Continuity of derivations on some semiprime Banach algebra

Garimella, Ramesh V.

doi:10.1090/S0002-9939-1987-0870787-6

Continuity of derivations on some semiprime Banach algebra
HTML articles powered by AMS MathViewer

by Ramesh V. Garimella

Proc. Amer. Math. Soc. 99 (1987), 289-292

DOI: https://doi.org/10.1090/S0002-9939-1987-0870787-6

PDF | Request permission

Abstract:

If every prime ideal is closed in a commutative semiprime Banach algebra with unit, then every derivation on it is continuous. Also if derivations are continuous on integral domains, then they are continuous on semiprime Banach algebras.

References

W. G. Bade and P. C. Curtis Jr., Prime ideals and automatic continuity problems for Banach algebras, J. Functional Analysis 29 (1978), no. 1, 88–103. MR 499936, DOI 10.1016/0022-1236(78)90048-4
Philip C. Curtis Jr., Derivations in commutative Banach algebras, Radical Banach algebras and automatic continuity (Long Beach, Calif., 1981), Lecture Notes in Math., vol. 975, Springer, Berlin-New York, 1983, pp. 328–333. MR 697597
Julian Cusack, Automatic continuity and topologically simple radical Banach algebras, J. London Math. Soc. (2) 16 (1977), no. 3, 493–500. MR 461136, DOI 10.1112/jlms/s2-16.3.493
J. Esterle, Elements for a classification of commutative radical Banach algebras, Radical Banach algebras and automatic continuity (Long Beach, Calif., 1981), Lecture Notes in Math., vol. 975, Springer, Berlin-New York, 1983, pp. 4–65. MR 697578
Sandy Grabiner, The nilpotency of Banach nil algebras, Proc. Amer. Math. Soc. 21 (1969), 510. MR 236700, DOI 10.1090/S0002-9939-1969-0236700-9
Sandy Grabiner, A formal power series operational calculus for quasi-nilpotent operators. II, J. Math. Anal. Appl. 43 (1973), 170–192. MR 358410, DOI 10.1016/0022-247X(73)90267-9
B. E. Johnson, Continuity of derivations on commutative algebras, Amer. J. Math. 91 (1969), 1–10. MR 246127, DOI 10.2307/2373262
A. Khosravi, Derivations on commutative Banach algebras, Proc. Amer. Math. Soc. 84 (1982), no. 1, 60–64. MR 633278, DOI 10.1090/S0002-9939-1982-0633278-9
I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260–264. MR 70061, DOI 10.1007/BF01362370