A counterexample to a conjecture of S. E. Morris

Feinstein, J.

doi:10.1090/S0002-9939-04-07382-4

A counterexample to a conjecture of S. E. Morris
HTML articles powered by AMS MathViewer

by J. F. Feinstein PDF

Proc. Amer. Math. Soc. 132 (2004), 2389-2397 Request permission

Abstract:

We give a counterexample to a conjecture of S. E. Morris by showing that there is a compact plane set $X$ such that $R(X)$ has no nonzero, bounded point derivations but such that $R(X)$ is not weakly amenable. We also give an example of a separable uniform algebra $A$ such that every maximal ideal of $A$ has a bounded approximate identity but such that $A$ is not weakly amenable.

References

W. G. Bade, P. C. Curtis Jr., and H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. (3) 55 (1987), no. 2, 359–377. MR 896225, DOI 10.1093/plms/s3-55_{2}.359
Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029, DOI 10.1007/978-3-642-65669-9
Andrew Browder, Introduction to function algebras, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0246125
H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs. New Series, vol. 24, The Clarendon Press, Oxford University Press, New York, 2000. Oxford Science Publications. MR 1816726
B. J. Cole, One point parts and the peak point conjecture, Ph.D. Thesis, Yale University, 1968.
J. F. Feinstein, Weak $(F)$-amenability of $R(X)$, Conference on Automatic Continuity and Banach Algebras (Canberra, 1989) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 21, Austral. Nat. Univ., Canberra, 1989, pp. 97–125. MR 1021999
Joel F. Feinstein, A nontrivial, strongly regular uniform algebra, J. London Math. Soc. (2) 45 (1992), no. 2, 288–300. MR 1171556, DOI 10.1112/jlms/s2-45.2.288
J. F. Feinstein, Trivial Jensen measures without regularity, Studia Math. 148 (2001), no. 1, 67–74. MR 1881440, DOI 10.4064/sm148-1-6
J. F. Feinstein and D. W. B. Somerset, Non-regularity for Banach function algebras, Studia Math. 141 (2000), no. 1, 53–68. MR 1782912, DOI 10.4064/sm-141-1-53-68
Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
Niels Groenbaek, A characterization of weakly amenable Banach algebras, Studia Math. 94 (1989), no. 2, 149–162. MR 1025743, DOI 10.4064/sm-94-2-149-162
Alfred P. Hallstrom, On bounded point derivations and analytic capacity, J. Functional Analysis 4 (1969), 153–165. MR 0243358, DOI 10.1016/0022-1236(69)90028-7
Barry Edward Johnson, Cohomology in Banach algebras, Memoirs of the American Mathematical Society, No. 127, American Mathematical Society, Providence, R.I., 1972. MR 0374934
T. W. Körner, A cheaper Swiss cheese, Studia Math. 83 (1986), no. 1, 33–36. MR 829896, DOI 10.4064/sm-83-1-37-45
S. E. Morris, Bounded derivations from uniform algebras, Ph.D. thesis, University of Cambridge, 1993.
Edgar Lee Stout, The theory of uniform algebras, Bogden & Quigley, Inc., Publishers, Tarrytown-on-Hudson, N.Y., 1971. MR 0423083
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