Rotation invariant ideals in subalgebras of 𝐿^{∞}

Gorkin, Pamela

doi:10.1090/S0002-9939-1985-0796441-5

Rotation invariant ideals in subalgebras of $L^ \infty$
HTML articles powered by AMS MathViewer

by Pamela Gorkin

Proc. Amer. Math. Soc. 95 (1985), 32-36

DOI: https://doi.org/10.1090/S0002-9939-1985-0796441-5

PDF | Request permission

Abstract:

In this paper, it is shown that the only (nontrivial) finitely generated rotation invariant ideals in ${H^\infty }$ are ${z^n}{H^\infty }$ for some positive integer $n$. Using results about function algebras, it is shown that other rotation invariant ideals exist. Rotation invariant ideals of other subalgebras of ${L^\infty }$ are also studied.

References

Sheldon Axler, Factorization of $L^{\infty }$ functions, Ann. of Math. (2) 106 (1977), no. 3, 567–572. MR 461142, DOI 10.2307/1971067
E. F. Collingwood, On sets of maximum indetermination of analytic functions, Math. Z. 67 (1957), 377–396. MR 95951, DOI 10.1007/BF01258870
Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
S. C. Power, On ideals of nest subalgebras of $C^\ast$-algebras, Proc. London Math. Soc. (3) 50 (1985), no. 2, 314–332. MR 772716, DOI 10.1112/plms/s3-50.2.314
Walter Rudin, Invariant means on $L^{\infty }$, Studia Math. 44 (1972), 219–227. MR 304975, DOI 10.4064/sm-44-3-219-227
Donald Sarason, Algebras of functions on the unit circle, Bull. Amer. Math. Soc. 79 (1973), 286–299. MR 324425, DOI 10.1090/S0002-9904-1973-13144-1
Edgar Lee Stout, The theory of uniform algebras, Bogden & Quigley, Inc., Publishers, Tarrytown-on-Hudson, N.Y., 1971. MR 0423083
Thomas H. Wolff, Two algebras of bounded functions, Duke Math. J. 49 (1982), no. 2, 321–328. MR 659943

Some theorems on vanishing mean oscillation