A simple constructive proof of an analogue of the corona theorem

von Renteln, Michael

doi:10.1090/S0002-9939-1981-0624918-8

A simple constructive proof of an analogue of the corona theorem
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by Michael von Renteln PDF

Proc. Amer. Math. Soc. 83 (1981), 299-303 Request permission

Abstract:

We present a simple and constructive proof (i.e. a proof without using Gelfand theory) of an analogue of the Corona theorem for the Wiener algebras $W$ and ${W^ + }$ of absolutely convergent Fourier and Taylor series respectively, also the disc algebra $A(\overline D )$ and the subalgebras ${A^n}(\overline D )$ of functions whose $n$th derivatives extend continuously to $\overline D = \{ z:\left | z \right | \leqslant 1\}$.

References

Lennart Carleson, On bounded analytic functions and closure problems, Ark. Mat. 2 (1952), 283–291. MR 52506, DOI 10.1007/BF02590884
Paul J. Cohen, A note on constructive methods in Banach algebras, Proc. Amer. Math. Soc. 12 (1961), 159–163. MR 124515, DOI 10.1090/S0002-9939-1961-0124515-4
V. I. Gavrilov and G. D. Lëvšina, Linear functionals over Lipschitz spaces of holomorphic functions in the unit disc, Dokl. Akad. Nauk SSSR 239 (1978), no. 1, 30–33 (Russian). MR 0473799
A. L. Matheson, Closed ideals in rings of analytic functions satisfying a Lipschitz condition, Banach spaces of analytic functions (Proc. Pelczynski Conf., Kent State Univ., Kent, Ohio, 1976) Lecture Notes in Math., Vol. 604, Springer, Berlin, 1977, pp. 67–72. MR 0463926
D. J. Newman, A simple proof of Wiener’s $1/f$ theorem, Proc. Amer. Math. Soc. 48 (1975), 264–265. MR 365002, DOI 10.1090/S0002-9939-1975-0365002-8
Jürgen Spilker, A simple proof of an analogue of Wiener’s $1/f$ theorem, Arch. Math. (Basel) 32 (1979), no. 3, 265–266. MR 541624, DOI 10.1007/BF01238498