. Wilken, Lebesgue measure of parts for $R(X)$, Proc. Amer. Math. Soc. 18 (1967), 508–512. MR 216297, DOI 10.1090/S0002-9939-1967-0216297-8