Subharmonic functions and uniform algebras

Kumagai, Donna

doi:10.1090/S0002-9939-1980-0548077-4

Subharmonic functions and uniform algebras
HTML articles powered by AMS MathViewer

by Donna Kumagai PDF

Proc. Amer. Math. Soc. 78 (1980), 23-29 Request permission

Abstract:

Recently Aupetit and Wermer [J. Functional Anal. 28 (1978), 386-400] have shown conditions under which analytic structure exists in the spectrum space of a uniform algebra. Their work makes critical use of subharmonicity properties of certain classes of functions. In this paper, we develop a technique which offers an alternate and unified proof for subharmonicity of the functions in their paper assuming Basener’s generalized Shilov boundary conjecture. Our technique uses the Oka-Wermer method applied to the n-fold tensor product of the algebra. We exhibit further applications of our main result including a special case which holds for all uniform algebras.

References

John Wermer, Banach algebras and several complex variables, 2nd ed., Graduate Texts in Mathematics, No. 35, Springer-Verlag, New York-Heidelberg, 1976. MR 0394218
Bernard Aupetit and John Wermer, Capacity and uniform algebras, J. Functional Analysis 28 (1978), no. 3, 386–400. MR 496966, DOI 10.1016/0022-1236(78)90095-2
John Wermer, Subharmonicity and hulls, Pacific J. Math. 58 (1975), no. 1, 283–290. MR 393567
Richard F. Basener, A generalized Shilov boundary and analytic structure, Proc. Amer. Math. Soc. 47 (1975), 98–104. MR 352990, DOI 10.1090/S0002-9939-1975-0352990-9

Boundaries for product algebras

Richard F. Basener, A condition for analytic structure, Proc. Amer. Math. Soc. 36 (1972), 156–160. MR 308789, DOI 10.1090/S0002-9939-1972-0308789-X
Errett Bishop, Holomorphic completions, analytic continuation, and the interpolation of semi-norms, Ann. of Math. (2) 78 (1963), 468–500. MR 155016, DOI 10.2307/1970537
G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039